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Q-Edge: Symmetry-Reduced Quantum Simulation of Structured Extreme Dependence

Hongrui Zhang, Paolo Recchia, Ying Chen

arXiv:2609.03706v1quant-phstat.ML

TL;DR

High-dimensional multivariate-extreme simulation is challenged by the combinatorial growth of dependence representations. Q-Edge uses a lossless symmetry-orbit representation before quantum learning, and the reported experiments show improved statistical accuracy with fewer qubits and effective degrees of freedom.

  • Problem

    The angular dependence space grows combinatorially, and it remains unclear whether symmetry can reduce it before estimation without discarding information relevant to structured targets.

    Source: §1.1.2 · p. 3, §2.1 · p. 6

  • Method

    Q-Edge maps symmetry-equivalent angular configurations to a lossless orbit-space representation and trains an unconstrained quantum generative model on the reduced space.

    Source: §1 · p. 2, §1.1.4 · p. 4

  • Results

    Numerical experiments show improved statistical accuracy with fewer qubits and fewer effective degrees of freedom.

    Source: §5 · p. 19

  • Takeaways & Limitations

    For exchangeable structured extremes, orbit-space learning makes the target representation equivalent and lower-dimensional while preserving it exactly.

    Source: §5 · p. 19

    The experiments use exact statevector simulation, and robustness to hardware noise and finite sampling remains unestablished.

    Source: §5 · p. 19

Abstract

from arXiv · show

High-dimensional simulation of multivariate extremes is fundamentally limited by the combinatorial complexity of dependence, often more than by the scarcity of extreme observations. We show that symmetry admits a lossless orbit-space representation that preserves structured extreme dependence while replacing an exponentially large dependence space with a compact set of symmetry classes. Based on this principle, we develop Q-Edge (Quantum Extreme Dependence Engine), a symmetry-reduced quantum framework that operates directly in orbit space, enabling scalable simulation and digital twins of structured extreme systems. By transferring symmetry into the data representation rather than the quantum circuit, Q-Edge allows unconstrained quantum generative models to exploit dramatically reduced state spaces. For a 30-dimensional problem, approximately 1.6 million angular states collapse to 256 orbit states, reducing the required quantum representation from about 21 qubits to 8. Our results establish a general computational principle for scalable quantum simulation of structured extreme dependence.

1 Introduction

The paper frames scalable simulation of multivariate extremes as a dependence-representation problem: symmetry can identify equivalent configurations and preserve structured dependence in a smaller orbit space. Q-Edge applies this representation before quantum learning, allowing an unconstrained quantum model to operate on fewer states.

1 Introduction

  • Extreme-event simulation depends on reproducing joint dependence, yet the angular state space grows combinatorially and makes high-dimensional structured dependence difficult to learn.The paper asks whether the dependence space itself can be reduced without losing information relevant to structured targets.Source: §1 · p. 1, §1.1.2 · p. 3

1.1.1 Can modern generative models faithfully simulate structured extreme dependence?

  • Modern generative models improve flexibility for extremes, but typically learn on the original dependence space rather than directly reducing its growth.The gap remains even as methods address heavy tails, angular distributions, and geometric structure.Source: §1.1.1 · pp. 2–3

1.1.2 Can the intrinsic representation complexity of multivariate extremes be reduced?

  • The unresolved representation question is whether symmetry-equivalent angular configurations can be combined exactly before model estimation.Existing approaches generally model configurations separately unless they impose parametric or graphical restrictions.Source: §1.1.2 · p. 3

1.1.3 Can symmetry reduce quantum learning complexity?

  • Symmetry-aware quantum circuits constrain the architecture, while unconstrained quantum generators retain the full state-space complexity; Q-Edge asks whether symmetry can be used before learning instead.This targets reduced complexity without constraining the quantum model itself.Source: §1.1.3 · pp. 3–4

