1.1. Scope and contributions

The contribution is principally a software toolkit that implements a preliminary, configurable audit framework whose mathematical primitives are derived from a companion theoretical study [26] and are stated here in self-contained form. The framework is preliminary in that the four components and their composite serve as practical proxies rather than fully validated invariants, and we say so throughout. Concretely, we provide a reproducible implementation of the four-component pipeline together with all sector-projector constructions, the generator-sum compliance defect, and a reference ansatz library spanning hardware-efficient [5], equivariant [17,18,20], and symmetry-broken constructions.

We audit five ansätze under both sector-confined and multi-sector trajectory regimes and compare against expressibility [8], gradient variance [9], and Meyer-Wallach Q [12], identifying a regime where the standard diagnostics cluster equivariant and symmetry-broken ansätze together while the audit profile separates them through the fluctuation and compliance components, with a unitary-level check that sharpens the separation. Every reported quantity is replicated across 20 seeds and across , with confidence intervals. The induced ranking is shown to be robust to the composite weights and penalty through a sweep over 200 weight vectors and a grid of penalties. A downstream study on symmetry-matched tasks shows that task performance depends on the alignment between the task and the symmetry of the architecture. We explicitly state the implementation’s limitations, including the orbit decomposition used for , the generator-sum nature of the compliance proxy, and the saturation of the multiplicity-space contribution to .

The remainder of the paper is organised as follows. Section 2 describes the audit framework, the supported groups, the reference ansätze, the trajectory regimes, the comparative baselines, and the experimental design. Section 3 presents the empirical results. Section 4 interprets them and states the limitations, and Section 5 summarises the contribution. Appendix A gives the mathematical definitions, the sector-projector constructions, and the unit-test battery, and Appendix B documents the software API with a worked example.

2.1. Ethics statement

This study did not involve human participants, animal experimentation, or the collection of personal data. All experiments used numerical simulations of parameterised quantum circuits implemented in publicly available open-source scientific Python libraries. No hardware experiments on human or animal subjects were performed. Therefore, no institutional ethics approval was required to undertake this study.

2.2. Overview of the audit framework

PsiAudit is a Python package and a reproducible Python notebook for symmetry-aware ansatz auditing. The package is built on the standard scientific Python stack [27–30] and runs end to end on a CPU in approximately one hour at the configuration used in this paper.

The audit pipeline takes as input a parameterised ansatz that maps a parameter vector and an initial state to an output state , a target symmetry group G drawn from , a trajectory specification that produces an ordered ensemble of T states, a component-weight vector with , and a compliance penalty parameter . The pipeline returns four normalised audit components on [0,1], namely the sector-occupation entropy , the operational cross-sector coherence proxy , the sectoral fluctuation amplitude , and the generator-sum compliance factor , together with the configurable composite audit score . It also writes a one-page audit report, raw CSV tables of the audit and comparative-diagnostic outputs, sector-occupation heatmaps, and a metadata JSON recording configuration, package versions, and random seeds.

2.3. The composite score and its components

We define the configurable composite audit index as the symmetry-compliance-weighted gated combination,

(1)

in which , , and are derived from the state-trajectory ensemble produced by on , and is the generator-sum compliance factor derived from the ansatz operator structure. The first two terms reward sector spread and cross-sector coherence, respectively. The third term rewards stability , but is gated by so that the stability contribution is active only when the sector structure itself is activated. Under the gated form, a sector-confined trajectory with collapses to regardless of or . Practitioners who care more about sector spread than about coherence, or who wish to remove the compliance penalty, can change the weights or set without modifying the underlying audit components.

We use the conventional values and throughout the empirical study. These values give roughly equal influence to the three positive components when all three are non-trivial, while letting the compliance proxy attenuate the composite for non-equivariant generator-sum structures. The four-component profile and the composite are reported together so that the reader can inspect each axis directly. We do not claim that these weights are optimal in any task-specific sense. They are a convention.

