Search Results for « Quantum query complexity »

Quantum channel discrimination has been studied from an information-theoretic perspective, wherein one is interested in the optimal decay rate of error probabilities as a function of the number of unknown channel accesses. In this paper, we study the query complexity of quantum channel discrimination, wherein the goal is to determine the minimum number of channel uses needed to reach a desired error probability. To this end, we show that the query complexity of binary channel discrimination depends logarithmically on the inverse error probability and inversely on the negative logarithm of the (geometric and Holevo) channel fidelity. As a special case of these findings, we precisely characterize the query complexity of discriminating two classical channels and two classical-quantum channels. Furthermore, by obtaining an optimal characterization of the sample complexity of quantum hypothesis testing, including prior probabilities, we provide a more precise characterization of query complexity when the error probability does not exceed a fixed threshold. We also provide lower and upper bounds on the query complexity of binary asymmetric channel discrimination and multiple quantum channel discrimination. For the former, the query complexity depends on the geometric Rényi and Petz Rényi channel divergences, while for the latter, it depends on the negative logarithm of the (geometric and Uhlmann) channel fidelity. For multiple channel discrimination, the upper bound scales as the logarithm of the number of channels.

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 S n . 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.

We study the quantum query complexity of constant-sized subgraph containment. Such problems include determining whether an $ n $-vertex graph contains a triangle, clique or star of some size. For a general subgraph $ H $ with $ k $ vertices, we show that $ H $ containment can be solved with quantum query complexity $ O(n^{2-\frac{2}{k}-g(H)}) $, with $ g(H) $ a strictly positive function of $ H $. This is better than $ \tilde{O}\s{n^{2-2/k}} $ by Magniez et al. These results are obtained in the learning graph model of Belovs.

This paper considers the query complexity of the functions in the family F_{N,M} of N-variable Boolean functions with onset size M, i.e., the number of inputs for which the function value is 1, where 1<= M <= 2^{N}/2 is assumed without loss of generality because of the symmetry of function values, 0 and 1. Our main results are as follows: (1) There is a super-linear gap between the average-case and worst-case quantum query complexities over F_{N,M} for a certain range of M. (2) There is no super-linear gap between the average-case and worst-case randomized query complexities over F_{N,M} for every M. (3) For every M bounded by a polynomial in N, any function in F_{N,M} has quantum query complexity Theta (sqrt{N}). (4) For every M=O(2^{cN}) with an arbitrary large constant c<1, any function in F_{N,M} has randomized query complexity Omega (N).

Aaronson and Ambainis (SICOMP `18) showed that any partial function on $N$ bits that can be computed with an advantage $δ$ over a random guess by making $q$ quantum queries, can also be computed classically with an advantage $δ/2$ by a randomized decision tree making ${O}_q(N^{1-\frac{1}{2q}}δ^{-2})$ queries. Moreover, they conjectured the $k$-Forrelation problem -- a partial function that can be computed with $q = \lceil k/2 \rceil$ quantum queries -- to be a suitable candidate for exhibiting such an extremal separation. We prove their conjecture by showing a tight lower bound of $\widetildeΩ(N^{1-1/k})$ for the randomized query complexity of $k$-Forrelation, where the advantage $δ= 2^{-O(k)}$. By standard amplification arguments, this gives an explicit partial function that exhibits an $O_ε(1)$ vs $Ω(N^{1-ε})$ separation between bounded-error quantum and randomized query complexities, where $ε>0$ can be made arbitrarily small. Our proof also gives the same bound for the closely related but non-explicit $k$-Rorrelation function introduced by Tal (FOCS `20). Our techniques rely on classical Gaussian tools, in particular, Gaussian interpolation and Gaussian integration by parts, and in fact, give a more general statement. We show that to prove lower bounds for $k$-Forrelation against a family of functions, it suffices to bound the $\ell_1$-weight of the Fourier coefficients between levels $k$ and $(k-1)k$. We also prove new interpolation and integration by parts identities that might be of independent interest in the context of rounding high-dimensional Gaussian vectors.

Quantum span program algorithms for function evaluation sometimes have reduced query complexity when promised that the input has a certain structure. We design a modified span program algorithm to show these improvements persist even without a promise ahead of time, and we extend this approach to the more general problem of state conversion. As an application, we prove exponential and superpolynomial quantum advantages in average query complexity for several search problems, generalizing Montanaro&amp;apos;s Search with Advice [Montanaro, TQC 2010].

We introduce the concept of quantum polymorphisms to the complexity theory of quantum constraint satisfaction. Via this notion, we build an algebraic framework of reductions between quantum CSPs, and we establish a Galois connection between quantum polymorphism minions and quantum relational constructions. By leveraging a contextuality property of quantum polymorphisms, we fully characterise the existence of commutativity gadgets for relational structures, introduced by Ji as a method for achieving quantum soundness of classical CSP reductions. Prior to our work, only a partial classification was known for a subclass of Boolean languages and for non-Boolean languages meeting specific structural conditions [Culf--Mastel, FOCS'25]. As an application of our framework, we prove that the quantum CSPs parameterised by odd cycles and the quantum CSP expressing quantum satisfiability of Siggers clauses are undecidable.

Quantum query complexity underpins much of our understanding of quantum speedups, yet nearly all existing lower and upper bounds are derived for simple Boolean oracles. Many practical tasks-noisy database search, stochastic optimization, adaptive learning, and time-varying systems-are best modeled by richer oracle classes. This paper systematically extends quantum query complexity theory to four non-standard oracle models: (1) stochastic oracles that return random outputs drawn from a distribution; (2) multi-solution oracles encoding a solution set of size k; (3) time-dependent oracles whose internal state evolves between queries; and (4) relational oracles encoding structured constraints rather than simple membership. Using generalized adversary methods, polynomial method techniques, and hybrid arguments, we derive tight (up to polylog factors) lower bounds for each model and construct matching algorithms where possible. For instance, we show that searching for any one of k marked items in an unstructured database still admits a Grover-like quadratic speedup, but with optimal complexity Θ(N/k), and that stochastic noise in oracle outputs degrades the advantage gracefully as a function of the noise parameter. For certain time-dependent and relational models, we prove that no better-than-classical speedup is possible, clarifying the limits of quantum advantage in adaptive settings. These results provide a more realistic foundation for quantum algorithms targeting noisy search, stochastic optimization, and constraint satisfaction, and they highlight the precise structural features that enable or preclude query-based quantum speedups.