Quantum Convolutional Neural Network for Phase Recognition in Two Dimensions

TL;DR

This work constructs a QCNN that can perform phase recognition in two dimensions and correctly identify the phase transition from a toric code phase with topological order to the paramagnetic phase.

Abstract

Quantum convolutional neural networks (QCNNs) are quantum circuits for characterizing complex quantum states. They have been proposed for recognizing quantum phases of matter at low sampling cost and have been designed for condensed matter systems in one dimension. Here we construct a QCNN that can perform phase recognition in two dimensions and correctly identify the phase transition from a Toric Code phase with $\mathbb{Z}_2$-topological order to the paramagnetic phase. The network also exhibits a noise threshold up to which the topological order is recognized. Furthermore, it captures correlations between all stabilizer elements of the Toric Code, which cannot be accessed by direct measurements. This increases the threshold for errors leading to such correlations and allows for correctly identifying the topological phase in the presence of strong correlated errors. Our work generalizes phase recognition with QCNNs to higher spatial dimensions and intrinsic topological order, where exploration and characterization via classical numerics become challenging.

Figures (14)

• Figure 1: Schematic of phase recognition via a QCNN for two phases with characteristic states $\ket{\psi_1}$ and $\ket{\psi_2}$. Other states $\ket{\phi_j^k}$ in the respective phases can be generated by local unitary perturbations $\ket{\phi_j^k}=U_{k,\mathrm{E}}\ket{\psi_j}$ and the phase can, for each input state, be identified by undoing the respective perturbation $U_{k,\mathrm{E}}$ via sufficient error correction.
• Figure 2: QCNN structure. The first lattice on the input side shows the Toric Code lattice with dimension $d=4\times 4$, which is periodic in vertical and horizontal direction. The qubits on the respective sublattices are represented by gray (horizontal edges) and green circles (vertical edges). Examples of plaquette ($A_p$) and vertex ($B_v$) operators are highlighted in blue and orange. The circuit of the convolution maps the stabilizers of the input lattice to measurements on individual qubits of the respective colors, i.e., blue (green) qubits carry the measurement of the plaquette (vertex) stabilizer that is anchored on the qubits to their left. This convolution evolves error patterns on the input state and brings them into a form that can be corrected by the local operations in the pooling if the input was in the topological phase. Iterated pooling procedures reduce the lattice dimension (green squares) with every step until only two output qubits are left (green dots) and read out.
• Figure 3: QCNN output for a variation of the magnetic field strength $h_Z$, where $h_X = 0$, and for different depths $d$ of the QCNN, corresponding to the number of pooling layers $U_\mathrm{P}^{(l)}$. Each data point represents 2000 samples from the MPS representation of the corresponding ground state with the bond dimension $\chi=1250$ for an infinite cylinder with the periodic dimension $l_2 = 9$. The shaded region corresponds to the topologically ordered phase and the vertical line to critical magnetic field strength $h^*_Z=0.34$. The inset shows the output for different samples with higher resolution around the phase transition, which are calculated with a bond dimension of $\chi=1500$. For all samples, we observe increased steepness at the phase transition in the QCNN output with increasing QCNN depth corresponding to greater precision in the phase recognition.
• Figure 4: QCNN output for incoherent Pauli noise perturbing the Toric Code ground state as in \ref{['PauliNoise']} with $p_X=p_Y=0$. Via tracking the evolution of Pauli strings on the lattice under the convolution and the pooling layers we can classically simulate the QCNN output for large system sizes. For this plot, we calculate the results for about 9.6 Million qubits, which allows us to employ more pooling layers compared to the MPS samples. As for the case of the magnetic field, we find a transition in the QCNN output that becomes more pronounced with increasing depth of the QCNN. The red vertical line marks the error threshold of $p_\mathrm{th}=2.28\%$.
• Figure 5: Output of (a) uncorrelated pooling and (b) correlated pooling for the magnetic field strength $h_Y$ in the presence of Pauli-Y noise with $p_Y=3\%$ and for different depths $d$. Each data point represents 2000 samples from the MPS representation of the corresponding ground state with (a) the bond dimension $\chi=1000$ and the periodic dimension $l_2 = 9$, and (b) the bond dimension $\chi=1200$ and the periodic dimension $l_2 = 6$. This comparison shows that the correlated pooling scheme can precisely detect the first-order phase transition (vertical red line), whereas topological states close the phase transition are misclassified for uncorrelated pooling, see the inset.