The sector-occupation entropy measures how widely the trajectory occupies the symmetry sectors of G. For a target group with K sectors and an averaged trajectory occupation vector , the normalised Shannon entropy is,

(2)

The normalisation places on [0, 1]. A trajectory that visits a single sector has , while a trajectory that visits all sectors uniformly has .

The cross-sector coherence proxy combines two contributions. The inter-sector Frobenius mass measures coherence between distinct symmetry sectors. For each state in the trajectory, the inter-sector contribution is,

(3)

clipped to [0,1], where denotes the Frobenius norm, and the factor of two implements the empirical normalisation adopted by the implementation. The intra-sector multiplicity-space contribution measures off-diagonal coherence within a symmetry sector and its multiplicity copies, when the underlying representation has non-trivial multiplicity. For target groups whose physically relevant sectors have unit multiplicity in the analysed setting, takes the empirical clip value. The combined cross-sector coherence proxy is,

(4)

in which the overline denotes the mean over trajectory steps. We label this quantity an operational coherence proxy rather than a representation-theoretic coherence measure. The empirical normalisation in Equation (4) can saturate at the clip cap in sectors with non-trivial multiplicity, and we observe this saturation in our experiments at . As a result, the discriminative power of in the present implementation comes primarily from . A dimension-aware normalisation that scales by rather than by a fixed factor of one half would yield a non-saturating multiplicity contribution and is a natural next step for the package.

The sectoral fluctuation measures the step-to-step variability of sectoral occupation along the trajectory. For each step t, the inverse participation ratio summarises how concentrated the sector occupation is at that step. The fluctuation component is,

(5)

where denotes the standard deviation over trajectory steps. Larger values indicate greater step-to-step variability in how concentrated the sectoral occupation is, which is associated with symmetry-broken dynamics that allow the trajectory to fluctuate across the sector ladder. Equivariant ansatz dynamics in regimes where the ansatz preserves the relevant sectors tend to produce small . The composite in Equation (1) uses the complementary quantity , which we interpret as a stability contribution.

The compliance factor measures how closely the gate-generator structure of the ansatz respects the target group action. We use a generator-sum formulation. Let be a generator set of the Lie algebra of G, or in the discrete-group case, a complete set of group generators. Given an effective ansatz Hamiltonian defined as the sum of the gate generators in the parameterised circuit, the normalised commutator defect is,

(6)

and the generator-sum compliance factor is then . We emphasise that this is a generator-sum proxy rather than a circuit-level equivariance check. A generator sum can look symmetric even when individual circuit instances do not commute with the target symmetry, for example through gate ordering or fixed non-infinitesimal local rotations that cancel under the linear sum but not under the unitary product. The score should therefore be read as a fast structural proxy. To complement it, PsiAudit also reports a unitary-level compliance check for the natural-group cases as a standard part of the audit output. This second diagnostic is more expensive because it samples realised circuit unitaries across a parameter ensemble, but it directly tests whether the implemented circuit commutes with the target action rather than relying on the generator sum; therefore, it is the more stringent of the two compliance measures. The channel-level version required for noisy circuits described by quantum channels is developed in a related study [31].

2.4. Supported target groups, the hamming-weight orbit audit, and reference ansätze

PsiAudit currently supports three target groups, chosen to span Abelian, non-Abelian Lie, and finite-discrete symmetries. The Abelian group U(1) generates phase rotations and admits a particle-number-preserving sector decomposition on n qubits. The non-Abelian Lie group SU(2) generates global spin rotations and admits a total-spin sector decomposition through Schur-Weyl duality [32]. The symmetric group generates permutations of the qubits. For U(1), the package implements particle-number projectors with sector labels . For SU(2), the package implements total-spin sector projectors with sector labels assembled from the Schur-Weyl decomposition. For , the package implements a Hamming-weight orbit decomposition. We describe the audit against throughout as an Hamming-weight orbit audit. The orbit decomposition correctly identifies the trivial-irrep occupation pattern and is sufficient for the audit primitives implemented here, but it does not separate the multiplicity space of the standard irrep, the sign irrep, or higher Young-diagram irreps [22,33]. The full implementation details for all three groups are stated in Appendix A.

