From 8767554e4e6ed6e0f46b15f401493b86d9bb4e4e Mon Sep 17 00:00:00 2001
From: SciPost Editorial Administration <edadmin@scipost.org>
Date: Mon, 24 Mar 2025 11:01:53 +0200
Subject: [PATCH] Include PauliStrings.jl-1.5
---
PauliStrings.jl-1.5/LICENSE | 21 +
PauliStrings.jl-1.5/Project.toml | 25 +
PauliStrings.jl-1.5/README.md | 1 +
PauliStrings.jl-1.5/docs/Manifest.toml | 1186 +++++++++++++++++
PauliStrings.jl-1.5/docs/Project.toml | 6 +
PauliStrings.jl-1.5/docs/make.jl | 29 +
.../docs/src/assets/custom.css | 3 +
.../docs/src/assets/favicon.ico | Bin 0 -> 93818 bytes
.../docs/src/assets/favicon.png | Bin 0 -> 11172 bytes
PauliStrings.jl-1.5/docs/src/assets/logo.svg | 95 ++
PauliStrings.jl-1.5/docs/src/circuits.md | 78 ++
PauliStrings.jl-1.5/docs/src/constructing.md | 84 ++
PauliStrings.jl-1.5/docs/src/docstrings.md | 6 +
PauliStrings.jl-1.5/docs/src/documentation.md | 298 +++++
PauliStrings.jl-1.5/docs/src/evolution.md | 97 ++
PauliStrings.jl-1.5/docs/src/gif.gif | Bin 0 -> 475361 bytes
PauliStrings.jl-1.5/docs/src/index.md | 209 +++
PauliStrings.jl-1.5/docs/src/lanczos.md | 70 +
.../docs/src/lanczos_example.png | Bin 0 -> 27635 bytes
.../docs/src/time_evolve_example.png | Bin 0 -> 19987 bytes
PauliStrings.jl-1.5/docs/src/toffoli.png | Bin 0 -> 36432 bytes
PauliStrings.jl-1.5/docs/src/translation.md | 119 ++
.../examples/lanczos_example.jl | 50 +
.../examples/lanczos_example.png | Bin 0 -> 27635 bytes
.../examples/time_evolve_example.jl | 70 +
.../examples/time_evolve_example.png | Bin 0 -> 19987 bytes
PauliStrings.jl-1.5/gif.gif | Bin 0 -> 475361 bytes
PauliStrings.jl-1.5/lanczos_example.png | Bin 0 -> 27635 bytes
PauliStrings.jl-1.5/src/PauliStrings.jl | 42 +
PauliStrings.jl-1.5/src/circuits.jl | 274 ++++
PauliStrings.jl-1.5/src/construction.jl | 146 ++
PauliStrings.jl-1.5/src/entropy.jl | 8 +
PauliStrings.jl-1.5/src/equivalence.jl | 27 +
PauliStrings.jl-1.5/src/graph.jl | 25 +
PauliStrings.jl-1.5/src/io.jl | 342 +++++
PauliStrings.jl-1.5/src/lanczos.jl | 49 +
PauliStrings.jl-1.5/src/moments.jl | 141 ++
PauliStrings.jl-1.5/src/operations.jl | 494 +++++++
PauliStrings.jl-1.5/src/operator.jl | 146 ++
PauliStrings.jl-1.5/src/operatorts1d.jl | 206 +++
PauliStrings.jl-1.5/src/random.jl | 68 +
PauliStrings.jl-1.5/src/time_evolution.jl | 39 +
PauliStrings.jl-1.5/src/truncation.jl | 216 +++
PauliStrings.jl-1.5/test/algorithms.jl | 35 +
PauliStrings.jl-1.5/test/circuits.jl | 68 +
PauliStrings.jl-1.5/test/examples.jl | 165 +++
PauliStrings.jl-1.5/test/operator.jl | 71 +
PauliStrings.jl-1.5/test/operatorts1d.jl | 30 +
PauliStrings.jl-1.5/test/runtests.jl | 18 +
PauliStrings.jl-1.5/test/truncation.jl | 21 +
PauliStrings.jl-1.5/time_evolve_example.png | Bin 0 -> 19987 bytes
PauliStrings.jl-1.5/toffoli.png | Bin 0 -> 36432 bytes
52 files changed, 5078 insertions(+)
create mode 100644 PauliStrings.jl-1.5/LICENSE
create mode 100644 PauliStrings.jl-1.5/Project.toml
create mode 120000 PauliStrings.jl-1.5/README.md
create mode 100644 PauliStrings.jl-1.5/docs/Manifest.toml
create mode 100644 PauliStrings.jl-1.5/docs/Project.toml
create mode 100644 PauliStrings.jl-1.5/docs/make.jl
create mode 100755 PauliStrings.jl-1.5/docs/src/assets/custom.css
create mode 100644 PauliStrings.jl-1.5/docs/src/assets/favicon.ico
create mode 100644 PauliStrings.jl-1.5/docs/src/assets/favicon.png
create mode 100644 PauliStrings.jl-1.5/docs/src/assets/logo.svg
create mode 100644 PauliStrings.jl-1.5/docs/src/circuits.md
create mode 100644 PauliStrings.jl-1.5/docs/src/constructing.md
create mode 100644 PauliStrings.jl-1.5/docs/src/docstrings.md
create mode 100644 PauliStrings.jl-1.5/docs/src/documentation.md
create mode 100644 PauliStrings.jl-1.5/docs/src/evolution.md
create mode 100644 PauliStrings.jl-1.5/docs/src/gif.gif
create mode 100644 PauliStrings.jl-1.5/docs/src/index.md
create mode 100644 PauliStrings.jl-1.5/docs/src/lanczos.md
create mode 100644 PauliStrings.jl-1.5/docs/src/lanczos_example.png
create mode 100644 PauliStrings.jl-1.5/docs/src/time_evolve_example.png
create mode 100644 PauliStrings.jl-1.5/docs/src/toffoli.png
create mode 100644 PauliStrings.jl-1.5/docs/src/translation.md
create mode 100644 PauliStrings.jl-1.5/examples/lanczos_example.jl
create mode 100644 PauliStrings.jl-1.5/examples/lanczos_example.png
create mode 100644 PauliStrings.jl-1.5/examples/time_evolve_example.jl
create mode 100644 PauliStrings.jl-1.5/examples/time_evolve_example.png
create mode 100644 PauliStrings.jl-1.5/gif.gif
create mode 100644 PauliStrings.jl-1.5/lanczos_example.png
create mode 100644 PauliStrings.jl-1.5/src/PauliStrings.jl
create mode 100644 PauliStrings.jl-1.5/src/circuits.jl
create mode 100644 PauliStrings.jl-1.5/src/construction.jl
create mode 100644 PauliStrings.jl-1.5/src/entropy.jl
create mode 100644 PauliStrings.jl-1.5/src/equivalence.jl
create mode 100644 PauliStrings.jl-1.5/src/graph.jl
create mode 100644 PauliStrings.jl-1.5/src/io.jl
create mode 100644 PauliStrings.jl-1.5/src/lanczos.jl
create mode 100644 PauliStrings.jl-1.5/src/moments.jl
create mode 100644 PauliStrings.jl-1.5/src/operations.jl
create mode 100644 PauliStrings.jl-1.5/src/operator.jl
create mode 100644 PauliStrings.jl-1.5/src/operatorts1d.jl
create mode 100644 PauliStrings.jl-1.5/src/random.jl
create mode 100644 PauliStrings.jl-1.5/src/time_evolution.jl
create mode 100644 PauliStrings.jl-1.5/src/truncation.jl
create mode 100644 PauliStrings.jl-1.5/test/algorithms.jl
create mode 100644 PauliStrings.jl-1.5/test/circuits.jl
create mode 100644 PauliStrings.jl-1.5/test/examples.jl
create mode 100644 PauliStrings.jl-1.5/test/operator.jl
create mode 100644 PauliStrings.jl-1.5/test/operatorts1d.jl
create mode 100644 PauliStrings.jl-1.5/test/runtests.jl
create mode 100644 PauliStrings.jl-1.5/test/truncation.jl
create mode 100644 PauliStrings.jl-1.5/time_evolve_example.png
create mode 100644 PauliStrings.jl-1.5/toffoli.png
diff --git a/PauliStrings.jl-1.5/LICENSE b/PauliStrings.jl-1.5/LICENSE
new file mode 100644
index 0000000..9048d53
--- /dev/null
+++ b/PauliStrings.jl-1.5/LICENSE
@@ -0,0 +1,21 @@
+MIT License
+
+Copyright (c) 2024 Nicolas Loizeau <nicolas.loizeau@nbi.ku.dk> and contributors
+
+Permission is hereby granted, free of charge, to any person obtaining a copy
+of this software and associated documentation files (the "Software"), to deal
+in the Software without restriction, including without limitation the rights
+to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
+copies of the Software, and to permit persons to whom the Software is
+furnished to do so, subject to the following conditions:
+
+The above copyright notice and this permission notice shall be included in all
+copies or substantial portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
+IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
+FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
+AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
+LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
+OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
+SOFTWARE.
diff --git a/PauliStrings.jl-1.5/Project.toml b/PauliStrings.jl-1.5/Project.toml
new file mode 100644
index 0000000..9511ff9
--- /dev/null
+++ b/PauliStrings.jl-1.5/Project.toml
@@ -0,0 +1,25 @@
+name = "PauliStrings"
+uuid = "f07625cc-80e4-4099-a362-36a3484d9bcc"
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+
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diff --git a/PauliStrings.jl-1.5/README.md b/PauliStrings.jl-1.5/README.md
new file mode 120000
index 0000000..8785db1
--- /dev/null
+++ b/PauliStrings.jl-1.5/README.md
@@ -0,0 +1 @@
+docs/src/index.md
\ No newline at end of file
diff --git a/PauliStrings.jl-1.5/docs/Manifest.toml b/PauliStrings.jl-1.5/docs/Manifest.toml
new file mode 100644
index 0000000..d918864
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/Manifest.toml
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diff --git a/PauliStrings.jl-1.5/docs/Project.toml b/PauliStrings.jl-1.5/docs/Project.toml
new file mode 100644
index 0000000..825e44e
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/Project.toml
@@ -0,0 +1,6 @@
+[deps]
+Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
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+
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diff --git a/PauliStrings.jl-1.5/docs/make.jl b/PauliStrings.jl-1.5/docs/make.jl
new file mode 100644
index 0000000..77f69bd
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/make.jl
@@ -0,0 +1,29 @@
+using Documenter, PauliStrings
+using PauliStrings.Circuits
+
+makedocs(
+ format = Documenter.HTML(prettyurls = get(ENV, "CI", nothing) == "true",
+ assets = [
+ "assets/custom.css",
+ "assets/favicon.ico",
+ ],
+ analytics = "G-L0F0JFQ15V"
+ ),
+ sitename = "PauliStrings.jl",
+ authors = "Nicolas Loizeau",
+ pages = [
+ "Getting started" => "index.md",
+ "Tutorials" => ["Constructing operators"=>"constructing.md",
+ "Lanczos"=>"lanczos.md",
+ "Time evolution"=>"evolution.md",
+ "Translation symmetry"=>"translation.md",
+ "Circuits"=>"circuits.md"],
+ "Documentation" => "documentation.md",
+ "Index" => "docstrings.md"]
+)
+
+deploydocs(
+ repo = "github.com/nicolasloizeau/PauliStrings.jl.git",
+ devbranch = "main",
+ branch = "gh-pages",
+)
diff --git a/PauliStrings.jl-1.5/docs/src/assets/custom.css b/PauliStrings.jl-1.5/docs/src/assets/custom.css
new file mode 100755
index 0000000..824d9b6
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/src/assets/custom.css
@@ -0,0 +1,3 @@
+div#documenter .docs-sidebar .docs-logo > img {
+ max-height: 8em;
+}
diff --git a/PauliStrings.jl-1.5/docs/src/assets/favicon.ico b/PauliStrings.jl-1.5/docs/src/assets/favicon.ico
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diff --git a/PauliStrings.jl-1.5/docs/src/circuits.md b/PauliStrings.jl-1.5/docs/src/circuits.md
new file mode 100644
index 0000000..84a2f9e
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/src/circuits.md
@@ -0,0 +1,78 @@
+# Quantum circuits
+
+The module `Circuits` provides a way to construct and simulate circuits.
+Initialise an empty circuit with `Circuit(n)` where `n` is the number of qubits.
+Then, add gates to the circuit with `push!(circuit, gate, qubits...)` where `gate` is a string representing the gate and `qubits` are the qubits the gate acts on.
+For example, a CNOT gate with control qubit 1 and target qubit 2 is added to the circuit `c` with `push!(c, "CNOT", 1, 2)`.
+Depolarizing noise can be added to the circuit with `push!(c, "Noise")`. The noise amplitude can be set with `c.noise_amplitude=p`.
+
+
+Lets construct a CCX gate (Toffoli gate) out of CNOT and single qubit gates.
+
+
+```@example circuits
+using PauliStrings
+using PauliStrings.Circuits
+
+function noisy_toffoli()
+ c = Circuit(3)
+ push!(c, "H", 3)
+ push!(c, "CNOT", 2, 3); push!(c, "Noise")
+ push!(c, "Tdg", 3)
+ push!(c, "CNOT", 1, 3); push!(c, "Noise")
+ push!(c, "T", 3)
+ push!(c, "CNOT", 2, 3); push!(c, "Noise")
+ push!(c, "Tdg", 3)
+ push!(c, "CNOT", 1, 3); push!(c, "Noise")
+ push!(c, "T", 2)
+ push!(c, "T", 3)
+ push!(c, "CNOT", 1, 2); push!(c, "Noise")
+ push!(c, "H", 3)
+ push!(c, "T", 1)
+ push!(c, "Tdg", 2)
+ push!(c, "CNOT", 1, 2); push!(c, "Noise")
+ return c
+end
+```
+
+We can plot the circuit using [QuantumCircuitDraw.jl](https://github.com/nicolasloizeau/QuantumCircuitDraw.jl):
+```julia
+using QuantumCircuitDraw
+c = noisy_toffoli()
+paulistrings_plot(c)
+```
+
+
+
+
+The circuit can be compiled to a unitary matrix with `compile(c)`.
+Before compiling, set `c.noise_amplitude` to the desired noise amplitude and `c.max_strings` to the number of strings to keep at each step.
+Compile will multiply the gates in the circuit from left to right, apply the noise using [`add_noise`](@ref) and trim the operator at each step using [`trim`](@ref).
+
+Lets check our Toffoli gate against the built-in `CCX` gate in `Circuits`:
+
+```@example circuits
+using PauliStrings.Circuits
+c = noisy_toffoli()
+c.noise_amplitude = 0
+U1 = compile(c)
+U2 = CCXGate(3,1,2,3)
+println(opnorm(U1-U2))
+```
+
+We can also compute expectation values of states in the computational basis with `expect(c, state_out)` and `expect(c, state_in, state_out)` .
+Let's compute the expectation values of the output states of the Toffoli gate when the input state is $|111\rangle$:
+```@example circuits
+in_state = "111"
+out_states = ["000", "001", "010", "011", "100", "101", "110", "111"]
+p = [expect(c, in_state, out_state) for out_state in out_states]
+using Plots
+bar(out_states, p, legend=false, xlabel="out state", ylabel="<out|U|in>")
+```
+
+Same as above but seting the noise amplitude to 0.05:
+```@example circuits
+c.noise_amplitude = 0.05
+p = [expect(c, in_state, out_state) for out_state in out_states]
+bar(out_states, p, legend=false, xlabel="out state", ylabel="<out|U|in>")
+```
diff --git a/PauliStrings.jl-1.5/docs/src/constructing.md b/PauliStrings.jl-1.5/docs/src/constructing.md
new file mode 100644
index 0000000..ec533eb
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/src/constructing.md
@@ -0,0 +1,84 @@
+# Constructing operators
+
+Start by importing PauliStrings:
+```@example constructing
+using PauliStrings
+import PauliStrings as ps
+```
+
+To construct an operator we first need to declare an empty operator of $N$ qubits:
+```julia
+H = Operator(N)
+```
+For the moment, `PauliStrings.jl` supports a maximum of 64 qubits.
+
+We can add a term of the form $J X_i$ by doing
+```julia
+H += J, "X", i
+```
+
+and a term of the form $J X_iX_j$ by doing
+```julia
+H += J, "X", i, "X", j
+```
+
+Similarly, we add a term of the form $J X_iX_jX_k$ by doing
+```julia
+H += J, "X", i, "X", j, "X", k
+```
+etc.
+
+## 1D transverse Ising model
+Let's construct the Hamiltonian of a [1D transverse Ising model](https://en.wikipedia.org/wiki/Transverse-field_Ising_model)
+$$H=-J(\sum_{<i,j>}Z_i Z_j +g \sum_i X_i)$$
+
+```@example constructing
+function ising1D(N, J, g)
+ H = Operator(N)
+ for j in 1:(N - 1)
+ H += "Z",j,"Z",j+1
+ end
+ H += "Z",1,"Z",N # periodic boundary condition
+ for j in 1:N
+ H += g,"X",j
+ end
+ return -J*H
+end
+```
+Note that the first qubit starts at index 1, following Julia's 1-based index.
+
+Operators can be printed in strings format with the `println` function:
+```@example constructing
+println(ising1D(3, 1, 0.5))
+```
+
+
+## 2D transverse Ising model
+
+
+Here we construct a 2D Ising model on a square lattice of L*L sites, with no periodic boundary conditions.
