Showing papers on "Quantum error correction published in 1995"
TL;DR: U(2) gates are derived, which derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number of unitary operations on arbitrarily many bits.
Abstract: We show that a set of gates that consists of all one-bit quantum gates (U(2)) and the two-bit exclusive-or gate (that maps Boolean values (x,y) to (x,x ⊕y)) is universal in the sense that all unitary operations on arbitrarily many bits n (U(2 n )) can be expressed as compositions of these gates. We investigate the number of the above gates required to implement other gates, such as generalized Deutsch-Toffoli gates, that apply a specific U(2) transformation to one input bit if and only if the logical AND of all remaining input bits is satisfied. These gates play a central role in many proposed constructions of quantum computational networks. We derive upper and lower bounds on the exact number of elementary gates required to build up a variety of two- and three-bit quantum gates, the asymptotic number required for n-bit Deutsch-Toffoli gates, and make some observations about the number required for arbitrary n-bit unitary operations.
3,731 citations
TL;DR: In the mid-1990s, theorists devised methods to preserve the integrity of quantum bits\char22{}techniques that may become the key to practical quantum computing on a large scale.
Abstract: In the mid-1990s, theorists devised methods to preserve the integrity of quantum bits---techniques that may become the key to practical quantum computing on a large scale.
3,668 citations
TL;DR: The operation of a two-bit "controlled-NOT" quantum logic gate is demonstrated, which, in conjunction with simple single-bit operations, forms a universal quantum logic Gate for quantum computation.
Abstract: We demonstrate the operation of a two-bit "controlled-NOT" quantum logic gate, which, in conjunction with simple single-bit operations, forms a universal quantum logic gate for quantum computation. The two quantum bits are stored in the internal and external degrees of freedom of a single trapped atom, which is first laser cooled to the zero-point energy. Decoherence effects are identified for the operation, and the possibility of extending the system to more qubits appears promising.
1,377 citations
IBM1
TL;DR: A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit.
Abstract: A proof is given, which relies on the commutator algebra of the unitary Lie groups, that quantum gates operating on just two bits at a time are sufficient to construct a general quantum circuit. The best previous result had shown the universality of three-bit gates, by analogy to the universality of the Toffoli three-bit gate of classical reversible computing. Two-bit quantum gates may be implemented by magnetic resonance operations applied to a pair of electronic or nuclear spins. A ``gearbox quantum computer'' proposed here, based on the principles of atomic-force microscopy, would permit the operation of such two-bit gates in a physical system with very long phase-breaking (i.e., quantum-phase-coherence) times. Simpler versions of the gearbox computer could be used to do experiments on Einstein-Podolsky-Rosen states and related entangled quantum states.
1,073 citations
TL;DR: A simple quantum logic gate, the quantum controlled-NOT, is described, and two possible physical realizations of the gate are discussed, one based on Ramsey atomic interferometry and the other on the selective driving of optical resonances of two subsystems undergoing a dipole-dipole interaction.
Abstract: Quantum logic gates provide fundamental examples of conditional quantum dynamics. They could form the building blocks of general quantum information processing systems which have recently been shown to have many interesting nonclassical properties. We describe a simple quantum logic gate, the quantum controlled-NOT, and analyze some of its applications. We discuss two possible physical realizations of the gate, one based on Ramsey atomic interferometry and the other on the selective driving of optical resonances of two subsystems undergoing a dipole-dipole interaction.
849 citations
TL;DR: This work uses the theory of continuous measurement to analyze the effects of decoherence on a realistic model of a quantum computer based on cavity QED and shows howDecoherence affects the computation, and methods to prevent it.
Abstract: We use the theory of continuous measurement to analyze the effects of decoherence on a realistic model of a quantum computer based on cavity QED. We show how decoherence affects the computation, and methods to prevent it.
635 citations
TL;DR: This work identifies a 2-bit quantum gate that is sufficient to build any quantum logic network and proposes an explicit construction of this gate, which is based on cavity QED techniques and may be realizable with current technology.
