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 method, which the authors call Quantum Feature Selection (QFS), exploits analog quantum simulation with arrays of neutral atoms excited to Rydberg states. In these processors, individual atoms are trapped by optical tweezers and act as two-level systems, with a ground state standing for zero and a highly excited Rydberg state standing for one. Crucially, atoms in Rydberg states interact through van der Waals forces that fall off with the sixth power of distance. That fixed physical law, normally a constraint, becomes the computational resource: by rearranging the atoms in a plane, the researchers can program the interaction structure of an optimization problem directly into the geometry of the array.

The mapping works in two steps. First, the relevance of each feature, quantified by its mutual information with the target variable, is encoded as a site-dependent local detuning, a laser parameter that makes certain atoms energetically more likely to be excited. Features with high relevance receive stronger detunings, biasing their atoms toward the Rydberg state. Second, pairwise redundancy between features, measured as mutual information between the features themselves, is translated into spatial distance. The team applies an inverse-sixth-root transform to the normalized redundancy matrix, so that highly redundant pairs of features are placed close together, where their strong interaction penalizes the configuration in which both are simultaneously selected. Weakly correlated features are pushed apart, where they behave nearly independently.

Because a general redundancy matrix cannot usually be embedded exactly in two dimensions, the researchers use multidimensional scaling (MDS), a classical technique that finds atomic positions best approximating the target distances. The resulting layout is then rescaled so that the closest atom pairs sit near the Rydberg blockade radius, the critical distance at which joint excitation is suppressed. An annealing protocol follows: the global Rabi frequency ramps up and down smoothly, a global detuning initializes the system, and the local detunings encoding relevance switch on in the latter half of the evolution. The dynamics steer the array toward low-energy configurations that balance relevance against redundancy, and repeated projective measurements yield bitstrings, each one a candidate feature subset.

The post-processing stage is deliberately conservative. Only the lowest-energy ten percent of measured bitstrings are retained, and features are ranked by their average Rydberg excitation density across this ensemble. A greedy construction then accepts a feature only if its normalized redundancy with all previously accepted features falls below a threshold, set to 0.7 in the reference experiments, ensuring that the final subset is not merely relevant but internally non-redundant. When the threshold rule cannot fill a requested subset size, the remaining slots are filled by density ranking, a completion step the authors flag explicitly so that large subsets are not misread as strict independent sets.

Evaluated in simulation on Amazon Braket, using a noise-free Schrödinger equation solver and 10,000 measurement shots per run, QFS was tested on three real-world binary classification benchmarks: Adult Income, Bank Marketing, and Telco Churn. The comparison set was demanding, including mutual-information ranking, Random Forest and XGBoost feature importances, L1-regularized logistic selection, and two classical optimizers of exactly the same relevance-redundancy objective, an exhaustive search called Exact-Q and a greedy heuristic called Greedy-Q. Across ten stratified train/test splits with a Random Forest downstream classifier, QFS achieved competitive, dataset-dependent performance. On Adult Income, QFS, Exact-Q, Greedy-Q, and XGBoost importance all landed within roughly three ten-thousandths of a point in mean AUC, while on Telco Churn every method was essentially tied within split-to-split variability.

The authors are candid about where the method falls short. On Bank Marketing, tree-based importance rankings achieved slightly higher mean AUC than the quantum-derived subsets, and QFS sometimes reached its best performance at larger subset sizes, a consequence of the evaluation protocol rather than an intrinsic preference of the sampler for compact sets. The geometric embedding itself introduces moderate distortion: the mean relative reconstruction error of the MDS layouts ranged from about 0.22 to 0.26 across the three datasets. Yet a stability analysis over 50 random embedding seeds showed that this distortion does not produce arbitrary subsets. Bank Marketing was invariant across seeds for subset sizes up to five, and Adult Income stabilized for sizes of three or more, with mean Jaccard similarity above 0.96.

