MATLAB library for the discrete variables quantum state and quantum process tomography using root approach and maximum likelihood estimation. The library contains a set of tools for quantum state and quantum process reconstruction by the complementary measurements results, estimation of statistical adequacy and theoretical analysis of reconstruction fidelity. Full product documentation is available here.
- Getting Started
- Definitions, algorithms and data format
- Quantum state tomography
- Quantum process tomography
- License
Required toolboxes:
- Statistics toolbox
- Symbolic Math Toolbox (optional)
To install the library clone the repository or download and unpack zip-archive. Run the startup script before use.
>> rt_startup
For the entities definitions, algorithms description and required data format see full product documentation.
Examples directory of the project contains a set of examples that show basic features of the library. Below we briefly review the quantum state tomography example.
Consider an example of a pure ququart state reconstruction using mutually-unbiased bases measurement protocol.
dim = 4; % System dimension
r_true = 1; % True state rank
dm_true = rt_randstate(dim, 'Rank', r_true); % True state
nshots = 1e3; % Total sample size
proto = rt_proto_measurement('mub', 'dim', dim); % Generate measurements operators
The rt_experiment class allows one to store and simulate tomography data.
ex = rt_experiment(dim, 'state'); % Generate experiment instance
ex.set_data('proto', proto, 'nshots', nshots); % Set measurements protocol
ex.simulate(dm_true); % Simulate measurement data
The reconstruction is performed using the rt_dm_reconstruct function. By default the state rank is estimated automatically using the adequacy criteria.
dm_rec = rt_dm_reconstruct(ex, 'Display', 10);
Fidelity = rt_fidelity(dm_rec, dm_true);
fprintf('Fidelity: %.6f\n', Fidelity);
Output:
=== Automatic rank estimation ===
=> Try rank 1
Optimization: fixed point iteration method
Iteration 33 Delta 8.0049e-09
=> Rank 1 is statistically significant at significance level 0.05. Procedure terminated.
Fidelity: 0.996887
Using the fiducial approach and the theoretical infidelity distribution one can use the reconstruction result to estimate the guaranteed reconstruction fidelity. In the following code we estimate the 95%-probability fidelity bound . That means that we get the fidelity
or higher with probability 95%.
d = rt_bound(dm_rec, ex); % Calculate variances
Fidelity95 = 1 - rt_gchi2inv(0.95, d); % Get fidelity bound
fprintf('Fiducial 95%% fidelity bound: %.6f\n', Fidelity95);
Output:
Fiducial 95% fidelity bound: 0.992890
The following code plots the infidelity distribution based on the true state and shows the fidelity of reconstructed state.
d = rt_bound(dm_true, ex);
[p, df] = rt_gchi2pdf([], d);
figure; hold on; grid on;
plot(df, p, 'LineWidth', 1.5, 'DisplayName', 'Theory');
plot([1,1] * (1 - Fidelity), ylim, 'LineWidth', 1.5, 'DisplayName', 'Reconstruction');
xlabel('$$1-F$$', 'Interpreter', 'latex');
legend('show');
The theoretical infidelity mean and variance could be obtained from the variance vector.
sum(d) % Mean infidelity
2*sum(d.^2) % Infidelity variance
1-sum(d) % Mean fidelity
The measurements protocol for the quantum process tomography could be generated by the combination of preparation and measurements protocols.
proto_prep = rt_proto_preparation('tetra');
proto_meas = rt_proto_measurement('mub', 'dim', dim);
proto = rt_proto_process(proto_prep, proto_meas);
The reconstruction of a quantum process is performed in a very similar way to the quantum state reconstruction.
chi = rt_chi_reconstruct(ex);
The fidelity bound estimation is also done in a similar way.
d = rt_bound(chi, ex);
All code found in this repository is licensed under GPL v3