arXiv subjects

Gerald Gilbert

Publications and source records attributed to Gerald Gilbert.

30 records · Page 2Linked to original sources

arXiv2013

Imperfect Construction of Microclusters

Microclusters are the basic building blocks used to construct cluster states capable of supporting fault-tolerant quantum computation. In this paper, we explore the consequences of errors on microcluster construction using two error models. To quantify the effect of the errors we calculate the fidelity of the constructed microclusters and the fidelity with which two such microclusters can be fused together. Such simulations are vital for gauging the capability of an experimental system to achieve fault tolerance.

quant-ph↗

arXiv2012

Superoperator Analysis of Entanglement in a Four-Qubit Cluster State

In this paper we utilize superoperator formalism to explore the entanglement evolution of four-qubit cluster states in a number of decohering environments. A four-qubit cluster state is a resource for the performance of an arbitrary single logical qubit rotation via measurement based cluster state quantum computation. We are specifically interested in the relationship between entanglement evolution and the fidelity with which the arbitrary single logical qubit rotation can be implemented in the presence of decoherence as this will have important experimental ramifications. We also note the exhibition of entanglement sudden death (ESD) and ask how severely its onset affects the utilization of the cluster state as a means of implementing an arbitrary single logical qubit rotation.

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arXiv2010

Storing Small Photonic Cluster States in a Dephasing Environment

We consider the effects of decoherence on the entanglement of photonic cluster states. Large photonic cluster states can be built by fusing together smaller photonic cluster states via probabilistic fusion operations. For this construction process it is necessary to store these smaller cluster states in some way so as to have them available for attempted fusion operations. While in storage the photonic cluster states may undergo dephasing. The effects of dephasing on small, primitive cluster states is explored here with the aim of determining how to locally rotate the qubits of the cluster state so as to lose the least amount of entanglement due to the dephasing process.

quant-ph↗

arXiv2008

A Universal Operator Theoretic Framework for Quantum Fault Tolerance

In this paper we introduce a universal operator theoretic framework for quantum fault tolerance. This incorporates a top-down approach that implements a system-level criterion based on specification of the full system dynamics, applied at every level of error correction concatenation. This leads to more accurate determinations of error thresholds than could previously be obtained. This is demonstrated both formally and with an explicit numerical example. The basis for our approach is the Quantum Computer Condition (QCC), an inequality governing the evolution of a quantum computer. We show that all known coding schemes are actually special cases of the QCC. We demonstrate this by introducing a new, operator theoretic form of entanglement assisted quantum error correction, which incorporates as special cases all known error correcting protocols, and is itself a special case of the QCC.

quant-ph↗

arXiv2007

Reliable Final Computational Results from Faulty Quantum Computation

In this paper we extend both standard fault tolerance theory and Kitaev's model for quantum computation, combining them so as to yield quantitative results that reveal the interplay between the two. Our analysis establishes a methodology that allows us to quantitatively determine design parameters for a quantum computer, the values of which ensure that an overall computation of interest yields a correct *final result* with some prescribed probability of success, as opposed to merely ensuring that the desired *final quantum state* is obtained. As a specific example of the practical application of our approach, we explicitly calculate the number of levels of error correction concatenation needed to achieve a correct final result for the overall computation with some prescribed success probability. Since our methodology allows one to determine parameters required in order to achieve the correct final result for the overall quantum computation, as opposed to merely ensuring that the desired final quantum state is produced, our method enables the determination of complete quantum computational resource requirements associated to the actual solution of practical problems.

quant-ph↗

arXiv2005

Efficient Construction of Photonic Quantum Computational Clusters

We demonstrate a method of creating photonic two-dimensional cluster states that is considerably more efficient than previously proposed approaches. Our method uses only local unitaries and type-I fusion operations. The increased efficiency of our method compared to previously proposed constructions is obtained by identifying and exploiting local equivalence properties inherent in cluster states.

