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:

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

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,
)

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_measurement.utils.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_measurement.utils.utils_trace import expectation_trace
exp_reconstructed_state = expectation_trace(state, observable)

tensor([[-1.9903]])

References

Hsin-Yuan Huang, Richard Kueng and John Preskill, Predicting Many Properties of a Quantum System from Very Few Measurements (2020) ↩↩
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 ↩
Aniket Rath. Probing entanglement on quantum platforms using randomized measurements. Physics [physics]. Université Grenoble Alpes [2020-..], 2023. English. ffNNT : 2023GRALY072ff. fftel-04523142 ↩↩