To evaluate the audit toolkit, we provide five reference ansatz families that span hardware-efficient, equivariant, and symmetry-broken constructions. The hardware-efficient family is a hardware-efficient ansatz of the kind introduced for variational quantum eigensolvers [5]. It uses arbitrary single-qubit rotations interleaved with a fixed two-qubit entangling pattern, has no symmetry constraint, and serves as the generic baseline. The U(1)-equivariant family uses a particle-number-preserving gate set consisting of Givens-style hopping rotations. Its generator sum commutes with the number operator, so the family is U(1)-equivariant at the generator-sum level [17,20]. The SU(2)-equivariant family uses Heisenberg-style two-qubit interaction generators together with global spin rotations. Its generator sum commutes with the total-spin generators , so the family is SU(2)-equivariant at the generator-sum level [18,19]. The -equivariant family follows a parameter-input form whose generators commute with all qubit permutations [18,22]. The U(1)-broken family takes the U(1)-equivariant base and appends a fixed local-X rotation on each qubit at the end of every layer. This is a U(1)-symmetry-breaking inserted at the gate level. It serves as the near-control against which the U(1)-equivariant ansatz is compared.

2.5. Trajectory regimes and comparative diagnostics

The audit framework supports two complementary trajectory regimes that exercise different aspects of the sector structure. In the sector-confined regime, which we refer to as Regime A, the trajectory starts from the computational-basis vacuum . This state is supported entirely in the k = 0 sector for U(1) and in the j = n/2 sector for SU(2). Under any ansatz in which the generators all commute with the relevant target generators, the trajectory remains confined to that sector. This regime is faithful to the typical operational scenario in variational quantum algorithms, where the input state is a single computational-basis reference. In the multi-sector regime, which we refer to as Regime B, we use a product-state initialisation generated by independent single-qubit y-axis rotations applied to the vacuum. At trajectory step t, the initial state is,

(7)

We use a product-state initialisation rather than a symmetric Dicke superposition. A product state with independent rotation angles has non-zero overlap with multiple computational-basis strings and therefore with multiple U(1) sectors. The same product state is not, in general, supported within a single SU(2) total-spin sector, so it activates more than one j when audited against SU(2). The angles in Equation (7) are bounded between approximately 0.16 and 1.41 radians, with a typical value near , which gives each qubit an excitation probability of roughly 0.15. For n = 6 this places the expected occupation peak in the k = 1 U(1) sector rather than at the binomial k = n/2 peak that would be obtained from a fully symmetric Dicke superposition. We chose this initialisation because it is the simplest product-state construction that activates multiple sectors across all three target groups, as required by the audit components. A more symmetry-faithful initialisation that uses a properly normalised symmetric Dicke superposition is straightforward to add as an alternative trajectory builder. The expected occupation patterns under Equation (7) are described and validated in Appendix A.

Three standard diagnostics serve as the comparative baselines against . Expressibility is computed as the symmetric Kullback-Leibler divergence between the empirical fidelity distribution of the ansatz and the Haar fidelity distribution on the relevant Hilbert space, following the protocol of [8]. We use 200 random parameter samples per ansatz and 50 fidelity-distribution bins. Gradient variance is computed as the variance of under random initialisation [9,11], with 80 finite-difference random initialisations under central finite difference with . Meyer-Wallach Q is the average single-qubit linear-entropy entanglement of the output state averaged over 200 random parameter samples [12]. These three diagnostics provide the standard comparison surface against which and its components are read.

2.6. Experimental design and statistical analysis

The audit grid at n = 6 uses circuit depth L = 2 and trajectory length T = 40, with both Regime A and Regime B evaluated for each of the five reference ansätze against all three target groups. The scaling experiment uses at the same depth and trajectory length in Regime B against the natural target group of each ansatz. The comparative diagnostics are computed at n = 6 from 200 random parameter samples for expressibility and Meyer-Wallach Q and from 80 finite-difference initialisations for gradient variance.

