An S-box is a function that maps an n-bit input to an m-bit output, and in ciphers like the Advanced Encryption Standard it is the primary source of nonlinearity. When analysts model a cipher algebraically, the equations describing the S-box are replicated for every occurrence of the box inside the cipher, so the number and structure of those equations directly shape the size of the full model. The simplest description writes out explicit algebraic normal form equations for each output bit, but that is not always the most efficient formulation. Instead, the researchers work in the Boolean ring, where a relation holds for a given S-box if it evaluates to zero at every valid input-output pair, and a system of relations uniquely describes the box only if it both accepts all valid pairs and rejects every incorrect output.

Why does this matter for quantum computing? Formulating cryptanalytic problems as quadratic unconstrained binary optimization, or QUBO, remains an active research direction tied to quantum annealing. The computational cost of solving QUBO or Ising problems depends heavily on the number of variables involved. Each quadratic monomial in an equation system must be replaced by a shared auxiliary variable during linearization, and each parity condition needs binary-encoded counter bits. The package therefore evaluates candidate equation systems using a cost measure that sums the counts of linearization variables and parity-counter bits. Minimizing this measure reduces the auxiliary-variable count in the adopted construction, although the authors are careful to note that it does not guarantee lower practical solution difficulty, and their software neither constructs the QUBO matrix nor solves the resulting problem.

The payoff from careful modeling can be dramatic. For the AES S-box, replacing the original 39-equation system with a 13-equation system proposed in earlier work by the same group reduces the number of auxiliary variables from 299 to 106 per S-box, shrinking the AES-128 QUBO model from 68,600 to 30,026 logical variables. Those earlier results, obtained through laborious manual construction for a single cipher, motivated the generalization now embodied in SBoxQUBOSearch. The new package extends the underlying theory of implicit-relation construction and variance-guided basis transformations beyond AES, wrapping the methods in optimized standalone programs, versioned data schemas, reproducible job files, and a partitioning scheme that lets searches run on independent machines without any communication between them.

The software is organized as three command-line programs driven by JSON job files. The first, eqgen, generates equations: explicit ones via an in-place Möbius transform of the truth tables, and implicit relations by building a catalog of all monomials up to a chosen degree, evaluating them at every valid input-output pair, and extracting relations from the left nullspace of the resulting evaluation matrix through packed-row Gaussian elimination. The second program, search-global, then lexicographically enumerates subsets of the generated relations of a specified size, checking each candidate system for exactness using a bit-packed evaluation table in which a bitwise AND with the selected-equation mask vanishes precisely when a pair satisfies all chosen equations. The third, search-local, explores equivalent bases of a chosen system by applying invertible transformations over the two-element field, guided by an LFSR-inspired state machine that uses the variance of equation lengths to steer the traversal.

The exhaustive search is where the tool shows both its power and its limits. For the AES S-box, with 39 quadratic relations available, no exact system of eight, nine, ten, or eleven equations exists; the search confirmed this by testing billions of subsets. Exact twelve-equation systems first appeared among roughly 3.9 billion candidates, and at thirteen equations the search found 611,109 exact systems, the best of which achieved a cost of 109. That unpartitioned run took 13.2 hours on a laptop-class processor, but splitting the enumeration across eight independent partitions reproduced all 611,109 systems and the same eight minimizers, with each partition finishing in roughly 2.5 hours. For the lightweight ciphers ARADI and Ascon, the searches were nearly instant: four equations suffice for ARADI with a cost of just 8, while Ascon yielded 42 exact five-equation systems, the best scoring 19.

The 16-bit MK3 S-box pushed the method to its practical boundary. Exhaustive enumeration of sixteen-equation subsets from 79 relations would require examining roughly 2.16 times ten to the sixteenth power combinations, which is infeasible, and the evaluation table alone occupies 64 gigabytes. The researchers instead used a deletion strategy, removing one equation at a time and verifying with search-global that exactness survived, eventually establishing 28 as an upper bound on the minimum number of equations needed. Verified deletions followed by local optimization reduced the auxiliary-variable count from 1014 to 501. For Ascon, the combination of subset search and local optimization cut the count from 110 to 17, and for ARADI subset search alone reduced it from 71 to 8.

The local optimization stage proved remarkably reliable across hundreds of runs. Fifty sampled AES systems were each evaluated under three configurations, and every one of the 150 runs improved the cost measure, with a median gain of five points and the best system reaching 105. For Ascon, all 42 exact systems were tested under five configurations, improving in 91 percent of the 210 runs with a median gain of two and a best value of 17. For MK3, all 120 runs across two subset sizes improved, with median gains of four. The variance-guided heuristic works because the cost measure includes a logarithmic term, meaning bases that mix very short and very long equations can be advantageous; since an invertible basis change preserves the space spanned by the equations, the linearization-variable count stays fixed and all improvement comes from trimming the counter-bit term.

