Commit
This commit does not belong to any branch on this repository, and may belong to a fork outside of the repository.
Abstractions for quantum information on qubits (#251)
* created abstractions for quantum information on qubits * changed how equality is handled and added tests * code review * cleanup on types and added tests * code review and composition for chi and ptm * better caching * code review for isapprox vs ==
- Loading branch information
Showing
4 changed files
with
410 additions
and
1 deletion.
There are no files selected for viewing
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Original file line number | Diff line number | Diff line change |
---|---|---|
@@ -0,0 +1,312 @@ | ||
module pauli | ||
|
||
export PauliBasis, PauliTransferMatrix, DensePauliTransferMatrix, | ||
ChiMatrix, DenseChiMatrix | ||
|
||
import Base: == | ||
import Base: isapprox | ||
import Base: * | ||
|
||
using ..bases, ..spin, ..superoperators | ||
using ..operators: identityoperator, AbstractOperator | ||
using ..superoperators: SuperOperator | ||
using ..operators_dense: DenseOperator | ||
using ..spin: sigmax, sigmay, sigmaz | ||
using SparseArrays: sparse | ||
using LinearAlgebra: tr | ||
|
||
""" | ||
PauliBasis(num_qubits::Int) | ||
Basis for an N-qubit space where `num_qubits` specifies the number of qubits. | ||
The dimension of the basis is 2²ᴺ. | ||
""" | ||
mutable struct PauliBasis{B<:Tuple{Vararg{Basis}}} <: Basis | ||
shape::Vector{Int} | ||
bases::B | ||
function PauliBasis(num_qubits::Int) | ||
shape = [2 for _ in 1:num_qubits] | ||
bases = Tuple(SpinBasis(1//2) for _ in 1:num_qubits) | ||
return new{typeof(bases)}(shape, bases) | ||
end | ||
end | ||
==(pb1::PauliBasis, pb2::PauliBasis) = length(pb1.bases) == length(pb2.bases) | ||
|
||
""" | ||
Base class for Pauli transfer matrix classes. | ||
""" | ||
abstract type PauliTransferMatrix{B1<:Tuple{PauliBasis, PauliBasis}, B2<:Tuple{PauliBasis, PauliBasis}} end | ||
|
||
|
||
""" | ||
DensePauliTransferMatrix(B1, B2, data) | ||
DensePauliTransferMatrix stored as a dense matrix. | ||
""" | ||
mutable struct DensePauliTransferMatrix{B1<:Tuple{PauliBasis, PauliBasis}, | ||
B2<:Tuple{PauliBasis, PauliBasis}, | ||
T<:Matrix{Float64}} <: PauliTransferMatrix{B1, B2} | ||
basis_l::B1 | ||
basis_r::B2 | ||
data::T | ||
function DensePauliTransferMatrix(basis_l::BL, basis_r::BR, data::T) where {BL<:Tuple{PauliBasis, PauliBasis}, | ||
BR<:Tuple{PauliBasis, PauliBasis}, | ||
T<:Matrix{Float64}} | ||
if length(basis_l[1])*length(basis_l[2]) != size(data, 1) || | ||
length(basis_r[1])*length(basis_r[2]) != size(data, 2) | ||
throw(DimensionMismatch()) | ||
end | ||
new{BL, BR, T}(basis_l, basis_r, data) | ||
end | ||
end | ||
|
||
PauliTransferMatrix(ptm::DensePauliTransferMatrix{B, B, Matrix{Float64}}) where B <: Tuple{PauliBasis, PauliBasis} = ptm | ||
|
||
function *(ptm0::DensePauliTransferMatrix{B, B, Matrix{Float64}}, | ||
ptm1::DensePauliTransferMatrix{B, B, Matrix{Float64}}) where B <: Tuple{PauliBasis, PauliBasis} | ||
return DensePauliTransferMatrix(ptm0.basis_l, ptm1.basis_r, ptm0.data*ptm1.data) | ||
end | ||
|
||
""" | ||
Base class for χ (process) matrix classes. | ||
""" | ||
