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arXiv:2607.08517

Quantum Communication Lower Bounds for Search Problems via Matrix Discrepancy

Minbo Gao, Chenghua Liu, Guangxu Yang, Tianyi Zhang

quant-phSubmitted Jul 9, 2026• Announced Jul 10, 2026• 0 citations• Revised Oct 4, 2026 (v2)
v1 · Jul 9, 2026v2 · Oct 4, 2026
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Abstract

We study one-way quantum communication lower bounds for search problems. Unlike decision problems, search problems can have many valid outputs, which pose a fundamental barrier to standard quantum lower-bound techniques. We overcome this by developing a novel method based on matrix discrepancy, which allows us to bound the output measurements of a quantum protocol jointly. As applications of our method, we establish the first tight quantum lower bounds for two fundamental search problems in some natural parameter regimes: collision finding and triangle finding. For collision finding, we prove a tight $Ω(N^{1/4})$ one-way quantum communication lower bound. Previously, the best-known quantum communication lower bound for collision finding was $Ω(N^{1/12})$ due to Göös and Jain (RANDOM 2022), and no stronger bound was known even under the one-way restriction. For triangle finding in graph streams, we prove a one-pass quantum streaming space lower bound of $Ω\left(\sqrt{Δ_V}\right)$ for graphs with $m$ edges, $Θ(m)$ triangles, and constant $Δ_E$, where $Δ_V$ and $Δ_E$ denote the maximum number of triangles sharing a common vertex and edge, respectively, under the condition that $1\le Δ_V\le m^{2/3}$. This constitutes the first nontrivial quantum space lower bound in this regime, matching the classical upper bound of Jayaram and Kallaugher (RANDOM 2021) up to logarithmic factors. Notably, our method also recovers the classical lower bound of Kallaugher and Price (SODA 2017) through an entirely different argument, avoiding their Boolean-Hidden-Matching reduction that breaks down for quantum protocols.

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Random Order in Quantum Streaming: Replenishment and Robust Lower Bounds

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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 $r$ independent matching requests requires $Ω(r)$ qubits for $r\le N^{1/2-δ}$ and every fixed $δ\in(0,1/2)$, even with simultaneous revelation and arbitrary joint decoding. For sequential requests, the linear bound extends through $r=Θ(\sqrt N)$. 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 $Ω(n)$ 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.

quant-pharXiv:2610.06727•Oct 6, 2026•0 citations

A Tight Quantum Algorithm for Multiple Collision Search

Xavier Bonnetain, Johanna Loyer, André Schrottenloher +1 more

Searching for collisions in random functions is a fundamental computational problem, with many applications in symmetric and asymmetric cryptanalysis. When one searches for a single collision, the known quantum algorithms match the query lower bound. This is not the case for the problem of finding multiple collisions, despite its regular appearance as a sub-component in sieving-type algorithms. At EUROCRYPT 2019, Liu and Zhandry gave a query lower bound $Ω(2^{m/3 + 2k/3})$ for finding $2^k$ collisions in a random function with m-bit output. At EUROCRYPT 2023, Bonnetain et al. gave a quantum algorithm matching this bound for a large range of $m$ and $k$, but not all admissible values. Like many previous collision-finding algorithms, theirs is based on the MNRS quantum walk framework, but it chains the walks by reusing the state after outputting a collision. In this paper, we give a new algorithm that tackles the remaining non-optimal range, closing the problem. Our algorithm is tight (up to a polynomial factor) in queries, and also in time under a quantum RAM assumption. The idea is to extend the chained walk to a regime in which several collisions are returned at each step, and the ``walks'' themselves only perform a single diffusion layer.

arXiv:2509.13909•Sep 17, 2025•0 citations

Geometric optimization for quantum communication

Chengkai Zhu, Mao, Hongyu, Kun Fang +1 more

Determining the ultimate limits of quantum communication, such as the quantum capacity of a channel and the distillable entanglement of a shared state, remains a central challenge in quantum information theory, primarily due to the phenomenon of superadditivity. This work develops Riemannian optimization methods to establish significantly tighter, computable two-sided bounds on these fundamental quantities. For upper bounds, our method systematically searches for state and channel extensions that minimize known information-theoretic bounds. We achieve this by parameterizing the space of all possible extensions as a Stiefel manifold, enabling a universal search that overcomes the limitations of ad-hoc constructions. Combined with an improved upper bound on the one-way distillable entanglement based on a refined continuity bound on quantum conditional entropy, our approach yields new state-of-the-art upper bounds on the quantum capacity of the qubit depolarizing channel for large values of the depolarizing parameter, strictly improving the previously best-known bounds. For lower bounds, we introduce Riemannian optimization methods to compute multi-shot coherent information. We establish lower bounds on the one-way distillable entanglement by parameterizing quantum instruments on the unitary manifold, and on the quantum capacity by parameterizing code states with a product of unitary manifolds. Numerical results for noisy entangled states and different channels demonstrate that our methods successfully unlock superadditive gains, improving previous results. Together, these findings establish Riemannian optimization as a principled and powerful tool for navigating the complex landscape of quantum communication limits. Furthermore, we prove that amortization does not enhance the channel coherent information, thereby closing a potential avenue for improving capacity lower bounds in general.

