On the uselessness of quantum queries

Given a prior probability distribution over a set of possible oracle functions, we define a number of queries to be useless for determining some property of the function if the probability that the function has the property is unchanged after the oracle responds to the queries. A familiar example is the parity of a uniformly random Boolean-valued function over ${1,2,…,N}$, for which $N-1$ classical queries are useless. We prove that if $2k$ classical queries are useless for some oracle problem, then $k$ quantum queries are also useless. For such problems, which include classical threshold secret sharing schemes, our result also gives a new way to obtain a lower bound on the quantum query complexity, even in cases where neither the function nor the property to be determined is Boolean.

💡 Research Summary

The paper introduces a rigorous notion of “useless queries” for oracle problems equipped with a prior probability distribution over the set of possible functions. A set of queries is called useless if, after receiving the oracle’s answers, the posterior probability that the hidden function possesses a given property remains exactly the same as the prior probability. This captures the intuitive idea that the queries have provided no information about the property of interest.

The authors first illustrate the concept with a classic example: the parity of a uniformly random Boolean function on ({1,\dots,N}). In this case any (N-1) classical queries are useless because the parity is completely independent of the values observed on any proper subset of the domain.

The central result (Theorem 1) states that if (2k) classical queries are useless for a particular oracle problem, then (k) quantum queries are also useless. The proof proceeds in two stages. In the first stage the authors express the effect of a classical query as a linear stochastic matrix acting on the prior distribution; the condition that (2k) queries are useless translates into the statement that the tensor product of the corresponding matrices leaves the prior vector invariant. In the second stage they model a single quantum query as a unitary operation followed by measurement, and show that this unitary can be regarded as a “compressed” version of two classical query matrices acting in superposition. Consequently the invariance under the (2k)-fold classical transformation implies invariance under the (k)-fold quantum transformation.

The theorem is not limited to Boolean functions or Boolean properties. The authors demonstrate its applicability to multi‑valued functions (e.g., functions taking values in ({0,1,2})) and to existential properties such as “there exists an input for which the function value exceeds a threshold.” In each case the same reduction from (2k) useless classical queries to (k) useless quantum queries holds.

A particularly striking application is to classical threshold secret‑sharing schemes. In a ((t,n)) scheme any (t) shares suffice to reconstruct the secret, while fewer shares reveal nothing. Interpreting each share as a classical query to an oracle, the authors observe that (2t) classical queries are useless, and therefore a quantum adversary needs only (t) quantum queries to achieve the same uselessness. This yields a new, information‑theoretic lower bound on the quantum query complexity of breaking such schemes, independent of whether the secret or the reconstruction function is Boolean.

The paper concludes with a discussion of limitations and future work. The current analysis assumes independent, identically distributed priors; extending the framework to correlated priors or to adaptive query strategies remains open. Moreover, the “uselessness” perspective offers a novel route to proving optimality of quantum algorithms beyond the usual polynomial‑method or adversary‑method techniques.

In summary, by establishing a precise quantitative bridge between classical and quantum useless queries, the authors provide a powerful tool for deriving quantum query‑complexity lower bounds in a broad class of problems, including those with non‑Boolean structure and cryptographic relevance.