Quantum framework for parameterizing partial differential equations via diagonal block-encoding

TL;DR

A quantum-algorithmic framework for parameterizing partial differential equations with block-encodings of diagonal matrices used to represent spatially varying coefficients with structured, potentially complicated profiles is studied.

Abstract

We study a quantum-algorithmic framework for parameterizing partial differential equations (PDEs). For a broad class of problems in which the discretized parameter field admits a diagonal representation, block-encodings of diagonal matrices, or diagonal block-encodings, can be used to represent spatially varying coefficients with structured, potentially complicated profiles. This encoding enables efficient quantum simulation of forward PDEs and extends naturally to parameter-dependent settings. Such simulations are a key primitive for quantum algorithms for PDE-constrained optimization, where the goal is to identify optimal design parameters. We illustrate the framework numerically through forward simulation and parameter design for the two-dimensional wave equation with a Gaussian parameter profile.

Figures (4)

• Figure 1: The value of spatially varying coefficient $c(x,y)$ in the wave equation. (a) The original two-dimensional Gaussian profile $c(x,y)$, (b) its degree-three Fourier series approximation $c_{(3,3)}^F(x,y)$, and (c) its degree-$10$ Fourier series approximation $c_{(10,10)}^F(x,y)$. The lower-order approximation (b) was used in the numerical simulation to reduce the scale of the quantum circuit simulation.
• Figure 2: The value of $c_{(3,3)}^F(x,y)^{-1} \partial u(t,x,y)/\partial t$ at time $t = 1.0$.
• Figure 3: Numerical simulation results for the parameter design problem. (a) Target region (yellow). (b) Objective function $\mathcal{F}(\xi_x,\xi_y)$ evaluated by matrix computation. (c) Objective function $\mathcal{F}(\xi_x,\xi_y)$ evaluated by quantum circuit simulation. In (c), the block-encoded matrix yields $(2/\alpha_\mathrm{for}^2) \mathcal{F}(\xi_x,\xi_y) - 1$ according to sato2025Explicit, where $\alpha_\mathrm{for}$ is the normalization constant for the forward simulation of the wave equation. For ease of comparison with (b), we plot the rescaled value $\mathcal{F}(\xi_x,\xi_y)$.
• Figure 4: A quantum circuit for block-encoding of $M_j$. Let $[k] = \{0, ..., k-1\}$ and an integer $j \in [2^k]$ represented by an $k$-bit string as $j = j_{k-1}...j_{0}$, i.e., $j = \sum_{i \in [k]} j_i 2^i$ with $j_i \in \{0,1\}$. We denote $\mathrm{AND}_j$ for an oracle $\ket{x}_S \ket{0}_{f_j} \mapsto \ket{x}_S \ket{f(x)}_{f_j}$ where $f(x) = \land_{i=1}^k (1 \oplus x_i \oplus j_i)$. $\mathrm{SWAP}_{0,j}$ is a unitary $U = \ketbra{0}{j} + \ketbra{j}{0} + \sum_{i \notin \{0,j\}} \ketbra{i}{i}$. The multi-qubit-controlled $X \otimes I^{\otimes a-1}$ is applied for encoding a zero to the top-left block on the irrelevant selector basis by using the fact that $\bra{0^a} X \otimes I^{\otimes a-1} \ket{0^a} = 0$. The inverse of $\mathrm{AND}_j$ and $\mathrm{AND}_0$ at the end of the circuit makes the flag qubits clean.

Theorems & Definitions (18)

• Theorem 3: Query complexity for a block-encoding of $A^{\mathrm{(2nd)}}$
• Theorem 4: Gate complexity for a block-encoding of $A^{\mathrm{(2nd)}}$
• Theorem 6: Query complexity for a block-encoding of $A^{\mathrm{(1st)}}$
• Theorem 7: Gate complexity for a block-encoding of $A^{\mathrm{(1st)}}$
• Lemma 8: Product of two block-encoded matrices, gilyen2019Quantum
• Lemma 9: Linear combination of block-encoded matrices
• proof
• Corollary 10
• Lemma 11
• proof