Beyond the headline numbers, the package is designed as a reproducible, reusable stage in a larger cryptanalytic pipeline. Versioned JSON and JSONL schemas define the persistent interface between stages, streaming output accommodates enormous result collections, and the file holding all 611,109 exact AES systems compresses from 124.7 megabytes to 4.45 megabytes. An 85-test Python suite independently validates equations, metrics, partitioning, and input handling. The authors position the tool as an upstream component rather than a solver: its outputs can feed QUBO models, algebraic attacks, SAT, SMT, and MILP formulations, or Gröbner-basis engines such as PolyBoRi. Current limitations are clearly stated, including support only for relations of degree at most two and memory requirements that scale with the size of the S-box domain. Released under the MIT license with all experimental artifacts, SBoxQUBOSearch turns what was once a bespoke, cipher-specific craft into a systematic search, offering cryptographers a practical head start whenever they need to squeeze a cipher’s innermost component into the variable budget of a quantum annealer.

Subject of Research: Automated search for exact algebraic S-box equation systems to minimize QUBO model size in cryptanalysis

Article Title: SBoxQUBOSearch: Exhaustive subset search and local basis optimization for exact algebraic S-box models

Article References: Burek, E., & Misztal, M. (2026). SBoxQUBOSearch: Exhaustive subset search and local basis optimization for exact algebraic S-box models. SoftwareX, 36, Article 103113. https://doi.org/10.1016/j.softx.2026.103113

Image Credits: AI Generated

DOI: 10.1016/j.softx.2026.103113

Keywords: SBoxQUBOSearch, S-box, cryptography, QUBO, quantum annealing, AES, Ascon, ARADI, MK3, algebraic cryptanalysis, Boolean equations, open-source software

The new method, called SymLQA, is described in a paper published in Quantum Information Processing by Wenjie Sun, Zhigang Wang, and colleagues at the University of Electronic Science and Technology of China, Tsinghua University, and partner institutions. Rather than attacking the higher-order terms head-on, the researchers exploit a structural property of the problems they target: planar face-flux optimization. These problems arise naturally in lattice gauge models, where spins live on the edges of a graph and the physically meaningful quantities are fluxes around closed loops, or faces. In such settings, the seemingly complicated HUBO energy function hides a much simpler structure that only becomes visible when the graph is treated as a geometric object rather than a bag of coupled variables.

The first step of the SymLQA pipeline is a careful topological extraction. Using a rotation system, a standard combinatorial device that records the cyclic order of edges around each vertex, together with half-edge traversal, the algorithm identifies all the bounded faces of an embedded planar graph. This is the computational equivalent of tracing every enclosed region of a map drawn on a flat sheet. The implementation is rigorous enough that it satisfies Euler’s identity, the classic relation among vertices, edges, and faces, on every embedding tested, and it successfully extracts irregular faces containing up to sixteen boundary edges. That robustness matters because real-world planar instances are rarely neat square grids; they are irregular, lopsided, and full of awkward boundary shapes.

Once the faces are known, the transformation at the heart of SymLQA begins. Each face is assigned a new binary variable defined as the product of the edge spins along its boundary, a quantity the authors call a classical face-flux variable. This move mirrors a deep idea from lattice gauge theory, where fluxes around plaquettes, rather than the underlying link variables, often carry the essential physics. In the face-flux variables, the original edge-spin HUBO, with its face fields and face-flux interaction terms, becomes a sparse objective containing only one-body and two-body terms. Crucially, this reduction requires no auxiliary variables at all, which distinguishes it from the usual penalty-based encodings that inflate problem size and introduce fragile constraint weights.

The reduction would be of limited use if solutions in the new variables could not be translated back. SymLQA handles this with a reconstruction step based on arithmetic over the finite field GF(2), the two-element field where addition is equivalent to exclusive-or. For connected open planar embeddings, the GF(2) reconstruction maps every assignment of the dual, or face, variables back to a consistent assignment of the original edge spins, and the mapping preserves the objective value exactly. The consequence is mathematically clean: the minimum of the primal problem and the minimum of the dual problem coincide. The annealer can therefore search the transformed, quadratic landscape with the full confidence that whatever optimum it finds corresponds to a genuine optimum of the original hard problem.

To find out whether this elegant reformulation actually pays off in practice, the team benchmarked SymLQA against three formidable baselines: a momentum-based native local quantum annealing applied directly to the HUBO, a gauged variant of local quantum annealing, and classical simulated annealing on the primal formulation. The test bed consisted of independently generated certified frustrated-loop instances, a family of benchmark problems whose ground states are known in advance, which allows success or failure to be verified without ambiguity. Frustrated loops are notoriously treacherous for annealers because competing interactions create rugged energy landscapes riddled with local minima, making them a demanding and honest yardstick for any new solver.

