PaperlayerView exact revision on arXiv

Source-linked AI summary

Do Quantum AIs Dream in Paths? Path-Integral Slow Thinking through Grover Interference

Xiansheng Cai, Xiu-Hao Deng, Kun Chen

arXiv:2609.05842v1quant-phcs.AIcs.LG

TL;DR

Classical trajectory-level training can favor increasing single-attempt success without protecting alternative reasoning paths, while coherence alone does not select useful interference. The paper trains quantum reasoning dynamics for Grover-amplified inference, whose finite-depth optimum occurs at an interior success probability; its reported scaling comparison is a controlled finite-task result.

  • Problem

    Outcome-only classical repetition remains monotonic in single-attempt success, while coherence alone does not explain how to preserve and exploit diverse reasoning trajectories.

    Source: §I · p. 2, §IV · p. 6

  • Method

    The policy is trained on the success probability after a chosen number of Grover rounds, using verifier reflections to act on its prepared trajectory distribution.

    Source: §I · p. 3, §VII · p. 11

  • Results

    The fitted scaling exponents differ by 1.116 (95% bootstrap interval [0.979, 1.306]), a controlled finite-task separation rather than an asymptotic query-complexity result.

    Source: REPLICATION AND SCALING · p. 27

  • Takeaways & Limitations

    The finite-round interior target can preserve multiple successful routes while increasing the probability of obtaining the correct answer.

    Source: §VII · p. 11

    Training and full coherent-circuit evaluation use exact classical calculation, which does not establish quantum hardware advantage and omits physical implementation resources.

    Source: SCOPE, PHYSICAL RESOURCES, AND… · p. 27

Abstract

from arXiv · show

Reinforcement learning with verifiable rewards enables large language models to think slowly, but the same training can induce policy collapse: probability concentrates onto a few successful trajectories and exploratory diversity erodes. We ask whether quantum AI can realize slow thinking differently. We formulate slow thinking as coherent dynamics over reasoning trajectories, a discrete path integral in which action sequences coexist in superposition and recombine before measurement. In our trainable realization, an exact verifier partitions the ensemble into collective accepted and rejected components that interfere under Grover amplitude amplification. A finite Grover evolution is maximized when the pre-amplification success probability lies at an analytically determined value below one, so inference itself defines an interior training target and removes the monotonic pressure toward unit success. In exact statevector simulations of a 2x3 sliding puzzle, Grover training reaches accuracy 0.95 on a 32-question training set at one round, against 0.73 for the strongest classical control. On held-out questions specialization has a cost: an untrained uniform policy read out through the same amplification remains the strongest reference on this solution-dense benchmark, and quantum training preserves far more held-out accuracy than classical training - at four rounds with matched circuit applications the two quantum models reach 3.2 and 3.9 times the strongest classical controls. The number of training questions supported by fixed-size policies trained at each amplification budget also grows faster with the budget than with matched classical repetition. These results establish a Grover-based realization of path-integral slow thinking: the interior target preserves exploratory path diversity, and ensemble-level interference converts it into verified performance.

I. INTRODUCTION

The paper asks whether quantum dynamics can support slow reasoning without the loss of trajectory diversity associated with classical trajectory-level reinforcement learning. It proposes coherent path-based inference with Grover amplification and reports stronger held-out performance than matched classical controls on a sliding-puzzle benchmark.

I. INTRODUCTION

  • The paper frames quantum slow thinking as a path integral in which multiple reasoning histories coexist in superposition and can recombine before measurement.Unlike classical sampling, coherent dynamics can combine trajectory amplitudes, although coherence alone does not select useful interference or ensure path diversity.Source: §I · pp. 1–2
  • Classical trajectory-level RLVR can reinforce an initially favored strategy until competing routes disappear, leaving less diversity for exploring alternative solutions.The paper identifies this concentration as policy collapse and notes that additional test-time computation may then repeat variations of essentially the same path.Source: §I · p. 2
  • The trainable quantum models prepare coherent ensembles of action sequences, then use a verifier and Grover reflections to recombine accepted and rejected components.The objective is the success probability after the chosen number of rounds, so the policy is trained for the coherent inference it will undergo.Source: §I · pp. 2–3
  • Finite Grover evolution peaks at an analytically determined pre-amplification success probability below one, providing an interior training target that can retain multiple reasoning paths.This changes the training pressure from monotonically pushing each problem toward unit success.Source: §I · p. 3
  • At four Grover rounds, QuCoNet and the neural quantum AI model achieve respectively 3.2 and 3.9 times the held-out accuracy of the strongest classical controls.The comparison matches reasoning-circuit applications; held-out accuracy is mean success probability on question–answer pairs not used for training.Source: §I · p. 3