1.1.4 A unified answer through symmetry-reduced representation

  • Q-Edge partitions angular configurations into symmetry orbits and trains an unconstrained quantum circuit Born machine directly on the reduced probability space.The orbit representation is presented as exact, and symmetry is transferred to the data representation rather than encoded in the circuit.Source: §1.1.4 · p. 4
  • The strategy reduces representation complexity exactly and lowers quantum resources by avoiding separate learning of symmetry-equivalent configurations.The framework connects extreme-value theory, symmetry-aware representation, and quantum generative modelling.Source: §1.1.4 · p. 4

1.2 Structure of the Paper

  • The paper proceeds from the orbit-space framework to quantum implementation, then evaluates dependence structures and discusses implications and future work.The listed empirical settings include nonlinear, heterogeneous, multimodal, geometrically constrained, and high-dimensional cases.Source: §1.2 · pp. 4–5

2 A Representation Principle for Structured Extreme Dependence

The section derives an exact representation principle for structured extreme dependence: discretized angular configurations can be grouped by exchangeability into symmetry orbits. For exchangeable distributions, learning in orbit space preserves statistical information while reducing the representation to the number of distinct orbits.

2.1 The Representation Bottleneck

  • The discretized angular state space grows combinatorially with dimension and resolution, so improving approximation fidelity rapidly increases the configurations that must be represented.The section identifies this growth as a representation bottleneck that arises before learning or optimization.Source: §2.1 · p. 6

2.2 The Combinatorial Barrier

  • State preparation and optimization become costly as every configuration is represented independently, even though permutations can encode identical dependence patterns.Arbitrary state preparation requires O(2^n) gates; larger registers also exacerbate barren plateaus and NISQ hardware constraints.Source: §2.2 · p. 7

2.3 Exchangeability and the Orbit-Space Representation

  • Under exchangeability, coordinate permutations leave probability assignments unchanged, so configurations related by permutations form orbits representing the same dependence pattern.An exchangeable distribution is constant within each orbit and is determined by one probability per orbit; its intrinsic dimension is governed by the orbit count.Source: §2.3 · p. 8

2.4 Lossless Orbit-Space Representation

  • The orbit map is a bijection between orbit distributions and exchangeable distributions, preserving total variation and well-defined f-divergences, including the training KL divergence.Thus, orbit-space learning is an exact reformulation, not lossy compression; the reduction in representation complexity follows when many configurations share an orbit.Source: §2.4 · p. 9

2 A Representation Principle for Structured Extreme Dependence

  • Q-Edge removes symmetry-induced redundancy before learning and operates directly on the smaller orbit simplex.The representation principle supplies the basis for the quantum learning framework developed next.Source: §2 · p. 5, §2.4 · p. 9

3 Q-Edge: A Symmetry-Aware Quantum Representation

Q-Edge moves symmetry into a lossless orbit-space representation, reducing the dependence state space before quantum learning while preserving structured extreme dependence. Its pipeline separately models radial severity and angular dependence, and uses symmetry to make the learning problem compatible with hardware-efficient circuits.

3.1 Representing Extremes

  • Q-Edge models angular dependence separately from radial magnitude, with radius-dependent circuit parameters that approach a radius-independent limit as r → ∞.The stated continuity result guarantees nearby radial magnitudes produce nearby angular distributions.Source: §3.1 · pp. 10–11

3.2 Representing the Angular Simplex

  • The augmented grid assigns every computational basis state to one simplex cell, avoiding unused states and penalty terms while yielding a statistically consistent discretization.The discretization error vanishes asymptotically under the stated conditions.Source: §3.2 · p. 11

3.3 The Fundamental Representation Barrier

  • With fewer trainable parameters than the full simplex’s intrinsic dimension, a quantum circuit cannot represent every distribution, regardless of ansatz, optimizer, or training algorithm.For polynomially many parameters, the stated lower bound on approximation error approaches 1/8 as n grows.Source: §3.3 · p. 12

3.4 Escaping the Barrier through Symmetry

  • The lossless orbit-space construction learns exchangeable dependence with an unconstrained circuit and reaches every distribution an equivariant circuit can, using no more parameters and sometimes fewer qubits.It uses strictly fewer qubits whenever the orbit count K is smaller than 2^n.Source: §3.4 · p. 12

