Measurement methods

Sample-based measurements techniques are fundamental tools for the prediction and estimation of a quantum state as the result of NISQ programs executions. Their resource efficient implementation is a current and active research field. Currently, quantum state tomography is implemented in qadence-measurement.

Quantum state tomography

The fundamental task of quantum state tomography is to learn an approximate classical description of an output quantum state described by a density matrix \(\rho\), from repeated measurements of copies on a chosen basis. To do so, \(\rho\) is expanded in a basis of observables (the tomography step) and for a given observable \(\hat{\mathcal{O}}\), the expectation value is calculated with \(\langle \hat{\mathcal{O}} \rangle=\textrm{Tr}(\hat{\mathcal{O}}\rho)\). A number of measurement repetitions in a suitable basis is then required to estimate \(\langle \hat{\mathcal{O}} \rangle\).

The main drawback is the scaling in measurements for the retrieval of the classical expression for a \(n\)-qubit quantum state as \(2^n \times 2^n\), together with a large amount of classical post-processing.

For an observable expressed as a Pauli string \(\hat{\mathcal{P}}\), the expectation value for a state \(|\psi \rangle\) can be derived as:

\[ \langle \hat{\mathcal{P}} \rangle=\langle \psi | \hat{\mathcal{P}} |\psi \rangle=\langle \psi | \hat{\mathcal{R}}^\dagger \hat{\mathcal{D}} \hat{\mathcal{R}} |\psi \rangle \]

The operator \(\hat{\mathcal{R}}\) diagonalizes \(\hat{\mathcal{P}}\) and rotates the state into an eigenstate in the computational basis. Therefore, \(\hat{\mathcal{R}}|\psi \rangle=\sum\limits_{z}a_z|z\rangle\) and the expectation value can finally be expressed as:

\[ \langle \hat{\mathcal{P}} \rangle=\sum_{z,z'}\langle z |\bar{a}_z\hat{\mathcal{D}}a_{z'}|z'\rangle = \sum_{z}|a_z|^2(-1)^{\phi_z(\hat{\mathcal{P}})} \]

Using tomography

In Qadence, running a tomographical experiment is made simple by defining a Measurements object that captures all options for execution:

from torch import tensor
from qadence import hamiltonian_factory, BackendName, DiffMode, NoiseHandler
from qadence import chain, kron, X, Z, QuantumCircuit, QuantumModel
from qadence_measurement.protocol import Measurements
from qadence_measurement.utils.types import MeasurementProtocol

blocks = chain(
    kron(X(0), X(1)),
    kron(Z(0), Z(1)),
)

# Create a circuit and an observable.
circuit = QuantumCircuit(2, blocks)
observable = hamiltonian_factory(2, detuning=Z)

# Create a model.
model = QuantumModel(
    circuit=circuit,
    observable=observable,
    backend=BackendName.PYQTORCH,
    diff_mode=DiffMode.GPSR,
)

# Define a measurement protocol by passing the shot budget as an option.
tomo_options = {"n_shots": 100000}
tomo_measurement = Measurements(protocol=MeasurementProtocol.TOMOGRAPHY, options=tomo_options)

# Get the exact expectation value.
exact_values = model.expectation()

# Run the tomography experiment.
estimated_values_tomo = tomo_measurement(model=model)

Exact expectation value = tensor([[-2.]])
Estimated expectation value tomo = tensor([[-2.]], grad_fn=<TransposeBackward0>)

Getting measurements

If we are interested in accessing the measurements for computing different quantities of interest other than the expectation values, we can access the measurement data via data as follows:

measurements_tomo = tomo_measurement.data

MeasurementData(samples=[[[OrderedCounter({'11': 100000})], [OrderedCounter({'11': 100000})]]], unitaries=tensor([]))