Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.29672

The Kikuchi Hierarchy is Sharp for kkkXOR

Alexander Schmidhuber, Matthew B. Hastings

cs.DScs.CCquant-phSubmitted Jul 31, 2026• Announced Aug 3, 2026
arXivHTMLPDF

Abstract

Planted noisy $k$XOR and the strong refutation of random $k$XOR are governed by a conjectured trade-off between signal strength and time: Level $\ell$ of the Kikuchi hierarchy should achieve the smooth curve \begin{equation*} m\ \gtrsim\ ρ^{-2}n^{k/2}/\ell^{k/2-1}\ \text{clauses} \quad\Longleftrightarrow\quad \text{solvable in time }n^{O(\ell)}, \end{equation*} where $ρ$ is the bias of the planted signal or, for refutation, the target advantage. However, every spectral analysis of sparse $k$XOR to date loses polylogarithmic factors against this curve, a loss that enters the exponent of the running time. We show that a normalized variant of the Kikuchi hierarchy achieves the sharp conjectured trade-off, with no logarithmic loss, at every arity $k\ge3$. At the scale above, our algorithms achieve strong detection, weak recovery, and strong refutation; an additional cleanup step boosts weak recovery to exact recovery, and the refutation certificates yield sum-of-squares proofs of degree $O_k(\ell)$. We also prove matching lower bounds in the same model. The inference and refutation upper bounds transfer to more general planting laws and predicates. Finally, we give a quantum algorithm that achieves a quartic speedup over the classical spectral algorithms for detection and weak recovery. The proofs rest on two key ingredients: a normalization of the sparse Kikuchi matrix, and a sharp count of the closed walks in its trace expansion. We use a closely related trace-walk count to prove Feige's 2008 hypergraph Moore bound conjecture in a companion paper.

Comments
Companion paper to arXiv:2607.26028

Related papers

Hypergraph cycle enumeration for efficient inference of planted k-XOR with smooth trade-off

Arman Babakhani, Enrico Fontana, Junhyung Lyle Kim +2 more

We introduce hypergraph cycle enumeration (HCE), a simple spectral framework for detecting and recovering planted signals in noisy hypergraph inference problems. We study HCE on one representative problem from this class, namely planted noisy k-XOR, also known as sparse Learning Parity with Noise (LPN). Both theoretically and empirically, HCE achieves comparable recovery behavior to Kikuchi method, matching Kikuchi hierarchy. We identify parameter regimes where HCE outperforms Kikuchi method, showing that neither algorithm is universally better. We also show a subquadratic quantum speedup for HCE by replacing one of the algorithm's steps with a quantum walk. Our classical and quantum algorithms do not refute the quartic quantum speedup of quantum Kikuchi method, which remains valid for the practically relevant regime of large size and high noise strength.

quant-pharXiv:2610.40941•Nov 2, 2026

Spectral Certificates and Non-commutative Sum-of-Squares Lower Bounds for Hamiltonians

Nicholas Kocurek, Chinmay Nirkhe

A central question in quantum many-body physics is estimating the ground energy of a $k$-local Hamiltonian system. In this work, we present a spectral technique for certifying a lower bound on the ground energy of a random $n$-qubit Hamiltonian system defined as the sum of signed $k$-local Pauli operators. In particular, we prove that for any constant $\ell$, there exists an efficiently computable length $n^{O(\ell)}$ certificate that is always a lower bound on the ground energy with the promise that, with high probability over the random Hamiltonian distribution, the certificate value is an $\varepsilon$-good approximation of the true ground energy when the number of terms is sufficiently large. Second, we show by construction that this technique, while successful on average over random Hamiltonian systems, can fail to produce good certificates on worst-case instances. Our spectral technique for producing these certificates comes from extending classical results on $k$-XOR refutations to $k$-local Pauli Hamiltonians by crafting a quantum variant of the Kikuchi matrix for CSP refutations. To show the limitations of this technique, we prove non-commutative Sum-of-Squares lower bounds for worst-case signed $k$-local Pauli operators. More generally, we explore how the non-commutative Sum-of-Squares relaxation can be understood as augmenting the standard Sum-of-Squares relaxation with the commutation relations between the Pauli operators. We instantiate the resulting framework with a modification to prior quantum code-based NLTS Hamiltonians that yields stronger complexity guarantees for the low-energy space; our Hamiltonian family satisfies simultaneously (1) $Ω(\log n)$-circuit depth lower bounds for all low-energy states, (2) constant-gap NP-hardness to approximate the ground energy, and (3) a constant-gap non-commutative Sum-of-Squares integrality gap up to $Ω(n)$-levels.

