Path Integration on a Quantum Computer
J. F. Traub
and H. Woz
Received September 19, 2002; accepted November 18, 2002
We study path integration on a quantum computer that performs quantum sum-
mation. We assume that the measure of path integration is Gaussian, with the
eigenvalues of its covariance operator of order j
with k > 1. For the Wiener
measure occurring in many applications we have k ¼ 2. We want to compute an e-
approximation to path integrals whose integrands are at least Lipschitz. We prove:
Path integration on a quantum computer is tractable.
Path integration on a quantum computer can be solved roughly e
faster than on a classical computer using randomization, and exponentially
faster than on a classical computer with a worst case assurance.
The number of quantum queries needed to solve path integration is roughly the
square root of the number of function values needed on a classical computer
using randomization. More precisely, the number of quantum queries is at
most 4.46 e
. Furthermore, a lower bound is obtained for the minimal
number of quantum queries which shows that this bound cannot be sig-
The number of qubits is polynomial in e
. Furthermore, for the Wiener
measure the degree is 2 for Lipschitz functions, and the degree is 1 for
KEY WORDS: Quantum computation; quantum summation; path integration;
quantum queries; quantum speedup; number of qubits.
PACS: 03.67.Lx; 31.15Kb; 31.15.p; 02.70.c.
Although quantum computers currently exist only as prototypes in the
laboratory, we believe it is important to study theoretical aspects of quan-
1570-0755/02/1000-0365/0 # 2003 Plenum Publishing Corporation
Computer Science, Columbia University.
Computer Science, Columbia University and Institute of Applied Mathematics, University of
To whom correspondence should be addressed. E-mail: email@example.com
Quantum Information Processing, Vol. 1, No. 5, October 2002 (# 2003)