Symmetry-protected dissipative preparation of matrix product states

Leo Zhou, Soonwon Choi, and Mikhail D. Lukin

Phys. Rev. A 104, 032418 – Published 22 September 2021

Abstract

We propose and analyze a method for efficient dissipative preparation of matrix product states that exploits their symmetry properties. Specifically, we construct an explicit protocol that makes use of driven-dissipative dynamics to prepare a many-body quantum state that features symmetry-protected topological order and nontrivial edge excitations. The preparation protocol is protected from errors that respect the symmetry, allowing for robust experimental implementation without fine-tuned control. Numerical simulations show that the preparation time scales polynomially in system size $n$ . Furthermore, we demonstrate that this scaling can be improved to $O ({log}^{2} n)$ by using parallel preparation of individual segments and fusing them via quantum feedback. A concrete scheme using excitation of trapped neutral atoms into the Rydberg state via electromagnetically induced transparency is proposed and generalizations to a broader class of matrix product states are discussed.

Received 5 October 2020
Accepted 14 October 2020

DOI:https://doi.org/10.1103/PhysRevA.104.032418

Physics Subject Headings (PhySH)

Optical pumping Quantum control Quantum feedback Quantum state engineering Quantum stochastic processes Topological phases of matter

Matrix product states Symmetries in condensed matter

Quantum Information, Science & TechnologyAtomic, Molecular & OpticalInterdisciplinary PhysicsCondensed Matter, Materials & Applied PhysicsStatistical Physics & Thermodynamics

Authors & Affiliations

Leo Zhou ^*, Soonwon Choi^*, and Mikhail D. Lukin

Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

^*These authors contributed equally to this work.

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Vol. 104, Iss. 3 — September 2021

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Images

Figure 1
Visualization of incoherent quantum jumps as random walks on a directed graph in Hilbert space $H$ . Here $G$ is the subspace of steady states that do not undergo quantum jumps. In the absence of the dashed arrow $c_{μ^{*}}$ , the subspace of three states $S$ is closed under a quantum jump, allowing a mixed steady state to form. The presence of $c_{μ^{*}}$ eliminates this possibility.
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Figure 2
(a) and (b) Numerical simulation of $L_{MP}$ for $Γ = 1$ with a maximally mixed initial state, using exact diagonalization (ED) for system size up to $n = 8$ . Both the energy density $〈 H_{AKLT} 〉 / (n - 1)$ and the state-preparation fidelity $F$ in the long-time regime are fitted to an exponential function (dashed lines). (c) A log-log plot of the fitted preparation time to achieve $F = 0.9$ from simulations using ED and time-evolving block decimation (TEBD) algorithms, as a function of system size, up to $n = 25$ . Error bars are 90% confidence intervals.
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Figure 3
(a) Scheme for preparing AKLT states in parallel to achieve logarithmic scaling. Many short chains of spins in AKLT states are prepared initially, and adjacent chains are connected probabilistically in parallel. Failures are addressed with quantum feedback on every other segment before reattempting the connections. (b) Illustration of the connection algorithm, where the success of each attempt occurs with probability $p$ , while failure can be corrected by discarding two spins and reattempting. Note that only one success is necessary among all the attempts. On average, $1 / p$ attempts are sufficient to obtain a successful connection, with a constant overhead of $2 (1 - p) / p$ spins.
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Figure 4
(a) Atomic level diagrams for Rydberg EIT implementation of our jump operators, where $| r 〉$ is a Rydberg level with strong interaction and $| e 〉$ is an excited level with short lifetime $1 / γ$ . The lower three levels encode the spin-1 particles. (b) Pulse sequence to engineer $L_{MP}$ . (c) Effects of the finite dephasing time $T_{2}$ of spin-1 levels and long-range interaction. We use steady-state fidelity $F_{SS}$ from numerical simulations to calculate effective temperature $T_{eff}$ in units of $Δ_{gap}$ , the energy gap of $H_{AKLT}$ .
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Figure 5
Preparation time in the parallelized protocol with detector efficiency $1 - η = 0.2$ and dark count rate $r = 25$ Hz, using jump operators of the form $c = {| + + 〉}_{} 〈 + + |$ (method 1). We also assume a quantum jump rate $τ_{0}^{- 1} = 1$ MHz, ideal case success probability $p = \frac{2}{9}$ , time to discard atoms and reset edges $τ_{r} = τ_{0}$ , and target final error $E = 10^{- 4}$ .
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