Explicit block encodings of boundary value problems for many-body elliptic operators
arXiv:2407.18347 · doi:10.22331/q-2025-06-04-1764
Abstract
Simulation of physical systems is one of the most promising use cases of future digital quantum computers. In this work we systematically analyze the quantum circuit complexities of block encoding the discretized elliptic operators that arise extensively in numerical simulations for partial differential equations, including high-dimensional instances for many-body simulations. When restricted to rectangular domains with separable boundary conditions, we provide explicit circuits to block encode the many-body Laplacian with separable periodic, Dirichlet, Neumann, and Robin boundary conditions, using standard discretization techniques from low-order finite difference methods. To obtain high-precision, we introduce a scheme based on periodic extensions to solve Dirichlet and Neumann boundary value problems using a high-order finite difference method, with only a constant increase in total circuit depth and subnormalization factor. We then present a scheme to implement block encodings of differential operators acting on more arbitrary domains, inspired by Cartesian immersed boundary methods. We then block encode the many-body convective operator, which describes interacting particles experiencing a force generated by a pair-wise potential given as an inverse power law of the interparticle distance. This work provides concrete recipes that are readily translated into quantum circuits, with depth logarithmic in the total Hilbert space dimension, that block encode operators arising broadly in applications involving the quantum simulation of quantum and classical many-body mechanics.
arXiv commentPreparation for publication with quantum journal
References in corpus (37)
- Elementary gates for quantum computation
- Hamiltonian Simulation by Qubitization
- Optimal Hamiltonian Simulation by Quantum Signal Processing
- Efficient quantum algorithms for simulating sparse Hamiltonians
- Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics
- A new quantum ripple-carry addition circuit
- Preconditioned quantum linear system algorithm
- Halving the cost of quantum addition
- High-order quantum algorithm for solving linear differential equations
- Efficient quantum algorithm for dissipative nonlinear differential equations
- PennyLane: Automatic differentiation of hybrid quantum-classical computations
- Quantum algorithm for linear differential equations with exponentially improved dependence on precision
- Quantum algorithms and the finite element method
- Quantum algorithm and circuit design solving the Poisson equation
- High-precision quantum algorithms for partial differential equations
- Quantum Algorithm for Simulating the Wave Equation
- Efficient phase-factor evaluation in quantum signal processing
- Quantum spectral methods for differential equations
- Black-box quantum state preparation without arithmetic
- Optimal polynomial based quantum eigenstate filtering with application to solving quantum linear systems
- Improved quantum algorithms for linear and nonlinear differential equations
- Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions
- Quantum simulation of partial differential equations via Schrodingerisation: technical details
- Towards provably efficient quantum algorithms for large-scale machine-learning models
- Linear combination of Hamiltonian simulation for nonunitary dynamics with optimal state preparation cost
- A quantum algorithm to solve nonlinear differential equations
- Time-marching based quantum solvers for time-dependent linear differential equations
- Bounding the costs of quantum simulation of many-body physics in real space
- Quantum simulation of real-space dynamics
- Potential quantum advantage for simulation of fluid dynamics
- Time complexity analysis of quantum difference methods for linear high dimensional and multiscale partial differential equations
- On efficient quantum block encoding of pseudo-differential operators
- Block-encoding dense and full-rank kernels using hierarchical matrices: applications in quantum numerical linear algebra
- Explicit Quantum Circuits for Block Encodings of Certain Sparse Matrices
- Efficient explicit gate construction of block-encoding for Hamiltonians needed for simulating partial differential equations
- Explicit Error Bounds for Carleman Linearization
- Generalized Quantum Signal Processing