Quantum Scrambling Born Machine

Marcin Płodzień

quant-phFeb 19, 2026

Quantum Scrambling Born Machine

Authors: Marcin Płodzień

Organizations: Qilimanjaro Quantum Tech, Carrer de Veneçuela 74, 08019 Barcelona, Spain

Abstract

Quantum generative modeling, where the Born rule naturally defines probability distributions through measurement of parameterized quantum states, is a promising near-term application of quantum computing. We propose a Quantum Scrambling Born Machine in which a fixed entangling unitary -- acting as a scrambling reservoir -- provides multi-qubit entanglement, while only single-qubit rotations are optimized. We consider three entangling unitaries -- a Haar random unitary and two physically realizable approximations, a finite-depth brickwork random circuit and analog time evolution under nearest-neighbor spin-chain Hamiltonians -- and show that, for the benchmark distributions and system sizes considered, once the entangler produces near-Haar-typical entanglement the model learns the target distribution with weak sensitivity to the scrambler's microscopic origin. Finally, promoting the Hamiltonian couplings to trainable parameters casts the generative task as a variational Hamiltonian problem, with performance competitive with representative classical generative models at matched parameter count.

Figures & tables

Figure 1: Architecture of the quantum scrambling Born machine. An $N$ -qubit register, initialized in $\ket{0}^{\otimes N}$ , is processed through $L$ repeated layers; each layer applies trainable single-qubit rotations ( $\hat{R}_{x},\hat{R}_{z}$ before, $\hat{R}_{y}$ after) surrounding a fixed scrambling unitary $\hat{U}_{S}$ that generates entanglement. After $L$ layers, $N_{A}$ ancilla qubits are traced out and the remaining $n=N-N_{A}$ system qubits are measured in the computational basis, producing a probability distribution via the Born rule [Eq. ( 2 )]. Only the rotation angles are optimized; the scrambler is kept fixed. Three scrambler types are considered: (a) Haar random unitary, (b) brickwork random quantum circuit (RQC) of depth $K$ , (c) analog Hamiltonian evolution $e^{-i\tau\hat{H}}$ .

Figure 2: Born machine performance with a Haar-random scrambler ( $N=8$ , $N_{\mathrm{shots}}=5000$ ). (a) Converged KLD vs. number of layers $L$ for $N_{A}=0,1,2$ ancilla qubits. The KLD decreases monotonically with $L$ (inset, log scale), with each additional ancilla qubit lowering the KLD. (b)–(e) Target distribution (gray) vs. learned Born-machine output (blue) at $N_{A}=2$ for $L=2,4,6,8$ , illustrating progressive convergence from a coarse to a near-exact reproduction of the multimodal target. Shaded bands: $\pm 1\sigma$ over 20 independent realizations (each with a fresh Haar draw and random initial rotation parameters).

Figure 3: Finite-depth brickwork random circuit scrambler ( $N=8$ , $N_{\mathrm{shots}}=5000$ ). Converged KLD vs. number of layers $L$ for brickwork depths $K=1$ – $6$ at (a) $N_{A}=0$ , (b) $N_{A}=1$ , (c) $N_{A}=2$ . At $K=1$ the circuit generates only nearest-neighbor entanglement and the KLD saturates well above the Haar-scrambler level (red dashed); by $K\geq 3\approx N/2$ every curve collapses onto the Haar level, consistent with the theoretical expectation that brickwork circuits of depth $O(N)$ converge to approximate 2-designs and reproduce the entanglement properties of a full Haar-random unitary [ 6 ] . Ancilla qubits lower the KLD. Mean $\pm 1\sigma$ over 20 random circuit instances.

Figure 4: Analog Hamiltonian scramblers ( $N=8$ , $N_{\mathrm{shots}}=5000$ ). Converged KLD vs. evolution time $\tau$ for $L=1$ – $8$ layers (colored lines). Columns: Ancilla qubits $N_{A}=0,1,2$ . Top row (a–c): Transverse-field Ising model [Eq. ( 9 )]—the KLD drops at intermediate $\tau\geq 0.1$ for $L>5$ . Bottom row (d–f): XX model with transverse field [Eq. ( 10 )]—the KLD drops at $\tau\geq 0.1$ . Both Hamiltonians for $L\geq 5$ saturate at the Haar-scrambler KLD (red dashed) for $N_{A}=1,2$ ancilla qubits. The role of ancilla qubits is most visible for shallow circuits ( $L<3$ ) for both Hamiltonians.

