When numerically simulating the unitary time evolution of an infinite-dimensional quantum system, one is usually led to treat the Hamiltonian H as an "infinite-dimensional matrix" by expressing it in some orthonormal basis of the Hilbert space, and then truncate it to some finite dimensions. However, the solutions of the Schrödinger equations generated by the truncated Hamiltonians need not converge, in general, to the solution of the Schrödinger equation corresponding to the actual Hamiltonian.In this paper we demonstrate that, under mild assumptions, they converge to the solution of the Schrödinger equation generated by a specific Hamiltonian which crucially depends on the particular choice of basis: the Friedrichs extension of the restriction of H to the space of finite linear combinations of elements of the basis. Importantly, this is generally different from H itself; in all such cases, numerical simulations will unavoidably reproduce the wrong dynamics in the limit, and yet there is no numerical test that can reveal this failure, unless one has the analytical solution to compare with.As a practical demonstration of such results, we consider the quantum particle in the box, and we show that, for a wide class of bases (which include associated Legendre polynomials as a concrete example) the dynamics generated by the truncated Hamiltonians will always converge to the one corresponding to the particle with Dirichlet boundary conditions, regardless the initial choice of boundary conditions. Other such examples are discussed.
When numerically simulating the unitary time evolution of an infinite-dimensional quantum system, one is usually led to treat the Hamiltonian H as an "infinite-dimensional matrix" by expressing it in some orthonormal basis of the Hilbert space, and then truncate it to some finite dimensions. However, the solutions of the Schrödinger equations generated by the truncated Hamiltonians need not converge, in general, to the solution of the Schrödinger equation corresponding to the actual Hamiltonian.In this paper we demonstrate that, under mild assumptions, they converge to the solution of the Schrödinger equation generated by a specific Hamiltonian which crucially depends on the particular choice of basis: the Friedrichs extension of the restriction of H to the space of finite linear combinations of elements of the basis. Importantly, this is generally different from H itself; in all such cases, numerical simulations will unavoidably reproduce the wrong dynamics in the limit, and yet there is no numerical test that can reveal this failure, unless one has the analytical solution to compare with.As a practical demonstration of such results, we consider the quantum particle in the box, and we show that, for a wide class of bases (which include associated Legendre polynomials as a concrete example) the dynamics generated by the truncated Hamiltonians will always converge to the one corresponding to the particle with Dirichlet boundary conditions, regardless the initial choice of boundary conditions. Other such examples are discussed.
We show unexpected behaviour in simulations of generalized squeezing performed with finite-dimensional truncations of the Fock space: even for extremely large dimension of the state space, the results depend on whether the truncation dimension is even or odd. This situation raises the question whether the simulation results are physically meaningful. We demonstrate that, in fact, the two truncation schemes correspond to two well-defined, distinct unitary evolutions whose generators are defined on different subsets of the infinite-dimensional Fock space. This is a consequence of the fact that the generalized squeezing Hamiltonian is not self-adjoint on states with finite excitations, but possesses multiple self-adjoint extensions. Furthermore, we present results on the spectrum of the squeezing Hamiltonians corresponding to even and odd truncation size that elucidate the properties of the two different self-adjoint extensions corresponding to the even and odd truncation scheme. To make the squeezing operator applicable to a physical system, we must regularize it by other terms that depend on the specifics of the experimental implementation. We show that the addition of a Kerr interaction term in the Hamiltonian leads to uniquely converging simulations, with no dependence on the parity of the truncation size, and demonstrate that the Kerr term indeed renders the Hamiltonian self-adjoint and thus physically interpretable.
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.
We generalize the Linear Combination of Hamiltonian Simulation (LCHS) formula [An, Liu, Lin, Phys. Rev. Lett. 2023] to simulate time-evolution operators in infinite-dimensional spaces, including scenarios involving unbounded operators. This extension, named Inf-LCHS for short, bridges the gap between finite-dimensional quantum simulations and the broader class of infinite-dimensional quantum dynamics governed by partial differential equations (PDEs). Furthermore, we propose two sampling methods by integrating the infinite-dimensional LCHS with Gaussian quadrature schemes (Inf-LCHS-Gaussian) or Monte Carlo integration schemes (Inf-LCHS-MC). We demonstrate the applicability of the Inf-LCHS theorem to a wide range of non-Hermitian dynamics, including linear parabolic PDEs, queueing models (birth-or-death processes), Schrödinger equations with complex potentials, Lindblad equations, and black hole thermal field equations. Our analysis provides insights into simulating general linear dynamics using a finite number of quantum dynamics and includes cost estimates for the corresponding quantum algorithms.
