One of the main challenges in numerical simulation of quantum dynamics is the prohibitive cost in the semi-classical regime, in which the de Broglie wave length is small compared with the characteristic length scale and the solution is highly oscillatory. For the von-Neumann equation for mixed-state quantum dynamics, this difficulty is overcome by using the Weyl variables, under which the solution is not oscillatory. Furthermore, we use the integral representation of the potential difference, which robustly captures the classical limit as the semi-classical parameter approaches zero. By using exact Hermite moments and quantum singular value transformation to treat the polynomial and sparse coordinate matrices of the dense projected smooth non-polynomial potentials respectively, we obtain a quantum algorithm efficient for {\it all} ranges of wave lengths, with complexity {\it polynomial} in the spatial dimension and discretization and query bounds containing {\it no} negative powers of the small wavelength. Thus it can capture the correct physical observables even if the spatial grid does not resolve the frequency, hence defying the Nyquist-Shannon sampling theorem.
One of the main challenges in numerical simulation of quantum dynamics is the prohibitive cost in the semi-classical regime, in which the de Broglie wave length is small compared with the characteristic length scale and the solution is highly oscillatory. For the von-Neumann equation for mixed-state quantum dynamics, this difficulty is overcome by using the Weyl variables, under which the solution is not oscillatory. Furthermore, we use the integral representation of the potential difference, which robustly captures the classical limit as the semi-classical parameter approaches zero. By using exact Hermite moments and quantum singular value transformation to treat the polynomial and sparse coordinate matrices of the dense projected smooth non-polynomial potentials respectively, we obtain a quantum algorithm efficient for {\it all} ranges of wave lengths, with complexity {\it polynomial} in the spatial dimension and discretization and query bounds containing {\it no} negative powers of the small wavelength. Thus it can capture the correct physical observables even if the spatial grid does not resolve the frequency, hence defying the Nyquist-Shannon sampling theorem.
Solving the time-dependent Schrödinger equation is an important application area for quantum algorithms. We consider Schrödinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter ℏ , in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schrödinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of ℏ and the precision ε are obtained. It is found that the number of required qubits, m , scales only logarithmically with respect to ℏ . When the solution has bounded derivatives up to order ℓ , the symmetric Trotting method has gate complexity O ( ( ε ℏ ) − 1 2 p o l y l o g ( ε − 3 2 ℓ ℏ − 1 − 1 2 ℓ ) ) , provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with p o l y ( m ) operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of ℏ . The gate complexity in this case is reduced to O ( ε − 1 2 p o l y l o g ( ε − 3 2 ℓ ℏ − 1 ) ) , with ℓ again indicating the smoothness of the solution.
The Helmholtz equation is a prototypical model for time-harmonic wave propagation. Numerical solutions become increasingly challenging as the wave number k grows, due to the equation's elliptic yet noncoercive character and the highly oscillatory nature of its solutions, with wavelengths scaling as 1/k. These features lead to strong indefiniteness and large system sizes. We present a quantum algorithm for solving such indefinite problems, built upon the Schrödingerization framework. This approach reformulates linear differential equations into Schrödinger-type systems by capturing the steady state of damped dynamics. A warped phase transformation lifts the original problem to a higher-dimensional formulation, making it compatible with quantum computation. To suppress numerical pollution, the algorithm incorporates asymptotic dispersion correction. It achieves a query complexity of O(κ2polylogε−1), where κ is the condition number and ε the desired accuracy. For the Helmholtz equation, a simple preconditioner further reduces the complexity to O(κpolylogε−1). Our constructive extension to the quantum setting is broadly applicable to all indefinite problems.
Simulating Markovian open quantum systems in the semiclassical regime poses a grand challenge for computational physics, as the highly oscillatory nature of the dynamics imposes prohibitive resolution requirements on traditional grid-based methods. To overcome this barrier, this paper introduces an efficient Frozen Gaussian Sampling (FGS) algorithm based on the Wigner-Fokker-Planck phase-space formulation. The proposed algorithm exhibits two transformative advantages. First, for the computation of physical observables, its sampling error is independent of the semiclassical parameter ε, thus fundamentally breaking the prohibitive computational scaling faced by grid methods in the semiclassical limit. Second, its mesh-free nature entirely eliminates the boundary-induced instabilities that constrain long-time grid-based simulations. Leveraging these capabilities, the FGS algorithm serves as a powerful investigatory tool for exploring the long-time behavior of open quantum systems. Specifically, we provide compelling numerical evidence for the existence of steady states in strongly non-harmonic potentials-a regime where rigorous analytical results are currently lacking.