Figure 5: Trainable-Hamiltonian Born machine applied to the correlated 2D Gaussian of Eq. ( 6 ) with $\rho=0.9$ ( $N=10$ , $N_{A}=2$ , $n_{x}=n_{y}=4$ qubits per register). The nearest-neighbor coupling $J_{xx}^{(\ell)}$ and local fields $h_{x,i}^{(\ell)},h_{y,i}^{(\ell)},h_{z,i}^{(\ell)}$ are independently optimized in each of $L=10$ trainable layers ( $\tau=0.5$ per layer), yielding 310 trainable parameters. Averages over 20 realizations. (a) Target distribution $p(x,y)$ . (b) Learned distribution $q_{\boldsymbol{\theta}}(x,y)$ . (c) Conditional slices $p(y\,|\,x)$ at three values of $x$ (dashed lines in panels a,b): target (black) vs. learned (red).

Figure 6: Trainable-Hamiltonian Born machine applied to the multimodal 2D target of Eq. ( 7 ) ( $N=10$ , $N_{A}=2$ , $n_{x}=n_{y}=4$ qubits per register). The nearest-neighbor coupling $J_{xx}^{(\ell)}$ and local fields $h_{x,i}^{(\ell)},h_{y,i}^{(\ell)},h_{z,i}^{(\ell)}$ are independently optimized in each of $L=10$ trainable layers ( $\tau=0.5$ per layer), yielding 310 trainable parameters. Averages over 20 realizations. (a) Target distribution. (b) Learned distribution. (c) Conditional slices through the peak locations: target (black) vs. learned (red). All four modes are captured without mode collapse.

Figure 7: Top panel (a): KLD vs. training epoch for the trainable-Hamiltonian QSBM (310 params), a generative adversarial network (GAN), a variational autoencoder (VAE), and a restricted Boltzmann machine (RBM) on the multimodal target of Fig. 6 , with matched parameter budgets ( $\approx 310$ ) and $50\,000$ training epochs (mean $\pm 1\sigma$ over 20 realizations; log–log scale). Bottom panels: representative learned densities from the best-performing seed of each classical model. (b) GAN (302 params; mean KLD: $1.74\pm 1.30$ ). (c) VAE (306 params; mean KLD: $0.11\pm 0.01$ ). (d) RBM (308 params; mean KLD: $0.045\pm 0.008$ ). The QSBM (310 params) achieves a mean KLD of $0.030\pm 0.005$ .

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: Gradient variance

\mathrm{Var}[\partial_{k}\mathcal{L}]

vs. system size

N

for the QSBM (RQC scrambler), the hardware-efficient ansatz (HEA, CNOT ladder), and the fully parameterized QCBM (FQCBM,

R_{ZZ}

entangler) at

L=4

. Markers with error bars: numerical data; lines: exponential fits; thin dashed:

2^{-N}

reference. The QSBM and HEA track the

2^{-N}

reference; the FQCBM decays more slowly. Averages over independent seeds.

Figure 9: Comparison of the QSBM (brickwork RQC scrambler, depth

K=2N

), the hardware-efficient ansatz (HEA, CNOT ladder), and the fully parameterized QCBM (FQCBM, trainable

R_{ZZ}

gates) on the multimodal target [Eq. ( 5 )] at

N=8

. (a) Converged KLD vs. number of layers

L

; lines show the median over 20 seeds and shaded bands the interquartile range. The HEA attains its best KLD at

L=3

and degrades for deeper circuits, whereas the QSBM decreases monotonically and matches or surpasses it for

L\geq 5

at identical parameter count

3NL

. (b) Median converged KLD vs. total trainable parameters. The FQCBM reaches the lowest KLD owing to its trainable two-qubit gates, at the cost of

(3N+N-1)L

parameters.

Figure 10: Ancilla effect at fixed output register size

n=8

for

N_{A}=0,1,2

(total system sizes

N_{\mathrm{total}}=8,9,10

) using the Haar scrambler. Increasing

N_{A}

systematically reduces the KLD at fixed target resolution, demonstrating that the mixed-state rank

2^{N_{A}}

provides a genuine expressivity resource. Averages over 20 realizations.

Figure 11: Comparison of integrable (

\hat{H}_{1}

, TFIM) and non-integrable (

\hat{H}_{2}

, MFIM;

\hat{H}_{3}

, mixed-field XXZ) scramblers at

N=8

N_{A}=1

. (a) Converged KLD vs.

L

at representative

\tau

0.5

for

\hat{H}_{1}

\hat{H}_{2}

and

0.22

for

\hat{H}_{3}

. (b) Converged KLD vs.

\tau

L=6

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

Arunava Majumder, Marius Krumm, Hendrik Poulsen Nautrup, Hans J. Briegel

University of Innsbruck, Department of Theoretical Physics, Technikerstraße 21a, A-6020 Innsbruck, Austria.

Quantum Generative Modeling Quantum Machine Learning Generative Modeling

Open paper AskPDF arXiv

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

Ω(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.

Abstract

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