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Robust Quantum Algorithms and Polynomials

Harry Buhrman, Ilan Newman, Hein Röhrig, Ronald de Wolf

Published Jan 1, 2005• 18 citations
OpenAlex

Abstract

We study the complexity of robust quantum algorithms. These still work with high probability if the n input bits are noisy. We exhibit a robust quantum algorithm that recovers the complete input with high probability using O(n) queries. This implies that every n-bit function can be quantum computed robustly with O(n) queries, which contrasts with Feige et al.’s Ω(n log n) classical bound for PARITY. We also give similar bounds on the degrees of multilinear polynomials that robustly approximate Boolean functions. 1

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We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main results are * For every n-bit Boolean function f there is an n-variate polynomial p of degree O(n) that robustly approximates it, in the sense that p(x) remains close to f(x) if we slightly vary each of the n inputs of the polynomial. * There is an O(n)-query quantum algorithm that robustly recovers n noisy input bits. Hence every n-bit function can be quantum computed with O(n) queries in the presence of noise. This contrasts with the classical model of Feige et al., where functions such as parity need Theta(n*log n) queries. We give several extensions and applications of these results.

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We study the complexity of robust quantum algorithms. These still work with high probability if the n input bits are noisy. We exhibit a robust quantum algorithm that recovers the complete input with high probability using O(n) queries. This implies that every n-bit function can be quantum computed robustly with O(n) queries, which contrasts with Feige et al.’s Ω(n log n) classical bound for PARITY. We also give similar bounds on the degrees of multilinear polynomials that robustly approximate Boolean functions. 1
We define and study the complexity of robust polynomials for Boolean functions and the related fault-tolerant quantum decision trees, where input bits are perturbed by noise. We compare several different possible definitions. Our main results are * For every n-bit Boolean function f there is an n-variate polynomial p of degree O(n) that robustly approximates it, in the sense that p(x) remains close to f(x) if we slightly vary each of the n inputs of the polynomial. * There is an O(n)-query quantum algorithm that robustly recovers n noisy input bits. Hence every n-bit function can be quantum computed with O(n) queries in the presence of noise. This contrasts with the classical model of Feige et al., where functions such as parity need Theta(n*log n) queries. We give several extensions and applications of these results.
We examine the number of queries to input variables that a quantum algorithm requires to compute Boolean functions on {0,1} N in the black-box model. We show that the exponential quantum speed-up obtained for partial functions (i.e., problems involving a promise on the input) by Deutsch and Jozsa, Simon, and Shor cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with small error probability using T black-box queries, then there is a classical deterministic algorithm that computes f exactly with O ( Ts 6 ) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.
The degree of a polynomial representing (or approximating) a function f is a lower bound for the quantum query complexity of f. This observation has been a source of many lower bounds on quantum algorithms. It has been an open problem whether this lower bound is tight. This is why Boolean functions are needed with a high number of essential variables and a low polynomial degree. Unfortunately, it is a well-known problem to construct such functions. The best separation between these two complexity measures of a Boolean function was exhibited by Ambai- nis [5]. He constructed functions with polynomial degree M and number of variables Omega(M2). We improve such a separation to become exponential. On the other hand, we use a computerized exhaustive search to prove tightness of this bound.
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}/sup N/ in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T/sup 6/) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.