Measurement protocols

Sample-based measurement protocols 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-protocols.

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_protocols import Measurements, 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([]))

Classical shadows

A much less resource demanding protocol based on classical shadows has been proposed¹. It combines ideas from shadow tomography² and randomized measurement protocols ³ capable of learning a classical shadow of an unknown quantum state \(\rho\). It relies on deliberately discarding the full classical characterization of the quantum state, and instead focuses on accurately predicting a restricted set of properties that provide efficient resources for the study of the system.

A random measurement consists of applying random unitary rotations before a fixed measurement on each copy of a state. Appropriately averaging over these measurements produces an efficient estimator for the expectation value of an observable. This protocol therefore creates a robust classical representation of the quantum state or classical shadow. The captured measurement information is then reuseable for multiple purposes, i.e. any observable expected value and available for noise mitigation postprocessing.

A classical shadow is therefore an unbiased estimator of a quantum state \(\rho\). Such an estimator is obtained with the following procedure¹: first, apply a random unitary gate \(U\) to rotate the state: \(\rho \rightarrow U \rho U^\dagger\) and then perform a basis measurement to obtain a \(n\)-bit measurement \(|\hat{b}\rangle \in \{0, 1\}^n\). Both unitary gates \(U\) and the measurement outcomes \(|\hat{b}\rangle\) are stored on a classical computer for postprocessing \(U^\dagger |\hat{b}\rangle\langle \hat{b}|U\), a classical snapshot of the state \(\rho\). The whole procedure can be seen as a quantum channel \(\mathcal{M}\) that maps the initial unknown quantum state \(\rho\) to the average result of the measurement protocol:

\[ \mathbb{E}[U^\dagger |\hat{b}\rangle\langle \hat{b}|U] = \mathcal{M}(\rho) \Rightarrow \rho = \mathbb{E}[\mathcal{M}^{-1}(U^\dagger |\hat{b}\rangle\langle \hat{b}|U)] \]

It is worth noting that the single classical snapshot \(\hat{\rho}=\mathcal{M}^{-1}(U^\dagger |\hat{b}\rangle\langle \hat{b}|U)\) equals \(\rho\) in expectation: \(\mathbb{E}[\hat{\rho}]=\rho\) despite \(\mathcal{M}^{-1}\) not being a completely positive map. Repeating this procedure \(N\) times results in an array of \(N\) independent, classical snapshots of \(\rho\) called the classical shadow:

\[ S(\rho, N) = \{ \hat{\rho}_1=\mathcal{M}^{-1}(U_1^\dagger |\hat{b}_1\rangle\langle \hat{b}_1|U_1),\cdots,\hat{\rho}_N=\mathcal{M}^{-1}(U_N^\dagger |\hat{b}_N\rangle\langle \hat{b}_N|U_N)\} \]

Running classical shadows

Along the same lines as the example before, estimating the expectation value using classical shadows in Qadence only requires to pass the right set of parameters to the Measurements object:

# Classical shadows are defined up to some accuracy and confidence.
from qadence_protocols.measurements.utils_shadow.data_acquisition import number_of_samples

shadow_options = {"accuracy": 0.1, "confidence": 0.1}
N, K = number_of_samples(observable, shadow_options["accuracy"], shadow_options["confidence"])
shadow_measurement = Measurements(protocol=MeasurementProtocol.SHADOW, options=shadow_options)

# Run the shadow experiment.
estimated_values_shadow = shadow_measurement(model=model)

Estimated expectation value shadow = tensor([[-2.]])

