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
for result in qstdata1.analysis_results():
    print(result)

AnalysisResult
- name: state
- value: DensityMatrix([[ 0.47151693+0.j        , -0.00455729+0.00569661j,
                -0.00878906+0.0069987j , -0.00585938-0.44970703j],
               [-0.00455729-0.00569661j,  0.01839193+0.j        ,
                -0.0078125 +0.01220703j, -0.00195313+0.02067057j],
               [-0.00878906-0.0069987j , -0.0078125 -0.01220703j,
                 0.01969401+0.j        , -0.00358073-0.01090495j],
               [-0.00585938+0.44970703j, -0.00195313-0.02067057j,
                -0.00358073+0.01090495j,  0.49039714+0.j        ]],
              dims=(2, 2))
- quality: unknown
- extra: <9 items>
- device_components: ['Q0', 'Q1']
- verified: False
AnalysisResult
- name: state_fidelity
- value: 0.9306640624999993
- quality: unknown
- extra: <9 items>
- device_components: ['Q0', 'Q1']
- verified: False
AnalysisResult
- name: positive
- value: True
- quality: unknown
- extra: <9 items>
- device_components: ['Q0', 'Q1']
- verified: False

Tomography Results

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

state_result = qstdata1.analysis_results("state")
print(state_result.value)

DensityMatrix([[ 0.47151693+0.j        , -0.00455729+0.00569661j,
                -0.00878906+0.0069987j , -0.00585938-0.44970703j],
               [-0.00455729-0.00569661j,  0.01839193+0.j        ,
                -0.0078125 +0.01220703j, -0.00195313+0.02067057j],
               [-0.00878906-0.0069987j , -0.0078125 -0.01220703j,
                 0.01969401+0.j        , -0.00358073-0.01090495j],
               [-0.00585938+0.44970703j, -0.00195313-0.02067057j,
                -0.00358073+0.01090495j,  0.49039714+0.j        ]],
              dims=(2, 2))

We can also visualize the density matrix:

from qiskit.visualization import plot_state_city
plot_state_city(qstdata1.analysis_results("state").value, 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")
print("State Fidelity = {:.5f}".format(fid_result.value))

State Fidelity = 0.93066

Additional state metadata

Additional data is stored in the tomography under the "state_metadata" field. This includes

• eigvals: the eigenvalues of the fitted state

• trace: the trace of the fitted state

• positive: Whether the eigenvalues are all non-negative

• positive_delta: the deviation from positivity given by 1-norm of negative eigenvalues.

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.

state_result.extra

{'trace': 1.0000000000000016,
 'eigvals': array([0.93124602, 0.05002387, 0.01617965, 0.00255046]),
 'raw_eigvals': array([0.93124602, 0.05002387, 0.01617965, 0.00255046]),
 'rescaled_psd': False,
 'fitter_metadata': {'fitter': 'linear_inversion',
  'fitter_time': 0.008013248443603516},
 'conditional_probability': 1.0,
 'positive': True,
 'experiment': 'StateTomography',
 'run_time': None}

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")

# Print result
print(bad_state_result)

# Show extra data
bad_state_result.extra

AnalysisResult
- name: state
- value: DensityMatrix([[ 0.36891386+0.00000000e+00j,  0.09699391+2.15097594e-02j,
                -0.02722458+2.40686779e-02j, -0.0333696 -4.39944025e-01j],
               [ 0.09699391-2.15097594e-02j,  0.04015393+1.73472348e-18j,
                -0.0021221 +1.40441051e-02j, -0.00801283-1.06558694e-01j],
               [-0.02722458-2.40686779e-02j, -0.0021221 -1.40441051e-02j,
                 0.0073675 +0.00000000e+00j, -0.01580262+2.45045540e-02j],
               [-0.0333696 +4.39944025e-01j, -0.00801283+1.06558694e-01j,
                -0.01580262-2.45045540e-02j,  0.5835647 +0.00000000e+00j]],
              dims=(2, 2))
- quality: unknown
- extra: <9 items>
- device_components: ['Q0', 'Q1']
- verified: False

{'trace': 1.0000000000000004,
 'eigvals': array([0.95550198, 0.04449802, 0.        , 0.        ]),
 'raw_eigvals': array([ 1.04714348,  0.13613953,  0.01725243, -0.20053544]),
 'rescaled_psd': True,
 'fitter_metadata': {'fitter': 'linear_inversion',
  'fitter_time': 0.004986286163330078},
 'conditional_probability': 1.0,
 'positive': True,
 'experiment': 'StateTomography',
 'run_time': None}

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")
    print(state_result2)
    print("\nextra:")
    for key, val in state_result2.extra.items():
        print(f"- {key}: {val}")

