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arXiv:quant-ph/0511200 · Algorithmica

A New Quantum Lower Bound Method, with Applications to Direct Product Theorems and Time-Space Tradeoffs

Andris Ambainis, Robert Špalek, Ronald de Wolf

Published Sep 12, 2007• 37 citations
arXivHTMLPDF
DOI
10.1007/s00453-007-9022-9

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A strong direct product theorem for quantum query complexity

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We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2-sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1-sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the time-space tradeoff of this algorithm is close to optimal.
Abstract. We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices [Formula: see text], including the discrete Fourier transform matrix, we prove that quantum circuits with at most [Formula: see text] input queries and [Formula: see text] qubits of memory require [Formula: see text] to compute matrix-vector product [Formula: see text] for [Formula: see text]. We similarly prove that matrix multiplication for [Formula: see text] binary matrices requires [Formula: see text]. Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound. We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity—the sum of the space per layer of a circuit. We also consider Boolean (i.e., AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for [Formula: see text] Boolean matrix multiplication to [Formula: see text] from [Formula: see text]. Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem that was the basis for prior work. To obtain our tight lower bounds for linear algebra problems, we require much stronger bounds than strong direct product theorems. We obtain these bounds by adding a new bucketing method to the quantum recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.