Qubit-based quantum simulators naturally target two-level systems, whereas many quantum many-body problems are intrinsically d level. Encodings from qudits to qubits then enlarge the Hilbert space and can introduce unphysical states that interfere with variational optimization. We formulate a variational framework for encoded d -level models that suppresses these illegitimate states with penalty terms and benchmark it for spin-1 and spin-3/2 bilinear-biquadratic Heisenberg chains. We compare binary encoding, which minimizes the qubit overhead, with symmetry encoding, which preserves the relevant spin symmetries and enables symmetry-conserving ansätze. Although binary encoding is more qubit efficient, its hardware-efficient ansatz is harder to train and less effective at exploiting conserved quantities. In contrast, symmetry encoding requires more qubits but reaches substantially higher fidelities, converges faster, and exhibits better trainability than the binary hardware-efficient ansatz. These results identify symmetry-preserving encodings as a practical route to simulating higher-spin models on existing qubit platforms.

Abstract Wegner duality is essential for ℤ 2 lattice gauge theory, yet the duality on non-trivial topologies has remained implicit. We extend Wegner duality to arbitrary topology and dimension, obtaining a new class of Ising models, in which topology is encoded in non-local domain-wall patterns. Without the overhead of gauge constraints, simulating this model on an L × L torus requires only L 2 qubits with two-body couplings, halving the conventional four-body coupled 2L 2 qubits, enabling full experimental realization of ℤ 2 lattice gauge theory on near-term devices.

Combinatorial optimization is widely regarded as a primary application for near-term quantum processors, although a definitive demonstration of the practical quantum advantage remains elusive. Recent studies have reported that both gate-based quantum circuits and quantum annealers can outperform state-of-the-art classical heuristics on multi-objective optimization (MO-MaxCut) problems. However, these studies did not fully account for the substantial pre- and post-processing overheads intrinsic to quantum solvers, leading to incomplete comparisons between quantum and classical approaches. In this work, we re-examine the same benchmark suite using GPU-based quantum-annealing-inspired algorithms (QAIAs), which, analogously to quantum processors, generate probabilistic samples and thus serve as formidable classical contenders. Our results show that QAIAs can sample candidate solutions approximately two orders of magnitude faster than previously studied quantum processors. In terms of end-to-end runtime, QAIAs also surpass industry-leading classical solvers, thereby establishing themselves as the superior performers among the quantum and classical solvers evaluated thus far for the MO-MaxCut instances.

A pivotal task for quantum computing is to speed up the solution of problems that are both classically intractable and practically valuable. Among these, combinatorial optimization problems have attracted tremendous attention due to their broad applicability and natural fitness to Ising Hamiltonians. Here we propose a quantum-sampling strategy, on the basis of which we design an algorithm to accelerate the solution of the ground states of the Ising model, a class of Nondeterministic Polynomial time (NP)-hard problems in combinatorial optimization. The algorithm employs a shallow-circuit quantum-sampling subroutine to navigate the energy landscape. Using up to 104 superconducting qubits, we experimentally demonstrate that this algorithm outputs favorable solutions compared with even a highly optimized classical simulated-annealing algorithm, and we illustrate the path toward quantum speedup based on the time-to-solution metric relative to simulated annealing under serial execution. Our results indicate a promising alternative to classical heuristics for combinatorial optimization, for which quantum advantage may become possible on near-term superconducting quantum processors.

Quantum circuit simulators running on classical computers offer a vital platform for designing, testing, and optimizing quantum algorithms, driving innovation despite limited access to real quantum hardware. However, their scalability is inherently constrained by exponential memory and computational overhead, which restricts accurate simulation of large-scale quantum circuits and often results in approximate output distributions. Here, we propose an exact sampling algorithm that integrates tensor network contraction techniques with a Markov process, wherein a classical state evolves according to the local structure of the quantum circuit. As a demonstration, we target the challenge of generating samples from ideal and noisy QAOA circuits with up to 476 qubits, incorporating both depolarizing and amplitude damping noise models. These results enable further validation of several assumptions and conjectures at a scale previously out of reach, significantly expanding the scope of classical simulation in quantum algorithm research.

