arXiv subjects

Ruslan Shaydulin

Publications and source records attributed to Ruslan Shaydulin.

At least 19 recordsLinked to original sources

arXiv2026

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

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-ph↗

arXiv2026

The Compendium of Super-Quadratic Computational Quantum Advantages

This compendium presents a structured collection of computational problems with credible prospects for realizing meaningful quantum advantage, focused on describing problem characteristics rather than specific algorithms. It is intended for non-quantum researchers and practitioners with strong computational backgrounds, particularly domain experts, seeking to assess the relevance of quantum methods to real-world problems. The compendium focuses on enumerating core problems for which there is good evidence that a quantum computer can realize more than a quadratic speedup in offline non-relativized time-complexity (e.g., as opposed to in oracular settings, streaming models, or communication advantages). Entries characterize the structural constraints that real-world instances must satisfy to realize those advantages in practice. The intention is to provide a reference that can be leveraged to connect quantum algorithms to practical use cases.

quant-ph↗

arXiv2026

Provable Classical and Quantum Local Algorithms for Max-$k$-Cut and Quantum Advantage at Moderate Girth

Broadening the study of quantum optimization algorithms from binary to $k$-element alphabets has been shown to open new avenues for potential quantum advantage. A quantum advantage claim for approximate optimization requires showing that, under the same assumptions, an efficient quantum algorithm provably achieves a better performance than can be proven for the best known efficient classical algorithms. We study local classical and quantum algorithms for Max-$k$-Cut on $d$-regular graphs of girth $g$. We advance classical algorithms for this problem by developing a local vector algorithm based on the explicit vector construction of Thompson, Parekh, and Marwaha (TPM). Our algorithm gives the best provable cut fraction guarantee among known efficient classical algorithms on regular graphs for $k\geq3$. To evaluate the performance of QAOA under identical assumptions of girth and regularity, we develop tensor network techniques for general $d$ and $k \geq 2$. In addition, using an equivalence to a coupled qudit--boson system, we compute the QAOA performance in the infinite-degree limit. Together, these techniques give provable guarantees on QAOA performance on large graphs. Despite the improvements we introduce to the classical algorithm, QAOA achieves a better cut fraction guarantee for depths $p\geq 9$, corresponding to girth $g\geq 20$, for both finite- and infinite-degree regimes. Thus, we obtain an apparent quantum advantage from applying QAOA to the Max-$k$-Cut problem.

quant-ph↗

arXiv2026

Cycle Codes and Decoded Quantum Interferometry

Decoded Quantum Interferometry (DQI) reduces optimization problems with two-variable constraints to decoding cycle codes. For one such problem, namely MaxCut, prior work showed that DQI achieves a nontrivial satisfaction fraction guarantee only on linear-girth graphs, for which MaxCut is classically easy. However, these no-go results rely on minimum distance assumptions that do not represent true decodability thresholds for common noise channels, thus underestimating actual DQI performance. To estimate the true performance, we derive DQI satisfaction guarantees in the presence of imperfect decoding for fixed instances and generalize prior results for random instances. We then study homological cycle codes over arbitrary finite fields, and show that minimum-weight decoding is NP-hard for every field of size $q>2$, in contrast with known results for binary cycle codes. On the Linial--Simkin ensemble of regular graphs with logarithmic girth, we prove maximum-likelihood recovery bounds and a strong converse for a family of non-uniform additive noise channels. Moreover, LP decoders provide polynomial-time recovery of a positive fraction of randomly located errors with arbitrary values. These results allow us to prove upper and lower bounds on the approximate optima achievable by DQI when restricted to classical decoders. These bounds rule out quantum advantage in the regimes we analyze but also identify a family of regular Max-$k$-Cut instances on which DQI efficiently achieves a nontrivial satisfaction fraction guarantee.

