We provide an exact quantum query algorithm that identifies uncorrupted codewords from a degree-d generalized Reed-Muller code of length qn over the finite field of size q. When d is constant, the algorithm needs 𝒪(nd-1) quantum queries, which is optimal. Classically, Ω(nd) queries are necessary to accomplish this task, even with constant probability of error admitted. Our work extends a result by Montanaro about learning multilinear polynomials.
We provide an exact quantum query algorithm that identifies uncorrupted codewords from a degree-d generalized Reed-Muller code of length qn over the finite field of size q. When d is constant, the algorithm needs 𝒪(nd-1) quantum queries, which is optimal. Classically, Ω(nd) queries are necessary to accomplish this task, even with constant probability of error admitted. Our work extends a result by Montanaro about learning multilinear polynomials.
In this note we study the number of quantum queries required to identify an unknown multilinear polynomial of degree d in n variables over a finite field F_q. Any bounded-error classical algorithm for this task requires Omega(n^d) queries to the polynomial. We give an exact quantum algorithm that uses O(n^(d-1)) queries for constant d, which is optimal. In the case q=2, this gives a quantum algorithm that uses O(n^(d-1)) queries to identify a codeword picked from the binary Reed-Muller code of order d.
Our task of quantum list decoding for a classical block code is to recover from a given quantumly corrupted codeword a short list containing all messages whose codewords have high "presence" in this quantumly corrupted codeword. All known families of efficiently quantum list decodable codes, nonetheless, have exponentially-small message rate. We show that certain generalized Reed-Solomon codes concatenated with Hadamard codes of polynomially-small rate and constant codeword alphabet size have efficient quantum list decoding algorithms, provided that target codewords should have relatively high presence in a given quantumly corrupted codeword.
We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field $\F$ with d elements. The hidden functions of the generalized problem are not restricted to be linear but can also be m-variate polynomial functions of total degree n≥2. The problem of identifying hidden m-variate polynomials of degree less or equal to n for fixed n and m is hard on a classical computer since Ω(d) black-box queries are required to guarantee a constant success probability. In contrast, we present a quantum algorithm that correctly identifies such hidden polynomials for all but a finite number of values of d with constant probability and that has a running time that is only polylogarithmic in d.
In the search with wildcards problem [Ambainis, Montanaro, Quantum Inf.~Comput.'14], one's goal is to learn an unknown bit-string x∈{−1,1}n. An algorithm may, at unit cost, test equality of any subset of the hidden string with a string of its choice. Ambainis and Montanaro showed a quantum algorithm of cost O(nlogn) and a near-matching lower bound of Ω(n). Belovs [Comput.~Comp.'15] subsequently showed a tight O(n) upper bound.
We consider a natural generalization of this problem, parametrized by a subset Q⊆2[n], where an algorithm may test whether xS=b for an arbitrary S∈Q and b∈{−1,1}S of its choice, at unit cost. We show near-tight bounds when Q is any of the following collections: bounded-size sets, contiguous blocks, prefixes, and only the full set.
All of these results are derived using a framework that we develop. Using symmetries of the task at hand we show that the quantum query complexity of learning x is characterized, up to a constant factor, by an optimization program, which is succinctly described as follows: `maximize over all odd functions f:{−1,1}n→R the ratio of the maximum value of f to the maximum (over T∈Q) standard deviation of f on a subcube whose free variables are exactly T.'
To the best of our knowledge, ours is the first work to use the primal version of the negative-weight adversary bound (which is a maximization program typically used to show lower bounds) to show new quantum query upper bounds without explicitly resorting to SDP duality.
We study the query complexity of quantum learning problems in which the oracles form a group G of unitary matrices. In the simplest case, one wishes to identify the oracle, and we find a description of the optimal success probability of a t -query quantum algorithm in terms of group characters. As an application, we show that Ω(n) queries are required to identify a random permutation in Sn . More generally, suppose H is a fixed subgroup of the group G of oracles, and given access to an oracle sampled uniformly from G , we want to learn which coset of H the oracle belongs to. We call this problem coset identification and it generalizes a number of well-known quantum algorithms including the Bernstein-Vazirani problem, the van Dam problem and finite field polynomial interpolation. We provide character-theoretic formulas for the optimal success probability achieved by a t -query algorithm for this problem. One application involves the Heisenberg group and provides a family of problems depending on n which require n+1 queries classically and only 1 query quantumly.
Given a classical error-correcting block code, the task of quantum list decoding is to produce from any quantumly corrupted codeword a short list containing all messages whose codewords exhibit high "presence" in the quantumly corrupted codeword.Efficient quantum list decoders have been used to prove a quantum hardcore property of classical codes.However, the code rates of all known families of efficiently quantum list-decodable codes are, unfortunately, too small for other practical applications.To improve those known code rates, we prove that a specific code family of polynomially small code rate over a fixed code alphabet, obtained by concatenating generalized Reed-Solomon codes as outer codes with Hadamard codes as inner codes, has an efficient quantum list-decoding algorithm if its codewords have relatively high codeword presence in a given quantumly corrupted codeword.As an immediate application, we use the quantum list decodability of this code family to solve a certain form of quantum search problems in polynomial time.When the codeword presence becomes smaller, in contrast, we show that the quantum list decodability of generalized Reed-Solomon codes with high confidence is closely related to the efficient solvability of the following two problems: the noisy polynomial interpolation problem and the bounded distance vector problem.Moreover, assuming that NP BQP, we also prove that no efficient quantum list decoder exists for the generalized Reed-Solomon codes.