3.5 Symmetry as an escape: the orbit-space construction

  • Replacing the full simplex with the symmetric simplex lowers the parameter requirement to one tied to the number of symmetry orbits.For highly symmetric problems, the orbit count is smaller than 2^n, reducing the target distribution’s intrinsic dimension.Source: §3.5 · p. 13
  • For D=30 and N=6, about 1.6 × 10^6 angular cells reduce to 256 = 2^8 orbit states.The passage describes this as an example of block symmetry.Source: §3.5 · p. 13

4 Results

The results test whether orbit-space representation improves structured-extreme reconstruction and scales to high dimension. Q-Edge improves accuracy over direct-cell quantum learning, performs strongly against a parameter-matched VAE on complex targets, and compresses a 30-dimensional problem to an eight-qubit representation.

4 Results

  • The evaluations cover five radius-conditioned S3-symmetric targets in three dimensions and a 30-dimensional S3-symmetric Dirichlet target, using RMSE, symmetric KL divergence, and TVD.The comparisons use Direct-QCBM with the same circuit architecture and training procedure, and a parameter-matched conditional VAE where appropriate.Source: §4 · p. 14, §4.2 · p. 15, §4.3 · p. 18

4.1 Does Orbit-Space Representation Overcome the Representation Barrier?

  • Across five targets, Q-Edge reduces RMSE by 30%, symmetric KL divergence by 53%, and TVD by 36% versus Direct-QCBM.The models share circuit architecture and optimization procedure; Q-Edge learns on the orbit simplex, while Direct-QCBM learns on the discretized simplex.Source: §4.1 · pp. 14–15, Table 2 · p. 15

4.2 Can Orbit-Space Quantum Learning Outperform Classical Generative Models?

  • For complex targets, Q-Edge reduces distributional errors by up to 89% and improves pointwise reconstruction accuracy by up to 95% versus the parameter-matched VAE.The VAE does better on the Dirichlet target, which belongs to its native decoder family; both methods are already near-perfect there.Source: §4.2 · pp. 15–16
  • Q-Edge preserves multimodal and curved-support dependence better than the VAE, whose Dirichlet-family decoder smooths complex targets toward simpler distributions.As radius increases, Q-Edge errors remain consistently low while VAE errors rise when target geometry departs from the Dirichlet family.Source: §4.2 · p. 16, Figure 3 · p. 17, Figure 4 · p. 17

4.3 Does Orbit-Space Compression Enable Scalable Quantum Learning?

  • 1,628,616 direct partition cells (~21 qubits) collapse to 256 orbit states (8 qubits), a 6,362× reduction while preserving reconstruction quality.The 30-dimensional comparison reports lower per-point RMSE and TVD for Q-Edge, while the VAE retains a slight symmetric-KL advantage because the target belongs to its Dirichlet decoder family.Source: §4.3 · p. 18

5 Discussion

Q-Edge’s central contribution is to match the learning representation to symmetry in structured extreme dependence, reducing the effective problem size without constraining the quantum circuit. The discussion reports exact compression and improved statistical accuracy, while limiting its conclusions to known symmetry and classically simulable experiments.

5 Discussion

  • 1,628,616 angular cells (about 21 qubits) compress to 256 orbit states (8 qubits), a 6,362× reduction in the 30-dimensional example.Table 4 compares the orbit-space QCBM with a parameter-matched S3 baseline.Source: §5 · p. 19, Table 4 · p. 19
  • Q-Edge addresses a representation mismatch: exchangeable extremes occupy a symmetry-constrained manifold, so learning over orbit space turns an exponentially large problem into an equivalent lower-dimensional one.Source: §5 · p. 19
  • Orbit-space compression preserves the target distribution exactly while reducing the effective learning dimension and improving statistical accuracy despite fewer qubits and effective degrees of freedom.Source: §5 · p. 19
  • The proposed quantum benefit is for intrinsically compressible structured probability spaces, not arbitrary high-dimensional distributions; the principle may extend to other generative tasks with group symmetries.Source: §5 · p. 19
  • The experiments use exact statevector simulation and remain classically simulable, and the framework assumes the underlying symmetry is known exactly.Source: §5 · p. 19
  • The authors frame orbit-space compression as a general mechanism for matching quantum representations to structured probability spaces, with possible applications beyond extreme-value modelling.Source: §5 · p. 19

A.1 Experiment Setup

The experiments train an orbit-space quantum Born machine on radius-conditioned, exchangeable targets and compare it with a parameter-matched conditional VAE. Training uses an exact symmetric-KL objective, while the 30-dimensional baseline remains symmetry-constrained and uses more parameters than the quantum model.