Robustness tests of the classical extraction stage added further reassurance. Varying the retained low-energy fraction from 5 to 30 percent shifted the best AUC only marginally on a reference split, and injecting controlled noise into the measured excitation densities, mimicking readout fluctuations or calibration drift, left predictive performance largely intact even when the exact identity of the selected subsets drifted at small cardinalities. The authors interpret this as evidence that the pipeline’s predictive output is more robust than its exact subset choices, though they stress that the simulations remain noise-free: decoherence, atom loss, finite-temperature atomic motion, readout errors, and detuning miscalibration were all excluded and must be addressed in future device-level studies.

What makes the work notable is less a claim of quantum advantage than the physical interpretability of the encoding. Every term of the statistical objective has a hardware counterpart: relevance lives in local detuning amplitudes, redundancy in the sixth-power interaction law, and the relevance-redundancy trade-off emerges from the relative energy scales of the two. The main contribution, the authors write, is a neutral-atom implementation of a relevance-redundancy feature-selection objective together with a quantitative characterization of its geometric and post-processing approximations. Scaling up will require one atom per feature, site-resolved detuning control, and embedding strategies, such as sparsification or community-aware layouts, that preserve the dominant redundancy structure without forcing every pairwise relation into a single plane.

The study points toward a broader program in which analog quantum processors tackle the combinatorial core of data preprocessing rather than the models themselves. Future work, the team says, should focus on experimental validation on real neutral-atom devices, noisy analog Hamiltonian simulation, improved weighted and graph-aware embeddings, dynamic detuning optimization, and larger benchmarks where the scaling of analog sampling can be measured directly. Extensions to multiclass problems, regression, and unsupervised settings are also on the table. For now, QFS stands as a proof of concept that a laser-driven array of atoms, arranged according to the statistical anatomy of a dataset, can select features as well as established classical heuristics, and that the path from information theory to atomic physics can be traversed without losing the meaning along the way.

Subject of Research: Analog quantum feature selection using neutral-atom Rydberg processors

Article Title: Analog quantum feature selection with neutral-atom quantum processors

Article References: Orquín-Marqués, J. J., Flores-Garrigós, C., Gomez Cadavid, A., Simen, A., Solano, E., Hegade, N. N., Martín-Guerrero, J. D., & Vives-Gilabert, Y. (2026). Analog quantum feature selection with neutral-atom quantum processors. Quantum Machine Intelligence, 8(2), Article 108. https://doi.org/10.1007/s42484-026-00456-8

Image Credits: AI Generated

DOI: 10.1007/s42484-026-00456-8

Keywords: neutral atoms, Rydberg arrays, quantum feature selection, analog quantum simulation, mutual information, quantum optimization, machine learning, multidimensional scaling, Rydberg blockade, QUBO, Amazon Braket, Quantum Machine Intelligence

Omnisolver itself is a plugin-based framework that provides a uniform Python and command-line interface to QUBO and Ising solvers, integrated with the widely used dimod ecosystem. Each solver arrives as an independent package that registers itself through an entry point and a small specification file, from which the framework automatically derives the command-line interface and input-output handling. Solvers can therefore be swapped without touching the surrounding workflow. The exhaustive-search plugin discussed in the update exposes two sampler classes: a single-node, single-GPU sampler and a distributed multi-GPU sampler built on the Ray framework, which fixes a small number of variables, enumerates all assignments of those frozen spins, and dispatches each resulting subproblem as an independent Ray task to a GPU worker. The performance-critical kernels are written in CUDA and C++ and exposed to Python through Cython.

The benchmark hardware was a cluster of eight NVIDIA H100 GPUs with 96 gigabytes of memory each, spread across two nodes connected by standard Ethernet. On dense random Ising instances with couplings drawn uniformly from the interval [-1, 1], the single-GPU sampler certified ground states up to N=54 spins, with measured runtimes growing by a factor of almost exactly two per added variable, the expected signature of the O(2^N) enumeration. The distributed sampler, using three fixed variables so that eight subproblems map one-to-one onto the eight GPUs, pushed the certified frontier to N=60, a run that took roughly 3.15 days and landed within 0.3 percent of the doubling-rule extrapolation. Projected to the same size, the single-GPU path would have needed about 25 days, confirming nearly ideal linear scaling.

