Learning quantum symmetries with interactive quantum-classical variational algorithms

TL;DR

The paper tackles the problem of efficiently discovering all symmetries of an unknown quantum state prepared by a black-box oracle. It develops an interactive hybrid quantum-classical approach where a parameterized quantum circuit generates candidate symmetry operators and a classical neural network regularizes the search by memorizing explored symmetry manifolds. The key contributions include a verifiable symmetry test via SWAP$_{\epsilon}$ (and near-term KL-based alternatives), a variational quantum generator for symmetries, and a dynamic classical-regularization scheme that prevents repetitive discoveries and enables termination with a full symmetry set. The framework shows robustness to noise, scalability to moderate system sizes, and practical adaptations for near-term devices, with demonstrated insights on simple circuit states, Rydberg chains, and Ising-type models that highlight its potential as a data-driven probe of quantum structures and phases.

Abstract

A symmetry of a state $\vert ψ\rangle$ is a unitary operator of which $\vert ψ\rangle$ is an eigenvector. When $\vert ψ\rangle$ is an unknown state supplied by a black-box oracle, the state's symmetries provide key physical insight into the quantum system; symmetries also boost many crucial quantum learning techniques. In this paper, we develop a variational hybrid quantum-classical learning scheme to systematically probe for symmetries of $\vert ψ\rangle$ with no a priori assumptions about the state. This procedure can be used to learn various symmetries at the same time. In order to avoid re-learning already known symmetries, we introduce an interactive protocol with a classical deep neural net. The classical net thereby regularizes against repetitive findings and allows our algorithm to terminate empirically with all possible symmetries found. Our scheme can be implemented efficiently on average with non-local SWAP gates; we also give a less efficient algorithm with only local operations, which may be more appropriate for current noisy quantum devices. We simulate our algorithm on representative families of states, including cluster states and ground states of Rydberg and Ising Hamiltonians. We also find that the numerical query complexity scales well with qubit size.

Figures (13)

• Figure 1: Schematic overview of symmetry learning scheme. A variational quantum circuit minimizes a loss function to generate symmetries, at the same time training a classical neural network to recognize the path of potential symmetry operators it searches. The classical net then alerts the quantum circuit when it is searching for symmetries similar to those already found, so that the quantum circuit can redirect its search accordingly.
• Figure 2: Depiction of the symmetry learning algorithm. The quantum net (red) involves a variational quantum circuit $C_{L, d}({\boldsymbol\theta})$ on $L$ qubits and with block-depth $d$ measured by a loss function $\mathcal{L}_{\vert \psi \rangle}({\boldsymbol\theta})$. ${\boldsymbol\theta}$ is varied until $C_{L, d}({\boldsymbol\theta})$ represents a symmetry. At the end of each epoch, a classical 3-dimensional convolutional deep net (CNet; blue) learns the loss function along the path just explored by the QNet. During future epochs, the CNet informs the QNet as to whether it has already explored its current path, and hence whether it needs to leave its current path to an as-yet explored region. The CNet structure matches the parameter structure of the QNet. The first layer convolves over each set of 3 parameters per qubit, and the second layer convolves over every parameter; the result is inputted into a 3-layer fully connected network.
• Figure 3: Verification circuit $V$ implementing a $\mathop{\mathrm{SWAP}}\nolimits_\epsilon$ test. The central operation is a swap of the two $L$-qubit registers (bottom registers), controlled by the ancillary qubit (top register). By running $V$$O(1/\epsilon^2)$ times, we can determine the overlap $|\langle \psi \vert U \vert \psi \rangle|^2$ up to error $\epsilon$. We estimate $U$ to be a symmetry of $\vert \psi \rangle$ if the overlap is at least $1 - O(\epsilon)$.
• Figure 4: Example of a universal parameterized quantum circuit on $L = 3$ qubits. The cross-hairs are controlled-NOT gates and the $R$ gates are single-qubit rotations, parameterized by ${\boldsymbol\theta}$. There are $d$ layers of $\binom{L}{2}$ controlled-NOT gates between each pair of qubits, sandwiched by the rotation gates.
• Figure 5: Restricted quantum circuit family, of $d$ layers, appropriate for current noisy devices capable of local operations. Cross-hairs represent controlled-NOT gates, while $R$ gates are single-qubit rotations, parameterized by ${\boldsymbol\theta}$.