The results are striking. On regular grids up to 32 by 32 and on irregular planar grids, SymLQA reached the known ground state in every tested run, maintaining a perfect success probability. The baselines told a very different story: their success probabilities fell rapidly as the instances grew, a familiar signature of gradients drowning in higher-order terms and of generic thermal dynamics failing to navigate the landscape. SymLQA’s advantage was not confined to a single coupling regime either. Across four independently sampled coupling regimes, the method retained unit success probability, suggesting that the improvement stems from the structural reformulation itself rather than from a lucky interaction between the algorithm and one particular class of random instances.

The speedup numbers are equally dramatic. At a grid size of 12 by 12, SymLQA achieved a batch time to solution at the 99 percent confidence level, abbreviated TTS99, of just 0.137 seconds. Native local quantum annealing needed 22.35 seconds to reach the same reliability, roughly 160 times slower, while primal simulated annealing required 8.85 seconds, about 65 times slower. Time to solution is a standard metric in the annealing community because it combines the probability of finding the optimum with the cost of each run, rewarding algorithms that are both accurate and consistently repeatable. A two-order-of-magnitude gap on this metric is not an incremental gain; it is the kind of separation that changes which problems are considered practically solvable.

What makes SymLQA especially interesting is that it is, at its core, a classical pipeline. The face extraction, the flux-variable transformation, and the GF(2) reconstruction are all classical preprocessing and postprocessing steps wrapped around an annealing solver. This positions the work squarely within the growing field of quantum-inspired optimization, where insights from physics and from quantum hardware motivate classical algorithms that can run on ordinary computers today. The connection to quantum Z2 lattice gauge formulations of HUBO problems, explored in recent work by other groups, shows that gauge-theoretic structure is emerging as a general resource for taming higher-order optimization, and SymLQA demonstrates how to exploit that structure with full topological awareness on planar graphs.

The implications extend beyond a single benchmark family. Planar optimization structures appear in routing and network design, in grid-based physical models, and in any setting where constraints or costs are naturally associated with regions rather than individual links. By converting edge-spin HUBO models with polynomially suppressed gradients into sparse Ising objectives with clean one- and two-body terms, SymLQA makes an entire problem class friendly to the growing ecosystem of Ising machines, digital annealers, and quantum annealers. The work also carries a broader lesson for the field: before throwing more hardware or more sophisticated dynamics at a hard optimization problem, it can pay enormously to ask whether the problem’s hidden topology already contains the key to simplifying it. In this case, a map’s faces turned out to be the secret ingredient that turned an intractable-feeling search into one solved, reliably, in a fraction of a second.

Subject of Research: Topology-aware local quantum annealing for planar face-flux HUBO optimization problems

Article Title: SymLQA: topology-aware local quantum annealing for planar face-flux HUBO problems

Article References: Sun, W., Wang, Z., Hu, J., Yu, L., Chen, G., Wang, L., Liu, H., & Li, X. (2026). SymLQA: topology-aware local quantum annealing for planar face-flux HUBO problems. Quantum Information Processing, 25(10), Article 323. https://doi.org/10.1007/s11128-026-05345-4

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05345-4

Keywords: quantum annealing, HUBO, Ising model, lattice gauge theory, combinatorial optimization, planar graphs, face-flux variables, simulated annealing, topology, GF(2) reconstruction, time to solution, spin glass

The package addresses a class of problems that sits at the heart of both physics and computer science: finding the lowest-energy configuration of spins on a lattice, the classic task of computing a ground state. These problems are computationally brutal in general, and they matter well beyond condensed matter theory. Many scheduling, routing and circuit-design challenges can be mapped onto Ising models, which is why quantum annealers and specialized Ising machines have attracted so much attention. Tensor networks offer a classical alternative, compressing the exponentially large space of spin configurations into structured objects that can be manipulated with controlled approximations. SpinGlassPEPS.jl implements this approach using projected entangled-pair-style constructions adapted to quasi-two-dimensional connectivity, including the Chimera graphs used by D-Wave quantum hardware.

The consolidation alone changes the user experience substantially. The original release consisted of four packages, SpinGlassTensors, SpinGlassNetworks, SpinGlassExhaustive and SpinGlassEngine, whose version numbers were kept in sync but whose interfaces gradually drifted apart. Renamed symbols lacked deprecation paths, leaving users with breakages and no guidance. Version 2.0.1 folds all four into internal modules of one package, so a single command, adding SpinGlassPEPS, installs the complete solver. The public surface shrinks from roughly 240 re-exported symbols to about 80 documented ones, while lower-level kernels remain accessible through the submodules for those who need them. It is a classic engineering trade: less clutter at the top, the same power underneath.

Two correctness fixes deserve particular attention because they affected results. The corner_matrix function, a core building block of the contraction scheme, failed on every SiteTensor input and permuted the trailing dimensions of VirtualTensor outputs. Meanwhile, the exhaustive GPU search launched too few execution blocks and returned incomplete spectra for problems with ten or more spins. Both defects are now covered by regression tests against independent dense references on CPU and GPU, including a deterministic ten-spin test that compares all 1024 GPU state codes and energies, produced by two 512-thread blocks, against complete CPU enumeration. The release also bundles the benchmark instances behind the original publication’s figures, which had previously been referenced but not distributed, closing a reproducibility gap that plagues much of computational science.

