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

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

Time Stamp: July 11, 2023 5:48 AM

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 $xmapsto e^{i(x A + B)/hbar}$. 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.

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Featured image: Preliminary numerical results indicate that the new method is better than the state-of-the art. Here is a plot showing, for one randomly chosen parameterized quantum evolution, the error (bias) in estimating the derivatives on the vertical, and the parameter value on the horizontal. The green dots show the error of the new method, the red dots give the state of the art. (Other performance variables are fixed and equal for both methods.)

Popular summary

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.

► BibTeX data

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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).

The above citations are from SAO/NASA ADS (last updated successfully 2023-07-14 10:03:06). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2023-07-14 10:03:04).

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