+
+
+```julia
+function ising2D(L, J, g)
+ H = ps.Operator(L*L)
+ for x in 1:L-1
+ for y in 1:L
+ # convert x,y to qubit index
+ i = L*(y-1)+x
+ j = L*(y-1)+(x+1)
+ # horizontal interaction terms
+ H += ('Z',i,'Z',j)
+ # convert x,y to qubit index
+ i = L*(x-1)+y
+ j = L*x+y
+ # vertical interaction terms
+ H += ('Z',i,'Z',j)
+ end
+ end
+ for j in 1:L*L
+ H += g,"X",j
+ end
+ return -J*H
+end
+```
diff --git a/PauliStrings.jl-1.5/docs/src/docstrings.md b/PauliStrings.jl-1.5/docs/src/docstrings.md
new file mode 100644
index 0000000..ff46fb1
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/src/docstrings.md
@@ -0,0 +1,6 @@
+
+
+# Index
+
+```@index
+```
diff --git a/PauliStrings.jl-1.5/docs/src/documentation.md b/PauliStrings.jl-1.5/docs/src/documentation.md
new file mode 100644
index 0000000..f78068f
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/src/documentation.md
@@ -0,0 +1,298 @@
+
+# Documentation
+
+
+## Basics
+```@docs
+Operator(N::Int)
+```
+
+```@docs
+Operator(o::OperatorTS1D)
+```
+
+```@docs
+OperatorTS1D(o::Operator; full=true)
+```
+
+```@docs
+Base.length(o::Operator)
+```
+
+```@docs
+eye(N::Int)
+```
+
+```@docs
+Base.:+(o::Operator, args::Tuple{Number, Vararg{Any}})
+Base.:+(o::Operator, args::Tuple{Vararg{Any}})
+Base.:+(o::Operator, term::Tuple{Number, String})
+Base.:+(o::Operator, term::String)
+```
+
+
+## Operations
+
+
+```@docs
+Base.:+(o1::Operator, o2::Operator)
+Base.:+(o::Operator, a::Number)
+Base.:+(a::Number, o::Operator)
+```
+
+
+```@docs
+Base.:*(o1::Operator, o2::Operator)
+Base.:*(o::Operator, a::Number)
+Base.:*(a::Number, o::Operator)
+```
+
+```@docs
+Base.:-(o::Operator)
+Base.:-(o1::Operator, o2::Operator)
+Base.:-(o::Operator, a::Real)
+Base.:-(a::Real, o::Operator)
+```
+
+```@docs
+com(o1::Operator, o2::Operator; epsilon::Real=0, maxlength::Int=1000)
+```
+
+```@docs
+diag(o::Operator)
+```
+
+```@docs
+trace(o::Operator)
+```
+
+```@docs
+opnorm(o::Operator)
+```
+
+```@docs
+dagger(o::Operator)
+```
+
+```@docs
+ptrace(o::Operator, keep::Vector{Int})
+```
+
+
+## Power and moments
+```@docs
+Base.:^(o::Operator, k::Int)
+```
+```@docs
+oppow(o::Operator, k::Int)
+```
+```@docs
+trace_product(o1::Operator, o2::Operator; scale=0)
+```
+```@docs
+trace_product(A::Operator, k::Int, B::Operator, l::Int; scale=0)
+```
+```@docs
+trace_product(A::Operator, k::Int; scale=0)
+```
+```@docs
+moments(H::Operator, kmax::Int; start=1, scale=0)
+```
+
+
+
+
+## Random operators
+```@docs
+rand_local1(N::Int)
+```
+```@docs
+rand_local2(N::Int)
+```
+```@docs
+rand_local1_TS1D(N::Int)
+```
+```@docs
+rand_local2_TS1D(N::Int)
+```
+
+
+
+## Truncation and noise
+```@docs
+truncate(o::Operator, N::Int; keepnorm::Bool = false)
+```
+```@docs
+trim(o::Operator, N::Int; keepnorm::Bool = false, keep::Operator=Operator(N))
+```
+```@docs
+prune(o::Operator, alpha::Real; keepnorm::Bool = false)
+```
+```@docs
+cutoff(o::Operator, epsilon::Real; keepnorm::Bool = false)
+```
+```@docs
+add_noise(o::Operator, g::Real)
+```
+```@docs
+k_local_part(o::Operator, k::Int; atmost=false)
+```
+
+## Algorithms
+```@docs
+lanczos(H::Operator, O::Operator, steps::Int, nterms::Int; keepnorm=true, maxlength=1000, returnOn=false)
+lanczos(H::OperatorTS1D, O::OperatorTS1D, steps::Int, nterms::Int; keepnorm=true, maxlength=1000, returnOn=false)
+```
+
+```@docs
+rk4(H::Operator, O::Operator, dt::Real; hbar::Real=1, heisenberg=false, M=2^20, keep::Operator=Operator(N))
+```
+
+```@docs
+rk4(H::Function, O::Operator, dt::Real, t::Real; hbar::Real=1, heisenberg=false)
+```
+
+```@docs
+equivalence_class(A1::Union{Operator64,Operator128}, H::Union{Operator64,Operator128})
+```
+
+```@docs
+frustration_graph(o::Operator)
+```
+## Construction
+```@docs
+all_strings(N::Int)
+```
+```@docs
+all_k_local(N::Int, k::Int)
+```
+```@docs
+all_x(N::Int)
+```
+```@docs
+all_y(N::Int)
+```
+```@docs
+all_z(N::Int)
+```
+```@docs
+set_coefs(o::Operator, coefs::Vector{T}) where T <: Number
+```
+
+## Circuits
+
+```@docs
+Circuit(N::Int; max_strings=2^30, noise_amplitude=0)
+```
+```@docs
+push!(c::Circuit, gate::String, sites::Real...)
+```
+```@docs
+pushfirst!(c::Circuit, gate::String, sites::Real...)
+```
+
+```@docs
+XGate(N::Int, i::Int)
+```
+```@docs
+PhaseGate(N::Int, i::Int, theta::Real)
+```
+
+```@docs
+CXGate(N::Int, i::Int, j::Int)
+```
+
+```@docs
+SwapGate(N::Int, i::Int, j::Int)
+```
+```@docs
+CSXGate(N::Int, i::Int, j::Int)
+```
+
+```@docs
+CCXGate(N::Int, i::Int, j::Int, k::Int)
+```
+```@docs
+MCZGate(N::Int, sites::Int...)
+```
+```@docs
+grover_diffusion(N::Int, sites::Int...)
+```
+```@docs
+compile(c::Circuit)
+```
+```@docs
+expect(c::Circuit, state::String)
+```
+```@docs
+expect(c::Circuit, in_state::String, out_state::String)
+```
+
+
+
+
+## Tools
+```@docs
+compress(o::Operator)
+```
+```@docs
+op_to_strings(o::Operator)
+```
+```@docs
+get_coefs(o::Operator)
+```
+```@docs
+op_to_dense(o::Operator)
+```
+```@docs
+shift_left(O::Operator)
+```
+```@docs
+xcount(v::Unsigned, w::Unsigned)
+```
+```@docs
+ycount(v::Unsigned, w::Unsigned)
+```
+```@docs
+zcount(v::Unsigned, w::Unsigned)
+```
+
+```@docs
+is_ts(o::Operator)
+```
+
+```@docs
+get_coef(o::Operator, v::Unsigned, w::Unsigned)
+```
+
+```@docs
+get_pauli(o::Operator, i::Int)
+```
+
+
+## Low level
+
+```@docs
+com(v1::Unsigned, w1::Unsigned, v2::Unsigned, w2::Unsigned)
+```
+
+```@docs
+string_to_vw(pauli::String)
+```
+
+```@docs
+vw_to_string(v::Unsigned, w::Unsigned, N::Int)
+```
+
+```@docs
+vw_in_o(v::Unsigned, w::Unsigned, o::Operator)
+```
+
+```@docs
+Base.push!(o::Operator, c::Number, v::Unsigned, w::Unsigned)
+```
+
+## Index
+
+```@index
+```
diff --git a/PauliStrings.jl-1.5/docs/src/evolution.md b/PauliStrings.jl-1.5/docs/src/evolution.md
new file mode 100644
index 0000000..4bc391d
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/src/evolution.md
@@ -0,0 +1,97 @@
+# Time evolution
+
+
+Time evolution with PauliStrings.jl is done in the Heisenberg picture because pure states are rank 1 density matrices and low rank density matrices cannot be efficiently encoded as a sum of Pauli strings ([Loizeau 2024](https://www.pnas.org/doi/abs/10.1073/pnas.2308006120)).
+
+
+The advantage of working with Pauli strings is that noisy systems can be efficiently simulated in this representation ([Schuster 2024](https://arxiv.org/abs/2407.12768)). Depolarizing noise makes long strings decay, so we can make the simulations tractable by combining noise with truncation.
+
+
+Let's time evolve operator $Z_1$ in the chaotic spin chain
+$$H = \sum_i X_iX_{i+1}-1.05 Z_i +h_X X_i.$$
+First we construct the Hamiltonian:
+
+```julia
+using PauliStrings
+import PauliStrings as ps
+```
+
+```julia
+function chaotic_chain(N::Int)
+ H = ps.Operator(N)
+ # XX interactions
+ for j in 1:(N - 1)
+ H += "X",j,"X",j+1
+ end
+ H += "X",1,"X",N # close the chain
+ # fields
+ for j in 1:N
+ H += -1.05,"Z",j
+ H += 0.5,"X",j
+ end
+ return H
+end
+```
+We initialize a Hamiltonian and the $Z_1$ operator on a 32 spins system.
+
+```julia
+N = 32 # system size
+H = chaotic_chain(N) # Hamiltonian
+O = ps.Operator(N) # operator to time evolve
+O += "Z", 1 # Z on site 1
+```
+
+Now we write a function that will time evolve operator O under Hamiltonian H and return some observable. Here we are interested in recording the correlator
+$$
+S(t) = \frac{1}{2^N} \text{Tr} [Z_1(t)Z_1(0)]
+$$.
+We will time evolve O by integrating Von Neuman's equation $i \frac{dO}{dt}=-[H,O]$ with Runge-Kutta ([`rk4`](@ref)). At each time step we do 3 things :
+
+1. Perform a [`rk4`](@ref) step
+2. [`add_noise`](@ref) that makes long strings decay
+3. [`trim`](@ref) O by keeping only M strings with the largest weight
+
+```julia
+# heisenberg evolution of the operator O using rk4
+# return tr(O(0)*O(t))/tr(O(t)^2)
+# M is the number of strings to keep at each step
+# noise is the amplitude of depolarizing noise
+using ProgressBars
+
+function evolve(H, O, M, times, noise)
+ echo = []
+ O0 = deepcopy(O)
+ dt = times[2]-times[1]
+ for t in ProgressBar(times)
+ push!(echo, ps.trace(O*ps.dagger(O0))/ps.trace(O0*O0))
+ # perform one step of rk4, keep only M strings, do not discard O0
+ O = ps.rk4(H, O, dt; heisenberg=true, M=M, keep=O0)
+ # add depolarizing oise
+ O = ps.add_noise(O, noise*dt)
+ # keep the M strings with the largest weight. Do not discard O0
+ O = ps.trim(O, M; keep=O0)
+ end
+ return real.(echo)
+end
+```
+
+Now we can actually time evolve O for different trim values and plot the result:
+
+```julia
+# time evolve O for different trim values
+times = range(0, stop=5, step=0.05)
+noise = 0.01
+for trim in (10,12,14)
+ S = evolve(H, O, 2^trim, times, noise)
+ loglog(times, S) #plot S(t)
+end
+
+legend()
+title("N=$N")
+xlabel("t")
+ylabel(L"tr$(Z_1(0)*Z_1(t))$")
+savefig("time_evolve_example.png")
+show()
+```
+
+
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diff --git a/PauliStrings.jl-1.5/docs/src/index.md b/PauliStrings.jl-1.5/docs/src/index.md
new file mode 100644
index 0000000..1b719d0
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/src/index.md
@@ -0,0 +1,209 @@
+
+[](https://github.com/nicolasloizeau/PauliStrings.jl/actions/workflows/CI.yml?query=branch%3Amain)
+[](https://nicolasloizeau.github.io/PauliStrings.jl/dev)
+[](https://nicolasloizeau.github.io/PauliStrings.jl/stable)
+[](https://arxiv.org/abs/2410.09654)
+
+
+# Getting started
+PauliStrings.jl is a Julia package for many-body quantum mechanics with Pauli string represented as binary integers.
+It is particularly adapted for running Lanczos, time evolving noisy systems and simulating spin systems on arbitrary graphs.
+Paper : [https://arxiv.org/abs/2410.09654](https://arxiv.org/abs/2410.09654)
+
+
+
+
+
+## [Documentation](https://nicolasloizeau.github.io/PauliStrings.jl/dev/)
+
+
+## Installation
+You can install the package using Julia's package manager
+```julia
+using Pkg; Pkg.add(PauliStrings)
+```
+Or
+```julia
+] add PauliStrings
+```
+
+## Initializing an operator
+
+Import the library and initialize a operator of 4 qubits
+```julia
+using PauliStrings
+import PauliStrings as ps
+H = ps.Operator(4)
+```
+
+
+Add a Pauli string to the operator
+```julia
+H += "XYZ1"
+H += "1YZY"
+```
+```
+julia> H
+(1.0 - 0.0im) XYZ1
+(1.0 - 0.0im) 1YZY
+```
+
+Add a Pauli string with a coefficient
+```julia
+H += -1.2, "XXXZ" # coefficient can be complex
+```
+
+Add a 2-qubit string coupling qubits i and j with X and Y:
+```julia
+H += 2, "X", i, "Y", j # with a coefficient=2
+H += "X", i, "Y", j # with a coefficient=1
+```
+
+Add a 1-qubit string:
+```julia
+H += 2, "Z", i # with a coefficient=2
+H += "Z", i # with a coefficient=1
+H += "S+", i
+```
+
+Supported sites operators are `X`, `Y`, `Z`, `Sx`$=X/2$, `Sy`$=Y/2$, `Sz`$=Z/2$, `S+`$=(X+iY)/2$, `S-`$=(X-iY)/2$.
+
+## Basic Algebra
+The Operator type supports the +,-,* operators with other Operators and Numbers:
+```julia
+H3 = H1*H2
+H3 = H1+H2
+H3 = H1-H2
+H3 = H1+2 # adding a scalar is equivalent to adding the unit times the scalar
+H = 5*H # multiply operator by a scalar
+```
+Trace : `ps.trace(H)`
+
+Frobenius norm : `ps.opnorm(H)`
+
+Conjugate transpose : `ps.dagger(H)`
+
+Number of terms: `length(H)`
+
+Commutator: `ps.com(H1, H2)`. This is much faster than `H1*H2-H2*H1`
+
+
+## Print and export
+`print` shows a list of terms with coefficients e.g:
+```julia
+julia> println(H)
+(10.0 - 0.0im) 1ZZ
+(5.0 - 0.0im) 1Z1
+(15.0 + 0.0im) XYZ
+(5.0 + 0.0im) 1YY
+```
+
+Export a list of strings with coefficients:
+```julia
+coefs, strings = ps.op_to_strings(H)
+```
+
+## Truncate, Cutoff, Trim, Noise
+`ps.truncate(H,M)` removes Pauli strings longer than M (returns a new Operator)
+`ps.cutoff(H,c)` removes Pauli strings with coefficient smaller than c in absolute value (returns a new Operator)
+`ps.trim(H,N)` keeps the first N trings with higest weight (returns a new Operator)
+`ps.prune(H,alpha)` keeps terms with probability 1-exp(-alpha*abs(c)) (returns a new Operator)
+
+`ps.add_noise(H,g)` adds depolarizing noise that make each strings decay like $e^{gw}$ where $w$ is the length of the string. This is useful when used with `trim` to keep the number of strings manageable during time evolution.
+
+
+## Time evolution
+
+`ps.rk4(H, O, dt; hbar=1, heisenberg=false)` performs a step of Runge Kutta and returns the new updated O(t+dt)
+
+H can be an Operator, or a function that takes a time and return an Operator. In case H is a function, a time also needs to be passed to `rk4(H, O, dt, t)`. O is an Observable or a density matrix to time evolve.
+If evolving an observable in the Heisenberg picture, set `heisenberg=true`.
+
+An example is in `time_evolve_example.jl`.
+The following will time evolve O in the Heisenberg picture. At each step, we add depolarizing noise and trim the operator to keep the number of strings manageable
+```julia
+function evolve(H, O, M, times, noise)
+ dt = times[2]-times[1]
+ for t in times
+ O = ps.rk4(H, O, dt; heisenberg=true, M=M) # perform one step of rk4, keep only M strings
+ O = ps.add_noise(O, noise*dt) # add depolarizing noise
+ O = ps.trim(O, M) # keep the M strings with the largest weight
+ end
+ return O
+end
+```
+
+Time evolution of the spin correlation function $\textup{Tr}(Z_1(0)Z_1(t))$ in the chaotic spin chain.
+Check time_evolve_example.jl to reproduce the plot.
+
+
+## Lanczos
+Compute lanczos coefficients
+```julia
+bs = ps.lanczos(H, O, steps, nterms)
+```
+`H` : Hamiltonian
+
+`O` : starting operator
+
+`nterms` : maximum number of terms in the operator. Used by trim at every step
+
+Results for X in XX from https://journals.aps.org/prx/pdf/10.1103/PhysRevX.9.041017 :
+
+
+
+
+## Circuits
+The module `Circuits` provides an easy way to construct and simulate circuits.
