Realizing quantum convolutional neural networks on a superconducting quantum processor to recognize quantum phases

Abstract

Quantum computing crucially relies on the ability to efficiently characterize the quantum states output by quantum hardware. Conventional methods which probe these states through direct measurements and classically computed correlations become computationally expensive when increasing the system size. Quantum neural networks tailored to recognize specific features of quantum states by combining unitary operations, measurements and feedforward promise to require fewer measurements and to tolerate errors. Here, we realize a quantum convolutional neural network (QCNN) on a 7-qubit superconducting quantum processor to identify symmetry-protected topological (SPT) phases of a spin model characterized by a non-zero string order parameter. We benchmark the performance of the QCNN based on approximate ground states of a family of cluster-Ising Hamiltonians which we prepare using a hardware-efficient, low-depth state preparation circuit. We find that, despite being composed of finite-fidelity gates itself, the QCNN recognizes the topological phase with higher fidelity than direct measurements of the string order parameter for the prepared states.

Fig. 1. Concept of the quantum phase recognition experiment.

a Phase diagram displaying the numerically calculated expectation value of the string order parameter $⟨\mathcal{S}⟩=⟨Z_1{X}_2{X}_4{X}_6{Z}_7⟩$ for ground states ρ of a cluster-Ising Hamiltonian (Eq. (1)) in the parameter space spanned by h₁ and h₂ for N = 7. The white dashed lines indicate the phase boundaries between the symmetry-protected topological (SPT) phase and the paramagnetic and antiferromagnetic phases, respectively. b An unknown state ρ drawn from the phase diagram in a is processed by a QCNN to recognize the phase to which it belongs. The QCNN consists of convolutional layers (C) decomposed into two-qubit gates (orange), of pooling layers (P) implemented as single-qubit operations conditioned on intermediate measurement outcomes (purple), a fully-connected circuit layer (FC), and the measurement of a single output qubit yielding outcome y.

Fig. 2. Variational ground state preparation.

a Variational quantum circuit parametrized with 19 rotation angles θ used to prepare approximate ground states of the cluster-Ising Hamiltonian H. b Rotation angles θ_i found by an optimization algorithm on a conventional computer for three example states {h₁, h₂} in the paramagnetic (PM) {1.1, 1.4}, SPT {0.0, − 0.2}, and antiferromagnetic (AF) {0.8, − 1.4} phase. c Measured expectation values of the indicated operators (solid bars) along the qubit array in comparison to the simulated values (wire frames) for the three states prepared using the rotation angles in b. d Measured string order parameters $⟨\mathcal{S}⟩$ for all prepared variational states vs. Hamiltonian parameters h₁ and h₂. Open circles indicate the three example states presented in b and c.

Fig. 3. Quantum phase recognition circuits.

a Quantum circuit for the case in which the qubits are measured in the indicated basis, directly after executing the state preparation circuit U(θ), to evaluate $\mathcal{S}=Z_1{X}_2{X}_4{X}_6{Z}_7$. b QCNN circuit consisting of a convolutional layer (C) of CZ gates (orange), and a pooling (P) and fully-connected (FC) layer implemented as a measurement in the X basis with outcome x, followed by a Boolean function f(x), here, represented by a logic circuit expressed in terms of AND and XOR gates (purple). An example of an X (Z) error occurring on qubit six (four) and its propagation through the QCNN is highlighted in red (blue).

Fig. 4. Performance of the QCNN.

a Probability to sample bitstrings x after having applied the convolutional layer (compare Fig. 3b) for the two indicated Hamiltonian parameter sets {h₁, h₂}. Bitstrings mapped onto 1 (0) by the function f(x) are colored in orange (purple) and the expected probabilities from a Kraus operator simulation in the corresponding light color, whereas the ideal values are depicted as black wire frames. b Expectation value $⟨\mathcal{S}⟩=⟨Z_1{X}_2{X}_4{X}_6{Z}_7⟩$ measured directly after variational state preparation (compare Fig. 3a) vs. h₂ for fixed h₁ = 0.2, in comparison to the ideal values (solid line) and simulated values taking decoherence into account (dashed line). The SPT phase is indicated in light gray. c Expectation value $2⟨y⟩-1$ measured after applying the QCNN for the same parameters as in b compared to values extracted from a Kraus operator simulation of the QCNN circuit (dashed line) and the ideal value (solid line).