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
703dec8e	state	StateTomography	[Q0, Q1]	DensityMatrix([[ 0.47428385+0.j , 0.00...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9176219048772116, 0.04067262563748829, 0.02...	[0.9176219048772116, 0.04067262563748829, 0.02...	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
d9088a61	state_fidelity	StateTomography	[Q0, Q1]	0.916992	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
1e0f693c	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([[ 0.47428385+0.j        ,  0.00504557-0.00585938j,
                -0.00927734+0.00472005j, -0.01220703-0.44628906j],
               [ 0.00504557+0.00585938j,  0.03580729+0.j        ,
                -0.00146484+0.00390625j,  0.01611328-0.00504557j],
               [-0.00927734-0.00472005j, -0.00146484-0.00390625j,
                 0.02278646+0.j        ,  0.00113932+0.00488281j],
               [-0.01220703+0.44628906j,  0.01611328+0.00504557j,
                 0.00113932-0.00488281j,  0.4671224 +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 = 0.91699

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: [0.9176219  0.04067263 0.02494114 0.01676433]
trace: 1.000000000000002
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([[ 0.49554763+0.00000000e+00j,  0.02318321+3.33568434e-02j,
                 0.07331708-8.46287217e-02j, -0.04052301-3.32048009e-01j],
               [ 0.02318321-3.33568434e-02j,  0.05549517+1.73472348e-18j,
                -0.00871813-6.82190403e-03j,  0.07686302-5.61177129e-03j],
               [ 0.07331708+8.46287217e-02j, -0.00871813+6.82190403e-03j,
                 0.02618034+0.00000000e+00j,  0.03849224-6.09543085e-02j],
               [-0.04052301+3.32048009e-01j,  0.07686302+5.61177129e-03j,
                 0.03849224+6.09543085e-02j,  0.42277685-6.93889390e-18j]],
              dims=(2, 2))
quality: unknown
backend: aer_simulator_from(fake_perth)
run_time: None
trace: 1.0000000000000009
eigvals: [0.81840385 0.18159615 0.         0.        ]
raw_eigvals: [ 0.88152896  0.24472126  0.02361771 -0.14986793]
rescaled_psd: True
fitter_metadata: {'fitter': 'linear_inversion', 'fitter_time': 0.001786947250366211}
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([[ 0.48075468+0.00000000e+00j,  0.01615475-9.60437643e-03j,
                -0.01065275-5.96450514e-03j,  0.02164237-4.43944170e-01j],
               [ 0.01615475+9.60437643e-03j,  0.03706996+0.00000000e+00j,
                -0.00393331-4.12493023e-04j,  0.00821493+4.38033927e-04j],
               [-0.01065275+5.96450514e-03j, -0.00393331+4.12493023e-04j,
                 0.02446304+0.00000000e+00j, -0.01118789+1.13100594e-02j],
               [ 0.02164237+4.43944170e-01j,  0.00821493-4.38033927e-04j,
                -0.01118789-1.13100594e-02j,  0.45771232+0.00000000e+00j]],
              dims=(2, 2))
quality: unknown
backend: aer_simulator_from(fake_perth)
run_time: None
trace: 1.00000000671447
eigvals: [0.91448509 0.04593354 0.03001426 0.0095671 ]
raw_eigvals: [0.91448508 0.04593354 0.03001426 0.0095671 ]
rescaled_psd: False
fitter_metadata: {'fitter': 'cvxpy_gaussian_lstsq', 'cvxpy_solver': 'SCS', 'cvxpy_status': ['optimal'], 'psd_constraint': True, 'trace_preserving': True, 'fitter_time': 0.03195619583129883}
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
68640a97	state	StateTomography	[Q0]	DensityMatrix([[ 0.97265625+0.j , -0.02...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9735452158042912, 0.02645478419570962]	[0.9735452158042912, 0.02645478419570962]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
d5ad2714	state_fidelity	StateTomography	[Q0]	0.972656	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
78e60f72	positive	StateTomography	[Q0]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
b0a6249c	state	StateTomography	[Q1]	DensityMatrix([[ 0.83691406+0.j , -0.01...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9699427370940645, 0.030057262905936244]	[0.9699427370940645, 0.030057262905936244]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
7d03052f	state_fidelity	StateTomography	[Q1]	0.969563	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
4d894964	positive	StateTomography	[Q1]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
b56652d8	state	StateTomography	[Q2]	DensityMatrix([[0.5078125 +0.j , 0.0058...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9668990147145383, 0.03310098528546265]	[0.9668990147145383, 0.03310098528546265]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
eadca843	state_fidelity	StateTomography	[Q2]	0.966797	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
bab316b1	positive	StateTomography	[Q2]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
c9355ee9	state	StateTomography	[Q3]	DensityMatrix([[ 0.17871094+0.j , -0.01...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.958682996133913, 0.04131700386608829]	[0.958682996133913, 0.04131700386608829]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
04d61c85	state_fidelity	StateTomography	[Q3]	0.958515	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
04f67e91	positive	StateTomography	[Q3]	True	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
dbec1949	state	StateTomography	[Q4]	DensityMatrix([[ 0.02441406+0.j , -0.02...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9765244765825014, 0.02347552341749992]	[0.9765244765825014, 0.02347552341749992]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
a70c36ea	state_fidelity	StateTomography	[Q4]	0.975586	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
b4ace17f	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
68640a97	state	StateTomography	[Q0]	DensityMatrix([[ 0.97265625+0.j , -0.02...	unknown	aer_simulator_from(fake_perth)	None	1.0	[0.9735452158042912, 0.02645478419570962]	[0.9735452158042912, 0.02645478419570962]	False	{'fitter': 'linear_inversion', 'fitter_time': ...	1.0	True
d5ad2714	state_fidelity	StateTomography	[Q0]	0.972656	unknown	aer_simulator_from(fake_perth)	None	None	None	None	None	None	None	None
78e60f72	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¶