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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. Qadence offers two main measurement protocols: quantum state tomography and classical shadows.

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:

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

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
from qadence import Parameter, chain, kron, RX, RY, Z, QuantumCircuit, QuantumModel
from qadence.measurements import Measurements

# Define parameters for a circuit.
theta1 = Parameter("theta1", trainable=False)
theta2 = Parameter("theta2", trainable=False)
theta3 = Parameter("theta3", trainable=False)
theta4 = Parameter("theta4", trainable=False)

blocks = chain(
    kron(RX(0, theta1), RY(1, theta2)),
    kron(RX(0, theta3), RY(1, theta4)),
)

values = {
    "theta1": tensor([0.5]),
    "theta2": tensor([1.5]),
    "theta3": tensor([2.0]),
    "theta4": tensor([2.5]),
}

# 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=Measurements.TOMOGRAPHY, options=tomo_options)

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

# Run the tomography experiment.
estimated_values_tomo = model.expectation(
    values=values,
    measurement=tomo_measurement,
)

Exact expectation value = tensor([[-1.4548]])
Estimated expectation value tomo = tensor([[-1.4584]])

Classical shadows

Recently, a much less resource demanding protocol based on classical shadows has been proposed1. It combines ideas from shadow tomography2 and randomized measurement protocols 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 protocols 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 v \(U^\dagger |\hat{b}\rangle\langle \hat{b}|U\), 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:

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

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.
shadow_options = {"accuracy": 0.1, "confidence": 0.1}  # Shadow size N=54400.
shadow_measurement = Measurements(protocol=Measurements.SHADOW, options=shadow_options)

# Run the experiment with classical shadows.
estimated_values_shadow = model.expectation(
    values=values,
    measurement=shadow_measurement,
)

Estimated expectation value shadow = tensor([[-1.4365]])

References

  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 ↩