+Construct a Toffoli gate out elementary gates:
+
+
+```julia
+using PauliStrings
+using PauliStrings.Circuits
+
+function noisy_toffoli()
+ c = Circuit(3)
+ push!(c, "H", 3)
+ push!(c, "CNOT", 2, 3); push!(c, "Noise")
+ push!(c, "Tdg", 3)
+ push!(c, "CNOT", 1, 3); push!(c, "Noise")
+ push!(c, "T", 3)
+ push!(c, "CNOT", 2, 3); push!(c, "Noise")
+ push!(c, "Tdg", 3)
+ push!(c, "CNOT", 1, 3); push!(c, "Noise")
+ push!(c, "T", 2)
+ push!(c, "T", 3)
+ push!(c, "CNOT", 1, 2); push!(c, "Noise")
+ push!(c, "H", 3)
+ push!(c, "T", 1)
+ push!(c, "Tdg", 2)
+ push!(c, "CNOT", 1, 2); push!(c, "Noise")
+ return c
+end
+```
+
+
+Compute the expectation value $<110|U|111>$:
+```julia
+c = noisy_toffoli()
+expect(c, "111", "110")
+```
+
+
+
+## Contact
+For questions and suggestions : `nicolas.loizeau@nbi.ku.dk`
+
+## Citation
+```
+@misc{loizeau2024,
+ title={Quantum many-body simulations with PauliStrings.jl},
+ author={Nicolas Loizeau and J. Clayton Peacock and Dries Sels},
+ year={2024},
+ eprint={2410.09654},
+ archivePrefix={arXiv},
+ primaryClass={quant-ph},
+ url={https://arxiv.org/abs/2410.09654},
+}
+```
diff --git a/PauliStrings.jl-1.5/docs/src/lanczos.md b/PauliStrings.jl-1.5/docs/src/lanczos.md
new file mode 100644
index 0000000..aa39fbc
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/src/lanczos.md
@@ -0,0 +1,70 @@
+# Lanczos
+
+
+Here we show how to use PauliStrings.jl to run the recursion method and compute [`lanczos`](@ref) coefficients.
+We will focus on reproducing the X in XX results from figure 2 of [Parker 2019](https://journals.aps.org/prx/abstract/10.1103/PhysRevX.9.041017).
+
+Start by importing PauliStrings:
+```julia
+using PauliStrings
+import PauliStrings as ps
+```
+
+Define the XX Hamiltonian $$H = \sum_i X_iX_{i+1}+Y_iY_{i+1}$$ on a 1D chain with periodic boundary conditions
+```julia
+function XX(N)
+ H = ps.Operator(N)
+ for j in 1:(N - 1)
+ H += "X",j,"X",j+1
+ H += "Z",j,"Z",j+1
+ end
+ H += "X",1,"X",N
+ H += "Z",1,"Z",N
+ return H
+end
+```
+and the X operator $$\sum_{i=1}^N X_i$$
+```julia
+function X(N)
+ H = ps.Operator(N)
+ for j in 1:N
+ H += "X",j
+ end
+ return H
+end
+```
+
+Initialize a Hamiltonian and an operator:
+```julia
+N = 50 # system size
+H = XX(N) # Hamiltonian
+O = X(N) # operator
+```
+
+Compute and plot the [`lanczos`](@ref) coefficients for different truncations.
+For each level of truncation, we keep only $$2^p$$ terms with the highest weight at each step of the [`lanczos`](@ref) algorithm.
+```julia
+ioff() # pyplot
+# nterms is the max pauli string length
+for p in (14,16,18,20)
+ @time bs = ps.lanczos(H, O, 20, 2^p; keepnorm=true)
+ plot(bs, label="trim: 2^$p")
+end
+legend()
+ylabel(L"$b_n$")
+xlabel(L"$n$")
+title("X in XX, N=$N spins")
+savefig("lanczos_example.png")
+show()
+```
+
+
+
+# Tacking advantage of translation symmetry
+If the problem is 1D and has translation symmetry, we can take advantage of the symmetry to save time and memory. Just pass $$O$$ and $$H$$ as [`OperatorTS1D`](@ref) to [`lanczos`](@ref). For example :
+```julia
+Hts = OperatorTS1D(H)
+Ots = OperatorTS1D(O)
+bs = ps.lanczos(Hts, Ots, 20, 2^p; keepnorm=true)
+```
+Check the [translation symmetry tutorial](@ref translation).
diff --git a/PauliStrings.jl-1.5/docs/src/lanczos_example.png b/PauliStrings.jl-1.5/docs/src/lanczos_example.png
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diff --git a/PauliStrings.jl-1.5/docs/src/translation.md b/PauliStrings.jl-1.5/docs/src/translation.md
new file mode 100644
index 0000000..b0c785c
--- /dev/null
+++ b/PauliStrings.jl-1.5/docs/src/translation.md
@@ -0,0 +1,119 @@
+# [Translation symmetry in 1D with [`OperatorTS1D`](@ref)] (@id translation)
+
+Here we will show how to take advantage of translation symmetry to save time and memory in PauliStrings.jl.
+Consider the 1D Ising Hamiltonian with periodic boundary conditions
+$$H=-J(\sum_{i}Z_i Z_{i+1} +g \sum_i X_i)$$.
+There is no need to actually store all the Pauli strings in this case. H is fully specified by just $$-JZ_1Z_2$$ and $$-Jg X_1$$ and the fact that it's translation symmetric.
+
+In general, a 1D translation symmetric operator can be written as $$\sum_i T_i H_0$$ where $$T_i$$ is the i-sites translation operator and $$H_0$$ is the local operator that generates $$H$$. $$H_0$$ can be chosen so that it's only composed of Pauli strings that start on the first site.
+
+In PauliStrings.jl, the structure [`OperatorTS1D`](@ref) lets you manipulate operators in this format. If your problem is 1D translation symmetric, [`OperatorTS1D`](@ref) will be much faster than [`Operator`](@ref).
+
+## Construction
+
+There are two ways of constructing an [`OperatorTS1D`](@ref) :
+### From the full translation symmetric $$H$$
+First we construct the full [`Operator`](@ref):
+```julia
+function ising1D(N, J, g)
+ H = Operator(N)
+ for j in 1:(N - 1)
+ H += "Z",j,"Z",j+1
+ end
+ H += "Z",1,"Z",N #periodic boundary condition
+ for j in 1:N
+ H += g,"X",j
+ end
+ return -J*H
+end
+H = ising1D(N, J, g)
+```
+then convert it to an [`OperatorTS1D`](@ref)
+```julia
+Hts = OperatorTS1D(H)
+```
+If `H` is not translation symmetric, then an error will be returned
+### From the local generator $$H_0$$
+Construct $$H_0$$ using [`Operator`](@ref):
+```julia
+H = Operator(N)
+H += -J, "Z",1,"Z",2
+H += -J*g,"X",1
+```
+then convert it to an [`OperatorTS1D`](@ref)
+```julia
+Hts = OperatorTS1D(H, full=false)
+```
+Note that here, we need to set `full=false` in order to specify that we are not passing the full Hamiltonian but just its local generator.
+
+## Manipulation
+All the operations defined on [`Operator`](@ref) are also defined on [`OperatorTS1D`](@ref).
+
+Construct a simple translation invariant operator on 4 sites:
+```julia
+H = Operator(N)
+H += "X", 1
+H += "Z", 1,"Z", 2
+
+Hts = OperatorTS1D(H, full=false)
+
+println(H)
+println(Operator(Hts))
+```
+```
+(1.0 + 0.0im) X111
+(1.0 + 0.0im) ZZ11
+
+(1.0 + 0.0im) 1ZZ1
+(1.0 + 0.0im) 1X11
+(1.0 + 0.0im) X111
+(1.0 + 0.0im) Z11Z
+(1.0 + 0.0im) 11ZZ
+(1.0 + 0.0im) 11X1
+(1.0 + 0.0im) ZZ11
+(1.0 + 0.0im) 111X
+```
+Note that only the local generator is printed when printing [`OperatorTS1D`](@ref), not the full operator.
+
+Multiplication:
+```
+julia> Hts*Hts
+(1.0 + 0.0im) X1X1
+(2.0 + 0.0im) Z1Z1
+(2.0 + 0.0im) 1111
+(1.0 + 0.0im) ZZZZ
+(2.0 + 0.0im) XZZ1
+(2.0 + 0.0im) XX11
+(2.0 + 0.0im) ZZX1
+```
+
+Addition:
+```
+julia> Hts+Hts
+(2.0 + 0.0im) X111
+(2.0 + 0.0im) ZZ11
+```
+
+etc.
+
+## Example: computing $$Tr(H^k)$$
+As an example of performance gains of using [`OperatorTS1D`](@ref) instead of [`Operator`](@ref) we compute the 8th moment ($$Tr(H^8)$$) of a 30 spin system.
+
+Using the function defined above we construct a Ising Hamiltonian and convert it to an [`OperatorTS1D`](@ref):
+```
+N = 30
+k = 8
+H = ising1D(N, 1)
+Hts = OperatorTS1D(H)
+```
+then we compute the kth moment ([`trace_product`](@ref)) using both [`Operator`](@ref) and [`OperatorTS1D`](@ref) :
+```
+julia> @time println(trace_product(H, k))
+1.1904927790006272e18 + 0.0im
+ 80.697013 seconds (28.91 k allocations: 111.213 MiB, 0.07% gc time, 0.04% compilation time)
+
+julia> @time println(trace_product(Hts, k))
+1.190492779000627e18 + 0.0im
+ 1.951678 seconds (37.09 k allocations: 36.165 MiB, 2.00% gc time, 2.01% compilation time)
+```
+[`OperatorTS1D`](@ref) is 40 times faster in this case.
diff --git a/PauliStrings.jl-1.5/examples/lanczos_example.jl b/PauliStrings.jl-1.5/examples/lanczos_example.jl
new file mode 100644
index 0000000..f9577e6
--- /dev/null
+++ b/PauliStrings.jl-1.5/examples/lanczos_example.jl
@@ -0,0 +1,50 @@
+
+
+# lanczos example from :
+# https://journals.aps.org/prx/abstract/10.1103/PhysRevX.9.041017
+# figure 2, "X in XX"
+
+using PauliStrings
+import PauliStrings as ps
+using PyPlot
+
+# XX hamiltonian: 1d chain with XX+YY interraction
+function XX(N)
+ H = ps.Operator(N)
+ for j in 1:(N - 1)
+ H += "X",j,"X",j+1
+ H += "Z",j,"Z",j+1
+ end
+ return H
+end
+
+# X local operator: X operator on each site
+function X(N)
+ H = ps.Operator(N)
+ for j in 1:N
+ H += "X",j
+ end
+ return H
+end
+
+
+N = 50 # system size
+H = XX(N) #hamiltonian
+O = X(N) #operator
+
+ioff()#pyplot
+
+# nterms is the max pauli string length
+for p in (14,16,18,20)
+ @time bs = ps.lanczos(H, O, 20, 2^p; keepnorm=true)
+ plot(bs, label="trim: 2^$p")
+end
+
+
+
+legend()
+ylabel(L"$b_n$")
+xlabel(L"$n$")
+title("X in XX, N=$N spins")
+savefig("lanczos_example.png")
+show()
diff --git a/PauliStrings.jl-1.5/examples/lanczos_example.png b/PauliStrings.jl-1.5/examples/lanczos_example.png
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diff --git a/PauliStrings.jl-1.5/examples/time_evolve_example.jl b/PauliStrings.jl-1.5/examples/time_evolve_example.jl
new file mode 100644
index 0000000..498282a
--- /dev/null
+++ b/PauliStrings.jl-1.5/examples/time_evolve_example.jl
@@ -0,0 +1,70 @@
+
+# example of time evolving Z_1 on a chaotic spin chain
+
+
+using PauliStrings
+import PauliStrings as ps
+using PyPlot
+using ProgressBars
+
+
+# build a chaotic spin chain with periodic bc
+function chaotic_chain(N::Int)
+ H = ps.Operator(N)
+ # XX interractions
+ for j in 1:(N - 1)
+ H += "X",j,"X",j+1
+ end
+ H += "X",1,"X",N # close the chain
+ # fields
+ for j in 1:N
+ H += -1.05,"Z",j
+ H += 0.5,"X",j
+ end
+ return H
+end
+
+
+N = 32 # system size
+H = chaotic_chain(N) #hamiltonian
+O = ps.Operator(N) #operator to time evolve
+O += "Z", 1 # Z on site 1
+
+ioff()
+
+
+# heisenberg evolution of the operator O using rk4
+# return tr(O(0)*O(t))/tr(O(t)^2)
+# M is the number of strings to keep at each step
+# noise is the amplitude of depolarizing noise
+function evolve(H, O, M, times, noise)
+ echo = []
+ O0 = deepcopy(O)
+ dt = times[2]-times[1]
+ for t in ProgressBar(times)
+ push!(echo, ps.trace(O*ps.dagger(O0))/ps.trace(O0*O0))
+ #preform one step of rk4, keep only M strings, do not discard O0
+ O = ps.rk4(H, O, dt; heisenberg=true, M=M, keep=O0)
+ #add depolarizingn oise
+ O = ps.add_noise(O, noise*dt)
+ # keep the M strings with the largest weight. Do not discard O0
+ O = ps.trim(O, M; keep=O0)
+ end
+ return real.(echo)
+end
+
+plt.cla()
+# time evolve O for different trim values
+times = range(0, stop=5, step=0.05)
+noise = 0.01
+for trim in (10,12,14)
+ S = evolve(H, O, 2^trim, times, noise)
+ loglog(times, S) #plot S(t)
+end
+
+legend()
+title("N=$N")
+xlabel("t")
+ylabel(L"tr$(Z_1(0)*Z_1(t))$")
+savefig("time_evolve_example.png")
+show()
diff --git a/PauliStrings.jl-1.5/examples/time_evolve_example.png b/PauliStrings.jl-1.5/examples/time_evolve_example.png
new file mode 100644
index 0000000000000000000000000000000000000000..b319fc93fc23100b76545a7b0dc54f6f5f2d4541
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diff --git a/PauliStrings.jl-1.5/src/PauliStrings.jl b/PauliStrings.jl-1.5/src/PauliStrings.jl
new file mode 100644
index 0000000..d54fa0e
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/PauliStrings.jl
@@ -0,0 +1,42 @@
+module PauliStrings
+
+export Operator, OperatorTS1D, Operator64, Operator128, OperatorTS1D64, OperatorTS1D128
+export trace, opnorm, eye, dagger, com, add, compress, ptrace, shift_left, shift, com
+export diag, xcount, ycount, zcount
+export truncate, trim, cutoff, prune, add_noise, k_local_part, participation
+export rand_local1, rand_local2
+export lanczos, rk4, norm_lanczos, rotate_lower
+export op_to_strings, vw_to_string, string_to_vw, tring_to_dense, op_to_dense, get_pauli, push!, vw_in_o
+export get_coefs, get_coef
+export trace_product, oppow, trace_product_pow, trace_exp, moments
+export OperatorTS1D, resum, rand_local1_TS1D, rand_local2_TS1D, is_ts
+export all_strings, set_coefs, all_z, all_x, all_y, all_k_local
+export equivalence_class
+export frustration_graph
+export renyi_entropy
+export Circuits
+
+
+using Random
+using LinearAlgebra
+using ProgressBars
+using Dictionaries
+
+rng = MersenneTwister(0)
+
+include("operator.jl")
+include("operatorts1d.jl")
+include("io.jl")
+include("operations.jl")
+include("lanczos.jl")
+include("truncation.jl")
+include("random.jl")
+include("time_evolution.jl")
+include("moments.jl")
+include("construction.jl")
+include("equivalence.jl")
+include("graph.jl")
+include("entropy.jl")
+include("circuits.jl")
+
+end
diff --git a/PauliStrings.jl-1.5/src/circuits.jl b/PauliStrings.jl-1.5/src/circuits.jl
new file mode 100644
index 0000000..b50bece
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/circuits.jl
@@ -0,0 +1,274 @@
+
+
+module Circuits
+export CCXGate, TGate, TdgGate, HGate, SwapGate, PhaseGate, SXGate
+export Sgate, SXGate
+export CXGate, CYGate, CZGate, CNOTGate
+export MCZGate
+export CSXGate, CSXdgGate
+export Circuit, compile, expect
+export grover_diffusion
+export XGate, YGate, ZGate
+using PauliStrings
+
+single_gates = ["X", "Y", "Z", "H", "S", "T", "Tdg", "Phase"]
+two_gates = ["CNOT", "Swap", "CX", "CY", "CZ", "CCX", "CSX", "CSXdg"]
+other = ["CCX", "Noise", "MCZ"]
+
+allowed_gates = vcat(single_gates, two_gates, other)
+
+
+
+mutable struct Circuit
+ N::Int
+ gates::Vector{Tuple{String,Vector{Real}}}
+ max_strings::Int
+ noise_amplitude::Real
+end
+
+
+"""
+ Circuit(N::Int; max_strings=2^30, noise_amplitude=0)
+
+Creates an empty quantum circuit with `N` qubits. `max_strings` is the maximum number of strings to keep in the operator. `noise_amplitude` is the amplitude of the noise gate.
+"""
+Circuit(N::Int; max_strings=2^30, noise_amplitude=0) = Circuit(N, [], max_strings, noise_amplitude)
+
+
+
+"""
+ push!(c::Circuit, gate::String, sites::Real...)
+
+Adds a gate to the circuit `c`. The gate is specified by a string `gate` and a list of sites `sites`.
+The gates have the same naming convention as in Qiskit.
+Allowed gates are: "X", "Y", "Z", "H", "S", "T", "Tdg", "Phase", "CNOT", "Swap", "CX", "CY", "CZ", "CCX", "CSX", "CSXdg", "MCZ", "Noise".
