arXiv subjects

Alexander Schmidhuber

Publications and source records attributed to Alexander Schmidhuber.

15 recordsLinked to original sources

arXiv2026

The Kikuchi Hierarchy is Sharp for $k$XOR

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.

cs.DS↗

arXiv2026

A Spectral Proof of the Hypergraph Moore Bound

A nonempty subfamily of a $k$-uniform hypergraph is an \emph{even cover} if every vertex lies in an even number of its hyperedges; for $k=2$ these are edge-disjoint unions of cycles, so the minimum size of an even cover is the natural hypergraph analogue of girth. We prove Feige's 2008 conjecture on the hypergraph Moore bound: there are absolute constants $A$ and $C$ (independent of $k$) such that for every $k\ge3$ and every $1\le\ell\le n$, any $k$-uniform hypergraph on $n$ vertices with more than $C\,n^{k/2}/\ell^{k/2-1}$ hyperedges contains an even cover of size at most $A\,\ell\log(en/\ell)$. Our proof is based on sharp spectral bounds for Kikuchi matrices, which we expect to be of independent interest; we apply them to the refutation of random constraint satisfaction problems in a companion paper.

math.CO↗

arXiv2026

Phase estimation with partially randomized time evolution

Quantum phase estimation combined with Hamiltonian simulation is the most promising algorithmic framework to computing ground state energies on quantum computers. Its main computational overhead derives from the Hamiltonian simulation subroutine. In this paper we use randomization to speed up product formulas, one of the standard approaches to Hamiltonian simulation. We propose new partially randomized Hamiltonian simulation methods in which some terms are kept deterministically and others are randomly sampled. We perform a detailed resource estimate for single-ancilla phase estimation using partially randomized product formulas for benchmark systems in quantum chemistry and obtain orders-of-magnitude improvements compared to other simulations based on product formulas. When applied to the hydrogen chain, we have numerical evidence that our methods exhibit asymptotic scaling with the system size that is competitive with the best known qubitization approaches.

quant-ph↗

arXiv2026

Provable quantum speedups for computing persistence in topological data analysis

Topological data analysis (TDA) aims to extract noise-robust features from a data set by examining the number and persistence of holes in its topology. We provide an efficient quantum algorithm for a computational problem closely related to a core task in TDA -- determining whether a given hole persists across different length scales. Further, we prove the problem itself is $\mathsf{BQP}_1$-hard, implying that a classical solution is extremely unlikely; this stands in contrast to all previous quantum approaches to TDA, where the problems were also intractable for quantum computers, or where a rigorous proof of classical hardness still remains open. This result implies an {exponential} quantum speedup for this problem under standard complexity-theoretic assumptions. Our approach relies on encoding the persistence of a hole in a variant of the guided sparse Hamiltonian problem, where the guiding state is constructed from a harmonic representative of the hole.

quant-ph↗

arXiv2026

Adiabatic Quantum Phase Estimation

Quantum phase estimation (QPE) is a central algorithmic primitive that estimates eigenvalues of a Hamiltonian up to precision $ε$ in Heisenberg-limited time $T=Θ(1/ε)$. Standard gate-based implementations of QPE require deep controlled time-evolution circuits and are not native to analog hardware. Here, we present a simple adiabatic protocol for QPE that achieves (up to logarithmic factors) the optimal Heisenberg-limited scaling $T = O\left( \frac{1}ε \log\left(δ^{-1}\right)\right)$ in both the precision $ε$ and failure probability $δ$. By encoding eigenvalues in populations of computational basis states rather than complex phases, our approach is naturally robust against certain dephasing errors. The adiabatic protocol only requires the ability to couple a single ancilla qubit to the system Hamiltonian as well as pairwise couplings within the ancilla register.

quant-ph↗

arXiv2026

The free energy limit of the SYK model at high temperature

The Sachdev-Ye-Kitaev (SYK) model is a disordered quantum mean-field model studied in condensed matter physics and the holographic theory of black holes. Its structural properties can be derived heuristically using a combination of the replica method and path integration techniques. Analyzing it mathematically rigorously, however, turned out to be notoriously difficult, even for basic questions such as computing the annealed free energy. In this paper we rigorously compute the free energy limit (annealed and quenched) for this model at high enough but constant temperature. Our results are in numerical agreement with the results derived by physics methods. Remarkably, though, our method of proof is novel and is different from the physics approach. It is based on (a) the theory of the component structure of sparse random graphs and (b) a variant of the cavity method, used widely in prior rigorous and heuristic treatments of classical spin glasses.