Training and the orbit-space encoding

  • The orbit-space ansatz has 3n(L + 1) parameters and uses an unconstrained circular-CNOT entangler in simulation; hardware runs use an equivalent cyclic-CZ layer, while the equivariant baseline requires an invariant full-CZ entangler.Source: §A.1 · p. 23

Training and objective

  • Training minimizes symmetric KL over 41 representative radii of adaptive radial bins; with exact statevector evaluation, driving this divergence to zero also drives MMD and total variation to zero.The stated bound is one-way: small MMD need not imply small KL.Source: §A.1 · p. 23

Target distributions (2-simplex)

  • The five radius-conditioned targets are S3-invariant on the 2-simplex and include Dirichlet, logistic, logistic-normal, and multimodal or geometric forms.Their radial behavior spans vertex-favouring, barycentre-concentrated, edge-midpoint, and annular structures.Source: §A.1 · p. 23

Implementation and metrics

  • The orbit-space QCBM uses 6 qubits for the 2-simplex experiment and 8 qubits for D = 30; the D = 30 conditional VAE has 6,596 parameters versus 3,456 for the quantum model.The VAE uses a three-dimensional latent variable that is active on every target.Source: §A.1 · p. 24
  • The conditional VAE uses a Dirichlet decoder and is evaluated by Monte Carlo marginalization on the target grid; performance is reported with RMSE, symmetric KL divergence, and total variation distance.Source: §A.1 · p. 24

A.2 Execution on quantum hardware

Hardware execution reproduces the learned angular measure on two coupling topologies, with post-mitigation fidelity limited mainly by device error. The appendix also specifies the circuit, radius-dependent parameterization, Born output, and distance measures used to describe the procedure.

A.2 Execution on quantum hardware

  • The two devices execute the same trained angles, but the heavy-hexagonal layout requires routing and has a 1.7× two-qubit-gate overhead relative to the square lattice.The square lattice compiles to 16 two-qubit gates at depth 56, versus 27 gates at depth 71 on heavy-hex.Source: §A.2 · p. 24
  • Readout unfolding followed by depolarizing correction improves the hardware distributions, but the device contribution dominates over sampling and model error after mitigation.The readout correction uses a measured 16 × 16 confusion matrix; depolarizing strength is estimated separately at each radius from mirror circuits.Source: §A.2 · p. 25
  • The heavy-hexagonal device’s final total variation distance differs by only 1.22× from the square-lattice result, as stronger readout mitigation offsets much of its routing overhead.Readout mitigation removes 29% of the raw residual on heavy-hex and 14% on the square lattice.Source: §A.2 · pp. 25–26
  • 3.9× and 4.7× the shot-matched baseline in total variation are the post-mitigation results for the square and heavy-hexagonal devices, respectively.Their absolute total variation distances are 1.8 × 10^-2 and 2.27 × 10^-2; the device contributes 93% and 96% of the respective error budgets.Source: §A.2 · p. 25, Table 6 · p. 26
  • The square-lattice post-mitigation residual varies 2.3× across r and correlates with the angle schedule at Pearson −0.94, compared with 1.1× and −0.48 on heavy-hex.The authors interpret the stronger square-lattice spread as an angle-dependent coherent component surviving isotropic correction.Source: §A.2 · p. 26
  • The hardware comparison is limited because runs were separated in time and each configuration was run once, so calibration drift cannot be excluded and residual patterns warrant replication.Source: §A.2 · p. 26

B.1 Setup and Notation

  • A parameterized circuit starts with Hadamards, applies YZY parameterized blocks on each qubit, and inserts circular-CNOT entangling layers; its parameter count is 3n(L + 1).Source: §B.1 · p. 28

B.2 Polynomial Radius Parameterization

  • Radius-dependent parameters combine T + 1 learned vectors with basis functions gi(r) = 1/(1 + r)^i, giving (T + 1)Np trainable real parameters.Source: §B.2 · p. 29

B.5 Main Result

The section establishes uniform and radius-dependent Lipschitz bounds for the circuit’s Born distribution. The radius-dependent bound is tighter, with equality conditions specified by the polynomial parameters.