II. CLASSICAL AND QUANTUM REASONING TRAJECTORIES

The section contrasts statistical trajectories with quantum path amplitudes. Classical runs sample histories, whereas quantum dynamics can coherently recombine histories before measurement.

II. CLASSICAL AND QUANTUM REASONING TRAJECTORIES

  • An autoregressive language model assigns a trajectory probability through conditional probabilities of successive reasoning states given the input and preceding prefix.The conditionals may depend on the complete prefix, so the formulation makes no Markov approximation.Source: §II · p. 4
  • A quantum reasoning circuit assigns complex transition amplitudes to successive states, whose product forms the amplitude for a complete trajectory.The trajectory is represented as a basis state, and answer probabilities are obtained using a measurement associated with the answer.Source: §II · p. 4
  • Off-diagonal terms permit interference when histories are coherently recombined, but the Grover realization keeps individual trajectories orthogonal with fresh which-path records.Its verifier and prepared-state reflections act on collective accepted and rejected components; pairwise interference between recorded paths does not contribute to reported success probabilities or path statistics.Source: §II · p. 4

III. SLIDING-PUZZLE BENCHMARK AND QUANTUM REASONING MODELS

The paper uses a finite configuration network to study verified multistep reasoning, then implements its routing dynamics in table-based and neural quantum models. The benchmark supplies exact transitions and an exact verifier.

III. SLIDING-PUZZLE BENCHMARK AND QUANTUM REASONING MODELS

  • The benchmark is a 3-regular graph of 120 reachable configurations, with three reversible actions: V, L, and R.Each complete L or R cycle counts as one action and one graph edge.Source: §III · p. 4, Figure 2 · p. 5

III. SLIDING- PUZZLE BENCHMARK AND QUANTUM REASONING MODELS

  • Each question supplies start and target configurations 3–6 graph steps apart; an exact verifier accepts first arrival at the target within eight actions.Training and held-out sets are disjoint, with 64 question–answer pairs reserved for held-out evaluation in each pool.Source: §III · p. 4
  • The sliding puzzle isolates branching, delayed verification, and potentially multiple trajectories to the same outcome while keeping transitions and verification exact.The paper presents this network as a controlled benchmark for measuring trajectory exploration, policy collapse, and interference.Source: §III · p. 4
  • QuCoNet uses a trainable local unitary coin at each node, while the neural quantum model generates the coin from the question and preceding action sequence.Both propagate amplitudes over action sequences, and each move is recorded in a fresh which-path register.Source: §III · p. 5

IV. RLVR AND POLICY COLLAPSE

The section explains why classical RLVR can improve single-attempt success while eroding alternative reasoning routes. Its objective rewards total accepted probability but does not protect how that probability is distributed among successful paths.

IV. RLVR AND POLICY COLLAPSE

  • The classical training objective uses the total accepted trajectory weight as expected reward, and exact calculations evaluate each question’s success probability and gradient.The setup applies to classical CoNet, QuCoNet, and the neural quantum model; the comparison distinguishes their objectives and inference dynamics rather than gradient-estimation accuracy.Source: §IV · p. 6
  • GRPO compares terminal rewards among sampled trajectories for the same question and uses the group mean as a reference instead of a separately trained critic.In this finite trajectory space, the paper computes the complete sum and its gradient exactly.Source: §IV · p. 6
  • Because the loss improves whenever success probability rises and is indifferent to its distribution across successful routes, it pushes toward unit success without protecting alternatives.The objective supplies no counterpressure when one successful route gains probability at the expense of others.Source: §IV · p. 6
  • With finite policy capacity, easier reinforcement of an already successful route can reduce sampling of competing routes and erode diversity for unfamiliar problems.The paper states that collapse is not mathematically forced, but that the objective lacks a mechanism to protect alternative paths; exact evaluation also shows the benchmark effect is not solely a finite-sampling artifact.Source: §IV · p. 6

V. QUANTUM RLVR THROUGH GROVER INTERFERENCE

The paper treats slow reasoning as coherent evolution over complete trajectories, with a verifier and Grover reflections recombining accepted and rejected components before measurement. Because finite Grover evolution has an interior optimum, inference also defines the probability target used in training.