The most scientifically interesting addition is a discarded-weight diagnostic. Every truncating factorization in the contraction now records the relative weight it discards, accumulated in a task-local accumulator so that concurrent solves cannot mix statistics. A solve reports the sum and maximum of these discarded weights, along with how many truncations were forced by the bond-dimension cap rather than the singular-value tolerance, and how many of the offered singular values were retained. On a 128-spin instance, bond dimension 4 yields a total discarded weight of 3.1 times ten to the minus four, with all 18 truncations limited by the bond bound; at bond dimension 32 the figure drops to 5.6 times ten to the minus fourteen, with none of its four truncations bond-limited. In effect, users gain an internal warning light that was previously available only to those with an exact solution to compare against.

The diagnostics come with honest caveats, which is refreshing in a field where optimism often outruns rigor. Discarded weight measures truncation loss in cold contractions, not warm-start error: a warm start optimizes within a fixed bond dimension without a truncating factorization, so it can report essentially zero discarded weight despite a nonzero variational gap. The package warns when the two are combined. Nor does discarded weight reliably rank solution quality. In benchmarks on ten 2500-spin square-lattice instances, the median accumulated discarded weight peaked at inverse temperature 4, where the median energy error was smaller, while at inverse temperature 2 the discarded weight was near its minimum but the energy error was largest. The documentation now positions the diagnostic as a contraction indicator and optional preference filter, not a selection criterion.

Performance work runs throughout the release. A new beta_ladder feature evaluates an increasing schedule of the inverse temperature, which controls how strongly the solver’s branch probabilities favor low-energy states. When a retained boundary matrix product state has the dimensions required by the next rung, it warm-starts variational compression instead of building the target state exactly and truncating it. On a 2048-spin instance where boundary construction dominates, the two warmed rungs saw wall times fall by about 24 and 25 percent, with identical energies, translating to roughly 16 percent over the full ladder. Meanwhile, a sweep over eight lattice transformations now runs concurrently, with a calibration solve estimating peak memory and admitting tasks within a byte budget. On an Intel Xeon Platinum 8462Y+ processor, CPU speed-ups reached 3.1 times for a 36-spin case at eight concurrent solves, and identical energies were returned in every serial and concurrent configuration tested.

Perhaps the most surprising finding concerns hardware. The original publication recommended GPU execution for larger examples, but on the tested Xeon and NVIDIA H100 system, the GPU won in only one matched configuration: a 2048-spin sparse instance at bond dimension 32, where the CPU-to-GPU wall-clock ratio was 1.45. At 36 spins, CPU wall time was roughly 2 percent of GPU time. Profiling explains why: on a 128-spin solve, host-side CUDA API calls occupied 27 percent of the profiled interval while GPU activities occupied just 6.6 percent, meaning at least two-thirds of the time lay outside both categories. Even eliminating the entire measured CUDA API overhead would cap speed-up at about 1.4 times by Amdahl’s law. Faster tensor kernels alone cannot fix a problem that lives on the host side.

So the developers attacked host allocation instead. In the previous implementation, a function called branch_states accounted for 52.7 percent of the bytes allocated by a 128-spin solve, spawning tens of thousands of small vectors per call; a single matrix now stores branched configurations instead. Contraction temporaries were moved off the garbage-collected heap, cutting allocated bytes on a corresponding CPU solve from 90.9 to 32 gibibytes, a reduction of about two thirds. The post-change totals were 32.3 gibibytes on CPU and 24.1 gibibytes on GPU for a 2048-spin, bond-32 solve. These are the unglamorous numbers that determine whether a scientific tool feels responsive or sluggish, and they illustrate how modern performance engineering often means fighting the runtime environment rather than the mathematics.

Version 2.0.1, archived on Zenodo and released under the Apache License 2.0, arrives as debate intensifies over whether classical algorithms can match specialized quantum and analog Ising machines. A companion study comparing tensor-network approaches to quantum and classical Ising machines suggests the answer is nuanced, and honest tooling like this strengthens the comparison. With one public API, deprecation shims for renamed entry points, distributed benchmark instances and raw per-run data with runnable drivers, the release models what reproducible computational research should look like. The work was supported by the National Science Centre, Poland, and the package’s developers, Łukasz Pawela and Bartłomiej Gardas, have made clear-eyed limitations part of the documentation itself. For anyone using tensor networks to hunt ground states, the message is simple: the toolkit got faster, safer and considerably more transparent, and it now tells you when it might be wrong.

