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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