The simulation of multidimensional wave propagation with variable material parameters is a computationally intensive task, with applications from seismology to electromagnetics. While quantum computers offer a promising path forward, their algorithms are often analyzed in the abstract oracle model, which can mask the high gate-level complexity of implementing those oracles. We present a framework for constructing a quantum algorithm for the multidimensional wave equation with a variable speed profile. The core of our method is a decomposition of the system Hamiltonian into sets of mutually commuting Pauli strings, paired with a dedicated diagonalization procedure that uses Clifford gates to minimize simulation cost. Within this framework, we derive explicit bounds on the number of quantum gates required for Trotter–Suzuki-based simulation. Our analysis reveals significant computational savings for structured block-model speed profiles compared to general cases. Numerical experiments in three dimensions confirm the practical viability and performance of our approach. Beyond providing a concrete, gate-level algorithm for an important class of wave problems, the techniques introduced here for Hamiltonian decomposition and diagonalization enrich the general toolbox of quantum simulation.
The simulation of multidimensional wave propagation with variable material parameters is a computationally intensive task, with applications from seismology to electromagnetics. While quantum computers offer a promising path forward, their algorithms are often analyzed in the abstract oracle model, which can mask the high gate-level complexity of implementing those oracles. We present a framework for constructing a quantum algorithm for the multidimensional wave equation with a variable speed profile. The core of our method is a decomposition of the system Hamiltonian into sets of mutually commuting Pauli strings, paired with a dedicated diagonalization procedure that uses Clifford gates to minimize simulation cost. Within this framework, we derive explicit bounds on the number of quantum gates required for Trotter–Suzuki-based simulation. Our analysis reveals significant computational savings for structured block-model speed profiles compared to general cases. Numerical experiments in three dimensions confirm the practical viability and performance of our approach. Beyond providing a concrete, gate-level algorithm for an important class of wave problems, the techniques introduced here for Hamiltonian decomposition and diagonalization enrich the general toolbox of quantum simulation.
Simulating quantum many-body dynamics is important both for fundamental understanding of physics and practical applications for quantum information processing. Therefore, classical simulation methods have been developed so far. Specifically, the Trotter-Suzuki decomposition can analyze a highly complex quantum dynamics, if the number of qubits is sufficiently small so that main memory can store the state vector. However, simulation of quantum dynamics via Trotter-Suzuki decomposition requires a huge number of steps, each of which accesses the state vector, and hence the simulation time becomes impractically long. To settle this issue, we propose a technique to accelerate simulation of quantum dynamics via simultaneous diagonalization of mutually commuting Pauli groups, which is also attracting a lot of attention to reduce the measurement overheads in quantum algorithms. We group the Hamiltonian into mutually commuting Pauli strings, and each of them is diagonalized in the computational basis via a Clifford transformation. Since diagonal operators are applied on the state vector simultaneously with minimum memory access, this method successfully uses performance of highly parallel processors such as Graphics Processing Units (GPU). Compared to both CPU and GPU implementations using fast quantum circuit simulator “qulacs,” the numerical experiments have shown that our method provides a few tens of times acceleration.
The construction of quantum circuits to simulate Hamiltonian evolution is central to many quantum algorithms. State-of-the-art circuits are based on oracles whose implementation is often omitted, and the complexity of the algorithm is estimated by counting oracle queries. However, in practical applications, an oracle implementation contributes a large constant factor to the overall complexity of the algorithm. The key finding of this work is the efficient procedure for representation of a tridiagonal matrix in the Pauli basis, which allows one to construct a Hamiltonian evolution circuit without the use of oracles. The procedure represents a general tridiagonal matrix ${2}^{n}\ifmmode\times\else\texttimes\fi{}{2}^{n}$ by systematically determining all Pauli strings present in the decomposition, dividing them into commuting subsets. The efficiency is in the number of commuting subsets O(n). The method is demonstrated using the one-dimensional wave equation, verifying numerically that the gate complexity as a function of the number of qubits is lower than the oracle-based approach for n<15 and requires half the number of qubits. This method is applicable to other Hamiltonians based on the tridiagonal matrices.
Hamiltonian simulation represents an important module in a large class of quantum algorithms and simulations such as quantum machine learning, quantum linear algebra methods, and modeling for physics, material science, and chemistry. One of the most prominent methods for realizing the time-evolution unitary is via the Trotter-Suzuki decomposition. However, there is a large class of possible decompositions for the infinitesimal time-evolution operator as the order in which the Hamiltonian terms are implemented is arbitrary. We introduce a perspective for generating a low-depth Trotter-Suzuki decomposition assuming the standard Clifford+rz gate set by adapting ideas from quantum error correction. We map a given Trotter-Suzuki decomposition to a constrained path on a graph which we deem the Pauli frame graph (PFG). Each node of the PFG represents the set of possible Hamiltonian terms currently available to be applied, Clifford operations represent a move from one node to another, and so the graph distance represents the gate cost of implementing the decomposition. The problem of finding the optimal decomposition is then equivalent to solving a problem similar to the traveling salesman. Although this is an NP-hard problem, we demonstrate the simplest heuristic, greedy search, and compare the resulting two-qubit gate count and circuit depth to more standard methods for a large class of scientifically relevant Hamiltonians, both fermionic and bosonic, found in chemical, vibrational, and condensed-matter problems which naturally scale. We find, in nearly every case we study, the resulting depth and two-qubit gate counts are less than those provided by standard methods, by as much as an order of magnitude. We also find the method is efficient and amenable to parallelization, making it scalable for problems of real interest.
Partial differential equations (PDEs) play a central role in scientific and engineering analysis, with applications spanning fluid dynamics, heat and mass transfer, electromagnetism, quantum mechanics, and financial modeling, where they are used to describe diffusion processes, wave propagation, and the evolution of complex systems over space and time. Solving multidimensional partial differential equations (PDEs) is a computationally challenging problem, even for the most advanced classical systems. Over the past decade, quantum computing has attracted significant interest as a potential approach for solving complex computational problems, including multidimensional PDEs. Although a variety of approaches have been proposed for solving PDEs, most of the existing techniques are based on variational quantum algorithms (VQAs). Despite being promising, these VQA-based approaches suffer from low accuracy, long execution times, and limited scalability. In this work, we propose a scalable and efficient quantum algorithm for solving multidimensional PDEs. Our algorithm has two variants. One variant is based on the finite difference method (FDM), classical-to-quantum (C2Q) encoding, and numerical instantiation, whereas the other is based on FDM, C2Q, and column-by-column decomposition (CCD). We have also evaluated our algorithm using several practical case studies; namely, Poisson, heat, Black–Scholes, and Navier–Stokes equations. The results show that our proposed approach achieves higher accuracy, greater scalability, and faster execution time than the VQA-based approaches. We validated these findings on both noise-free and noisy simulators, as well as on a hardware emulator and real IBM quantum hardware.
We explore the utilization of higher-order discretization techniques in optimizing the gate count needed for quantum computer based solutions of partial differential equations. To accomplish this, we present an efficient approach for decomposing d-band diagonal matrices into Pauli strings that are grouped into mutually commuting sets. Using numerical simulations of the one-dimensional wave equation, we show that higher-order methods can reduce the number of qubits necessary for discretization, similar to the classical case, although they do not decrease the number of Trotter steps needed to preserve solution accuracy. This result has important consequences for the practical application of quantum algorithms based on Hamiltonian evolution.
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.