Subject of Research: A tensor-network software package for solving Ising-like optimization problems on quasi-two-dimensional graphs

Article Title: Version 2.0.1 – SpinGlassPEPS.jl: Tensor-network package for Ising-like optimization on quasi-two-dimensional graphs

Article References: Pawela, Ł., & Gardas, B. (2026). Version 2.0.1 – SpinGlassPEPS.jl: Tensor-network package for Ising-like optimization on quasi-two-dimensional graphs. SoftwareX, 36, Article 103027. https://doi.org/10.1016/j.softx.2026.103027

Image Credits: AI Generated

DOI: 10.1016/j.softx.2026.103027

Keywords: tensor networks, Ising optimization, SpinGlassPEPS.jl, Julia language, ground state computation, GPU computing, high-performance computing, combinatorial optimization, reproducibility, software engineering, spin glasses, quantum annealing

The research, conducted by Seon-Geun Jeong, Dinh Cong Mai, Dae-Il Noh, Quoc-Viet Pham, and Won-Joo Hwang, addresses a fundamental problem that has haunted quantum annealing since its commercialization by D-Wave Systems. Quantum annealing is a heuristic realization of adiabatic quantum computation, in which a quantum system is slowly evolved from an easily prepared initial state toward a final Hamiltonian whose ground state encodes the solution to a combinatorial optimization problem. In theory, if the evolution is slow enough, the system remains in its instantaneous ground state and emerges with the optimal answer. In practice, finite annealing times, thermal fluctuations, and hardware imperfections turn the process into a statistical sampling exercise rather than a guaranteed convergence procedure.

The core difficulty lies in what physicists call minor embedding. D-Wave processors, from the earlier Chimera topology through the Pegasus generation to the newest Zephyr architecture, arrange superconducting flux qubits in sparse but structured connectivity graphs. The Zephyr topology, for example, supports up to 7440 qubits in a graph with maximum degree 20, a substantial improvement over its predecessors, but still nowhere near fully connected. When a logical problem graph, such as a fully connected clique representing interactions among many variables, must be mapped onto this hardware, each logical variable is represented by a chain of ferromagnetically coupled physical qubits. The longer these chains become, the more vulnerable they are to what researchers call chain breaks, events in which the physical qubits within a chain disagree, destroying the correspondence between the logical and physical representations of the problem.

Previous studies of embedding have been largely empirical, measuring clique sizes, approximation ratios, and chain break fractions across hardware generations without offering a predictive theoretical framework. The new work changes that by extending the integrated control error (ICE) model, a statistical description of local field and coupler errors commonly used in D-Wave documentation, into a rigorous analytic tool. The researchers distinguish between a moment-level uncorrelated ICE baseline, which requires only that errors have zero mean and finite variance with negligible covariance terms, and a Gaussian closure that converts accumulated variance into a closed-form expression for chain break probability using the complementary error function. This careful separation clarifies exactly which results depend on which assumptions, providing a transparent baseline against which real hardware behavior can be measured.

At the heart of the framework is a deceptively simple variance law. When errors on individual qubits and couplers are independent, the variance of the accumulated perturbation along a chain grows linearly with chain length. Since clique embeddings on Zephyr hardware produce chain lengths that grow approximately linearly with problem size, verified experimentally with a fitted slope of about 0.124 per logical variable and a coefficient of determination of 0.974, the model predicts precisely how embedding overhead translates into noise overhead. The framework further derives a design rule: to maintain a fixed target break probability, the intra-chain coupling strength, known as chain strength, should scale as the square root of chain length under the independent-noise assumption. This square-root law has long been treated as the reference for chain-strength tuning in practice.

To test these predictions, the team ran systematic experiments on the D-Wave Advantage2 system with Zephyr topology, embedding fully connected random QUBO instances ranging from 5 to 105 logical variables and sweeping annealing times from 5 to 200 microseconds and chain strengths from 0.1 to 2.5. Each data point aggregated 20,000 samples across 10 independent QPU replicates. The calibrated baseline model successfully captured the dominant growth trend of the chain break fraction with embedding size, and held-out validation splits showed that the fit was not merely memorizing calibration points. Fitted effective parameters remained within comparable ranges across annealing schedules, and longer annealing times systematically suppressed chain breaks while leaving the fundamental scaling with embedding size intact.

Then came the surprise. When the researchers extracted the empirical critical chain strength, the minimum chain strength needed to keep chain breaks below a given tolerance, and examined how it scales with average chain length, the data told a different story than the square-root law. The fitted power-law exponent ranged from 0.805 at a permissive 5 percent chain-break tolerance to 1.006 at a strict 1 percent tolerance, with all 95 percent bootstrap confidence intervals decisively excluding the independent-noise prediction of 0.5. In other words, real hardware accumulates errors faster than independent noise would allow, requiring chain strength to grow substantially more steeply, approaching linear growth under strict tolerance requirements, than the idealized baseline suggests.

