Worst-case to average-case reductions are a cornerstone of complexity theory, providing a bridge between worst-case hardness and average-case computational difficulty. While recent works have demonstrated such reductions for fundamental problems using deep tools from ad- ditive combinatorics, these approaches often suffer from substantial complexity and suboptimal overheads. In this work, we focus on the quantum setting, and provide a new reduction for the Matrix-Vector Multiplication problem that is more efficient, and conceptually simpler than previous constructions. By adapting hardness self-amplification techniques to the quantum do- main, we obtain a quantum worst-case to average-case reduction with improved dependence on the success probability, laying the groundwork for broader applications in quantum fine-grained complexity.
Worst-case to average-case reductions are a cornerstone of complexity theory, providing a bridge between worst-case hardness and average-case computational difficulty. While recent works have demonstrated such reductions for fundamental problems using deep tools from ad- ditive combinatorics, these approaches often suffer from substantial complexity and suboptimal overheads. In this work, we focus on the quantum setting, and provide a new reduction for the Matrix-Vector Multiplication problem that is more efficient, and conceptually simpler than previous constructions. By adapting hardness self-amplification techniques to the quantum do- main, we obtain a quantum worst-case to average-case reduction with improved dependence on the success probability, laying the groundwork for broader applications in quantum fine-grained complexity.
We study the problem of constructing worst-case algorithms from average-case algorithms. Prior to this work, such reductions were only known for a small number of specific problems or restricted computational models. In contrast, we show that for quantum computation, all linear problems admit worst-case to average-case reductions. Specifically, we provide an explicit and efficient transformation of quantum algorithms that are only correct on a small (even sub-constant) fraction of their inputs into ones that are correct on all inputs. En route, we obtain a tight Ω(n2) lower bound on the average-case quantum query complexity of the Matrix-Vector Multiplication problem.
We study the problem of designing worst-case to average-case reductions for quantum algorithms. For all linear problems, we provide an explicit and efficient transformation of quantum algorithms that are only correct on a small (even sub-constant) fraction of their inputs into ones that are correct on all inputs. This stands in contrast to the classical setting, where such results are only known for a small number of specific problems or restricted computational models. En route, we obtain a tight Ω(n2) lower bound on the average-case quantum query complexity of the Matrix-Vector Multiplication problem. Our techniques strengthen and generalise the recently introduced additive combinatorics framework for classical worst-case to average-case reductions (STOC 2022) to the quantum setting. We rely on quantum singular value transformations to construct quantum algorithms for linear verification in superposition and learning Bogolyubov subspaces from noisy quantum oracles. We use these tools to prove a quantum local correction lemma, which lies at the heart of our reductions, based on a noise-robust probabilistic generalisation of Bogolyubov's lemma from additive combinatorics.
We study quantum random self-reductions for linear problems over finite fields. Let M∈Fn×n be an arbitrary matrix, and let O be an oracle that agrees with the linear map x↦Mx on an ε-fraction of inputs x∼Fn. Given coherent access to O and coherent entry access to M, we give a uniform quantum reduction that computes Mx on any prescribed input x with probability at least 2/3 in time O(nT1/3), for n≤T≤n3/2 and constant field size and ε, where T is the cost of one coherent query to O. In particular, when T=O(n), the reduction runs in time O(n4/3), improving the O(n3/2+T) reduction of Asadi, Golovnev, Gur, Shinkar, and Subramanian (SODA 2024).
Our reduction uses the Bogolyubov--Ruzsa subspace guaranteed by additive combinatorics, but it avoids learning this subspace explicitly, which was computationally expensive for the previous reduction; in particular, it does not recover a basis for its orthogonal complement. The main technical step is to decompose the inputs into sparse pieces and find a vector that lies outside the Bogolyubov--Ruzsa subspace via a quantum search based on amplitude amplification. This yields a tunable tradeoff between the cost of querying the average-case oracle and the cost of verifying matrix-vector products.
The linear problem specified by an n × n matrix M over a finite field is the problem of computing the product of M and a given vector x. We present optimal error-tolerant random self-reductions (also known as worst-case to average-case reductions) for all linear problems: Given a linear-size circuit that computes M x on an ε-fraction of inputs x for a positive constant ε, we construct a randomized linear-size circuit that computes M x for all inputs x with high probability. This resolves the open problem posed by Asadi, Golovnev, Gur, Shinkar, and Subramanian (SODA’24), who presented quantum n1.5-time random self-reductions for all linear problems. Somewhat surprisingly, we also demonstrate the quantum advantage of their quantum reduction over classical uniform algorithms, by proving that any classical subquadratic-time random self-reduction requires the advice complexity of Ω(log(1/ε) · logn), as long as the field size is at most 1/ε. We complement this advice complexity lower bound by presenting (1) a random self-reduction with the optimal advice complexity of O(log(1/ε) · logn) and (2) a uniform random self-reduction over a large finite field.
A fundamental pursuit in complexity theory concerns reducing worst-case problems to average-case problems. There exist complexity classes such as PSPACE that admit worst-case to average-case reductions. However, for many other classes such as NP, the evidence so far is typically negative, in the sense that the existence of such reductions would cause collapses of the polynomial hierarchy(PH). Basing cryptographic primitives, e.g., the average-case hardness of inverting one-way permutations, on NP-completeness is a particularly intriguing instance. As there is evidence showing that classical reductions from NP-hard problems to breaking these primitives result in PH collapses, it seems unlikely to base cryptographic primitives on NP-hard problems. Nevertheless, these results do not rule out the possibilities of the existence of quantum reductions. In this work, we initiate a study of the quantum analogues of these questions. Aside from formalizing basic notions of quantum reductions and demonstrating powers of quantum reductions by examples of separations, our main result shows that if NP-complete problems reduce to inverting one-way permutations using certain types of quantum reductions, then coNP ⊆ QIP( 2 ) .
BosonSampling and Random Circuit Sampling are important both as a theoretical tool for separating quantum and classical computation, and as an experimental means of demonstrating quantum speedups. Prior works have shown that average-case hardness of sampling follows from certain unproven conjectures about the hardness of computing output probabilities, such as the Permanent-of-Gaussians Conjecture (PGC), which states that e−nlogn−n−O(logn) additive-error estimates to the output probability of most random BosonSampling experiments are #P-hard. Prior works have only shown weaker average-case hardness results that do not imply sampling hardness. Proving these conjectures has become a central question in quantum complexity. In this work, we show that e−nlogn−n−O(nδ) additive-error estimates to output probabilities of most random BosonSampling experiments are #P-hard for any δ>0, exponentially improving on prior work. In the process, we circumvent all known barrier results for proving PGC. The remaining hurdle to prove PGC is now “merely” to show that the O(nδ) in the exponent can be improved to O(logn). We also obtain an analogous result for Random Circuit Sampling. We then show, for the first time, a hardness of average-case classical sampling result for BosonSampling, under an anticoncentration conjecture. Specifically, we prove the impossibility of multiplicative-error sampling from random BosonSampling experiments with probability 1−2−O~(N1/3) for input size N, unless the Polynomial Hierarchy collapses. This exponentially improves upon the state-of-the-art. To do this, we introduce new proof techniques which tolerate exponential loss in the worst-to-average-case reduction. This opens the possibility to show the hardness of average-case sampling without ever proving PGC.