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

AnalysisResult
- name: state
- value: DensityMatrix([[ 4.90840670e-01+0.j        ,  1.17603726e-02+0.0034719j ,
                -5.21816654e-03-0.00913745j, -1.70649022e-02-0.46041266j],
               [ 1.17603726e-02-0.0034719j ,  1.60143192e-02+0.j        ,
                -4.43545342e-03+0.00623454j,  8.34532835e-05+0.00435506j],
               [-5.21816654e-03+0.00913745j, -4.43545342e-03-0.00623454j,
                 2.20125936e-02+0.j        , -6.12722262e-03+0.00167128j],
               [-1.70649022e-02+0.46041266j,  8.34532835e-05-0.00435506j,
                -6.12722262e-03-0.00167128j,  4.71132417e-01+0.j        ]],
              dims=(2, 2))
- quality: unknown
- extra: <9 items>
- device_components: ['Q0', 'Q1']
- verified: False

extra:
- trace: 1.0000000013561003
- eigvals: [9.41888690e-01 3.57349546e-02 2.23597004e-02 1.66549820e-05]
- raw_eigvals: [9.41888689e-01 3.57349546e-02 2.23597003e-02 1.66549820e-05]
- rescaled_psd: False
- fitter_metadata: {'fitter': 'cvxpy_gaussian_lstsq', 'cvxpy_solver': 'SCS', 'cvxpy_status': ['optimal'], 'psd_constraint': True, 'trace_preserving': True, 'fitter_time': 0.02200007438659668}
- conditional_probability: 1.0
- positive: True
- experiment: StateTomography
- run_time: None

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()

for result in pardata.analysis_results():
    print(result)

AnalysisResult
- name: state
- value: DensityMatrix([[0.97949219+0.j        , 0.03222656+0.00976562j],
               [0.03222656-0.00976562j, 0.02050781+0.j        ]],
              dims=(2,))
- quality: unknown
- extra: <9 items>
- device_components: ['Q0']
- verified: False
AnalysisResult
- name: state_fidelity
- value: 0.9794921875000003
- quality: unknown
- extra: <9 items>
- device_components: ['Q0']
- verified: False
AnalysisResult
- name: positive
- value: True
- quality: unknown
- extra: <9 items>
- device_components: ['Q0']
- verified: False
AnalysisResult
- name: state
- value: DensityMatrix([[0.84667969+0.j     , 0.02734375+0.34375j],
               [0.02734375-0.34375j, 0.15332031+0.j     ]],
              dims=(2,))
- quality: unknown
- extra: <9 items>
- device_components: ['Q1']
- verified: False
AnalysisResult
- name: state_fidelity
- value: 0.9882075139637592
- quality: unknown
- extra: <9 items>
- device_components: ['Q1']
- verified: False
AnalysisResult
- name: positive
- value: True
- quality: unknown
- extra: <9 items>
- device_components: ['Q1']
- verified: False
AnalysisResult
- name: state
- value: DensityMatrix([[0.48046875+0.j        , 0.015625  +0.48632812j],
               [0.015625  -0.48632812j, 0.51953125+0.j        ]],
              dims=(2,))
- quality: unknown
- extra: <9 items>
- device_components: ['Q2']
- verified: False
AnalysisResult
- name: state_fidelity
- value: 0.9863281250000006
- quality: unknown
- extra: <9 items>
- device_components: ['Q2']
- verified: False
AnalysisResult
- name: positive
- value: True
- quality: unknown
- extra: <9 items>
- device_components: ['Q2']
- verified: False
AnalysisResult
- name: state
- value: DensityMatrix([[0.15039063+0.j      , 0.00585938+0.328125j],
               [0.00585938-0.328125j, 0.84960938+0.j      ]],
              dims=(2,))
- quality: unknown
- extra: <9 items>
- device_components: ['Q3']
- verified: False
AnalysisResult
- name: state_fidelity
- value: 0.9792305724057271
- quality: unknown
- extra: <9 items>
- device_components: ['Q3']
- verified: False
AnalysisResult
- name: positive
- value: True
- quality: unknown
- extra: <9 items>
- device_components: ['Q3']
- verified: False
AnalysisResult
- name: state
- value: DensityMatrix([[ 0.01367188+0.j        , -0.03027344+0.01464844j],
               [-0.03027344-0.01464844j,  0.98632812+0.j        ]],
              dims=(2,))
- quality: unknown
- extra: <9 items>
- device_components: ['Q4']
- verified: False
AnalysisResult
- name: state_fidelity
- value: 0.9863281249999989
- quality: unknown
- extra: <9 items>
- device_components: ['Q4']
- verified: False
AnalysisResult
- name: positive
- value: True
- quality: unknown
- extra: <9 items>
- device_components: ['Q4']
- verified: False

View component experiment analysis results:

for i, expdata in enumerate(pardata.child_data()):
    state_result_i = expdata.analysis_results("state")
    fid_result_i = expdata.analysis_results("state_fidelity")

    print(f'\nPARALLEL EXP {i}')
    print("State Fidelity: {:.5f}".format(fid_result_i.value))
    print("State: {}".format(state_result_i.value))

References

[1]

J Smolin, JM Gambetta, G Smith, Phys. Rev. Lett. 108, 070502 (2012), open access.

See also

• API documentation: StateTomography