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arXiv:1309.2680 · Physical Review Letters

Self-Correcting Quantum Memories Beyond the Percolation Threshold

Matthew B. Hastings, Grant H. Watson, Roger G. Melko

Published Feb 20, 2014• 27 citations
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Abstract

We analyze several high dimensional generalizations of the toric code at a nonzero temperature. We find that in a large enough dimension, there can be a distinct separation between the critical temperature Tc, given by thermodynamic singularities, and the percolation temperature Tp, given by the percolation of defects. We argue that the regime Tp<T<Tc is a range of temperatures where a self-correcting quantum memory can operate despite having percolating defects. We present analytic arguments and numerical evidence in support of this scenario, including a mean-field treatment and Monte Carlo simulations. Near Tc, simulations observe a large hysteretic behavior, which may have applications by allowing the self-correcting phase to survive in a "superheated" regime.

DOI
10.1103/physrevlett.112.070501

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We analyze several high dimensional generalizations of the toric code at a nonzero temperature. We find that in a large enough dimension, there can be a distinct separation between the critical temperature Tc, given by thermodynamic singularities, and the percolation temperature Tp, given by the percolation of defects. We argue that the regime Tp<T<Tc is a range of temperatures where a self-correcting quantum memory can operate despite having percolating defects. We present analytic arguments and numerical evidence in support of this scenario, including a mean-field treatment and Monte Carlo simulations. Near Tc, simulations observe a large hysteretic behavior, which may have applications by allowing the self-correcting phase to survive in a "superheated" regime.
We introduce two-dimensional toric code models with additional long-range loop interactions. Our models preserve the topologically ordered ground states of the ordinary toric code, while confining anyons at zero temperature. For sufficiently long-ranged interactions, we argue that this confinement persists at finite temperatures, strongly constraining anyon mobility and giving rise to finite-temperature topological order. This topological phase acts as a self-correcting quantum memory and exhibits long-range entanglement. For shorter-ranged interactions, signatures of topological order survive only up to an exponentially large length scale at finite temperature, yielding a partially self-correcting quantum memory.
The ability to store information is of fundamental importance to any computer, be it classical or quantum. To identify systems for quantum memories, which rely, analogously to classical memories, on passive error protection (``self-correction''), is of greatest interest in quantum information science. While systems with topological ground states have been considered to be promising candidates, a large class of them was recently proven unstable against thermal fluctuations. Here, we propose two-dimensional (2D) spin models unaffected by this result. Specifically, we introduce repulsive long-range interactions in the toric code and establish a memory lifetime polynomially increasing with the system size. This remarkable stability is shown to originate directly from the repulsive long-range nature of the interactions. We study the time dynamics of the quantum memory in terms of diffusing anyons and support our analytical results with extensive numerical simulations. Our findings demonstrate that self-correcting quantum memories can exist in 2D at finite temperatures.
A big open question in the quantum information theory concerns the feasibility of a self-correcting quantum memory. A quantum state recorded in such memory can be stored reliably for a macroscopic time without need for active error correction, if the memory is in contact with a cold enough thermal bath. Here we report analytic and numerical evidence for self-correcting behavior in the quantum spin lattice model known as the 3D cubic code. We prove that its memory time is at least L(cβ), where L is the lattice size, β is the inverse temperature of the bath, and c>0 is a constant coefficient. However, this bound applies only if the lattice size L does not exceed a critical value which grows exponentially with β. In that sense, the model can be called a partially self-correcting memory. We also report a Monte Carlo simulation indicating that our analytic bounds on the memory time are tight up to constant coefficients. To model the readout step we introduce a new decoding algorithm, which can be implemented efficiently for any topological stabilizer code. A longer version of this work can be found in Bravyi and Haah, arXiv:1112.3252.
We analyze a time-continuous version of a cellular automaton decoder for the toric code in the form of a Lindblad master equation. In this setting, a self-correcting quantum memory becomes a thermodynamical phase of the steady state, which manifests itself through the steady state being topologically ordered. We compute the steady state phase diagram, finding a competition between the error correction rate and the update rate for the classical field of the cellular automaton. Strikingly, we find that self-correction of errors is possible even in situations where conventional quantum error correction does not have a finite threshold.
When interacting spin systems possess non-zero magnetization or topological entanglement entropy below the transition temperature, they often serve as classical or quantum self-correcting memory. In particular, their memory time grows exponentially in the system size due to polynomially growing energy barrier, as in the 2D Ising model and 4D Toric code. Here, we argue that this is not always the case. We demonstrate that memory time of classical clock model (a generalization of ferromagnet to q-state spins) may be polynomially long even when the system possesses finite magnetization. This weak violation of the Arrhenius law occurs above the percolation temperature, but below the transition temperature, a regime where excitation droplets percolate the entire lattice, yet the system retains a finite magnetization. We present numerical evidences for polynomial scaling as well as analytical argument showing that energy barrier is effectively suppressed and is only logarithmically divergent. We also suggest an intriguing possibility of experimentally observing the precession of magnetization vectors at experimentally relevant time scale.
We explore the possibility of passive error correction in the toric code model. We first show that even coherent dynamics, stemming from spin interactions or the coupling to an external magnetic field, lead to logical errors. We then argue that Anderson localization of the defects, arising from unavoidable fluctuations of the coupling constants, provides a remedy. This protection is demonstrated by using general analytical arguments that are complemented with numerical results which show that self-correcting memory can in principle be achieved in the limit of a nonzero density of identical defects.