Quantum Scrambling Born Machine

TL;DR

The paper addresses quantum generative modeling by introducing the Quantum Scrambling Born Machine (QSBM), which uses a fixed entangling reservoir to create Haar-like entanglement while learning is confined to single-qubit rotations. It evaluates three scramblers—a Haar random unitary, brickwork random circuits, and analog Hamiltonian evolution—and finds that once Haar-like entanglement is achieved, learning performance is robust to the scrambler’s microscopic origin; tracing out ancillas enhances expressivity through mixed states. A further extension promotes Hamiltonian parameters to trainable variables, showing a piecewise-constant evolution can reproduce target distributions with a mean $D_{\mathrm{KL}}$ competitive with classical models at similar parameter counts. Overall, QSBM offers a parameter-efficient, hardware-friendly approach to near-term quantum generative modeling, with potential practical impact on quantum-assisted data synthesis and probabilistic modeling.

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 (7)

• 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. \ref{['eq:born_rule']}]. 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 shifting the curve downward by roughly one order of magnitude. (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 $K \ge 3 \approx N/2$ 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 brandao2016local. Ancilla qubits shift the entire KLD surface downward by roughly one order of magnitude per ancilla. 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. \ref{['eq:H_tfim']}]---the KLD drops at intermediate $\tau \ge 0.1$ for $L>5$. Bottom row (d--f): XX model with transverse field [Eq. \ref{['eq:H_xx']}]---the KLD drops at $\tau\ge0.1$. Both Hamiltonians for $L \ge 5$ 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. \ref{['eq:2d_gauss']} 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).