A fault-tolerant one-way quantum computer

We describe a fault-tolerant one-way quantum computer on cluster states in three dimensions. The presented scheme uses methods of topological error correction resulting from a link between cluster states and surface codes. The error threshold is 1.4% for local depolarizing error and 0.11% for each source in an error model with preparation-, gate-, storage- and measurement errors.Comment: 26 page

Fault-tolerant quantum computation

Recently, it was realized that use of the properties of quantum mechanics might speed up certain computations dramatically. Interest in quantum computation has since been growing. One of the main difficulties of realizing quantum computation is that decoherence tends to destroy the information in a superposition of states in a quantum computer, thus making long computations impossible. A futher difficulty is that inaccuracies in quantum state transformations throughout the computation accumulate, rendering the output of long computations unreliable. It was previously known that a quantum circuit with t gates could tolerate O(1/t) amounts of inaccuracy and decoherence per gate. We show, for any quantum computation with t gates, how to build a polynomial size quantum circuit that can tolerate O(1/(log t)^c) amounts of inaccuracy and decoherence per gate, for some constant c. We do this by showing how to compute using quantum error correcting codes. These codes were previously known to provide resistance to errors while storing and transmitting quantum data.Comment: Latex, 11 pages, no figures, in 37th Symposium on Foundations of Computing, IEEE Computer Society Press, 1996, pp. 56-6

Topological quantum memory

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.Comment: 39 pages, 21 figures, REVTe

Quantum information and statistical mechanics: an introduction to frontier

This is a short review on an interdisciplinary field of quantum information science and statistical mechanics. We first give a pedagogical introduction to the stabilizer formalism, which is an efficient way to describe an important class of quantum states, the so-called stabilizer states, and quantum operations on them. Furthermore, graph states, which are a class of stabilizer states associated with graphs, and their applications for measurement-based quantum computation are also mentioned. Based on the stabilizer formalism, we review two interdisciplinary topics. One is the relation between quantum error correction codes and spin glass models, which allows us to analyze the performances of quantum error correction codes by using the knowledge about phases in statistical models. The other is the relation between the stabilizer formalism and partition functions of classical spin models, which provides new quantum and classical algorithms to evaluate partition functions of classical spin models.Comment: 15pages, 4 figures, to appear in Proceedings of 4th YSM-SPIP (Sendai, 14-16 December 2012

Symmetry protected topological order at nonzero temperature

We address the question of whether symmetry-protected topological (SPT) order can persist at nonzero temperature, with a focus on understanding the thermal stability of several models studied in the theory of quantum computation. We present three results in this direction. First, we prove that nontrivial SPT order protected by a global on-site symmetry cannot persist at nonzero temperature, demonstrating that several quantum computational structures protected by such on-site symmetries are not thermally stable. Second, we prove that the 3D cluster state model used in the formulation of topological measurement-based quantum computation possesses a nontrivial SPT-ordered thermal phase when protected by a global generalized (1-form) symmetry. The SPT order in this model is detected by long-range localizable entanglement in the thermal state, which compares with related results characterizing SPT order at zero temperature in spin chains using localizable entanglement as an order parameter. Our third result is to demonstrate that the high error tolerance of this 3D cluster state model for quantum computation, even without a protecting symmetry, can be understood as an application of quantum error correction to effectively enforce a 1-form symmetry.Comment: 42 pages, 10 figures, comments welcome; v2 published versio

Multi-party Quantum Computation

We investigate definitions of and protocols for multi-party quantum computing in the scenario where the secret data are quantum systems. We work in the quantum information-theoretic model, where no assumptions are made on the computational power of the adversary. For the slightly weaker task of verifiable quantum secret sharing, we give a protocol which tolerates any t < n/4 cheating parties (out of n). This is shown to be optimal. We use this new tool to establish that any multi-party quantum computation can be securely performed as long as the number of dishonest players is less than n/6.Comment: Masters Thesis. Based on Joint work with Claude Crepeau and Daniel Gottesman. Full version is in preparatio

Multiple Particle Interference and Quantum Error Correction

The concept of multiple particle interference is discussed, using insights provided by the classical theory of error correcting codes. This leads to a discussion of error correction in a quantum communication channel or a quantum computer. Methods of error correction in the quantum regime are presented, and their limitations assessed. A quantum channel can recover from arbitrary decoherence of x qubits if K bits of quantum information are encoded using n quantum bits, where K/n can be greater than 1-2 H(2x/n), but must be less than 1 - 2 H(x/n). This implies exponential reduction of decoherence with only a polynomial increase in the computing resources required. Therefore quantum computation can be made free of errors in the presence of physically realistic levels of decoherence. The methods also allow isolation of quantum communication from noise and evesdropping (quantum privacy amplification).Comment: Submitted to Proc. Roy. Soc. Lond. A. in November 1995, accepted May 1996. 39 pages, 6 figures. This is now the final version. The changes are some added references, changed final figure, and a more precise use of the word `decoherence'. I would like to propose the word `defection' for a general unknown error of a single qubit (rotation and/or entanglement). It is useful because it captures the nature of the error process, and has a verb form `to defect'. Random unitary changes (rotations) of a qubit are caused by defects in the quantum computer; to entangle randomly with the environment is to form a treacherous alliance with an enemy of successful quantu

Phase Structure of the Random-Plaquette Z_2 Gauge Model: Accuracy Threshold for a Toric Quantum Memory

We study the phase structure of the random-plaquette Z_2 lattice gauge model in three dimensions. In this model, the "gauge coupling" for each plaquette is a quenched random variable that takes the value \beta with the probability 1-p and -\beta with the probability p. This model is relevant for the recently proposed quantum memory of toric code. The parameter p is the concentration of the plaquettes with "wrong-sign" couplings -\beta, and interpreted as the error probability per qubit in quantum code. In the gauge system with p=0, i.e., with the uniform gauge couplings \beta, it is known that there exists a second-order phase transition at a certain critical "temperature", T(\equiv \beta^{-1}) = T_c =1.31, which separates an ordered(Higgs) phase at T<T_c and a disordered(confinement) phase at T>T_c. As p increases, the critical temperature T_c(p) decreases. In the p-T plane, the curve T_c(p) intersects with the Nishimori line T_{N}(p) at the certain point (p_c, T_{N}(p_c)). The value p_c is just the accuracy threshold for a fault-tolerant quantum memory and associated quantum computations. By the Monte-Carlo simulations, we calculate the specific heat and the expectation values of the Wilson loop to obtain the phase-transition line T_c(p) numerically. The accuracy threshold is estimated as p_c \simeq 0.033.Comment: 24 pages, 14 figures, some clarification