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, andâ€¦ (More)

The discovery of quantum error correction has greatly improved the long-term prospects for quantum computing technology. Encoded quantum information can be protected from errors that arise due toâ€¦ (More)

We prove a new version of the quantum threshold theorem that applies to concatenation of a quantum code that corrects only one error, and we use this theorem to derive a rigorous lower bound on theâ€¦ (More)

The new field of quantum error correction has developed spectacularly since its origin less than two years ago. Encoded quantum information can be protected from errors that arise due to uncontrolledâ€¦ (More)

This paper prove the security of the Bennett-Brassard (BB84) quantum key distribution protocol in the case where the source and detector are under the limited control of an adversary. This proofâ€¦ (More)

We formulate a universal characterization of the many-particle quantum entanglement in the ground state of a topologically ordered two-dimensional medium with a mass gap. We consider a disk in theâ€¦ (More)

We formulate a scheme for fault-tolerant quantum computation that works effectively against highly biased noise, where dephasing is far stronger than all other types of noise. In our scheme, theâ€¦ (More)

Following a suggestion of A. Kitaev, we explore the connection between fault-tolerant quantum computation and nonabelian quantum statistics in two spatial dimensions. A suitably designed spin systemâ€¦ (More)

We prove the security of a quantum key distribution scheme based on transmission of squeezed quantum states of a harmonic oscillator. Our proof employs quantum error-correcting codes that encode aâ€¦ (More)