Combined Error Correction Techniques for Quantum Computing Architectures (original) (raw)
Related papers
2001
Proposals for physical systems to act as quantum computers are inevitably plagued by the unavoidable coupling with the environment (bath) that causes decoherence, and by technological difficulties connected with the controllability of quantum states. Several techniques exist for achieving reliable quantum computation by countering the effects of decoherence. At this time, however, not one, by itself, will serve as a panacea for error correction. In this paper, we introduce a method that combines system-bath decoupling operations with error avoidance or active error correction in order to address these major concerns. By using an empirical approach to error correction based on experimental data, and an efficient set of decoupling operations that will serve to protect encoded quantum information, we are able to propose a comprehensive method for reducing the adverse effects of decoherence, in particular in scalable solid state quantum computing devices. Our method has the added benefit of significantly reducing design constraints associated with certain difficult-to-implement single-qubit operations in these devices. We illustrate our results by applying them to quantum dot quantum computing proposals.
Experimental Implementation of a Concatenated Quantum Error-Correcting Code
Physical Review Letters, 2005
Concatenated coding provides a general strategy to achieve the desired level of noise protection in quantum information storage and transmission. We report the implementation of a concatenated quantum error-correcting code able to correct against phase errors with a strong correlated component. The experiment was performed using liquid-state nuclear magnetic resonance techniques on a four spin subsystem of labeled crotonic acid. Our results show that concatenation between active and passive quantum error correcting codes offers a practical tool to handle realistic noise contributed by both independent and correlated errors.
Protecting quantum information encoded in decoherence-free states against exchange errors
Physical Review A, 2000
The exchange interaction between identical qubits in a quantum information processor gives rise to unitary two-qubit errors. It is shown here that decoherence free subspaces (DFSs) for collective decoherence undergo Pauli errors under exchange, which however do not take the decoherence free states outside of the DFS. In order to protect DFSs against these errors it is sufficient to employ a recently proposed concatenated DFS-quantum error correcting code scheme [D.A. Lidar, D. Bacon and K.B. Whaley, Phys. Rev. Lett. 82, 4556 (1999)].
Phys Rev a, 2001
Decoherence-free subspaces (DFSs) shield quantum information from errors induced by the interaction with an uncontrollable environment. Here we study a model of correlated errors forming an Abelian subgroup (stabilizer) of the Pauli group (the group of tensor products of Pauli matrices). Unlike previous studies of DFSs, this type of error does not involve any spatial symmetry assumptions on the system-environment interaction. We solve the problem of universal, fault-tolerant quantum computation on the associated class of DFSs. We do so by introducing a hybrid DFS quantum error-correcting-code approach, where errors that arise due to departure of the codewords from the DFS are corrected actively.
Quantum error correction is a set of methods to protect quantum information-that is, quantum states-from unwanted environmental interactions (decoherence) and other forms of noise. The information is stored in a quantum error-correcting code, which is a subspace in a larger Hilbert space. This code is designed so that the most common errors move the state into an error space orthogonal to the original code space while preserving the information in the state. It is possible to determine whether an error has occurred by a suitable measurement and to apply a unitary correction that returns the state to the code space, without measuring (and hence disturbing) the protected state itself. In general, codewords of a quantum code are entangled states. No code that stores information can protect against all possible errors; instead, codes are designed to correct a specific error set, which should be chosen to match the most likely types of noise. An error set is represented by a set of operators that can multiply the codeword state. Most work on quantum error correction has focused on systems of quantum bits, or qubits, which are two-level quantum systems. These can be physically realized by the states of a spin-1/2 particle, the polarization of a single photon, two distinguished levels of a trapped atom or ion, the current states of a microscopic superconducting loop, or many other physical systems. The most widely-used codes are the stabilizer codes, which are closely related to classical linear codes. The code space is the joint +1 eigenspace of a set of commuting Pauli operators on n qubits, called stabilizer generators; the error syndrome is determined by measuring these operators, which allows errors to be diagnosed and corrected. A stabilizer code is characterized by three parameters [[ , , ]], where n is the number of physical qubits, k is the number of encoded logical qubits, and d is the minimum distance of the code (the smallest number of simultaneous qubit errors that can transform one valid codeword into another). Every useful code has n > k; this physical redundancy is necessary to detect and correct errors without disturbing the logical state. Quantum error correction is used to protect information in quantum communication (where quantum states pass through noisy channels) and quantum computation (where quantum states are transformed through a sequence of imperfect computational steps in the presence of environmental decoherence to solve a computational problem). In quantum computation, error correction is just one component of fault-tolerant design. Other approaches to error mitigation in quantum systems include decoherence-free subspaces, noiseless subsystems, and dynamical decoupling.
Quantum error correction beyond qubits
Nature Physics, 2009
Quantum computation and communication rely on the ability to manipulate quantum states robustly and with high fidelity. Thus, some form of error correction is needed to protect fragile quantum superposition states from corruption by socalled decoherence noise. Indeed, the discovery of quantum error correction (QEC) 1,2 turned the field of quantum information from an academic curiosity into a developing technology. Here we present a continuous-variable experimental implementation of a QEC code, based upon entanglement among 9 optical beams 3 . In principle, this 9-wavepacket adaptation of Shor's original 9qubit scheme 1 allows for full quantum error correction against an arbitrary single-beam (singleparty) error.
Effects of noise on quantum error correction algorithms
Physical Review A, 1997
It has recently been shown that there are efficient algorithms for quantum computers to solve certain problems, such as prime factorization, which are intractable to date on classical computers. The chances for practical implementation, however, are limited by decoherence, in which the effect of an external environment causes random errors in the quantum calculation. To combat this problem, quantum error correction schemes have been proposed, in which a single quantum bit (qubit) is "encoded" as a state of some larger number of qubits, chosen to resist particular types of errors. Most such schemes are vulnerable, however, to errors in the encoding and decoding itself. We examine two such schemes, in which a single qubit is encoded in a state of n qubits while subject to dephasing or to arbitrary isotropic noise. Using both analytical and numerical calculations, we argue that error correction remains beneficial in the presence of weak noise, and that there is an optimal time between error correction steps, determined by the strength of the interaction with the environment and the parameters set by the encoding.
Overview of quantum error prevention and leakage elimination
Journal of Modern Optics, 2004
Quantum error prevention strategies will be required to produce a scalable quantum computing device and are of central importance in this regard. Progress in this area has been quite rapid in the past few years. In order to provide an overview of the achievements in this area, we discuss the three major classes of error prevention strategies, the abilities of these methods and the shortcomings. We then discuss the combinations of these strategies which have recently been proposed in the literature. Finally we present recent results in reducing errors on encoded subspaces using decoupling controls. We show how to generally remove mixing of an encoded subspace with external states (termed leakage errors) using decoupling controls. Such controls are known as "leakage elimination operations" or "LEOs." * Electronic address: mbyrd@physics.siu.edu † Electronic address: lwu@chem.utoronto.ca ‡ Electronic address: dlidar@chem.utoronto.ca
Physical Review Letters, 1997
In this paper we study a model quantum register R made of N replicas (cells) of a given finitedimensional quantum system S. Assuming that all cells are coupled with a common environment with equal strength we show that, for N large enough, in the Hilbert space of R there exists a linear subspace CN which is dynamically decoupled from the environment. The states in CN evolve unitarily and are therefore decoherence-dissipation free. The space CN realizes a noiseless quantum code in which information can be stored, in principle, for arbitrarily long time without being affected by errors.
Theory of quantum error-correcting codes
Physical Review A, 1997
Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-errorcorrecting quantum codes, provided that the interaction is dominated by an identity component.