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Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity p o l y ( 1 / ϵ ) , where ϵ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be p o l y ( d , log ( 1 / ϵ ) ) , where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.
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Quantum simulation in the semi-classical regime
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.
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- Award ID(s):
- 2111221
- PAR ID:
- 10346227
- Date Published:
- Journal Name:
- Quantum
- Volume:
- 6
- ISSN:
- 2521-327X
- Page Range / eLocation ID:
- 739
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.22331/q-2022-06-17-739