The primary observational unit is one ansatz-by-target-group-by-regime combination at a fixed (n, L, T) triple, for which the pipeline reports , , , , and along with the three comparative diagnostics and the unitary-level commutator deviation, as described below. To characterise variability rather than report a single realisation, every quantity in the study is replicated across twenty independent random seeds that control ansatz initialisation, trajectory sampling, and the random-parameter draws used by the diagnostics. We report the mean across seeds, along with its standard deviation and a 95% confidence interval obtained by bootstrap resampling over the seeds. A single audit run is deterministic once its seed is fixed, so each replicate is reproducible exactly on the same hardware, and the spread across replicates reflects genuine seed-to-seed variation rather than numerical noise. The unitary-level commutator deviation is a structural property of the circuit that does not vary across seeds, so it is evaluated once and reported with the rest of the audit. No reported audit quantity therefore rests on a single realisation. The structural correctness of the underlying primitives is checked independently of these experiments by a unit-test battery. These tests verify the idempotence and completeness of the sector projectors, the vanishing of the compliance defect for the equivariant constructions against their target generators, the calibration of the entanglement diagnostic on reference states, and the normalisation bounds of the components. The full test battery passes at every system size used in this study.

The scope of inference is the five reference ansätze, three target groups, two regimes, system sizes up to n = 8, and the configuration parameters listed above. Cross-family comparisons among the reference ansätze are read descriptively rather than as estimates for the broader population of equivariant constructions. Beyond the multi-seed replication, we probe two further questions that bear directly on how the composite should be trusted. We test how sensitive the ranking of the five ansätze is to the composite weights and the compliance penalty by recomputing over two hundred randomly drawn weight vectors and a grid of penalty values, measuring rank agreement against the convention setting through Kendall’s . We also measure the wall-clock cost of one audit as a function of qubit count up to n = 10, which characterises the computational reach of the method. The audit components are evaluated without forming the dense density matrix or circuit unitary. For pure states, the Frobenius-norm quantities required by the components can be obtained from projector and gate actions on the state vector, which keeps the audit tractable across the range studied here.

The downstream compatibility study uses its own training protocol, which is kept simple so that the result reflects structural compatibility rather than optimiser tuning. Each task is a balanced binary problem with 12 examples per class at n = 6, classified using the normalised total-Z expectation with a logistic loss. Each ansatz is trained on each task using simultaneous-perturbation stochastic approximation, which estimates the gradient from two loss evaluations per step regardless of the parameter count, and is run for 60 update steps at a learning rate of 0.3 with a perturbation size of 0.05. To characterise variability, every ansatz-and-task cell is trained from eight independent random initialisations, and we report the mean accuracy, loss, and convergence rate, along with their standard deviations, across these eight runs. The convergence rate is the fraction of the total loss reduction achieved within the first 10 steps, so a higher value indicates faster early learning. These training seeds are independent of the twenty seeds used for the audit replication, since the two studies probe different objects, the audit profile in one case and the trained model behaviour in the other.

3.1. Sector-occupation trajectories reveal architecture-specific structure

Fig 1 shows the sector-occupation heatmaps that the pipeline records for each ansatz at n = 6 in Regime B. The hardware-efficient ansatz mixes broadly across all U(1) sectors with occupation that drifts slowly and never concentrates. The U(1)-equivariant ansatz keeps its occupation near the low-k end of the ladder, reflecting the product-state initialisation of Equation (7) whose typical angles give each qubit an excitation probability near 0.15 and place the peak in the k = 1 sector, and because its generators commute with the number operator, it preserves this pattern across the trajectory. The SU(2)-equivariant ansatz, audited against SU(2), is strongly stationary and dominated by the highest-spin j = 3 sector with residual support in j = 2, the latter because the product-state input is not fully contained in j = 3 and the ansatz preserves whatever spin content it is given. The -equivariant ansatz occupies a broad range of Hamming-weight orbits with visible step-to-step variation, reflecting the richer orbit ladder open to a permutation-equivariant construction, while the U(1)-broken ansatz develops a pronounced and shifting concentration near k = 1 that is the signature of an explicit symmetry violation which nonetheless preserves much of the average occupation structure.