B.5 Main Result

  • Theorem 6 establishes a uniform Lipschitz bound for the Born distribution generated by the polynomially parameterized circuit.Theorem 7 gives a tighter radius-dependent refinement.Source: §B.5 · p. 30
  • The radius-dependent bound is strictly tighter when a higher-order parameter is nonzero, except when both radii are zero; otherwise, the bounds coincide.They coincide when all higher-order parameters vanish, including when T = 1.Source: §B.5 · p. 30

B.6 Proof

The section bounds how circuit outputs vary with the radial input, derives adaptive radial bins, and describes the circuit and grid encodings used for learning. It also states scope limits for approximation and discretization.

B.6 Proof

  • The radius-dependent Lipschitz bound is tighter than the uniform bound, except when both radii are zero or all higher-order parameters vanish.The proof composes bounds along the chain from radial input to parameters, quantum state, and Born distribution.Source: §B.6 · pp. 30, 33

B.7 Consequences

  • The uniform bound yields closed-form equal-error radial bins whose boundary positions depend on the domain and bin count, not circuit parameters.Circuit parameters affect the achieved error level, while the relaxed denominator bound discards algebraic structure retained by the exact bound.Source: §B.7 · pp. 34–35
  • For r ≥ Rmax, the design procedure assigns the base distribution P∞ with error at most ε; each interior bin also incurs error at most ε.The base distribution is defined at r = ∞.Source: §B.7 · pp. 33, 36

C.1 Circuit Architecture

  • The specified circuit uses n qubits, L entangling layers, YZY single-qubit blocks, and circular CNOT layers, producing a Born distribution over bit strings.Its parameter count is Np = 3n(L + 1); with L = 2n, Np = 6n^2 + 3n.Source: §C.1 · p. 36

C.2 Target Distributions and Grid Bijection

  • The target is discretized on an augmented Voronoi grid; direct encoding matches grid points to 2^n basis states, while orbit-space encoding matches orbits to 2^n′ states.The two cardinality constraints are generally incompatible; the discretization is stated to converge in L1/total variation as resolution increases.Source: §C.2 · p. 36
  • As n increases, most distributions in the probability simplex cannot be approximated by the reachable set B.Source: §C.2 · p. 36

C.3 Metric Entropy of the Reachable Set

This section characterizes the metric entropy of the circuit’s reachable distributions using covers of its parameter domain and compares that set with the full probability simplex.

C.3 Metric Entropy of the Reachable Set

  • The reachable set B admits a covering-number bound obtained by covering the parameter domain and mapping the cover through the circuit’s Lipschitz output map.The parameter-domain cover gives an ε-cover of B in total variation.Source: §C.3 · p. 37
  • The argument relies on structural properties of parameterized gates and applies to the circular-CNOT and full-CZ ansätze with independent parameters.For a shared-parameter equivariant ansatz, the Lipschitz constant acquires orbit-size weights while the qualitative conclusions remain unchanged.Source: §C.3 · p. 37
  • The section also gives a covering-number proposition for the full probability simplex, providing a comparison with the reachable set.Source: §C.3 · p. 37

C.4 Main Hardness Theorem

The section characterizes approximation limits of parameter-limited quantum Born models on a high-dimensional simplex, including the difficulty of approximating most uniformly drawn distributions and constructions for representing simplex distributions. It also establishes partition-based convergence and identification results for absolutely continuous measures, with quantum basis states mapped bijectively to partition cells and stronger conditions needed for almost-everywhere density convergence.

