“Proper” Shift Rules for Derivatives of Perturbed-Parametric Quantum Evolutions

Dirk Oliver Theis

Theoretical Computer Science, University of Tartu, Estonia

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Abstract

Banchi & Crooks (Quantum, 2021) have given methods to estimate derivatives of expectation values depending on a parameter that enters via what we call a "perturbed" quantum evolution $x\mapsto e^{i(x A + B)/\hbar}$. Their methods require modifications, beyond merely changing parameters, to the unitaries that appear. Moreover, in the case when the $B$-term is unavoidable, no exact method (unbiased estimator) for the derivative seems to be known: Banchi & Crooks's method gives an approximation.
In this paper, for estimating the derivatives of parameterized expectation values of this type, we present a method that only requires shifting parameters, no other modifications of the quantum evolutions (a "proper" shift rule). Our method is exact (i.e., it gives analytic derivatives, unbiased estimators), and it has the same worst-case variance as Banchi-Crooks's.
Moreover, we discuss the theory surrounding proper shift rules, based on Fourier analysis of perturbed-parametric quantum evolutions, resulting in a characterization of the proper shift rules in terms of their Fourier transforms, which in turn leads us to non-existence results of proper shift rules with exponential concentration of the shifts. We derive truncated methods that exhibit approximation errors, and compare to Banchi-Crooks's based on preliminary numerical simulations.

In attempts to use current-day or near-future quantum devices for meaningful computations, the variational hybrid quantum-classical approach is widely pursued. It consists in parameterizing the quantum evolution and then optimizing these parameters in a loop, alternating between quantum and classical computation.

Another approach consists in mapping a computational problem to a Hamiltonian that can be realized on quantum hardware. For example, for modeling the Maximum Stable Set problem on cold-atom quantum devices, the Rydberg blockade may serve as a way to partially realize the stability constraints.

Attempts are, of course, under way to combine the two approaches.

For optimizing the parameters, the variational approach typically employs estimators of the gradient, and these estimators should have small bias and small variance. In the digital quantum computing world — i.e., quantum circuits containing (parameterized) gates gates — estimating the gradients is well understood, and based on so-called 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑠ℎ𝑖𝑓𝑡 𝑟𝑢𝑙𝑒𝑠. But when combining the digital with the analog, the situation arises that the parameterized part of the Hamiltonian does not commute with other parts.
Think of choosing as one of the parameters the Rabi frequency, say locally to a single atom, in an array of Rydberg atoms: The Rabi term does not commute with the Rydberg blockade terms. Many more examples exist. In these situations, the known shift-rule theory breaks down.
In our paper, we propose a new method for estimating derivatives for these situations. Our method works along the known shift-rule paradigm, and improves upon the state of the art in reducing the bias of the estimator.

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► References

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Cited by

[1] Roeland Wiersema, Dylan Lewis, David Wierichs, Juan Carrasquilla, and Nathan Killoran, "Here comes the $\mathrm{SU}(N)$: multivariate quantum gates and gradients", arXiv:2303.11355, (2023).

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