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arXiv2026

Decoupling of the QAOA into independent spin-boson systems and high-depth performance on pure and mixed spin glasses

The Quantum Approximate Optimization Algorithm (QAOA) is regarded as a promising candidate for near-term quantum advantage in combinatorial optimization, yet our ability to study it at scale is limited. Exact recursive formulas have been introduced for predicting QAOA performance on large spin glasses, but the computational cost of evaluating them grows exponentially with the number of QAOA layers, preventing the study of QAOA in the promising high-depth regime. In this work, we show that for the task of computing QAOA energy on any mixed dense spin glass problem, spins approximately decouple into independent spin-boson systems. In the infinite-size limit, the decoupling becomes exact, establishing the spin-boson mapping as a natural framework for the many-body physics of QAOA. This generalized spin-boson mapping gives a recursive procedure for computing QAOA energies that can be executed at modest cost using tensor networks. As a numerical application, we optimize QAOA on pure and mixed spin glasses at depths intractable for prior techniques, observing that higher-degree problems require more layers for a comparable approximation ratio while angle optimization becomes more challenging. While decoupling enables the evaluations of QAOA energies at large size and depth, it does not enable strong simulation of QAOA; a quantum computer is required to sample the bitstring corresponding to the predicted energy.

quant-ph↗

arXiv2026

Quantum counterdiabatic driving enhanced by two-stage local control

Counterdiabatic driving can suppress diabatic losses during adiabatic ground state preparation. As its implementation necessitates the generation of an adiabatic gauge potential, which requires knowledge of the spectral gap of instantaneous Hamiltonians and involves highly non-local controls in many-body systems, local counterdiabatic driving with approximate adiabatic gauge potential has been widely adopted. In this work, using transverse-field Ising model as an example, we present an in-depth study of the performance of local counterdiabatic protocols and provide a closed-form analysis of ground-state-fidelity optimization. We then propose a two-stage protocol based on local counterdiabatic and simple local single-body control to further improve the performance. The practical implementation of these protocols does not require diagonalization of instantaneous Hamiltonians or spectral gaps, and only additional local single-body driving is involved. To benchmark their performance, we experimentally implement digitized adiabatic quantum evolution in a trapped-ion system. We characterize the quality of the prepared states and explore the scaling behavior with system size up to 14 qubits. Our demonstration of quantum shortcut to adiabaticity opens a path towards preparing ground states of complex systems with accessible local controls.

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arXiv2026

Verifiable Random Sampling

Verifiable random functions (VRF) underpin a wide range of applications that require publicly verifiable evaluations of a pseudorandom function on a given input. However, once the public key is published, the induced function is fixed and is a deterministic function of the input. This determinism can enable collusion and grinding-style attacks in which adversaries precompute and selectively exploit favorable input-output pairs. To address these limitations, we introduce the formal notion of verifiable random sampling (VRS). We propose a concrete VRS construction based on random quantum circuit sampling (RCS) executable on today's quantum computing devices. VRS supports multiparty protocols in which the verifier's final output is a sample that is statistically close to a specified target distribution, while remaining publicly verifiable. We model the construction and prove its security within the constructive cryptography (CC) framework, thereby ensuring composability with other cryptographic protocols. Overall, our results provide a mechanism for verifiable random sampling that simultaneously guarantees sample freshness and public verifiability, enabling applications that require unpredictable, fresh randomness while preserving fairness through public verifiability.

cs.CR↗

arXiv2026

qReduMIS: A Quantum-Informed Reduction Algorithm for the Maximum Independent Set Problem

We propose and implement a quantum-informed reduction algorithm for the maximum independent set problem that integrates classical kernelization techniques with information extracted from quantum devices. Our larger framework consists of dedicated application, algorithm, and hardware layers, and easily generalizes to the maximum weight independent set problem. In this hybrid quantum-classical framework, which we call qReduMIS, the quantum computer is used as a co-processor to inform classical reduction logic about frozen vertices that are likely (or unlikely) to be in large independent sets, thereby opening up the reduction space after removal of targeted subgraphs. We systematically assess the performance of qReduMIS based on experiments with up to 231 qubits run on Rydberg quantum hardware available through Amazon Braket. Our experiments show that qReduMIS can help address fundamental performance limitations faced by a broad set of (quantum) solvers including Rydberg quantum devices. We outline implementations of qReduMIS with alternative platforms, such as superconducting qubits or trapped ions, and we discuss potential future extensions.