Abstract: We identify a 2-bit quantum gate that is sufficient to build any quantum logic network. The existence of such a 2-bit universal gate considerably simplifies the search for physical realizations of quantum computational networks. We propose an explicit construction of this gate, which is based on cavity QED techniques and may be realizable with current technology.
437 citations
TL;DR: An implementation of a quantum computer to solve Deutsch's problem, which requires exponential time on a classical computer but only linear time with quantum parallelism, is proposed by using a dual-rail quantum-bit representation as a simple form of error correction.
Abstract: We propose an implementation of a quantum computer to solve Deutsch's problem, which requires exponential time on a classical computer but only linear time with quantum parallelism. By using a dual-rail quantum-bit representation as a simple form of error correction, our machine can tolerate some amount of decoherence and still give the correct result with high probability. The design that we employ also demonstrates a signature for quantum parallelism which unambiguously distinguishes the desired quantum behavior from the merely classical. The experimental demonstration of our proposal using quantum optical components calls for the development of several key technologies common to single photonics.
364 citations
TL;DR: In this article, a reduction mapping of the quantum density matrix is proposed for mixed quantum-classical systems. But the quantum decoherence problem is not addressed in this paper, instead, the classical paths are restricted to a single path among all the quantum paths.
Abstract: We address the issue of quantum decoherence in mixed quantum‐classical simulations. We demonstrate that restricting the classical paths to a single path among all the quantum paths affects a coarse graining of the quantum paths. Such coarse graining causes the quantum paths to lose coherence as the various possible classical paths associated with each quantum state diverge. This defines a reduction mapping of the quantum density matrix, and we derive a quantum master equation suitable for mixed quantum‐classical systems. The equation includes two terms: first, the ordinary quantum Liouvillian which is parametrized by a single classical path, and second, a quantum decoherence term that includes both a coherence time and length scale which are determined by the dynamics of the classical paths. Model calculations for electronic coherence loss in nonadiabatic mixed quantum‐classical dynamics are presented as examples. For a model charge transfer chemical reaction with nonadiabatic transitions, application of ...
357 citations
TL;DR: Here it is shown how the decoherence process degrades the interference pattern that emerges from the quantum factoring algorithm, a problem of practical significance for cryptographic applications.
Abstract: It is known that quantum computers can dramatically speed up the task of finding factors of large numbers, a problem of practical significance for cryptographic applications. Factors of an L -digit number can be found in ∼ L 2 time [compared to ∼exp( L 1/3 ) time] by a quantum computer, which simultaneously follows all paths corresponding to distinct classical inputs, obtaining the solution from the coherent quantum interference of the alternatives. Here it is shown how the decoherence process degrades the interference pattern that emerges from the quantum factoring algorithm. For a quantum computer performing logical operations, an exponential decay of quantum coherence is inevitable. However, even in the presence of exponential decoherence, quantum computation can be useful as long as a sufficiently low decoherence rate can be achieved to allow meaningful results to be extracted from the calculation.
234 citations
Book•
01 Jan 1995
TL;DR: This work discusses how local and non-local coherence are generated and how these may affect the function of composite systems andumerical examples concern quantum trajectories, quantum noise and quantum parallelism.
Abstract: The superposition principle makes quantum networks behave very differently from their
classical counterparts: We discuss how local and non-local coherence are generated and
how these may affect the function of composite systems. Numerical examples concern
quantum trajectories, quantum noise and quantum parallelism.
TL;DR: In this article, the authors discuss the quantum interference, superposition states of light and the nonclassical effects of quantum coherences between coherent states, and associate quantum correlations with quantum interference in the phase space.