That scaling was quantified explicitly. Below a crossover between N=40 and N=42, splitting the work across eight GPUs is actually slower than using one, because kernel-launch and Ray-scheduling overheads dominate the tiny subproblems. Efficiency then climbs steeply, from 43 percent at N=44 through 75 percent at N=46 and 93 percent at N=48, and saturates at the ideal eightfold speedup from N=52 onward. A complementary experiment held the problem fixed and varied the machine: strong scaling at N=50 reached 2.00 times on two GPUs, 3.99 on four and 7.86 on eight, while weak scaling kept wall-clock time flat to within 0.2 percent as the instance grew from 50 to 53 spins alongside the device count. Notably, crossing the node boundary between the two hosts produced no measurable slowdown at all, consistent with the search phase involving no inter-worker communication whatsoever.

The most delicate engineering problem was numerical. The fast ground-state path operates in single-precision float32 arithmetic, which halves the size of the energy buffers resident on the GPU and effectively buys working-set size rather than raw speed, since the enumeration turns out to be limited by memory traffic and integer bookkeeping rather than floating-point throughput. But single precision accumulates roundoff drift over long incremental updates. The update counters this with compensated incremental updates, periodic re-anchoring of the energy from scratch, and a final refresh of the best-state buffer, all activated automatically whenever the kernel sees at least 40 variables. After these fixes, the deviation between the reported energies and independent float64 recomputation stayed below 10^-5 across all twenty measured runs, corresponding to about eight significant digits. Interestingly, repeating the search in full double precision showed it to be only about 2 percent slower, and both precisions returned identical configurations at every tested size.

Verification is layered and deliberately paranoid. Every returned configuration is re-evaluated from scratch in float64 on the host. On the eight sizes where both samplers are feasible, the single-GPU and distributed routes, which sweep the configuration space in genuinely different patterns, returned bit-identical ground states with energies agreeing to within 8.2 times 10^-6. Small instances were cross-checked against dimod’s exact CPU solver, and the QUBO and Ising code paths, which coincide up to a host-side transformation, produced runtimes matching to within 0.2 percent. The authors are careful about terminology: because the fast path uses floating-point arithmetic, their minima are described as empirically cross-validated rather than formal certificates for arbitrary real-valued coefficients.

The real payoff comes when the certified optima are turned against the heuristics they are meant to judge. The team distributed a discrete simulated-bifurcation solver, a relative of the Simulated Bifurcation Machine that has served as the classical benchmark in recent studies of quantum runtime advantage, and ran it against every stored brute-force result from N=38 to N=60. On the dense random family it matched the certified optimum on all twenty instances, with bit-identical configurations and energies agreeing to 6 times 10^-14 after independent recomputation. A second comparison using simulated annealing across twenty instances drawn from four algorithmically distinct coupling families, uniform, bimodal, Gaussian and sparse, likewise reached the certified optimum on every one, with the rare disagreements traced to genuine ground-state degeneracy rather than optimization error.

The story changes dramatically on a harsher landscape. On the Wishart ensemble, a fully connected zero-field family with a tunable ruggedness parameter, the simulated-bifurcation heuristic underwent a sharp collapse. It recovered the certified optimum on every instance at ruggedness values of 0.45 and above, but on only 3 of 20 instances at 0.3, 1 of 20 at 0.25, and none at all at 0.2, terminating at configurations up to 0.6 percent above optima that were known only because brute force had found them. These are wrong, near-optimal answers that nothing short of exhaustive certification can expose. The cost asymmetry is striking: certification took about 2 seconds per instance on a single H100, less than one heuristic run, and time-to-solution analysis showed that relying on the heuristic was between roughly 600 and 1800 times slower than simply certifying the answer directly on the hardest instances.

The authors are candid about limits. Exhaustive enumeration remains exponential no matter how many GPUs are thrown at it: a back-of-the-envelope projection to 16,384 GPUs, a partition within reach of leadership-class supercomputers, would cut the N=60 certification from about 3.15 days to roughly 2.2 minutes and place the practical ceiling for a one-month budget near N=74, with the 64-bit configuration word imposing a hard bound at N=64 plus the number of fixed variables. A hypothetical fourfold increase to 65,536 GPUs would buy only two additional spins, a stark illustration of the tyranny of exponential complexity. The controller-side merge cost was measured directly up to 65,536 subproblems, showing that the merge arithmetic itself is not the bottleneck, and a hierarchical tree merge is available and automatically engaged beyond 64 subproblems.

