Signatures of Topological Symmetries on a Noisy Quantum Simulator

Abstract

Topological symmetries, invertible and otherwise, play a fundamental role in the investigation of quantum field theories. Despite their ubiquitous importance across a multitude of disciplines ranging from string theory to condensed matter physics, controlled realizations of models exhibiting these symmetries in physical systems are rare. Quantum simulators based on engineered solid-state devices provide a novel alternative to conventional condensed matter systems for realizing these models. In this work, eigenstates of impurity Hamiltonians and loop operators associated with the topological symmetries for the Ising conformal field theory in two space-time dimensions are realized on IBM's Kingston simulator. The relevant states are created on the quantum device using a hybrid quantum-classical algorithm. The latter is based on a variation of the quantum approximate optimization algorithm ansatz combined with the quantum natural gradient optimization method. Signatures of the topological symmetry are captured by measuring correlation functions of different qubit operators with results obtained from the quantum device in reasonable agreement with those obtained from classical computations. The current work demonstrates the viability of noisy quantum simulators as platforms for investigating low-dimensional quantum field theories with direct access to observables that are often difficult to probe in conventional condensed matter experiments.

Figures (4)

• Figure 1: (a) A topological symmetry/defect line (red dashed line) in a 2D CFT on a torus. (b) Schematic of the impurity Hamiltonian for the Ising case. The green and blue lines correspond to the ferromagnetic interaction and the transverse field for the qubits (gray circles) respectively. The orange line indicates a variable ferromagnetic coupling parameterized by $b$ with $b = 1(0)$ corresponding to the periodic (open) chain. The red hatched box corresponds to the impurity part of the Hamiltonian, $H_d(v)$, at site $j$ and $j+1$ [see Eqs. (\ref{['eq:H']}, \ref{['eq:H_d']})]. (c) Representation of the variational quantum circuit optimization scheme used to realize the ground states of the Hamiltonian in Eq. \ref{['eq:H']}. After initialization of the qubits in the $\left| \rightarrow \right>^{\otimes L}$ state, $N$ layers of the unitary operators (orange dashed boxes) are applied. The parameters of the circuit are iteratively optimized using the quantum natural gradient (QNG) optimization method.
• Figure 2: Circuits for the evaluation of the gradient of the cost function [panel (a)] and the Fubini-Study matrix element [panels (b), (c)] required for the QNG optimization update [Eq. \ref{['eq:QNG_update']}]. (a) For the computation of the gradient $\partial{\cal L}/\partial\Theta_p$, first the portion of the circuit until the rotation by $\Theta_p$ (denoted by $U^p_<$) is applied to the qubits initialized to $|\psi_0\rangle$. This is followed by a controlled-$\tilde{O}_p$ rotation with an ancilla qubit, initialized to $|+\rangle$, as control and the remaining gates of the circuits (denoted by $U^p_>$). Here, $\tilde{O}_p = -iO_p$. Finally, a controlled unitary rotation is performed by the $j^{\rm th}$ term of the Hamiltonian, $h_j$. Averaging over the X-measurements of the ancilla qubit yields the contribution to the gradient from the $j^{\rm th}$ term. The total gradient is the sum of the different such contributions. (b) Computation of the overlaps $\langle\partial\psi_f/\partial\Theta_p|\psi_f\rangle$. In this case, after the application of $U^p_<$ and the controlled-$\tilde{O}_p$, average is performed over the Y-basis measurements of the ancilla qubit. (c) Computation of the overlaps $\langle\partial\psi_f/\partial\Theta_p|\partial\psi_f/\partial\Theta_q\rangle$. After application of $U^p_<$ and controlled-$\tilde{O}_p$, the portion of the circuit until the rotation by the angle $\Theta_q$ is applied (denoted by ${}^p_>U^q_<$), followed by a controlled-$iO_q$. Averaging over the X-measurement results yields the relevant overlap. See Secs. S1 and S2 of the Supplementary Material for more details.
• Figure 3: (a) Schematic of an open Ising chain with 12 qubits and the impurity between sites $j, j+1$ with $j = 6$. The topological defect is introduced (removed) by tuning the parameter $v$ [Eq. \ref{['eq:H']}] to $\infty(0)$. (b) Ground state energies obtained from measurement of single and two-qubit correlation functions corresponding to the different terms of the Hamiltonian [Eq. \ref{['eq:H']}] using ZNE and 5 runs, each with 1024 shots. For comparison, exact results are also shown. (c) Raw measurement data for the correlation function $\langle Z_1Z_r\rangle$ for a 12-qubit chain with $v = 0$ and 4 shown using orange circles and blue squares respectively. For comparison, the results computed using exact diagonalization are also shown. In contrast to the $v = 0$ case where the correlation function exhibits a power-law decay characteristic of a critical theory, for $v = 4$, the correlation function drops abruptly to zero as the defect location is traversed. The data was obtained averaging over 10 runs with 8192 shots per run. For panels (b, c), the circuit parameters for the realization of the ground state were computed classically using QNG optimization method. The so-obtained circuit was then implemented on ibm_kingston followed by relevant measurements (see main text for more details).
• Figure 4: (a) Schematic of the Ising chain with periodic boundary conditions with an impurity between sites $j, j+1$ [Eq. \ref{['eq:H']}]. The model reduces to the periodic (duality-twisted) Ising chain as $v\rightarrow0(\infty)$. (b) Quantum circuit for the measurement of the topological symmetry operator, $\bar{Y}$. After preparing the qubits in the ground state of $H(v = 0)$ using circuit parameters obtained from QNG-based optimization, controlled braid-operators $g_j$-s [Eq. \ref{['eq:g']}] are applied with an ancilla qubit as the control. Averaging over X-measurement of the ancilla yields the desired expectation value. (c) Comparison of measurement results for $|\langle\bar{Y}\rangle|$ [Eq. \ref{['eq:Yb']}] with exact predictions. The orange circles (crosses) denote experimental data (exact results) for the expectation value of $\bar{Y}$. (d) Comparison of the ground state energies obtained from the measurement of one and two-qubit correlation functions with exact results for $L = 8, 10, 12$. The experimental data is obtained using ZNE and averaging over 5 runs of the experiment on the ibm_kingston simulator with each run containing 1024 shots. See main text for details regarding the measurement protocol.