Haplotype phasing, the process of resolving parental allele inheritance patterns in diploid genomes, is critical for precision medicine and population genetics, yet the underlying optimization is NP-hard, posing a scalability challenge. To address this, we introduce QHap, a haplotype phasing tool that leverages quantum-inspired optimization. By reformulating haplotype phasing as a Max-Cut problem and deploying a GPU-accelerated ballistic simulated bifurcation solver, QHap accelerates phasing while maintaining accuracy comparable to established phasing tools. On the highly polymorphic human major histocompatibility complex region, QHap demonstrates 4- to 20-fold acceleration with zero switch error across multiple long read sequencing platforms. The framework implements two strategies: a read-based method for regional phasing, and a single nucleotide polymorphism-based method that, through quality-weighted probabilistic edge construction, efficiently scales to chromosome-scale tasks. Integration of chromatin conformation capture data extends phase block contiguity by up to 15-fold, enabling near-chromosome-spanning haplotype reconstruction. QHap demonstrates that quantum-inspired algorithms operating on classical hardware offer a promising approach to addressing the growing computational demands of sequencing data, establishing a new paradigm for applying physics-inspired optimization to fundamental challenges in computational genomics.

The Ising model is ubiquitous in various optimization problems but notoriously difficult to solve due to combinatorial explosion. In view of this, Hamiltonian reduction is a useful preprocessing technique for reducing the effective problem size before applying heuristic solvers. However, existing reduction techniques mainly target second-order Ising models, whereas many pseudo-Boolean formulations naturally contain higher-order interactions. In this work, we generalize the concept of non-separable groups to arbitrary-order Ising-like models and develop a Hamiltonian reduction framework that iteratively detects and merges constrained spin groups into single variables. We benchmark the reduction on synthetic hypergraphs and higher-order network datasets, and evaluate its integration with downstream order-reduction and solver workflows. Our results establish a foundation for Hamiltonian reduction in higher-order Ising-like optimization problems.

Simulated Bifurcation (SB) algorithms, inspired by quantum annealing, can efficiently solve large-scale combinatorial optimization problems on classical hardware, often outperforming traditional approaches such as simulated annealing. However, their tendency to be trapped in local optima limits global solution quality. In this work, we introduce Tabu-Enhanced Simulated Bifurcation (TESB), an improved SB variant that incorporates a Tabu Search-inspired mechanism. By leveraging a dynamic penalty guided by early search history, TESB can naturally avoid revisiting suboptimal regions. On Max-Cut benchmarks, TESB achieves up to a three-order-of-magnitude reduction in Time-to-Solution compared to standard SB. When applied to particle track reconstruction in high-energy physics, TESB identifies lower-energy configurations on problems exceeding 100,000 spin variables, demonstrating enhanced scalability and performance across a wide range of combinatorial tasks.

Multi-objective combinatorial optimization requires identifying Pareto-optimal trade-off solutions among conflicting objectives, often making it more demanding than its single-objective counterpart. Although quantum multi-objective optimization methods have begun to emerge, most existing quantum optimization workflows are still built around single-objective or fixed-scalarization settings. Building on existing weighted-sum QAOA approaches to quantum multi-objective optimization, we propose QEMOO, a quantum-enhanced multi-objective optimization framework that combines Pareto-based selection and warm-started QAOA sampling in a multi-round protocol under the same total shot budget. We further introduce a PBI-inspired adaptive direction-update scheme to improve coverage in strongly conflicting benchmark regimes. Across three benchmark stages, QEMOO improves Pareto-front hypervolume over the single-pass weighted-sum QAOA baseline under matched shot budgets, suggesting a practical route toward shot-efficient quantum-assisted multi-objective optimization and its future applications.

Recent demonstrations of quantum computational advantage have been driven largely by sampling problems. A prominent model, boson sampling, involves sampling from the output distribution of a linear optical network. However, its classical hardness hinges on two plausible yet less-studied conjectures: the average-case hardness of approximating Gaussian permanents, and the permanent anti-concentration conjecture (PACC). The PACC is a purely mathematical assertion regarding the distributional properties of random Gaussian matrices. While the typical magnitude of the permanent has been established for discrete random matrices, the complex Gaussian case, which governs transition amplitudes in linear optical networks, has remained open. Here, we establish a weak anti-concentration bound by upper-bounding the probability that a random Gaussian permanent is superexponentially smaller than its standard deviation. Tightening this bound to an inverse-polynomial fraction would prove the original PACC. As a corollary, we establish the typical magnitude of Gaussian permanents, on par with Tao and Vu's seminal result for Bernoulli matrices. Combined with the Aaronson-Arkhipov framework, our result implies that classically simulating boson sampling to within a superexponentially small total variation distance would collapse the polynomial hierarchy, assuming the remaining conjectures hold.

