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

Xiansheng Cai; Xiu-Hao Deng; Kun Chen

quant-phSep 5, 2026

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

Authors: Xiansheng Cai, Xiu-Hao Deng, Kun Chen

Organizations: Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China · Shenzhen International Quantum Academy, Shenzhen 518048, China · Shenzhen Branch, Hefei National Laboratory, Shenzhen 518048, China

Abstract

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.

Figures & tables

Figure 1: Coherent inference preserves multiple reasoning trajectories. (a) The table-based quantum concept-network model (QuCoNet) records each move in a fresh auxiliary register, called a which-path register. Measuring these records reduces the reasoning circuit to the corresponding classical Markov chain, whose probabilities are computed exactly; retaining the records coherently prepares all trajectories for Grover amplification in Fig. 4 (a) (Supplemental Material). (b),(c) Routing for one training question (training-set size

B=32

, question-pool seed

1

; start, green circle; target, purple star). Nodes are ordered by distance from the start, and edge width gives the success-probability flow through that edge. Single-attempt training concentrates the classical walker on one route. Grover-based training retains a branching ensemble of paths distinguished by orthogonal which-path records at the interior target. The inverse participation ratio (IPR) is the effective number of paths; panel (c) shows a favorable case, while the accuracy-weighted mean IPR is

1.6

q=3

, where

q

counts reasoning-circuit applications. The panels show path weights, not interference between the displayed paths.

Figure 2: The sliding puzzle as a network benchmark for slow thinking. Top: the three reversible graph moves.

V

swaps the two central cells;

L

and

R

move the hole through the left or right corners using three elementary tile slides. Each complete

L

R

cycle counts as one action and one edge of the configuration graph. Restricting the hole to the middle column and the tile permutation to one parity class gives

N=120

reachable configurations, each with

K=3

available actions. Bottom: a question–answer pair specifies start and target nodes in this graph. The target is supplied to the model, whose output is a trajectory connecting the two nodes. The example requires four actions. The verifier accepts on first arrival at the target within the reasoning horizon

M=8

; the node numbers label graph configurations.

Figure 3: Slow thinking as a walk on a concept network. Nodes denote coarse-grained concepts, later given an operational definition as effective predictive states, and directed edges denote learnable local transitions. A single hop is a fast System-1 operation [ 5 ] . Repeatedly applying the same local dynamics constructs a trajectory from a question

Q

through intermediate states to an answer, realizing System-2 slow thinking as evolution in time. Red arrows highlight one such trajectory.

Figure 4: Grover-based training produces an interior success target and preserves accuracy at matched inference cost. (a) The classical single-attempt objective increases monotonically with the pre-amplification success probability

p

. The Grover success probability instead reaches its first maximum at

p=p^{*}(q)

; arrows show the corresponding training directions. (b) Distributions of

p_{i}

for

B=32

, computed exactly over

32

question-pool seeds. Classical single-attempt training drives both classical and quantum models toward

p_{i}=0

1

, whereas Grover-based training concentrates them near

p^{*}(q)

. Curves are smoothed histograms of the displayed data. (c) Mean training accuracy versus the number

B

of training question–answer pairs at

q=3

. Grover-based training exceeds the best result among

27

classical controls granted the same three circuit applications, at every tested

B

. Bands span the minimum and maximum values across question-pool seeds; control definitions and plotting details are given in the Supplemental Material.

applications $q$	3	5	7	9
QuCoNet	0.042(20)	0.147(34)	0.272(39)	0.407(45)
neural quantum AI	0.073(17)	0.193(25)	0.340(30)	0.496(43)
best classical	0.071(23)	0.091(27)	0.109(24)	0.126(25)
uniform policy + Grover	0.077(4)	0.204(9)	0.367(15)	0.544(20)

Table 1: Held-out accuracy at matched inference cost. The learned systems were trained on

B=32

question–answer pairs and evaluated with

q

circuit applications. Entries are means over independent replicas; parentheses give replica-to-replica standard deviations in the last digits. The two quantum-model rows use

32

replicas. For the tabular models, replicas vary the question pools while fixing the optimization seed; for the neural model, the question-pool and model-initialization seeds vary together. The classical row reports the best result among

27

classical controls at each

q

and data split, using

32

replicas per control. Each quantum model used the same fixed

q

for every question. The uniform-policy row provides a diversity-preserving reference rather than a learned solution. Complete control definitions and training-set results are given in the Supplemental Material.