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Quantum Query Lower Bounds for Triangle-Listing and Spanners

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This paper gives quantum query lower bounds for two relational graph problems, triangle listing and explicit multiplicative spanner construction, in the general graph query model, where quantum adjacency, degree and neighborhood queries are all available in arbitrary superposition. Both results are obtained by reductions via intermediate multi-block search problems. We extend the quantum query recording framework by Zhandry (CRYPTO 2019) and Hamoudi and Magniez (ToCT 2023) to handle a recording architecture for bidirectional oracles and give a generic blockwise soundness framework, which for arbitrary families of local accepting projectors, gives an exact operator-norm characterization of their maximum overlap with the subspace of bounded weight records. Using the intermediate search problems, we exhibit a family of $n$-vertex graphs with $Θ(n)$ triangles on which listing any constant fraction of the triangles requires $Ω(n^{3/2-o(1)})$ quantum queries. This is the first nontrivial quantum lower bound for triangle listing, a question raised by Jiang and Peng (ICML 2026). Further, we show that, for every fixed $k\ge 7$, constructing a multiplicative $k$-spanner requires $Ω(n^{1+\frac{1}{2μ_k}})$ quantum queries, where $μ_k=k/3+O(1)$. For $k\in\{7,8\}$ the bound is $Ω(n^{5/4})$, which matches what would be implied by an unproven instance of the Erdős girth conjecture, and for large $k$ the exponent $1+\frac{3}{2k}$ exceeds the $1+\frac{4}{3k}$ implied by the provable high girth dense graphs due to Lazebnik, Ustimenko and Woldar, 1995.

quant-pharXiv:2609.37091•Sep 30, 2026•0 citations

The Quantum and Classical Streaming Complexity of Quantum and Classical Max-Cut

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We investigate the space complexity of two graph streaming problems: Max-Cut and its quantum analogue, Quantum Max-Cut. Previous work by Kapralov and Krachun [STOC `19] resolved the classical complexity of the \emph{classical} problem, showing that any $(2 - \varepsilon)$-approximation requires $Ω(n)$ space (a $2$-approximation is trivial with $\textrm{O}(\log n)$ space). We generalize both of these qualifiers, demonstrating $Ω(n)$ space lower bounds for $(2 - \varepsilon)$-approximating Max-Cut and Quantum Max-Cut, even if the algorithm is allowed to maintain a quantum state. As the trivial approximation algorithm for Quantum Max-Cut only gives a $4$-approximation, we show tightness with an algorithm that returns a $(2 + \varepsilon)$-approximation to the Quantum Max-Cut value of a graph in $\textrm{O}(\log n)$ space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using $\textrm{o}(n)$ space. We prove our lower bounds through the techniques of Boolean Fourier analysis. We give the first application of these methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next. To this end, we show how Fourier-analytic techniques may be used to understand the application of a quantum channel.

arXiv:2206.00213•Jun 1, 2022•0 citations

Tight bounds for hybrid quantum-classical query algorithms

Andris Ambainis, András Gilyén, Martins Kokainis

We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most $q$ oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model: (i) (Unbiased) Phase estimation up to a precision $ε$ requires $Θ(\frac{1}{q ε^2})$ queries; (ii) (Unbiased) Amplitude estimation up to a precision $ε$ requires $Θ(\frac{p(1-p)}{q ε^2})$ queries; (iii) Search among $N$ items requires $Θ(\frac{N}{q})$ queries; (iv) Two level AND-OR tree (AND of $m$ ORs, with $n$ inputs to each OR) requires $Θ(\frac{nm}{q})$ queries. All of these bounds are optimal up to a constant factor, for all $q$ from 1, corresponding to the classical complexity, to $Q(f)$, the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor. Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.