V. QUANTUM RLVR THROUGH GROVER INTERFERENCE

  • Quantum slow thinking uses a fixed-size learned policy to prepare reasoning trajectories, then spends additional inference-time Grover rounds rather than adding trainable parameters.The same finite-depth success map is used for both inference and training.Source: §V · p. 7
  • Earlier quantum reinforcement-learning work used quantum deliberation, trainable policies, or amplitude amplification, while this approach couples coherent reasoning inference to the learned trajectory distribution.The mechanism specifically relies on Grover interference, coherent access to the reasoning circuit and its inverse, and a verifier that marks accepted trajectories without measurement; hardware implementation also requires coherent verification and gradient estimation.Source: §V · pp. 7–8
  • A coherent verifier marks accepted trajectories without measurement, and Grover reflections rotate collective accepted and rejected components before final readout.The trajectories remain orthogonal records; interference acts between the two collective sectors, not pairwise between individual paths.Source: §V · p. 7
  • The finite-depth Grover success map reaches its first maximum at a nonunit initial success probability, reversing the training gradient above that interior target.Below the target, increasing the initial probability improves final success; just above it, Grover rotation overshoots.Source: §V · p. 7
  • Training optimizes the success probability after coherent inference, so each question is trained toward the Grover target rather than toward unit pre-amplification success.The adaptive protocol stops at the Grover round nearest the first maximum when limited to at most q circuit applications.Source: §V · pp. 7–8

A. Grover-based training preserves path diversity

Grover-based training concentrates pre-amplification success probabilities near the analytically predicted interior target, whereas single-attempt training drives probabilities toward extremes. This target is accompanied by greater successful-path diversity, although fixed-depth inference can overshoot on some training questions.

A. Grover-based training preserves path diversity

  • Grover-objective training peaked near the predicted p∗(q) for every tested depth, while single-attempt training made question probabilities bimodal near zero or one.The extreme-probability behavior also occurred in CoNet and both quantum models, so quantum parameterization alone did not prevent collapse.Source: A. Grover-based training preserves… · p. 8
  • Adaptive inference produced mean training accuracy above 0.95 at every tested q, but fixed-depth evaluation reduced training accuracy to 0.77 at q = 9.Questions with larger initial success probabilities reached their maxima earlier and were rotated past them under fixed depth.Source: A. Grover-based training preserves… · p. 8
  • At B = 32, the accuracy-weighted effective number of successful paths rose from 1.6 at q = 3 to 5.7 at q = 9 under Grover-based training.The inverse participation ratio measures path diversity, not pairwise interference between recorded paths.Source: A. Grover-based training preserves… · p. 8

B. Performance at matched inference cost

At matched circuit-application counts, the quantum models generally outperform the strongest classical control on held-out questions at larger inference depths. Held-out accuracy also increases with the depth used during training, while the q = 3 training comparison favors Grover-based training across tested dataset sizes.

B. Performance at matched inference cost

  • The matched-cost comparison counts q reasoning-circuit applications for both methods: Grover inference uses one preparation plus two applications per round, while classical inference samples q trajectories.Verifier evaluations and reflections are reported separately, and the comparison excludes hardware costs such as reversible computation, trajectory memory, and fault tolerance.Source: B. Performance at matched inference cost · p. 8
  • When evaluated at a common depth, held-out accuracy still increased with the q value used during training, indicating the trend was not only an effect of deeper evaluation circuits.Each Table I entry used the same fixed q for every question, without using pre-amplification success probability to choose when to stop.Source: B. Performance at matched inference cost · p. 8
  • At q = 3, Grover-based training exceeded the best of 27 classical controls in mean training accuracy at every tested training-set size.Figure 4(c) plots mean training accuracy against the number of question–answer pairs B.Source: Figure 4 · p. 9
  • At B = 32 and q = 9, QuCoNet reached held-out accuracy 0.407 and the neural quantum AI model 0.496, versus 0.126 for the strongest classical control.The two quantum models achieved 3.2 and 3.9 times the classical control's held-out accuracy; at q = 3, QuCoNet remained below the control, while both quantum models were higher from q = 5 onward.Source: B. Performance at matched inference cost · p. 8, Table I · p. 9

