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arXiv:2610.03645

Exponential quantum space advantage in random data streams

Adam Bouland, Matthew Ding, Siddhartha Jain

quant-phcs.CCSubmitted Oct 2, 2026• Announced Oct 5, 2026• 0 citations
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Abstract

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 $(x,f(x))$ pairs for a uniformly random function $f$ 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 $d$ and $n$ evaluation points, attaining $1/2 + Ω(\sqrt{d/n})$ fraction of satisfied OPI constraints via streaming requires $Ω(n)$ classical bits of memory but only $O(d\log n)$ 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.

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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.
Decoded quantum interferometry (DQI) is a polynomial-time quantum algorithm introduced by Jordan et al. (Nature 2025). For a natural optimization problem, known as optimal polynomial intersection (OPI), it achieves approximation guarantees in regimes where all known classical algorithms require exponential time. Besides time, space is another central resource: storing and manipulating a massive input can be very challenging, especially when logical qubits carry substantial fault-tolerant implementation overhead. This motivates the following question: does DQI yield quantum advantages in memory, and can we prove it unconditionally? We give an affirmative answer to this question in the streaming setting. In particular, we consider a natural generalization of OPI using Hermite interpolation and Hasse derivatives, which asks for a low-degree polynomial satisfying as many constraints on its values and derivatives as possible. As a concrete example, we show [Quantum efficiency.] An adaptation of the DQI algorithm produces a polynomial satisfying 93%93\%93% of the constraints; moreover, it only reads the input stream in one pass, uses polylogarithmic space, and has polylogarithmic computation time per stream entry. [Classical hardness.] Any classical algorithm that produces an answer satisfying just 76%76\%76% of the constraints requires polynomial space, even if it can read the input stream with polynomially many passes and can use unlimited time. Our result provides a complete tradeoff curve for the tunable parameters, and implies that DQI has provable quantum advantages for the original OPI problem.
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.
Decoded quantum interferometry (DQI) is a novel paradigm for tackling approximate optimization problems on quantum computers. This framework comes with strong performance guarantees and exploits a well-established duality between optimization and coding theory. A central question, however, is whether DQI can actually provably outperform all polynomial-time classical algorithms. In this work, we establish such an advantage in an oracle setting: we consider an optimization task called folded optimal polynomial intersection (folded OPI), where the acceptance sets are chosen randomly and accessed through membership oracles. We establish a strict gap between the approximation ratio achievable by any polynomial-time classical algorithm and the approximation ratio achieved by the DQI algorithm. Our proof builds on Jordan et al.'s DQI framework for approximate optimization and extends the classical lower-bound method underlying Yamakawa and Zhandry's exact-search oracle separation to approximation. Building on recent developments by Sun and Wootters, Horinaga and Yamakawa, and Jo, we further show that a modified version of the DQI algorithm achieves a strictly higher score guarantee on typical sampled folded OPI instances. As a concrete example, for code rate 0.30.30.3, DQI achieves expected scores of approximately 0.850.850.85. In contrast, exceeding the classical threshold of 0.650.650.65 by any fixed amount with constant probability on sampled instances requires super-polynomially many classical membership queries.
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.
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.
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.