C.4 Main Hardness Theorem

  • When α = Np/(M −1) < 1, some distribution in ∆M−1 has TV distance at least ε∗ from every distribution reachable by the PQC.Here Np is the parameter count and ε∗ is the theorem’s explicit lower bound.Source: §C.4 · pp. 38–39
  • For polynomially many parameters, the lower bound tends to 1/8 as the qubit count n grows, for all sufficiently large n.The corollary applies because M −1 = 2^n −1 grows exponentially while Np grows polynomially.Source: §C.4 · p. 39
  • For ε < ε∗, the probability that a uniformly drawn simplex distribution lies within TV distance ε of the reachable set decays exponentially in M.The bound follows from covering the reachable set and bounding the volume of TV-balls around the cover.Source: §C.4 · p. 40

D.1 Setup

  • The target distribution is represented by a probability measure on the simplex that is absolutely continuous with respect to its surface measure and has a density.The density is identified as the target density.Source: §D.1 · p. 41

D.2 Admissible Convex Partitions

  • The convergence framework applies to admissible partitions at resolutions h tending to zero, with axioms (A1)–(A4) governing the cells.The supplied definition specifies diameter and non-degeneracy conditions; the other axioms are not included in these passages.Source: §D.2 · p. 41, §D.2.1 · p. 41

D.2.2 Concrete realisation: level-set polytope partition

  • The level-set partition satisfies all four admissibility axioms, while Voronoi partitions support the distributional conclusions under (A1)–(A3) but may fail (A4).The abstract framework needs only connected cells rather than convexity for the cited results other than pointwise convergence.Source: §D.2.1 · p. 41, §D.2.2 · pp. 42, 44

D.3.1 Bijection from qubits to partition cells

  • An n-qubit QCBM assigns Born probabilities qk(θ) to the 2^n computational-basis outcomes indexed by k.The supplied passage gives qk(θ) = |⟨k|U(θ)|0⟩|².Source: §D.3.1 · p. 44
  • When the cell count is 2^n, basis states biject to partition cells, requiring no surplus states or penalty term.For the Voronoi realization, the correspondence is W_k = cell(ϕ^-1(k)).Source: §D.3.1 · pp. 44–45

D.3.2 Histograms as simple functions

  • The QCBM output and target define piecewise-constant density histograms, which coincide when q_k = µ(W_k) for every cell.The histogram densities are simple functions.Source: §D.3.2 · p. 45

D.4 Core Convergence Theorems

  • Admissible convex partitions with resolution h_n↓0 provide the setting for the core convergence theorems; the uniform partition W_N with h = 1/N qualifies.Source: §D.4 · p. 45

D.4.1 Inner–outer measure approximation

  • For every Borel set A, the inner and outer cell approximations converge to µ(A), with boundary-cell error vanishing as the boundary tube shrinks.The proof uses only (A1)–(A3), including the null-boundary condition.Source: §D.4.1 · p. 45, (Uses only (A1)–(A3).) · p. 45

D.4.2 Simple-function approximation of densities

  • For every g in L1, conditional expectations converge to g in L1 using only (A1)–(A3); for densities, almost-everywhere convergence additionally uses (A4).The L1 result also holds for signed functions without positivity or normalization assumptions.Source: §D.4.2 · p. 46

D.4.3 Uniqueness via cell histograms

  • For absolutely continuous probability measures, matching cell masses along resolutions h_n↓0 identifies the measures; partitions need not be nested, and (A4) is unnecessary.Absolute continuity is required; cell histograms may fail to distinguish mutually singular parts. The identification argument requires (A2) and shrinking maximum cell diameter.Source: §D.4.3 · p. 46

D.5 Main Error Decomposition

The error decomposition separates discretization from circuit expressivity: increasing grid resolution removes the former, while exact histogram matching removes the latter. The section then turns to symmetry-respecting circuits as a route for symmetric targets to escape generic hardness.

