arXiv subjects

Mark S. Byrd

Publications and source records attributed to Mark S. Byrd.

At least 37 records · Page 2Linked to original sources

arXiv2010

Casimir Invariants for Systems Undergoing Collective Motion

Dicke states are states of a collection of particles which have been under active investigation for several reasons. One reason is that the decay rates of these states can be quite different from a set of independently evolving particles. Another reason is that a particular class of these states are decoherence-free or noiseless with respect to a set of errors. These noiseless states, or more generally subsystems, can avoid certain types of errors in quantum information processing devices. Here we provide a method for calculating invariants of systems of particles undergoing collective motions. These invariants can be used to determine a complete set of commuting observables for a class of Dicke states as well as identify possible logical operations for decoherence-free/noiseless subsystems. Our method is quite general and provides results for cases where the constituent particles have more than two internal states.

quant-ph↗

arXiv2009

Compatible Transformations for a Qudit Decoherence-free/Noiseless Encoding

The interest in decoherence-free, or noiseless subsystems (DFS/NSs) of quantum systems is both of fundamental and practical interest. Understanding the invariance of a set of states under certain transformations is mutually associated with a better understanding of some fundamental aspects of quantum mechanics as well as the practical utility of invariant subsystems. For example, DFS/NSs are potentially useful for protecting quantum information in quantum cryptography and quantum computing as well as enabling universal computation. Here we discuss transformations which are compatible with a DFS/NS that is composed of d-state systems which protect against collective noise. They are compatible in the sense that they do not take the logical (encoded) states outside of the DFS/NS during the transformation. Furthermore, it is shown that the Hamiltonian evolutions derived here can be used to perform universal quantum computation on a three qudit DFS/NS. Many of the methods used in our derivations are directly applicable to a large variety of DFS/NSs. More generally, we may also state that these transformations are compatible with collective motions.

quant-ph↗

arXiv2008

Robust and Reliable Transfer of a Quantum State Through a Spin Chain

We present several protocols for reliable quantum state transfer through a spin chain. We use a simple two-spin encoding to achieve a remarkably high fidelity transfer for an arbitrary quantum state. The fidelity of the transfer also decreases very slowly with increasing chain length. We find that we can also increase the reliability by taking advantage of a local memory and/or confirm transfer using a second spin-chain.

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arXiv2008

General Depolarized Pure States: Identification and Properties

The Schmidt decomposition is an important tool in the study of quantum systems especially for the quantification of the entanglement of pure states. However, the Schmidt decomposition is only unique for bipartite pure states, and some multipartite pure states. Here a generalized Schmidt decomposition is given for states which are equivalent to depolarized pure states. Experimental methods for the identification of this class of mixed states are provided and some examples are discussed which show the utility of this description. A particularly interesting example provides, for the first time, an interpretation of the number of negative eigenvalues of the density matrix.

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arXiv2008

Methods for Producing Decoherence-Free States and Noiseless Subsystems Using Photonic Qutrits

We outline a proposal for a method of preparing an encoded two-state system (logical qubit) that is immune to collective noise acting on the Hilbert space of the states supporting it. The logical qubit is comprised of three photonic three-state systems (qutrits) and is generated by the process of spontaneous parametric down conversion. The states are constructed using linear optical elements along with three down-conversion sources, and are deemed successful by the simultaneous detection of six events. We also show how to select a maximally entangled state of two qutrits by similar methods. For this maximally entangled state we describe conditions for the state to be decoherence-free which do not correspond to collective errors.

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arXiv2007

Self-Protected Quantum Algorithms Based on Quantum State Tomography

Only a few classes of quantum algorithms are known which provide a speed-up over classical algorithms. However, these and any new quantum algorithms provide important motivation for the development of quantum computers. In this article new quantum algorithms are given which are based on quantum state tomography. These include an algorithm for the calculation of several quantum mechanical expectation values and an algorithm for the determination of polynomial factors. These quantum algorithms are important in their own right. However, it is remarkable that these quantum algorithms are immune to a large class of errors. We describe these algorithms and provide conditions for immunity.

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arXiv2006

Implications of Qudit Superselection rules for the Theory of Decoherence-free Subsystems

The use of d-state systems, or qudits, in quantum information processing is discussed. Three-state and higher dimensional quantum systems are known to have very different properties from two-state systems, i.e., qubits. In particular there exist qudit states which are not equivalent under local unitary transformations unless a selection rule is violated. This observation is shown to be an important factor in the theory of decoherence-free, or noiseless, subsystems. Experimentally observable consequences and methods for distinguishing these states are also provided, including the explicit construction of new decoherence-free or noiseless subsystems from qutrits. Implications for simulating quantum systems with quantum systems are also discussed.