The intended role of the software is therefore not to solve application-scale problems but to serve as a certification backend for research pipelines: a source of ground-truth optima against which heuristics, annealers and NISQ-era quantum algorithms can be calibrated, particularly for metrics such as the optimality gap that depend explicitly on the true ground-state energy. The entire reproduction package, including instances, raw results, benchmark drivers and even the heuristic solver, ships with the article, and much of the verification can be re-run on a laptop without any GPU. In a field where claims of quantum advantage increasingly rest on the quality of classical reference answers, a dependable, open, GPU-accelerated arbiter of global optima is a quietly essential piece of infrastructure, and this update makes it one that anyone can install, inspect and trust.

Subject of Research: A distributed GPU exhaustive-search backend for certifying ground states of Ising spin-glass and QUBO optimization problems

Article Title: Omnisolver: An extensible interface to Ising spin–glass and QUBO solvers – a numerically stabilized, distributed GPU exhaustive-search plugin

Article References: Jałowiecki, K., Pawłowski, J., Gardas, B., & Pawela, Ł. (2026). Omnisolver: An extensible interface to Ising spin–glass and QUBO solvers – a numerically stabilized, distributed GPU exhaustive-search plugin. SoftwareX, 36, Article 103031. https://doi.org/10.1016/j.softx.2026.103031

Image Credits: AI Generated

DOI: 10.1016/j.softx.2026.103031

Keywords: Ising spin glass, QUBO, combinatorial optimization, GPU computing, CUDA, exhaustive search, simulated bifurcation, quantum advantage benchmarking, Ray distributed computing, ground-state certification, H100 GPUs, Omnisolver

The Quantum Approximate Optimization Algorithm, known as QAOA, has become the workhorse of near-term quantum optimization. It works as a hybrid quantum-classical dance: a quantum processor prepares a candidate state, a classical optimizer tunes the circuit’s angles, and the two repeat until a good solution emerges. The catch is that QAOA’s performance hinges on a parameter called depth, denoted p, which counts how many alternating layers of problem and mixing operations the circuit contains. Deeper circuits generally produce better answers, but they demand more two-qubit gates, more classical optimization effort, and more tolerance of hardware noise. Until now, practitioners had to guess this depth in advance, with no reliable guidance.

Guessing wrong is costly in both directions. A circuit that is too shallow is underparameterized and cannot represent a high-quality solution, particularly as problems grow denser and more constrained. A circuit that is too deep burns scarce quantum resources on layers that add nothing, a serious problem on today’s noisy intermediate-scale quantum devices where every additional two-qubit gate multiplies the chance of error. The research team, led by Rakesh Saini with co-authors Nora Mohamed, Saif Al-Kuwari and Ahmed Farouk, set out to eliminate this guesswork entirely by letting the algorithm discover its own depth.

DDQAOA begins modestly, with a single QAOA layer, and monitors the optimization as it proceeds. The algorithm tracks the expectation value of the problem’s cost function, the quantity that measures how good the current quantum state is. When improvement stalls, the algorithm does not give up; instead it concludes that the current depth has exhausted its expressive capacity and adds one more layer. Two complementary convergence checks guard against false alarms. The first detects a plateau by verifying that the best energy has not improved beyond a small tolerance over a patience window of fifty iterations. The second examines the variance of recent energy values, distinguishing genuine convergence from mere oscillations around a local minimum. Only when one of these conditions is satisfied does the depth increase.

Crucially, when a new layer is added, the algorithm does not start its optimization from scratch. Building on the INTERP parameter-transfer protocol developed by Zhou and colleagues in 2020, DDQAOA interpolates the optimized parameters from the shallower circuit to seed the deeper one. For the transition from one to two layers, the team applies transfer coefficients of 1.2 for the cost angle gamma and 0.8 for the mixer angle beta, values motivated by the structure of adiabatic quantum evolution. For deeper transitions, the method constructs smooth interpolants, switching from linear interpolation to cubic splines once four or more layers are available. This warm-starting places each new optimization inside the basin of attraction of a good solution, dramatically accelerating convergence.