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Fig 1. Sector-occupation trajectories at n = 6 in Regime B.

Each row shows the sector-probability trajectory of one reference ansatz, audited against its natural target group. Rows from top to bottom show the hardware-efficient ansatz against U(1), the U(1)-equivariant ansatz against U(1), the SU(2)-equivariant ansatz against SU(2), the -equivariant ansatz against the Hamming-weight orbit decomposition, and the U(1)-broken ansatz against U(1). Within each row, columns index trajectory steps and rows index symmetry sectors. The colour scale shows sector probabilities from 0 to 1. Equivariant ansätze produce stable sectoral occupation patterns inherited from the product-state initialisation in Equation (7), while the U(1)-broken ansatz develops a visibly drifting concentration.

https://doi.org/10.1371/journal.pone.0353739.g001

These panels motivate the use of the composite index. A statistic that summarised only sectoral spread would call the hardware-efficient and -equivariant ansätze similar, while one that tracked only stability would call the U(1)- and SU(2)-equivariant ansätze similar. The pipeline records all four behaviours in a single pass and combines them, weighted by generator-sum compliance, into the configurable composite in Equation (1).

3.2. The composite score separates architectures that standard diagnostics cluster together

Fig 2 compares the gated against the three standard diagnostics for the five ansätze at n = 6 in Regime B, with on the vertical axis and the expressibility KL, gradient variance, and Meyer-Wallach Q in turn on the horizontal axis. Two findings stand out.

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Fig 2. Corrected versus standard quantum machine learning diagnostics at n = 6 in Regime B.

Each panel plots the corrected of Equation (1) audited against the natural target group of each ansatz on the vertical axis. The horizontal axes show the expressibility KL of [8] on the left, the gradient variance of [9] in the middle, and the Meyer-Wallach Q of [12] on the right. The five reference ansätze are coloured consistently across panels. Under each standard diagnostic, the U(1)-equivariant and U(1)-broken ansätze cluster closely together. Under the PsiAudit component profile, they separate, primarily through the sectoral fluctuation rather than through the composite alone.

https://doi.org/10.1371/journal.pone.0353739.g002

The first concerns the U(1)-equivariant and U(1)-broken ansätze, which share a base structure and differ only by a fixed local-X rotation appended to each layer in the broken variant. Under every standard diagnostic they sit close together, with expressibility KL values of 0.038 and 0.028, gradient variances of 0.007 and 0.012, and Meyer-Wallach Q values of 0.702 and 0.721, so by any one of them the two look almost interchangeable. The audit separates them. Their composites are close, 0.612 against 0.624 as means over twenty seeds, but the components differ cleanly. The equivariant ansatz is exactly deterministic with and , whereas the broken ansatz carries and a compliance that drops to . The unitary-level commutator deviation of Equation (8) gives the sharpest reading, scoring 0 exactly for the equivariant ansatz and 0.027 for the broken one, with corresponding unitary-level compliance . The circuit-level diagnostic and the trajectory-level fluctuation both expose the local-X breaking that the generator-sum gap of 0.028 is, by itself, too small to flag with confidence.

The second concerns the -equivariant ansatz, which sits at intermediate values under all three standard diagnostics, with expressibility KL 0.656 between the hardware-efficient 0.038 and the SU(2)-equivariant 5.980, gradient variance 0.325 below the SU(2)-equivariant 0.530 and above the U(1)-family range of 0.007 to 0.096, and Meyer-Wallach Q of 0.553. Audited against its natural orbit target it attains the highest composite of the five, with a mean of 0.744 over twenty seeds, and it is the only one with appreciable seed variation, a standard deviation of 0.072 and a 95% confidence interval from 0.713 to 0.774, where the others are deterministic or vary by less than 0.02. That spread reflects the richer orbit-ladder permutation equivariance that opens up on the product-state input. The high mean reflects broad orbit occupation and exact compliance with the natural target in the multi-sector regime, and is conditional on that target and regime rather than a statement of general superiority, a point we return to in the compatibility study below.