Abstract: Publisher Summary This chapter discusses the quantum interference, superposition states of light and the nonclassical effects. The superposition principle is at the heart of quantum mechanics. The coherent state forms a position-momentum, phase-space patch of minimum area, and is the quantum analog of the classical point in phase space. The chapter discusses how quantum interference between coherent states results in nonclassical effects, and associates quantum coherences between coherent states with quantum interference in the phase space. Schrodinger's argument is based on stability and invariance of Gaussian wavepackets of an isolated quantum harmonic oscillator. The effect of coarsening is identical to the influence of reservoirs on quantum interference. The chapter describes several methods proposed for the generation of quantum-mechanical superposition states of light. The methods for the detection of schrodinger cats include homodyne detection, DAP-QND detection scheme and optical homodyne tomography method. These methods provide a complete quantum mechanical description of the measured mode and therefore, can be applied for characterization of quantum-mechanical superposition states, providing these fields are directly measureable.
Proceedings Article•
27 Aug 1995
TL;DR: It is shown that any cryptosystem based on what is referred to as a ‘hidden linear form’ can be broken in quantum polynomial time and the notion of ‘junk bits’ is introduced which are helpful when performing classical computations that are not injective.
Abstract: Recently there has been a great deal of interest in the power of "Quantum Computers" [4, 15, 18]. The driving force is the recent beautiful result of Shor that shows that discrete log and factoring are solvable in random quantum polynomial time [15]. We use a method similar to Shor's to obtain a general theorem about quantum polynomial time. We show that any cryptosystem based on what we refer to as a 'hidden linear form' can be broken in quantum polynomial time. Our results imply that the discrete log problem is doable in quantum polynomial time over any group including Galois fields and elliptic curves. Finally, we introduce the notion of 'junk bits' which are helpful when performing classical computations that are not injective.
TL;DR: In this article, the problem of optimal measurement of complex amplitude for a quantum Markovian oscillator, loaded on a quantum wave communication line, is considered, and the optimal filtering of a quantum signal with the Gaussian white quantum noise can be described by a coherent Markovians linear filter corresponding to quantum Gaussian state diffusion.
Abstract: Time-continuous non-anticipating processes of nondemolition measurements in quantum systems are described. In particular, the notion of physically realisable quantum filter is introduced and the problem of its optimisation to obtain the best a posteriori quantum state is considered. The fact that the optimal filtering of a quantum Markovian Gaussian signal with the Gaussian white quantum noise can be described by a coherent Markovian linear filter corresponding to quantum Gaussian state diffusion is proved. As an example, the problem of optimal measurement of complex amplitude for a quantum Markovian oscillator, loaded on a quantum wave communication line, is considered.
01 Jan 1995
TL;DR: In this paper, Belavkin et al. proposed a Hamiltonian solution to Quantum Collapse, State Diffusion, and Spontaneous Localization, which is a solution to the problem of Quantum Measurement Problem of Collapse.
Abstract: Quantum States and Input-Output Processes: The World of Quantum Noise and the Fundamental Output Process V.P. Belavkin, et al. Geometry of Quantum States S.L. Braunstein, C.M. Caves. Quantum Measurement Problem of Collapse: Mathematical Characterizations of Measurement Statistics M. Ozawa. Stochastic Dynamics of Continuously Observed Quantum Systems P. Staszewski. Quantum Jumps, Diffusion, and Localization: A Hamiltonian Solution to Quantum Collapse, State Diffusion, and Spontaneous Localization V.P. Belavkin, O. Melsheimer. Dissipation and Reduction of Superconducting States Due to Spontaneous Localization M. Buffa, et al. Quantum Channels, Entropy, and Information: State Change, Complexity and Fractals in Quantum Systems M. Ohya. Information in Mary Quantum Optical Communications: An Inequality Providing an Upper Limit A. Vourdas. Quantum Detection, Estimation, and Filtering: Quantum Filtering of Markov Signals with White Quantum Noise V.P. Belavkin. Covariant POVmeasures for W*Dynamical Systems T. Breuer. Quantum Optics, Experiments and Simulation: Quantum Noise Eaters C. Fabre, et al. Stochastic Simulations of Dissipation in Quantum Optics: Quantum Superpositions B.M. Garraway, P.L. Knight. 40 additional articles. Index.