cond-mat.dis-nn↗

arXiv2025

Optimization by Decoded Quantum Interferometry

Achieving superpolynomial speedups for optimization has long been a central goal for quantum algorithms. Here we introduce Decoded Quantum Interferometry (DQI), a quantum algorithm that uses the quantum Fourier transform to reduce optimization problems to decoding problems. For approximating optimal polynomial fits over finite fields, DQI achieves a superpolynomial speedup over known classical algorithms. The speedup arises because the problem's algebraic structure is reflected in the decoding problem, which can be solved efficiently. We then investigate whether this approach can achieve speedup for optimization problems that lack algebraic structure but have sparse clauses. These problems reduce to decoding LDPC codes, for which powerful decoders are known. To test this, we construct a max-XORSAT instance where DQI finds an approximate optimum significantly faster than general-purpose classical heuristics, such as simulated annealing. While a tailored classical solver can outperform DQI on this instance, our results establish that combining quantum Fourier transforms with powerful decoding primitives provides a promising new path toward quantum speedups for hard optimization problems.

quant-ph↗

arXiv2025

Hamiltonian Decoded Quantum Interferometry

We introduce Hamiltonian Decoded Quantum Interferometry (HDQI), a quantum algorithm that utilizes coherent Bell measurements and the symplectic representation of the Pauli group to reduce Gibbs sampling and Hamiltonian optimization to classical decoding. For a signed Pauli Hamiltonian $H$ and any degree-$\ell$ polynomial ${P}$, HDQI prepares a purification of the density matrix $ρ_{P}(H) \propto {P}^2(H)$ by solving a combination of two tasks: decoding $\ell$ errors on a classical code defined by $H$, and preparing a pilot state that encodes the anti-commutation structure of $H$. Choosing $P(x)$ to approximate $\exp(-βx/2)$ yields Gibbs states at inverse temperature $β$; other choices prepare approximate ground states, microcanonical ensembles, and other spectral filters. For local Hamiltonians, the corresponding decoding problem is that of LDPC codes. Preparing the pilot state is always efficient for commuting Hamiltonians, but highly non-trivial for non-commuting Hamiltonians. Nevertheless, we prove that this state admits an efficient matrix product state representation for Hamiltonians whose anti-commutation graph decomposes into connected components of logarithmic size. We show that HDQI efficiently prepares Gibbs states at arbitrary temperatures for a class of physically motivated commuting Hamiltonians -- including the toric code and Haah's cubic code -- but we also develop a matching efficient classical algorithm for this task. For a non-commuting semiclassical spin glass and commuting stabilizer Hamiltonians with quantum defects, HDQI prepares Gibbs states up to a constant inverse-temperature threshold using polynomial quantum resources and quasi-polynomial classical pre-processing. These results position HDQI as a versatile algorithmic primitive and the first extension of Regev's reduction to non-abelian groups.

quant-ph↗

arXiv2025

Quartic quantum speedups for community detection

Community detection is a foundational problem in data science. Its natural extension to hypergraphs captures higher-order correlations beyond pairwise interactions. In this work, we develop a quantum algorithm for hypergraph community detection that achieves a quartic quantum speedup over the best known classical algorithm, along with superpolynomial savings in space. Our algorithm is based on the Kikuchi method, which we extend beyond previously considered problems such as Tensor PCA and $p$XORSAT to a broad family of generalized stochastic block models. To demonstrate (near) optimality of this method, we prove matching lower bounds (up to logarithmic factors) in the low-degree framework, showing that the algorithm saturates a smooth statistical-computational tradeoff. The quantum speedup arises from a quantized version of the Kikuchi method and is based on the efficient preparation of a guiding state correlated with the underlying community structure. Our work suggests that prior quantum speedups using the Kikuchi method are sufficiently robust to encompass a broader set of problems than previously believed; we conjecture that a quantity known as marginal order characterizes the existence of these quantum speedups.

quant-ph↗

arXiv2025

Classically estimating observables of noiseless quantum circuits

We present a classical algorithm based on Pauli propagation for estimating expectation values of arbitrary observables on random unstructured quantum circuits across all circuit architectures and depths, including those with all-to-all connectivity. We prove that for any architecture where each circuit layer is randomly sampled from a distribution invariant under single-qubit rotations, our algorithm achieves a small error $\varepsilon$ on all circuits except for a small fraction $δ$. The computational time is polynomial in qubit count and circuit depth for any small constant $\varepsilon, δ$, and quasi-polynomial for inverse-polynomially small $\varepsilon, δ$. Our results show that estimating observables of quantum circuits exhibiting chaotic and locally scrambling behavior is classically tractable across all geometries. We further conduct numerical experiments beyond our average-case assumptions, demonstrating the potential utility of Pauli propagation methods for simulating real-time dynamics and finding low-energy states of physical Hamiltonians.