3.3. Audit profiles are stable across seeds and system sizes

Fig 3 shows the four components and the gated against system size in Regime B; each point is the mean over twenty seeds. The ordering of the ansätze on every component is preserved across the tested range. The -equivariant ansatz holds the highest at every size, followed by the hardware-efficient, U(1)-broken, U(1)-equivariant, and SU(2)-equivariant ansätze, while falls slowly for all of them as the enlarging sector list dilutes inter-sector coherence and stays small for the equivariant ansätze and larger for the U(1)-broken near-control. The composites are stable across both seeds and sizes. The SU(2)- and U(1)-equivariant ansätze are exactly deterministic at 0.520 to 0.528 and 0.603 to 0.612 respectively, the U(1)-broken composite moves from 0.599 at n = 4 to 0.629 at n = 8 with its seed spread shrinking from 0.019 to 0.012, the hardware-efficient ansatz shows the largest size dependence as it rises from 0.587 to 0.687 with growing compliance, and the -equivariant composite stays near 0.74 while carrying the widest seed band at a standard deviation near 0.07.

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Fig 3. Scaling of the gated and its components across .

Each sub-panel shows one audit component on the vertical axis as a function of qubit count on the horizontal axis, with every point a mean over twenty seeds at fixed circuit depth L = 2 and trajectory length T = 40. Colours encode the five reference ansätze. The qualitative ordering of the ansätze on each component is preserved across the tested range of system sizes, and the seed-to-seed spread is small for every ansatz except the -equivariant family. Values are computed using the gated composite of Equation (1).

https://doi.org/10.1371/journal.pone.0353739.g003

Because each point aggregates 20 seeds, this stability reflects both sampling variation and system size. The narrow confidence bands on four of the five ansätze and the exact determinism of the two fully equivariant families indicate that the audit profile at n = 6 is representative of n = 4 and n = 8 under the same configuration, so a small-system audit can serve as a useful preliminary screen for deployment at slightly larger sizes in the same setting. We do not sweep trajectory length or depth, so we read this as a finite-size stability property at the studied depth rather than a claim of large-n universality. The wider band on the -equivariant composite is itself informative, since it singles out the one architecture whose audit is genuinely sensitive to the random trajectory and therefore the one for which a confidence interval matters most.

3.4. Sector-confinement collapses equivariant ansätze to zero in regime A

Fig 4 contrasts the two regimes, showing the composite audited against each ansatz’s natural group for the sector-confined trajectories starting from on the left and the multi-sector trajectories of Equation (7) on the right. In Regime A the U(1)- and SU(2)-equivariant ansätze collapse to exactly. The vacuum lies in the k = 0 sector for U(1) and the j = 3 sector for SU(2), so any ansatz whose generators commute with the target generators stays confined there, the entropy and coherence components vanish, the gate on the stability term forces the third component to vanish too, and the composite is identically zero. The -equivariant and U(1)-broken ansätze sit well above zero in Regime A, the former near 0.84 because its gate set does not preserve Hamming weight and spreads the vacuum across orbits, the latter near 0.43 because the deliberate breaking lets the trajectory leave the k = 0 sector. In Regime B, all five composites lie in a band between roughly 0.52 and 0.85.

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Fig 4. Sector-confinement collapse versus multi-sector audit at n = 6.

Bars show the gated composite of Equation (1) audited against the natural target group of each ansatz. The left panel shows Regime A, in which the trajectory starts from the computational-basis vacuum and equivariant ansätze remain sector-confined. The right panel shows Regime B, in which the trajectory starts from a product-state initialisation that spans multiple sectors. The U(1)-equivariant and SU(2)-equivariant ansätze collapse to in Regime A, while the -equivariant and U(1)-broken ansätze sit above zero in Regime A because their gate structure spreads the vacuum across multiple sectors or orbits. In Regime B, all five composites lie between approximately 0.52 and 0.85.