C. Controls isolate joint training and interference

Controls distinguish joint Grover training and inference from post-training amplification, while showing that an interior probability target can also be learned classically. Held-out performance nevertheless reflects specialization: an untrained uniform policy is a strong reference on this benchmark, and the training-capacity comparison favors coherent inference as budget grows.

C. Controls isolate joint training and interference

  • At B = 32 and q = 3, Grover-trained QuCoNet reached training accuracy 0.953, versus 0.729 for the strongest classical control and 0.919 for classical training followed by Grover amplification.The post-training amplification control's probabilities spread around the interior target rather than concentrating on it.Source: C. Controls isolate joint training and… · p. 9
  • A symmetric classical loss targeting p∗(q) matched Grover training's held-out accuracy with coherent inference, showing that the interior target—not the Grover loss form—preserved path diversity.The capped loss, which did not penalize probabilities above the target, reached training accuracy 0.706 after one fixed-depth Grover round.Source: C. Controls isolate joint training and… · p. 9
  • For fixed-size policies trained at each budget, coherent inference supported more question–answer pairs above the 0.95 accuracy threshold as q increased than matched classical best-of-q training and inference.Figure 5 compares the extracted B0(q) capacity values and reports that coherent capacity approximately follows 1/p∗(q), while classical capacity increases more slowly.Source: Figure 5 · p. 10
  • The untrained uniform policy had the highest held-out accuracy, but nearly identical training and held-out performance meant it represented no learned solution.Its probabilities were too small to reach a Grover maximum within tested budgets; at q = 9, trained quantum models retained 75–91% of its held-out accuracy, versus just under a quarter for the strongest classical control.Source: C. Controls isolate joint training and… · p. 10
  • The comparisons persisted across the tested stopping conventions, training-set sizes, and a second random-regular graph family, with the untrained model's lead narrowing on larger instances.On harder distance-7, M = 9 questions, the trained neural quantum AI model matched the untrained reference.Source: C. Controls isolate joint training and… · p. 10

D. Reasoning capacity

The number of training question–answer pairs supported by fixed-size policies increased with the Grover budget, with faster fitted growth than under matched classical repetition. The measured growth depends on the inference protocol and accuracy threshold.

D. Reasoning capacity

  • B0 fit q^1.74 for adaptive Grover inference, versus q^0.62 for matched classical best-of-q repetition.The fitted exponent difference was 1.12, with a 95% bootstrap interval of [0.98, 1.31].Source: D. Reasoning capacity · p. 10
  • B0 increased from 33 at q = 3 to 450 at q = 13 for adaptive Grover inference with mean accuracy above 0.95.The capacity approximately followed 1/p*(q), which grows as q^2 at large q.Source: D. Reasoning capacity · p. 10
  • The fitted exponents depend on threshold and inference protocol; at the 0.90 threshold, adaptive and fixed-depth exponents were 2.07 and 1.69.The audit pools gave corresponding exponents of 2.04 and 1.69; the 0.95 criterion was not reliably attainable under fixed depth.Source: D. Reasoning capacity · p. 10
  • Matched classical best-of-q capacity rose from 11 to 28 question–answer pairs over the measured range.Its finite-range fit was Bcl0 ∝ q^0.62.Source: D. Reasoning capacity · p. 10

VII. CONCLUSION AND OUTLOOK

The paper presents Grover amplitude amplification as a trainable form of slow thinking over reasoning trajectories. On the tested sliding-puzzle benchmark, quantum training preserves more held-out accuracy than matched classical controls, while known-target verification and physical implementation remain boundaries.