cs.CCCross-listedarXiv:2511.02264•Nov 5, 2025•0 citations

Quartic quantum speedups for planted inference

Alexander Schmidhuber, Ryan O’Donnell, Robin Kothari +1 more

We describe a quantum algorithm for the Planted Noisy kXOR problem (also known as sparse Learning Parity with Noise) that achieves a nearly quartic (4th power) speedup over the best known classical algorithm while also only using logarithmically many qubits. Our work generalizes and simplifies prior work of Hastings [Has20], by building on his quantum algorithm for the Tensor Principal Component Analysis (PCA) problem. We achieve our quantum speedup using a general framework based on the Kikuchi Method (recovering the quartic speedup for Tensor PCA), and we anticipate it will yield similar speedups for further planted inference problems. These speedups rely on the fact that planted inference problems naturally instantiate the Guided Sparse Hamiltonian problem. Since the Planted Noisy kXOR problem has been used as a component of certain cryptographic constructions, our work suggests that some of these are susceptible to super-quadratic quantum attacks.

Society for Industrial and Applied Mathematics eBooks•Jan 1, 2025•1 citation

A Near-Optimal Joint Lower Bound for Sparse Quantum Linear System Solvers

Dhrumil Patel

Quantum linear system solvers form one of the central algorithmic primitives in quantum computing, with applications ranging from differential equations and optimization to machine learning. Their cost is commonly measured through query complexity, which counts the number of oracle calls needed to access the input matrix. In the sparse-access model, this complexity is governed by three parameters: the condition number $κ$, the sparsity $s$ of the input matrix, and the target precision $\varepsilon$. The dependence on $κ$ and $\varepsilon$ is already well understood through the lower bound $Ω(κ\log(1/\varepsilon))$, which matches the best known scaling in these parameters. Once the sparsity $s$ is included, however, the expected lower bound has long been conjectured to be $Ω(κ\sqrt{s}\log(1/\varepsilon))$. Recent work by Mori et al. [Quantum Sci. Tech. 11 035063 (2026)] made an important step towards this goal by establishing the lower bound $Ω(κ\sqrt{s})$ for constant error $\varepsilon$. In this work, we complete the picture and prove the full joint lower bound $Ω(κ\sqrt{s}\log(1/\varepsilon))$ in the sparse-access model, thereby establishing the anticipated dependence on all three parameters simultaneously.

quant-pharXiv:2609.22445•Sep 22, 2026•0 citations

Accelerating Classical and Quantum Tensor PCA

Matthew B. Hastings

Spectral methods are a leading approach for tensor PCA with a ``spiked" Gaussian tensor. The methods use the spectrum of a linear operator in a vector space with exponentially high dimension and in Ref. 1 it was shown that quantum algorithms could then lead to an exponential space saving as well as a quartic speedup over classical. Here we show how to accelerate both classical and quantum algorithms quadratically, while maintaining the same quartic separation between them. That is, our classical algorithm here is quadratically faster than the original classical algorithm, while the quantum algorithm is eigth-power faster than the original classical algorithm. We then give a further modification of the quantum algorithm, increasing its speedup over the modified classical algorithm to the sixth power. We only prove these speedups for detection, rather than recovery, but we give a strong plausibility argument that our algorithm achieves recovery also. Note added: After this paper was prepared, A. Schmidhuber pointed out to me Ref. 3. This improves the best existing bounds on the spectral norm of a certain random operator. Because the norm of this operator enters into the runtime, with this improvement on the norm, we no longer have a provable polynomial speedup. Our results are phrased in terms of certain properties of the spectrum of this operator (not merely the largest eigenvalue but also the density of states). So, if these properties still hold, the speedup still holds. Rather than modify the paper, I have left it unchanged but added a section at the end discussing the needed property of density of states and considering for which problems (there are several problems for which this kind of quartic quantum speedup has been used and the techniques here will likely be applicable to several of them) the property is likely to hold.