+"""
+function Base.push!(c::Circuit, gate::String, sites::Real...)
+ @assert gate in allowed_gates "Unknown gate: $(gate)"
+ push!(c.gates, (gate, collect(sites)))
+end
+
+"""
+ pushfirst!(c::Circuit, gate::String, sites::Real...)
+
+Adds a gate to the beginning of the circuit `c`.
+"""
+function Base.pushfirst!(c::Circuit, gate::String, sites::Real...)
+ @assert gate in allowed_gates "Unknown gate: $(gate)"
+ pushfirst!(c.gates, (gate, collect(sites)))
+end
+
+
+
+function XYZGate(N::Int, i::Int, type::String)
+ O = Operator(N)
+ O += type, i
+ return O
+end
+
+
+
+"""
+ XGate(N::Int, i::Int)
+ YGate(N::Int, i::Int)
+ ZGate(N::Int, i::Int)
+ HGate(N::Int, i::Int)
+ SGate(N::Int, i::Int)
+ TGate(N::Int, i::Int)
+ TdgGate(N::Int, i::Int)
+ SXGate(N::Int, i::Int)
+
+Creates a single qubit gate acting on qubit `i` of a `N` qubit system.
+"""
+XGate(N::Int, i::Int) = XYZGate(N, i, "X")
+YGate(N::Int, i::Int) = XYZGate(N, i, "Y")
+ZGate(N::Int, i::Int) = XYZGate(N, i, "Z")
+HGate(N::Int, i::Int) = (XGate(N, i) + ZGate(N, i)) / sqrt(2)
+SGate(N::Int, i::Int) = PhaseGate(N, i, pi / 2)
+TGate(N::Int, i::Int) = PhaseGate(N, i, pi / 4)
+TdgGate(N::Int, i::Int) = dagger(TGate(N::Int, i::Int))
+SXGate(N::Int, i::Int) = ((1-1im)*XGate(N, i) + (1+1im)*eye(N))/2
+
+"""
+ PhaseGate(N::Int, i::Int, theta::Real)
+
+Creates a phase gate acting on qubit `i` of a `N` qubit system with phase `theta`.
+"""
+PhaseGate(N::Int, i::Int, theta::Real) = (eye(N) + ZGate(N, i)) / 2 + (eye(N) - ZGate(N, i)) / 2 * exp(1im * theta)
+
+
+function CXYZGate(N::Int, i::Int, j::Int, type::String)
+ O = Operator(N)
+ O -= "Z", i, type, j
+ O += type, j
+ O += "Z", i
+ O += eye(N)
+ return O / 2
+end
+
+
+"""
+ CXGate(N::Int, i::Int, j::Int)
+ CYGate(N::Int, i::Int, j::Int)
+ CZGate(N::Int, i::Int, j::Int)
+ CNOTGate(N::Int, i::Int, j::Int)
+
+Creates a two qubit gate with control qubit `i` and target qubit `j` of a `N` qubit system.
+"""
+CXGate(N::Int, i::Int, j::Int) = CXYZGate(N, i, j, "X")
+CYGate(N::Int, i::Int, j::Int) = CXYZGate(N, i, j, "Y")
+CZGate(N::Int, i::Int, j::Int) = CXYZGate(N, i, j, "Z")
+CNOTGate(N::Int, i::Int, j::Int) = CXGate(N, i, j)
+
+
+"""
+ SwapGate(N::Int, i::Int, j::Int)
+
+Creates a swap gate between qubits `i` and `j` of a `N` qubit system.
+"""
+function SwapGate(N::Int, i::Int, j::Int)
+ O = Operator(N)
+ O += "X", i, "X", j
+ O += "Y", i, "Y", j
+ O += "Z", i, "Z", j
+ O += eye(N)
+ return O / 2
+end
+
+"""
+ CSXGate(N::Int, i::Int, j::Int)
+ CSXdgGate(N::Int, i::Int, j::Int)
+
+Controlled sqrt X gate and its dagger
+"""
+function CSXGate(N::Int, i::Int, j::Int)
+ O = Operator(N)
+ O += -1+1im, "Z", i, "X", j
+ O += 1-1im, "Z", i
+ O += 1-1im, "X", j
+ O += eye(N)*(3+1im)
+ return O / 4
+end
+CSXdgGate(N::Int, i::Int, j::Int) = dagger(CSXGate(N, i, j))
+
+"""
+ CCXGate(N::Int, i::Int, j::Int, k::Int)
+
+Tofolli gate with control qubits `i` and `j` and target qubit `k` of a `N` qubit system.
+"""
+function CCXGate(N::Int, i::Int, j::Int, k::Int)
+ O = Operator(N)
+ O += "Z", i, "Z", j, "X", k
+ O -= "Z", i, "X", k
+ O -= "Z", j, "X", k
+ O += "X", k
+ O -= "Z", i, "Z", j
+ O += "Z", i
+ O += "Z", j
+ O += eye(N) * 3
+ return O / 4
+end
+
+"""
+ MCZGate(N::Int, sites::Int...)
+
+Creates a multi-controlled Z gate acting on `sites` qubits of a `N` qubit system.
+"""
+function MCZGate(N::Int, sites::Int...)
+ sites = collect(sites)
+ U = eye(N)-2*all_z(N, sites)/2^N
+ for i in sites
+ U = XGate(N, i) * U * XGate(N, i)
+ end
+ return compress(U)
+end
+
+MCZGate(N::Int) = MCZGate(N, 1:N...)
+
+"""
+ grover_diffusion(N::Int, sites::Int...)
+
+Creates the Grover diffusion operator acting on `sites` qubits of a `N` qubit system.
+"""
+function grover_diffusion(N::Int, sites::Int...)
+ println(sites)
+ U = MCZGate(N, sites...)
+ for i in sites
+ U = HGate(N, i) * U * HGate(N, i)
+ end
+ return compress(U)
+end
+
+
+
+"""
+ compile(c::Circuit)
+
+Compiles the quantum circuit `c` into a unitary operator. Applies the gates in the order they were added.
+Applies noise gates if present and trim the operator to `c.max_strings` strings at each step.
+"""
+function compile(c::Circuit)
+ U = eye(c.N)
+ for (gate, args) in c.gates # args is a vector of the form [site1, site2, ... other args]
+ if gate == "Noise"
+ U = add_noise(U, c.noise_amplitude)
+ elseif gate in allowed_gates
+ O = eval(Symbol(gate * "Gate"))(c.N, args...)
+ U = O * U
+ else
+ error("Unknown gate: $gate")
+ end
+ U = compress(U)
+ U = trim(U, c.max_strings)
+ end
+ return U
+end
+
+
+function trace_zpart(o::Operator)
+ s = 0
+ for i in 1:length(o)
+ if xcount(o.v[i], o.w[i]) == 0 && ycount(o.v[i], o.w[i]) == 0
+ s += o.coef[i]
+ end
+ end
+ return s*2^o.N
+end
+
+
+"""
+ expect(c::Circuit, state::String)
+
+Computes the expectation value of the state `state` after applying the circuit `c` to the state `|0>^N`.
+"""
+function expect(c::Circuit, state::String)
+ @assert c.N == length(state) "State length does not match circuit size"
+ c2 = deepcopy(c)
+ for i in 1:c.N
+ if state[i] == '1'
+ push!(c2, "X", i)
+ end
+ end
+ U = compile(c2)
+ return real(trace_zpart(U))/2^c.N
+end
+
+"""
+ expect(c::Circuit, in_state::String, out_state::String)
+
+Computes the expectation value of the state `out_state` after applying the circuit `c` to the state `in_state`.
+"""
+function expect(c::Circuit, in_state::String, out_state::String)
+ @assert c.N == length(in_state) "State length does not match circuit size"
+ @assert c.N == length(out_state) "State length does not match circuit size"
+ c2 = deepcopy(c)
+ for i in 1:c.N
+ if in_state[i] == '1'
+ pushfirst!(c2, "X", i)
+ end
+ end
+ return expect(c2, out_state)
+end
+
+
+end
diff --git a/PauliStrings.jl-1.5/src/construction.jl b/PauliStrings.jl-1.5/src/construction.jl
new file mode 100644
index 0000000..aa24183
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/construction.jl
@@ -0,0 +1,146 @@
+
+"""
+ all_strings(N::Int)
+
+Return the sum of all the strings supported on N spins, with coeficients 1
+"""
+function all_strings(N::Int)
+ O = Operator(N)
+ for i in 0:2^N-1
+ for j in 0:2^N-1
+ push!(O.v, i)
+ push!(O.w, j)
+ push!(O.coef, (1im)^ycount(i, j))
+ end
+ end
+ return O
+end
+
+
+"""
+ all_k_local(N::Int, k::Int)
+
+Return the sum of all the k-local strings supported on N spins, with coeficients 1.
+These are k-local only strings, not including strings shorter than k.
+
+```
+julia> all_k_local(2, 1)
+(1.0 + 0.0im) X1
+(1.0 + 0.0im) 1X
+(1.0 + 0.0im) Z1
+(1.0 - 0.0im) Y1
+(1.0 + 0.0im) 1Z
+(1.0 - 0.0im) 1Y
+```
+"""
+function all_k_local(N::Int, k::Int)
+ O = Operator(N)
+ for i in 0:2^N-1
+ for j in 0:2^N-1
+ if pauli_weight(i, j) == k
+ push!(O.v, i)
+ push!(O.w, j)
+ push!(O.coef, (1im)^ycount(i, j))
+ end
+ end
+ end
+ return O
+end
+
+
+"""
+ all_z(N::Int)
+
+Return the sum of all the strings supported on N spins with only z and with coeficients 1
+
+```
+julia> all_z(2)
+(1.0 + 0.0im) 11
+(1.0 + 0.0im) Z1
+(1.0 + 0.0im) 1Z
+(1.0 + 0.0im) ZZ
+```
+"""
+function all_z(N::Int)
+ O = Operator(N)
+ for i in 0:2^N-1
+ push!(O.v, i)
+ push!(O.w, 0)
+ push!(O.coef, 1)
+ end
+ return O
+end
+
+function all_z(N::Int, bits::Vector{Int})
+ O = Operator(N)
+ mask = sum(1 << (b-1) for b in bits)
+ for i in 0:2^N-1
+ if (i & ~mask) == 0
+ push!(O.v, i)
+ push!(O.w, 0)
+ push!(O.coef, 1)
+ end
+ end
+ return O
+end
+
+
+
+"""
+ all_x(N::Int)
+
+Return the sum of all the strings supported on N spins with only x and with coeficients 1
+
+```
+julia> all_x(2)
+(1.0 + 0.0im) 11
+(1.0 + 0.0im) X1
+(1.0 + 0.0im) 1X
+(1.0 + 0.0im) XX
+```
+"""
+function all_x(N::Int)
+ O = Operator(N)
+ for i in 0:2^N-1
+ push!(O.v, 0)
+ push!(O.w, i)
+ push!(O.coef, 1)
+ end
+ return O
+end
+
+
+"""
+ all_y(N::Int)
+
+Return the sum of all the strings supported on N spins with only y and with coeficients 1
+
+```
+julia> all_y(2)
+(1.0 + 0.0im) 11
+(1.0 - 0.0im) Y1
+(1.0 - 0.0im) 1Y
+(1.0 - 0.0im) YY
+```
+"""
+function all_y(N::Int)
+ O = Operator(N)
+ for i in 0:2^N-1
+ push!(O.v, i)
+ push!(O.w, i)
+ push!(O.coef, (1im)^ycount(i, i))
+ end
+ return O
+end
+
+"""
+ Base.push!(o::Operator, c::Number, v::Unsigned, w::Unsigned)
+
+Add string c,(v,w) to operator o. Note that c is not the coeficient in front of the pauli string
+but the coeficient in front of the real string. A factor (1im)^ycount(v,w) relate the two.
+"""
+function Base.push!(o::Operator, c::Number, v::Unsigned, w::Unsigned)
+ push!(o.v, v)
+ push!(o.w, w)
+ push!(o.coef, c)
+end
diff --git a/PauliStrings.jl-1.5/src/entropy.jl b/PauliStrings.jl-1.5/src/entropy.jl
new file mode 100644
index 0000000..6b59e1f
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/entropy.jl
@@ -0,0 +1,8 @@
+
+
+function renyi_entropy(o::Operator, alpha::Int)
+ o = compress(o)
+ c = get_coefs(o)
+ c /= sum(abs.(c))
+ return log(sum(abs.(c) .^ alpha)) / (1 - alpha)
+end
diff --git a/PauliStrings.jl-1.5/src/equivalence.jl b/PauliStrings.jl-1.5/src/equivalence.jl
new file mode 100644
index 0000000..06bcd6c
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/equivalence.jl
@@ -0,0 +1,27 @@
+
+"""
+ equivalence_class(A1::Union{Operator64,Operator128}, H::Union{Operator64,Operator128})
+
+Equivalence class of A1 with respect to Hamiltonian H, as defined in [https://arxiv.org/abs/2406.13026](https://arxiv.org/abs/2406.13026) Algorithm 1.
+The equivalence class is returned as an operator with coeficients 1
+Use [`get_pauli`](@ref) to access individual strings.
+"""
+function equivalence_class(A1::Union{Operator64,Operator128}, H::Union{Operator64,Operator128})
+ length(A1) != 1 && error("A1 needs to be an operator with a single Pauli string")
+ A = deepcopy(A1)
+ D = 1
+ counter = 0
+ while counter < D
+ Vv = A.v[counter+1]
+ Vw = A.w[counter+1]
+ for i in 1:length(H)
+ a, Av, Aw = com(Vv, Vw, H.v[i], H.w[i])
+ if a != 0 && !vw_in_o(Av, Aw, A)
+ push!(A, (1im)^ycount(Av, Aw), Av, Aw)
+ D += 1
+ end
+ end
+ counter += 1
+ end
+ return A
+end
diff --git a/PauliStrings.jl-1.5/src/graph.jl b/PauliStrings.jl-1.5/src/graph.jl
new file mode 100644
index 0000000..f2b271a
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/graph.jl
@@ -0,0 +1,25 @@
+
+using SparseArrays
+"""
+ frustration_graph(o::Operator)
+
+Construct the frustration graph, or anticommutation graph of an operator o. Returns a sparse adjacency matrix.
+Each vertex represents a pauli string of the operator, and two vertices are connected by an edge if the corresponding pauli strings anticommute.
+The vertices are labeled by the index of the pauli string in the operator. Use [`get_pauli`](@ref) to access individual strings.
+"""
+function frustration_graph(o::Operator)
+ o = Operator(o)
+ n = length(o)
+ G = spzeros(Int, n, n)
+ for i in 1:n
+ for j in 1:n
+ v = o.v[i] ⊻ o.v[j]
+ w = o.w[i] ⊻ o.w[j]
+ k = (-1)^count_ones(o.v[i] & o.w[j]) - (-1)^count_ones(o.w[i] & o.v[j])
+ if k != 0
+ G[i, j] = 1
+ end
+ end
+ end
+ return G
+end
diff --git a/PauliStrings.jl-1.5/src/io.jl b/PauliStrings.jl-1.5/src/io.jl
new file mode 100644
index 0000000..51d5d70
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/io.jl
@@ -0,0 +1,342 @@
+
+import LinearAlgebra as la
+
+
+"""number of Y in a pauli string"""
+function getdelta(pauli::String)
+ return count(c -> c == 'Y', pauli)
+end
+
+"""
+get v and w from ref
+https://journals.aps.org/pra/abstract/10.1103/PhysRevA.68.042318
+"""
+function getvw(pauli::String)
+ v::UInt = 0
+ w::UInt = 0
+ for k in 1:length(pauli)
+ if pauli[k] == 'X'
+ w += 2^(k - 1)
+ end
+ if pauli[k] == 'Z'
+ v += 2^(k - 1)
+ end
+ if pauli[k] == 'Y'
+ w += 2^(k - 1)
+ v += 2^(k - 1)
+ end
+ end
+ return v, w
+end
+
+"""
+ string_to_vw(pauli::String)
+
+Convert a string to two integers (v,w)
+"""
+string_to_vw(pauli::String) = getvw(pauli)
+
+
+"""
+ set_coefs(o::Operator, coefs::Vector{T}) where T <: Number
+
+Sets the coefficient of `o` to `coefs`. Inplace.
+
+```julia
+A = Operator(4)
+A += 2, "1XXY"
+A += 3, "11Z1"
+```
+```
+julia> A
+(3.0 + 0.0im) 11Z1
+(2.0 - 0.0im) 1XXY
+julia> set_coefs(A, [5,6])
+julia> A
+(5.0 + 0.0im) 11Z1
+(6.0 - 0.0im) 1XXY
+```
+"""
+function set_coefs(o::Operator, coefs::Vector{T}) where {T<:Number}
+ length(o) != length(coefs) && error("length(o) != length(coefs)")
+ for i in 1:length(o)
+ o.coef[i] = (1im)^ycount(o.v[i], o.w[i]) * coefs[i]
+ end
+end
+
+"""
+add a pauli string term to an operator
+"""
+function add_string(o::Operator, pauli::String, J::Number)
+ v, w = getvw(pauli)
+ c = (1im)^getdelta(pauli) * J
+ push!(o.v, v)
+ push!(o.w, w)
+ push!(o.coef, c)
+end
+
+function string_from_inds(ind::Vector{Int})
+ l::Vector{Char} = []
+ paulis = ['1', 'X', 'Y', 'Z']
+ for i in ind
+ push!(l, paulis[i+1])
+ end
+ return join(l)
+end
+
+
+function Base.:+(o::Operator, term::Tuple{Number,Char,Int,Char,Int})
+ o1 = deepcopy(o)
+ J, Pi, i, Pj, j = term
+ pauli = fill('1', o1.N)
+ pauli[i] = Pi
+ pauli[j] = Pj
+ pauli = join(pauli)
+ add_string(o1, pauli, J)
+ return compress(o1)
+end
+
+
+function Base.:+(o::Operator, term::Tuple{Number,Char,Int})
+ o1 = deepcopy(o)
+ J, Pi, i = term
+ pauli = fill('1', o1.N)
+ pauli[i] = Pi
+ pauli = join(pauli)
+ add_string(o1, pauli, J)
+ return compress(o1)
+end
+
+Base.:+(o::Operator, term::Tuple{Char,Int,Char,Int}) = o + (1, term...)