It has been recently proven that the bound states of the one-dimensional Schr\"odinger equation H${\mathrm{\ensuremath{\psi}}}^{(\mathit{i})}$=E(i)${\mathrm{\ensuremath{\psi}}}^{(\mathit{i})}$ in [0,\ensuremath{\infty}) can be approximated by those of the corresponding Dirichlet eigenproblem HR${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{i})}$=ER(i)${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{i})}$ in a finite box [0,R] when R\ensuremath{\rightarrow}\ensuremath{\infty}. In this paper, we use a set of mathematical criteria that guarantee the correct calculation of expectation values in the frame of Hilbert spaces, to show that correct expectation values of most physical operators S are obtained by using the equation ${\mathrm{lim}}_{\mathit{R}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{i})}$,S${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{j})}$〉= 〈${\mathrm{\ensuremath{\psi}}}^{(\mathit{i})}$,S${\mathrm{\ensuremath{\psi}}}^{(\mathit{j})}$〉, which includes operators not relatively form bounded by the Hamiltonian H. It is shown that standard numerical methods supply approximate wave functions ${\mathrm{\ensuremath{\psi}}}_{\mathit{R}\mathit{m}}^{(\mathit{i})}$ for which the equation ${\mathrm{lim}}_{\mathit{m}\ensuremath{\rightarrow}\mathrm{\ensuremath{\infty}}}$〈${\mathrm{\ensuremath{\psi}}}_{\mathit{R}\mathit{m}}^{(\mathit{i})}$,S${\mathrm{\ensuremath{\psi}}}_{\mathit{R}\mathit{m}}^{(\mathit{j})}$〉= 〈${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{i})}$,S${\mathrm{\ensuremath{\psi}}}_{\mathit{R}}^{(\mathit{j})}$〉 holds true. Thus, combining the previous limits we obtain a general approach to compute correct expectation values. It is shown how this approach agrees with the results found by other authors in special cases. In the particular case of computing approximate wave functions ${\mathrm{\ensuremath{\psi}}}_{\mathit{R}\mathit{m}}^{(\mathit{i})}$ with the Ritz method, we show, analytically and numerically, that correct convergence toward the expectation values of high power of moment operators rk holds even when the basis functions have an analytic structure that substantially differs from that of the exact wave function.
We are interested in the simulation of open quantum systems governed by the Lindblad master equation in an infinite-dimensional Hilbert space. To simulate the solution of this equation, the standard approach involves two sequential approximations: first, we truncate the Hilbert space to derive a differential equation in a finite-dimensional subspace. Then, we use discrete time-step to obtain a numerical solution to the finite-dimensional evolution. In this paper, we establish bounds for these two approximations that can be explicitly computed to guarantee the accuracy of the numerical results. Through numerical examples, we demonstrate the efficiency of our method, empirically highlighting the tightness of the upper bound. While adaptive time-stepping is already a common practice in the time discretization of the Lindblad equation, we extend this approach by showing how to dynamically adjust the truncation of the Hilbert space. This enables fully adaptive simulations of the density matrix. For large-scale simulations, this approach can significantly reduce computational time and relieves users of the challenge of selecting an appropriate truncation. Furthermore, as a special case, our method naturally applies to Hamiltonian (unitary) dynamics.
For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates { q i }. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement p i → ( ħ /i)(∂/∂ q i ). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ (∂Ψ/∂ t ) = H Ψ which requires as initial condition the knowledge of the wave function Ψ({ q i }, t ) at some initial time t 0 . (Boundary conditions on Ψ({ q i }, t ) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψ ν ( q ), is a basis for the Hilbert space.