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.

翻译：量子计算机被提出作为高效求解非线性微分方程（DEs）的潜在解决方案，该任务是众多科技与科学领域的基础问题。然而，实现这一目标的关键里程碑在于设计硬件感知的协议，以高效利用有限的可用量子资源。本文聚焦于源自科学机器学习的有前景的变分方法：可微分量子电路（DQC），并特别关注其电路评估次数的成本。在混合量子/经典协议中，减少量子电路评估次数尤为重要，因为每个周期中与量子硬件接口和运行所需的时间对总耗时的影响，远超过相对廉价的经典后处理开销。在此，我们提出并测试了两种用于求解非线性微分方程的样本高效协议，实现了量子电路评估次数的指数级节省。这些协议基于“先测量”方法重新设计从DQC中提取信息的过程，通过引入类似于随机测量工具箱（即经典阴影）的工程化代价算子。在一维和二维微分方程的基准模拟中，我们实现了高达约100倍的电路评估次数减少。因此，我们的协议有望利用现有量子硬件解锁更大规模、更具挑战性的非线性微分方程演示。