abstract type ChiMatrix{B1<:Tuple{PauliBasis, PauliBasis}, B2<:Tuple{PauliBasis, PauliBasis}} end | ||
|
||
""" | ||
DenseChiMatrix(b, b, data) | ||
DenseChiMatrix stored as a dense matrix. | ||
""" | ||
mutable struct DenseChiMatrix{B1<:Tuple{PauliBasis, PauliBasis}, | ||
B2<:Tuple{PauliBasis, PauliBasis}, | ||
T<:Matrix{ComplexF64}} <: PauliTransferMatrix{B1, B2} | ||
basis_l::B1 | ||
basis_r::B2 | ||
data::T | ||
function DenseChiMatrix(basis_l::BL, basis_r::BR, data::T) where {BL<:Tuple{PauliBasis, PauliBasis}, | ||
BR<:Tuple{PauliBasis, PauliBasis}, | ||
T<:Matrix{ComplexF64}} | ||
if length(basis_l[1])*length(basis_l[2]) != size(data, 1) || | ||
length(basis_r[1])*length(basis_r[2]) != size(data, 2) | ||
throw(DimensionMismatch()) | ||
end | ||
new{BL, BR, T}(basis_l, basis_r, data) | ||
end | ||
end | ||
|
||
ChiMatrix(chi_matrix::DenseChiMatrix{B, B, Matrix{ComplexF64}}) where B <: Tuple{PauliBasis, PauliBasis} = chi_matrix | ||
|
||
""" | ||
A dictionary that represents the Pauli algebra - for a pair of Pauli operators | ||
σᵢσⱼ information about their product is given under the key "ij". The first | ||
element of the dictionary value is the Pauli operator, and the second is the | ||
scalar multiplier. For example, σ₀σ₁ = σ₁, and `"01" => ("1", 1)`. | ||
""" | ||
const pauli_multiplication_dict = Dict( | ||
"00" => ("0", 1.0+0.0im), | ||
"23" => ("1", 0.0+1.0im), | ||
"30" => ("3", 1.0+0.0im), | ||
"22" => ("0", 1.0+0.0im), | ||
"21" => ("3", -0.0-1.0im), | ||
"10" => ("1", 1.0+0.0im), | ||
"31" => ("2", 0.0+1.0im), | ||
"20" => ("2", 1.0+0.0im), | ||
"01" => ("1", 1.0+0.0im), | ||
"33" => ("0", 1.0+0.0im), | ||
"13" => ("2", -0.0-1.0im), | ||
"32" => ("1", -0.0-1.0im), | ||
"11" => ("0", 1.0+0.0im), | ||
"03" => ("3", 1.0+0.0im), | ||
"12" => ("3", 0.0+1.0im), | ||
"02" => ("2", 1.0+0.0im), | ||
) | ||
|
||
""" | ||
multiply_pauli_matirices(i4::String, j4::String) | ||
A function to algebraically determine result of multiplying two | ||
(N-qubit) Pauli matrices. Each Pauli matrix is represented by a string | ||
in base 4. For example, σ₃⊗σ₀⊗σ₂ would be "302". The product of any pair of | ||
Pauli matrices will itself be a Pauli matrix multiplied by any of the 1/4 roots | ||
of 1. | ||
""" | ||
cache_multiply_pauli_matrices() = begin | ||
local pauli_multiplication_cache = Dict() | ||
function _multiply_pauli_matirices(i4::String, j4::String) | ||
if (i4, j4) ∉ keys(pauli_multiplication_cache) | ||
pauli_multiplication_cache[(i4, j4)] = mapreduce(x -> pauli_multiplication_dict[prod(x)], | ||
(x,y) -> (x[1] * y[1], x[2] * y[2]), | ||
zip(i4, j4)) | ||
end | ||
return pauli_multiplication_cache[(i4, j4)] | ||
end | ||
end | ||
multiply_pauli_matirices = cache_multiply_pauli_matrices() | ||
|
||
function *(chi_matrix0::DenseChiMatrix{B, B, Matrix{ComplexF64}}, | ||
chi_matrix1::DenseChiMatrix{B, B, Matrix{ComplexF64}}) where B <: Tuple{PauliBasis, PauliBasis} | ||
|
||
num_qubits = length(chi_matrix0.basis_l[1].shape) | ||
sop_dim = 2 ^ prod(chi_matrix0.basis_l[1].shape) | ||
ret = zeros(ComplexF64, (sop_dim, sop_dim)) | ||
|
||
for ijkl in Iterators.product(0:(sop_dim-1), | ||
0:(sop_dim-1), | ||
0:(sop_dim-1), | ||
0:(sop_dim-1)) | ||
i, j, k, l = ijkl | ||
if (chi_matrix0.data[i+1, j+1] != 0.0) & (chi_matrix1.data[k+1, l+1] != 0.0) | ||
