Quantum algorithms for viscosity solutions to nonlinear Hamilton-Jacobi equations based on an entropy penalisation method

TL;DR

The paper addresses the quantum computation of viscosity solutions for nonlinear Hamilton-Jacobi equations with convex Hamiltonians, where caustics and long-time behavior pose major challenges. By combining Gomes–Valdinoci entropy penalisation with a discrete-time Cole-Hopf–like transform, the authors obtain a linear, global-in-time formulation that maps nonlinear viscosity dynamics to linear parabolic evolution. They develop both analog and digital quantum simulation schemes for the resulting linear PDEs and design quantum protocols to extract pointwise values, gradients, the minimum, and function values at the minimiser without full tomography. This framework enables efficient quantum access to physically relevant observables in high-dimensional settings, with potential impact on front propagation, mean-field games, and optimal control. Overall, the work provides a robust, scalable route to harness quantum computing for a broad class of nonlinear PDEs through linearisation while preserving essential nonlinear dynamics at the viscosity level.

Abstract

We present a framework for efficient extraction of the viscosity solutions of nonlinear Hamilton-Jacobi equations with convex Hamiltonians. These viscosity solutions play a central role in areas such as front propagation, mean-field games, optimal control, machine learning, and a direct application to the forced Burgers' equation. Our method is based on an entropy penalisation method proposed by Gomes and Valdinoci, which generalises the Cole-Hopf transform from quadratic to general convex Hamiltonians, allowing a reformulation of viscous Hamilton-Jacobi dynamics by a discrete-time linear dynamics which approximates a linear heat-like parabolic equation, and can also extend to continuous-time dynamics. This makes the method suitable for quantum simulation. The validity of these results hold for arbitrary nonlinearity that correspond to convex Hamiltonians, and for arbitrarily long times, thus obviating a chief obstacle in most quantum algorithms for nonlinear partial differential equations. We provide quantum algorithms, both analog and digital, for extracting pointwise values, gradients, minima, and function evaluations at the minimiser of the viscosity solution, without requiring nonlinear updates or full state reconstruction.

Figures (3)

• Figure 1: This is the quantum algorithm for simulating $|u(t)\rangle$ using Schrödingerisation, and it is also an algorithm for estimating the normalisation constant $\|\boldsymbol{u}(t)\|$. This normalisation factor is estimated by measuring the probability when $\xi>0$ is detected for the ancilla state (flag for success). Here the algorithm is expressed in the language of analog quantum simulation for simplicity, and it can easily be extended to the discrete-variable setting by discretising all the states and operators, and in the digital case amplitude amplification can be employed to quadratically boost the success probability.
• Figure 2: Schematic diagram of analog quantum protocol for extracting the gradient $|g_{ka}|=\partial S_{\nu}(t, x_a)/\partial x_k$, when given copies of the coherent state $|\alpha_0\rangle$ (easy to prepare) and the state $|u(t)\rangle$. Here $u(t,x)$ is related to $S_{\nu}(t,x)$ by the Cole-Hopf transformation as outlined in \ref{['sec:heatquantumsimulation']}. The coherent state is an ancilla qumode that acts as a measurement probe, and we proceed to use a 'pointer-variable' measurement. Given the initial state $|u(t)\rangle|\alpha_0\rangle$, we apply a Gaussian operation $\exp(2i\kappa\nu \hat{p}_k \otimes \hat{p})$, where $\hat{p}_k$ acts on the $|u(t)\rangle$ register and $\hat{p}$ acts on the coherent state register and $\kappa \ll 1$. Then projecting the resulting first register onto the desired position $|x_a\rangle \langle x_a|$, the resulting ancilla qumode will have its coherence shifted by a value that is $-i$ times $\partial S_{\nu}(t, x_a)/\partial x_k$. To extract this coherence, the expectation value of the ancilla mode with respect to $\hat{x}$ and $\hat{p}$ are taken. Then we can estimate the real and imaginary components of $g_{ak}$ by measuring the quadratures $\langle \hat{x}\rangle_{\alpha}=\kappa\text{Im}(g_{ka})$, $\langle \hat{p}\rangle_{\alpha}=-\kappa \text{Re}(g_{ka})$, and thus recover $|g_{ka}|$.
• Figure 3: Schematic diagram of digital quantum protocol for extracting the digital approximation of the gradient $\partial S_{\nu}(t, x=x_a/\partial x_k)$ by $|\hat{g}_{ka}|$, when given copies of the single-qubit ancilla state $|0\rangle$ and the discrete-variable state $|u(t)\rangle$. Given the initial state $|u(t)\rangle|0\rangle$, we apply the operation $\exp(2i \kappa \nu \hat{P}_k \otimes \sigma_x)$, where $\hat{P}_k$ acts on the $|u(t)\rangle$ register and $\sigma_x$ acts on the qubit ancilla and $\kappa \ll 1$. By projecting the resulting first register onto the desired position $|x_a\rangle \langle x_a|$, the resulting qubit state can be used to estimate $|\hat{g}_{jk}|$ by measuring $\langle \sigma_z\rangle$ of the resulting ancilla qubit.

Theorems & Definitions (40)

• Lemma 1
• Remark 2
• Lemma 3
• proof
• Remark 4
• Lemma 5
• Remark 6
• Lemma 7
• proof
• Lemma 8