VII. CONCLUSION AND OUTLOOK

  • The Grover realization treats reasoning as a discrete path integral, with the horizon setting available histories and ensemble-level interference shaping answer probabilities.Finite Grover evolution can maximize final success while pre-amplification success remains below one, avoiding pressure to solve every problem almost certainly before quantum evolution.Source: §VII · p. 11
  • At four Grover rounds, the two quantum models reached 3.2 and 3.9 times the held-out accuracy of the strongest matched classical controls.They retained 75–91% of the untrained diversity-preserving reference's held-out accuracy, which none of the trained policies surpassed on this solution-dense benchmark.Source: §VII · p. 11
  • Fixed-size policy capacity grew approximately as q^1.74 under coherent quantum inference over the tested range, compared with q^0.62 under classical repetition.The paper describes these as finite-range empirical fits.Source: §VII · p. 11
  • The benchmark's exact verifier relies on known target answers; generic question answering would need a learned evaluator coherently converted into a phase.A physical realization would also require coherent memory, reversible evaluator implementation, and control of noise and fault-tolerance costs.Source: §VII · p. 11
  • Richer quantum interference is proposed as a future direction, but its potential advantages remain an open question.The present two-sector mechanism leaves path-sensitive interference unused.Source: §VII · p. 11

Supplemental Material for “Do Quantum AIs Dream in Paths? Path-Integral Slow Thinking through Grover

The supplemental material documents the derivation and empirical foundation of the paper's approach.

Supplemental Material

  • The supplement derives the trajectory formulation and Grover-induced interior target, proves monotonicity for terminal-outcome-only classical sampling, and documents task, models, controls, validation, scaling, and reproducibility.Source: Supplemental Material for “Do Quantum… · p. 14

CLASSICAL TRAJECTORY MEASURES AND COHERENT PATH SUMS

The section contrasts classical probability over sampled reasoning trajectories with quantum amplitudes over coherent path sums. It then distinguishes general path interference from the Grover construction's collective accepted-versus-rejected rotation.

CLASSICAL TRAJECTORY MEASURES AND COHERENT PATH SUMS

  • A classical autoregressive policy assigns conditional probabilities along a complete trajectory, with each factor allowed to depend on the full prefix rather than a Markov state.The answer probability is a statistical sum over trajectories that produce that answer.Source: CLASSICAL TRAJECTORY MEASURES AND… · p. 14
  • A quantum circuit assigns complex transition amplitudes, and the outcome probability is a double path sum over complete trajectory records and the measurement operator.The state-preparation circuit is denoted A, and My is the positive measurement operator for outcome y.Source: CLASSICAL TRAJECTORY MEASURES AND… · p. 14
  • In the general measurement formulation, diagonal terms represent a classical mixture while off-diagonal terms encode coherent recombination of histories.Source: CLASSICAL TRAJECTORY MEASURES AND… · p. 14
  • In the Grover construction, the verifier phases the collective accepted sector and the A†/A reflection turns its contrast with rejected paths into a rotation of sector weights.Fresh which-path registers keep individual action sequences orthogonal, so pairwise path interference is not required for the studied success probabilities or path-resolved observables.Source: CLASSICAL TRAJECTORY MEASURES AND… · p. 14

TASKS, MODELS, AND TRAINING PROTOCOL

This section specifies the puzzle benchmark, model and training protocols, coherent inference construction, and the scope of classical controls and inference stopping. It examines path diversity and held-out accuracy across Grover training, classical objectives, and the uniform policy, while noting the limits of the uniform policy’s advantage on harder instances.

A. Reasoning tasks and data splits

  • The task uses a 120-node, degree-3 component of the 2×3 puzzle graph; questions have shortest-path distance 3–6 and must be solved within 8 moves.Each pool draws 192 question–answer pairs: the first B form training data and the final 64 are held out, with no overlap; tabular results generally pool 32 question-pool seeds.Source: A. Reasoning tasks and data splits · pp. 14–15

B. Classical and quantum walkers

  • QuCoNet uses six real coin parameters per node, while CoNet uses stochastic logits; the models match graph interface, horizon, data size, and optimization protocol, not parameter count.QuCoNet has 720 parameters versus CoNet’s 360 logits, and its fresh path-record registers make training position marginals follow the classical Markov chain.Source: B. Classical and quantum walkers · pp. 15–16

