sc-qmt

A repository with the code for quantum measurement tomography (QMT), quantum state tomography (QST) and self-consistent tomography, following the paper Self-consistent quantum measurement tomography based on semidefinite programming (Cattaneo et al., Phys. Rev. Res. 5, 033154 (2023) (https://journals.aps.org/prresearch/abstract/10.1103/PhysRevResearch.5.033154)). All the results of the paper can be found and reproduced here.

Installation

Clone the repository using

git clone https://github.com/Algorithmiq/sc-qmt.git

Create a Conda environment from the sc-qmt.yaml file in the root of the repository:
```
conda env create -f sc-qmt.yaml
```
Note that this environment is called sc-qmt. The new environment will contain Poetry and the packages needed to build the C extensions of our dependencies.
Activate the new environment:
```
conda activate sc-qmt
```
Install project and dependencies for local development:
```
poetry install
```

Usage

Reproducing the plots of the paper

The data for the plots of the paper are stored in the folder data_folder. The notebook Figures.ipynb can be used to produce and save the plots.

Using semidefinite programming as a fitting method for QMT

First, we import all the functions for quantum tomography via SDP from sc-qmt.py:

from sc_qmt import *

Then, let us simulate a QMT experiment by preparing a set of input states given by the Pauli eigenstates rhoPaulis for the SIC-POVM with effects given by the list M:

theta = 2 * np.pi / 3
phi = 4 * np.pi / 3 

v_0 = np.array([1,0])
v_1 = np.array([1, np.sqrt(2)]) / np.sqrt(3)
v_2 = np.array([1, np.exp(theta * 1j) * np.sqrt(2)]) / np.sqrt(3)
v_3 = np.array([1, np.exp(phi * 1j) * np.sqrt(2)]) / np.sqrt(3)

M = [] # list of effects of the SIC-POVM
M.append(np.outer(v_0.conj().T, v_0)/2)
M.append(np.outer(v_1.conj().T, v_1)/2)
M.append(np.outer(v_2.conj().T, v_2)/2)
M.append(np.outer(v_3.conj().T, v_3)/2)

rhoPaulis = [np.array([[1, 0],
       [0, 0]]), np.array([[0, 0],
       [0, 1]]), np.array([[0.5+0.j, 0.5+0.j],
       [0.5+0.j, 0.5+0.j]]), np.array([[ 0.5+0.j, -0.5+0.j],
       [-0.5-0.j,  0.5+0.j]]), np.array([[0.5+0.j , 0. +0.5j],
       [0. -0.5j, 0.5+0.j ]]), np.array([[0.5+0.j , 0. -0.5j],
       [0. +0.5j, 0.5+0.j ]])] # list of Pauli eigenstates

The experimental frequencies of the QMT experiment with 10000 shots can be generated through the following function:

num_shots = 10000
exp_freq = generate_frequencies(num_shots, M, rhoPaulis)

Then, we can process the data using a semidefinite program and we reconstruct the POVM effects:

process_results = run_SDP_QMT(rhoPaulis, exp_freq, SDPtype = 'singleDelta') # single-delta SDP
reconstructed_effects = process_results[1] # list of output effects
SDP_delta = process_results[0] # infinite distance between experimental and reconstructed probability distribution

Examples

Some examples of how to use the code for the different tomographic tasks discussed in the paper can be found in the files script_*.py. The examples include measurement tomography with and without noise on the input states and self-consistent tomography through the see-saw method.

Authors and citation

If you find this code useful, please consider citing the paper Cattaneo et al., Phys. Rev. Res. 5, 033154 (2023).

BibTeX record:

@article{PhysRevResearch.5.033154,
  title = {Self-consistent quantum measurement tomography based on semidefinite programming},
  author = {Cattaneo, Marco and Rossi, Matteo A. C. and Korhonen, Keijo and Borrelli, Elsi-Mari and Garc\'{\i}a-P\'erez, Guillermo and Zimbor\'as, Zolt\'an and Cavalcanti, Daniel},
  journal = {Phys. Rev. Res.},
  volume = {5},
  issue = {3},
  pages = {033154},
  numpages = {14},
  year = {2023},
  month = {Sep},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevResearch.5.033154},
  url = {https://link.aps.org/doi/10.1103/PhysRevResearch.5.033154}
}