Note

This is the documentation for the current state of the development branch of Qiskit Experiments. The documentation or APIs here can change prior to being released.

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
e0e46e2f	state	StateTomography	[Q0, Q1]	DensityMatrix([[ 0.45882161+0.00000000e+00j, -...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9122088926634183, 0.0510064800557372, 0.026...	[0.9122088926634183, 0.0510064800557372, 0.026...	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
e5dc296d	state_fidelity	StateTomography	[Q0, Q1]	0.911621	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
f9e5a50b	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([[ 0.45882161+0.00000000e+00j, -0.00341797-1.09049479e-02j,
                 0.01595052+1.38777878e-17j,  0.00732422-4.43359375e-01j],
               [-0.00341797+1.09049479e-02j,  0.02913411+0.00000000e+00j,
                -0.01025391+5.85937500e-03j, -0.00455729+1.36718750e-02j],
               [ 0.01595052-1.38777878e-17j, -0.01025391-5.85937500e-03j,
                 0.03434245+0.00000000e+00j,  0.00634766-3.09244792e-03j],
               [ 0.00732422+4.43359375e-01j, -0.00455729-1.36718750e-02j,
                 0.00634766+3.09244792e-03j,  0.47770182+0.00000000e+00j]],
              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 = 0.91162

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: [0.91220889 0.05100648 0.02698489 0.00979974]
trace: 1.0000000000000009
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([[ 0.4438714 +0.j        , -0.03260482-0.02214907j,
                 0.06683993-0.07176459j, -0.03857341-0.36891678j],
               [-0.03260482+0.02214907j,  0.06449703+0.j        ,
                -0.00910865-0.00424095j,  0.00994102+0.12215717j],
               [ 0.06683993+0.07176459j, -0.00910865+0.00424095j,
                 0.02536626+0.j        ,  0.03485139-0.07653968j],
               [-0.03857341+0.36891678j,  0.00994102-0.12215717j,
                 0.03485139+0.07653968j,  0.46626531+0.j        ]],
              dims=(2, 2))
quality: unknown
backend: aer_simulator_from(fake_perth)
run_time: None
trace: 1.0000000000000062
eigvals: [0.86134225 0.13865775 0.         0.        ]
raw_eigvals: [ 0.99199801  0.26931351  0.0365791  -0.29789062]
rescaled_psd: True
fitter_metadata: {'fitter': 'linear_inversion', 'fitter_time': 0.00392460823059082}
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([[ 0.46311523+0.j        , -0.0118114 -0.01463194j,
                 0.00244049-0.00233384j, -0.02233021-0.4436323j ],
               [-0.0118114 +0.01463194j,  0.0270951 +0.j        ,
                -0.00212236-0.00915219j,  0.00648153-0.00704437j],
               [ 0.00244049+0.00233384j, -0.00212236+0.00915219j,
                 0.03084326+0.j        ,  0.01257865+0.01197155j],
               [-0.02233021+0.4436323j ,  0.00648153+0.00704437j,
                 0.01257865-0.01197155j,  0.47894642+0.j        ]],
              dims=(2, 2))
quality: unknown
backend: aer_simulator_from(fake_perth)
run_time: None
trace: 1.0000000000660818
eigvals: [0.91574382 0.05053039 0.02556549 0.00816031]
raw_eigvals: [0.91574382 0.05053039 0.02556549 0.00816031]
rescaled_psd: False
fitter_metadata: {'fitter': 'cvxpy_gaussian_lstsq', 'cvxpy_solver': 'SCS', 'cvxpy_status': ['optimal'], 'psd_constraint': True, 'trace_preserving': True, 'fitter_time': 0.038671255111694336}
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
b708d58a	state	StateTomography	[Q0]	DensityMatrix([[ 0.97363281+0.j , -0.00...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9743943217769155, 0.02560567822308542]	[0.9743943217769155, 0.02560567822308542]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
f97db52c	state_fidelity	StateTomography	[Q0]	0.973633	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
874fc5ef	positive	StateTomography	[Q0]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
a8189338	state	StateTomography	[Q1]	DensityMatrix([[0.84179688+0.j , 0.0224...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.977727747424388, 0.022272252575612828]	[0.977727747424388, 0.022272252575612828]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
a3c31adb	state_fidelity	StateTomography	[Q1]	0.977159	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
1988fa99	positive	StateTomography	[Q1]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
49bdd81e	state	StateTomography	[Q2]	DensityMatrix([[0.48046875+0.j , 0.0009...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9603764627201941, 0.03962353727980669]	[0.9603764627201941, 0.03962353727980669]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
c5fc19a4	state_fidelity	StateTomography	[Q2]	0.959961	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
2b337211	positive	StateTomography	[Q2]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
21e34943	state	StateTomography	[Q3]	DensityMatrix([[0.16015625+0.j , 0.03125...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9680496069545541, 0.03195039304544672]	[0.9680496069545541, 0.03195039304544672]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
f6d06d26	state_fidelity	StateTomography	[Q3]	0.966801	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
7a10467f	positive	StateTomography	[Q3]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
9fb382ee	state	StateTomography	[Q4]	DensityMatrix([[ 0.04101563+0.j , -0.01...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9602397227717225, 0.039760277228278254]	[0.9602397227717225, 0.039760277228278254]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
28df80b3	state_fidelity	StateTomography	[Q4]	0.958984	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
ae3e441b	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
b708d58a	state	StateTomography	[Q0]	DensityMatrix([[ 0.97363281+0.j , -0.00...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9743943217769155, 0.02560567822308542]	[0.9743943217769155, 0.02560567822308542]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
f97db52c	state_fidelity	StateTomography	[Q0]	0.973633	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
874fc5ef	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¶