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Local 3D spin Hamiltonian as a thermally stable surface code

Fabio L. Pedrocchi, Adrian Hutter, James R. Wootton, Daniel Loss

Published Sep 24, 2012• 2 citations
OpenAlex

Abstract

We study a 2D toric code embedded in a 3D Heisenberg ferromagnet (FM) in a broken-symmetry state at finite temperature. Stabilizer operators of the toric code are locally coupled to individual spins of the FM. The effects of the low-energy modes of the FM lead to an energy penalty for anyons that grows linearly with L, the linear size of the toric code, and thus to a lifetime of the quantum memory growing exponentially with L. We study the backaction of the toric code onto the FM both analytically and with a Monte-Carlo simulation and show a tilting of the spins close to the code after a time tr independent of L. When t > tr two scenarios are conceivable. Either magnetic pulses are applied to the FM at constant time intervals tr in order to refresh the FM and thus maintain a O(L) energy penalty for the anyons, or the spins of the FM reach the new equilibrium position and the chemical potential scales with ln(L) only. In both scenarios, this provides a stable quantum memory with strictly local bounded-strength interactions in three dimensions.

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We study a 2D toric code embedded in a 3D Heisenberg ferromagnet (FM) in a broken-symmetry state at finite temperature. Stabilizer operators of the toric code are locally coupled to individual spins of the FM. The effects of the low-energy modes of the FM lead to an energy penalty for anyons that grows linearly with L, the linear size of the toric code, and thus to a lifetime of the quantum memory growing exponentially with L. We study the backaction of the toric code onto the FM both analytically and with a Monte-Carlo simulation and show a tilting of the spins close to the code after a time tr independent of L. When t > tr two scenarios are conceivable. Either magnetic pulses are applied to the FM at constant time intervals tr in order to refresh the FM and thus maintain a O(L) energy penalty for the anyons, or the spins of the FM reach the new equilibrium position and the chemical potential scales with ln(L) only. In both scenarios, this provides a stable quantum memory with strictly local bounded-strength interactions in three dimensions.
We propose and study a model of a quantum memory that features self-correcting properties and a lifetime growing arbitrarily with system size at nonzero temperature. This is achieved by locally coupling a two-dimensional $L\ifmmode\times\else\texttimes\fi{}L$ toric code to a three-dimensional (3D) bath of bosons hopping on a cubic lattice. When the stabilizer operators of the toric code are coupled to the displacement operator of the bosons, we solve the model exactly via a polaron transformation and show that the energy penalty to create anyons grows linearly with LLL. When the stabilizer operators of the toric code are coupled to the bosonic density operator, we use perturbation theory to show that the energy penalty for anyons scales with ln(L)\mathrm{ln}(L)ln(L). For a given error model, these energy penalties lead to a lifetime of the stored quantum information growing, respectively, exponentially and polynomially with LLL. Furthermore, we show how to choose an appropriate coupling scheme in order to hinder the hopping of anyons (and not only their creation) with energy barriers that are of the same order as the anyon creation gaps. We argue that a toric code coupled to a 3D Heisenberg ferromagnet realizes our model in its low-energy sector. Finally, we discuss the delicate issue of the stability of topological order in the presence of perturbations. While we do not derive a rigorous proof of topological order, we present heuristic arguments suggesting that topological order remains intact when perturbative operators acting on the toric code spins are coupled to the bosonic environment.
We construct self-correcting 3D surface codes over Zq\mathbb Z_qZq​ with local dimension qqq growing with system size. The model preserves the local geometry and stabilizer support of the ordinary 3D surface code and remains translation invariant in the bulk, while modifying only the stabilizer Pauli powers. We use the Lee metric to characterize error weight and consider local Metropolis dynamics with single-qudit X±1X^{\pm1}X±1 and Z±1Z^{\pm1}Z±1 updates, each changing the energy by at most a constant independent of qqq and system size NNN. We prove that, below the critical temperature, when the local dimension scales as q=Θ(N)q=Θ(N)q=Θ(N), the memory times in both error sectors grow at least as exp⁡[Ω(N1/3)]\exp\left[Ω(N^{1/3})\right]exp[Ω(N1/3)]. Our numerical results further support these analytical results. Together, these results establish growing local Hilbert-space dimension as a new resource for designing self-correcting quantum memories while retaining simple local geometry and local thermal dynamics.
We explore the feasibility of a quantum self-correcting memory based on 3D spin Hamiltonians with topological quantum order in which thermal diffusion of topological defects is suppressed by macroscopic energy barriers. To this end we characterize the energy landscape of stabilizer code Hamiltonians with local bounded-strength interactions which have a topologically ordered ground state but do not have stringlike logical operators. We prove that any sequence of local errors mapping a ground state of such a Hamiltonian to an orthogonal ground state must cross an energy barrier growing at least as a logarithm of the lattice size. Our bound on the energy barrier is tight up to a constant factor for one particular 3D spin Hamiltonian.
We study the two-dimensional toric-code Hamiltonian with effective long-range interactions between its anyonic excitations induced by coupling the toric code to external fields. It has been shown that such interactions allow an arbitrary increase in the lifetime of the stored quantum information by making LLL, the linear size of the memory, larger [Chesi et al., Phys. Rev. A 82, 022305 (2010)]. We show that for these systems the choice of boundary conditions (open boundaries as opposed to periodic boundary conditions) is not a mere technicality; the influence of anyons produced at the boundaries becomes in fact dominant for large enough LLL. This influence can be either beneficial or detrimental. In particular, we study an effective Hamiltonian proposed by Pedrocchi et al. [Phys. Rev. B 83, 115415 (2011)] that describes repulsion between anyons and anyon holes. For this system, we find a lifetime of the stored quantum information that grows exponentially in L2{L}^{2}L2 for both periodic and open boundary conditions, although the exponent in the latter case is found to be less favorable. However, LLL is upper bounded through the breakdown of the perturbative treatment of the underlying Hamiltonian.
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 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.