EXACT EVALUATION AND PATH SUPPORT

  • At B = 32, Grover-trained policies retain more effective routes as q rises: accuracy-weighted training IPR is 1.6, 2.8, 4.2, and 5.7 for q = 3, 5, 7, and 9.Single-attempt and repeated-sampling training approach one effective path. IPR is invariant under amplification and diagnoses path support, not interference among routes.Source: EXACT EVALUATION AND PATH SUPPORT · p. 16, Figure S1 · p. 16

CLASSICAL SAMPLE-ACCESS MONOTONICITY

  • For sound verifiers and success-maximizing procedures using only independent completed samples’ binary verdicts, increasing single-attempt success probability cannot reduce overall success.The result does not cover coherent access to amplitudes, trajectory-content-dependent rules, or procedures that change the generator using extra semantic information.Source: CLASSICAL SAMPLE-ACCESS MONOTONICITY · pp. 17–18

AMPLIFICATION OBJECTIVE AND STOPPING RULE

  • Grover training uses the interior target p*(q) = sin^2[π/(2q)], while the optimal-stop objective avoids deliberate overshoot; held-out main-text results instead use fixed-depth inference.At B = 32, fixed-depth held-out accuracies for q = 3, 5, 7, 9 are 0.042, 0.147, 0.272, 0.407, each within 0.012 of the optimal-stop values.Source: AMPLIFICATION OBJECTIVE AND STOPPING… · p. 18

A. State preparation and reversible shift

  • The reasoning circuit uses reversible shifts that record actions in fresh registers; its 787,320-dimensional position-and-record space embeds in 23 logical qubits before verifier and readout ancillas.The reverse-edge labels make each shift a permutation of the joint basis, and the resulting eight-step circuit is run coherently for amplitude amplification.Source: A. State preparation and reversible… · pp. 20–21, COHERENT INFERENCE CIRCUIT · p. 20

B. First-arrival oracle without measurement

  • A reversible first-arrival predicate accepts paths that reach the target by step M; full-state verification confirms the coherent accepted-subspace weight matches absorbing-state evaluation.The oracle acts on the complete recorded path, while Grover’s rotation depends on total accepted probability rather than pairwise interference among individual trajectories.Source: B. First-arrival oracle without… · p. 21, C. Full-state verification · p. 21

CLASSICAL CONTROLS AND DIAGNOSTIC QUESTIONS

  • At B = 32 and three circuit applications, the one-round system reaches 0.953 training accuracy versus 0.729 for the strongest classical control.The exact-gradient and sampled classical models reach 0.657 and 0.659, respectively, when granted three attempts. The control audit finds an accuracy–coverage trade-off: no single control achieves both high accuracy and broad coverage.Source: CLASSICAL CONTROLS AND DIAGNOSTIC… · pp. 21–22, Table S3 · p. 22
  • The uniform walker’s held-out accuracy after coherent inference exceeded every trained system, while it matched none on training questions.This reference is specific to the small, solution-dense puzzle: its mean first-arrival probability falls with graph size and question difficulty, and its lead reaches parity at N = 480.Source: CLASSICAL CONTROLS AND DIAGNOSTIC… · pp. 22–23
  • The strongest classical hybrid reached 0.919 training accuracy, below Grover-trained 0.953; capped and neural classical policies also overshot their targets under coherent inference.At q = 7 and 9, controls trained with large-k objectives reached 0.255 and 0.299 after inference, versus 0.272 and 0.407 for models trained at those larger q values.Source: CLASSICAL CONTROLS AND DIAGNOSTIC… · p. 23
  • Neither coin unitarity nor entropy regularization prevented collapse or supplied the question-wise stationary point needed for coherent inference.The single-attempt-trained coin policy had p = 1.000 and IPR = 1.04, versus p = 0.250 and IPR = 3.56 under the one-round loss; entropy bonuses changed matched-cost accuracy by only a few points or degraded it.Source: CLASSICAL CONTROLS AND DIAGNOSTIC… · p. 23
  • The capped classical control required about 5 × 10^5 independent verified attempts to match Grover-trained 0.953 training accuracy after amplification at B = 32.Its one-sided objective has zero gradient for questions whose probabilities drift above the cap.Source: CLASSICAL CONTROLS AND DIAGNOSTIC… · p. 23
  • The two-sided classical loss tracked Grover training on held-out accuracy, reaching 0.479 versus 0.407 at q = 9, while retaining at least as many successful routes.Its accuracy-weighted training IPR at q = 9 was 9.31 versus 5.79 for native Grover training; the target c = p∗(q) is supplied by quantum analysis, and the estimator’s sample complexity was not evaluated.Source: CLASSICAL CONTROLS AND DIAGNOSTIC… · pp. 23–24