Figure 5: Coherent inference increases the number of question–answer pairs supported by a policy of fixed size trained at each budget. (a) Mean training accuracy versus the number

B

of training question–answer pairs at matched values of

q

. Solid curves show Grover-based training and inference; dashed curves show classical best-of-

q

(the matched protocol using

q

independent trajectories) training and inference with the same

720

-parameter table-based policy. The dotted line at

0.95

defines

B_{0}(q)

as the largest supported training-set size. Error bars are central

68\%

bootstrap intervals from

16

shared question pools. Panel (a) shows

q=3,7,13

; all six odd values from

q=3

13

are used in panel (b). (b) The extracted

B_{0}(q)

values and finite-range power-law fits. The coherent result approximately follows

1/p^{*}(q)

, whereas the classical capacity increases more slowly. Complete sampling and fitting procedures are given in the Supplemental Material.

Figure S1: Effective successful-path number on training (left) and held-out (right) questions. Grover-based training retains multiple effective paths, whereas single-attempt and best-of-two objectives approach the single-path reference (dotted). The inverse participation ratio is weighted by each question’s success probability.

Figure S2: Success-probability flow for one training question (

B=32

, question-pool seed

1

; start, green circle; target, purple star), with nodes ordered left to right by graph distance from the start and each edge’s width giving its share of the question’s total success probability. Left: the single-attempt-trained CoNet policy collapses onto a single route (

p=0.993

\mathrm{IPR}=1.10

). Middle: the QuCoNet policy of identical architecture, trained with the single-attempt objective, collapses in the same way (

p=1.000

\mathrm{IPR}=1.04

), so the collapse comes from the objective, not the substrate. Right: the same QuCoNet policy trained for one Grover round holds the question at

p^{*}(3)=1/4

with a branching ensemble (

p=0.250

\mathrm{IPR}=3.56

), colored by orthogonal path-record channel. The inverse participation ratio counts effective routes and is not evidence of interference among them.

Figure S3: Exact training-set distributions of the per-question success probability after Grover-based training for

q=3,5,7,9

(

B=32

32

question-pool seeds each). Dashed lines mark the predicted interior optimum

p^{*}(q)=\sin^{2}[\pi/(2q)]

of Eq. ( S9 ); panel titles give the intended number of circuit applications and its interior optimum. The distribution concentrates at

p^{*}(q)

in every panel; smaller peaks at

q\geq 5

lie near higher-order maxima requiring fewer than

q

circuit applications.

	$q^{\prime}{=}3$	$q^{\prime}{=}5$	$q^{\prime}{=}7$	$q^{\prime}{=}9$
trained at $q=3$	0.042	0.067	0.091	0.111
trained at $q=5$	0.069	0.147	0.215	0.263
trained at $q=7$	0.076	0.176	0.272	0.349
trained at $q=9$	0.079	0.192	0.309	0.407

Table S1: Cross-depth evaluation on held-out questions at

B=32

: rows are training objectives and columns are the fixed inference depths, using all

q^{\prime}

circuit applications; the optimal-stop rule raises the diagonal by at most

0.012

and the off-diagonal cells, which overshoot more often, by up to

0.03

. The

q=5

–

9

entries at

q^{\prime}{=}3

lie within one seed standard deviation of one another.

Figure S4: Performance at increasing coherent inference depth. The Grover-trained model is evaluated at its matched number

q

of circuit applications under the

G_{\leq q}

rule, which stays within

0.026

of the fixed-depth protocol on held-out questions. The exact best-of-two and capped

c{=}0.25

walkers are evaluated with nine attempts on training (a) and held-out (b) questions across training-set size

B

on the sliding puzzle; the best-performing control in Table S3 at each

q

is slightly stronger. Grover-based objectives at larger

q

improve throughout and the ordering with

q

stays monotone, while these fixed-depth classical references do not separate. The held-out panel (b) is the full-

B

view of the

B{=}32

ladder in Table I of the main text. The dedicated

16

-pool scan used for the main-text capacity fit is reported in Sec. SIX .

Figure S5: Classical sampling and coherent inference use the same local walk. Left: CoNet samples one action from

\pi(\cdot\mid x)

at each node and returns one trajectory per attempt. Middle: QuCoNet applies the local coin operator

C_{x}

and a reversible shift

S

at each step. The shift moves to the neighboring node while recording the action in a fresh which-path register, giving

\mathcal{A}=\prod_{s}SC_{x}

. Measuring these records reproduces the CoNet trajectory distribution. Right: when the same reasoning circuit is kept coherent, the verifier reflection

I-2P_{\mathrm{acc}}

gives the accepted and rejected components opposite phases. The reflection about the prepared state converts this phase difference into amplitude amplification. After

r

rounds the accepted-sector angle is

(2r+1)\vartheta

. No pairwise interference between individual path records is required.

