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Annual Symposium on Foundations of Computer Science

A Quantum Advantage for a Natural Streaming Problem

John Kallaugher

Published Feb 1, 2022• 6 citations
Publisher

Abstract

Data streaming, in which a large dataset is received as a “stream” of updates, is an important model in the study of space-bounded computation. Starting with the work of Le Gall [SPAA '06], it has been known that quantum streaming algorithms can use asymptotically less space than their classical counterparts for certain problems. However, so far, all known examples of quantum advantages in streaming are for problems that are either specially constructed for that purpose, or require many streaming passes over the input. We give a one-pass quantum streaming algorithm for one of the best-studied problems in classical graph streaming-the triangle counting problem. Almost-tight parametrized upper and lower bounds are known for this problem in the classical setting; our algorithm uses polynomially less space in certain regions of the parameter space, resolving a question posed by Jain and Nayak in 2014 on achieving quantum advantages for natural streaming problems.

DOI
10.1109/focs52979.2021.00091

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Data streaming, in which a large dataset is received as a “stream” of updates, is an important model in the study of space-bounded computation. Starting with the work of Le Gall [SPAA '06], it has been known that quantum streaming algorithms can use asymptotically less space than their classical counterparts for certain problems. However, so far, all known examples of quantum advantages in streaming are for problems that are either specially constructed for that purpose, or require many streaming passes over the input. We give a one-pass quantum streaming algorithm for one of the best-studied problems in classical graph streaming-the triangle counting problem. Almost-tight parametrized upper and lower bounds are known for this problem in the classical setting; our algorithm uses polynomially less space in certain regions of the parameter space, resolving a question posed by Jain and Nayak in 2014 on achieving quantum advantages for natural streaming problems.
Data streaming, in which a large dataset is received as a "stream" of updates, is an important model in the study of space-bounded computation. Starting with the work of Le Gall [SPAA `06], it has been known that quantum streaming algorithms can use asymptotically less space than their classical counterparts for certain problems. However, so far, all known examples of quantum advantages in streaming are for problems that are either specially constructed for that purpose, or require many streaming passes over the input. We give a one-pass quantum streaming algorithm for one of the best studied problems in classical graph streaming - the triangle counting problem. Almost-tight parametrized upper and lower bounds are known for this problem in the classical setting; our algorithm uses polynomially less space in certain regions of the parameter space, resolving a question posed by Jain and Nayak in 2014 on achieving quantum advantages for natural streaming problems.
How can random order change the role of quantum memory in streaming? Later classical input can restore the usefulness of a quantum state consumed by earlier queries. We call this replenishment. We construct an artificial problem based on Hidden Matching, with repeated coordinate data and online matching requests. It admits a one-pass quantum algorithm using polylogarithmic space in uniformly random order, but unconditionally requires polynomial space both classically in random order and quantumly when all updates precede the requests. To prove the quantum lower bound, we strengthen the consumability bounds of Gilboa, Jain, and McClean for Multiple Hidden Matching. Without prior entanglement, any quantum encoding supporting rrr independent matching requests requires Ω(r)Ω(r)Ω(r) qubits for r≤N1/2−δr\le N^{1/2-δ}r≤N1/2−δ and every fixed δ∈(0,1/2)δ\in(0,1/2)δ∈(0,1/2), even with simultaneous revelation and arbitrary joint decoding. For sequential requests, the linear bound extends through r=Θ(N)r=Θ(\sqrt N)r=Θ(N​). We also adapt Kallaugher's triangle-counting algorithm to uniformly random streams in which every edge is repeated equally often. Rebuilding the quantum sketch and resampling the classical estimator improve its expected-space bound in suitable parameter regimes. Finally, we extend the robust Noisy Gap Cycle framework of Assadi and Sundaresan to quantum streaming. A quantum communication lower bound for Block Hidden XOR yields an Ω(n)Ω(n)Ω(n) space lower bound in random edge order for large enough cycles, with consequences for several graph problems. Thus random order can enable replenishment of small quantum representations, while substantial space requirements persist for other tasks.
Data streaming, in which a large dataset is received as a stream of updates, is an important model in the study of space-bounded computation. Starting with the work of Le Gall [SPAA `06], it has been known that quantum streaming algorithms can use asymptotically less space than their classical counterparts for certain problems. However, so far, all known examples of quantum advantages in streaming are for problems that are either specially constructed for that purpose, or require many streaming passes over the input. We give a one-pass quantum streaming algorithm for one of the best studied problems in classical graph streaming - the triangle counting problem. Almost-tight parametrized upper and lower bounds are known for this problem in the classical setting; our algorithm uses polynomially less space in certain regions of the parameter space, resolving a question posed by Jain and Nayak in 2014 on achieving quantum advantages for natural streaming problems.
We show two unconditional quantum space advantages in the random-order streaming model. First, we show the Yamakawa--Zhandry Code Intersection problem admits exponential quantum advantage in the streaming model when its inputs are streamed in random order. This means quantum computers exhibit exponential space advantage even when simply receiving (x,f(x))(x,f(x))(x,f(x)) pairs for a uniformly random function fff in a uniformly random order. Our lower bound is shown using density-restoring partitions as in the work of Göös, Gur, Jain, and Li (STOC 2025) combined with a convex potential, similar to the work of Raz (J.ACM 2018) on parity learning and its generalization by Garg, Raz, and Tal (STOC 2018). Second, we use our framework to show quantum space advantage for the Optimal Polynomial Intersection (OPI) problem in certain regimes via a streaming version of the Decoded Quantum Interferometry algorithm (Nature 2025; arXiv:2510.10967). In particular, we show that for degree ddd and nnn evaluation points, attaining 1/2+Ω(d/n)1/2 + Ω(\sqrt{d/n})1/2+Ω(d/n​) fraction of satisfied OPI constraints via streaming requires Ω(n)Ω(n)Ω(n) classical bits of memory but only O(dlog⁡n)O(d\log n)O(dlogn) qubits. This yields provable quantum advantage in a "low-rate" regime when the number of evaluation points is much larger than the degree, with a space advantage that can be as large as exponential in certain parameter settings.
Near-term quantum devices with limited qubits motivate the study of space-bounded quantum computation in the data stream model. We show that Shannon entropy estimation exhibits an exponential separation between quantum and classical space complexity in this setting. Technically, we develop a two-stage quantum streaming algorithm based on a quantum procedure with an explicitly constructed oracle derived from the streaming input. This algorithm achieves logarithmic space complexity in the accuracy parameter over the data stream, whereas any classical streaming algorithm under the same pass complexity requires polynomial space. In sharp contrast, existing results for Shannon entropy estimation in the quantum query model achieve only a quadratic speedup. Our work establishes a natural problem with practical applications in computer networking that admits an exponential quantum space advantage, revealing a fundamental gap between quantum query complexity and streaming space complexity.
While the search for quantum advantage typically focuses on speed­ups in execution time, quantum algorithms also offer the potential for advantage in space complexity. Previous work has shown such advantages for data stream problems, in which elements arrive and must be processed sequentially without random access, but these have been restricted to specially-constructed problems Le Gall, SPAA ‘06 or polynomial advantage Kallaugher, FOCS ‘21. We show an exponential quantum space advantage for the maximum directed cut problem. This is the first known exponential quantum space advantage for any natural streaming problem. This also constitutes the first unconditional exponential quantum resource advantage for approximating a discrete optimization problem in any setting.