Dynamical phase error in interacting topological quantum memories

Open Access

Dynamical phase error in interacting topological quantum memories

L. Coopmans, S. Dooley, I. Jubb, K. Kavanagh, and G. Kells

Phys. Rev. Research 3, 033105 – Published 30 July 2021

Abstract

A local Hamiltonian with topological quantum order (TQO) has a robust ground-state degeneracy that makes it an excellent quantum memory candidate. This memory can be corrupted however if part of the state leaves the protected ground-state manifold and returns later with a dynamically accrued phase error. Here we analyze how TQO suppresses this process and use this to quantify the degree to which spectral densities in different topological sectors are correlated. We provide numerical verification of our results by modeling an interacting p-wave superconducting wire.

Received 15 January 2021
Accepted 15 July 2021

DOI:https://doi.org/10.1103/PhysRevResearch.3.033105

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Anyons Dirac fermions Majorana bound states Quantum information processing Quantum memories Quasiparticles & collective excitations Symmetry protected topological states Topological quantum computing Topological superconductors

1-dimensional spin chains Nanowires Superconducting devices

Kitaev model Lattice models in condensed matter

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

L. Coopmans ^1,2,*, S. Dooley ¹, I. Jubb ¹, K. Kavanagh ^1,3, and G. Kells ¹

¹Dublin Institute for Advanced Studies, School of Theoretical Physics, 10 Burlington Road, D04 C932, Dublin 4, Ireland
²School of Physics, Trinity College Dublin, College Green, D02, Dublin 2, Ireland
³Department of Theoretical Physics, Maynooth University, Maynooth, W23, County Kildare, Ireland

^*coopmanl@tcd.ie

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Vol. 3, Iss. 3 — July - September 2021

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Images

Figure 1
(a) Schematic view of a 2D topological memory consisting of 4 anyons. Information is stored nonlocally, and any local excitation occurring near one anyon must propagate through the system for an error to occur. (b) Slight energy mismatches between higher energy states in different topological sectors open up an apparent relative phase error loophole when a local process couples the protected ground states to excited states. (c) As a concrete model we consider a symmetry-protected topological quantum memory consisting of two p-wave superconducting wires separated by a potential barrier. In this setup 4 Majorana zero modes occur at the domain boundaries, as indicated by the purple lines numbered 1–4 at the bottom. Local fluctuating noise can be realized, for example, by oscillating the left domain boundary, or as a quasi-particle tunneling into the system.
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Figure 2
Time-averaged rate of phase error as a function of interaction driven ground-state splitting. TDVP can pick up an interaction driven phase rotation (error) resulting from finite-size ground-state splitting of order $〈 δ E 〉 \sim 10^{- 10}$ . For system sizes where no appreciable splitting occurs we do not detect any systematic phase rotation. Inset: same data as in the main figure plotted against interaction strength (in units of $w$ ). In these simulations we oscillated the left boundary wall, see Appendix pp1, with frequency $ω = 1$ and maximum velocity $v_{max} = 0.1$ . The time-averaging was done over an $O (1)$ multiple of the oscillation period. We also set uniform parameters $μ = - 1.5$ and $Δ = 0.7$ .
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Figure 3
Phase error for two oscillating walls. In this simulation we set uniform parameters $w = 1$ , $Δ = 0.8$ , $μ = - 1.2$ , $L = 70$ , $d t = π / 60$ , and a bond dimension of $χ = 50$ . Both walls were oscillated with frequency $ω = 1$ , and maximum velocity $v_{max} = 0.1$ .
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Figure 4
Averaged rate of qubit-loss versus frequency of the left boundary wall for 4 different interaction strengths. The frequency covers a wide regime from slow almost adiabatic oscillations, $ω ≪ E_{gap}$ , to high (nonadiabatic) frequencies $ω ≫ E_{gap}$ . The resonance peak shifts to lower frequencies with increasing interaction strength, corresponding to an expected reduction in the gap. Moreover across all frequencies the rate of qubit-loss increases with interaction strength. For this plot we set $L = 50$ , $v_{max} = 0.1$ , $w = 1$ , $μ = - 1$ , and $Δ = 0.5$ .
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Figure 5
The single particle Green's functions (a) $| G^{e} (1, x, t) |$ and (b) the difference $| D | = | G^{e} (1, x, t) - G^{o} (1, x, t) |$ , which shows that the two correlators are the same up to exponentially small corrections for a time $T^{*}$ . (c) Introducing a cut-off in time, $Θ (t) = e^{- . 03 | t |}$ , leads to spectral density $F [Θ \times G^{a}] (k, ω)$ that is weighted around positive group velocities. The dashed line shows the free single-particle dispersion for comparison. The resolution of $F [Θ \times G^{e}]$ is set by $1 / L$ for momenta and $1 / T^{*} \sim v / L$ for frequency. At this resolution there are only exponentially small differences between $F [Θ \times G^{e}]$ and $F [Θ \times G^{o}]$ .
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Figure 6
(a) The difference $| G^{e} (x, t) - G^{o} (x, t) |$ is negligible up to a time $t = T^{*}$ . (b) As a result, $F (Θ \times G^{e}) \approx F (Θ \times G^{o})$ for a cut-off function $Θ (t)$ that drops off after $T^{*}$ . (c) The total momentum resolved 2-particle spectral density of the noninteracting system ( $u_{x} = 0$ ) for a system size of $L = 100$ .
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