Entanglement-assisted Hamiltonian dynamics learning

TL;DR

This work proposes an entanglement-assisted learning strategy that couples a single randomly initialized auxiliary qubit to the learning system at an intermediate stage of the training process, and the interplay between randomization and entanglement significantly enhances the learning performance of the protocol.

Abstract

Approximating the dynamics given by a complex many-body Hamiltonian with a simpler effective model lies at the interface of quantum Hamiltonian learning and quantum simulation. In this context, quantum generative adversarial networks (QGANs) have been shown to outperform standard Trotter-based approximations. However, their performance is often hindered by training plateaus and local minima that become increasingly severe with system size. To overcome these limitations, we propose an entanglement-assisted learning strategy that couples a single randomly initialized auxiliary qubit to the learning system at an intermediate stage of the training process. The interplay between randomization and entanglement significantly enhances the learning performance of the protocol.

Figures (4)

• Figure 1: (a) Layout of ancilla-assisted QGANs. The auxiliary ancilla (dotted line) is embedded in the generator $G$. The input state is given by Eq. \ref{['eq:bipartite_extra']}. (b) The generator is composed of several layers, with each layer consisting of single-qubit rotations $X$, $Z$, and two-qubit interactions $ZZ$. (c) Sketch of ancilla-assisted learning, where $U_T$, $U_G$, and $U_{Ga}$ are, respectively, the operators of the target, the original generator, and the ancilla-expanded generator, with $V$ a new redundancy not used for training. The white areas indicate zeros. The expanded $U_{Ga}$ is trained against a block diagonal target until the top-left generator subspace approximates the target.
• Figure 2: Ancilla-assisted learning when the time evolution operator, $e^{-i\sigma_z^1\sigma_z^2\sigma_z^3}$, is generated with a hardware-efficient ansatz. (a) Ancilla configurations for each layer. (b) Fidelity $F_i$ versus training iteration. In a typical run, a learning plateau is signalled by a fidelity saturating at $F_i<<1$. The plateau is overcome after adding an ancilla at iteration $i=3000$ (vertical dashed line) and letting the learning proceed. (c) The average maximum fidelity, $F^{avg}_{max}$, is plotted against the ancilla configurations. The dotted horizontal line indicates the reference fidelity achieved in the absence of an ancilla. (c1) Ancilla is added in the middle of training. The green dots (brown squares) denote the fidelity when the parameters of the ancilla are initiated randomly (set to zero). (c2) Ancilla is added from the beginning of training with random initial parameters.
• Figure 3: Randomised restart approach, when the time evolution operator $e^{-i\sigma_z^1\sigma_z^2\sigma_z^3}$ is targeted. Maximum reached fidelity ($F^{avg}_{max}$), given by Eq. \ref{['eq:F_choi']}, averaged over 100 attempts, for different randomization ratios we choose. The generator is trained first for 3000 iterations, and then after some of the parameters are randomized, for another 3000 iterations. The dotted lines represent the reference fidelity achieved when no randomization is done.
• Figure 4: Expressivity analysis of the ancilla-assisted learning protocol. Optimization of expressivity corresponds to a larger rank of the Jacobian (see \ref{['eq:jacobian']}). In the figure, we enumerate the parameters corresponding to the circuit $C(\text{a-1})$. Other circuits ($C(\text{a-2})-C(\text{a-4})$) are treated similarly.

Theorems & Definitions (1)

• Proposition 1