A PARAMETER-SHARING NEURAL QUANTUM AI MODEL

A parameter-sharing transformer applies Grover-based training to coherent reasoning trajectories, retaining an interior success-probability target and strong held-out performance. Replication and scaling results favor coherent inference on tested tasks, while exact classical simulation and the task’s verifier and horizon constrain the demonstrated scope.

A PARAMETER-SHARING NEURAL QUANTUM AI MODEL

  • The two-layer, four-head prefix-conditioned transformer shares 29,190 parameters across questions and emits the six entries of the walk’s unitary coin.Success probabilities are evaluated exactly by a level-synchronous prefix-tree sweep, validated against independent enumeration to 4×10−8.Source: A PARAMETER-SHARING NEURAL QUANTUM AI… · p. 24
  • At q=3, Grover training places every neural-policy question in the interior band, with mean initial success p = 0.2493±0.0036; single-attempt and best-of-two objectives concentrate at the boundary.At q≥5, the neural means lie above, rather than exactly at, their analytic optima.Source: A PARAMETER-SHARING NEURAL QUANTUM AI… · p. 24, Table S4 · p. 25
  • At q = 3, 5, 7, 9, the neural Grover policy reaches 1.0, 2.1, 2.9, 3.6 times the strongest classical control’s held-out accuracy; it is at parity at q=3 and ahead from q=5.The comparison is against the strongest control across parameterizations; the ratios are 0.073 ± 0.017 versus 0.070 ± 0.026 at q=3.Source: A PARAMETER-SHARING NEURAL QUANTUM AI… · p. 25, Figure S6 · p. 25
  • At fixed 29k parameters, q=3 training accuracy remains above the capacity threshold through B=256, while held-out accuracy rises from 0.073 to 0.092.The reported training accuracy at B=256 is 0.976; the last two sweep points use four replicas.Source: Figure S6 · p. 25, A PARAMETER-SHARING NEURAL QUANTUM AI… · p. 26

REPLICATION AND SCALING

  • On random 3-regular graphs, the held-out matched-cost ratio grows from approximately 1.8 to 3.1 as N increases from 120 to 480.At fixed B, the training-set gap narrows as graph size increases.Source: REPLICATION AND SCALING · p. 26, Figure S7 · p. 26
  • At the 0.95 capacity threshold, B0 scales as q^1.739 for coherent training versus q^0.624 for matched classical training; B0/Bcl grows from 2.91 to 15.89 across the tested range.The authors characterize this as a controlled finite-task separation, not an asymptotic query-complexity result.Source: REPLICATION AND SCALING · p. 27
  • For a fixed base-probability distribution, the finite-range query-count fits have slopes 1.31 for classical verified sampling and 0.63 for phase-matched amplitude amplification against 1/p.These are finite-range diagnostics of standard asymptotic query maps, not an end-to-end hardware speedup.Source: REPLICATION AND SCALING · p. 27, Figure S8 · p. 28

SCOPE, PHYSICAL RESOURCES, AND REPRODUCIBILITY

  • The results come from exact classical calculations, not quantum hardware; implementation requires full-horizon coherence and record storage, while noise risks erasing the benefit and is not assigned a robustness threshold.The demonstrated task is Markovian, fixed-horizon, and uses an exact binary verifier; learned or probabilistic verification and richer language-policy features are outside scope.Source: SCOPE, PHYSICAL RESOURCES, AND… · pp. 27–28
Loading 2609.05842v1…