To explain this deviation, the researchers proposed a correlated-variance extension, adding a term to the variance that scales superlinearly with chain length. When fitted directly to the critical chain strength data, this extension recovered a correlation exponent of approximately 1.8, in excellent agreement with the prediction that the exponent should equal twice the empirical chain-strength exponent. Model comparison using the Akaike information criterion favored the correlated extension over the independent baseline at all three tolerance thresholds. The physical origin of this correlated component remains an open question, with candidates including calibration drift, low-frequency flux noise, crosstalk between neighboring qubits, systematic offsets in programmed Hamiltonians, or non-Gaussian error tails, all of which are known to plague analog quantum processors.

The practical implications are significant for anyone deploying quantum annealing at scale. The framework provides quantitative guidance for embedding-aware chain-strength selection, replacing heuristic sweet-spot searches with principled scaling predictions. It also enables more realistic embedding-aware simulation of quantum annealers, allowing researchers to anticipate how reliability will degrade as problems grow before ever touching the hardware. Under time-matched budget comparisons, the quantum annealer delivered solution quality competitive with simulated annealing, the open-source PuLP solver, and the commercial Gurobi optimizer on the tested dense instances, matching certified Gurobi optima for problem sizes up to 60 logical variables, though the authors are careful to frame this as a controlled benchmark rather than a claim of quantum speedup.

Looking forward, the authors emphasize that their model is a first-order baseline, not a complete microscopic theory of annealing hardware. The fitted parameters should be interpreted as effective, schedule-dependent quantities rather than universal hardware constants, and broader validation across independently generated instances, alternative embeddings, gauge transformations, and future device generations remains essential work. Yet the message is clear and compelling: the noise that limits quantum annealing is not a collection of independent random insults but carries a correlated structure that grows with problem size, and any serious effort to scale quantum annealing toward commercially relevant optimization must confront that correlation head-on. By turning embedding overhead from an empirical nuisance into a quantifiable, predictable quantity, this work provides the analytical foundation on which the next generation of noise-aware quantum annealing algorithms and hardware-conscious mitigation strategies can be built.

Subject of Research: Embedding-aware noise modeling of quantum annealing on superconducting quantum annealers

Article Title: Embedding-aware noise modeling of quantum annealing

Article References: Jeong, S.-G., Cong Mai, D., Noh, D.-I., Pham, Q.-V., & Hwang, W.-J. (2026). Embedding-aware noise modeling of quantum annealing. Quantum Information Processing, 25(10), Article 315. https://doi.org/10.1007/s11128-026-05293-z

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05293-z

Keywords: quantum annealing, quantum computing, D-Wave, chain breaks, minor embedding, integrated control error, noise modeling, combinatorial optimization, Zephyr topology, quantum information, Embedding-aware, noise

The core idea behind the new scheme is simple to state but technically demanding to implement. In a second-order Ising model, every term in the energy function involves at most two spins, and reduction algorithms can look for groups of spins that are locked together — for example, spins that always take opposite values, or spins whose joint behavior is fully determined by a single effective variable. Such groups are called non-separable, because the spins inside them cannot be treated independently without changing the problem. Merging a non-separable group into one variable reduces the number of spins the solver must handle, often without any loss of accuracy. Until now, however, this machinery had been developed almost exclusively for quadratic Hamiltonians, leaving the many problems that naturally involve three-body, four-body, or even higher-order interactions without an equivalent tool.

Higher-order interactions are not an exotic curiosity; they are the rule rather than the exception in many formulations. When engineers express constraints such as all-different, cardinality limits, or logical clauses in pseudo-Boolean form, terms coupling three or more binary variables appear almost immediately. The authors of the new study generalize the notion of non-separable groups so that it applies to these arbitrary-order Hamiltonians. Their framework iteratively scans the Hamiltonian, detects clusters of spins that are constrained to move as a unit, and merges each cluster into a single variable, updating the energy function at every step so that the reduced problem remains an exact representation of the original. Because the procedure is iterative, reductions found early in the process can trigger further reductions later, compounding the savings in problem size.

Technically, the scheme must contend with subtleties that do not arise in the quadratic case. With higher-order terms, the energy landscape can contain degeneracies and symmetries that make it nontrivial to decide which spins are truly inseparable and which merely appear coupled at first glance. The researchers’ algorithm carefully tracks the structure of interactions across the hypergraph of the problem, where hyperedges connect groups of spins of any size. By examining how these hyperedges overlap, the algorithm identifies minimal constrained groups whose internal degrees of freedom can be fixed or absorbed. The team has released a reference implementation, called GH_minimal, as open-source code, allowing other researchers to integrate the reduction step into existing optimization pipelines.

To evaluate the method, the authors benchmarked it on synthetic hypergraphs as well as on real higher-order network datasets, including structures modeled on scientific collaboration networks and other empirically observed systems. These benchmarks matter because the performance of any reduction technique depends on the topology of the problem: sparse, tree-like structures tend to offer many opportunities for merging, while dense random structures may resist reduction. By testing across a spectrum of graph densities and higher-order interaction patterns, the team could characterize when their approach delivers large gains and when it offers more modest improvements. They also examined how well the reduced Hamiltonians integrate with downstream workflows, including order-reduction procedures that convert high-order terms into quadratic form and heuristic solvers such as simulated annealing, quantum annealing, and related quantum heuristic algorithms.