D.5 Main Error Decomposition

  • As Nn →∞, the discretization error tends to zero; if the QCBM exactly matches each grid histogram, the expressivity error vanishes and the total-variation error tends to zero.The stated convergence requires both increasing resolution and exact matching for this guarantee.Source: §D.5 · p. 47
  • Discretization error is controlled by grid resolution, whereas expressivity error depends on whether the circuit can represent the target histogram.The expressivity error is bounded away from zero for generic targets, while the orbit-space construction makes the reachable set dense in the symmetric simplex for SD-symmetric targets.Source: §D.5 · p. 47

E The Equivariance Escape

  • Symmetric distributions can escape generic hardness when the circuit respects the symmetry.The section frames symmetry-respecting circuits as the condition for this escape.Source: E The Equivariance Escape · p. 47

E.1 Group Actions on the Grid

Coordinate permutations partition the augmented grid into symmetry orbits, with centroid types preserving the orbit structure of their base points. For symmetric distributions, these orbits determine the effective number of free probabilities.

E.1 Group Actions on the Grid

  • Permuting coordinates preserves each augmented-grid component, so an orbit cannot contain points from distinct components.The grid includes lattice points and selected centroid types under the SD action.Source: §E.1 · p. 48
  • For each centroid type, the centroid map gives a bijection between base-point orbits and centroid orbits, preserving orbit sizes and stabilizers.The correspondence is stated for the admissible base points and centroid set.Source: §E.1 · p. 48
  • 20 symmetry orbits represent 85 augmented-grid points when D = 3 and N = 7.The example combines 8 lattice orbits with 7 and 5 orbits for the two centroid types.Source: §E.1 · pp. 49–50

E.2 Symmetric Distributions and Effective Dimension

  • An SD-symmetric distribution has OD(QN) −1 free parameters because it is constant on each orbit and its probabilities sum to one.The symmetric Dirichlet distribution is an example and is likewise determined by one probability weight per orbit, minus normalization.Source: §E.2 · p. 50

E.3 Equivariant Circuits and the Symmetry Escape

For SD-symmetric targets, equivariance restricts the approximation problem to the lower-dimensional symmetric simplex rather than the full distribution space. The section presents orbit-space construction as a second route to symmetry-respecting QCBMs and reports that it dominates the equivariant-circuit route.

E.3 Equivariant Circuits and the Symmetry Escape

  • An SD-equivariant circuit with a Π-invariant initial state produces an SD-symmetric Born distribution.This is the symmetry condition underlying the reduced approximation problem.Source: §E.3 · pp. 51–52
  • For SD-symmetric targets, an SD-equivariant circuit reduces the approximation space from ∆M−1 to ∆sym, whose dimension is OD(QN) −1.A necessary condition for covering the symmetric target space is Np ≥ OD(QN) −1.Source: §E.3 · p. 52
  • For fixed D, pD(N) grows polynomially with resolution, while for D > N, pD(N) equals the partition number p(N) and is independent of D.In both regimes, OD(QN) is exponentially smaller than |QN| for large D; for moderate N, the stated parameter requirement is satisfiable with Np = O(n2).Source: §E.3 · p. 52

E.4 Constructive Approaches to Symmetry-Respecting QCBM

  • The section compares equivariant circuits with the introduced orbit-space construction and proves that the latter dominates the former.The two routes are Strategy A, the established equivariant-circuit approach, and Strategy B, the orbit-space construction.Source: §E.4 · p. 52

E.4.1 Strategy A (Baseline): SD-Equivariant Circuit via Full Entanglement

Strategy A builds symmetry into a quantum circuit through an encoding-compatible qubit-permutation action and shared parameters. The approach yields symmetric Born distributions under its assumptions, but the required encoding compatibility can fail for generic settings.

E.4.1 Strategy A (Baseline): SD-Equivariant Circuit via Full Entanglement

  • Strategy A requires an encoding bijection that intertwines the SD action on grid points with a qubit-permutation action.This is a nontrivial combinatorial constraint that can fail for generic (D, N) when |QN| = 2n.Source: §E.4.1 · pp. 52–53
  • When τ(SD) = Sn, the invariant subspace is spanned by Dicke states and has dimension n+1; compatibility would then require K = n+1, which must also be a power of two.The paper says this is rarely satisfiable and notes Strategy B requires no encoding-compatibility assumption.Source: §E.4.1 · p. 53
  • The full CZ entanglement layer is invariant under every qubit permutation, giving the full-entanglement circuit structural symmetry group Sn.The result follows from applying CZ gates to all unordered qubit pairs.Source: §E.4.1 · p. 53
  • Under encoding compatibility, restricting circuit parameters to the H-invariant subspace makes the circuit SD-equivariant and its Born distribution SD-symmetric.Here H = τ(SD), and the stated result also assumes an invariant initial state.Source: §E.4.1 · p. 54