quant-ph↗

arXiv2026

Optimization Using Locally-Quantum Decoders

It was pointed out in [JSW+25] that widely-studied optimization problems such as D-regular max-k-XORSAT can be reduced to decoding of LDPC codes, using quantum algorithms related to Regev's reduction. LDPC codes have very good decoders, such as Belief Propagation (BP), and this therefore makes D-regular max-k-XORSAT an enticing target for this class of quantum algorithms. However, BP was found insufficient to achieve quantum advantage. Here, we develop an intrinsically quantum decoding technique, which decodes classical LDPC codes subject to coherent superpositions of bit flip errors. For average-case instances of D-regular max-k-XORSAT drawn from Gallager's ensemble, this quantum decoder strongly outperforms classical belief propagation at many values of k and D. For some (k,D) the approximate optima achievable using this decoder surpass both Prange's algorithm and simulated annealing. However, we stop short of achieving quantum advantage because we identify an enhancement to Prange's algorithm that recovers a precise tie, much as a precise tie was observed between the standard version of Prange's algorithm and a more limited version of locally-quantum decoding in [CT24].

quant-ph↗

arXiv2026

Quantum Approximate Optimization of Integer Graph Problems and Surpassing Semidefinite Programming for Max-k-Cut

Quantum algorithms for binary optimization problems have been the subject of extensive study. However, the application of quantum algorithms to integer optimization problems remains comparatively unexplored. In this paper, we study the Quantum Approximate Optimization Algorithm (QAOA) applied to integer problems on graphs, with each integer variable encoded in a qudit. We derive a general iterative formula for depth-$p$ QAOA expectation on high-girth $d$-regular graphs of arbitrary size. The cost of evaluating the formula is exponential in the QAOA depth $p$ but does not depend on the graph size. Evaluating this formula for Max-$k$-Cut problem for $p\leq 4$, we identify parameter regimes ($k=3$ with degree $d \leq 10$ and $k=4$ with $d \leq 40$) in which QAOA outperforms the Frieze-Jerrum semi-definite programming (SDP) algorithm, which provides the best worst-case guarantee on the approximation ratio. To strengthen the classical baseline, we introduce a new heuristic algorithm based on the degree-of-saturation that achieves strong results on the \texttt{GSet} benchmark with quasi-linear runtime in the number of edges. It empirically outperforms both the Frieze-Jerrum algorithm and shallow-depth QAOA on regular graphs. Nevertheless, we provide numerical evidence that QAOA may overtake this heuristic at depth $p\leq 20$. Our results show that moving beyond binary to integer optimization problems can open up new avenues for quantum advantage.

quant-ph↗

arXiv2026

Iterative Interpolation Schedules for Quantum Approximate Optimization Algorithm

Quantum Approximate Optimization Algorithm (QAOA) is a promising quantum heuristic with empirical evidence of speedup over classical state-of-the-art for some problems. QAOA uses a parameterized circuit with $p$ layers, where higher $p$ yields better solutions, but requires optimizing $2p$ independent parameters, which is challenging at large $p$. We present an iterative interpolation method that exploits the smoothness of optimal parameter schedules by expressing them in a basis of orthogonal functions, generalizing the work of Zhou et al. By optimizing a small number of basis coefficients and iteratively increasing both circuit depth and coefficient count until convergence, our method constructs high-quality schedules for large $p$. We provide theoretical justification using Jackson's theorem and Lipschitz continuity to bound the required number of basis coefficients for a given accuracy. Our approach achieves better performance with fewer optimization steps than existing methods across three benchmark problems: the Sherrington-Kirkpatrick (SK) model, portfolio optimization, and Low Autocorrelation Binary Sequences (LABS). For the largest LABS instance, we achieve near-optimal merit factors with schedules exceeding 1000 layers, an order of magnitude beyond previous methods. Additionally, we observe that a mild growth in QAOA depth suffices to solve the SK model exactly, a result of independent theoretical interest.

quant-ph↗

arXiv2026

Iceberg Beyond the Tip: Co-Compilation of a Quantum Error Detection Code and a Quantum Algorithm

The rapid progress in quantum hardware is expected to make them viable tools for the study of quantum algorithms in the near term. The timeline to useful algorithmic experimentation can be accelerated by techniques that use many noisy shots to produce an accurate estimate of the observable of interest. One such technique is to encode the quantum circuit using an error detection code and discard the samples for which an error has been detected. An underexplored property of error-detecting codes is the flexibility in the circuit encoding and fault-tolerant gadgets, which enables their co-optimization with the algorthmic circuit. However, standard circuit optimization tools cannot be used to exploit this flexibility as optimization must preserve the fault-tolerance of the gadget. In this work, we focus on the $[[k+2, k, 2]]$ Iceberg quantum error detection code, which is tailored to trapped-ion quantum processors. We design new flexible fault-tolerant gadgets for the Iceberg code, which we then co-optimize with the algorithmic circuit for the quantum approximate optimization algorithm (QAOA) using tree search. By co-optimizing the QAOA circuit and the Iceberg gadgets, we achieve an improvement in QAOA success probability from $44\%$ to $65\%$ and an increase in post-selection rate from $4\%$ to $33\%$ at 22 algorithmic qubits, utilizing 330 algorithmic two-qubit gates and 744 physical two-qubit gates on the Quantinuum H2-1 quantum computer, compared to the previous state-of-the-art hardware demonstration. Furthermore, we demonstrate better-than-unencoded performance for up to 34 algorithmic qubits, employing 510 algorithmic two-qubit gates and 1140 physical two-qubit gates.

