Resource-efficient quantum simulation of transport phenomena via Hamiltonian embedding

TL;DR

This work addresses the challenge of efficiently simulating transport phenomena on quantum hardware by integrating Schrödingerization for non-unitary-to-unitary mapping with Hamiltonian embedding to realize sparse, tensor-product-structured Hamiltonians. The authors provide a hardware-aware end-to-end framework that combines spatial discretization, embedding-based Hamiltonian simulation, and Richardson extrapolation to achieve nearly linear-in-time scaling with polylogarithmic dependence on the precision, along with rigorous guarantees of speedups for structured problems. They demonstrate the approach through linear and nonlinear transport PDEs, including a first real-device demonstration of a 2D advection equation on a trapped-ion computer, and provide extensive resource analyses comparing encoding schemes (one-hot, unary, circulant unary) to optimize circuit depth and gate counts. The results show substantial circuit-depth reductions and point to practical pathways for validating transport PDE solvers on intermediate-term quantum hardware, while noting challenges in state preparation and noise mitigation for larger-scale problems.

Abstract

Transport phenomena play a key role in a variety of application domains, and efficient simulation of these dynamics remains an outstanding challenge. While quantum computers offer potential for significant speedups, existing algorithms either lack rigorous theoretical guarantees or demand substantial quantum resources, preventing scalable and efficient validation on realistic quantum hardware. To address this gap, we develop a comprehensive framework for simulating classes of transport equations, offering both rigorous theoretical guarantees -- including exponential speedups in specific cases -- and a systematic, hardware-efficient implementation. Central to our approach is the Hamiltonian embedding technique, a white-box approach for end-to-end simulation of sparse Hamiltonians that avoids abstract query models and retains near-optimal asymptotic complexity. Empirical resource estimates indicate that our approach can yield an order-of-magnitude (e.g., $42\times$) reduction in circuit depth given favorable problem structures. We then apply our framework to solve linear and nonlinear transport PDEs, including the first experimental demonstration of a 2D advection equation on a trapped-ion quantum computer.

Figures (7)

• Figure 1: Schematic of quantum PDE solvers via Hamiltonian embedding.
• Figure 2: Circuit diagram for Schrödingerization. The unitaries $U_{\text{state prep}}^{(p)}$ and $U_{\text{state prep}}^{(u)}$ denote state preparation circuits for the auxiliary variable $p$ and the solution $u$, respectively.
• Figure 3: Quantum simulation of the 2D linear advection equation.Left: (a) Each row shows the measured probability distribution for simulation times $T \in \{0, 0.05, 0.10, 0.15, 0.20\}$ (left to right). A high resolution classical simulation is shown in the top row for reference. The middle row shows results obtained from a noiseless circuit simulation, and the bottom row shows the results obtained from the Aria-1 quantum computer. Right: Empirical estimates of the two-qubit circuit depth (b) and gate count (c) required to simulate the advection equation with the upwind differences and Schrödingerization.
• Figure 4: Quantum simulation of nonlinear transport dynamics.Left: (a) Numerical simulations of \ref{['eqn:nonlinear_pde']} for times $T\in\{0,0.25,0.50,0.75,1.0\}$ using the method of characteristics (top) and the level set method (bottom). The initial condition is $u_0(x_1,x_2)=0.4(\sin(2\pi x_1)+\sin(2\pi x_2))$. We discretize using $N_x=N_q=128$ grid points for each dimension. Right: Empirical estimates of the two-qubit circuit depth (b) and gate counts (c) for simulating \ref{['eqn:nonlinear_pde']} using $N_q$ grid points for $q$, where $N_q\in\{8, 16, 32, 64, 128\}$.
• Figure 5: Landscape of the best encoding scheme for simulating a diagonal operator $P^K$ for varying system size $N$ and polynomial degree $K$, determined by the total number of 1- and 2-qubit gates. The yellow region shows the regime where standard binary uses fewer gates, while the teal region shows where the one-hot code uses fewer gates. For linear polynomials, the standard binary code outperforms the one-hot code for system size up to $N=256$. For higher degree polynomials, the one-hot code tends to use fewer gates, unless the system size is very large.

Theorems & Definitions (26)

• Definition 1: Hamiltonian embeddings
• Remark 1
• Theorem 1: PDE solvers using Hamiltonian embedding, informal
• Definition 2: One-hot code
• Proposition 1: One-hot embedding without penalty
• Definition 3: Unary code
• Proposition 2: Unary embedding without penalty
• Remark 2
• Definition 4: Circulant unary code
• Proposition 3: Circulant unary embedding without penalty