quant-pharXiv:2610.06803•Oct 6, 2026•0 citations
We study one-way quantum communication lower bounds for search problems. Unlike decision problems, search problems can have many valid outputs, which pose a fundamental barrier to standard quantum lower-bound techniques. We overcome this by developing a novel method based on matrix discrepancy, which allows us to bound the output measurements of a quantum protocol jointly.
As applications of our method, we establish the first tight quantum lower bounds for two fundamental search problems in some natural parameter regimes: collision finding and triangle finding. For collision finding, we prove a tight Ω(N1/4)Ω(N^{1/4})Ω(N1/4) one-way quantum communication lower bound. Previously, the best-known quantum communication lower bound for collision finding was Ω(N1/12)Ω(N^{1/12})Ω(N1/12) due to Göös and Jain (RANDOM 2022), and no stronger bound was known even under the one-way restriction. For triangle finding in graph streams, we prove a one-pass quantum streaming space lower bound of Ω(ΔV)Ω\left(\sqrt{Δ_V}\right)Ω(ΔV​​) for graphs with mmm edges, Θ(m)Θ(m)Θ(m) triangles, and constant ΔEΔ_EΔE​, where ΔVΔ_VΔV​ and ΔEΔ_EΔE​ denote the maximum number of triangles sharing a common vertex and edge, respectively, under the condition that 1≤ΔV≤m2/31\le Δ_V\le m^{2/3}1≤ΔV​≤m2/3. This constitutes the first nontrivial quantum space lower bound in this regime, matching the classical upper bound of Jayaram and Kallaugher (RANDOM 2021) up to logarithmic factors. Notably, our method also recovers the classical lower bound of Kallaugher and Price (SODA 2017) through an entirely different argument, avoiding their Boolean-Hidden-Matching reduction that breaks down for quantum protocols.
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.
Searching for collisions in random functions is a fundamental computational problem, with many applications in symmetric and asymmetric cryptanalysis. When one searches for a single collision, the known quantum algorithms match the query lower bound. This is not the case for the problem of finding multiple collisions, despite its regular appearance as a sub-component in sieving-type algorithms. At EUROCRYPT 2019, Liu and Zhandry gave a query lower bound Ω(2m/3+2k/3)Ω(2^{m/3 + 2k/3})Ω(2m/3+2k/3) for finding 2k2^k2k collisions in a random function with m-bit output. At EUROCRYPT 2023, Bonnetain et al. gave a quantum algorithm matching this bound for a large range of mmm and kkk, but not all admissible values. Like many previous collision-finding algorithms, theirs is based on the MNRS quantum walk framework, but it chains the walks by reusing the state after outputting a collision. In this paper, we give a new algorithm that tackles the remaining non-optimal range, closing the problem. Our algorithm is tight (up to a polynomial factor) in queries, and also in time under a quantum RAM assumption. The idea is to extend the chained walk to a regime in which several collisions are returned at each step, and the ``walks'' themselves only perform a single diffusion layer.
Determining the ultimate limits of quantum communication, such as the quantum capacity of a channel and the distillable entanglement of a shared state, remains a central challenge in quantum information theory, primarily due to the phenomenon of superadditivity. This work develops Riemannian optimization methods to establish significantly tighter, computable two-sided bounds on these fundamental quantities. For upper bounds, our method systematically searches for state and channel extensions that minimize known information-theoretic bounds. We achieve this by parameterizing the space of all possible extensions as a Stiefel manifold, enabling a universal search that overcomes the limitations of ad-hoc constructions. Combined with an improved upper bound on the one-way distillable entanglement based on a refined continuity bound on quantum conditional entropy, our approach yields new state-of-the-art upper bounds on the quantum capacity of the qubit depolarizing channel for large values of the depolarizing parameter, strictly improving the previously best-known bounds. For lower bounds, we introduce Riemannian optimization methods to compute multi-shot coherent information. We establish lower bounds on the one-way distillable entanglement by parameterizing quantum instruments on the unitary manifold, and on the quantum capacity by parameterizing code states with a product of unitary manifolds. Numerical results for noisy entangled states and different channels demonstrate that our methods successfully unlock superadditive gains, improving previous results. Together, these findings establish Riemannian optimization as a principled and powerful tool for navigating the complex landscape of quantum communication limits. Furthermore, we prove that amortization does not enhance the channel coherent information, thereby closing a potential avenue for improving capacity lower bounds in general.