To test their method rigorously, the researchers turned to the Constrained Shortest Path Problem, an NP-hard challenge that asks for the cheapest route between two points in a network while respecting a limit on resource consumption, such as fuel or time. This problem is far more than an academic curiosity: it appears as a building block in air-cargo route planning, flight scheduling, airline crew pairing, and aircraft tail assignment. The team encoded the problem as a Quadratic Unconstrained Binary Optimization formulation, folding the source, target, flow-conservation and resource constraints into penalty terms, and then converted the result into an Ising Hamiltonian whose ground state encodes the optimal path.

The benchmark was substantial: 100 randomly generated problem instances each at the 10-qubit and 16-qubit scales, plus 20 additional instances requiring 22 qubits on complete five-node graphs. Against fixed-depth QAOA baselines at depths 3, 5, 10 and 15, DDQAOA delivered striking results. At 10 and 16 qubits it outperformed every fixed-depth baseline, achieving median approximation ratios of roughly 0.97 and 0.99 respectively, with the smallest variability of any method. At 22 qubits it matched the best deep circuits, statistically indistinguishable from depth-10 QAOA and within 0.1 percent of depth-15, while attaining the highest median success probability of all methods tested.

The resource savings are where the approach truly shines. Because CNOT gates are the dominant source of noise on real quantum hardware, cumulative two-qubit gate usage is the currency of practical quantum optimization. Standard QAOA at depth 15, which achieved results close to DDQAOA, consumed 217 percent, 159.3 percent and 315 percent more CNOT gates at the 10-, 16- and 22-qubit scales respectively. Measured per circuit, DDQAOA used 3.17, 2.59 and 4.15 times fewer cumulative CNOTs than the deepest baseline across the three problem sizes. The algorithm’s gate count grows stepwise, from a single layer’s worth of gates up to the depth-10 equivalent, allocating quantum resources only as the optimization landscape requires.

Perhaps the most scientifically satisfying finding concerns the parameters themselves. In fixed-depth QAOA, the optimized gamma and beta angles typically show no discernible pattern across layers. DDQAOA, by contrast, consistently produced monotonically increasing gamma values and beta values converging toward zero, exactly the structure predicted by adiabatic theory, in which the cost Hamiltonian’s influence grows and the mixer’s fades as the system approaches the ground state. Across 100 diverse problem instances, with the classical optimizer free to deviate after initialization, this adiabatic signature emerged reliably, suggesting the interpolation strategy guides the search toward structured, physically meaningful solution manifolds rather than random corners of parameter space.

The authors are careful to note the limits of the current work. All experiments ran on classical simulators using the PennyLane framework with the Adam optimizer, and the success probabilities, while far above the random-guessing baseline of one over two to the power N, remain small in absolute terms because sampling the exact ground state among exponentially many bitstrings is an inherently stringent criterion. Future work will target larger instances and validation on real quantum hardware, where noise resilience and connectivity constraints will provide the ultimate test. Still, by removing the need to choose circuit depth in advance while matching or beating hand-tuned deep circuits at a fraction of the gate cost, DDQAOA offers a practical, NISQ-aware route to applying quantum optimization to genuinely constrained industrial problems, and a compelling demonstration that sometimes the best way to go deep is to earn each layer.

Subject of Research: A dynamic-depth variant of the quantum approximate optimization algorithm for solving the NP-hard constrained shortest path problem on near-term quantum devices.

Article Title: Dynamic depth quantum approximate optimization algorithm for solving constrained shortest path problem

Article References: Saini, R., Mohamed, N., Al-Kuwari, S., & Farouk, A. (2026). Dynamic depth quantum approximate optimization algorithm for solving constrained shortest path problem. Quantum Machine Intelligence, 8(2), Article 104. https://doi.org/10.1007/s42484-026-00442-0

Image Credits: AI Generated

DOI: 10.1007/s42484-026-00442-0

Keywords: quantum computing, QAOA, DDQAOA, constrained shortest path problem, combinatorial optimization, NISQ devices, circuit depth, parameter transfer, Ising Hamiltonian, QUBO, adiabatic evolution, CNOT gates

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