quant-ph↗

arXiv2026

Spin-Boson Mapping of the Quantum Approximate Optimization Algorithm

The Quantum Approximate Optimization Algorithm (QAOA) achieves monotonically improving performance with circuit depth $p$, yet the study of the high-depth regime has been obstructed by the exponential in $p$ cost of existing exact evaluation techniques. In this Letter, we prove that, in the infinite-size limit, the depth-$p$ QAOA state for the Sherrington-Kirkpatrick (SK) model converges to the state of a spin coupled to $p$ bosonic modes. We simulate the spin-boson system using matrix product states and provide numerical evidence that QAOA obtains a $(1-ε)$ approximation to the optimal energy of the SK model with circuit depth $O(n/ε^{1.13})$ in the average case. The modest computational cost of our approach allows us to optimize QAOA parameters and observe that QAOA achieves $\varepsilon\lesssim 2.2\%$ at $p=160$ in the infinite-size limit, extending far beyond $p\leq 20$ accessible to prior exact methods. Our mapping provides a many-body route to study and optimize high-depth QAOA in regimes previously inaccessible to exact evaluation.

quant-ph↗

arXiv2026

Regularized Warm-Started Quantum Approximate Optimization and Conditions for Surpassing Classical Solvers on the Max-Cut Problem

Demonstrating quantum heuristics that outperform strong classical solvers on large-scale optimization remains an open challenge. Here we introduce Regularized Warm-Started QAOA (RWS-QAOA), which initializes qubits by minimizing expected energy with a regularizer that penalizes near-bitstring states, preventing QAOA from stalling. We further propose a protocol that yields fixed, instance-independent parameters, enabling RWS-QAOA to operate as a non-variational algorithm in which the quantum circuit parameters are fixed and only a classical warm starting step is instance-dependent. We evaluate RWS-QAOA on the Max-Cut problem for random regular graphs, where this protocol yields a constant-depth quantum circuit, across three complementary settings. First, on Quantinuum's trapped-ion processor, RWS-QAOA outperforms the classical algorithms with the best provable guarantees for Max-Cut on $3$-regular graphs, namely Goemans-Williamson and Halperin-Livnat-Zwick, on $96$-node instances. Second, tensor-network simulations on graphs with up to $N{=}10{,}000$ nodes show that depth-$6$ RWS-QAOA, achieving an average cut fraction of $0.9167$, surpasses the best classical heuristics under matched restrictions (no local-search post-processing and no iterative refinement). Third, we remove these restrictions and benchmark against the strongest unrestricted classical heuristics, including an optimized parallel Burer-Monteiro solver that improves upon the MQLib implementation. Even against this stronger baseline, we project that surface-code RWS-QAOA reaches a quantum-classical runtime crossover below $0.2$ seconds on $3{,}000$-node graphs with fewer than $1.3$ million physical qubits. Our results show that constant-depth quantum circuits combined with a classical warm start have a credible potential to surpass classical solvers on the Max-Cut problem when executed on future quantum computers.

