Topological quantum memory

TL;DR

Topological quantum memory analyzes surface codes (toric and planar) as robust, locality-friendly quantum memory and as a platform for fault-tolerant universal quantum computation. By mapping error recovery to a three-dimensional $Z_2$ lattice gauge theory with quenched disorder, the paper derives threshold phenomena and provides both analytical lower bounds and practical recovery strategies, including minimum-weight chains and overlapping recovery. It also details syndrome-measurement circuits, encoding/measurement protocols, and a fault-tolerant universal gate set, with additional exploration of a four-dimensional local-recovery limit. The results argue that surface codes offer scalable protection against decoherence with local quantum processing and polynomial classical support, making them a promising route for future quantum technologies.

Abstract

We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional Z_2 lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.

Figures (21)

• Figure 1: Check operators of the toric code. Each plaquette operator is a tensor product of $Z$'s acting on the four links contained in the plaquette. Each site operator is a tensor product of $X$'s acting on the four links that meet at the site.
• Figure 2: Cycles on the lattice. $(a)$ A homologically trivial cycle bounds a region that can be tiled by plaquettes. The corresponding tensor product of $Z$'s lies in the stabilizer of the toric code. $(b)$ A homologically nontrivial cycle is not a boundary. The corresponding tensor product of $Z$'s commutes with the stabilizer but is not contained in it. It is a logical operation that acts nontrivially in the code subspace.
• Figure 3: Basis for the operators that act on the two encoded qubits of the toric code. The logical operators $\bar{Z}_1$ and $\bar{Z}_2$ are tensor products of $Z$'s associated with the fundamental nontrivial cycles of the torus constructed from links of the lattice. The complementary operators $\bar{X}_1$ and $\bar{X}_2$ are tensor products of $X$'s associated with nontrivial cycles constructed from links of the dual lattice.
• Figure 4: The highly ambiguous syndrome of the toric code. The two site defects shown could arise from errors on either one of the two chains shown. In general, error chains with the same boundary generate the same syndrome, and error chains that are homologically equivalent act on the code space in the same way.
• Figure 5: A planar quantum code. $(a)$ At the top and bottom are the "plaquette edges" (or "rough edges") where there are three-qubit plaquette operators, and at the left and right are the "site edges" (or "smooth edges") where there are three-qubit site operators. The logical operation $\bar{Z}$ for the one encoded qubit is a tensor product of $Z$'s acting on a chain running from one rough edge to the other, and the logical operation $\bar{X}$ is a tensor product of $X$'s acting on a chain of the dual lattice running from one smooth edge to the other. For the lattice shown, the code's distance is $L=8$. $(b)$ Site and plaquette defects can appear singly, rather than in pairs. An isolated site defect arises from an error chain that ends at a rough edge, and an isolated plaquette defect arises from a dual error chain that ends at a smooth edge.