https://doi.org/10.1371/journal.pone.0353739.g004

The composite is therefore conditional on the input regime and is not an intrinsic architectural quantity. An equivariant ansatz that collapses to zero in Regime A is not defective. It is acting correctly under a degenerate input that does not exercise its symmetry structure, so a useful audit requires a regime that activates the relevant sectors, in keeping with the role of initialisation choices noted in the trainability and equivariant-learning literature [13–16,34]. The collapse is a designed property rather than an accident of the inputs, which a small ablation confirms. If the stability term is computed without the gate, so that it contributes through alone, the two confined equivariant ansätze no longer collapse but instead pick up a spurious floor of 0.250 in Regime A purely from the stability term acting on a degenerate trajectory. Gating that term by removes the artefact and reports that a confined trajectory carries no activated structure, so the gate is not a free parameter but the mechanism that gives the Regime A collapse its clean interpretation.

3.5. Component-level profile and robustness of the ranking

Fig 5 reports the full per-component summary at n = 6 in Regime B against the natural target groups, as means over twenty seeds. The hardware-efficient ansatz combines a high near 0.82, a moderate near 0.60, a near-zero of about 0.04, and compliance near 0.91 for a composite of 0.67. The U(1)-equivariant ansatz has , zero , exact compliance, and a deterministic composite of 0.61, while the SU(2)-equivariant ansatz has a smaller of 0.47 from its concentration on the highest-spin sector, zero , exact compliance, and a deterministic composite of 0.52. The -equivariant ansatz has the highest near 0.88 and the highest composite near 0.74, with exact orbit-level compliance, and is the only profile with a wide seed band. The U(1)-broken ansatz has , a visibly larger near 0.07, and a near-compliant of 0.97 that is small enough that the generator-sum proxy alone would not flag it with confidence, which is exactly what the unitary-level deviation of Equation (8) is for. The profile makes the design point concrete. The composite compresses a four-dimensional reading into one number, so a reader screening equivariant from non-equivariant constructions can consult but should not rely on it alone. A reader interested in distinguishing confined from multi-sector trajectories can consult , while a reader interested in the persistence of occupation can consult .

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Fig 5. Component-level audit summary at n = 6 in Regime B.

For each of the five reference ansätze, the bars show sector-occupation entropy , the operational cross-sector coherence proxy , the sectoral fluctuation , the generator-sum compliance , and the corrected composite audit index of Equation (1), evaluated against the natural target group of each ansatz. The target group is shown in brackets below the ansatz label. and separate the ansätze on different axes. is small for the equivariant ansätze and visibly larger for the symmetry-broken near-control.

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Because the composite combines the four components through a weight vector and a penalty, it is fair to ask whether the ranking it induces is an artefact of those choices. Recomputing over two hundred weight vectors drawn from a Dirichlet distribution over the three positive components and over a grid of penalties , and comparing each ranking against the convention setting through Kendall’s , the ordering proves highly stable. The median rank agreement across the weight draws is with a mean of 0.965, and the highest-scoring ansatz is unchanged in every draw, with reordering confined to adjacent lower-ranked ansätze under extreme weight vectors. Across the whole penalty grid the agreement is exact in this experiment, with at every tested value of . This reflects the fact that the five ansätze are well separated on the components the penalty acts through, so changing does not reorder them here, rather than a general guarantee that the penalty preserves order, since ansätze with different compliance defects could in principle reorder under a different penalty. The four-component profile remains the primary object, but the ordering produced by the composite does not depend on the particular weights and penalty adopted here.

3.6. Downstream compatibility is governed by symmetry structure

A recurring question for any diagnostic is whether it relates to task performance, and answering it for equivariant models needs care, since an ansatz equivariant to a symmetry cannot distinguish data differing only along that symmetry’s quotient, so no single fixed task is equally fair to all five architectures. The study that follows is not intended as a benchmark of classifier performance, but as a controlled check that the audit profile reflects the symmetry compatibility between the ansatz and the task. Rather than tune a single task, we defined three fixed binary tasks matched to the three target symmetries. These were a U(1) task separating states of different total excitation, an SU(2) task separating collective-spin configurations, and an task separating permutation-symmetric states of different collective angle. All three tasks were read out via the total magnetisation, and each ansatz was trained on each task using simultaneous-perturbation gradient descent. The tasks and readout are fixed in advance and not adjusted per ansatz, so the resulting accuracy matrix is a structural result rather than a constructed one.

