Single Qubit Tomography

This project has been made during the Quantum Computing Summer School 2021 at Los Alamos National Laboratory with the help of:

• Carleton Coffrin
• Marc Vuffray
• Andrey Lokhov
• Jon Nelson

and continued during my PhD.

Quick start

Install

In order to start using the code in this repository, first install the sqt package with the command

git clone https://github.com/nelimee/sqt.git
python -m pip install -e sqt/

Command line interface

The sqt package provides scripts that are installed along with the package.

sqt_bloch_tomography_submit

This script makes the circuit building and submission process easier by providing a simple interface to the user.

The script inputs can be listed with the --help option:

>>> sqt_bloch_tomography_submit --help
usage: sqt_bloch_tomography_submit [-h] [--equidistant-points EQUIDISTANT_POINTS] [--hub HUB] [--group GROUP] [--project PROJECT] [--backup-dir BACKUP_DIR] [--rep-delay REP_DELAY] [--shots SHOTS] [--delay-dt DELAY_DT]
                                   [--max-qubits MAX_QUBITS] [--local-backend] [--noisy-simulator]
                                   backend approximate_point_number {pauli,tetrahedral,equidistant}

Execute the circuits to perform state tomography for approximately uniformly distributed quantum states over the Bloch sphere.

positional arguments:
  backend               Backend to perform tomography on.
  approximate_point_number
                        Approximate number of points used to cover the Bloch sphere. The actual number of points used might vary a little bit from this value. Each point will be tomographied independently.
  {pauli,tetrahedral,equidistant}
                        Name of the tomography basis used to perform quantum state tomography.

options:
  -h, --help            show this help message and exit
  --equidistant-points EQUIDISTANT_POINTS
                        If basis is 'equidistant', the number of approximately equidistant projectors that will be used. Else, this option is ignored.
  --hub HUB             Hub of your IBMQ provider. Defaults to 'ibm-q' available to all users.
  --group GROUP         Group of your IBMQ provider. Defaults to 'open' available to all users.
  --project PROJECT     Project of your IBMQ provider. Defaults to 'main' available to all users.
  --backup-dir BACKUP_DIR
                        Directory used to save the data needed to post-process job results.
  --rep-delay REP_DELAY
                        Delay between each shot. Default to the backend default value.
  --shots SHOTS         Number of shots performed for each circuit.
  --delay-dt DELAY_DT   Duration (in dt) of the delay to insert before state tomography.
  --max-qubits MAX_QUBITS
                        Maximum number of qubits that should be used.
  --local-backend       If present, the backend used is a local one.
  --noisy-simulator     If present, the given IBMQ backend will be used to initialise a noisy simulator. Implies '--local-backend'.

Here are some examples of usage:

# Submiting the quantum circuits to tomography approximately 100 different (and
# approximately evenly-spread on the Bloch sphere) 1-qubit states using the 
# tetrahedral basis (4 measurements) on ibm_algiers.
>>> sqt_bloch_tomography_submit --hub ibm-q --group open --project main ibm_algiers 100 tetrahedral
# In the following, the argument ibm_algiers is ignored due to the option
# --local-backend being given.
>>> sqt_bloch_tomography_submit ibm_algiers 100 tetrahedral --local-backend
# In the following, a noisy simulator initialised with ibm_algiers current calibrations will
# be used.
>>> sqt_bloch_tomography_submit --hub ibm-q --group open --project main ibm_algiers 100 tetrahedral --noisy-simulator

You can of course use a custom provider, change the rep_delay value for backends that implement this feature, specify a maximum number of qubits to use (useful for the simulators and can be used to limit the number of qubits used on a given backend) or change the directory that will be used to store the backup file.

The backup file is the main output of this script: it contains all the needed information for sqt_bloch_tomography_recover.

sqt_bloch_tomography_recover

This script takes as input a backup file created with sqt_bloch_tomography_submit and reconstructs the density matrices representing the tomographied quantum state with the provided reconstruction method.

The script inputs can be listed with the --help option:

>>> sqt_bloch_tomography_recover --help
usage: sqt_bloch_tomography_recover [-h]
                                    backup_filepath {mle,pauli,lssr,grad}
                                    [{mle,pauli,lssr,grad} ...]

Post-process the job result and compute the reconstructed density matrices.

positional arguments:
  backup_filepath       Backup file path that has been saved during the job submission.
  {mle,pauli,lssr,grad}
                        Post-processing method used to reconstruct the density matrices.

optional arguments:
  -h, --help            show this help message and exit

Here are some examples of usage:

# Replace the "[file with .pkl extension]" with the backup file
# obtained with sqt_bloch_tomography_submit
>>> sqt_bloch_tomography_recover [file with .pkl extension] grad
>>> sqt_bloch_tomography_recover [file with .pkl extension] lssr grad mle

Using the sqt package directly

The sqt package can be used to implement quantum state tomography. It has been specifically designed for efficient 1-qubit tomography but might be improved in the future for efficient multi-qubit tomography. Moreover, sqt only supports Qiskit for the moment. More frameworks might come if there is a need.

Using sqt is quite simple and the different steps are explained below.

1. Pick your tomography basis

sqt implements 3 different 1-qubit tomography basis that you can choose from:

1. The pauli basis that measure along the X, Y and Z axis.
2. The tetrahedral basis that implements the SIC-POVM basis described here.
3. The equidistant basis that perform measurements in a user-defined number of projectors that are approximately equidistant from each other. This basis has not been used yet and might be useful when redundancy is desired.

from sqt.basis.equidistant import EquidistantMeasurementBasis
from sqt.basis.pauli import PauliMeasurementBasis
from sqt.basis.tetrahedral import TetrahedralMeasurementBasis

pauli_basis = PauliMeasurementBasis()
tetrahedral_basis = TetrahedralMeasurementBasis()
equidistant_basis = EquidistantMeasurementBasis(approximative_point_number=10)

# Define the basis that will be used in this README:
basis = tetrahedral_basis

2. Construct the tomography circuits

Once the basis has been picked, a simple call to one_qubit_tomography_circuits will generate all the necessary quantum circuits to perform quantum tomography.

from sqt.circuits import one_qubit_tomography_circuits

circuit = None  # Replace with a QuantumCircuit instance to tomography
                # No measurements should be appended to this circuit!
tomography_circuits = one_qubit_tomography_circuits(
    circuit,         # The circuit to tomography
    basis,           # The basis used for tomography
    # Number of qubits the tomography circuits should be repeated on.
    qubit_number=7,
)

Parallel execution of 1-qubit tomography circuits is natively supported by sqt and is provided as simple keywords.

3. Post process the results

Now that the quantum circuits returned by one_qubit_tomography_circuits have been executed, they should be processed in order to recover the density matrix.

from sqt.fit.grad import post_process_tomography_results_grad
from sqt.fit.mle import post_process_tomography_results_mle

backend = None # Fill this with the backend of your choice
result = backend.run(tomography_circuits).result()

density_matrices = post_process_tomography_results_mle(
    result, 
    # Warning: this is the original circuit without basis change nor
    #          measurements at the end. Do not change it between the
    #          call to one_qubit_tomography_circuits and here!
    circuit, 
    basis,
    qubit_number=7
)