This paper gives quantum query lower bounds for two relational graph problems, triangle listing and explicit multiplicative spanner construction, in the general graph query model, where quantum adjacency, degree and neighborhood queries are all available in arbitrary superposition. Both results are obtained by reductions via intermediate multi-block search problems. We extend the quantum query recording framework by Zhandry (CRYPTO 2019) and Hamoudi and Magniez (ToCT 2023) to handle a recording architecture for bidirectional oracles and give a generic blockwise soundness framework, which for arbitrary families of local accepting projectors, gives an exact operator-norm characterization of their maximum overlap with the subspace of bounded weight records. Using the intermediate search problems, we exhibit a family of nnn-vertex graphs with Θ(n)Θ(n)Θ(n) triangles on which listing any constant fraction of the triangles requires Ω(n3/2−o(1))Ω(n^{3/2-o(1)})Ω(n3/2−o(1)) quantum queries. This is the first nontrivial quantum lower bound for triangle listing, a question raised by Jiang and Peng (ICML 2026). Further, we show that, for every fixed k≥7k\ge 7k≥7, constructing a multiplicative kkk-spanner requires Ω(n1+12μk)Ω(n^{1+\frac{1}{2μ_k}})Ω(n1+2μk​1​) quantum queries, where μk=k/3+O(1)μ_k=k/3+O(1)μk​=k/3+O(1). For k∈{7,8}k\in\{7,8\}k∈{7,8} the bound is Ω(n5/4)Ω(n^{5/4})Ω(n5/4), which matches what would be implied by an unproven instance of the Erdős girth conjecture, and for large kkk the exponent 1+32k1+\frac{3}{2k}1+2k3​ exceeds the 1+43k1+\frac{4}{3k}1+3k4​ implied by the provable high girth dense graphs due to Lazebnik, Ustimenko and Woldar, 1995.
We investigate the space complexity of two graph streaming problems: Max-Cut and its quantum analogue, Quantum Max-Cut. Previous work by Kapralov and Krachun [STOC `19] resolved the classical complexity of the \emph{classical} problem, showing that any (2−ε)(2 - \varepsilon)(2−ε)-approximation requires Ω(n)Ω(n)Ω(n) space (a 222-approximation is trivial with O(log⁡n)\textrm{O}(\log n)O(logn) space). We generalize both of these qualifiers, demonstrating Ω(n)Ω(n)Ω(n) space lower bounds for (2−ε)(2 - \varepsilon)(2−ε)-approximating Max-Cut and Quantum Max-Cut, even if the algorithm is allowed to maintain a quantum state. As the trivial approximation algorithm for Quantum Max-Cut only gives a 444-approximation, we show tightness with an algorithm that returns a (2+ε)(2 + \varepsilon)(2+ε)-approximation to the Quantum Max-Cut value of a graph in O(log⁡n)\textrm{O}(\log n)O(logn) space. Our work resolves the quantum and classical approximability of quantum and classical Max-Cut using o(n)\textrm{o}(n)o(n) space. We prove our lower bounds through the techniques of Boolean Fourier analysis. We give the first application of these methods to sequential one-way quantum communication, in which each player receives a quantum message from the previous player, and can then perform arbitrary quantum operations on it before sending it to the next. To this end, we show how Fourier-analytic techniques may be used to understand the application of a quantum channel.
We study hybrid quantum-classical algorithms in the query model. The computation consists of quantum subroutines that make at most qqq oracle queries; between subroutines, all qubits are measured and discarded. We prove matching upper and lower bounds for several problems in this model: (i) (Unbiased) Phase estimation up to a precision εεε requires Θ(1qε2)Θ(\frac{1}{q ε^2})Θ(qε21​) queries; (ii) (Unbiased) Amplitude estimation up to a precision εεε requires Θ(p(1−p)qε2)Θ(\frac{p(1-p)}{q ε^2})Θ(qε2p(1−p)​) queries; (iii) Search among NNN items requires Θ(Nq)Θ(\frac{N}{q})Θ(qN​) queries; (iv) Two level AND-OR tree (AND of mmm ORs, with nnn inputs to each OR) requires Θ(nmq)Θ(\frac{nm}{q})Θ(qnm​) queries. All of these bounds are optimal up to a constant factor, for all qqq from 1, corresponding to the classical complexity, to Q(f)Q(f)Q(f), the unrestricted quantum query complexity of the respective problem. We also derive a hybrid lower bound for distinguishing two distributions specified by a state-preparation unitary; this bound is tight up to a logarithmic factor. Besides specific lower bounds, an important contribution is developing methods for proving lower bounds on hybrid quantum algorithms (which have been very ad-hoc up to now). The first two bounds follow from a common framework: we analyze probability distributions over measurement transcripts and bound a progress measure that captures their distinguishability. The AND-OR bound requires a more delicate argument, combining two progress measures that track the information available to the algorithm classically and in quantum form, respectively.