The results establish, for the first time, a systematic foundation for Hamiltonian reduction in higher-order Ising-like optimization. This matters practically because modern hardware solvers impose strict constraints. Quantum annealers such as D-Wave’s machines natively implement only quadratic interactions on a fixed chip topology, so any high-order term must be replaced by auxiliary spins and couplings — a gadget construction that inflates the qubit count. Every spin removed before this step translates directly into hardware resources saved. Similarly, coherent Ising machines and simulated bifurcation algorithms, which have recently been extended to handle higher-order cost functions directly, still benefit from smaller effective problem sizes, since solution quality and convergence speed typically degrade as problem dimension grows.

The new work connects to a rich history. Andrew Lucas’s influential 2014 catalog showed that a wide range of NP-hard problems admit Ising formulations, while Boros and Hammer’s theory of pseudo-Boolean optimization provided the mathematical language for handling nonlinear binary objectives. Preprocessing techniques for quadratic unconstrained binary optimization, such as those developed by Gueye and Michelon, and the FastHARE reduction algorithm for large-scale quantum annealing, demonstrated the practical value of shrinking Hamiltonians before solving. The contribution of Mao and colleagues is to lift this entire toolkit into the higher-order regime, where the combinatorial structure is richer and the potential savings are correspondingly larger. Their generalization of non-separable groups provides a unified lens: what previously required separate analyses for pairwise couplings now follows from a single framework that treats interactions of any order on equal footing.

The implications extend to some of the most active frontiers in quantum computing research. Quantum approximate optimization algorithms have recently been demonstrated on higher-order Ising models, including whole-chip QAOA experiments on heavy-hex lattices, and alternative formulations based on quantum Z2 lattice gauge theory have shown speedups for high-order unconstrained binary optimization. All of these approaches face the same fundamental bottleneck: the gap between the natural high-order formulation of a problem and the limited connectivity and interaction order of available hardware. A preprocessing layer that reduces the effective problem size before compilation could improve the resource efficiency of the entire pipeline, potentially allowing larger, more realistic optimization instances to be tackled on near-term quantum and quantum-inspired platforms.

There are, of course, limits to what any reduction scheme can achieve. The computational complexity of Ising spin glass problems guarantees that no preprocessing step will make hard instances universally easy — Barahona’s classic results showed that even planar spin glasses are computationally hard in general. The value of reduction lies instead in the substantial fraction of practical instances that contain exploitable structure. Real-world optimization problems, unlike worst-case adversarial constructions, frequently encode constraints that force groups of variables to move together, and it is precisely this structure that the new algorithm harvests. The benchmark results on synthetic and real higher-order networks suggest that the technique can yield meaningful reductions on problems with realistic topology, though the authors note that raw benchmark data cannot be fully released due to intellectual property and confidentiality policies.

The research was supported by the National Natural Science Foundation of China and the Chinese Academy of Sciences, and the code is freely available for the community to build upon. As quantum heuristic solvers mature — from superconducting annealers running thousands of qubits to optical parametric oscillator networks and digital simulated-bifurcation chips — the surrounding software ecosystem of problem formulation, reduction, and compilation becomes just as important as the hardware itself. By extending Hamiltonian reduction to arbitrary-order Ising-like models, this work fills a conspicuous gap in that ecosystem, and it may well become a standard preprocessing step in the workflows that carry combinatorial optimization problems onto the quantum and quantum-inspired machines of the coming decade.

Subject of Research: Hamiltonian reduction for arbitrary-order Ising-like optimization problems in quantum heuristic solvers

Article Title: A reduction scheme for general-order Ising-like Hamiltonians in quantum heuristic solvers

Article References: Mao, C., Mosharev, P., Wang, Y., & Yung, M.-H. (2026). A reduction scheme for general-order Ising-like Hamiltonians in quantum heuristic solvers. Quantum Information Processing, 25(10), Article 319. https://doi.org/10.1007/s11128-026-05339-2

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05339-2

Keywords: Ising model, combinatorial optimization, Hamiltonian reduction, quantum annealing, pseudo-Boolean optimization, quantum heuristic algorithms, higher-order interactions, non-separable groups, QUBO, quantum computing, simulated annealing, network datasets

Researchers Ernesto Acosta and Carlos Cano of the University of Granada, together with Guillermo Botella of Complutense University of Madrid, have reformulated the parameter-optimization problem of variational quantum algorithms as a Quadratic Unconstrained Binary Optimization problem, the native language of quantum annealers. Their work, published as volume 8, article 101 of the journal, demonstrates that the same machines designed to find low-energy configurations of combinatorial optimization problems can also serve as scalable optimizers for gate-based quantum models, opening a hybrid pathway that combines gate-model quantum computing, quantum annealing, and classical control in a single training loop.