TL;DR: In this article, a quantum state diffusion with a moving basis (MQSD) method is proposed for numerical simulation of open quantum systems, which takes full advantage of the localization of quantum states into wavepackets occupying small regions of classical phase space.
Abstract: Numerical simulation of individual open quantum systems has proven advantages over density operator computations. Quantum state diffusion with a moving basis (MQSD) provides a practical numerical simulation method which takes full advantage of the localization of quantum states into wavepackets occupying small regions of classical phase space. Following and extending the original proposal of Percival, Alber and Steimle (1995), we show that MQSD can provide a further gain over ordinary QSD and other quantum trajectory methods of many orders of magnitude in computational space and time. Because of these gains, it is even possible to calculate an open quantum system trajectory when the corresponding isolated system is intractable. MQSD is particularly advantageous where classical or semiclassical dynamics provides an adequate qualitative picture but is numerically inaccurate because of significant quantum effects. The principles are illustrated by computations for the quantum Duffing oscillator and for second-harmonic generation in quantum optics. Potential applications in atomic and molecular dynamics, quantum circuits and quantum computation are suggested.
TL;DR: In this article, the authors use a clock to read zero when a light signal is sent from A towards B, where a mirror is stationed to reflect the light signal back to A. The measurement can be carried out in the following way: a clock is put a t point A, and when the signal arrives at A, one deduces that the length AB is given by e = ct /2, where c denotes the speed of light.
Abstract: We start by recalling the fundamental nature of space-time distance measurements.'\" In quantum mechanics, one specifies a space-time point by its coordinates, without bothering to give a prescription as to how these coordinates are to be measured. However, general relativity ordains that coordinates d o not have any meaning independent of observations; in other words, according to relativity, a coordinate system is defined only by explicitly carrying out space-time distance measurements. We will pay heed to this dictum of general relativity and will follow Salecker and Wigner' to use clocks and light signals to measure distances (see reference 5): Suppose we want to measure the length between two spatially separated pointsA and B. The measurement can be carried out in the following way. A clock is put a t point A . Set the clock to read zero when a light signal is sent fromA towards B, where a mirror is stationed to reflect the light signal back to A . From the reading of the clock, to be denoted by t , when the light signal arrives at A, one deduces that the length AB is given by e = ct /2 , where c denotes the speed of light. There are two major sources of errors in the length measurement: one comes from the uncertainty principle of quantum mechanics and the other is due to space-time curvature effects. First, we note that the clock is not absolutely stationary, its spread in speed being given by the uncertainty principle of quantum mechanics: -J. A. Wheeler (private correspondence)
TL;DR: It is shown that quantum-nondemolition-mediated feedback is able to preserve the interference fringes of the superposition of macroscopically distinguishable quantum states.
Abstract: It is shown that quantum-nondemolition-mediated feedback is able to preserve the interference fringes of the superposition of macroscopically distinguishable quantum states.
TL;DR: In this article, an account of noise in electrical circuits and its relation to quantum noise connected with Heisenberg's uncertainty principle is given, where experiments on squeezing in optical fibers and the attendant noise reduction are presented.
Abstract: An account is given of noise in electrical circuits and its relation to quantum noise connected with Heisenberg’s uncertainty principle. A phase-insensitive amplifier must introduce noise (most commonly the spontaneous emission noise) in order to satisfy the constraint imposed by quantum mechanics on the simultaneous measurement of two observables represented by noncommuting operators. Nonstationary, phase-sensitive amplifiers can employ squeezed radiation and reduce the noise below that of phase-insensitive amplifiers. Experiments on squeezing in optical fibers and the attendant noise reduction are presented. The squeezing apparatus can also be employed as a quantum nondemolition measurement of photon number. A thought experiment of a two-slit interferometer is described, in which the fringes are washed out progressively with the accuracy of the photon number determination in one of its arms. Implications of measurements of this kind for the resolution of the Einstein–Podolsky–Rosen paradox are discussed.
