Preconditioned Quantum Linear System Algorithm

Abstract

We describe a quantum algorithm that generalizes the quantum linear system algorithm [Harrow et al., Phys. Rev. Lett. 103, 150502 (2009)] to arbitrary problem specifications. We develop a state preparation routine that can initialize generic states, show how simple ancilla measurements can be used to calculate many quantities of interest, and integrate a quantum-compatible preconditioner that greatly expands the number of problems that can achieve exponential speedup over classical linear systems solvers. To demonstrate the algorithm's applicability, we show how it can be used to compute the electromagnetic scattering cross section of an arbitrary target exponentially faster than the best classical algorithm.The potential power of quantum computing was first described by Feynman, who showed that the exponential growth of the Hilbert space of a quantum computer allows efficient simulations of quantum systems, whereas a classical computer would be overwhelmed [1]. Shor extended the applicability of quantum computing when he developed a quantum factorization algorithm that also provides exponential speedup over the best classical algorithm [2].More recently, Harrow et al.[3] demonstrated a quantum algorithm for solving a linear system of equations that, for well-conditioned matrices, gives exponential speedup over the best classical method. In that paper, the authors demonstrated how to invert a sparse matrix to solve the quantum linear system A|x = |b . The requirements for achieving exponential speedup were: 1) the elements of A be efficiently computable via a blackbox oracle; 2) the matrix A must to be sparse, or efficiently decomposable into sparse form; 3) the condition number of A must scale as polylog N where N is the size of the linear system. As presented, the algorithm had three features that made it difficult to apply to generic problem specifications and achieve the promised exponential speedup. These included: State preparation -preparing the generic state |b is an unsolved problem [4][5][6][7][8], and no mention on how one might do this was provided. Solution readoutsince the solution is stored in a quantum state |x , measurement of it is impractical. The authors suggested that it could be used to calculate some expectation values of an arbitrary operator x|R|x . However, no measurement procedure was specified, and estimating x|R|x is not trivial in general. Condition number -in order for the quantum algorithm to achieve exponential speedup, the condition number can scale at most poly logarithmically with the size of the matrix A. This is a very strict condition that greatly limits the class of problems that can achieve exponential speedup.In this letter, we provide solutions to these three problems, greatly expanding the applicability of the Quantum Linear Systems Algorithm (QLSA). In addition, we show how our new techniques enable the first start-to-finish application of the QLSA to a problem of broad interest and importance. Namely, we show how to solve for the scattering cross section of an arbit...

Quantum Spectral Methods for Differential Equations

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