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
a8dc6dee	state	StateTomography	[Q0, Q1]	DensityMatrix([[ 0.46500651+0.j , -0.01...	None	aer_simulator_from(fake_perth)	None	1.0	[0.9118390623289906, 0.051634993656863096, 0.0...	[0.9118390623289906, 0.051634993656863096, 0.0...	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
783c8836	state_fidelity	StateTomography	[Q0, Q1]	0.910645	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
8bb4d093	positive	StateTomography	[Q0, Q1]	True	None	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([[ 0.46500651+0.j        , -0.01204427+0.00748698j,
                 0.00227865-0.01123047j,  0.0078125 -0.44140625j],
               [-0.01204427-0.00748698j,  0.02360026+0.j        ,
                -0.01367188+0.00390625j, -0.01822917+0.00634766j],
               [ 0.00227865+0.01123047j, -0.01367188-0.00390625j,
                 0.03792318+0.j        ,  0.01822917+0.00846354j],
               [ 0.0078125 +0.44140625j, -0.01822917-0.00634766j,
                 0.01822917-0.00846354j,  0.47347005+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 = 0.91064

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: [0.91183906 0.05163499 0.02271696 0.01380898]
trace: 1.0000000000000013
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([[ 0.35133904+0.00000000e+00j, -0.05460501-1.30768269e-02j,
                -0.0296055 +1.47087084e-01j,  0.11114873-3.27070794e-01j],
               [-0.05460501+1.30768269e-02j,  0.09319463-3.46944695e-18j,
                 0.03814685-5.05548020e-02j,  0.06972563+4.37987876e-02j],
               [-0.0296055 -1.47087084e-01j,  0.03814685+5.05548020e-02j,
                 0.10005139+1.73472348e-18j, -0.1102921 -2.17247025e-02j],
               [ 0.11114873+3.27070794e-01j,  0.06972563-4.37987876e-02j,
                -0.1102921 +2.17247025e-02j,  0.45541494+2.77555756e-17j]],
              dims=(2, 2))
quality: None
backend: aer_simulator_from(fake_perth)
run_time: None
trace: 1.000000000000006
eigvals: [0.81115115 0.1671573  0.02169155 0.        ]
raw_eigvals: [ 0.88335141  0.23935756  0.09389182 -0.21660079]
rescaled_psd: True
fitter_metadata: {'fitter': 'linear_inversion', 'fitter_time': 0.004080772399902344}
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([[ 4.58614058e-01+0.j        ,  5.98255698e-03+0.00125795j,
                 6.42806612e-05+0.00897406j,  5.12200875e-03-0.43504751j],
               [ 5.98255698e-03-0.00125795j,  2.70696280e-02+0.j        ,
                -2.08977388e-03+0.00247464j,  2.05446931e-04+0.00176623j],
               [ 6.42806612e-05-0.00897406j, -2.08977388e-03-0.00247464j,
                 4.29273330e-02+0.j        , -1.34110440e-02+0.00402144j],
               [ 5.12200875e-03+0.43504751j,  2.05446931e-04-0.00176623j,
                -1.34110440e-02-0.00402144j,  4.71388981e-01+0.j        ]],
              dims=(2, 2))
quality: None
backend: aer_simulator_from(fake_perth)
run_time: None
trace: 0.9999999898537167
eigvals: [0.90043866 0.04404499 0.03406292 0.02145343]
raw_eigvals: [0.90043867 0.04404499 0.03406292 0.02145343]
rescaled_psd: False
fitter_metadata: {'fitter': 'cvxpy_gaussian_lstsq', 'cvxpy_solver': 'SCS', 'cvxpy_status': ['optimal'], 'psd_constraint': True, 'trace_preserving': True, 'fitter_time': 0.032004356384277344}
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
acd5171b	state	StateTomography	[Q0]	DensityMatrix([[0.96972656+0.j , 0. ...	None	aer_simulator_from(fake_perth)	None	1.0	[0.9700553520560977, 0.029944647943903127]	[0.9700553520560977, 0.029944647943903127]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
e5198d37	state_fidelity	StateTomography	[Q0]	0.969727	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
2c05c5b0	positive	StateTomography	[Q0]	True	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
e21d9bbe	state	StateTomography	[Q1]	DensityMatrix([[0.86328125+0.j , 0.0214...	None	aer_simulator_from(fake_perth)	None	1.0	[0.999259400242468, 0.000740599757532906]	[0.999259400242468, 0.000740599757532906]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
081d1f15	state_fidelity	StateTomography	[Q1]	0.998566	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
05dd38d8	positive	StateTomography	[Q1]	True	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
1a13c39f	state	StateTomography	[Q2]	DensityMatrix([[ 0.49023438+0.j , -0.018...	None	aer_simulator_from(fake_perth)	None	1.0	[0.9614141553594923, 0.038585844640508626]	[0.9614141553594923, 0.038585844640508626]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
916698fc	state_fidelity	StateTomography	[Q2]	0.960938	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
a7f3d3f2	positive	StateTomography	[Q2]	True	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
6864b2e6	state	StateTomography	[Q3]	DensityMatrix([[0.16503906+0.j , 0.0117...	None	aer_simulator_from(fake_perth)	None	1.0	[0.9738512302435445, 0.026148769756456525]	[0.9738512302435445, 0.026148769756456525]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
920a299b	state_fidelity	StateTomography	[Q3]	0.973706	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
70fc0549	positive	StateTomography	[Q3]	True	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
48056f55	state	StateTomography	[Q4]	DensityMatrix([[ 0.02734375+0.j , -0.0156...	None	aer_simulator_from(fake_perth)	None	1.0	[0.9731724969966684, 0.02682750300333219]	[0.9731724969966684, 0.02682750300333219]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
f826e349	state_fidelity	StateTomography	[Q4]	0.972656	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
795cca59	positive	StateTomography	[Q4]	True	None	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
acd5171b	state	StateTomography	[Q0]	DensityMatrix([[0.96972656+0.j , 0. ...	None	aer_simulator_from(fake_perth)	None	1.0	[0.9700553520560977, 0.029944647943903127]	[0.9700553520560977, 0.029944647943903127]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
e5198d37	state_fidelity	StateTomography	[Q0]	0.969727	None	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
2c05c5b0	positive	StateTomography	[Q0]	True	None	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¶