i4, j4, k4, l4 = map(x -> string(x, base=4, pad=2), ijkl) | ||
|
||
pauli_product_ik = multiply_pauli_matirices(i4, k4) | ||
pauli_product_lj = multiply_pauli_matirices(l4, j4) | ||
|
||
ret[parse(Int, pauli_product_ik[1], base=4)+1, | ||
parse(Int, pauli_product_lj[1], base=4)+1] += (pauli_product_ik[2] * pauli_product_lj[2] * chi_matrix0.data[i+1, j+1] * chi_matrix1.data[k+1, l+1]) | ||
end | ||
end | ||
return DenseChiMatrix(chi_matrix0.basis_l, chi_matrix0.basis_r, ret / 2^num_qubits) | ||
end | ||
|
||
|
||
# TODO MAKE A GENERATOR FUNCTION | ||
""" | ||
pauli_operators(num_qubits::Int) | ||
Generate a list of N-qubit Pauli operators. | ||
""" | ||
function pauli_operators(num_qubits::Int) | ||
pauli_funcs = (identityoperator, sigmax, sigmay, sigmaz) | ||
po = [] | ||
for paulis in Iterators.product((pauli_funcs for _ in 1:num_qubits)...) | ||
basis_vector = reduce(⊗, f(SpinBasis(1//2)) for f in paulis) | ||
push!(po, basis_vector) | ||
end | ||
return po | ||
end | ||
|
||
""" | ||
pauli_basis_vectors(num_qubits::Int) | ||
Generate a matrix of basis vectors in the Pauli representation given a number | ||
of qubits. | ||
""" | ||
function pauli_basis_vectors(num_qubits::Int) | ||
po = pauli_operators(num_qubits) | ||
sop_dim = 4 ^ num_qubits | ||
return mapreduce(x -> sparse(reshape(x.data, sop_dim)), (x, y) -> [x y], po) | ||
end | ||
|
||
""" | ||
PauliTransferMatrix(sop::DenseSuperOperator) | ||
Convert a superoperator to its representation as a Pauli transfer matrix. | ||
""" | ||
function PauliTransferMatrix(sop::DenseSuperOperator{B, B, Matrix{ComplexF64}}) where B <: Tuple{PauliBasis, PauliBasis} | ||
num_qubits = length(sop.basis_l[1].bases) | ||
pbv = pauli_basis_vectors(num_qubits) | ||
sop_dim = 4 ^ num_qubits | ||
data = Matrix{Float64}(undef, (sop_dim, sop_dim)) | ||
data .= real.(pbv' * sop.data * pbv / √sop_dim) | ||
return DensePauliTransferMatrix(sop.basis_l, sop.basis_r, data) | ||
end | ||
|
||
SuperOperator(unitary::DenseOperator{B, B, Matrix{ComplexF64}}) where B <: PauliBasis = spre(unitary) * spost(unitary') | ||
SuperOperator(sop::DenseSuperOperator{B, B, Matrix{ComplexF64}}) where B <: Tuple{PauliBasis, PauliBasis} = sop | ||
|
||
""" | ||
SuperOperator(ptm::DensePauliTransferMatrix) | ||
Convert a Pauli transfer matrix to its representation as a superoperator. | ||
""" | ||
function SuperOperator(ptm::DensePauliTransferMatrix{B, B, Matrix{Float64}}) where B <: Tuple{PauliBasis, PauliBasis} | ||
num_qubits = length(ptm.basis_l[1].bases) | ||
pbv = pauli_basis_vectors(num_qubits) | ||
sop_dim = 4 ^ num_qubits | ||
data = Matrix{ComplexF64}(undef, (sop_dim, sop_dim)) | ||
data .= pbv * ptm.data * pbv' / √sop_dim | ||
return DenseSuperOperator(ptm.basis_l, ptm.basis_r, data) | ||
end | ||
|
||
""" | ||
PauliTransferMatrix(unitary::DenseOperator) | ||
Convert an operator, presumably a unitary operator, to its representation as a | ||
Pauli transfer matrix. | ||
""" | ||
PauliTransferMatrix(unitary::DenseOperator{B, B, Matrix{ComplexF64}}) where B <: PauliBasis = PauliTransferMatrix(SuperOperator(unitary)) | ||
|
||
""" | ||
ChiMatrix(unitary::DenseOperator) | ||
Convert an operator, presumably a unitary operator, to its representation as a χ matrix. | ||
