Explicit block-encoding for partial differential equation-constrained optimization

TL;DR

This work addresses PDE-constrained optimization by proposing a fully quantum algorithm that coherently couples a quantum PDE solver with a quantum optimizer via explicit block-encoding of the objective. The method builds BE's for the objective using a forward-simulation oracle and LCU/LC-SVT techniques, enabling end-to-end optimization without costly quantum-state tomography. Under strong convexity, the authors show polynomial speedups in the PDE system size $N$, exponential speedups in spatial dimension $d$, and a space advantage over classical approaches, with the speedups inherited from the underlying PDE solver. The framework is validated on two applications: parameter calibration in the Black-Scholes equation and material parameter design in a wave equation, illustrating the practicality and potential quantum advantages of the approach. Overall, this work outlines a coherent, bottleneck-free quantum pipeline for PDE-constrained optimization and provides concrete complexity and application results that highlight potential quantum benefits for large-scale, high-dimensional PDE problems.

Abstract

Partial differential equation (PDE)-constrained optimization, where an optimization problem is subject to PDE constraints, arises in various applications such as design, control, and inference. Solving such problems is computationally demanding because it requires repeatedly solving a PDE and using its solution within an optimization process. In this paper, we first propose a fully coherent quantum algorithm for solving PDE-constrained optimization problems. The proposed method combines a quantum PDE solver that prepares the solution vector as a quantum state, and a quantum optimizer that assumes oracle access to a quantized objective function. The central idea is the explicit construction of the oracle in a form of block-encoding for the objective function, which coherently uses the output of a quantum PDE solver. This enables us to avoid classical access to the full solution that requires quantum state tomography canceling out the potential quantum speedups. We also derive the overall computational complexity of the proposed method with respect to parameters for optimization and PDE simulation, where quantum speedup is inherited from the underlying quantum PDE solver. We numerically demonstrate the validity of the proposed method by applications, including a parameter calibration problem in the Black-Scholes equation and a material parameter design problem in the wave equation. This work presents the concept of composing quantum subroutines so that the weakness of one (i.e., prohibitive readout overhead) is neutralized by the strength of another (i.e., coherent oracle access), toward a bottleneck-free quantum algorithm.

Figures (13)

• Figure 1: Conceptual diagram of the proposed method (right), compared with the conventional classical approach (left). $\text{HS}(\hat{\mathcal{F}}(u(t;\xi))) := \mathcal{T}e^{-\int_0^S (e^{-\nu s'} \nabla_\xi^2/2 + e^{\nu s'} \hat{\mathcal{F}}) \mathrm{d} s'} \ket{\psi_0}$ is Hamiltonian simulation algorithm of the time-dependent Hamiltonian with the objective function as a potential term, where $\nu$ is a constant parameter.
• Figure 2: Quantum circuit for block-encoding of $\mathcal{F}_\mathrm{quad}$.
• Figure 3: Quantum circuit for block-encoding of $\mathcal{F}_\mathrm{err}$.
• Figure 4: The objective function values computed by the block-encoding and the forward simulation. The horizontal axis represents the index of the discretized design variables, i.e., decimal value of the basis $\ket{\xi}$.
• Figure 5: The history of the expected value of the objective function (red line) with its standard deviation (purple region). The horizontal axis represents the number of iterations of the product formula. The black dashed lines represent the optimal value $\mathcal{F}(7 / 15)$.