Quantum Error Correction via Convex Optimization

We show that the problem of designing a quantum information error correcting procedure can be cast as a bi-convex optimization problem, iterating between encoding and recovery, each being a semidefinite program. For a given encoding operator the problem is convex in the recovery operator. For a given method of recovery, the problem is convex in the encoding scheme. This allows us to derive new codes that are locally optimal. We present examples of such codes that can handle errors which are too strong for codes derived by analogy to classical error correction techniques.Comment: 16 page

Empirical Determination of Bang-Bang Operations

Strong and fast "bang-bang" (BB) pulses have been recently proposed as a means for reducing decoherence in a quantum system. So far theoretical analysis of the BB technique relied on model Hamiltonians. Here we introduce a method for empirically determining the set of required BB pulses, that relies on quantum process tomography. In this manner an experimenter may tailor his or her BB pulses to the quantum system at hand, without having to assume a model Hamiltonian.Comment: 14 pages, 2 eps figures, ReVTeX4 two-colum

Universal control of quantum subspaces and subsystems

We describe a broad dynamical-algebraic framework for analyzing the quantum control properties of a set of naturally available interactions. General conditions under which universal control is achieved over a set of subspaces/subsystems are found. All known physical examples of universal control on subspaces/systems are related to the framework developed here.Comment: 4 Pages RevTeX, Some typos fixed, references adde

Universal quantum control in irreducible state-space sectors: application to bosonic and spin-boson systems

We analyze the dynamical-algebraic approach to universal quantum control introduced in P. Zanardi, S. Lloyd, quant-ph/0305013. The quantum state-space $\cal H$ encoding information decomposes into irreducible sectors and subsystems associated to the group of available evolutions. If this group coincides with the unitary part of the group-algebra \CC{\cal K} of some group $\cal K$ then universal control is achievable over the ${\cal K}$-irreducible components of $\cal H$. This general strategy is applied to different kind of bosonic systems. We first consider massive bosons in a double-well and show how to achieve universal control over all finite-dimensional Fock sectors. We then discuss a multi-mode massless case giving the conditions for generating the whole infinite-dimensional multi-mode Heisenberg-Weyl enveloping-algebra. Finally we show how to use an auxiliary bosonic mode coupled to finite-dimensional systems to generate high-order non-linearities needed for universal control.Comment: 10 pages, LaTeX, no figure

Dynamical Decoupling Using Slow Pulses: Efficient Suppression of 1/f Noise

The application of dynamical decoupling pulses to a single qubit interacting with a linear harmonic oscillator bath with $1/f$ spectral density is studied, and compared to the Ohmic case. Decoupling pulses that are slower than the fastest bath time-scale are shown to drastically reduce the decoherence rate in the $1/f$ case. Contrary to conclusions drawn from previous studies, this shows that dynamical decoupling pulses do not always have to be ultra-fast. Our results explain a recent experiment in which dephasing due to $1/f$ charge noise affecting a charge qubit in a small superconducting electrode was successfully suppressed using spin-echo-type gate-voltage pulses.Comment: 5 pages, 3 figures. v2: Many changes and update

Encoded Recoupling and Decoupling: An Alternative to Quantum Error Correcting Codes, Applied to Trapped Ion Quantum Computation

A recently developed theory for eliminating decoherence and design constraints in quantum computers, ``encoded recoupling and decoupling'', is shown to be fully compatible with a promising proposal for an architecture enabling scalable ion-trap quantum computation [D. Kielpinski et al., Nature 417, 709 (2002)]. Logical qubits are encoded into pairs of ions. Logic gates are implemented using the Sorensen-Molmer (SM) scheme applied to pairs of ions at a time. The encoding offers continuous protection against collective dephasing. Decoupling pulses, that are also implemented using the SM scheme directly to the encoded qubits, are capable of further reducing various other sources of qubit decoherence, such as due to differential dephasing and due to decohered vibrational modes. The feasibility of using the relatively slow SM pulses in a decoupling scheme quenching the latter source of decoherence follows from the observed 1/f spectrum of the vibrational bath.Comment: 12 pages, no figure

The <i>Castalia</i> mission to Main Belt Comet 133P/Elst-Pizarro

We describe Castalia, a proposed mission to rendezvous with a Main Belt Comet (MBC), 133P/Elst-Pizarro. MBCs are a recently discovered population of apparently icy bodies within the main asteroid belt between Mars and Jupiter, which may represent the remnants of the population which supplied the early Earth with water. Castalia will perform the first exploration of this population by characterising 133P in detail, solving the puzzle of the MBC’s activity, and making the first in situ measurements of water in the asteroid belt. In many ways a successor to ESA’s highly successful Rosetta mission, Castalia will allow direct comparison between very different classes of comet, including measuring critical isotope ratios, plasma and dust properties. It will also feature the first radar system to visit a minor body, mapping the ice in the interior. Castalia was proposed, in slightly different versions, to the ESA M4 and M5 calls within the Cosmic Vision programme. We describe the science motivation for the mission, the measurements required to achieve the scientific goals, and the proposed instrument payload and spacecraft to achieve these

Quantitative Treatment of Decoherence

We outline different approaches to define and quantify decoherence. We argue that a measure based on a properly defined norm of deviation of the density matrix is appropriate for quantifying decoherence in quantum registers. For a semiconductor double quantum dot qubit, evaluation of this measure is reviewed. For a general class of decoherence processes, including those occurring in semiconductor qubits, we argue that this measure is additive: It scales linearly with the number of qubits.Comment: Revised version, 26 pages, in LaTeX, 3 EPS figure

Rotational Surfaces in L3\mathbb{L}^3 and Solutions in the Nonlinear Sigma Model

The Gauss map of non-degenerate surfaces in the three-dimensional Minkowski space are viewed as dynamical fields of the two-dimensional O(2,1) Nonlinear Sigma Model. In this setting, the moduli space of solutions with rotational symmetry is completely determined. Essentially, the solutions are warped products of orbits of the 1-dimensional groups of isometries and elastic curves in either a de Sitter plane, a hyperbolic plane or an anti de Sitter plane. The main tools are the equivalence of the two-dimensional O(2,1) Nonlinear Sigma Model and the Willmore problem, and the description of the surfaces with rotational symmetry. A complete classification of such surfaces is obtained in this paper. Indeed, a huge new family of Lorentzian rotational surfaces with a space-like axis is presented. The description of this new class of surfaces is based on a technique of surgery and a gluing process, which is illustrated by an algorithm.Comment: PACS: 11.10.Lm; 11.10.Ef; 11.15.-q; 11.30.-j; 02.30.-f; 02.40.-k. 45 pages, 11 figure

Quantum Computing in the Presence of Detected Spontaneous Emission

A new method for quantum computation in the presence of detected spontaneous emission is proposed. The method combines strong and fast (dynamical decoupling) pulses and a quantum error correcting code that encodes $n$ logical qubits into only $n+1$ physical qubits. Universal fault-tolerant quantum computation is shown to be possible in this scheme using Hamiltonians relevant to a range of promising proposals for the physical implementation of quantum computers.Comment: 7 pages, no figures. This version corrects an error in the description of spontaneous emission in the quantum jumps picture. As a consequence the error correcting code and some aspects of the preparation, computation, and recovery operations have been modified. The main conclusions of the published paper remain intact. An erratum will be published shortly in Phys. Rev. A, detailing all the corrections required in the published paper. The present version includes all these corrections in the body of the pape