Representational separation between unitary and channel quantum generative models via shared classical randomness at shallow depth

arXiv: 2608.05110v1 [quant-ph], 5 Aug 2026

The paper addresses the representational power of quantum generative models—specifically quantum circuit Born machines (QCBMs)—under realistic near-term hardware constraints of shallow depth and bounded local connectivity. As the authors state: Near-term quantum hardware limits circuit depth and often imposes geometrically local connectivity for quantum generative models, restricting the output distributions accessible to shallow unitary Born models.

While introducing stochasticity into a unitary quantum Born model (yielding a channel model) was previously known to improve empirical generative performance and, in a restricted small-scale architecture, to represent a strictly larger family of distributions than its unitary counterpart, the question of whether such randomness provides a provable separation at fixed shallow depth for arbitrarily large systems has remained open.

The central question posed in the paper is:

"Can the family of output distributions generated by shallow local quantum circuits be strictly enlarged without introducing additional quantum resources in the form of circuit depth, long-range entangling gates, or adaptive mid-circuit measurements and feedforward?"

The main answer is affirmative: shared classical randomness—a comparatively weak resource from entanglement theory—is sufficient to establish a strict, scalable representational separation over the corresponding shallow unitary Born model.

Unitary Born model: The baseline is a one-dimensional nearest-neighbor brickwall circuit on N qubits with depth D, U(theta) = U D(theta D) U 1(theta 1), acting on a fixed product input state rho in = j=1 N rho j and measured in the computational basis, producing P U(theta)(x) = Tr[x x U(theta) rho in U(theta)].

Channel model with shared randomness: The unitary circuit is augmented by inserting a stochastic Pauli string P M = j in M Q j, Q j in X,Z, whose support M includes spatially separated qubits. A single classical Bernoulli variable s in 0,1 controls whether the entire Pauli string is applied:

Thus, all local Pauli operators supported on M are controlled by the same random variable s, producing a spatially correlated stochastic Pauli operation.

The corresponding channel is E theta,p(rho in) = (1-p)U(0)(theta) rho inU(0)(theta) + p,U(1)(theta) rho inU(1)(theta). The construction generalizes to multiple Pauli strings at multiple insertion slots. The paper defines the minimal correlated channel model (Definition 1), in which a single stochastic Pauli string acts on two distinct sites a,b in M.

MBQC realization: The paper shows that this shared randomness is natively realizable in measurement-based quantum computation (MBQC). In standard MBQC, measurement outcomes induce random Pauli byproducts; the authors show that classical feedforward can be chosen so that several qubits share the same effective byproduct bit s l in 0,1 , producing a correlated stochastic Pauli string Z M s l = j in M Z j s l. This yields a channel model E(theta,p) where the channel character of the model is therefore generated entirely classically. The paper also introduces an anti-correction mechanism allowing the effective byproduct probability to span the full interval [0,1], and an endpoint Pauli-correction procedure that removes final Pauli byproducts while retaining the branch-dependent intermediate angle flips.

The paper first establishes a weak inclusion (Observation 2): if the parameter domain includes p=0, then P unit(D) P ch(D). The central result is that this inclusion is strict at fixed shallow depth.

Key mechanism: In a shallow local unitary circuit, spatially separated output regions have disjoint backward light cones (Lemma 8). For a product input state, disjoint light cones imply factorized two-site marginal (Lemma 10): Cov rho ab(Z a, Z b) = 0 for all parameters when D < dist(a,b)/4. By contrast, the shared classical random bit correlates the local responses at the two distant sites, producing a nonzero two-point covariance in the channel model.

Theorem 4 (informal) / Theorem 15 (formal): For a one-dimensional nearest-neighbor brickwall architecture on N 6 qubits, with the minimal correlated channel model acting on two distant qubits a,b separated by dist(a,b), there exists a shallow channel model E E(theta,p) with output distribution Q such that for all depths 1 Q P unit(D 0)

and, moreover, for a constant delta > 0:

theta TV! (Q, P U(theta)) delta

where TV(times) is the total variation distance. The explicit gap is delta = 1 over 6 p(1-p) a b > 0, with a, b the branch-dependent changes in local expectation values. In particular, with a = 1 and b = N, this gives a linear-depth requirement D = N-1 over 4 for nearest-neighbor circuit. Thus, if dist(a,b) = (N), any purely unitary nearest-neighbor model that represents Q requires linear depth, D = (N).

Corollary 5: For any finite-range local circuit architecture, correlated stochastic Pauli operations can generate shallow-depth distributions that are not accessible to the corresponding purely unitary setup at shallow depth. The required unitary depth is governed by the graph distance between the correlated regions (Corollary 16).

Corollary 17: The separation is specifically due to shared randomness: if the two sites are controlled by independent random variables instead of a single shared variable, the covariance vanishes, and no covariance-based separation is obtained.