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arXiv2005

Quantum Computer Condition: Stability, Classical Computation and Norms

The Quantum Computer Condition (QCC) provides a rigorous and completely general framework for carrying out analyses of questions pertaining to fault-tolerance in quantum computers. In this paper we apply the QCC to the problem of fluctuations and systematic errors in the values of characteristic parameters in realistic systems. We show that fault-tolerant quantum computation is possible despite variations in these parameters. We also use the QCC to explicitly show that reliable classical computation can be carried out using as input the results of fault-tolerant, but imperfect, quantum computation. Finally, we consider the advantages and disadvantages of the superoperator and diamond norms in connection with application of the QCC to various quantum information-theoretic problems.

quant-ph↗

arXiv2005

Comment on "Operator Quantum Error Correction"

The attempt to equate operator quantum error correction (quant-ph/0504189v1) with the quantum computer condition (quant-ph/0507141) in version two of quant-ph/0504189 is shown to be invalid.

quant-ph↗

arXiv2005

A Theory of Physical Quantum Computation: The Quantum Computer Condition

In this paper we present a new unified theoretical framework that describes the full dynamics of quantum computation. Our formulation allows any questions pertaining to the physical behavior of a quantum computer to be framed, and in principle, answered. We refer to the central organizing principle developed in this paper, on which our theoretical structure is based, as the *Quantum Computer Condition* (QCC), a rigorous mathematical statement that connects the irreversible dynamics of the quantum computing machine, with the reversible operations that comprise the quantum computation intended to be carried out by the quantum computing machine. Armed with the QCC, we derive a powerful result that we call the *Encoding No-Go Theorem*. This theorem gives a precise mathematical statement of the conditions under which fault-tolerant quantum computation becomes impossible in the presence of dissipation and/or decoherence. In connection with this theorem, we explicitly calculate a universal critical damping value for fault-tolerant quantum computation. In addition we show that the recently-discovered approach to quantum error correction known as "operator quantum error-correction" (OQEC) is a special case of our more general formulation. Our approach furnishes what we will refer to as "operator quantum fault-tolerance" (OQFT). In particular, we show how the QCC allows one to derive error thresholds for fault tolerance in a completely general context. We prove the existence of solutions to a class of time-dependent generalizations of the Lindblad equation. Using the QCC, we also show that the seemingly different circuit, graph- (including cluster-) state, and adiabatic paradigms for quantum computing are in fact all manifestations of a single, universal paradigm for all physical quantum computation.

quant-ph↗

arXiv1992

All Or Nothing: On the Small Fluctuations of Two-Dimensional String-Theoretic Black Holes

A comprehensive analysis of small fluctuations about two-dimensional string-theoretic and string-inspired black holes is presented. It is shown with specific examples that two-dimensional black holes behave in a radically different way from all known black holes in four dimensions. For both the $SL(2,R)/U(1)$ black hole and the two-dimensional black hole coupled to a massive dilaton with constant field strength, it is shown that there are a {\it continuous infinity} of solutions to the linearized equations of motion, which are such that it is impossible to ascertain the classical linear response. It is further shown that the two-dimensional black hole coupled to a massive, linear dilaton admits {\it no small fluctuations at all}. We discuss possible implications of our results for the Callan-Giddings-Harvey-Strominger black hole.

hep-th↗

arXiv1991

"the Instability of String-Theoretic Black Holes"

It is demonstrated that static, charged, spherically--symmetric black holes in string theory are classically and catastrophically unstable to linearized perturbations in four dimensions, and moreover that unstable modes appear for arbitrarily small positive values of the charge. This catastrophic classical instability dominates and is distinct from much smaller and less significant effects such as possible quantum mechanical evaporation. The classical instability of the string--theoretic black hole contrasts sharply with the situation which obtains for the Reissner--Nordström black hole of general relativity, which has been shown by Chandrasekhar to be perfectly stable to linearized perturbations at the event horizon.

hep-th↗

arXiv1991

On the Perturbations of String-Theoretic Black Holes

The perturbations of string-theoretic black holes are analyzed by generalizing the method of Chandrasekhar. Attention is focussed on the case of the recently considered charged string-theoretic black hole solutions as a representative example. It is shown that string-intrinsic effects greatly alter the perturbed motions of the string-theoretic black holes as compared to the perturbed motions of black hole solutions of the field equations of general relativity, the consequences of which bear on the questions of the scattering behavior and the stability of string-theoretic black holes. The explicit forms of the axial potential barriers surrounding the string-theoretic black hole are derived. It is demonstrated that one of these, for sufficiently negative values of the asymptotic value of the dilaton field, will inevitably become negative in turn, in marked contrast to the potentials surrounding the static black holes of general relativity. Such potentials may in principle be used in some cases to obtain approximate constraints on the value of the string coupling constant. The application of the perturbation analysis to the case of two-dimensional string-theoretic black holes is discussed.

hep-th↗