Combined Error Correction Techniques for Quantum Computing Architectures

Proposals for quantum computing devices are many and varied. They each have unique noise processes that make none of them fully reliable at this time. There are several error correction/avoidance techniques which are valuable for reducing or eliminating errors, but not one, alone, will serve as a panacea. One must therefore take advantage of the strength of each of these techniques so that we may extend the coherence times of the quantum systems and create more reliable computing devices. To this end we give a general strategy for using dynamical decoupling operations on encoded subspaces. These encodings may be of any form; of particular importance are decoherence-free subspaces and quantum error correction codes. We then give means for empirically determining an appropriate set of dynamical decoupling operations for a given experiment. Using these techniques, we then propose a comprehensive encoding solution to many of the problems of quantum computing proposals which use exchange-type interactions. This uses a decoherence-free subspace and an efficient set of dynamical decoupling operations. It also addresses the problems of controllability in solid state quantum dot devices.Comment: Contribution to Proceedings of the 2002 Physics of Quantum Electronics Conference", to be published in J. Mod. Optics. This paper provides a summary and review of quant-ph/0205156 and quant-ph/0112054, and some new result

Schaffenrathʼs Inscription Column in Pisani rov, Postojnska jama

Napisi lahko pripomorejo k razjasnitvi nezadostno dokumentirane zgodovine odkrivanja glavnih rovov Postojnske jame. Ta čas je bil povezan s tremi osebami: Josipom Jeršinovičem plemenitim Löwengreif, Alojzom Schaffenrathom in grofom Francem Hohenwartom. Temelječ na sodobnih zapisih Schaffenratha (1834), Hohenwarta (1830, 1832a,b) in Schmidla (1854), avtorji razpravljajo o okoliščinah in času njihovih raziskovanj glavne jame ter menijo, da ta ni bila odkrita do prihoda nadvojvode Ferdinanda avgusta 1819. Eden najstarejših napisov iz tega časa je na kapniškem stebru v Pisanem rovu, 90 m od tam, kjer se odcepi od glavnega rova. Tu je Schaffenrath 1825 zapisal imeni Löwengreifa, Gospodaritscha in svoje. Ta steber je morda edino mesto v Postojnski jami, kjer so vsa tri imena skupaj. Če upoštevamo razmeroma pozne raziskave glavnega rova, je letnica 1825 morda leto odkritja tega dela jame. To potrjuje tudi dejstvo, da tega dela jame ni na prvem objavljenem zemljevidu (Bronn, 1826, temelječ na zemljevidu Foÿker/Schaffenrath iz okoli 1821). 1832 je bil odprt notranji del Pisanega rova in imenovan v čast nadvojvode Janeza. Na steber so dodali še več napisov, več pa jih je tudi dalje po rovu. Iz 1836 je podpis J(ozef) Hauer, to je paleontolog in oče Franca plemenitega Hauerja. Tudi Anton Perko, mlajši brat Ivana Andreja, je zapustil svoje ime. I.A. Perko je podpisan 1892 v Rovu brez imena, v letu preden so on, njegov brat in drugi v Trstu ustanovili študentsko jamarsko društvo “Hades”. Raziskovanje in dokumentiranje zgodovinskih napisov lahko pomaga pri rekonstrukciji in razlagi zgodovine raziskav in odkrivanj te najpomembnejše turistične jame.Inscriptions may help to clarify the incompletely documented early history of the discovery of the main passages in Postojnska jama. This period is associated with three people: Josef (Josip) Jeršinovič Ritter von Löwengreif, Alois Schaffenrath, and Franz Graf von Hohenwart. Based on the contemporary writings of Schaffenrath (1834), Hohenwart (1830, 1832a,b) and Schmidl (1854) the authors discuss the circumstances and timing of the exploration of the main cave, suggesting that the main passage was not discovered until after the visit of Erzherzog Ferdinand in August 1819. One of the earliest inscriptions from that period is found on a column in Pisani rov, 90 m from its branch from the main passage. Here Schaffenrath left in 1825 the names of Löwengreif, of Gospodaritsch, and of himself. This column may be the only site in Postojnska jama featuring all three names in one place. In view of the rather late exploration of the main passage, the date 1825 may be the discovery date of this section of the cave since it does not appear on the earliest map published (Bronn, 1826, based on a map of Foÿker/Schaffenrath ca. 1821). In 1832 the back part of Pisani rov was opened and named in honour of Erzherzog Johann. Several more inscriptions were placed on the column. Further down the passage a few more inscriptions exist. One was dated 1836 by J(ozef) Hauer, a paleontologist and the father of Franz Ritter von Hauer. Also Anton Perko, the younger brother of Ivan Andrej Perko left his name. I.A. Perko signed as well, but in the Rov brez imena, in the year 1892, a year before he, his brother and others founded the student caversʼ club “Hades” in Trieste. Search and documentation of historic inscriptions may therefore aid in reconstructing the exploration and visitation history of this most important show cave

