Quantum Tomography

PastaQ.tomography — Function

tomography(data::Matrix{Pair{String, Int}}, sites::Vector{<:Index};
           method::String = "linear_inversion",
           fillzeros::Bool = true,
           trρ::Number = 1.0,
           max_iters::Int = 10000,
           kwargs...)

tomography(data::Matrix{Pair{String,Pair{String, Int}}}, sites::Vector{<:Index};
           method::String="linear_inversion", kwargs...)

Run full quantum tomography for a set of input measurement data data. If the input data consists of a list of Pair{String, Int}, it is interpreted as quantum state tomography. Each data point is a single measurement outcome, e.g. "X" => 1 to refer to a measurement in the X basis with binary outcome 1. If instead the input data is a collection of Pair{String,Pair{String, Int}}, it is interpreted as quantum process tomography, with each data-point corresponding to having input a given state to the channel, followed by a measurement in a basis, e.g. "X+" => ("Z" => 0) referring to an input $|+\rangle$ state, followed by a measurement in the Z basis with outcome 0.

There are three methods to perform tomography (we show state tomography here as an example):

method = "linear_inversion" (or "LI"): optimize a variational density matrix $\rho$ (or Choi matrix)1

method = "least_squares" (or "LS"):
method = "maximum_likelihood" (or "ML"):

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tomography(train_data::AbstractMatrix, L::LPDO;
           optimizer::Optimizer,
           observer! = nothing,
           kwargs...)

Run quantum process tomography using a variational model $L$ to fit train_data. Depending on the type of train_data, run state or process tomography (see full tomography docs for the data format).

For quantum state tomography, optimize the average Kullbach-Leibler (KL) divergence for projective measurements in a set of bases:

\[C(\theta) = -\frac{1}{|D|}\sum_{k=1}^{|D|} \log P(x_k^{(b)})\]

where the cost function is computed as

\[C(\theta) = -\frac{1}{|D|}\sum_{k=1}^{|D|} \log |\langle x_k|U_b|\psi(\theta)\rangle|^2\]

for input MPS variational wavefunction, and

\[C(\theta) = -\frac{1}{|D|}\sum_{k=1}^{|D|} \log \langle x_k|U_b \rho(\theta) U_b^\dagger|x_k\rangle\]

for input LPDO variational density operators. Here $U_b$ is the depth-1 unitary that rotates the variational state into the measurement basis $b$ for outcome $x^{(b)}$.

For quantum process tomography, optimize the KL divergence for the process probability distribution

\[C(\theta) = -\frac{1}{|D|}\sum_{k=1}^{|D|} \log P(x_k^{(b)}|\xi)\]

where $\xi$ is the input state to the channel. The cost function is computed as

\[C(\theta) = -\frac{1}{|D|}\sum_{k=1}^{|D|} \log |\langle x_k|U_b|\tilde\Phi(\theta)\rangle|^2\]

for input MPO variational unitary operator (trated as a MPS $|\Phi\rangle$ after appropriate vectorization), and

\[C(\theta) = -\frac{1}{|D|}\sum_{k=1}^{|D|} \log \langle x|U_b \tilde\Lambda(\theta) U_b^\dagger|x\rangle\]

for input LPDO variational Choi matrix. Here we refer to $\tilde\Phi$ and $\tilde\Lambda$ respectively as the projection of the unitary operator or Choi matrix into the input state $|\xi\rangle$ to the channel.

Keyword arguments:

train_data: pairs of preparation/ (basis/outcome): ("X+"=>"X"=>0, "Z-"=>"Y"=>1, "Y+"=>"Z"=>0, …).
L: variational model (MPO/LPDO).
optimizer: optimizer object for stochastic optimization (from Optimisers.jl).
observer!: if provided, keep track of training metrics (from Observers.jl).
batch_size: number of samples used for one gradient update.
epochs: number of training iterations.
target: target quantum state/process for distance measures (e.g. fidelity).
test_data: data for computing cross-validation.
observer_step = 1: how often the Observer is called to record measurements.
outputlevel = 1: amount of information printed on screen during training.
outputpath: if provided, save metrics on file.
savestate: if true, save the variational state on file.
print_metrics = []: print these metrics on screen during training.
earlystop: if true, use pre-defined early-stop function. A function can also be passed as earlystop, in which case if the function (evaluated at each iteration) returns true, the training is halted.

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