MBQC instantiation (Corollary 18): On an N times 3 cluster state with computational depth D=2, correlating the boundary byproducts Z 1 s 1 and Z N s 1 yields nonzero endpoint covariance Cov E(Z 1, Z N) = beta not equal to 0, with beta = 4p(1-p)s 2,1s 2,Ns 1,1c 1,2c 1,N-1s 1,N 4 alpha 2 alpha, and hence theta TV(Q, P U(theta)) beta/6 against every purely unitary model of depth 1 < D < (N-1)/2.

The authors stress that the separation "does not rely on Pauli operators acting directly on the final measurement outcomes which can be viewed as trivial classical post-processing. Instead... the shared randomness remains encoded in correlated intermediate non-Clifford angle-flips that alter the output distribution non-trivially."

The paper defines branch-peaked mixture distributions (Definition 6):

P(x) = sum s in S pi(s) [delta s 1[x = x s] + (1-delta s) R s(x)]

where each branch s has a dominant bit string x s, a peak weight delta s, and a normalized residual distribution R s. This class is naturally suited to the correlated channel learning model, since different stochastic branches in the channel model can concentrate output probabilities around different dominant bit strings.

The authors numerically compare the unitary and correlated channel models with N=6 qubits and depth D=2. The target distribution is a two-branch mixture,

Y(x) = 0.4,Y 0(x) + 0.6,Y 1(x)

with Y 0 peaked at x 0 = 000000 (epsilon 0 = 0.15) and Y 1 peaked at x 1 = 111111 (epsilon 1 = 0.25). The channel model augments the unitary circuit with the stochastic Pauli string i=1 6 Z i s 1 controlled by a shared bit s 1 about Bernoulli(p 1).

Using the squared maximum mean discrepancy (MMD) loss over 300 epochs and 20 independent random initializations, the authors find:

The correlated channel consistently achieves substantially lower losses than the unitary model and exhibits less variation across random initializations.

The box plots in Fig. 5 show the distribution of minimum MMD losses—the unitary model achieves noticeably higher minima with more spread, while the channel model achieves substantially lower losses. The authors note: The analytical separation, rather than the numerical experiment, establishes the representational result.

The paper concludes:

"In this work, we demonstrated both analytically and numerically that shared classical randomness can strictly enlarge the set of output distributions accessible to shallow local quantum generative models. The separation does not arise from increasing the quantum depth, adding ancillas, or using long-range entangling gates. Instead, a single stochastic Pauli string, supported on spatially separated qubits and inserted at an intermediate circuit layer, is sufficient to create a channel model whose output distribution cannot be reproduced by the corresponding shallow local unitary Born model."

They further note that unitary circuits with linear depth typically suffer from expressivity-induced barren plateaus, whereas shallow circuits do not, and that extending the shallow unitary model with shared randomness requires only a few additional trainable parameters (often a single probability parameter). The implication for hardware is:

classically coordinated stochastic Pauli operations can be a cheap and useful resource for generative modeling on devices where quantum connectivity is limited.

The authors emphasize: These results should not be interpreted as a quantum advantage claim over classical computation. Instead, it is a separation between two quantum model classes under the same shallow local quantum resources. The additional resource is classically cheap: a shared random variable controlling local Pauli operations simultaneously.

Open directions include characterizing which target distributions benefit from shared randomness, optimizing placement and support of stochastic Pauli strings on realistic hardware graphs, studying trainability of the tunable probabilities, and exploring the broader connection between measurement-induced randomness, adaptive feedforward, and stochastic quantum generative modeling.

The paper positions itself relative to prior work on stochastic or non-unitary variational quantum models. It notes that other works (e.g., Wu et al. on randomness-enhanced quantum neural networks, Coyle et al. on density quantum neural networks, Heredge et al. on LCU schemes, Wen et al. on trainable quantum channels, and various noise-based approaches) rely on generic claims that randomness helps or use unitary-eigenvalue restrictions. The authors state:

"our contribution is not the generic claim that 'randomness helps', nor an empirical optimization effect, but strict complexity-theoretical shallow-depth separations for quantum generative models enabled by shared randomness."

• Augment shallow unitary Born models with shared classical randomness. Insert a single stochastic Pauli string P M = j in M Q j, controlled by one Bernoulli variable s, at an intermediate circuit layer. This adds only one probability parameter but strictly enlarges the representable output distribution family at fixed shallow depth and fixed local connectivity.

• Use correlated stochastic Pauli operations, not independent noise. The improvement comes from sharing the same classical bit across spatially separated qubits. Independent per-qubit random variables do not create the needed two-point covariance. Therefore design the generative model so one random variable simultaneously controls local Pauli operations on distant sites.