This dissertation is concerned with a variety of problems in quantum computing and the spectral theory of quantum-mechanical channels and Hamiltonians. We begin with the problem of quantum gate synthesis. The challenge is to find a fixed, finite set S of quantum gates such that, given a quantum circuit component U acting on n qubits and target precision ε, there is a short, efficiently computable composition of gates from S that approximates U to precision ε. This problem has already led to a rich literature in the case n=1 with deep connections to modern number theory. In this dissertation we present the state-of-the-art algorithm for a single qubit, achieving length 7 log (1/ε). The techniques used include the lattice structure of certain dense subgroups of PU(2) and Lenstra's algorithm for integer programming in convex domains. The next topic considered is scientific computing algorithms intended to run on a quantum computer. Specifically, we study a framework for computing functions of normal matrices using a quantum computer, with the objective of controlling the ancilla overhead, probability of success, and end precision. This arises as a special case of the multivariate quantum eigenvalue transform, for computing function of multiple commuting Hermitian matrices. The techniques applied include approximation of regular functions by low-degree polynomials and modern quantum primitives like quantum signal processing and linear combinations of unitaries. The third component is a spectral gap comparison result for Davies generators, a class of Lindbladians used for Gibbs state preparation and which are a well-studied example of quantum Markov dynamics, and which contain a canonical classical Markov chain. The main result shows that if the spectrum of the Hamiltonian driving the dynamics is free of long arithmetic progressions, then the spectral gaps of the quantum and classical Markov chains are comparable. Thus, classical techniques may be leveraged to control the behavior of the quantum dynamics. The proof is based on transferring any "off-diagonal" eigenvector of the Davies generator to an observable which commutes with the Hamiltonian and has a comparable Lindbladian Rayleigh quotient. Finally, we study the infimum of the spectrum, or ground state energy, of a discrete Schrödinger operator on the scaled integer grid in n dimensions, parameterized by a nonnegative potential V and a small frequency parameter θ. We relate this ground state energy to that of a corresponding continuous semiclassical Schrödinger operator on the full n-dimensional space with parameter θ, arising from the same choice of potential, and show that these quantities are equal up to vanishing factors depending on the regularity of V. The proofs are elementary and rely on sampling and interpolation to map low-energy functions for the discrete operator on the scaled grid to low-energy functions for the continuous operator on the full space, and vice versa. Coauthors on these works were J. Basso, T. R. Blackman, Y. Borns-Weil, I. Detherage, S. Ganguly, N. Srivastava, T. Saffat, A. Sinclair, and T.-D. Vuong.
Nonrelativistic quantum chemistry is based on the Schrödinger equation in continuous, unbounded space. Computational approaches on classical and quantum computers often truncate the spatial domain and introduce a finite-dimensional basis, reducing the continuum problem to a finite-dimensional one. There is therefore a fundamental question: to what extent does this approximation capture the original continuum problem? In this work, we develop a general Sobolev-Galerkin framework for estimating spatial discretization errors in quantum simulation with unbounded Hamiltonians. We apply this framework to many-body Coulomb systems and establish both periodization and Galerkin error estimates for wave-function dynamics and ground-state energies. For dynamics, the finite-basis projection error of the exact wave function does not capture the full basis-discretization error: an additional dynamical projection error enters our bound at a different rate, and we introduce an inner-product argument to control this contribution. Interestingly, the spatial convergence rates for wave-function dynamics and ground-state energies differ quantitatively. For ground-state energies, we establish a sharp M−1 convergence rate at fixed cell size, where M is the number of one-particle basis functions. We also show that a uniform M−1 convergence rate is impossible for the corresponding real-time wave-function error with general H2 initial data. The analysis relies on many-body Sobolev estimates with explicit polynomial dependence on system size. Combining these estimates with state-of-the-art quantum algorithms, we obtain, to our knowledge, the first rigorous end-to-end complexity bounds for quantum simulation of molecular and electronic dynamics with all spatial errors explicitly controlled. The number of basis functions is determined directly from the continuum problem, without regularization.
Abstract. Partial differential equation (PDE) models with multiple temporal/spatial scales are prevalent in several disciplines such as physics, engineering, and many others. These models are of great practical importance but notoriously difficult to solve due to prohibitively small mesh and time step sizes limited by the scaling parameter and CFL condition. Another challenge in scientific computing could come from curse-of-dimensionality. In this paper, we aim to provide a quantum algorithm, based on either direct approximations of the original PDEs or their homogenized models, for prototypical multiscale problems inPDEs, including elliptic, parabolic, and hyperbolic PDEs. To achieve this, we will lift these problems to higher dimensions and leverage the recently developed Schrödingerization based quantum simulation algorithms to efficiently reduce the computational cost of the resulting high-dimensional and multiscale problems. We will examine the error contributions arising from discretization, homogenization, and relaxation, and analyze and compare the complexities of these algorithms in order to identify the best algorithms in terms of complexities for different equations in different regimes.