Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away -we should regard it as a significant step toward the more powerful quantum technologies of the future. Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.

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 the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator for the degrees of freedom in the interior. The von Neumann entropy of , a measure of the entanglement of the interior and exterior variables, has the form S L ÿ , where the ellipsis represents terms that vanish in the limit L ! 1. We show that ÿ is a universal constant characterizing a global feature of the entanglement in the ground state. Using topological quantum field theory methods, we derive a formula for in terms of properties of the superselection sectors of the medium. In a quantum many-body system at zero temperature, a quantum phase transition may occur as a parameter varies in the Hamiltonian of the system. The two phases on either side of a quantum critical point may be characterized by different types of quantum order; the quantum correlations among the microscopic degrees of freedom have qualitatively different properties in the two phases. Yet in some cases, the phases cannot be distinguished by any local order parameter.For example, in two spatial dimensions a system with a mass gap can exhibit topological order [1]. The quantum entanglement in the ground state of a topologically ordered medium has global properties with remarkable consequences. For one thing, the quasiparticle excitations of the system (anyons) exhibit an exotic variant of indistinguishable particle statistics. Furthermore, in the infinite-volume limit the ground-state degeneracy depends on the genus (number of handles) of the closed surface on which the system resides.While it is clear that these unusual properties emerge because the ground state is profoundly entangled, up until now no firm connection has been established between topological order and any quantitative measure of entanglement. In this Letter we provide such a connection by relating topological order to von Neumann entropy, which quantifies the entanglement of a bipartite pure state.Specifically, we consider a disk in the plane, with a smooth boundary of length L, large compared to the correlation length. In the ground state, by tracing out all degrees of freedom in the exterior of the disk, we obtain a marginal density operator for the degrees of freedom in the interior. The von Neumann entropy S ÿtr log of this density operator, a measure of the entanglement of the interior and exterior variables, has the formwhere the ellipsis represents terms that vanish in the limit L ! 1. The coefficient , arising from short wavelength modes localized near the boundary, is nonuniversal and ultraviolet divergent [2], but ÿ (where is nonnegative) is a universal additive constant characterizing a global feature of the entanglement in the ground state. We call ÿ the topologi...

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 threshold͒, 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.