quant-ph↗

arXiv2025

Quartic quantum speedups for planted inference

We describe a quantum algorithm for the Planted Noisy $k$XOR problem (also known as sparse Learning Parity with Noise) that achieves a nearly quartic ($4$th power) speedup over the best known classical algorithm while also only using logarithmically many qubits. Our work generalizes and simplifies prior work of Hastings, 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 $k$XOR 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.

quant-ph↗

arXiv2025

Gate-tunable band-edge in few-layer MoS$_2$

Transition metal dichalcogenides (TMDs) have garnered significant research interest due to the variation in band-edge locations within the hexagonal Brillouin zone between single-layer and bulk configurations. In monolayers, the conduction band minima are centered at the $K$-points, whereas in multilayers, they shift to the $Q$-points, midway between the $Γ$ and $K$ points. In this study, we conduct magnetotransport experiments to measure the occupation in the $Q$ and $K$ valleys in fourlayer molybdenum disulfide (MoS$_2$). We demonstrate electrostatic tunability of the conduction band edge by combining our experimental results with a hybrid $k\cdot p$ tight-binding model that accounts for interlayer screening effects in a self-consistent manner. Furthermore, we extend our model to bilayer and trilayer MoS$_2$, reconciling prior experimental results and quantifying the tunable range of band edges in atomically thin TMDs.

cond-mat.mes-hall↗

arXiv2025

Measurement events relative to temporal quantum reference frames

The Page-Wootters formalism is a proposal for reconciling the background-dependent, quantum-mechanical notion of time with the background independence of general relativity. However, the physical meaning of this framework remains debated. In this work, we compare two consistent approaches to the Page-Wootters formalism to clarify the operational meaning of evolution and measurements with respect to a temporal quantum reference frame. The so-called "twirled observable" approach implements measurements as operators that are invariant with respect to the Hamiltonian constraint. The "purified measurement" approach instead models measurements dynamically by modifying the constraint itself. While both approaches agree in the limit of ideal clocks, a natural generalization of the purified measurement approach to the case of non-ideal, finite-resource clocks yields a radically different picture. We discuss the physical origin of this discrepancy and argue that these approaches describe operationally distinct situations. Moreover, we show that, for non-ideal clocks, the purified measurement approach yields a time non-local evolution equation, which can lead to non-unitary evolution. Moreover, it implies a fundamental limitation to the operational definition of the temporal order of events. Nevertheless, unitarity and definite temporal order can be restored if we assume that time is discrete.

quant-ph↗

arXiv2025

A quantum algorithm for Khovanov homology

Khovanov homology is a topological knot invariant that categorifies the Jones polynomial, recognizes the unknot, and is conjectured to appear as an observable in $4D$ supersymmetric Yang--Mills theory. Despite its rich mathematical and physical significance, the computational complexity of Khovanov homology remains largely unknown. To address this challenge, this work initiates the study of efficient quantum algorithms for Khovanov homology. We provide simple proofs that increasingly accurate additive approximations to the ranks of Khovanov homology are DQC1-hard, BQP-hard, and #P-hard, respectively. For the first two approximation regimes, we propose a novel quantum algorithm. Our algorithm is efficient provided the corresponding Hodge Laplacian thermalizes in polynomial time and has a sufficiently large spectral gap, for which we give numerical and analytical evidence. Our approach introduces a pre-thermalization procedure that allows our quantum algorithm to succeed even if the Betti numbers of Khovanov homology are much smaller than the dimensions of the corresponding chain spaces, overcoming a limitation of prior quantum homology algorithms. We introduce novel connections between Khovanov homology and graph theory to derive analytic lower bounds on the spectral gap.

math.GT↗

arXiv2024

Complexity-Theoretic Limitations on Quantum Algorithms for Topological Data Analysis

Quantum algorithms for topological data analysis (TDA) seem to provide an exponential advantage over the best classical approach while remaining immune to dequantization procedures and the data-loading problem. In this paper, we give complexity-theoretic evidence that the central task of TDA -- estimating Betti numbers -- is intractable even for quantum computers. Specifically, we prove that the problem of computing Betti numbers exactly is #P-hard, while the problem of approximating Betti numbers up to multiplicative error is NP-hard. Moreover, both problems retain their hardness if restricted to the regime where quantum algorithms for TDA perform best. Because quantum computers are not expected to solve #P-hard or NP-hard problems in subexponential time, our results imply that quantum algorithms for TDA offer only a polynomial advantage in the worst case. We support our claim by showing that the seminal quantum algorithm for TDA developed by Lloyd, Garnerone and Zanardi achieves a quadratic speedup over the best known classical approach in asymptotically almost all cases. Finally, we argue that an exponential quantum advantage can be recovered if the input data is given as a specification of simplices rather than as a list of vertices and edges.

quant-ph↗