Note that the option n_shots is by default 1, which means for one unitary, we sample only once. If we specify a higher number of shots, more samples are realized per unitary accordingly, and a different formula is used involving the Hamming distance denoted \(D\) (see Eq. 2.42 of Ref³): \(\(\hat{\rho}^{(r)} = 2^N \bigotimes_{i=1}^N \sum_{b_i} (-2)^{-D[b_i, b_i^{(r)}]} (U^\dagger |\hat{b_i}\rangle\langle \hat{b_i}|U)\)\)

Getting shadows

If we are interested in accessing the measurement data from shadows, we can access the measurement data via the manager attribute as follows:

measurements_shadows = shadow_measurement.data

Sampled unitary indices shape:  torch.Size([10200, 2])
Shape of batched measurements:  torch.Size([1, 10200, 2])

In the case of shadows, the measurement data is composed of two elements: - unitaries refers to the indices corresponding to the randomly sampled Pauli unitaries \(U\). It is returned as a tensor of shape (shadow_size, n_qubits). Its elements are integer values 0, 1, 2 corresponding respectively to X, Y, Z. - the second one, samples, refers to the bistrings (or probability vectors if n_shots is higher than 1) obtained by measurements of the circuit rotated depending on the sampled Pauli basis. It as returned as a tensor of batched measurements with shape (batch_size, shadow_size, n_qubits) or a tensor of shape (batch_size, shadow_size, \(2^n_qubits\)) depending on the value of n_shots.

Such a measurement data can be used directly for computing different quantities of interest other than the expectation values. For instance, we can do state reconstruction and use it to calculate another expectation value as follows:

# reconstruct state from snapshots
state = shadow_measurement.reconstruct_state()

# calculate expectations
from qadence_protocols.utils_trace import expectation_trace
exp_reconstructed_state = expectation_trace(state, observable)

tensor([[-2.0194]])

Robust shadows

Robust shadows ⁴ were built upon the classical shadow scheme but have the particularity to be noise-resilient. It involves an experimental simple calibration procedure based on the preparation of a all-zero state with very high fidelity. We then perform noisy randomized measurements to learn about the averaged, as known as `twirled'', effect of the noise, obtaining calibration coefficients for shadows. One can efficiently characterize and mitigate noises in the shadow estimation scheme, given only minimal assumptions on the experimental conditions. Such a procedure has been used in [^5] to estimate the Quantum Fisher information out of a quantum system. Note that robust shadows are equivalent to classical shadows in non-noisy settings by setting thecalibration` coefficients to \(\frac{1}{3}\) for each qubit, as shown below:

from qadence_protocols.measurements.calibration import zero_state_calibration
# Calibration coefficients are by default 1/3
calibration = zero_state_calibration(N, n_qubits=2, n_shots=100, backend=model.backend, noise=None)

robust_shadow_options = {"shadow_size": N, "shadow_medians": K, "calibration": calibration}
robust_shadow_measurement = Measurements(protocol=MeasurementProtocol.ROBUST_SHADOW, options=robust_shadow_options)
estimated_values_robust_shadow = robust_shadow_measurement(model=model)

Estimated expectation value shadow = tensor([[-2.0000]])

For an example of comparing robust shadows and classical shadows in noisy settings, please refer to the Robust shadow tomography tutorial.

References

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1. Hsin-Yuan Huang, Richard Kueng and John Preskill, Predicting Many Properties of a Quantum System from Very Few Measurements (2020) ↩↩

2. S. Aaronson. Shadow tomography of quantum states. In Proceedings of the 50th Annual A ACM SIGACT Symposium on Theory of Computing, STOC 2018, pages 325–338, New York, NY, USA, 2018. ACM ↩

3. Aniket Rath. Probing entanglement on quantum platforms using randomized measurements. Physics [physics]. Université Grenoble Alpes [2020-..], 2023. English. ffNNT : 2023GRALY072ff. fftel-04523142 ↩↩

4. Senrui Chen, Wenjun Yu, Pei Zeng, and Steven T. Flammia, Robust Shadow Estimation (2021) ↩

5. Vittorio Vitale, Aniket Rath, Petar Jurcevic, Andreas Elben, Cyril Branciard, and Benoît Vermersch, Robust Estimation of the Quantum Fisher Information on a Quantum Processor (2024) ↩