The matrix shows that compatibility, not raw capacity, decides which ansatz learns which task. Each accuracy is a mean over eight training runs, with zero spread for the clearly incompatible pairs and the deterministic equivariant cases and at most 0.10 elsewhere, so the pattern is stable rather than an optimisation artefact. The unconstrained hardware-efficient ansatz learns all three tasks, reaching perfect accuracy on the SU(2) and tasks and 0.83 on the U(1) task. The -equivariant ansatz solves the and U(1) tasks perfectly, since permutation equivariance subsumes the excitation structure that the U(1) task depends on, and reaches 0.67 on the SU(2) task. The U(1)-equivariant, SU(2)-equivariant, and U(1)-broken ansätze succeed only on the SU(2) task and sit at chance on the others, because the discriminating feature of those tasks lies where their structure either preserves or averages it away under a total-magnetisation readout. Downstream performance for an equivariant model is therefore irreducibly relative to the task, and an architecture is useful precisely on problems whose structure aligns with its symmetry.

A finer question is whether, among compatible pairs, a higher audit score goes with better learning. Fig 6 restricts attention to the nine pairs that train above chance and plots the composite against the two training outcomes. The composite tracks the convergence rate strongly, with a Spearman correlation of 0.89, while its correlation with final accuracy is weak at 0.26. Within the compatible set, a higher audit score is therefore associated with reaching the solution faster rather than with a better solution, consistent with the audit measuring how richly and stably an ansatz organises its trajectory rather than its ultimate fitting ceiling. The number of pairs is small, so we present this as indicative rather than as a calibrated predictor, but it shows that the audit carries information about learning behaviour beyond the binary fact of compatibility, in keeping with reading as a measure of symmetry-organised complexity rather than a task-agnostic quality score [35].

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Fig 6. Composite versus learning behaviour on compatible tasks at n = 6.

Each point is one ansatz-and-task pair that trains above chance, with the composite audit index on the horizontal axis. The left panel shows the convergence rate, defined as the fraction of the total loss reduction achieved within the first ten training steps, and the right panel shows the final classification accuracy. Within the compatible regime, the composite tracks the convergence rate closely, whereas its relationship with final accuracy is weak, indicating that a higher audit score is associated with faster learning rather than a higher performance ceiling. The number of compatible pairs is small, so the relationship is read as indicative rather than as a calibrated predictor.

https://doi.org/10.1371/journal.pone.0353739.g006

3.7. Audit cost across system size

The audit remains inexpensive over the small state-vector sizes considered here and was also timed up to n = 10. A full single audit takes about 0.08 seconds at n = 4, 0.58 at n = 6, 0.94 at n = 8, and 2.66 at n = 10 on a standard CPU, with projector construction adding a few seconds at n = 10. The growth tracks the Hilbert-space dimension, which reaches 1024 at n = 10, while the number of dense-sector projectors grows only linearly, reaching 28. Because the components are evaluated via projector and gate actions on the state vector rather than via dense operator products, the method remains tractable across this range on commodity hardware, and the dense state-vector representation, rather than the audit arithmetic, sets the ceiling. Reaching the larger systems that many quantum machine learning problems target would mean replacing that representation rather than changing the audit logic.

The occupation and coherence components are expectations of projector actions on states and so admit unbiased estimation from measurement samples at a cost that does not grow with dimension, while for ansätze and inputs of bounded entanglement, a tensor-network representation would let the same actions be evaluated without forming the full state vector. The unitary-level check is more demanding, since it averages a commutator norm over parameters and group elements, but it is expressible the same way and so admits the same treatment. Lifting the dense-simulation ceiling along these lines is the main implementation step toward auditing at near-term hardware scale, while leaving the definitions unchanged.