Finding optimal solutions to combinatorial optimization problems (COPs) is pivotal in both scientific and industrial domains. Considerable efforts have been invested on developing accelerated methods utilizing sophisticated models and advanced computational hardware. However, the challenge remains to achieve both high efficiency and broad generality in problem-solving. Here we propose a general method, free-energy machine (FEM), based on the ideas of free-energy minimization in statistical physics, combined with automatic differentiation and gradient-based optimization in machine learning. FEM flexibly addresses various COPs within a unified framework and efficiently leverages parallel computational devices such as graphics processing units. We benchmark FEM on diverse COPs including maximum cut, balanced minimum cut and maximum k-satisfiability, scaled to millions of variables, across synthetic and real-world instances. The findings indicate that FEM remarkably outperforms state-of-the-art algorithms tailored for individual COP in both efficiency and efficacy, demonstrating the potential of combining statistical physics and machine learning for broad applications.

In the noisy intermediate-scale quantum era, emerging classical-quantum hybrid optimization algorithms, such as variational quantum algorithms (VQAs), can leverage the unique characteristics of quantum devices to accelerate computations tailored to specific problems with shallow circuits. However, these algorithms encounter biases and iteration difficulties due to significant noise in quantum processors. These difficulties can only be partially addressed without error correction by optimizing hardware, reducing circuit complexity, or fitting and extrapolating. A compelling solution is applying probabilistic error cancellation (PEC), a quantum error mitigation technique that enables unbiased results without full error correction. Traditional PEC is challenging to apply in VQAs due to its variance amplification, contradicting iterative process assumptions. This paper proposes a novel noise-adaptable strategy that combines PEC with the quantum approximate optimization algorithm (QAOA). It is implemented through invariant sampling circuits (invariant-PEC, or IPEC) and substantially reduces iteration variance. This strategy marks the first successful integration of PEC and QAOA, resulting in efficient convergence. Moreover, we introduce adaptive partial PEC (APPEC), which modulates the error cancellation proportion of IPEC during iteration. We experimentally validate this technique on a superconducting quantum processor, cutting sampling cost by 90.1%. Notably, we find that dynamic adjustments of error levels via APPEC can enhance the ability to escape from local minima and reduce sampling costs. These results open promising avenues for executing VQAs with large-scale, low-noise quantum circuits, paving the way for practical quantum computing advancements.

Anomalous heat transfer (AHT), a process by which heat spontaneously flows from a cold system into a hot one, superficially contradicts the Clausius statement of the second law of thermodynamics. Here we provide a full classification of mechanisms of the AHT in nonequilibrium quantum systems from a quantum-information perspective. For initial states in local equilibrium, we find that the AHT can arise from three resources: initial correlation, intrasystem interaction, and intrasystem temperature inhomogeneity. In particular, for qubit systems, we prove that initial quantum coherence is necessary for AHT if the intersystem interactions are limited to the two-body type. We explicitly show the AHT dominated by each of the mechanisms in a three-qubit system. Our classification scheme may offer a guideline for developing high-efficiency quantum heat pump.

Information engines produce mechanical work through measurement and adaptive control. For information engines, the principal challenge lies in how to store the generated work to the external load. Here, we report an experimental demonstration where quantized mechanical motion serves as a quantum battery and gets charged in repeated cycles by a single trapped-ion information engine. This is enabled by a key technological advancement in rapid state discrimination, allowing us to suppress measurement-induced disturbances. Consequently, we were able to obtain an information-to-ergotropy conversion efficiency approaching 67% of the theoretical limit at the optimal temperature, along with a maximum information-to-work conversion efficiency of 70%. The experimental results substantiate that this approach can render trapped ions a promising platform for microscopic information engines with potential applications in the future upon scaling up.

An important and difficult problem in optimization is the high-order unconstrained binary optimization, which can represent many optimization problems more efficiently than quadratic unconstrained binary optimization, but how to quickly solve it has remained difficult. Here, we present an approach by mapping the high-order unconstrained binary optimization to quantum Z2 lattice gauge theory and propose the gauged local quantum annealing, which is the local quantum annealing protected by the gauge symmetry. We present the quantum algorithm and its corresponding quantum-inspired classical algorithm for this problem and achieve algorithmic speedup by using gauge symmetry. By running the quantum-inspired classical algorithm, we demonstrate that the gauged local quantum annealing reduces the computational time by one order of magnitude from that of the local quantum annealing.