The core insight behind the method is elegant. Variational quantum algorithms depend on a Hamiltonian expressed in terms of trainable rotation angles, and the exponential structure of the circuit’s unitary operator can be transformed, through a heuristic substitution that replaces imaginary phases with real-valued angles, into sums of real exponential terms. Those terms map naturally onto the quadratic binary cost functions that quantum annealers minimize. In other words, the mathematical structure of the ansatz itself provides the bridge: instead of estimating gradients with hundreds of circuit shots, the team encodes candidate parameter regions directly into QUBO form and asks the annealer to find the configuration with the lowest energy.

Because continuous rotation angles must be represented with discrete binary variables, the researchers developed a recursive refinement strategy that progressively narrows the search space. The parameter range is partitioned into segments, validation points are sampled within each partition, and the QUBO problem is solved to identify the most promising regions. The angle range is then rescaled — in their worked example, halved at each level — and centered on the best solution found so far, and the process repeats. This coarse-to-fine scheme approximates high-quality continuous solutions with a small number of discrete QUBO solves, trading resolution for tractability in a controlled way.

A key strength of the framework is its adaptability. The method exposes a rich set of configurable metaheuristic parameters, including the number of training levels, the number of partitions per angle, and the number of validation points per partition. These knobs allow practitioners to tune the balance between solution quality and computational cost according to the problem at hand and the computational resources available. The authors also integrate an adaptive metaheuristic optimization scheme rather than a fixed search rule, making the training procedure generalizable to arbitrary Hamiltonians rather than being tied to a specific circuit family or ansatz design.

To test the approach, the team benchmarked their adiabatic training scheme on publicly available classification datasets, including the Iris, Heart Disease, and Diabetes datasets from the UCI machine learning repository. The experimental evaluations show that the method achieves accuracy comparable to, and in some configurations better than, established classical and evolutionary optimizers, while significantly reducing computational overhead. The authors’ hyperparameter exploration reveals that the highest performance efficiency — accuracy gained per unit of training time — tends to occur at lower numbers of levels and partitions, meaning that modest configurations already capture most of the benefit before costs escalate.

The implications extend beyond raw benchmark numbers. Gradient-based training on near-term quantum hardware is plagued by noisy gradient estimates, since every partial derivative must be inferred from finite numbers of noisy circuit measurements. Meanwhile, barren plateaus, whose origins have been linked both to deep random circuits and to noise accumulation, can render gradients exponentially vanishing and classical optimizers effectively blind. A gradient-free, annealer-driven optimizer sidesteps both pathologies: it never estimates a derivative, and its search is driven by the energy landscape of the QUBO encoding rather than by local slope information that may not exist in any meaningful sense.

The study builds on a growing body of work connecting annealing-style hardware to machine learning. Previous research has shown that QUBO formulations can be used to train classical machine learning models, that Ising machines can serve as training engines for standard neural networks, and that universal adiabatic quantum computers can be harnessed for neural network training. An earlier preprint by the same team had already explored adiabatic training for variational quantum algorithms; the new journal publication consolidates that line of inquiry into a general, configurable framework with systematic experimental validation, code released through a public repository, and full experimental detail across multiple datasets and hyperparameter regimes.

What emerges is a compelling vision of near-term quantum computing as a genuinely hybrid enterprise. In the architecture the Granada and Madrid researchers describe, a gate-based quantum processor executes the variational circuit, a quantum annealer solves the training problem at each iteration, and a classical system orchestrates the recursive refinement and manages the workflow. Each platform contributes what it does best, and none has to shoulder the full burden alone. As annealing hardware scales and QUBO encodings grow more expressive, this division of labor could become a practical route to training quantum models that classical optimizers struggle to handle.

Challenges certainly remain. The heuristic substitution that enables real-valued QUBO encoding sacrifices strict unitarity, and the discretization of continuous parameters means solution precision depends on the depth of the refinement schedule. The authors’ own data show that pushing training levels and partitions higher does not always pay off, underscoring the need for careful configuration. Still, the demonstration that quantum annealers can train variational quantum algorithms with competitive quality and lower overhead marks a meaningful step. For a field hunting for any advantage in the noisy intermediate-scale era, recruiting one quantum machine to train another may prove to be one of the more inventive entries in the playbook.

Subject of Research: A QUBO-based method that uses quantum annealers to train variational quantum algorithms without gradients.

Article Title: QUBO-based training for VQAs on quantum annealers

Article References: Acosta, E., Botella, G., & Cano, C. (2026). QUBO-based training for VQAs on quantum annealers. Quantum Machine Intelligence, 8(2), Article 101. https://doi.org/10.1007/s42484-026-00441-1

Image Credits: AI Generated

DOI: 10.1007/s42484-026-00441-1

Keywords: quantum computing, variational quantum algorithms, quantum annealing, QUBO, optimization, barren plateaus, quantum machine learning, hybrid quantum computing, metaheuristics, NISQ era, gradient-free training, Hamiltonians