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We consider the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Our main results show that for a range of linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs, several of which are tight for every space bound, also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices A, including the discrete Fourier transform (DFT) matrix, we prove that quantum circuits with at most T input queries and S qubits of memory require T = Ω(n^2/S) to compute matrix-vector product Ax for x ∈ {0, 1}^n. We similarly prove that matrix multiplication for n × n binary matrices requires T = Ω(n^3/√S). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems with any space bound. We obtain matching lower bounds for the stronger notion of quantum cumulative memory complexity—the sum of the space per layer of a circuit. We also consider Boolean (i.e. AND-OR) matrix multiplication and matrix-vector products, improving the previous quantum time-space tradeoff lower bounds for n × n Boolean matrix multiplication to T = Ω(n^{2.5}/S^{1/4}) from T = Ω(n^{2.5}/S^{1/2}). Our improved lower bound for Boolean matrix multiplication is based on a new coloring argument that extracts more from the strong direct product theorem that was the basis for prior work. To obtain our tight lower bounds for linear algebra problems, we require much stronger bounds than strong direct product theorems. We obtain these bounds by adding a new bucketing method to the quantum recording-query technique of Zhandry that lets us apply classical arguments to upper bound the success probability of quantum circuits.
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This content will become publicly available on June 9, 2026
Fast Matrix Multiplication meets the Submodular Width
One fundamental question in database theory is the following: Given a Boolean conjunctive queryQ, what is the best complexity for computing the answer to Q in terms of the input database sizeN? When restricted to the class of combinatorial algorithms, it is known that the best known complexity for any queryQis captured by thesubmodular widthofQ. However, beyond combinatorial algorithms, certain queries are known to admit faster algorithms that often involve a clever combination of fast matrix multiplication and data partitioning. Nevertheless, there is no systematic way to derive and analyze the complexity of such algorithms for arbitrary queriesQ. In this work, we introduce a general framework that captures the best complexity for answering any Boolean conjunctive queryQusing matrix multiplication. Our framework unifies both combinatorial and non-combinatorial techniques under the umbrella of information theory. It generalizes the notion of submodular width to a new stronger notion called the ω-submodular widththat naturally incorporates the power of fast matrix multiplication. We describe a matching algorithm that computes the answer to any queryQin time corresponding to the ω-submodularwidth ofQ. We show that our framework recovers the best known complexities for Boolean queries that have been studied in the literature, to the best of our knowledge, and also discovers new algorithms for some classes of queries that improve upon the best known complexities.
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- PAR ID:
- 10627043
- Publisher / Repository:
- ACM SIGMOD/PODS
- Date Published:
- Journal Name:
- Proceedings of the ACM on Management of Data
- Volume:
- 3
- Issue:
- 2
- ISSN:
- 2836-6573
- Page Range / eLocation ID:
- 1 to 26
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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