Quantum State Tomography¶

Quantum tomography is an experimental procedure to reconstruct a description of part of a quantum system from the measurement outcomes of a specific set of experiments. In particular, quantum state tomography reconstructs the density matrix of a quantum state by preparing the state many times and measuring them in a tomographically complete basis of measurement operators.

Note

This tutorial requires the qiskit-aer and qiskit-ibm-runtime packages to run simulations. You can install them with python -m pip install qiskit-aer qiskit-ibm-runtime.

We first initialize a simulator to run the experiments on.

from qiskit_aer import AerSimulator
from qiskit_ibm_runtime.fake_provider import FakePerth

backend = AerSimulator.from_backend(FakePerth())

To run a state tomography experiment, we initialize the experiment with a circuit to prepare the state to be measured. We can also pass in an Operator or a Statevector to describe the preparation circuit.

import qiskit
from qiskit_experiments.framework import ParallelExperiment
from qiskit_experiments.library import StateTomography

# GHZ State preparation circuit
nq = 2
qc_ghz = qiskit.QuantumCircuit(nq)
qc_ghz.h(0)
qc_ghz.s(0)
for i in range(1, nq):
    qc_ghz.cx(0, i)

# QST Experiment
qstexp1 = StateTomography(qc_ghz)
qstdata1 = qstexp1.run(backend, seed_simulation=100).block_for_results()

# Print results
display(qstdata1.analysis_results(dataframe=True))

	name	experiment	components	value	quality	backend	run_time	trace	eigvals	raw_eigvals	rescaled_psd	fitter_metadata	conditional_probability	positive
9a4002b0	state	StateTomography	[Q0, Q1]	DensityMatrix([[ 0.4749349 +0.j , -0.00...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9137554587098121, 0.04180368599957844, 0.02...	[0.9137554587098121, 0.04180368599957844, 0.02...	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
0a328501	state_fidelity	StateTomography	[Q0, Q1]	0.913086	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
02b2147b	positive	StateTomography	[Q0, Q1]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None

Tomography Results¶

The main result for tomography is the fitted state, which is stored as a DensityMatrix object:

state_result = qstdata1.analysis_results("state", dataframe=True).iloc[0]
print(state_result.value)

DensityMatrix([[ 0.4749349 +0.j        , -0.00813802+0.01204427j,
                 0.00764974-0.00227865j,  0.00048828-0.44042969j],
               [-0.00813802-0.01204427j,  0.03222656+0.j        ,
                -0.00048828+0.00390625j, -0.00406901+0.00455729j],
               [ 0.00764974+0.00227865j, -0.00048828-0.00390625j,
                 0.02246094+0.j        ,  0.01432292-0.01432292j],
               [ 0.00048828+0.44042969j, -0.00406901-0.00455729j,
                 0.01432292+0.01432292j,  0.4703776 +0.j        ]],
              dims=(2, 2))

We can also visualize the density matrix:

from qiskit.visualization import plot_state_city
state = qstdata1.analysis_results("state", dataframe=True).iloc[0].value
plot_state_city(state, title='Density Matrix')

The state fidelity of the fitted state with the ideal state prepared by the input circuit is stored in the "state_fidelity" result field. Note that if the input circuit contained any measurements the ideal state cannot be automatically generated and this field will be set to None.

fid_result = qstdata1.analysis_results("state_fidelity", dataframe=True).iloc[0]
print("State Fidelity = {:.5f}".format(fid_result.value))

State Fidelity = 0.91309

Additional state metadata¶

Additional data is stored in the tomography under additional fields. This includes

eigvals: the eigenvalues of the fitted state
trace: the trace of the fitted state
positive: Whether the eigenvalues are all non-negative

If trace rescaling was performed this dictionary will also contain a raw_trace field containing the trace before rescaling. Futhermore, if the state was rescaled to be positive or trace 1 an additional field raw_eigvals will contain the state eigenvalues before rescaling was performed.

for col in ["eigvals", "trace", "positive"]:
    print(f"{col}: {state_result[col]}")

eigvals: [0.91375546 0.04180369 0.02965482 0.01478603]
trace: 1.0000000000000016
positive: True

To see the effect of rescaling, we can perform a “bad” fit with very low counts:

# QST Experiment
bad_data = qstexp1.run(backend, shots=10, seed_simulation=100).block_for_results()
bad_state_result = bad_data.analysis_results("state", dataframe=True).iloc[0]

# Print result
for key, val in bad_state_result.items():
    print(f"{key}: {val}")

name: state
experiment: StateTomography
components: [<Qubit(Q0)>, <Qubit(Q1)>]
value: DensityMatrix([[ 0.33450396+0.j        ,  0.05938614-0.04566114j,
                -0.11306592+0.05611775j,  0.05798364-0.39626276j],
               [ 0.05938614+0.04566114j,  0.0436272 +0.j        ,
                -0.06379144+0.00282676j,  0.09825567-0.06775048j],
               [-0.11306592-0.05611775j, -0.06379144-0.00282676j,
                 0.09861805+0.j        , -0.13320385+0.12088397j],
               [ 0.05798364+0.39626276j,  0.09825567+0.06775048j,
                -0.13320385-0.12088397j,  0.52325079+0.j        ]],
              dims=(2, 2))
quality: unknown
backend: aer_simulator_from(fake_perth)
run_time: None
trace: 1.0000000000000007
eigvals: [0.92575281 0.07424719 0.         0.        ]
raw_eigvals: [ 1.05200742  0.20050181 -0.06354245 -0.18896677]
rescaled_psd: True
fitter_metadata: {'fitter': 'linear_inversion', 'fitter_time': 0.0043561458587646484}
conditional_probability: 1.0
positive: True

Tomography Fitters¶

The default fitters is linear_inversion, which reconstructs the state using dual basis of the tomography basis. This will typically result in a non-positive reconstructed state. This state is rescaled to be positive-semidefinite (PSD) by computing its eigen-decomposition and rescaling its eigenvalues using the approach from Ref. [1].