quant-pharXiv:2602.10366•Feb 12, 2026•0 citations

Average-case hardness of Betti number estimation

Sergii Strelchuk, Sathyawageeswar Subramanian, Adam Wesołowski

We establish the average-case hardness of Betti number estimation on random clique complexes via a reduction from the planted clique problem. We further show that our reduction implies a series of hardness results for many problems in both classical and quantum Topological Data Analysis (qTDA). Under the classical planted clique conjecture, no randomized polynomial-time Betti number estimator achieves additive error below $\tfrac12$ with constant advantage. Under a new quantum planted clique conjecture that we introduce, the same conclusion holds for quantum polynomial-time algorithms. We also obtain related conditional hardness results for homology vanishing, additive approximations with larger error tolerances, preparation of simplex and harmonic states, cycle recovery, and counting eigenvalues at low energy. Our reduction clarifies the structural requirements for quantum advantage in TDA and provides a new lens to investigate the classical and quantum complexity of related problems.

quant-pharXiv:2609.12777•Sep 14, 2026•0 citations
Planted noisy kkkXOR and the strong refutation of random kkkXOR are governed by a conjectured trade-off between signal strength and time: Level ℓ\ellℓ of the Kikuchi hierarchy should achieve the smooth curve \begin{equation*}
m\ \gtrsim\ ρ^{-2}n^{k/2}/\ell^{k/2-1}\ \text{clauses}
\quad\Longleftrightarrow\quad
\text{solvable in time }n^{O(\ell)}, \end{equation*} where ρρρ is the bias of the planted signal or, for refutation, the target advantage. However, every spectral analysis of sparse kkkXOR to date loses polylogarithmic factors against this curve, a loss that enters the exponent of the running time.
We show that a normalized variant of the Kikuchi hierarchy achieves the sharp conjectured trade-off, with no logarithmic loss, at every arity k≥3k\ge3k≥3. At the scale above, our algorithms achieve strong detection, weak recovery, and strong refutation; an additional cleanup step boosts weak recovery to exact recovery, and the refutation certificates yield sum-of-squares proofs of degree Ok(ℓ)O_k(\ell)Ok​(ℓ). We also prove matching lower bounds in the same model. The inference and refutation upper bounds transfer to more general planting laws and predicates. Finally, we give a quantum algorithm that achieves a quartic speedup over the classical spectral algorithms for detection and weak recovery.
The proofs rest on two key ingredients: a normalization of the sparse Kikuchi matrix, and a sharp count of the closed walks in its trace expansion. We use a closely related trace-walk count to prove Feige's 2008 hypergraph Moore bound conjecture in a companion paper.
We introduce hypergraph cycle enumeration (HCE), a simple spectral framework for detecting and recovering planted signals in noisy hypergraph inference problems. We study HCE on one representative problem from this class, namely planted noisy k-XOR, also known as sparse Learning Parity with Noise (LPN). Both theoretically and empirically, HCE achieves comparable recovery behavior to Kikuchi method, matching Kikuchi hierarchy. We identify parameter regimes where HCE outperforms Kikuchi method, showing that neither algorithm is universally better. We also show a subquadratic quantum speedup for HCE by replacing one of the algorithm's steps with a quantum walk. Our classical and quantum algorithms do not refute the quartic quantum speedup of quantum Kikuchi method, which remains valid for the practically relevant regime of large size and high noise strength.
A central question in quantum many-body physics is estimating the ground energy of a kkk-local Hamiltonian system. In this work, we present a spectral technique for certifying a lower bound on the ground energy of a random nnn-qubit Hamiltonian system defined as the sum of signed kkk-local Pauli operators. In particular, we prove that for any constant ℓ\ellℓ, there exists an efficiently computable length nO(ℓ)n^{O(\ell)}nO(ℓ) certificate that is always a lower bound on the ground energy with the promise that, with high probability over the random Hamiltonian distribution, the certificate value is an ε\varepsilonε-good approximation of the true ground energy when the number of terms is sufficiently large. Second, we show by construction that this technique, while successful on average over random Hamiltonian systems, can fail to produce good certificates on worst-case instances. Our spectral technique for producing these certificates comes from extending classical results on kkk-XOR refutations to kkk-local Pauli Hamiltonians by crafting a quantum variant of the Kikuchi matrix for CSP refutations. To show the limitations of this technique, we prove non-commutative Sum-of-Squares lower bounds for worst-case signed kkk-local Pauli operators. More generally, we explore how the non-commutative Sum-of-Squares relaxation can be understood as augmenting the standard Sum-of-Squares relaxation with the commutation relations between the Pauli operators. We instantiate the resulting framework with a modification to prior quantum code-based NLTS Hamiltonians that yields stronger complexity guarantees for the low-energy space; our Hamiltonian family satisfies simultaneously (1) Ω(log⁡n)Ω(\log n)Ω(logn)-circuit depth lower bounds for all low-energy states, (2) constant-gap NP-hardness to approximate the ground energy, and (3) a constant-gap non-commutative Sum-of-Squares integrality gap up to Ω(n)Ω(n)Ω(n)-levels.