• Target branch-peaked mixture distributions. Parameterize the generative model as a mixture over branches indexed by the shared random bit s, where each branch concentrates probability around a dominant bit string x s. This matches the channel model's structure and makes the augmented model expressive exactly where unitary models fail.

• Incorporate covariance-based analytical guidance. Use the theoretical gap delta = 16 p(1-p) a b to detect when a target distribution is inaccessible to a shallow unitary model. For example, if the target has nonzero covariance between distant qubits a and b, a nearest-neighbor unitary model requires depth at least dist(a,b)/4, while the shared-randomness channel model achieves it at constant depth.

• Replace linear-depth unitary circuits with shallow channel circuits. Because shallow circuits avoid expressivity-induced barren plateaus and linear-depth unitary equivalents often suffer from them, the improved model should prefer the shallow channel architecture for trainability at scale.

• Implement the channel model in MBQC with classical feedforward. Use correlated byproduct bits on a cluster state, with anti-correction to tune the effective probability p and endpoint Pauli correction to remove final byproducts. This realizes the generative advantage using only classical coordination and shallow measurement depth, without extra quantum depth or long-range entangling gates.

• Use MMD loss with multiple initializations. Training the correlated channel model with squared maximum mean discrepancy loss yields lower minima and less variance across random initializations compared to the unitary model. Thus the improved training pipeline should track minimum MMD loss over several seeds, not a single run.

• Optimize placement of stochastic Pauli strings using graph distance. The required unitary depth to reproduce the channel distribution scales with the graph distance between correlated sites. Therefore an automated design procedure can choose the support M of the shared Pauli string to maximize correlation distance on the available hardware connectivity graph.

• Represent strictly more probability distributions than shallow unitary quantum generative models, using the same circuit depth, same local gates, no ancillas, no mid-circuit measurements, and only one additional classical probability parameter.

• Generate long-range correlated outputs at constant shallow depth. For example, on an N-qubit nearest-neighbor circuit, it can produce a distribution with nonzero covariance between qubit 1 and qubit N using depth D=2, whereas any purely unitary nearest-neighbor model needs D= (N) to reproduce that same distribution.

• Certify representational gaps. Given a target distribution and a candidate shallow unitary circuit, the system can compute or estimate the covariance between correlated sites. If the covariance is nonzero, it knows the unitary model must either increase depth beyond dist(a,b)/4 or be augmented with shared randomness.

• Train stably without barren-plateau blowup. Because the improved model stays shallow, it avoids expressivity-induced barren plateaus that plague the equivalent linear-depth unitary model, making gradient-based training feasible for larger systems.

• Learn branch-peaked mixtures effectively. The model can explicitly represent different branches of a target mixture with different dominant bit strings, sampling s first and then generating from the corresponding branch—exactly matching structured targets like the two-branch mixtures used in the numerical experiments.

• Deploy on measurement-based quantum hardware. On a cluster state, the system can use standard measurement randomness and classical feedforward to implement correlated stochastic Pauli operations, giving it the same representational advantage without requiring a deep gate-based circuit.

• Decide when shared randomness is needed. The system can compare the covariance generated by a shared random bit against independent random variables; if the target distribution's correlation structure is incompatible with independent noise, it automatically switches to the shared-control channel architecture.

• Guarantee approximate reproduction with a lower bound on fidelity. For a correctly chosen shared Pauli string and probability p, the system's output distribution Q is separated from every shallow unitary output by total variation distance at least delta>0, so no shallow unitary model can even approximate it well.

Near-term quantum hardware limits circuit depth and often imposes geometrically local connectivity for quantum generative models, restricting the output distributions accessible to shallow unitary Born models. Introducing stochasticity into a unitary quantum Born model can improve the empirical generative performance of the resulting channel model and, for a restricted small-scale architecture, has been proven to represent a strictly larger family of distributions than its unitary counterpart. However, whether such randomness provides a provable separation at fixed shallow depth for arbitrarily large systems has remained open. Here, we show that shared classical randomness, a comparatively weak resource from entanglement theory, is sufficient to establish such a strict scalable representational separation over the corresponding shallow unitary Born model. More specifically, we augment bounded-connectivity shallow unitary circuits, followed by computational-basis measurements, with spatially separated local Pauli operations, whose joint application is controlled by a single classically sampled random bit. The resulting shallow-depth channel model generates long-range correlations in the classical output distribution that no purely unitary shallow-depth model with bounded connectivity can reproduce. For one-dimensional nearest-neighbour architectures, reproducing such distributions with a purely unitary model can require depth (N) in the worst case. We further show that measurement-based quantum computation (MBQC) provides a natural implementation of the required shared classical randomness through suitable adaptation of the random measurement outcomes. Numerical experiments on MBQC-based generative models support the analytical results.