Symmetry and topology can provide for self-correcting quantum codes in a 3D topologically ordered spin lattice, a key insight for building quantum memories that correct their own errors without active control.
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 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.
In recent years, topological phases of matter have presented exciting new avenues to achieve scalable quantum computation. In this thesis, we investigate a class of quantum many-body spin models known as symmetry-protected topological (SPT) phases for use in quantum information processing and storage. We explore the fault-tolerant properties of SPT phases, and how they can be utilized in the design of a quantum computer. Of central importance in this thesis is the concept of quantum error-correction, which in addition to its importance in fault-tolerant quantum computation, is used to characterise the stability of topological phases at finite temperature. We begin with an introduction to quantum computation, quantum error correction, and topological phases of matter. We then focus on the fundamental question of whether symmetry-protected topological phases of matter can exist in thermal equilibrium; we prove that systems protected by global onsite symmetries cannot be ordered at nonzero temperature. Subsequently, we show that certain three-dimensional models with generalised higher-form symmetries can be thermally SPT ordered, and we relate this order to the ability to perform fault-tolerant measurement-based quantum computation. Following this, we assess feasibility of these phases as quantum memories, motivated by the fact that SPT phases in three dimensions can possess protected topological degrees of freedom on their boundary. We find that certain SPT ordered systems can be self-correcting, allowing quantum information to be stored for arbitrarily long times without requiring active error correction. Finally, we develop a framework to construct new schemes of fault-tolerant measurement-based quantum computation. As a notable example, we develop a cluster-state scheme that simulates the braiding and fusion of surface-code defects, offering novel alternative methods to achieve fault-tolerant universal quantum computation.