We consider the following data compression problem. Given a string $x \in \{0,1\}^m$ of Hamming weight at most $n$, compress it into a shorter string $y \in \{0,1\}^s$ so that any bit $x_i$ of $x$ can be retrieved without any error using at most $t$ quantum queries to the standard oracle encoding of $y$. If queries are allowed to be adaptive we show how optimal compression up to a logarithmic factor can be achieved. If the queries are required to be made non-adaptively, we show schemes whose space is optimal in its dependence on $m$ except for a logarithmic factor, and is at most quadratically worse when compared to the optimum in its dependence on $n$.
We consider the following data compression problem. Given a string x∈{0,1}m of Hamming weight at most n, compress it into a shorter string y∈{0,1}s so that any bit xi of x can be retrieved without any error using at most t quantum queries to the standard oracle encoding of y. If queries are allowed to be adaptive we show how optimal compression up to a logarithmic factor can be achieved. If the queries are required to be made non-adaptively, we show schemes whose space is optimal in its dependence on m except for a logarithmic factor, and is at most quadratically worse when compared to the optimum in its dependence on n.
Motivated by Johnson--Lindenstrauss dimension reduction, amplitude encoding, and the view of measurements as hash-like primitives, one might hope to compress an n-point approximate nearest neighbor (ANN) data structure into O(logn) qubits. We rule out this possibility in a broad quantum sketch model, the dataset P is encoded as an m-qubit state ρP, and each query is answered by an arbitrary query-dependent measurement on a fresh copy of ρP. For every approximation factor c≥1 and constant success probability p>1/2, we exhibit n-point instances in Hamming space {0,1}d with d=Θ(logn) for which any such sketch requires m=Ω(n) qubits, via a reduction to quantum random access codes and Nayak's lower bound. These memory lower bounds coexist with potential quantum query-time gains and in candidate-scanning abstractions of hashing-based ANN, amplitude amplification yields a quadratic reduction in candidate checks, which is essentially optimal by Grover/BBBV-type bounds.
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.
In the oracle identification problem, we are given oracle access to an unknown N-bit string x promised to belong to a known set C of size M and our task is to identify x. We present a quantum algorithm for the problem that is optimal in its dependence on N and M. Our algorithm considerably simplifies and improves the previous best algorithm due to Ambainis et al. Our algorithm also has applications in quantum learning theory, where it improves the complexity of exact learning with membership queries, resolving a conjecture of Hunziker et al. The algorithm is based on ideas from classical learning theory and a new composition theorem for solutions of the filtered γ2-norm semidefinite program, which characterizes quantum query complexity. Our composition theorem is quite general and allows us to compose quantum algorithms with input-dependent query complexities without incurring a logarithmic overhead for error reduction. As an application of the composition theorem, we remove all log factors from the best known quantum algorithm for Boolean matrix multiplication.
Hao, Huang, and Liu (STOC'26) recently showed that optimal quantum query complexity may require large workspace even for short-output problems, and asked whether a quantum query advantage over classical computation can itself require space. We resolve this question by exhibiting an explicit total Boolean function with quantum query complexity Q=Θ(M), and randomized query complexity R=Θ(M21/20), whereas, for every fixed 0<η<1/100 and S≤O(M1/100−η), its S-space quantum query complexity satisfies QS=ω(M21/20). Consequently, Q<R<QS, so the unrestricted quantum query advantage disappears under sufficiently small workspace. The separation is obtained through a one-bit filtered-parity construction and a space-sensitive quantum lower bound based on compressed-oracle capacity and a new parity-to-capacity inequality, which may be of independent interest.
We present a near-optimal quantum algorithm, up to logarithmic factors, for estimating the Shannon entropy in the quantum probability oracle model. Our approach combines the singular value separation algorithm with quantum amplitude amplification, followed by the application of quantum singular value transformation. On the lower bound side, we construct probability distributions encoded via Hamming weights in the oracle, establishing a tight query lower bound up to logarithmic factors. Consequently, our results show that the tight query complexity for estimating the Shannon entropy within ε-additive error is given by Θ~(εn).
The problem of minimizing the maximum of N convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring O(Nε−2/3+ε−8/3) queries to a first-order oracle to compute an ε-suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of N convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of O~(Nε−5/3+ε−8/3). On the other hand, we prove that quantum algorithms must take Ω~(Nε−2/3) queries to a first order quantum oracle, showing that our dependence on N is optimal up to poly-logarithmic factors.