quant-ph↗

arXiv2026

Fault-tolerant execution of error-corrected quantum algorithms

Scaling up quantum algorithms to tackle high-impact problems in science and industry requires quantum error correction and fault tolerance. While progress has been made in experimentally realizing error-corrected primitives, the end-to-end execution of logical quantum algorithms using only fault-tolerant (FT) components has remained out of reach. We demonstrate the FT and error-corrected execution of two quantum algorithms, the Quantum Approximate Optimization Algorithm (QAOA) and the Harrow-Hassidim-Lloyd (HHL) algorithm applied to the Poisson equation, on Quantinuum H2 and Helios trapped-ion quantum processors using the $[[7,1,3]]$ Steane code. For QAOA circuits on 5 and 6 logical qubits, we show performance improvements from increasing the number of QAOA layers and the number of $T$ gates used to approximate logical rotations, despite increased physical circuit complexity. We further show that QAOA circuits with up to 8 logical qubits and 9 logical $T$ gates perform similarly to unencoded circuits. For the largest QAOA circuits we run, with 12 logical (97 physical) qubits and 2132 physical two-qubit gates, we still observe better-than-random performance. Finally, we show that adding active QEC cycles and increasing the repeat-until-success limit of state preparation subroutines can improve the performance of a quantum algorithm, thereby demonstrating critical capabilities of scalable FT quantum computation. Our results are enabled by an FT logical $T$ gate implementation with an infidelity of $\sim 2.6(4)\times10^{-3}$ and dynamic circuits with measurement-dependent feedback. Our work demonstrates near-break-even performance of complex, error-corrected algorithmic quantum circuits using only FT components.

quant-ph↗

arXiv2026

Digital signatures with classical shadows on near-term quantum computers

Quantum mechanics provides cryptographic primitives whose security is grounded in hardness assumptions independent of those underlying classical cryptography. However, existing proposals require low-noise quantum communication and long-lived quantum memory, capabilities which remain challenging to realize in practice. In this work, we introduce a quantum digital signature scheme that operates with only classical communication, using the classical shadows of states produced by random circuits as public keys. We provide theoretical and numerical evidence supporting the conjectured hardness of learning the private key (the circuit) from the public key (the shadow). A key technical ingredient enabling our scheme is an improved state-certification primitive that achieves higher noise tolerance and lower sample complexity than prior methods. We realize this certification by designing a high-rate error-detecting code tailored to our random-circuit ensemble and experimentally generating shadows for 32-qubit states using circuits with $\geq 80$ logical ($\geq 582$ physical) two-qubit gates, attaining 0.90 $\pm$ 0.01 fidelity. With increased number of measurement samples, our hardware-demonstrated primitives realize a proof-of-principle quantum digital signature, demonstrating the near-term feasibility of our scheme.

quant-ph↗

arXiv2026

Q-CHOP: Quantum constrained Hamiltonian optimization

Combinatorial optimization problems that arise in science and industry typically have constraints. Yet the presence of constraints makes them challenging to tackle using both classical and quantum optimization algorithms. We propose a new quantum algorithm for constrained optimization, which we call quantum constrained Hamiltonian optimization (Q-CHOP). Our algorithm leverages the observation that for many problems, while the best solution is difficult to find, the worst feasible (constraint-satisfying) solution is known. The basic idea of Q-CHOP is to enforce a Hamiltonian constraint at all times, thereby restricting evolution to the subspace of feasible states, and slowly ``rotate'' an objective Hamiltonian to trace an adiabatic path from the worst feasible state to the best feasible state. Q-CHOP thereby assigns qualitatively distinct roles to the constraint and objective functions of a constrained optimization problem. We additionally propose a version of Q-CHOP that can start in any feasible state. Finally, we benchmark Q-CHOP against the commonly-used adiabatic algorithm of quantum annealing with an objective function that penalizes constraint violation, and find that Q-CHOP consistently performs significantly better on a wide range of problems, including textbook graph problems, knapsack problems, combinatorial auctions, and a real-world financial use case of bond exchange-traded fund basket optimization.

quant-ph↗

arXiv2025

Generalized Short Path Algorithms: Towards Super-Quadratic Speedup over Markov Chain Search for Combinatorial Optimization

We analyze generalizations of quantum algorithms based on the short path framework first proposed by Hastings~[\textit{Quantum} 2, 78 (2018)], which has been extended and shown by Dalzell~et~al.~[STOC~'23] to achieve super-Grover speedups for certain binary optimization problems. We demonstrate that, under some commonly satisfied technical conditions, an appropriate generalization can achieve super-quadratic speedups not only over unstructured search but also over a classical optimization algorithm that searches for the optimum by drawing samples from the stationary distribution of a Markov chain. We employ this framework to obtain algorithms for problems including variants of Max Bisection, Max Independent Set, and finding the ground states of the Antiferromagnetic Ising Model and the Sherrington-Kirkpatrick Model, whose runtimes are asymptotically faster than those obtainable with previous short path techniques. In certain cases, our algorithms achieve super-quadratic speedups compared to the best known classical algorithms with rigorously established runtimes.

quant-ph↗