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 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 robustness of quantum information stored in the degenerate ground space of a local, frustration-free Hamiltonian with commuting terms on a 2D spin lattice. On one hand, a macroscopic energy barrier separating the distinct ground states under local transformations would protect the information from thermal fluctuations. On the other hand, local topological order would shield the ground space from static perturbations. Here we demonstrate that local topological order implies a constant energy barrier, thus inhibiting thermal stability.
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 L. 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). For a given error model, these energy penalties lead to a lifetime of the stored quantum information growing, respectively, exponentially and polynomially with L. 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.
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.
This thesis addresses whether it is possible to build a robust memory device for quantum information. A three-dimensional gapped lattice spin model is found which demonstrates for the first time that a reliable quantum memory at finite temperature is possible, at least to some extent. When quantum information is encoded into a highly entangled ground state of this model and subjected to thermal errors, the errors remain easily correctable for a long time without any active intervention, because a macroscopic energy barrier keeps the errors well localized. As a result, stored quantum information can be retrieved faithfully for a memory time which grows exponentially with the square of the inverse temperature. In contrast, for previously known types of topological quantum storage in three or fewer spatial dimensions the memory time scales exponentially with the inverse temperature, rather than its square. This spin model exhibits a previously unexpected topological quantum order, in which ground states are locally indistinguishable, pointlike excitations are immobile, and the immobility is not affected by small perturbations of the Hamiltonian. The degeneracy of the ground state, though also insensitive to perturbations, is a complicated number-theoretic function of the system size, and the system bifurcates into multiple noninteracting copies of itself under real-space renormalization group transformations. The degeneracy, the excitations, and the renormalization group flow can be analyzed using a framework that exploits the spin model's symmetry and some associated free resolutions of modules over polynomial algebras.
Quantum error correction is necessary for the building of a useful fault-tolerant quantum computer. However, in practice, error correction requires a large amount of fault-tolerance overheads and fast decoding. One promising approach to overcome this challenge is to use codes with a macroscopic energy barrier, which can be used to build self-correcting quantum memory. So far, rigorous bounds on the energy barrier have typically been derived only for specific codes. Determining a lower bound for the energy barrier of codes is generally challenging due to two reasons: there are too many possible error paths (sequences of local errors) that can implement a fixed logical operator, and the logical operator itself is defined only modulo stabilizers. In this thesis, we prove tight bounds on the energy barrier that are applicable to any quantum code obtained from the hypergraph product of two classical codes. If the underlying classical codes are low-density parity-check codes (LDPC), the energy barrier of the quantum code is shown to be the minimum energy barrier of the underlying classical codes (and their transposes) up to an additive O(1) constant. Hypergraph product codes, at best, preserve the energy barrier of the underlying classical codes. Constructing a quantum code with a macroscopic energy barrier requires the underlying classical codes to feature an extensive energy barrier. However, determining the energy barrier for classical codes is challenging, although it is comparatively simpler than for quantum codes because of the absence of stabilizer effects. Utilizing a property of tensor product codes, we demonstrate that higher-dimensional hypergraph products may construct quantum codes with a macroscopic energy barrier, even when the underlying classical codes lack this property. Specifically, the 3D hypergraph product enhances the energy barrier of one type of logical operator.
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.