We describe a quantum algorithm for the Planted Noisy kXOR problem (also known as sparse Learning Parity with Noise) that achieves a nearly quartic (4th power) speedup over the best known classical algorithm while also only using logarithmically many qubits. Our work generalizes and simplifies prior work of Hastings [Has20], by building on his quantum algorithm for the Tensor Principal Component Analysis (PCA) problem. We achieve our quantum speedup using a general framework based on the Kikuchi Method (recovering the quartic speedup for Tensor PCA), and we anticipate it will yield similar speedups for further planted inference problems. These speedups rely on the fact that planted inference problems naturally instantiate the Guided Sparse Hamiltonian problem. Since the Planted Noisy kXOR problem has been used as a component of certain cryptographic constructions, our work suggests that some of these are susceptible to super-quadratic quantum attacks.
Quantum linear system solvers form one of the central algorithmic primitives in quantum computing, with applications ranging from differential equations and optimization to machine learning. Their cost is commonly measured through query complexity, which counts the number of oracle calls needed to access the input matrix. In the sparse-access model, this complexity is governed by three parameters: the condition number κκκ, the sparsity sss of the input matrix, and the target precision ε\varepsilonε. The dependence on κκκ and ε\varepsilonε is already well understood through the lower bound Ω(κlog⁡(1/ε))Ω(κ\log(1/\varepsilon))Ω(κlog(1/ε)), which matches the best known scaling in these parameters. Once the sparsity sss is included, however, the expected lower bound has long been conjectured to be Ω(κslog⁡(1/ε))Ω(κ\sqrt{s}\log(1/\varepsilon))Ω(κs​log(1/ε)). Recent work by Mori et al. [Quantum Sci. Tech. 11 035063 (2026)] made an important step towards this goal by establishing the lower bound Ω(κs)Ω(κ\sqrt{s})Ω(κs​) for constant error ε\varepsilonε. In this work, we complete the picture and prove the full joint lower bound Ω(κslog⁡(1/ε))Ω(κ\sqrt{s}\log(1/\varepsilon))Ω(κs​log(1/ε)) in the sparse-access model, thereby establishing the anticipated dependence on all three parameters simultaneously.
Spectral methods are a leading approach for tensor PCA with a ``spiked" Gaussian tensor. The methods use the spectrum of a linear operator in a vector space with exponentially high dimension and in Ref. 1 it was shown that quantum algorithms could then lead to an exponential space saving as well as a quartic speedup over classical. Here we show how to accelerate both classical and quantum algorithms quadratically, while maintaining the same quartic separation between them. That is, our classical algorithm here is quadratically faster than the original classical algorithm, while the quantum algorithm is eigth-power faster than the original classical algorithm. We then give a further modification of the quantum algorithm, increasing its speedup over the modified classical algorithm to the sixth power. We only prove these speedups for detection, rather than recovery, but we give a strong plausibility argument that our algorithm achieves recovery also. Note added: After this paper was prepared, A. Schmidhuber pointed out to me Ref. 3. This improves the best existing bounds on the spectral norm of a certain random operator. Because the norm of this operator enters into the runtime, with this improvement on the norm, we no longer have a provable polynomial speedup. Our results are phrased in terms of certain properties of the spectrum of this operator (not merely the largest eigenvalue but also the density of states). So, if these properties still hold, the speedup still holds. Rather than modify the paper, I have left it unchanged but added a section at the end discussing the needed property of density of states and considering for which problems (there are several problems for which this kind of quartic quantum speedup has been used and the techniques here will likely be applicable to several of them) the property is likely to hold.
We establish the average-case hardness of Betti number estimation on random clique complexes via a reduction from the planted clique problem. We further show that our reduction implies a series of hardness results for many problems in both classical and quantum Topological Data Analysis (qTDA). Under the classical planted clique conjecture, no randomized polynomial-time Betti number estimator achieves additive error below 12\tfrac1221​ with constant advantage. Under a new quantum planted clique conjecture that we introduce, the same conclusion holds for quantum polynomial-time algorithms. We also obtain related conditional hardness results for homology vanishing, additive approximations with larger error tolerances, preparation of simplex and harmonic states, cycle recovery, and counting eigenvalues at low energy. Our reduction clarifies the structural requirements for quantum advantage in TDA and provides a new lens to investigate the classical and quantum complexity of related problems.