Entanglement measures and approximate quantum error correction

It is shown that, if the loss of entanglement along a quantum channel is sufficiently small, then approximate quantum error correction is possible, thereby generalizing what happens for coherent information. Explicit bounds are obtained for the entanglement of formation and the distillable entanglement, and their validity naturally extends to other bipartite entanglement measures in between. Robustness of derived criteria is analyzed and their tightness compared. Finally, as a byproduct, we prove a bound quantifying how large the gap between entanglement of formation and distillable entanglement can be for any given finite dimensional bipartite system, thus providing a sufficient condition for distillability in terms of entanglement of formation.Comment: 7 pages, two-columned revtex4, no figures. v1: Deeply revised and extended version: different entanglement measures are separately considered, references are added, and some remarks are stressed. v2: Added a sufficient condition for distillability in terms of entanglement of formation; published versio

Encoded Universality for Generalized Anisotropic Exchange Hamiltonians

We derive an encoded universality representation for a generalized anisotropic exchange Hamiltonian that contains cross-product terms in addition to the usual two-particle exchange terms. The recently developed algebraic approach is used to show that the minimal universality-generating encodings of one logical qubit are based on three physical qubits. We show how to generate both single- and two-qubit operations on the logical qubits, using suitably timed conjugating operations derived from analysis of the commutator algebra. The timing of the operations is seen to be crucial in allowing simplification of the gate sequences for the generalized Hamiltonian to forms similar to that derived previously for the symmetric (XY) anisotropic exchange Hamiltonian. The total number of operations needed for a controlled-Z gate up to local transformations is five. A scalable architecture is proposed.Comment: 11 pages, 4 figure

Optimal estimation of two-qubit pure-state entanglement

We present optimal measuring strategies for the estimation of the entanglement of unknown two-qubit pure states and of the degree of mixing of unknown single-qubit mixed states, of which N identical copies are available. The most general measuring strategies are considered in both situations, to conclude in the first case that a local, although collective, measurement suffices to estimate entanglement, a non-local property, optimally.Comment: REVTEX, 9 pages, 1 figur

Three qubits can be entangled in two inequivalent ways

Invertible local transformations of a multipartite system are used to define equivalence classes in the set of entangled states. This classification concerns the entanglement properties of a single copy of the state. Accordingly, we say that two states have the same kind of entanglement if both of them can be obtained from the other by means of local operations and classical communcication (LOCC) with nonzero probability. When applied to pure states of a three-qubit system, this approach reveals the existence of two inequivalent kinds of genuine tripartite entanglement, for which the GHZ state and a W state appear as remarkable representatives. In particular, we show that the W state retains maximally bipartite entanglement when any one of the three qubits is traced out. We generalize our results both to the case of higher dimensional subsystems and also to more than three subsystems, for all of which we show that, typically, two randomly chosen pure states cannot be converted into each other by means of LOCC, not even with a small probability of success.Comment: 12 pages, 1 figure; replaced with revised version; terminology adapted to earlier work; reference added; results unchange

Asymptotic entanglement in a two-dimensional quantum walk

The evolution operator of a discrete-time quantum walk involves a conditional shift in position space which entangles the coin and position degrees of freedom of the walker. After several steps, the coin-position entanglement (CPE) converges to a well defined value which depends on the initial state. In this work we provide an analytical method which allows for the exact calculation of the asymptotic reduced density operator and the corresponding CPE for a discrete-time quantum walk on a two-dimensional lattice. We use the von Neumann entropy of the reduced density operator as an entanglement measure. The method is applied to the case of a Hadamard walk for which the dependence of the resulting CPE on initial conditions is obtained. Initial states leading to maximum or minimum CPE are identified and the relation between the coin or position entanglement present in the initial state of the walker and the final level of CPE is discussed. The CPE obtained from separable initial states satisfies an additivity property in terms of CPE of the corresponding one-dimensional cases. Non-local initial conditions are also considered and we find that the extreme case of an initial uniform position distribution leads to the largest CPE variation.Comment: Major revision. Improved structure. Theoretical results are now separated from specific examples. Most figures have been replaced by new versions. The paper is now significantly reduced in size: 11 pages, 7 figure