We propose a method for the stabilisation of quantum computations (including quantum state storage). The method is based on the operation of projection into SYM, the symmetric subspace of the full state space of R redundant copies of the computer. We describe an e#cient algorithm and quantum network e#ecting SYM--projection and discuss the stabilising e#ect of the proposed method in the context of unitary errors generated by hardware imprecision, and nonunitary errors arising from external environmental interaction. Finally, limitations of the method are discussed. 1 Introduction Any realistic model of computation must conform to certain requirements imposed not by the mathematical properties of the model but by the laws of physics. Computations which require an exponentially increasing precision or exponential amount of time, space, energy or any other physical resource are normally regarded as unrealistic and intractable. Any actual computational process is subject to unavoidab...
We propose a method for the stabilisation of quantum computations (including quantum state storage). The method is based on the operation of projection into
SYM, the symmetric subspace of the full state space of
R redundant copies of the computer. We describe an efficient algorithm and quantum network effecting
SYM--projection and discuss the stabilising effect of the proposed method in the context of unitary errors generated by hardware imprecision, and nonunitary errors arising from external environmental interaction. Finally, limitations of the method are discussed.
We propose a method for the stabilization of quantum computations (including quantum state storage). The method is based on the operation of projection into
SYM, the symmetric subspace of the full state space of R redundant copies of the computer. We describe an efficient algorithm and quantum network effecting
SYM--projection and discuss the stabilizing effect of the proposed method in the context of unitary errors generated by hardware imprecision, and nonunitary errors arising from external environmental interaction. Finally, limitations of the method are discussed.
Quantum state purification is the functionality that, given multiple copies of an unknown state, outputs a state with increased purity. This will be an essential building block for near- and middle-term quantum ecosystems before the availability of full fault tolerance, where one may want to suppress errors not only in expectation values but also in quantum states. We propose an effective state purification gadget with a moderate quantum overhead by projecting
M noisy quantum inputs to their symmetric subspace defined by a set of projectors forming a symmetric subgroup with order
M. Our method, applied in every short evolution over
M redundant copies of noisy states, can suppress both coherent and stochastic errors by a factor of
1/M, respectively. This reduces the circuit implementation cost
M times smaller than the state projection to the full symmetric subspace proposed by Barenco et al. more than two decades ago. We also show that our gadget purifies the depolarised inputs with probability
p to asymptotically
O(p2) with an optimal choice of
M when
p is small. The sampling cost scales
O(p−1) for small
p, which is also shown to be asymptotically optimal. Our method provides flexible choices of state purification depending on the hardware restrictions before fully fault-tolerant computation is available.
"The goal of this paper is to review the theoretical basis for achieving a faithful quantum information transmission and processing in the presence of noise. Initially, encoding and decoding, implementing gates and quantum error correction will be considered error-free. Finally, we shall relax this non-realistic assumption, introducing the quantum fault-tolerant concept. The existence of an error threshold permits us to conclude that there is no physical law preventing a quantum computer from being built. An error model based on the depolarising channel will be able to provide a simple estimate of the storage or memory computation error threshold: 5.2 10. The encoding is made by means of the [[7,1,3]] Calderbank-Shor-Steane quantum code, and Shor´s fault-tolerant method is used to measure the stabiliser´s generators."
A mathematical model of computation is an idealized abstraction. We design algorithms and perform analysis on the assumption that the mathematical operations we specify will be carried out exactly, and without error. Physical devices that implement an abstract model of computation are imperfect and of limited precision. For example, when a digital circuit is implemented on a physical circuit board, unwanted electrical noise in the environment may cause components to behave differently than expected, and may cause voltage levels (bit-values) to change. These sources of error must be controlled or compensated for, or else the resulting loss of efficiency may reduce the power of the information-processing device. If individual steps in a computation succeed with probability p, then a computation involving t sequential steps will have a success probability that decreases exponentially as pt. Although it may be impossible to eliminate the sources of errors, we can devise schemes to allow us to recover from errors using a reasonable amount of additional resources. Many classical digital computing devices use error-correcting codes to perform detection of and recovery from errors. The theory of error-correcting codes is itself a mathematical abstraction, but it is one that explicitly accounts for errors introduced by the imperfection and imprecision of realistic devices. This theory has proven extremely effective in allowing engineers to build computing devices that are resilient against errors. Quantum computers are more susceptible to errors than classical digital computers, because quantum mechanical systems are more delicate and more difficult to control. If large-scale quantum computers are to be possible, a theory of quantum error correction is needed. The discovery of quantum error correction has given researchers confidence that realistic large-scale quantum computing devices can be built despite the presence of errors. We begin by considering fundamental concepts for error correction in a classical setting. We will focus on three of these concepts: (a) the characterization of the error model, (b) the introduction of redundancy through encoding, and (c) an error recovery procedure. We will later see that these concepts generalize quite naturally for quantum error correction.
A quantum computer is a device capable of performing computational tasks that depend on characteristically quantum mechanical effects, in particular coherent quantum superposition. Such devices can efficiently perform classes of computation (e.g. factorisation) which are believed to be intractable on any classical computer. This makes it highly desirable to construct such devices. In this paper, we address the last remaining theoretical obstacle to such a construction, namely the problem of stability or error correction. This problem is more substantial in quantum computation than in classical computation because of the delicate nature of the interference phenomena on which quantum computation depends. We present a new, purely quantum mechanical method of error correction, which has no classical analogue, but can serve to stabilise coherent quantum computations. Like the classical methods, it utilises redundancy, but it does not depend on measuring intermediate results of the computation.>