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arXiv2004

Universal Leakage Elimination

``Leakage'' errors are particularly serious errors which couple states within a code subspace to states outside of that subspace thus destroying the error protection benefit afforded by an encoded state. We generalize an earlier method for producing leakage elimination decoupling operations and examine the effects of the leakage eliminating operations on decoherence-free or noiseless subsystems which encode one logical, or protected qubit into three or four qubits. We find that by eliminating the large class of leakage errors, under some circumstances, we can create the conditions for a decoherence free evolution. In other cases we identify a combination decoherence-free and quantum error correcting code which could eliminate errors in solid-state qubits with anisotropic exchange interaction Hamiltonians and enable universal quantum computing with only these interactions.

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arXiv2004

Overview of Quantum Error Prevention and Leakage Elimination

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.''

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arXiv2003

Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation

A parameterization of the density operator, a coherence vector representation, which uses a basis of orthogonal, traceless, Hermitian matrices is discussed. Using this parameterization we find the region of permissible vectors which represent a density operator. The inequalities which specify the region are shown to involve the Casimir invariants of the group. In particular cases, this allows the determination of degeneracies in the spectrum of the operator. The identification of the Casimir invariants also provides a method of constructing quantities which are invariant under {\it local} unitary operations. Several examples are given which illustrate the constraints provided by the positivity requirements and the utility of the coherence vector parameterization.

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arXiv2002

A Parametrization of Bipartite Systems Based on SU(4) Euler Angles

In this paper we give an explicit parametrization for all two qubit density matrices. This is important for calculations involving entanglement and many other types of quantum information processing. To accomplish this we present a generalized Euler angle parametrization for SU(4) and all possible two qubit density matrices. The important group-theoretical properties of such a description are then manifest. We thus obtain the correct Haar (Hurwitz) measure and volume element for SU(4) which follows from this parametrization. In addition, we study the role of this parametrization in the Peres-Horodecki criteria for separability and its corresponding usefulness in calculating entangled two qubit states as represented through the parametrization.

math-ph↗

arXiv2002

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.

quant-ph↗

arXiv2002

Combined encoding and decoupling solution to problems of decoherence and design in solid-state quantum computing

Proposals for scalable quantum computing devices suffer not only from decoherence due to the interaction with their environment, but also from severe engineering constraints. Here we introduce a practical solution to these major concerns, addressing solid state proposals in particular. Decoherence is first reduced by encoding a logical qubit into two qubits, then completely eliminated by an efficient set of decoupling pulse sequences. The same encoding removes the need for single-qubit operations, that pose a difficult design constraint. We further show how the dominant decoherence processes can be identified empirically, in order to optimize the decoupling pulses.

quant-ph↗

arXiv2002

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.

quant-ph↗

arXiv2001

Bang-Bang Operations from a Geometric Perspective

Strong, fast pulses, called ``bang-bang'' controls can be used to eliminate the effects of system-environment interactions. This method for preventing errors in quantum information processors is treated here in a geometric setting which leads to an intuitive perspective. Using this geometric description, we clarify the notion of group symmetrization as an averaging technique, and provide a geometric picture for evaluating errors due to imperfect bang-bang controls. This will provide additional support for the usefulness of such controls as a means for providing more reliable quantum information processing.

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arXiv2000

Bures Measures over the Spaces of Two and Three-Dimensional Density Matrices

Due to considerable recent interest in the use of density matrices for a wide variety of purposes, including quantum computation, we present a general method for their parameterizations in terms of Euler angles. We assert that this is of more fundamental importance than (as several people have remarked to us) ``just another parameterization of the density matrix.'' There are several uses to which this methodology can be put. One that has received particular attention is in the construction of certain distinguished (Bures) measures on the $(n^2 -1)$-dimensional convex sets of $n \times n$ density matrices.

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arXiv1999

Geometric Phases for Three State Systems

The adiabatic geometric phases for general three state systems are discussed. An explicit parameterization for space of states of these systems is given. The abelian and non-abelian connection one-forms or vector potentials that would appear in a three dimensional quantum system with adiabatic characteristics are given explicitly. This is done in terms of the Euler angle parameterization of SU(3) which enables a straight-forward calculation of these quantities and its immediate generalization.

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arXiv1999

Clifford Periodicity from Finite Groups

We deduce the periodicity 8 for the type of $Pin$ and $Spin$ representations of the orthogonal groups $O(n)$ from simple combinatorial properties of the finite Clifford groups generated by the gamma matrices. We also include the case of arbitrary signature $O(p,q)$. The changes in the type of representation can be seen as a rotation in the complex plane. The essential result is that adding a $(+)$ dimension performs a rotation by $π/4$ in the counter clock-wise sense, but for each $(-)$ sign in the metric, the rotation is clockwise.

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