+Base.:+(o::Operator, term::Tuple{Char,Int}) = o + (1, term...)
+
+
+"""
+ Base.:+(o::Operator, args::Tuple{Number, Vararg{Any}})
+ Base.:+(o::Operator, args::Tuple{Vararg{Any}})
+ Base.:+(o::Operator, term::Tuple{Number, String})
+ Base.:+(o::Operator, term::String)
+
+Main functions to contruct spin operators.
+Identical signatures are available for `-`.
+
+# Examples
+k-local terms can be added by adding a tuple to the operator.
+The first element of the tuple is an optional coefficient.
+The other element are couples (symbol,site) where symbol can be "X", "Y", "Z", "Sx", "Sy", "Sz", "S+", "S-" and site is an integer specifying the site on wich the symbol is acting.
+
+```julia
+A = Operator(4)
+A += 2, "X",1,"X",2
+A += 3, "Y",1,"X",2
+A += "X",3,"X",4
+A += 4,"Z",3
+A += 5.2,"X",1,"Y",2,"Z",3
+```
+```
+julia> A
+(4.0 + 0.0im) 11Z1
+(3.0 - 0.0im) YX11
+(1.0 + 0.0im) 11XX
+(2.0 + 0.0im) XX11
+(5.2 - 0.0im) XYZ1
+```
+
+Full strings can also be added:
+```julia
+A = Operator(4)
+A += 2, "1XXY"
+A += 2im, "11Z1"
+```
+```
+julia> A
+(0.0 + 2.0im) 11Z1
+(2.0 - 0.0im) 1XXY
+```
+"""
+function Base.:+(o::Operator, args::Tuple{Number,Vararg{Any}})
+ term = Operator(o.N) + 1
+ c = args[1]
+ for i in 2:2:length(args)
+ o2 = Operator(o.N)
+ symbol = args[i]::String
+ site = args[i+1]::Int
+ if occursin(symbol, "XYZ")
+ o2 += only(symbol), site
+ elseif occursin(symbol, "SxSySz")
+ o2 += 0.5, only(uppercase(symbol[2])), site
+ elseif symbol == "S+"
+ o2 += 0.5, 'X', site
+ o2 += 0.5im, 'Y', site
+ elseif symbol == "S-"
+ o2 += 0.5, 'X', site
+ o2 += -0.5im, 'Y', site
+ else
+ error("Allowed operators: X,Y,Z,Sx,Sy,Sz,S-,S+")
+ end
+ term *= o2
+ end
+ return compress(o + c * term)
+end
+Base.:+(o::Operator, args::Tuple{Vararg{Any}}) = o + (1, args...)
+Base.:-(o::Operator, args::Tuple{Number,Vararg{Any}}) = o + (-args[1], args[2:end]...)
+Base.:-(o::Operator, args::Tuple{Vararg{Any}}) = o + (-1, args...)
+
+
+
+function Base.:+(o::Operator, term::Tuple{Number,String})
+ o1 = deepcopy(o)
+ c, pauli = term
+ if o1.N != length(pauli)
+ error("The string needs to be of the same size as the operator")
+ end
+ add_string(o1, pauli, c)
+ return compress(o1)
+end
+
+Base.:+(o::Operator, term::String) = o + (1, term)
+Base.:-(o::Operator, term::String) = o + (-1, term)
+Base.:-(o::Operator, term::Tuple{Number,String}) = o + (-term[1], term[2])
+
+Base.:+(o::Operator, term::Tuple{Number,Vector{Int}}) = o + (term[1], string_from_inds(term[2]))
+Base.:-(o::Operator, term::Tuple{Number,Vector{Int}}) = o - (term[1], string_from_inds(term[2]))
+
+Base.:+(o::Operator, term::Vector{Int}) = o + (1, string_from_inds(term))
+Base.:-(o::Operator, term::Vector{Int}) = o - (1, string_from_inds(term))
+
+
+"""true if bit i of n is set"""
+function bit(n::Integer, i::Integer)
+ return (n & (1 << (i - 1))) != 0
+end
+
+"""
+ vw_to_string(v::Int, w::Unsigned, N::Unsigned)
+
+convert v,w to a string and a phase
+"""
+function vw_to_string(v::Unsigned, w::Unsigned, N::Int)
+ string::String = ""
+ phase::Complex{Float64} = 1
+ for i in 1:N
+ if !bit(v, i) && !bit(w, i)
+ string *= '1'
+ end
+ if !bit(v, i) && bit(w, i)
+ string *= 'X'
+ end
+ if bit(v, i) && !bit(w, i)
+ string *= 'Z'
+ end
+ if bit(v, i) && bit(w, i)
+ string *= 'Y'
+ phase *= 1im
+ end
+ end
+ return string, phase
+end
+
+
+
+"""print an operator"""
+function Base.show(io::IO, o::Operator)
+ for i in 1:length(o.v)
+ pauli, phase = vw_to_string(o.v[i], o.w[i], o.N)
+ c = o.coef[i] / phase
+ println(io, "($(round(c, digits=10))) ", pauli)
+ end
+end
+
+
+"""
+ get_coefs(o::Operator)
+
+Return the list of coefficient in front of each strings.
+"""
+function get_coefs(o::Operator)
+ return [o.coef[i] / (1im)^ycount(o.v[i], o.w[i]) for i in 1:length(o)]
+end
+
+
+"""
+ op_to_strings(o::Operator)
+
+takes an operator,
+return (coefs, strings)
+where coefs is a list of numbers and strings is a list of pauli string
+coefs[i] multiply strings[i]
+"""
+function op_to_strings(o::Operator)
+ strings::Vector{String} = []
+ coefs::Vector{Complex{Float64}} = []
+ for i in 1:length(o)
+ pauli, phase = vw_to_string(o.v[i], o.w[i], o.N)
+ push!(coefs, o.coef[i] / phase)
+ push!(strings, pauli)
+ end
+ return coefs, strings
+end
+
+s0 = [1 0; 0 1]
+sx = [0 1; 1 0]
+sy = [0 -im; im 0]
+sz = [1 0; 0 -1]
+pdict = Dict('1' => s0, 'X' => sx, 'Y' => sy, 'Z' => sz)
+
+function string_to_dense(v, w, N)
+ pauli, phase = vw_to_string(v, w, N)
+ tau = 1
+ for s in pauli
+ tau = la.kron(tau, pdict[s])
+ end
+ return tau, phase
+end
+
+"""
+ op_to_dense(o::Operator)
+ op_to_dense(o::OperatorTS1D)
+
+Convert an operator to a dense matrix.
+"""
+function op_to_dense(o::Operator)
+ dense = zeros(Complex, 2^o.N, 2^o.N)
+ for i in 1:length(o)
+ tau, phase = string_to_dense(o.v[i], o.w[i], o.N)
+ dense .+= tau * o.coef[i] / phase
+ end
+ return dense
+end
+
+
+
+"""
+ get_coef(o::Operator, v::Unsigned, w::Unsigned)
+ get_coef(o::Operator, v::Integer, w::Integer)
+
+Return the coefficient of the string v,w in o.
+"""
+function get_coef(o::Operator, v::Unsigned, w::Unsigned)
+ for i in 1:length(o)
+ if o.v[i] == v && o.w[i] == w
+ return o.coef[i] / (1im)^ycount(v, w)
+ end
+ end
+ return 0
+end
+get_coef(o::Operator, v::Integer, w::Integer) = get_coef(o, Unsigned(v), Unsigned(w))
+
+"""
+ get_pauli(o::Operator, i::Int)
+
+Return an operator that represent the i-th pauli string of `o'.
+Does not return the string multiplied by the coefficient. Only the string.
+"""
+function get_pauli(o::Operator, i::Int)
+ o2 = Operator(o.N)
+ push!(o2.v, o.v[i])
+ push!(o2.w, o.w[i])
+ push!(o2.coef, (1im)^ycount(o.v[i], o.w[i]))
+ return o2
+end
+
+op_to_dense(o::OperatorTS1D) = op_to_dense(Operator(o))
diff --git a/PauliStrings.jl-1.5/src/lanczos.jl b/PauliStrings.jl-1.5/src/lanczos.jl
new file mode 100644
index 0000000..cb9616a
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/lanczos.jl
@@ -0,0 +1,49 @@
+
+
+
+function norm_lanczos(O::Operator)
+ return opnorm(O) / sqrt(2^O.N)
+end
+
+
+"""
+ lanczos(H::Operator, O::Operator, steps::Int, nterms::Int; keepnorm=true, maxlength=1000, returnOn=false)
+ lanczos(H::OperatorTS1D, O::OperatorTS1D, steps::Int, nterms::Int; keepnorm=true, maxlength=1000, returnOn=false)
+
+Compute the first `steps` lanczos coeficients for Hamiltonian `H` and initial operator `O`
+
+At every step, the operator is trimed with [`trim`](@ref) and only `nterms` are kept.
+
+Using `maxlength` speeds up the commutator by only keeping terms of length <= `maxlength`
+
+Set `returnOn=false` to save the On's at each step. Then the function returns a pair of lists (bn, On).
+The first operators of the list On is O
+"""
+function lanczos(H::Operator, O::Operator, steps::Int, nterms::Int; keepnorm=true, maxlength=1000, returnOn=false, observer=false)
+ @assert typeof(H) == typeof(O)
+ @assert H.N == O.N
+ @assert observer === false || returnOn === false
+ O0 = deepcopy(O)
+ O0 /= norm_lanczos(O0)
+ LHO = com(H, O0)
+ b = norm_lanczos(LHO)
+ O1 = com(H, O0) / b
+ bs = [b]
+ returnOn && (Ons = [O0, O1])
+ (observer !== false) && (obs = [observer(O0), observer(O1)])
+ for n in ProgressBar(0:steps-2)
+ LHO = com(H, O1; maxlength=maxlength)
+ O2 = LHO - b * O0
+ b = norm_lanczos(O2)
+ O2 /= b
+ O2 = trim(O2, nterms; keepnorm=keepnorm)
+ returnOn && push!(Ons, O2)
+ (observer !== false) && push!(obs, observer(O2))
+ O0 = deepcopy(O1)
+ O1 = deepcopy(O2)
+ push!(bs, b)
+ end
+ (observer !== false) && return (bs, obs)
+ returnOn && (return bs, Ons)
+ return bs
+end
diff --git a/PauliStrings.jl-1.5/src/moments.jl b/PauliStrings.jl-1.5/src/moments.jl
new file mode 100644
index 0000000..537cd95
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/moments.jl
@@ -0,0 +1,141 @@
+
+
+using Base.Iterators
+
+
+"""
+ trace_product(o1::Operator, o2::Operator; scale=0)
+ trace_product(o1::OperatorTS1D, o2::OperatorTS1D; scale=0)
+
+Efficiently compute `trace(o1*o2)`. This is much faster than doing `trace(o1*o2)`.
+If `scale` is not 0, then the result is normalized such that trace(identity)=scale.
+"""
+function trace_product(o1::Operator, o2::Operator; scale=0)
+ (scale == 0) && (scale = 2^o1.N)
+ tr = 0
+ T = uinttype(o1)
+ o1v::Vector{T} = o1.v
+ o2v::Vector{T} = o2.v
+ o1w::Vector{T} = o1.w
+ o2w::Vector{T} = o2.w
+ for i in eachindex(o1v)
+ o1vi = o1v[i]
+ for j in eachindex(o2v)
+ if (o1vi == o2v[j] && o1w[i] == o2w[j])
+ tr += o1.coef[i] * o2.coef[j] * (-1)^count_ones(o1v[i] & o2w[j])
+ end
+ end
+ end
+ return tr * scale
+end
+function trace_product(o1::OperatorTS1D, o2::OperatorTS1D; scale=0)
+ (scale == 0) && (scale = 2^o1.N)
+ tr = 0
+ N = o1.N
+ T = uinttype(o1)
+ o1v::Vector{T} = o1.v
+ o2v::Vector{T} = o2.v
+ o1w::Vector{T} = o1.w
+ o2w::Vector{T} = o2.w
+ for k in 0:N-1
+ for j in eachindex(o2v)
+ o2vj = o2v[j]
+ o2wj = o2w[j]
+ v2 = rotate_lower(o2vj, N, k)
+ w2 = rotate_lower(o2wj, N, k)
+ for i in eachindex(o1v)
+ if (o1v[i] == v2) && (o1w[i] == w2)
+ tr += o1.coef[i] * o2.coef[j] * (-1)^count_ones(o1v[i] & w2)
+ end
+ end
+ end
+ end
+ return tr * scale * N
+end
+
+
+"""
+ oppow(o::Operator, k::Int)
+ oppow(o::OperatorTS1D, k::Int)
+
+kth power of o. Same as `^`.
+"""
+function oppow(o::Operator, k::Int)
+ # divide and conqueer is not faster
+ r = eye(o.N)
+ for i in 1:k
+ r *= o
+ end
+ return r
+end
+function oppow(o::OperatorTS1D, k::Int)
+ # divide and conqueer is not faster
+ r = OperatorTS1D(eye(o.N))
+ for i in 1:k
+ r *= o
+ end
+ return r
+end
+
+
+"""
+ Base.:^(o::Operator, k::Int)
+
+kth power of o. Same as `oppow`.
+"""
+Base.:^(o::Operator, k::Int) = oppow(o::Operator, k::Int)
+Base.:^(o::OperatorTS1D, k::Int) = oppow(o::OperatorTS1D, k::Int)
+
+"""
+ trace_product(A::Operator, k::Int, B::Operator, l::Int; scale=0)
+ trace_product(A::OperatorTS1D, k::Int, B::OperatorTS1D, l::Int; scale=0)
+
+Efficiently compute `trace(A^k*B^l)`. This is much faster than doing `trace(A^k*B^l)`.
+
+If `scale` is not 0, then the result is normalized such that trace(identity)=scale.
+"""
+function trace_product(A::Operator, k::Int, B::Operator, l::Int; scale=0)
+ @assert typeof(A) == typeof(B)
+ m = div(k + l, 2)
+ n = k + l - m
+ if k < m
+ C = oppow(A, k) * oppow(B, m - k)
+ D = oppow(B, n)
+ elseif k > m
+ C = oppow(A, m)
+ D = oppow(A, k - m) * oppow(B, l)
+ else
+ C = oppow(A, k)
+ D = oppow(B, l)
+ end
+ return trace_product(C, D; scale=scale)
+end
+
+"""
+ trace_product(A::Operator, k::Int; scale=0)
+
+Efficiently compute `trace(A^k)`. This is much faster than doing `trace(A^k)`.
+
+If `scale` is not 0, then the result is normalized such that trace(identity)=scale.
+"""
+function trace_product(A::Operator, k::Int; scale=0)
+ m = div(k, 2)
+ n = k - m
+ C = oppow(A, m)
+ D = oppow(A, n)
+ return trace_product(C, D; scale=scale)
+end
+
+
+"""
+ moments(H::Operator, kmax::Int; start=1, scale=0)
+ moments(H::OperatorTS1D, kmax::Int; start=1, scale=0)
+
+Compute the first kmax moments of H.
+start is the first moment to start from.
+
+If scale is not 0, then the result is normalized such that trace(identity)=scale.