TL;DR: In this paper, the exact action for a damped mechanical system was derived from the path integral formulation of the quantum Brownian motion problem developed by Schwinger and by Feynman and Vernon.
TL;DR: In this paper, a high-accuracy method for removing ''time slice'' errors in Trotter-approximated propagators is presented, which is simultaneously fast, high precision and computationally simple and can be applied to a wide variety of quantum propagators.
Abstract: Path integrals are a powerful method for calculating real time, finite temperature, and ground state properties of quantum systems. By exploiting some remarkable properties of the symmetric Trotter formula and the discrete Fourier transform, we arrive at a high-accuracy method for removing ``time slice'' errors in Trotter-approximated propagators. We provide an explicit demonstration of the method applied to the two-body density matrix of $^{4}\mathrm{He}$. Our method is simultaneously fast, high precision, and computationally simple and can be applied to a wide variety of quantum propagators.
TL;DR: The quantum Langevin equation is derived from the Feynman-Vernon forward-backward path integral for a density matrix of a quantum system in a thermal oscillator bath as discussed by the authors.
Posted Content•
TL;DR: A new technique based on the use of coding is proposed in order to detect and correct errors due to imperfect transmission lines in quantum cryptography or memories in quantum computers.
Abstract: Recent progress in quantum cryptography and quantum computers has given hope to their imminent practical realization. An essential element at the heart of the application of these quantum systems is a quantum error correction scheme. We propose a new technique based on the use of coding in order to detect and correct errors due to imperfect transmission lines in quantum cryptography or memories in quantum computers. We give a particular example of how to detect a decohered qubit in order to transmit or preserve with high fidelity the original qubit.
Posted Content•
TL;DR: The agenda of quantum algorithmic information theory, ordered ‘top-down,’ is the quantum halting amplitude, followed by the quantumgorithmic information content, which in turn requires the theory of quantum computation, based upon a model of universal quantum computer whose elementary unit is a two-port interferometer capable of arbitrary U(2) transformations.
Abstract: The agenda of quantum algorithmic information theory, ordered ‘top-down,’ is the quantum halting amplitude, followed by the quantum algorithmic information content, which in turn requires the theory of quantum computation. The fundamental atoms processed by quantum computation are the quantum bits which are dealt with in quantum information theory. The theory of quantum computation will be based upon a model of universal quantum computer whose elementary unit is a two-port interferometer capable of arbitrary U(2) transformations. Basic to all these considerations is quantum theory, in particular Hilbert space quantum mechanics.
TL;DR: The wave function of a single quantum system could be measured, therefore giving the wave function an ontological significance, i.e., physical reality in its own right.
Abstract: It is shown that a series of repeated weak quantum nondemolition measurements performed on a single quantum system gives no information about the wave function of the system. The physical explanation, based on the quantum Brownian motion and the continuous collapse of the wave function which originate in the projection postulate, is discussed in two specific examples.
TL;DR: In this article, a semi-classical implementation of the Schrodinger equation using Gaussian wave packet (GWP) dynamics has been proposed for quantum control in many-atom systems with sufficient accuracy and computational speed.
TL;DR: In this paper, the key ideas of the environment-induced decoherence approach are reviewed, and the application of decoboxing to the transition from quantum to classical in open quantum systems with chaotic classical analogs is described.
Abstract: The key ideas of the environment-induced decoherence approach are reviewed. Application of decoherence to the transition from quantum to classical in open quantum systems with chaotic classical analogs is described. The arrow of time is, in this context, a result of the information loss to the correlations with the environment. The asymptotic rate of entropy production (which is reached quickly, on the dynamical time scale) isindependent of the details of the coupling of the quantum system to the environment, and is set by the Lyapunov exponents. We also briefly outline theexistential interpretation of quantum mechanics, justifying the slogan «No information without representation».
TL;DR: In this article, a model of repeated quantum measurements of magnetic flux in superconducting circuits manifesting tunneling is discussed, where the perturbation due to the previous measurements is always present unless quantum non-destruction measurements are performed.