There are several other fitters are included (See API documentation for details). For example, if cvxpy is installed we can use the cvxpy_gaussian_lstsq() fitter, which allows constraining the fit to be PSD without requiring rescaling.

try:
    import cvxpy

    # Set analysis option for cvxpy fitter
    qstexp1.analysis.set_options(fitter='cvxpy_gaussian_lstsq')

    # Re-run experiment
    qstdata2 = qstexp1.run(backend, seed_simulation=100).block_for_results()

    state_result2 = qstdata2.analysis_results("state", dataframe=True).iloc[0]
    for key, val in state_result2.items():
        print(f"{key}: {val}")

except ModuleNotFoundError:
    print("CVXPY is not installed")

name: state
experiment: StateTomography
components: [<Qubit(Q0)>, <Qubit(Q1)>]
value: DensityMatrix([[ 0.47360887+0.j        ,  0.01651754-0.01645398j,
                 0.00528622-0.00449668j, -0.01767143-0.43915202j],
               [ 0.01651754+0.01645398j,  0.03051735+0.j        ,
                 0.00677056+0.00852105j, -0.00747277+0.00333155j],
               [ 0.00528622+0.00449668j,  0.00677056-0.00852105j,
                 0.03197703+0.j        , -0.01182688+0.0033077j ],
               [-0.01767143+0.43915202j, -0.00747277-0.00333155j,
                -0.01182688-0.0033077j ,  0.46389675+0.j        ]],
              dims=(2, 2))
quality: unknown
backend: aer_simulator_from(fake_perth)
run_time: None
trace: 0.9999999972176197
eigvals: [0.908474   0.05950643 0.02763186 0.0043877 ]
raw_eigvals: [0.90847401 0.05950643 0.02763186 0.0043877 ]
rescaled_psd: False
fitter_metadata: {'fitter': 'cvxpy_gaussian_lstsq', 'cvxpy_solver': 'SCS', 'cvxpy_status': ['optimal'], 'psd_constraint': True, 'trace_preserving': True, 'fitter_time': 0.037584781646728516}
conditional_probability: 1.0
positive: True

Parallel Tomography Experiment¶

We can also use the ParallelExperiment class to run subsystem tomography on multiple qubits in parallel.

For example if we want to perform 1-qubit QST on several qubits at once:

from math import pi
num_qubits = 5
gates = [qiskit.circuit.library.RXGate(i * pi / (num_qubits - 1))
         for i in range(num_qubits)]

subexps = [
    StateTomography(gate, physical_qubits=(i,))
    for i, gate in enumerate(gates)
]
parexp = ParallelExperiment(subexps)
pardata = parexp.run(backend, seed_simulation=100).block_for_results()

display(pardata.analysis_results(dataframe=True))

	name	experiment	components	value	quality	backend	run_time	trace	eigvals	raw_eigvals	rescaled_psd	fitter_metadata	conditional_probability	positive
503e0659	state	StateTomography	[Q0]	DensityMatrix([[0.96777344+0.j , 0.0175...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9682268402188912, 0.03177315978110963]	[0.9682268402188912, 0.03177315978110963]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
0bf4ab4b	state_fidelity	StateTomography	[Q0]	0.967773	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
da1fe881	positive	StateTomography	[Q0]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
471ee0e1	state	StateTomography	[Q1]	DensityMatrix([[0.82910156+0.j , 0.0097...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9607150300654657, 0.03928496993453533]	[0.9607150300654657, 0.03928496993453533]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
98e3e436	state_fidelity	StateTomography	[Q1]	0.960586	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
22cef7f8	positive	StateTomography	[Q1]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
04411412	state	StateTomography	[Q2]	DensityMatrix([[0.5234375+0.j , 0.00781...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9684253840187003, 0.03157461598130054]	[0.9684253840187003, 0.03157461598130054]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
96efb51e	state_fidelity	StateTomography	[Q2]	0.967773	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
15272b05	positive	StateTomography	[Q2]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
ce35f27a	state	StateTomography	[Q3]	DensityMatrix([[ 0.19238281+0.j , -0.013671...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9387157442028106, 0.061284255797190226]	[0.9387157442028106, 0.061284255797190226]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
5de0d93d	state_fidelity	StateTomography	[Q3]	0.938489	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
1ad60c4c	positive	StateTomography	[Q3]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
e1ba49f4	state	StateTomography	[Q4]	DensityMatrix([[0.03320313+0.j , 0. ...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9668224120237543, 0.033177587976246654]	[0.9668224120237543, 0.033177587976246654]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
c84e5c71	state_fidelity	StateTomography	[Q4]	0.966797	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
c9709ff8	positive	StateTomography	[Q4]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None

View experiment analysis results for one component:

results = pardata.analysis_results(dataframe=True)
display(results[results.components.apply(lambda x: x == ["Q0"])])

	name	experiment	components	value	quality	backend	run_time	trace	eigvals	raw_eigvals	rescaled_psd	fitter_metadata	conditional_probability	positive
503e0659	state	StateTomography	[Q0]	DensityMatrix([[0.96777344+0.j , 0.0175...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9682268402188912, 0.03177315978110963]	[0.9682268402188912, 0.03177315978110963]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
0bf4ab4b	state_fidelity	StateTomography	[Q0]	0.967773	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
da1fe881	positive	StateTomography	[Q0]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None

Quantum State Tomography¶

Tomography Results¶

Additional state metadata¶

Tomography Fitters¶

Parallel Tomography Experiment¶

References¶

See also¶