E.4.2 Strategy B (Route Proposed): Orbit-Space QCBM

This section develops Strategy B’s orbit-space QCBM for representing structured dependence and applies symmetry reduction to augmented grids with Dirichlet targets, whose equal-parameter blocks define the permutation symmetry used for orbit counting. The representation produces SD-symmetric full-grid distributions without circuit symmetry constraints, preserves total-variation geometry, and supports abstract reachability guarantees relative to Strategy A.

E.4.2 Strategy B (Route Proposed): Orbit-Space QCBM

  • For any circuit architecture, orbit-space QCBM produces SD-symmetric full-grid distributions, and its bijective map to the symmetric distributions preserves total variation exactly.The map divides each orbit's probability by that orbit's size; thus orbit-space TV optimization is metrically identical to optimization over symmetric full-grid distributions, without an expressivity assumption.Source: §E.4.2 · pp. 55, 56, 57, 60
  • With an SD-fixed grid point and individually equivariant Strategy A layers, Strategy B reaches every distribution Strategy A reaches at no greater abstract depth or parameter count, using log2 K ≤ n qubits.This compares abstract ansatz families; compiling the constructed layers into native gates may substantially increase circuit depth, so it does not assert a hardware-resource advantage.Source: §E.4.2 · p. 57
  • In the deep-circuit limit, Strategy B reaches every distribution reachable by equivariant Strategy A for any admissible entangler, without implying that Strategy A reaches all symmetric distributions.The result bounds Strategy A by the symmetric-distribution space; whether a particular entangler attains that entire space is a separate question.Source: §E.4.2 · p. 61

E.5 General Framework

  • Burnside’s lemma counts G-orbits by averaging fixed-point counts, and a generating function provides an efficient route to those counts without enumerating lattice solutions.The fixed-point count is for nonnegative integer solutions constrained by the cycle lengths of the group element.Source: §E.5 · pp. 61–62

E.5.1 Application to the Augmented Grid

  • For QCBM subgroup orbit counting, the relevant domain is the augmented grid QN rather than the natural lattice alone, and the framework extends to subgroup actions on that grid.The provided passage introduces a proposition on orbit counts for subgroups G ≤ SD acting on the augmented grid.Source: §E.5.1 · p. 62
  • Selecting individual G-orbits allows finer control of augmented orbit counts; a power-of-two count avoids inactive states in the quantum representation.For Dirichlet targets with G = Sn1 × · · · × Snr, selection need only respect G's block structure, not full symmetry.Source: §E.5.1 · pp. 62–63, §E.5.2 · p. 65
  • The orbit-space QCBM represents an augmented grid using its target-symmetry orbits, with no symmetry constraints on the circuit and compression ratio |QN|/OGP(QN).Choosing QN so its orbit count is a power of two makes the ceiling exact and avoids inactive states.Source: §E.5.1 · pp. 62–63

E.5.2 Application to Dirichlet Distributions

  • A Dirichlet distribution with r distinct parameter values has symmetry group equal to the direct product of permutation groups acting within the corresponding equal-parameter index blocks.Permutations across blocks with different parameter values are not included in this symmetry group.Source: §E.5.2 · pp. 63–64
  • For G = Sn1 × · · · × Snr, the orbit count on XN is obtained by summing products of partition counts across block-sum compositions of N.Each block contributes partitions of its sum into at most nk non-negative parts.Source: §E.5.2 · p. 64
  • At the symmetry extremes, the orbit count is pD(N) for full symmetry and the number of weak compositions of N into D parts when all parameters are distinct.The first case has one equal-parameter block; the second has D singleton blocks.Source: §E.5.2 · p. 65
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