Abstract An important and difficult problem in optimization is the high-order unconstrained binary optimization, which can represent many optimization problems more efficiently than quadratic unconstrained binary optimization, but how to quickly solve it has remained difficult. Here, we present an approach by mapping the high-order unconstrained binary optimization to quantum $${{\mathbb{Z}}}_{2}$$ Z 2 lattice gauge theory and propose the gauged local quantum annealing, which is the local quantum annealing protected by the gauge symmetry. We present the quantum algorithm and its corresponding quantum-inspired classical algorithm for this problem and achieve algorithmic speedup by using gauge symmetry. By running the quantum-inspired classical algorithm, we demonstrate that the gauged local quantum annealing reduces the computational time by one order of magnitude from that of the local quantum annealing.

Jet clustering or reconstruction is a crucial component at high-energy colliders, a procedure to identify sprays of collimated particles originating from the fragmentation and hadronization of quarks and gluons. It is a complicated combinatorial optimization problem and requires intensive computing resources. In this study, we formulate jet reconstruction as a quadratic unconstrained binary optimization (QUBO) problem and introduce novel quantum-annealing-inspired algorithms for clustering multiple jets in electron-positron collision events. One of these quantum-annealing-inspired algorithms, ballistic simulated bifurcation, overcomes problems previously observed in multijet clustering with quantum-annealing approaches. We find that both the distance defined in the QUBO matrix and the prediction power of the QUBO solvers have crucial impacts on the multijet clustering performance. This study opens up a new approach to globally reconstructing multijet beyond dijet in one go, in contrast to the traditional iterative method.

Applying quantum annealing or current quantum-/physics-inspired algorithms for MIMO detection always abandon the direct gray-coded bit-to-symbol mapping in order to obtain Ising form, leading to inconsistency errors. This often results in slow convergence rates and error floor, particularly with high-order modulations. We propose HOPbit, a novel MIMO detector designed to address this issue by transforming the MIMO detection problem into a higher-order unconstrained binary optimization (HUBO) problem while maintaining gray-coded bit-to-symbol mapping. The method then employs the simulated probabilistic bits (p-bits) algorithm to directly solve HUBO without degradation. This innovative strategy enables HOPbit to achieve rapid convergence and attain near-optimal maximum-likelihood performance in most scenarios, even those involving high-order modulations. The experiments show that HOPbit surpasses ParaMax by several orders of magnitude in terms of bit error rate (BER) in the context of 12-user massive and large MIMO systems even with computing resources. In addition, HOPbit achieves lower BER rates compared to other traditional detectors.

Contextuality, one of the strongest forms of quantum correlations, delineates the boundary between the quantum world and the classical one. Recent advances show that some translation-invariant contextuality witnesses are maximally violated by ground states and local observables of infinite one-dimensional translation-invariant Hamiltonians. However, these models all have local Hilbert space dimension larger than two, making the study of their ground states behavior difficult on current qubit-based platforms. In this work, we focus on the cost of simulating their 3-site reduced density matrices using qubit-based parameterized quantum circuits. The local approximations are purified then encoded into permutation-symmetric qubit states. By developing a universal set of permutation-symmetry preserving qubit-based gates, we assess the accuracy of simulating the purified local ground states against fixed classical and quantum resources. Results reveal that more contextual ground states with lower energy density are easier to simulate under identical resources.

The search for the optimal pair of active and protection paths in a network with Shared Risk Link Groups (SRLG) is a challenging but high-value problem in the industry that is inevitable in ensuring reliable connections on the modern Internet. We propose a new approach to solving this problem, with a novel use of statistical analysis of the distribution of paths with respect to their cost, which is an integral part of our innovation. The key idea in our algorithm is to employ iterative updates of cost bounds, allowing efficient pruning of suboptimal paths. This idea drives an efficacious exploration of the search space. We benchmark our algorithms against the state-of-the-art algorithms that exploit the alternative strategy of conflicting links exclusion, showing that our approach has the advantage of finding more feasible connections within a set time limit.