Check	value
$\max_{i}\|\langle\psi_{i}\|P_{\mathrm{acc}}\|\psi_{i}\rangle-p_{i}^{\rm enum}\|$	$3.6\times 10^{-7}$
$\max_{i}\|\\|P_{\mathrm{acc}}\mathcal{Q}\|\psi_{i}\rangle\\|^{2}-\sin^{2}(3\vartheta_{i})\|$	$1.5\times 10^{-5}$
$\max_{i}\|\\|P_{\mathrm{acc}}\mathcal{Q}^{2}\|\psi_{i}\rangle\\|^{2}-\sin^{2}(5\vartheta_{i})\|$	$1.3\times 10^{-5}$
Mean training $p$ / after one round	$0.243/0.970$

Table S2: Full-Hilbert-space verification. Maxima are over

32

training and

64

held-out questions. The after-round value in the last row is the mean of

G_{3}(p_{i})

over questions, not

G_{3}

of the mean.

objective	$q{=}3$	$q{=}5$	$q{=}7$	$q{=}9$	$q{=}3$	$q{=}5$	$q{=}7$	$q{=}9$
	training				held-out
single-attempt, REINFORCE	0.602	0.621	0.632	0.639	0.057	0.066	0.071	0.075
single-attempt, Adam	0.537	0.545	0.551	0.555	0.049	0.052	0.055	0.057
single-attempt, protocol-matched	0.533	0.540	0.544	0.547	0.045	0.048	0.050	0.052
single-attempt, exact gradient	0.535	0.541	0.545	0.548	0.047	0.050	0.052	0.053
best-of-2, sampled	0.659	0.701	0.716	0.723	0.053	0.060	0.065	0.068

Table S3: Control audit at

B=32

: best-of-

q

accuracy of the 27 classical objectives on training and held-out questions. Results pool

32

question-pool seeds for every family. “QuCoNet arch.” rows train a classical objective using the same local coin operators as QuCoNet. The main text’s “strongest classical control” is the best result in this table at each

q

and data split.

objective	initial success $p$	interior	held-out	ratio
Grover-based, $q=3$	$0.249\pm 0.004$	$1.00$	$0.073\pm 0.017$	$3.0\times$
Grover-based, $q=5$	$0.108\pm 0.010$	$1.00$	$0.193\pm 0.025$	$5.1\times$
Grover-based, $q=7$	$0.066\pm 0.007$	$1.00$	$0.340\pm 0.030$	$6.9\times$
Grover-based, $q=9$	$0.045\pm 0.004$	$1.00$	$0.496\pm 0.043$	$8.2\times$
single-attempt	$0.776\pm 0.081$	$0.05$	$0.008$	—
best-of-two	$0.915\pm 0.054$	$0.23$	$0.008$	—

Table S4: Parameter-sharing neural quantum AI model at

B=32

(

32

replicas), mean

\pm

standard deviation. “Interior” is the fraction of questions with

0.02<p_{i}<0.98

. For the Grover-based rows, “held-out” is accuracy at the matched

q

under the fixed-depth protocol, using all

q

circuit applications; the optimal-stop rule adds at most

0.003

, and the ratio is this held-out accuracy divided by the strongest same-architecture classical control’s best-of-

q

value; for the control rows, “held-out” is single-attempt accuracy (their best-of-

q

values appear in the text).

Figure S6: Grover-based training with a parameter-sharing neural quantum AI model (two-layer prefix-conditioned transformer,

29{,}190

shared parameters;

32

replicas unless noted). (a) At

B=32

, Grover-based training retains interior per-question probabilities near the predicted optima, whereas the single-attempt objective concentrates them near the boundaries. (b) At fixed model size, the training plateau persists through the tested range of

B

while held-out accuracy increases; the dotted line marks the

720

-parameter table’s capacity threshold at the

0.90

level of the separate four-pool audit (

B_{0}^{(0.9)}=40

; the sixteen-pool

0.95

definition of the main text gives

B_{0}(3)\approx 33

). (c) At matched

q

, the neural quantum AI model exceeds both the per-node table and the strongest same-architecture classical control.

Figure S7: Problem-size control on random

3

-regular graphs at

B=32

. The one-round system (plotted under the

G_{\leq 3}

optimal-stop rule; the fixed-depth values quoted in the text are up to

0.006

lower) is compared with the exact best-of-two-trained classical model at three and nine samples. The training-set difference narrows as

B/N

falls; the held-out matched-cost difference persists.

Figure S8: Finite-range check of the standard query scaling. Here one query is one application of the reasoning circuit

\mathcal{A}

or its inverse. The queries required to reach per-question success

0.9

are plotted against inverse base probability. Classical verified sampling approaches linear growth in

1/p

, whereas phase-matched amplitude amplification follows the square-root scale. The final reflection uses a phase adjustment chosen to reach the target success without overshooting it by an integer Grover round.

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