Stochastic gradient descent for hybrid quantum-classical optimization

Abstract

Within the context of hybrid quantum-classical optimization, gradient descent based optimizers typically require the evaluation of expectation values with respect to the outcome of parameterized quantum circuits. In this work, we explore the consequences of the prior observation that estimation of these quantities on quantum hardware results in a form of stochastic gradient descent optimization. We formalize this notion, which allows us to show that in many relevant cases, including VQE, QAOA and certain quantum classifiers, estimating expectation values with k measurement outcomes results in optimization algorithms whose convergence properties can be rigorously well understood, for any value of k. In fact, even using single measurement outcomes for the estimation of expectation values is sufficient. Moreover, in many settings the required gradients can be expressed as linear combinations of expectation values -- originating, e.g., from a sum over local terms of a Hamiltonian, a parameter shift rule, or a sum over data-set instances -- and we show that in these cases k-shot expectation value estimation can be combined with sampling over terms of the linear combination, to obtain ``doubly stochastic'' gradient descent optimizers. For all algorithms we prove convergence guarantees, providing a framework for the derivation of rigorous optimization results in the context of near-term quantum devices. Additionally, we explore numerically these methods on benchmark VQE, QAOA and quantum-enhanced machine learning tasks and show that treating the stochastic settings as hyper-parameters allows for state-of-the-art results with significantly fewer circuit executions and measurements.

Noise Robustness and Experimental Demonstration of a Quantum Generative Adversarial Network for Continuous Distributions

“…The generated distributions are plotted in Figure 5b, and it is evident that the training is unaffected by the number of circuit runs. This is in agreement with the result from previous works, [ 47,48 ] where it was shown that for various hybrid quantum‐classical optimization algorithms, the estimation of the expectation values can be done using very small number of measurements. The number of samples can be considered a hyper‐parameter of the algorithm that could be tuned or adjusted during the calculation.…”

Measuring Analytic Gradients of General Quantum Evolution with the Stochastic Parameter Shift Rule

“…As a result, we get ∂ x C(x) = C(x + π/4) − C(x − π/4), which is the so-called parameter shift rule, described in Fig. 1, often used for training quantum circuits [29,38,43]. Note that, with the formalism of the previous section,Ĥ = 0 corresponds to the use of the simpler parametric unitaries of Eq.…”

“…Figure 1: Parameter Shift Rule [29,38,43], only applicable to parametric gates as in Eq. (6) or, more generally, to parametrizations e ixV whereV has two distinct eigenvalues ±u.…”

“…We now introduce a simpler unbiased estimator of the gradient that requires significantly fewer operations to get a single sample. A similar technique has been developed in [16,43] for parametrizations as in Eq. (6), which was dubbed doubly stochastic gradient descend.…”

Application of linear regression algorithm and stochastic gradient descent in a machine‐learning environment for predicting biomass higher heating value

“…The proposed model uses hinge squared loss without regulation and with constant learning rate to cross‐entropy for binary classification 32 . The SGD has been used previously in energy‐modeling investigations such as energy‐entropy competition, 33 wind energy forecasting, 34 and quantum energy physics 35 …”