+"""
+function moments(H::Operator, kmax::Int; start=1, scale=0)
+ return [trace_product(H, k; scale=scale) for k in start:kmax]
+end
diff --git a/PauliStrings.jl-1.5/src/operations.jl b/PauliStrings.jl-1.5/src/operations.jl
new file mode 100644
index 0000000..f598c1f
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/operations.jl
@@ -0,0 +1,494 @@
+
+
+
+function uinttype(o::Operator)
+ if typeof(o) == Operator64 || typeof(o) == OperatorTS1D64
+ return UInt64
+ elseif typeof(o) == Operator128 || typeof(o) == OperatorTS1D128
+ return UInt128
+ else
+ error("Type not recognized")
+ end
+end
+
+function emptydict(o::Operator)
+ T = uinttype(o)
+ return UnorderedDictionary{Tuple{T,T},Complex{Float64}}()
+end
+
+
+
+"""
+ add(o1::Operator, o2::Operator)
+ Base.:+(o1::Operator, o2::Operator)
+ Base.:+(o::Operator, a::Number)
+ Base.:+(a::Number, o::Operator)
+
+Add two operators together or add a number to an operator
+
+# Example
+```julia
+A = Operator(4)
+A += "XYZ1"
+A += 1, "Y", 4
+B = Operator(4)
+B += 2, "Y", 2, "Y", 4
+B += 1, "Z", 3
+```
+
+```
+julia> A
+(1.0 - 0.0im) 111Y
+(1.0 - 0.0im) XYZ1
+
+julia> B
+(1.0 + 0.0im) 11Z1
+(2.0 - 0.0im) 1Y1Y
+
+julia> A+B
+(1.0 + 0.0im) 11Z1
+(2.0 - 0.0im) 1Y1Y
+(1.0 - 0.0im) 111Y
+(1.0 - 0.0im) XYZ1
+
+julia> A+5
+(1.0 - 0.0im) 111Y
+(1.0 - 0.0im) XYZ1
+(5.0 + 0.0im) 1111
+```
+"""
+function Base.:+(o1::Operator, o2::Operator)
+ @assert o1.N == o2.N "Adding operators of different dimention"
+ @assert typeof(o1) == typeof(o2) "Adding operators of different types"
+ o3 = typeof(o1)(o1.N)
+ o3.v = vcat(o1.v, o2.v)
+ o3.w = vcat(o1.w, o2.w)
+ o3.coef = vcat(o1.coef, o2.coef)
+ return compress(o3)
+end
+
+
+function Base.:+(o::Operator, a::Number)
+ o1 = deepcopy(o)
+ i = ione(o)
+ if i >= 0
+ o1.coef[ione(o)] += a
+ else
+ push!(o1.coef, a)
+ push!(o1.v, 0)
+ push!(o1.w, 0)
+ end
+ return o1
+end
+
+Base.:+(a::Number, o::Operator) = o + a
+
+"""
+ Base.:*(o1::Operator, o2::Operator)
+ Base.:*(o::Operator, a::Number)
+ Base.:*(a::Number, o::Operator)
+
+Multiply two operators together or an operator with a number
+
+# Example
+```julia
+A = Operator(4)
+A += "XYZ1"
+A += 1, "Y", 4
+B = Operator(4)
+B += 2, "Y", 2, "Y", 4
+B += 1, "Z", 3
+```
+
+```
+julia> A
+(1.0 - 0.0im) 111Y
+(1.0 - 0.0im) XYZ1
+
+
+julia> B
+(1.0 + 0.0im) 11Z1
+(2.0 - 0.0im) 1Y1Y
+
+julia> A*B
+(2.0 - 0.0im) X1ZY
+(1.0 - 0.0im) 11ZY
+(2.0 - 0.0im) 1Y11
+(1.0 - 0.0im) XY11
+
+julia> A*5
+(5.0 - 0.0im) 111Y
+(5.0 - 0.0im) XYZ1
+```
+"""
+function Base.:*(o1::Operator, o2::Operator)
+ @assert o1.N == o2.N "Multiplying operators of different dimention"
+ @assert typeof(o1) == typeof(o2) "Multiplying operators of different types"
+ d = emptydict(o1)
+ for i in 1:length(o1.v)
+ for j in 1:length(o2.v)
+ v = o1.v[i] ⊻ o2.v[j]
+ w = o1.w[i] ⊻ o2.w[j]
+ c = o1.coef[i] * o2.coef[j] * (-1)^count_ones(o1.v[i] & o2.w[j])
+ if isassigned(d, (v, w))
+ d[(v, w)] += c
+ else
+ insert!(d, (v, w), c)
+ end
+ end
+ end
+ return op_from_dict(d, o1.N)
+end
+
+
+function op_from_dict(d::UnorderedDictionary{Tuple{T,T},Complex{Float64}}, N::Int) where {T<:Unsigned}
+ o = Operator(N)
+ for (v, w) in keys(d)
+ push!(o.v, v)
+ push!(o.w, w)
+ push!(o.coef, d[(v, w)])
+ end
+ return o
+end
+
+
+
+function Base.:*(o::Operator, a::Number)
+ o1 = deepcopy(o)
+ o1.coef .*= a
+ return o1
+end
+
+Base.:*(a::Number, o::Operator) = o * a
+
+
+"""
+ Base.:/(o::Operator, a::Number)
+
+Divide an operator by a number
+"""
+function Base.:/(o::Operator, a::Number)
+ o1 = deepcopy(o)
+ o1.coef ./= a
+ return o1
+end
+
+"""
+ Base.:-(o::Operator)
+ Base.:-(o1::Operator, o2::Operator)
+ Base.:-(o::Operator, a::Real)
+ Base.:-(a::Real, o::Operator)
+
+Subtraction between operators and numbers
+"""
+Base.:-(o::Operator) = -1 * o
+Base.:-(o1::Operator, o2::Operator) = o1 + (-o2)
+Base.:-(o::Operator, a::Number) = o + (-a)
+Base.:-(a::Number, o::Operator) = a + (-o)
+
+
+"""
+ com(o1::Operator, o2::Operator; epsilon::Real=0, maxlength::Int=1000)
+ com(o1::OperatorTS1D, o2::OperatorTS1D; anti=false)
+
+Commutator of two operators. Set anti=true to compute the anti-commutator.
+
+# Example
+```
+julia> A = Operator(4)
+julia> A += "X111"
+julia> B = Operator(4)
+julia> B += "Z111"
+julia> B += "XYZ1"
+julia> com(A,B)
+(0.0 - 2.0im) Y111
+```
+"""
+function com(o1::Operator, o2::Operator; epsilon::Real=0, maxlength::Int=1000, anti=false)
+ s = 1
+ anti && (s = -1)
+ @assert o1.N == o2.N "Commuting operators of different dimention"
+ @assert typeof(o1) == typeof(o2) "Commuting operators of different types"
+ o3 = Operator(o1.N)
+ d = emptydict(o1)
+ for i in 1:length(o1.v)
+ for j in 1:length(o2.v)
+ v = o1.v[i] ⊻ o2.v[j]
+ w = o1.w[i] ⊻ o2.w[j]
+ k = (-1)^count_ones(o1.v[i] & o2.w[j]) - s * (-1)^count_ones(o1.w[i] & o2.v[j])
+ c = o1.coef[i] * o2.coef[j] * k
+ if (k != 0) && (abs(c) > epsilon) && pauli_weight(v, w) < maxlength
+ if isassigned(d, (v, w))
+ d[(v, w)] += c
+ else
+ insert!(d, (v, w), c)
+ end
+ end
+ end
+ end
+ for (v, w) in keys(d)
+ push!(o3.v, v)
+ push!(o3.w, w)
+ push!(o3.coef, d[(v, w)])
+ end
+ return o3
+end
+
+
+
+"""
+ com(v1::Unsigned, w1::Unsigned, v2::Unsigned, w2::Unsigned)
+
+Commutator of two pauli strings in integer representation
+Return k,v,w
+"""
+function com(v1::Unsigned, w1::Unsigned, v2::Unsigned, w2::Unsigned)
+ v = v1 ⊻ v2
+ w = w1 ⊻ w2
+ k = (-1)^count_ones(v1 & w2) - (-1)^count_ones(w1 & v2)
+ return k, v, w
+end
+
+
+"""
+ compress(o::Operator)
+
+Accumulate repeated terms and remove terms with a coeficient smaller than 1e-16
+"""
+function compress(o::Operator)
+ o1 = typeof(o)(o.N)
+ T = uinttype(o)
+ vw = Set{Tuple{T,T}}(zip(o.v, o.w))
+ d = UnorderedDictionary{Tuple{T,T},Complex{Float64}}(vw, zeros(length(vw)))
+ for i in 1:length(o)
+ v = o.v[i]
+ w = o.w[i]
+ d[(v, w)] += o.coef[i]
+ end
+ for (v, w) in keys(d)
+ if abs(d[(v, w)]) > 1e-16
+ push!(o1.v, v)
+ push!(o1.w, w)
+ push!(o1.coef, d[(v, w)])
+ end
+ end
+ return o1
+end
+
+
+"""return the index of the 1 string"""
+function ione(o::Operator)
+ for i in 1:length(o)
+ if o.v[i] == 0 && o.w[i] == 0
+ return i
+ end
+ end
+ return -1
+end
+
+"""
+ trace(o::Operator)
+ trace(o::OperatorTS1D)
+
+Trace of an operator
+
+# Example
+```
+julia> A = Operator(4)
+julia> A += 2,"1111"
+julia> A += 3,"XYZ1"
+julia> trace(A)
+32.0 + 0.0im
+```
+"""
+function trace(o::Operator)
+ t = 0
+ for i in 1:length(o.v)
+ if o.v[i] == 0 && o.w[i] == 0
+ t += o.coef[i]
+ end
+ end
+ return t * 2^o.N
+end
+
+
+"""
+ diag(o::Operator)
+ diag(o::OperatorTS1D)
+
+Diagonal of an operator. Keep the strings that only contain 1's or Z's.
+Return another operator.
+
+# Example
+```
+julia> A = Operator(4)
+julia> A += 2,"1111"
+julia> A += 3,"XYZ1"
+julia> A += 3,"Z11Z"
+julia> diag(A)
+(2.0 + 0.0im) 1111
+(3.0 + 0.0im) Z11Z
+```
+"""
+function diag(o::Operator)
+ o2 = Operator(o.N)
+ for i in 1:length(o)
+ v = o.v[i]
+ w = o.w[i]
+ if xcount(v, w) == 0 && ycount(v, w) == 0
+ push!(o2.coef, o.coef[i])
+ push!(o2.v, v)
+ push!(o2.w, w)
+ end
+ end
+ return o2
+end
+
+"""
+ ycount(v::Unsigned, w::Unsigned)
+
+Count the number of Y in a string
+"""
+ycount(v::Unsigned, w::Unsigned) = count_ones(v & w)
+
+"""
+ zcount(v::Unsigned, w::Unsigned)
+
+Count the number of Z in a string
+"""
+zcount(v::Unsigned, w::Unsigned) = count_ones(v & ~w)
+
+"""
+ xcount(v::Unsigned, w::Unsigned)
+
+Count the number of X in a string
+"""
+xcount(v::Unsigned, w::Unsigned) = count_ones(~v & w)
+
+
+
+"""
+ opnorm(o::Operator)
+ opnorm(o::OperatorTS1D)
+
+Frobenius norm
+
+# Example
+```
+julia> A = Operator(4)
+julia> A += 2,"X",2
+julia> A += 1,"Z",1,"Z",3
+julia> opnorm(A)
+8.94427190999916
+```
+"""
+function opnorm(o::Operator)
+ return norm(o.coef) * sqrt(2^o.N)
+end
+
+
+"""
+ dagger(o::Operator)
+ dagger(o::OperatorTS1D)
+
+Conjugate transpose
+
+# Example
+```julia
+A = Operator(3)
+A += 1im,"X",2
+A += 1,"Z",1,"Z",3
+```
+```
+julia> A
+
+(1.0 + 0.0im) Z1Z
+(0.0 + 1.0im) 1X1
+
+
+julia> dagger(A)
+(1.0 - 0.0im) Z1Z
+(0.0 - 1.0im) 1X1
+```
+"""
+function dagger(o::Operator)
+ o1 = deepcopy(o)
+ for i in 1:length(o1)
+ s = (-1)^count_ones(o1.v[i] & o1.w[i])
+ o1.coef[i] = s * conj(o1.coef[i])
+ end
+ return o1
+end
+
+
+
+
+"""
+v,w encode a string.
+return true if at least one index of keep is non unit in vw
+"""
+function tokeep(v::Unsigned, w::Unsigned, keep::Vector{Int})
+ for i in keep
+ if bit(v | w, i)
+ return true
+ end
+ end
+ return false
+end
+
+"""
+ ptrace(o::Operator, keep::Vector{Int})
+
+Partial trace.
+
+`keep` is list of qubits indices to keep starting at 1
+note that this still returns an operator of size N and doesnt permute the qubits
+this only gets rid of Pauli strings that have no support on keep
+and add their coeficient*2^N to the identity string
+
+# Example
+```julia
+A = Operator(5)
+A += "XY1XZ"
+A += "XY11Z"
+```
+```
+julia> ptrace(A, [3,4])
+(1.0 - 0.0im) XY1XZ
+(8.0 - 0.0im) 11111
+
+julia> ptrace(A, [1,5])
+(1.0 - 0.0im) XY1XZ
+(1.0 - 0.0im) XY11Z
+```
+"""
+function ptrace(o::Operator, keep::Vector{Int})
+ o2 = Operator(o.N)
+ NA = length(keep)
+ NB = o.N - NA
+ for i in 1:length(o)
+ if tokeep(o.v[i], o.w[i], keep)
+ push!(o2.v, o.v[i])
+ push!(o2.w, o.w[i])
+ push!(o2.coef, o.coef[i])
+ else
+ o2 += o.coef[i] * 2^NB / (1im)^count_ones(o.v[i] & o.w[i])
+ end
+ end
+ return o2
+end
+
+
+"""
+ vw_in_o(v::Unsigned, w::Unsigned, o::Operator)
+
+Return true is string (v,w) is in o
+"""
+function vw_in_o(v::Unsigned, w::Unsigned, o::Operator)
+ for i in 1:length(o)
+ if v == o.v[i] && w == o.w[i]
+ return true
+ end
+ end
+ return false
+end
diff --git a/PauliStrings.jl-1.5/src/operator.jl b/PauliStrings.jl-1.5/src/operator.jl
new file mode 100644
index 0000000..3dfe6c7
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/operator.jl
@@ -0,0 +1,146 @@
+
+abstract type Operator end
+abstract type OperatorTS1D <: Operator end
+
+"""
+operator as a sum of pauli string encoded like in
+https://journals.aps.org/pra/abstract/10.1103/PhysRevA.68.042318
+intialized as : O=Operator(N)
+where N is the number of qubits
+"""
+
+mutable struct Operator64 <: Operator
+ N::Int
+ v::Vector{UInt64}
+ w::Vector{UInt64}
+ coef::Vector{Complex{Float64}}
+end
+
+mutable struct Operator128 <: Operator
+ N::Int
+ v::Vector{UInt128}
+ w::Vector{UInt128}
+ coef::Vector{Complex{Float64}}
+end
+
+mutable struct OperatorTS1D64 <: OperatorTS1D
+ N::Int
+ v::Vector{UInt64}
+ w::Vector{UInt64}
+ coef::Vector{Complex{Float64}}
+end
+
+mutable struct OperatorTS1D128 <: OperatorTS1D
+ N::Int
+ v::Vector{UInt128}
+ w::Vector{UInt128}
+ coef::Vector{Complex{Float64}}
+end
+
+
+Operator64(N::Int) = Operator64(N, UInt64[], UInt64[], Complex{Float64}[])
+Operator128(N::Int) = Operator128(N, UInt128[], UInt128[], Complex{Float64}[])
+OperatorTS1D64(N::Int) = OperatorTS1D64(N, UInt64[], UInt64[], Complex{Float64}[])
+OperatorTS1D128(N::Int) = OperatorTS1D128(N, UInt128[], UInt128[], Complex{Float64}[])
+
+
+function Operator(N::Int, v::Vector{T}, w::Vector{T}, coef::Vector{Complex{Float64}}) where {T<:Unsigned}
+ if N <= 64
+ return Operator64(N, v, w, coef)
+ elseif N <= 128
+ return Operator128(N, v, w, coef)
+ else
+ error("N needs to be <= 128 qubits")
+ end
+end
+
+"""
+ Operator(N::Int)
+
+Initialize an empty operator on N qubits
+"""
+function Operator(N::Int)
+ if N <= 64
+ return Operator64(N)
+ elseif N <= 128
+ return Operator128(N)
+ else
+ error("N needs to be <= 128 qubits")
+ end
+end
+
+function OperatorTS1D(N::Int, v::Vector{T}, w::Vector{T}, coef::Vector{Complex{Float64}}) where {T<:Unsigned}
+ if N <= 64
+ return OperatorTS1D64(N, v, w, coef)
+ elseif N <= 128
+ return OperatorTS1D128(N, v, w, coef)
+ else
+ error("N needs to be <= 128 qubits")
+ end
+end
+
+
+function OperatorTS1D(N::Int)
+ if N <= 64
+ return OperatorTS1D64(N)
+ elseif N <= 128
+ return OperatorTS1D128(N)
+ else
+ error("N needs to be <= 128 qubits")
+ end
+end
+
+
+function Operator(pauli::String)
+ N = length(pauli)
+ v, w = string_to_vw(pauli)
+ return Operator(N, [v], [w], [(1.0im)^ycount(v, w)])
+end
+
+
+
+"""
+ Base.length(o::Operator)
+ Base.length(o::OperatorTS1D)
+
+Number of pauli strings in an operator
+
+# Example
+```
+julia> A = Operator(4)
+julia> A += "X111"
+julia> A += "XYZ1"
+julia> A += 2, "Y", 4
+julia> length(A)
+3
+```
+"""
+function Base.length(o::Operator)
+ return length(o.v)
+end
+
+"""
+ eye(N::Int)
+
+Identity operator on N qubits
+"""
+function eye(N::Int)
+ O = Operator(N)
+ return O + 1
+end
+
+"""number of non unit paulis in a string encoded by v,w"""
+function pauli_weight(v::Unsigned, w::Unsigned)
+ return count_ones(v | w)
+end
+
+
+"""returns the position of (v,w) in O. return 0 if (v,w) not in O"""
+function posvw(v, w, O)
+ for i in 1:length(O)
+ if O.v[i] == v && O.w[i] == w
+ return i
+ end
+ end
+ return 0
+end
diff --git a/PauliStrings.jl-1.5/src/operatorts1d.jl b/PauliStrings.jl-1.5/src/operatorts1d.jl
new file mode 100644
index 0000000..edae437
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/operatorts1d.jl
@@ -0,0 +1,206 @@
+
+using Dictionaries
+
+
+"""
+ OperatorTS1D(o::Operator; full=true)
+
+Initialize a 1D translation invariant operator from an Operator
+\$O=\\sum_i o_i O_i\$ where \$O_i=T_i(O_0)\$ and \$T_i\$ is the i-sites translation operator.
