# Path Integration on a Quantum Computer

@article{Traub2002PathIO,
title={Path Integration on a Quantum Computer},
author={Joseph F. Traub and Henryk Wozniakowski},
journal={Quantum Information Processing},
year={2002},
volume={1},
pages={365-388}
}
• Published 21 September 2001
• Computer Science
• Quantum Information Processing
AbstractWe 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 ε-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…

### Quantum algorithms and complexity for certain continuous and related discrete problems

• Computer Science, Mathematics
• 2005
The thesis shows that in both the randomized and quantum settings the curse of dimensionality is vanquished, i.e., the minimal number of function evaluations and/or quantum queries required to compute an approximation depends only polynomially on the reciprocal of the desired accuracy and has a bound independent of the number of variables.

### The Quantum Setting with Randomized Queries for Continuous Problems

It is proved that for path integration the authors have an exponential improvement for the qubit complexity over the quantum setting with deterministic queries, which limits the power of quantum computation for continuous problems.

### On the quantum and randomized approximation of linear functionals on function spaces

• B. Kacewicz
• Mathematics, Computer Science
Quantum Inf. Process.
• 2011
Lower bounds are provided on the power of quantum, randomized and deterministic algorithms for the exemplary problems, and some cases sharpness of the obtained results is compared.

### Mathematics on a Quantum Computer

In this chapter we gave but a taste of these developments. The quantum counting algorithm is especially noteworthy since it combines ideas from both Grover’s algorithm and phase estimation. Moreover,

### Quantum Algorithms and Complexity for Continuous Problems

• Computer Science
Encyclopedia of Complexity and Systems Science
• 2009
High-dimensional integration, path Integration, Feynman path integration, the smallest eigenvalue of a differential equation, approximation, partial differential equations, ordinary differential equations and gradient estimation are reported on.

### Numerical Analysis on a Quantum Computer

Having matching upper and lower complexity bounds for the quantum setting, this work is in a position to assess the possible speedups quantum computation could provide over classical deterministic or randomized algorithms for these numerical problems.

### Improved Upper Bounds on the Randomized and Quantum Complexity of Initial-Value Problems 1

This paper gives up the deterministic optimality of the basic algorithm, defining a new integral algorithm that is better suited for randomization and implementation of a quantum computer, and applies the optimal algorithms for summation of real numbers.

### Quantum Sub-Gaussian Mean Estimator

We present a new quantum algorithm for estimating the mean of a real-valued random variable obtained as the output of a quantum computation. Our estimator achieves a nearly-optimal quadratic speedup

### Quantum Algorithms and Complexity for Numerical Problems

This thesis designs an adiabatic quantum algorithm for the counting problem, and derives the optimal order of convergence, given e and the cost of the resulting algorithm, which is close to the best lower bound on query complexity known for the classical PAC learning model.

## References

SHOWING 1-10 OF 32 REFERENCES

### Quantum Summation with an Application to Integration

Developing quantum algorithms for computing the mean of sequences that satisfy a p-summability condition and for integration of functions from Lebesgue spaces Lp(0, 1]d, and proving lower bounds showing that the proposed algorithms are, in many cases, optimal within the setting of quantum computing.

### Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms

• Mathematics
• 2002
We survey old and new results about optimal algorithms for summation of finite sequences and for integration of functions from Holder or Sobolev spaces. First we discuss optimal deterministic and

### On tractability of path integration

• Computer Science, Mathematics
• 1996
The worst case complexity of path integration is studied, which is defined as the minimal number of the integrand evaluations needed to compute an approximation with error at most e and is considered with respect to a Gaussian measure, and for various classes of integrands.

### Quantum Complexity of Integration

• E. Novak
• Computer Science, Mathematics
J. Complex.
• 2001
It is known that quantum computers yield a speed-up for certain discrete problems. Here we want to know whether quantum computers are useful for continuous problems. We study the computation of the

### Algorithms for quantum computation: discrete logarithms and factoring

• P. Shor
• Computer Science
Proceedings 35th Annual Symposium on Foundations of Computer Science
• 1994
Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given.

### A fast quantum mechanical algorithm for database search

In early 1994, it was demonstrated that a quantum mechanical computer could efficiently solve a well-known problem for which there was no known efficient algorithm using classical computers, i.e. testing whether or not a given integer, N, is prime, in a time which is a finite power of o (logN) .

### Tractability of Approximation for Weighted Korobov Spaces on Classical and Quantum Computers

• Mathematics, Computer Science
Found. Comput. Math.
• 2004
The worst case, randomized, and quantum settings are considered and it is proved that strong tractability and tractability in the class \$\lall\$ are equivalent and this holds under the same assumption as for the class £lall in the worst case setting.

### A new algorithm and worst case complexity for Feynman-Kac path integration

• Computer Science, Mathematics
• 2000
A new algorithm is presented and an explicit bound on its cost to compute an e-approximation to the Feynman–Kac path integral is established, which is equal to the cost of the new algorithm and is given in terms of the complexity of a certain function approximation problem.

### The quantum query complexity of approximating the median and related statistics

• Computer Science, Mathematics
STOC '99
• 1999
The main ingredient in the proof is a polynomial degree lower bound for real multilinear polynomials that ``approximate'' symmetric partial boolean functions, which immediately yields lower bounds for the problems of approximating the kth-smallest element, approximates the mean of a sequence of numbers, and that of approximately counting the number of ones of a boolean function.