Isolated zero mode in a quantum computer from a duality twist

TL;DR

The paper addresses how dualities and noninvertible generalized symmetries can manifest as isolated zero modes bound to a twist defect in a Floquet transverse-field Ising chain. It develops a duality-twisted Floquet protocol, maps to Majorana fermions, and uses an efficient partial-trace autocorrelation measurement to detect a persistent zero mode on an IBM quantum computer, including translation of the twist and perturbation tests. The key finding is that the zero mode remains nondecaying in finite devices and under weak perturbations, indicating an infinite lifetime in the ideal case and finite-size robustness in practice. This work provides a practical framework for probing exotic topological defects in digitized quantum platforms and motivates further exploration of noninvertible symmetries in quantum materials.

Abstract

Investigating the interplay of dualities, generalized symmetries, and topological defects beyond theoretical models is an important challenge in condensed matter physics and quantum materials. A simple model exhibiting this physics is the transverse-field Ising model, which can host a topological defect that performs the Kramers-Wannier duality transformation. When acting on one point in space, this duality defect imposes the duality twisted boundary condition and binds a single zero mode. This zero mode is unusual as it lacks a localized partner in the same $\mathbb{Z}_2$ sector and has an infinite lifetime, even in finite systems. Using Floquet driving of a closed Ising chain with a duality defect, we generate this zero mode in a digital quantum computer. We detect the mode by measuring its associated persistent autocorrelation function using an efficient sampling protocol and a compound strategy for error mitigation. We also show that the zero mode resides at the domain wall between two regions related by a Kramers-Wannier duality transformation. Finally, we highlight the robustness of the isolated zero mode to integrability- and symmetry-breaking perturbations. Our findings provide a method for exploring exotic topological defects, associated with noninvertible generalized symmetries, in digitized quantum devices.

Figures (13)

• Figure 1: Implementation of the unitary \ref{['floquetunitary']} for a 4-qubit system in a quantum computer. The duality twist corresponds to modifying the Floquet TFIM by removing the transverse field on qubit $3$ and applying a twist exchange interaction $\sigma^z_2\sigma^x_3$ between qubits $2$ and $3$. Periodic boundary conditions have been imposed with a $\sigma^z_3\sigma^z_0$ exchange interaction.
• Figure 2: Signature of the zero mode at the defect site for the duality-twisted Floquet-TFIM. The label $q_i$ denotes the measurement at the $i$th qubit. Panel (a) shows the nondecaying autocorrelation for $\sigma^y_r$ at the defect site $r=L-1$, which overlaps with the zero mode. All other autocorrelations both at the twist defect $r=L-1$ [panel (b), (c)] and at a bulk site $j=-1+ L/2$ [panel (d), (e), (f)], far away from the twist defect, decay with stroboscopic time $n$. The chosen bulk qubit behaves qualitatively similarly to other bulk qubits. We present data from ibmq_kolkata device for $L=20$, the longest physical loop of qubits available in the device. There is good qualitative agreement between noiseless simulation and measurements on the noisy quantum device. The noiseless simulations were also performed for a smaller system of 12 qubits. The agreement between the two system sizes $L=12,20$ indicates that we are capturing the behavior in the thermodynamic limit. Date of demonstration at ibmq_kolkata device is 18th April, 2023.
• Figure 3: A twist defect that was originally on sites 2 and 3 (left) is moved to sites 1 and 2 (right) by a unitary transformation $CZ_{2,3}H_2$. Cyan denotes the site $r$ at which there is no magnetic field while the double red lines denote a twist interaction $\sigma^z_{r-1}\sigma^x_r$. Vertical lines denote a transverse field while solid lines between sites $j,j+1$ denote a $\sigma^z_j\sigma^z_{j+1}$ interaction. After the unitary transformation (right), note the Kramers-Wannier duality between sites 2,3 and the rest of the chain where the strength of the transverse field at site 3 is $J$ while the exchange interaction between qubits 2 and 3 is $g$. The unitary transformation shifts the twist but not the Majorana zero mode which stays localized at site 3. Thus the Majorana zero mode resides at the domain wall across which $g,J$ exchange roles.
• Figure 4: Unpaired Majorana in the ibmq_kolkata device for $L=20$ after a unitary translation that has moved the twist defect from sites $18,19$ to sites $17,18$. The Majorana stays localized at site 19 but now has an overlap with the longer Pauli string $\sigma^x_{18}\sigma^y_{19}$. Date of demonstration is 5th May, 2023.
• Figure 5: Signature of the Majorana zero mode at the defect site for duality-twisted interacting Floquet-TFIM. Panel (a) shows the nondecaying autocorrelation for $\sigma^y_r$ at the defect site $r=L-1$, which overlaps with the zero mode. All other autocorrelations both at the twist defect $r=L-1$ [panel (b), (c)] and at a bulk site $j=-1+ L/2$ [panel (d), (e), (f)], far away from the twist defect, decay with time. The chosen bulk qubit behaves qualitatively similarly to other bulk qubits. We present data from ibmq_kolkata device for $L=20$, the longest physical loop of qubits available in the device. There is good qualitative agreement between noiseless simulations and measurements on the noisy quantum device. The noiseless simulations were also performed for a smaller system of 12 qubits. The agreement between the two system sizes $L=12,20$ indicates that we capture the behavior in the thermodynamic limit for these times. At longer times, we expect a stronger $L$ dependence discussed in the main text. The label $q_i$ indicates measurement of the $i$th qubit. Date of demonstrations is 22nd April, 2023.