+Set full=true if passing \$O\$, an Operator that is supported on the whole chain (i.e converting from a translation symmetric [`Operator`](@ref))
+Set full=false if passing \$O_0\$,a local term o such that the full operator is \$O=\\sum_i o_i T_i(O_0)\$
+"""
+function OperatorTS1D(o::Operator; full=true)
+ if full && !is_ts(o)
+ error("o is not translation symmetric. If you want to initialize an OperatorTS1D only with its local part H_0, then set full=false")
+ end
+ o2 = shift_left(o)
+ full && (o2 /= o.N)
+ o3 = OperatorTS1D(o2.N, o2.v, o2.w, o2.coef)
+end
+
+
+
+"""
+ Operator(o::OperatorTS1D)
+
+Convert an OperatorTS1D to an Operator
+"""
+function Operator(o::OperatorTS1D; rs=true)
+ rs && (o = resum(o))
+ return Operator(o.N, o.v, o.w, o.coef)
+end
+
+
+
+
+"""
+ is_ts(o::Operator)
+
+return true if o is translation symmetric
+"""
+function is_ts(o::Operator)
+ for i in 1:o.N
+ if opnorm(o - shift(o, i)) / opnorm(o) > 1e-10
+ return false
+ end
+ end
+ return true
+end
+
+
+
+"""rotate left the first n bits of x by r"""
+function rotate_lower(x::Unsigned, n::Int, r::Int)
+ @assert r <= n
+ mask = (1 << n) - 1
+ lower_bits = x & mask
+ rotated_bits = (lower_bits >> r) | (lower_bits << (n - r))
+ rotated_bits &= mask
+ return (x & ~mask) | rotated_bits
+end
+
+
+"""rotate left the qubits of O by r"""
+function shift(o::Operator, r::Int)
+ o2 = deepcopy(o)
+ for i in 1:length(o)
+ o2.v[i] = rotate_lower(o2.v[i], o2.N, r)
+ o2.w[i] = rotate_lower(o2.w[i], o2.N, r)
+ end
+ return compress(o2)
+end
+
+"""shift the string v,w so it starts on site 1"""
+function shift_left(v, w, N)
+ l = 2^(N + 1)
+ v2 = v
+ w2 = w
+ for i in 0:N
+ v3 = rotate_lower(v, N, i)
+ w3 = rotate_lower(w, N, i)
+ if v3 | w3 + v3 & w3 / 100000 < l
+ v2 = v3
+ w2 = w3
+ l = v3 | w3 + v3 & w3 / 100000
+ end
+ end
+ return (v2, w2)
+end
+
+"""
+ shift_left(O::Operator)
+
+Shift evey string left so they start on site 1. This usefull for using translation symmetry in 1D systems
+# Example
+```julia
+A = Operator(4)
+A += "XYZ1"
+A += "11ZZ"
+A += "1XZY"
+A += "ZZ11"
+```
+```julia
+julia> shift_left(A)
+(1.0 - 0.0im) XZY1
+(1.0 - 0.0im) XYZ1
+(2.0 + 0.0im) ZZ11
+```
+"""
+function shift_left(O::Operator)
+ d = UnorderedDictionary{Tuple{Unsigned,Unsigned},Complex{Float64}}()
+ for i in 1:length(O)
+ v, w = shift_left(O.v[i], O.w[i], O.N)
+ c = O.coef[i]
+ if isassigned(d, (v, w))
+ d[(v, w)] += c
+ else
+ insert!(d, (v, w), c)
+ end
+ end
+ return op_from_dict(d, O.N)
+end
+
+shift1(O::Operator) = shift_left(O)
+
+
+function resum(o::OperatorTS1D)
+ o2 = Operator(o.N)
+ for i in 1:o.N
+ oi = Operator(o.N, o.v, o.w, o.coef)
+ o2 += shift(oi, i)
+ end
+ return o2
+end
+
+
+Base.:+(a::Number, o::OperatorTS1D) = OperatorTS1D(Operator(o, rs=false) + a / o.N; full=false)
+Base.:+(o::OperatorTS1D, a::Number) = a + o
+
+
+function Base.:*(o1::OperatorTS1D, o2::OperatorTS1D)
+ if o1.N != o2.N
+ error("Multiplying OperatorTS1D of different dimention")
+ end
+ N = o1.N
+ d = emptydict(o1)
+ for i in 1:length(o1.v)
+ for j in 1:length(o2.v)
+ for k in 0:N-1
+ v2 = rotate_lower(o2.v[j], N, k)
+ w2 = rotate_lower(o2.w[j], N, k)
+ v = o1.v[i] ⊻ v2
+ w = o1.w[i] ⊻ w2
+ c = o1.coef[i] * o2.coef[j] * (-1)^count_ones(o1.v[i] & w2)
+ if isassigned(d, (v, w))
+ d[(v, w)] += c
+ else
+ insert!(d, (v, w), c)
+ end
+ end
+ end
+ end
+ o = op_from_dict(d, N)
+ return OperatorTS1D(compress(shift_left(o)); full=false)
+end
+
+
+
+trace(o::OperatorTS1D) = trace(Operator(o; rs=false)) * o.N
+opnorm(o::OperatorTS1D) = opnorm(Operator(o; rs=false)) * sqrt(o.N)
+
+
+# diag(o::OperatorTS1D) = OperatorTS1D(diag(o.o); full=false)
+# compress(o::OperatorTS1D) = OperatorTS1D(compress(o.o); full=false)
+
+
+function com(o1::OperatorTS1D, o2::OperatorTS1D; epsilon::Real=0, maxlength::Int=1000, anti=false)
+ s = 1
+ anti && (s = -1)
+ if o1.N != o2.N
+ error("Multiplying OperatorTS1D of different dimention")
+ end
+ N = o1.N
+ d = emptydict(o1)
+ for i in 1:length(o1.v)
+ for j in 1:length(o2.v)
+ for k in 0:N-1
+ v2 = rotate_lower(o2.v[j], N, k)
+ w2 = rotate_lower(o2.w[j], N, k)
+ v = o1.v[i] ⊻ v2
+ w = o1.w[i] ⊻ w2
+ k = (-1)^count_ones(o1.v[i] & w2) - s * (-1)^count_ones(o1.w[i] & v2)
+ c = o1.coef[i] * o2.coef[j] * k
+ if (k != 0) && (abs(c) > epsilon) && pauli_weight(v, w) < maxlength
+ if isassigned(d, (v, w))
+ d[(v, w)] += c
+ else
+ insert!(d, (v, w), c)
+ end
+ end
+ end
+ end
+ end
+ o = op_from_dict(d, N)
+ return OperatorTS1D(compress(shift_left(o)); full=false)
+end
diff --git a/PauliStrings.jl-1.5/src/random.jl b/PauliStrings.jl-1.5/src/random.jl
new file mode 100644
index 0000000..23d6b3e
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/random.jl
@@ -0,0 +1,68 @@
+
+"""
+ rand_local2(N::Int)
+
+Random 2-local operator
+"""
+function rand_local2(N::Int)
+ o = Operator(N)
+ for i in 1:N
+ for j in 1:N
+ if i != j
+ for k in ['X', 'Y', 'Z']
+ for l in ['X', 'Y', 'Z']
+ o += (randn(rng, Float64), k, i, l, j)
+ end
+ end
+ end
+ end
+ end
+ return compress(o)
+end
+
+"""
+ rand_local2(N::Int)
+
+Random 1-local operator
+"""
+function rand_local1(N::Int)
+ o = Operator(N)
+ for i in 1:N
+ for k in ['X', 'Y', 'Z']
+ o += (randn(rng, Float64), k, i)
+ end
+ end
+ return compress(o)
+end
+
+
+"""
+ rand_local1_TS1D(N::Int)
+
+Random 1-local OperatorTS1D
+"""
+function rand_local1_TS1D(N::Int)
+ o = Operator(N)
+ for k in ['X', 'Y', 'Z']
+ o += (randn(rng, Float64), k, 1)
+ end
+ return OperatorTS1D(o; full=false)
+end
+
+"""
+ rand_local2_TS1D(N::Int)
+
+Random 2-local OperatorTS1D
+"""
+function rand_local2_TS1D(N::Int)
+ o = Operator(N)
+ for i in 2:N
+ for k in ['X', 'Y', 'Z']
+ for l in ['X', 'Y', 'Z']
+ o += (randn(rng, Float64), k, 1, l, i)
+ end
+ end
+ end
+ o = compress(o)
+ return OperatorTS1D(o; full=false)
+end
diff --git a/PauliStrings.jl-1.5/src/time_evolution.jl b/PauliStrings.jl-1.5/src/time_evolution.jl
new file mode 100644
index 0000000..9b9cc18
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/time_evolution.jl
@@ -0,0 +1,39 @@
+"""
+ rk4(H::Operator, O::Operator, dt::Real; hbar::Real=1, heisenberg=false, M=2^20, keep::Operator=Operator(N))
+
+Single step of Runge–Kutta-4 with time independant Hamiltonian.
+Returns O(t+dt).
+Set `heisenberg=true` for evolving an observable in the heisenberg picture.
+If `heisenberg=false` then it is assumed that O is a density matrix.
+"""
+function rk4(H::Operator, O::Operator, dt::Real; hbar::Real=1, heisenberg=false, M=2^20, keep::Operator=Operator(0))
+ (keep.N == 0) && (keep=Operator(O.N))
+ s = -1im
+ heisenberg && (s*=-1)
+ k1 = s/hbar*com(H, O)
+ k1 = trim(k1, M; keep=keep)
+ k2 = s/hbar*com(H, O+dt*k1/2)
+ k2 = trim(k2, M; keep=keep)
+ k3 = s/hbar*com(H, O+dt*k2/2)
+ k3 = trim(k3, M; keep=keep)
+ k4 = s/hbar*com(H, O+dt*k3)
+ k4 = trim(k4, M; keep=keep)
+ return O+(k1+2*k2+2*k3+k4)*dt/6
+end
+
+"""
+ rk4(H::Function, O::Operator, dt::Real, t::Real; hbar::Real=1, heisenberg=false)
+
+Single step of Runge–Kutta-4 with time dependant Hamiltonian.
+`H` is a function that takes a number (time) and returns an operator.
+"""
+function rk4(H::Function, O::Operator, dt::Real, t::Real; hbar::Real=1, heisenberg=false)
+ s = -1im
+ heisenberg && (s*=-1)
+ itime && (s*= 1im)
+ k1 = s/hbar*com(H(t), O)
+ k2 = s/hbar*com(H(t+dt/2), O+dt*k1/2)
+ k3 = s/hbar*com(H(t+dt/2), O+dt*k2/2)
+ k4 = s/hbar*com(H(t+dt), O+dt*k3)
+ return O+(k1+2*k2+2*k3+k4)*dt/6
+end
diff --git a/PauliStrings.jl-1.5/src/truncation.jl b/PauliStrings.jl-1.5/src/truncation.jl
new file mode 100644
index 0000000..bd067f8
--- /dev/null
+++ b/PauliStrings.jl-1.5/src/truncation.jl
@@ -0,0 +1,216 @@
+
+"""
+ truncate(o::Operator, max_lenght::Int; keepnorm::Bool = false)
+
+Remove all terms of length > max_lenght.
+Keep all terms of length <= max_lenght.
+i.e remove all M-local terms with M>max_lenght
+
+# Example
+```julia
+A = Operator(4)
+A += "X",1,"X",2
+A += "Z",1,"Z",2,"Z",4
+```
+```
+julia> A
+(1.0 + 0.0im) ZZ1Z
+(1.0 + 0.0im) XX11
+
+julia> ps.truncate(A,2)
+(1.0 + 0.0im) XX11
+```
+"""
+function PauliStrings.truncate(o::Operator, max_lenght::Int; keepnorm::Bool=false)
+ o2 = typeof(o)(o.N)
+ for i in 1:length(o)
+ v = o.v[i]
+ w = o.w[i]
+ if pauli_weight(v, w) <= max_lenght
+ push!(o2.coef, o.coef[i])
+ push!(o2.v, v)
+ push!(o2.w, w)
+ end
+ end
+ if keepnorm
+ return o2 * opnorm(o) / opnorm(o2)
+ end
+ return o2
+end
+
+
+
+"""
+ k_local_part(o::Operator, k::Int; atmost=false)
+
+Return the k-local part of o. I.e all the strings of lenght k.
+Set `atmost=true` to return the 'at most' k-local part, i.e all the strings of length <= k.
+
+# Example
+```julia
+A = Operator(4)
+A += "X",1,"X",2
+A += "Z",1,"Z",2,"Z",4
+A += "1X11"
+```
+```
+julia> A
+(1.0 + 0.0im) ZZ1Z
+(1.0 + 0.0im) 1X11
+(1.0 + 0.0im) XX11
+
+julia> k_local_part(A,2)
+(1.0 + 0.0im) XX11
+```
+"""
+function k_local_part(o::Operator, k::Int; atmost=false)
+ o2 = typeof(o)(o.N)
+ for i in 1:length(o)
+ v = o.v[i]
+ w = o.w[i]
+ l = pauli_weight(v, w)
+ if l == k || (atmost && l <= k)
+ push!(o2.coef, o.coef[i])
+ push!(o2.v, v)
+ push!(o2.w, w)
+ end
+ end
+ return o2
+end
+
+
+
+
+"""
+ trim(o::Operator, max_strings::Int; keepnorm::Bool = false, keep::Operator=Operator(N))
+
+Keep the first `max_strings` terms with largest coeficients.
+
+`keepnorm` is set to true to keep the norm of o.
+
+`keep` is an operator that specify a set of strings that cannot be removed
+
+# Example
+```julia
+A = Operator(4)
+A += 1,"XXXX"
+A += 2,"XX11"
+A += 3,"XX1X"
+A += 4,"ZZXX"
+B = Operator(4)
+B += 1,"XX11"
+B += 1,"XX1X"
+```
+```
+julia> trim(A,2)
+(4.0 + 0.0im) ZZXX
+(3.0 + 0.0im) XX1X
+
+julia> trim(A,2;keep=B)
+(4.0 + 0.0im) ZZXX
+(3.0 + 0.0im) XX1X
+(2.0 + 0.0im) XX11
+```
+"""
+function trim(o::Operator, max_strings::Int; keepnorm::Bool=false, keep::Operator=Operator(0))
+ if length(o) <= max_strings
+ return deepcopy(o)
+ end
+ # keep the N first indices
+ i = sortperm(abs.(o.coef), rev=true)[1:max_strings]
+ # add the string to keep in case there was a specified string to keep
+ if length(keep) > 0
+ for tau in 1:length(keep) #for each string tau in the keep operator
+ # we check if tau is in o and has been removed
+ j = posvw(keep.v[tau], keep.w[tau], o)
+ if !(j in i) && j != 0
+ push!(i, j)
+ end
+ end
+ end
+ o1 = typeof(o)(o.N, o.v[i], o.w[i], o.coef[i])
+ if keepnorm
+ return o1 * opnorm(o) / opnorm(o1)
+ end
+ return o1
+end
+
+"""
+ prune(o::Operator, alpha::Real; keepnorm::Bool = false)
+
+Keep terms with probability 1-exp(-alpha*abs(c)) where c is the weight of the term
+"""
+function prune(o::Operator, alpha::Real; keepnorm::Bool=false)
+ i = Int[]
+ for k in 1:length(o)
+ p = 1 - exp(-alpha * abs(o.coef[k]))
+ if rand() < p
+ push!(i, k)
+ end
+ end
+ o1 = typeof(o)(o.N, o.v[i], o.w[i], o.coef[i])
+ if keepnorm
+ return o1 * opnorm(o) / opnorm(o1)
+ end
+ return o1
+end
+
+"""
+ cutoff(o::Operator, epsilon::Real; keepnorm::Bool = false)
+
+Remove all terms with weight < epsilon
+
+# Example
+```julia
+A = Operator(4)
+A += 1,"XXXX"
+A += 2,"XX11"
+A += 3,"XX1X"
+A += 4,"ZZXX"
+```
+```
+julia> cutoff(A, 2.5)
+(3.0 + 0.0im) XX1X
+(4.0 + 0.0im) ZZXX
+```
+"""
+function cutoff(o::Operator, epsilon::Real; keepnorm::Bool=false)
+ o2 = typeof(o)(o.N)
+ for i in 1:length(o)
+ if abs(o.coef[i]) > epsilon
+ push!(o2.coef, o.coef[i])
+ push!(o2.v, o.v[i])
+ push!(o2.w, o.w[i])
+ end
+ end
+ if keepnorm
+ return o2 * opnorm(o) / opnorm(o2)
+ end
+ return o2
+end
+
+
+
+"""
+ add_noise(o::Operator, g::Real)
+
+Add depolarizing noise that make the long string decays. `g` is the noise amplitude.