""" | ||
function ChiMatrix(unitary::DenseOperator{B, B, Matrix{ComplexF64}}) where B <: PauliBasis | ||
num_qubits = length(unitary.basis_l.bases) | ||
pbv = pauli_basis_vectors(num_qubits) | ||
aj = pbv' * reshape(unitary.data, 4 ^ num_qubits) | ||
return DenseChiMatrix((unitary.basis_l, unitary.basis_l), (unitary.basis_r, unitary.basis_r), aj * aj' / (2 ^ num_qubits)) | ||
end | ||
|
||
""" | ||
ChiMatrix(sop::DenseSuperOperator) | ||
Convert a superoperator to its representation as a Chi matrix. | ||
""" | ||
function ChiMatrix(sop::DenseSuperOperator{B, B, Matrix{ComplexF64}}) where B <: Tuple{PauliBasis, PauliBasis} | ||
num_qubits = length(sop.basis_l) | ||
sop_dim = 4 ^ num_qubits | ||
po = pauli_operators(num_qubits) | ||
data = Matrix{ComplexF64}(undef, (sop_dim, sop_dim)) | ||
for (idx, jdx) in Iterators.product(1:sop_dim, 1:sop_dim) | ||
data[idx, jdx] = tr((spre(po[idx]) * spost(po[jdx])).data' * sop.data) / √sop_dim | ||
end | ||
return DenseChiMatrix(sop.basis_l, sop.basis_r, data) | ||
end | ||
|
||
""" | ||
PauliTransferMatrix(chi_matrix::DenseChiMatrix) | ||
Convert a χ matrix to its representation as a Pauli transfer matrix. | ||
""" | ||
function PauliTransferMatrix(chi_matrix::DenseChiMatrix{B, B, Matrix{ComplexF64}}) where B <: Tuple{PauliBasis, PauliBasis} | ||
num_qubits = length(chi_matrix.basis_l) | ||
sop_dim = 4 ^ num_qubits | ||
po = pauli_operators(num_qubits) | ||
data = Matrix{Float64}(undef, (sop_dim, sop_dim)) | ||
for (idx, jdx) in Iterators.product(1:sop_dim, 1:sop_dim) | ||
data[idx, jdx] = tr(mapreduce(x -> po[idx] * po[x[1]] * po[jdx] * po[x[2]] * chi_matrix.data[x[1], x[2]], | ||
+, | ||
Iterators.product(1:16, 1:16)).data) / sop_dim |> real | ||
end | ||
return DensePauliTransferMatrix(chi_matrix.basis_l, chi_matrix.basis_r, data) | ||
end | ||
|
||
""" | ||
SuperOperator(chi_matrix::DenseChiMatrix) | ||
Convert a χ matrix to its representation as a superoperator. | ||
""" | ||
function SuperOperator(chi_matrix::DenseChiMatrix{B, B, Matrix{ComplexF64}}) where B <: Tuple{PauliBasis, PauliBasis} | ||
return SuperOperator(PauliTransferMatrix(chi_matrix)) | ||
end | ||
|
||
""" | ||
ChiMatrix(ptm::DensePauliTransferMatrix) | ||
Convert a Pauli transfer matrix to its representation as a χ matrix. | ||
""" | ||
function ChiMatrix(ptm::DensePauliTransferMatrix{B, B, Matrix{Float64}}) where B <: Tuple{PauliBasis, PauliBasis} | ||
return ChiMatrix(SuperOperator(ptm)) | ||
end | ||
|
||
"""Equality for all varieties of superoperators.""" | ||
==(sop1::T, sop2::T) where T<:Union{DensePauliTransferMatrix, DenseSuperOperator, DenseChiMatrix} = sop1.data == sop2.data | ||
==(sop1::Union{DensePauliTransferMatrix, DenseSuperOperator, DenseChiMatrix}, sop2::Union{DensePauliTransferMatrix, DenseSuperOperator, DenseChiMatrix}) = false | ||
|
||
"""Approximate equality for all varieties of superoperators.""" | ||
function isapprox(sop1::T, sop2::T; kwargs...) where T<:Union{DensePauliTransferMatrix, DenseSuperOperator, DenseChiMatrix} | ||
return isapprox(sop1.data, sop2.data; kwargs...) | ||
end | ||
|
||
end # end module |
This file contains bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Oops, something went wrong.