Differential equation quantum solvers: engineering measurements to reduce cost

TL;DR

This work tackles the high cost of quantum-circuit evaluations in differentiable quantum circuit methods for solving nonlinear differential equations on near-term devices. It introduces two resource-efficient protocols—Trainable Observable (TO) and Flipped Shadow (FS)—that restructure information extraction: TO replaces parameter-dependent circuit scans with precomputed, parameter-free measurements, while FS uses classical shadows to batch-derive derivatives across many collocation points. Benchmark results on 1D and 2D DEs show up to around 100x reductions in circuit evaluations, with tradeoffs depending on the measurement-operator choice and grid size, indicating these methods can be complementary. Overall, the approaches substantially improve the practical implementability of DQC for SciML, enabling larger and more complex DE demonstrations on hardware-constrained quantum devices.

Abstract

Quantum computers have been proposed as a solution for efficiently solving non-linear differential equations (DEs), a fundamental task across diverse technological and scientific domains. However, a crucial milestone in this regard is to design protocols that are hardware-aware, making efficient use of limited available quantum resources. We focus here on promising variational methods derived from scientific machine learning: differentiable quantum circuits (DQC), addressing specifically their cost in number of circuit evaluations. Reducing the number of quantum circuit evaluations is particularly valuable in hybrid quantum/classical protocols, where the time required to interface and run quantum hardware at each cycle can impact the total wall-time much more than relatively inexpensive classical post-processing overhead. Here, we propose and test two sample-efficient protocols for solving non-linear DEs, achieving exponential savings in quantum circuit evaluations. These protocols are based on redesigning the extraction of information from DQC in a ``measure-first" approach, by introducing engineered cost operators similar to the randomized-measurement toolbox (i.e. classical shadows). In benchmark simulations on one and two-dimensional DEs, we report up to $\sim$ 100 fold reductions in circuit evaluations. Our protocols thus hold the promise to unlock larger and more challenging non-linear differential equation demonstrations with existing quantum hardware.

Figures (4)

• Figure 1: Solving the damped oscillator Eq. \ref{['eq:damped_osc']}. Training as detailed in main text. (a) The solution plotted as attained via DQC, FS and various TO models (dashed lines). "loc$k$" TO models include only $k$-local Pauli strings, whereas all indicates inclusion of all $4^N$ Pauli strings for the circuit. Exact solution for comparison plotted with a solid line. (b) The loss (solid lines) and measure of success at each iteration of the training for various models. Models identified by same colour as in (a). Solid vertical (horizontal) gray lines indicate as reference the maximum number of circuit executions $d_{\text{max}}$ invoked by any TO model (the baseline MoS performance attained by DQC within the shown iterations).
• Figure 2: Solving the stationary Burgers Eq. \ref{['eq:stnry_burgers']}. Training as detailed in main text. (a) The solution plotted as attained via DQC, FS and various TO models (dashed lines). Refer to Fig. \ref{['fig:dampedosc']} for details. Exact solution for comparison plotted with a solid line. (b) The loss (solid lines) and measure of success at each iteration of the training for various models. Models identified by same colour as in (a). Solid vertical (horizontal) gray lines as in Fig. \ref{['fig:dampedosc']}.
• Figure 3: Solving the coupled differential equations Eq. \ref{['eq:cpd_eqn']}. Training as detailed in main text. (a) The solutions $f$ and $g$ plotted, as attained via DQC, FS and various TO models (dashed lines). Refer to Fig. \ref{['fig:dampedosc']} for details. Exact solution for comparison plotted with a solid line. (b) The loss (solid lines) and measure of success at each iteration of the training for various models. Models identified by same colour as in (a). Solid vertical (horizontal) gray lines as in Fig. \ref{['fig:dampedosc']}.
• Figure 4: Solving higher dimensional DEs (Eq. \ref{['eq:hd_ex']}). (a) Plots of the known analytic solution (true) and error plots (red-white) of the measure of success (MoS) for various quantum models as introduced in Fig. \ref{['fig:coupled']} upon training - as detailed in main text. MoS is here evaluated as the square error between trained solution $f({x}, {y})$ and exact $\bar{f}({x}, {y})$ at each point in the training grid. (b) The loss (solid lines) and MoS (dashed lines) as a function of iteration throughout training for each model. Solid vertical (horizontal) gray lines as in Fig. \ref{['fig:dampedosc']}.