Physics – Quantum Physics
Scientific paper
2001-09-21
Quantum Information Processing 1(5), 365-388, Oct. 2002
Physics
Quantum Physics
24 pages; Revision of 9/2/02 includes a query lower bound and the upper bound of $4.22 \e^{-1}$ to compute an $\e$-approximati
Scientific paper
We study path integration on a quantum computer that performs quantum summation. We assume that the measure of path integration is Gaussian, with the eigenvalues of its covariance operator of order j^{-k} with k>1. For the Wiener measure occurring in many applications we have k=2. We want to compute an $\e$-approximation to path integrals whose integrands are at least Lipschitz. We prove: 1. Path integration on a quantum computer is tractable. 2. Path integration on a quantum computer can be solved roughly $\e^{-1}$ times faster than on a classical computer using randomization, and exponentially faster than on a classical computer with a worst case assurance. 3.The number of quantum queries is 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 $4.22 \e^{-1}$. Furthermore, a lower bound is obtained for the minimal number of quantum queries which shows that this bound cannot be significantly improved. 4.The number of qubits is polynomial in $\e^{-1}$. Furthermore, for the Wiener measure the degree is 2 for Lipschitz functions, and the degree is 1 for smoother integrands.
Traub Joseph F.
Woźniakowski Henryk
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