+# Example
+```julia
+A = add_noise(A, 0.1)
+```
+# Reference
+[https://arxiv.org/pdf/2407.12768](https://arxiv.org/pdf/2407.12768)
+"""
+function add_noise(o::Operator, g::Real)
+ o2 = deepcopy(o)
+ for i in 1:length(o)
+ o2.coef[i] *= exp(-pauli_weight(o.v[i], o.w[i]) * g)
+ end
+ return o2
+end
+
+
+function participation(o::Operator)
+ return sum(o.coef .^ 4) / sum(o.coef .^ 2)^2
+end
diff --git a/PauliStrings.jl-1.5/test/algorithms.jl b/PauliStrings.jl-1.5/test/algorithms.jl
new file mode 100644
index 0000000..3dccd3a
--- /dev/null
+++ b/PauliStrings.jl-1.5/test/algorithms.jl
@@ -0,0 +1,35 @@
+@testset "lanczos" begin
+ bs1 = ps.lanczos(XX(10), X(10), 10, 2^14)
+ bs2 = ps.lanczos(OperatorTS1D(XX(10)), OperatorTS1D(X(10)), 10, 2^14)
+ @test norm(bs1 .- [2.8284271247461903, 4.0, 4.898979485566357, 5.65685424949238, 6.324555320336759, 6.92820323027551, 7.483314773547883, 8.0, 8.487501570645295, 8.957117508915214]) < 1e-14
+ @test norm(bs1 .- bs2) < 1e-10
+end
+
+@testset "time_evolution" begin
+ @test norm(evolve_chaotic() - 0.09208935978092929) < 1e-10
+end
+
+@testset "moments" begin
+ for N in (10, 70)
+ O1 = rand_local2_M(N, 15)
+ O2 = rand_local2_M(N, 15)
+ @test opnorm(oppow(O1, 3) - O1 * O1 * O1) < 1e-9
+ @test abs(trace_product(O1, O2) - trace(O1 * O2)) < 1e-9
+ @test abs(trace_product(O1, 4) - trace(oppow(O1, 4))) < 1e-9
+ @test abs(trace_product(O1, 4, O2, 3) - trace(oppow(O1, 4) * oppow(O2, 3))) < 1e-8
+ end
+ N = 6
+ O1 = ising1D(N, 0.5)
+ O2 = ising1D(N, 1.1)
+ O1ts = OperatorTS1D(O1)
+ O2ts = OperatorTS1D(O2)
+ @test opnorm(oppow(O1ts, 3) - O1ts * O1ts * O1ts) < 1e-10
+ @test abs(trace_product(O1ts, 4) - trace_product(O1, 4)) < 1e-10
+ @test abs(trace_product(O1ts, 4, O2ts, 3) - trace_product(O1, 4, O2, 3)) < 1e-10
+end
+
+
+@testset "equivalence" begin
+ @test length(equivalence_class(Operator("Y11111"), XX(6))) == 72
+ @test length(equivalence_class(Operator("Y111Z1"), XX(6))) == 512
+end
diff --git a/PauliStrings.jl-1.5/test/circuits.jl b/PauliStrings.jl-1.5/test/circuits.jl
new file mode 100644
index 0000000..f76d783
--- /dev/null
+++ b/PauliStrings.jl-1.5/test/circuits.jl
@@ -0,0 +1,68 @@
+
+using PauliStrings.Circuits
+
+function ccx()
+ c = Circuit(3)
+ push!(c, "H", 3)
+ push!(c, "CNOT", 2, 3)
+ push!(c, "Tdg", 3)
+ push!(c, "CNOT", 1, 3)
+ push!(c, "T", 3)
+ push!(c, "CNOT", 2, 3)
+ push!(c, "Tdg", 3)
+ push!(c, "CNOT", 1, 3)
+ push!(c, "T", 2)
+ push!(c, "T", 3)
+ push!(c, "CNOT", 1, 2)
+ push!(c, "H", 3)
+ push!(c, "T", 1)
+ push!(c, "Tdg", 2)
+ push!(c, "CNOT", 1, 2)
+ return compile(c)
+end
+
+function swap()
+ c = Circuit(2)
+ push!(c, "CNOT", 1, 2)
+ push!(c, "CNOT", 2, 1)
+ push!(c, "CNOT", 1, 2)
+ return compile(c)
+end
+
+function swapH()
+ c = Circuit(2)
+ push!(c, "H", 1)
+ push!(c, "CZ", 1, 2)
+ push!(c, "H", 1)
+ push!(c, "H", 2)
+ push!(c, "CZ", 1, 2)
+ push!(c, "H", 1)
+ push!(c, "H", 2)
+ push!(c, "CZ", 1, 2)
+ push!(c, "H", 1)
+ return compile(c)
+end
+
+function cxH()
+ c = Circuit(2)
+ push!(c, "H", 2)
+ push!(c, "CZ", 2, 1)
+ push!(c, "H", 2)
+ return compile(c)
+end
+
+import LinearAlgebra: diag as ldiag
+@testset "circuits" begin
+ U1 = compress(ccx())
+ U2 = CCXGate(3, 1, 2, 3)
+ @test opnorm(U1 - U2) < 1e-10
+ @test isidentity(U1 * dagger(U2))
+ @test isunitary(TGate(1,1))
+ @test isunitary(U1)
+ @test isunitary(U2)
+ @test isidentity(swap() * SwapGate(2, 1, 2))
+ @test isidentity(swapH() * SwapGate(2, 1, 2))
+ @test isidentity(cxH() * CXGate(2, 1, 2))
+ @test opnorm(CZGate(2, 1, 2)-CZGate(2, 2, 1)) < 1e-10
+ @test ldiag(op_to_dense(MCZGate(3))) == [1,1,1,1,1,1,1,-1]
+end
diff --git a/PauliStrings.jl-1.5/test/examples.jl b/PauliStrings.jl-1.5/test/examples.jl
new file mode 100644
index 0000000..e239553
--- /dev/null
+++ b/PauliStrings.jl-1.5/test/examples.jl
@@ -0,0 +1,165 @@
+
+using ProgressBars
+using Random
+
+rng = MersenneTwister(0)
+
+function construction_example1()
+ A = Operator(4)
+ A += "X", 1
+ A += "Y", 2
+ A += 2, "Z", 3
+ return A
+end
+
+function construction_example2()
+ A = Operator(8)
+ A += 2, "X", 1, "X", 2
+ A += 1.5, "Y", 2, "Z", 3
+ A += "Z", 3, "Z", 8
+ return A
+end
+
+function construction_example3()
+ A = Operator(6)
+ A += "XYZ111"
+ A += 5, "11XX11"
+ return A
+end
+
+"""XX hamiltonian https://arxiv.org/abs/1812.08657"""
+function XX(N)
+ H = Operator(N)
+ for j in 1:(N-1)
+ H += ('X', j, 'X', j + 1)
+ H += ('Z', j, 'Z', j + 1)
+ end
+ H += ('X', N, 'X', 1)
+ H += ('Z', N, 'Z', 1)
+ return H
+end
+
+
+"""X operator on all sites"""
+function X(N)
+ H = Operator(N)
+ for j in 1:N
+ H += ('X', j)
+ end
+ return H
+end
+
+"""Y operator on all sites"""
+function Y(N)
+ H = Operator(N)
+ for j in 1:N
+ H += ('Y', j)
+ end
+ return H
+end
+
+"""Z operator on all sites"""
+function Z(N)
+ H = Operator(N)
+ for j in 1:N
+ H += ('Z', j)
+ end
+ return H
+end
+
+# build a chaotic spin chain with periodic bc
+function chaotic_chain(N::Int)
+ H = ps.Operator(N)
+ # XX interractions
+ for j in 1:(N-1)
+ H += "X", j, "X", j + 1
+ end
+ H += "X", 1, "X", N # close the chain
+ # fields
+ for j in 1:N
+ H += -1.05, "Z", j
+ H += 0.5, "X", j
+ end
+ return H
+end
+
+
+
+function ising1D(N, g)
+ H = Operator(N)
+ for j in 1:(N-1)
+ H += "Z", j, "Z", j + 1
+ end
+ H += "Z", 1, "Z", N #periodic boundary condition
+ for j in 1:N
+ H += g, "X", j
+ end
+ return -H
+end
+
+
+
+
+# heisenberg evolution of the operator O using rk4
+# return tr(O(0)*O(t))/tr(O(t)^2)
+# M is the number of strings to keep at each step
+# noise is the amplitude of depolarizing noise
+function evolve(H, O, M, times, noise)
+ echo = []
+ O0 = deepcopy(O)
+ dt = times[2] - times[1]
+ for t in ProgressBar(times)
+ push!(echo, ps.trace(O * ps.dagger(O0)) / ps.trace(O0 * O0))
+ #preform one step of rk4, keep only M strings, do not discard O0
+ O = ps.rk4(H, O, dt; heisenberg=true, M=M, keep=O0)
+ #add depolarizingn oise
+ O = ps.add_noise(O, noise * dt)
+ # keep the M strings with the largest weight. Do not discard O0
+ O = ps.trim(O, M; keep=O0)
+ end
+ return real.(echo)
+end
+
+function evolve_chaotic()
+ N = 20 # system size
+ H = chaotic_chain(N) #hamiltonian
+ O = ps.Operator(N) #operator to time evolve
+ O += "Z", 1 # Z on site 1
+ times = range(0, stop=5, step=0.05)
+ noise = 0.01
+ S = evolve(H, O, 2^8, times, noise)
+ return S[end]
+end
+
+
+"""
+random 2-local operator with N sites and M terms for use in tests
+"""
+function rand_local2_M(N::Int, M::Int)
+ o = Operator(N)
+ for i in 1:M
+ k = rand(['X', 'Y', 'Z'])
+ l = rand(['X', 'Y', 'Z'])
+ i = rand(1:N)
+ j = i
+ while j == i
+ j = rand(1:N)
+ end
+ o += (randn(rng, Float64), k, i, l, j)
+ end
+ return compress(o)
+end
+
+
+"""
+random 1-local operator with N sites and M terms for use in tests
+"""
+function rand_local1_M(N::Int, M::Int)
+ o = Operator(N)
+ for i in 1:M
+ k = rand(['X', 'Y', 'Z'])
+ i = rand(1:N)
+ o += (randn(rng, Float64), k, i)
+ end
+ return compress(o)
+end
diff --git a/PauliStrings.jl-1.5/test/operator.jl b/PauliStrings.jl-1.5/test/operator.jl
new file mode 100644
index 0000000..ec0b872
--- /dev/null
+++ b/PauliStrings.jl-1.5/test/operator.jl
@@ -0,0 +1,71 @@
+
+
+# tests for the Operator type and operations on operators
+
+
+@testset "io" begin
+ @test construction_example1().w == [0, 1, 2]
+ @test construction_example2().v == [6, 0, 132]
+ @test construction_example2().w == [2, 3, 0]
+ @test construction_example3().v == [0, 6]
+ @test construction_example3().w == [12, 3]
+ @test get_coef(construction_example1(), 4, 0) == 2
+ @test get_coef(construction_example1(), 0, 1) == 1
+ @test get_coef(construction_example1(), 2, 2) == 1
+end
+
+@testset "random" begin
+ N = 10
+ @test length(ps.rand_local1(N)) == 3 * N
+ @test ps.rand_local1(N).N == N
+ @test length(ps.rand_local2(N)) == 9 * N * (N - 1) / 2
+ @test ps.rand_local2(N).N == N
+end
+
+@testset "operators" begin
+ @test Operator128 <: Operator
+ @test Operator64 <: Operator
+ @test OperatorTS1D128 <: OperatorTS1D
+ @test OperatorTS1D64 <: OperatorTS1D
+ @test OperatorTS1D <: Operator
+ @test typeof(Operator(10)) <: Operator
+ @test typeof(Operator(70)) <: Operator
+ ising10 = ising1D(10, 1)
+ ising70 = ising1D(70, 1)
+ @test typeof(ising10) <: Operator
+ @test typeof(ising70) <: Operator
+ @test typeof(ising10) == Operator64
+ @test typeof(ising70) == Operator128
+ ising10ts = OperatorTS1D(ising10)
+ ising70ts = OperatorTS1D(ising70)
+ @test typeof(ising10ts) == OperatorTS1D64
+ @test typeof(ising70ts) == OperatorTS1D128
+ @test typeof(Operator(ising10ts)) == Operator64
+ @test typeof(Operator(ising70ts)) == Operator128
+end
+
+@testset "operations" begin
+ for N in (10, 10)
+ O1 = rand_local1_M(N, 20)
+ O2 = rand_local2_M(N, 20)
+ @test trace(O1 * O2) == 0
+ @test trace(O1 * O1) != 0
+ @test opnorm(O1) > 0
+ @test opnorm(dagger(XX(N)) - XX(N)) == 0
+ @test opnorm(dagger(X(N)) - X(N)) == 0
+ @test opnorm(dagger(O1) - O1) == 0
+ @test opnorm(dagger(O2) - O2) == 0
+ @test opnorm(com(O1, O2) - (O1 * O2 - O2 * O1)) <= 1e-10
+ @test opnorm(com(O1, O2, anti=true) - (O1 * O2 + O2 * O1)) <= 1e-10
+ @test opnorm(com(O1, eye(N))) == 0
+ @test opnorm(com(XX(N), Y(N))) == 0
+ @test opnorm(com(XX(N), X(N))) == opnorm(com(XX(N), Z(N)))
+ @test trace(diag(O1)) == trace(O1)
+ @test trace(diag(O2)) == trace(O2)
+ end
+ O = Operator(6)
+ O += "XYZZ1Y"
+ @test xcount(O.v[1], O.w[1]) == 1
+ @test ycount(O.v[1], O.w[1]) == 2
+ @test zcount(O.v[1], O.w[1]) == 2
+end
diff --git a/PauliStrings.jl-1.5/test/operatorts1d.jl b/PauliStrings.jl-1.5/test/operatorts1d.jl
new file mode 100644
index 0000000..fb52013
--- /dev/null
+++ b/PauliStrings.jl-1.5/test/operatorts1d.jl
@@ -0,0 +1,30 @@
+
+@testset "operatorts1d" begin
+ N = 6
+ O1 = ising1D(N, 0.5)
+ O2 = ising1D(N, 1.1)
+ O1ts = OperatorTS1D(O1)
+ O2ts = OperatorTS1D(O2)
+ eps = 1e-10
+ @test typeof(O1ts + O2ts) == typeof(O1ts)
+ @test typeof(O1ts - O2ts) == typeof(O1ts)
+ @test typeof(O1ts * O2ts) == typeof(O1ts)
+ @test typeof(O1ts - 1) == typeof(O1ts)
+ @test typeof(O1ts + 1) == typeof(O1ts)
+ @test typeof(O1ts * 2) == typeof(O1ts)
+ @test typeof(O1ts / 2) == typeof(O1ts)
+ @test typeof(OperatorTS1D(ising1D(70, 1))) == OperatorTS1D128
+ @test typeof(OperatorTS1D(ising1D(40, 1))) == OperatorTS1D64
+ @test opnorm(Operator(O1ts) - O1) < eps
+ @test opnorm(Operator(O2ts) - O2) < eps
+ @test opnorm(Operator(O2ts) - O2) < eps
+ @test opnorm(Operator(O1ts * O2ts) - O1 * O2) < eps
+ @test opnorm(Operator(O1ts * O2ts * O2ts) - O1 * O2 * O2) < eps
+ @test opnorm(Operator(O1ts + O2ts) - (O1 + O2)) < eps
+ @test opnorm(Operator(O1ts * 7) - O1 * 7) < eps
+ @test opnorm(Operator(O1ts + 7) - (O1 + 7)) < eps
+ @test is_ts(O1)
+ @test !is_ts(ps.rand_local2(N))
+ @test trace(O1ts) == trace(dagger(O1))
+ @test trace(O1ts) == trace(diag(O1ts))
+end
diff --git a/PauliStrings.jl-1.5/test/runtests.jl b/PauliStrings.jl-1.5/test/runtests.jl
new file mode 100644
index 0000000..9036e3a
--- /dev/null
+++ b/PauliStrings.jl-1.5/test/runtests.jl
@@ -0,0 +1,18 @@
+using PauliStrings
+import PauliStrings as ps
+using Test
+using LinearAlgebra: norm
+
+
+
+ishermitian(H::Operator) = opnorm(H-dagger(H)) < 1e-10
+isunitary(U::Operator) = opnorm(U * dagger(U) - eye(U.N)) < 1e-10
+isidentity(U::Operator) = opnorm(U - eye(U.N)) < 1e-10
+
+
+include("examples.jl")
+
+include("operator.jl")
+include("algorithms.jl")
+include("operatorts1d.jl")
+include("circuits.jl")
diff --git a/PauliStrings.jl-1.5/test/truncation.jl b/PauliStrings.jl-1.5/test/truncation.jl
new file mode 100644
index 0000000..f1044de
--- /dev/null
+++ b/PauliStrings.jl-1.5/test/truncation.jl
@@ -0,0 +1,21 @@
+
+@testset "truncation" begin
+ N = 8
+ O2 = ps.rand_local2(N)
+ @test opnorm(ps.truncate(O2, 1)) == 0
+ @test opnorm(ps.truncate(O2, 2)) == opnorm(O2)
+ @test length(ps.trim(O2, 10)) == 10
+ @test length(ps.prune(O2, 2)) <= length(ps.prune(O2, 20))
+ @test length(ps.cutoff(XX(N) + 0.1 * X(N), 0.5)) == length(XX(N))
+ @test opnorm(ps.cutoff(O2, 0.5)) <= opnorm(O2)
+ @test opnorm(ps.add_noise(O2, 0.5)) < opnorm(O2)
+ @test opnorm(ps.add_noise(O2, 0.5)) < opnorm(ps.add_noise(O2, 0.1))
+ @test opnorm(k_local_part(O2, 1) - ps.truncate(O2, 1)) == 0
+ O1 = ising1D(N, 0.5)
+ O1ts = OperatorTS1D(O1)
+ @test opnorm(ps.truncate(O1ts, 2)) == opnorm(O1)
+ @test length(ps.trim(O1ts, 1)) == 1
+ @test opnorm(ps.cutoff(O1ts, 0.8)) <= opnorm(O1ts)
+ @test opnorm(ps.add_noise(O1ts, 0.5)) < opnorm(O1ts)
+ @test opnorm(Operator(ps.add_noise(O1ts, 0.5)) - ps.add_noise(O1, 0.5)) < 1e-10
+end
diff --git a/PauliStrings.jl-1.5/time_evolve_example.png b/PauliStrings.jl-1.5/time_evolve_example.png
new file mode 100644
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--
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