skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Number of representations providing noiseless subsystems

Abstract

This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free (DF) subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function f{sub d}(n) which is the fraction of all d-dimensional quantum systems which preserve n bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.

Authors:
 [1]
  1. Department of Physics, Harvard University, 17 Oxford Street, Cambridge, Massachusetts 02138 (United States)
Publication Date:
OSTI Identifier:
20786291
Resource Type:
Journal Article
Resource Relation:
Journal Name: Physical Review. A; Journal Volume: 72; Journal Issue: 6; Other Information: DOI: 10.1103/PhysRevA.72.062328; (c) 2006 The American Physical Society; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
74 ATOMIC AND MOLECULAR PHYSICS; ALGEBRA; DISTRIBUTION; ERRORS; HILBERT SPACE; LIE GROUPS; QUANTUM COMPUTERS; QUANTUM DECOHERENCE

Citation Formats

Ritter, William Gordon. Number of representations providing noiseless subsystems. United States: N. p., 2005. Web. doi:10.1103/PHYSREVA.72.0.
Ritter, William Gordon. Number of representations providing noiseless subsystems. United States. doi:10.1103/PHYSREVA.72.0.
Ritter, William Gordon. Thu . "Number of representations providing noiseless subsystems". United States. doi:10.1103/PHYSREVA.72.0.
@article{osti_20786291,
title = {Number of representations providing noiseless subsystems},
author = {Ritter, William Gordon},
abstractNote = {This paper studies the combinatoric structure of the set of all representations, up to equivalence, of a finite-dimensional semisimple Lie algebra. This has intrinsic interest as a previously unsolved problem in representation theory, and also has applications to the understanding of quantum decoherence. We prove that for Hilbert spaces of sufficiently high dimension, decoherence-free (DF) subspaces exist for almost all representations of the error algebra. For decoherence-free subsystems, we plot the function f{sub d}(n) which is the fraction of all d-dimensional quantum systems which preserve n bits of information through DF subsystems, and note that this function fits an inverse beta distribution. The mathematical tools which arise include techniques from classical number theory.},
doi = {10.1103/PHYSREVA.72.0},
journal = {Physical Review. A},
number = 6,
volume = 72,
place = {United States},
year = {Thu Dec 15 00:00:00 EST 2005},
month = {Thu Dec 15 00:00:00 EST 2005}
}
  • Noiseless subsystems offer a general and efficient method for protecting quantum information in the presence of noise that has symmetry properties. A paradigmatic class of error models displaying nontrivial symmetries emerges under collective noise behavior, which implies a permutationally invariant interaction between the system and the environment. We expand our previous investigation of the noiseless subsystem idea [L. Viola et al., Science 293, 2059 (2001)] by reporting and analyzing NMR experiments that demonstrate the preservation of a qubit encoded in a three-qubit noiseless subsystem for general collective noise. A complete set of input states is used to determine the superoperatormore » for the implemented one-qubit process and to confirm that the fidelity of entanglement is improved for a large, noncommutative set of engineered errors. To date, this is the largest set of error operators that has been successfully corrected for by any quantum code.« less
  • We examine the problem of verifying the implementation of a desired noiseless subsystem from a quantum error-correcting perspective. General conditions are identified, under which the verification of error-correcting behavior against a linear set of errors E suffices for the verification of noiseless subsystems of an error algebra A contained in E. From a practical standpoint, our results imply that the verification of a noiseless subsystem need not require the explicit verification of noiseless behavior for all possible initial states of the syndrome subsystem.
  • We combine dynamical decoupling and universal control methods for open quantum systems with coding procedures. By exploiting a general algebraic approach, we show how appropriate encodings of quantum states result in obtaining universal control over dynamically generated noise-protected subsystems with limited control resources. In particular, we provide a constructive scheme based on two-body Hamiltonians for performing universal quantum computation over large noiseless spaces which can be engineered in the presence of arbitrary linear quantum noise.
  • The T-matrix formulation of electromagnetic scattering given previously by Waterman for the case of one scatterer is extended to the case of an arbitrary number of scatterers. The resulting total T matrix is expressed in terms of the individual T matrices by an iterative procedure. The essential tools used in the extension are the expansions associated with a translation of the origin for the spherical-wave solutions of Helmholtz's equation. The connection between these expansions and the unitary irreducible representations and associated local representations of the threedimensional Euclidean group E(3) is emphasized. Some applications to two spheres are given. (auth)