34 research outputs found
Quantum Stabilizer Codes and Classical Linear Codes
We show that within any quantum stabilizer code there lurks a classical
binary linear code with similar error-correcting capabilities, thereby
demonstrating new connections between quantum codes and classical codes. Using
this result -- which applies to degenerate as well as nondegenerate codes --
previously established necessary conditions for classical linear codes can be
easily translated into necessary conditions for quantum stabilizer codes.
Examples of specific consequences are: for a quantum channel subject to a
delta-fraction of errors, the best asymptotic capacity attainable by any
stabilizer code cannot exceed H(1/2 + sqrt(2*delta*(1-2*delta))); and, for the
depolarizing channel with fidelity parameter delta, the best asymptotic
capacity attainable by any stabilizer code cannot exceed 1-H(delta).Comment: 17 pages, ReVTeX, with two figure
Efficient Computations of Encodings for Quantum Error Correction
We show how, given any set of generators of the stabilizer of a quantum code,
an efficient gate array that computes the codewords can be constructed. For an
n-qubit code whose stabilizer has d generators, the resulting gate array
consists of O(n d) operations, and converts k-qubit data (where k = n - d) into
n-qubit codewords.Comment: 16 pages, REVTeX, 3 figures within the tex
Correcting the effects of spontaneous emission on cold trapped ions
We propose two quantum error correction schemes which increase the maximum
storage time for qubits in a system of cold trapped ions, using a minimal
number of ancillary qubits. Both schemes consider only the errors introduced by
the decoherence due to spontaneous emission from the upper levels of the ions.
Continuous monitoring of the ion fluorescence is used in conjunction with
selective coherent feedback to eliminate these errors immediately following
spontaneous emission events, and the conditional time evolution between quantum
jumps is removed by symmetrizing the quantum codewords.Comment: 19 pages; 2 figures; RevTex; The quantum codewords are extended to
achieve invariance under the conditional time evolution between jump
Entanglement required in achieving entanglement-assisted channel capacities
Entanglement shared between the two ends of a quantum communication channel
has been shown to be a useful resource in increasing both the quantum and
classical capacities for these channels. The entanglement-assisted capacities
were derived assuming an unlimited amount of shared entanglement per channel
use. In this paper, bounds are derived on the minimum amount of entanglement
required per use of a channel, in order to asymptotically achieve the capacity.
This is achieved by introducing a class of entanglement-assisted quantum codes.
Codes for classes of qubit channels are shown to achieve the quantum
entanglement-assisted channel capacity when an amount of shared entanglement
per channel given by, E = 1 - Q_E, is provided. It is also shown that for very
noisy channels, as the capacities become small, the amount of required
entanglement converges for the classical and quantum capacities.Comment: 9 pages, 2 figures, RevTex
Heating and decoherence suppression using decoupling techniques
We study the application of decoupling techniques to the case of a damped
vibrational mode of a chain of trapped ions, which can be used as a quantum bus
in linear ion trap quantum computers. We show that vibrational heating could be
efficiently suppressed using appropriate ``parity kicks''. We also show that
vibrational decoherence can be suppressed by this decoupling procedure, even
though this is generally more difficult because the rate at which the parity
kicks have to applied increases with the effective bath temperature.Comment: 13 pages, 5 figures. Typos corrected, references adde
Basic concepts in quantum computation
Section headings: 1 Qubits, gates and networks 2 Quantum arithmetic and
function evaluations 3 Algorithms and their complexity 4 From interferometers
to computers 5 The first quantum algorithms 6 Quantum search 7 Optimal phase
estimation 8 Periodicity and quantum factoring 9 Cryptography 10 Conditional
quantum dynamics 11 Decoherence and recoherence 12 Concluding remarksComment: 37 pages, lectures given at les Houches Summer School on "Coherent
Matter Waves", July-August 199
Resonant cancellation of off-resonant effects in a multilevel qubit
Off-resonant effects are a significant source of error in quantum
computation. This paper presents a group theoretic proof that off-resonant
transitions to the higher levels of a multilevel qubit can be completely
prevented in principle. This result can be generalized to prevent unwanted
transitions due to qubit-qubit interactions. A simple scheme exploiting dynamic
pulse control techniques is presented that can cancel transitions to higher
states to arbitrary accuracy.Comment: 4 pages, Revtex, submitted for publicatio
Topological defects: A problem for cyclic universes?
We study the behaviour of cosmic string networks in contracting universes,
and discuss some of their possible consequences. We note that there is a
fundamental time asymmetry between defect network evolution for an expanding
universe and a contracting universe. A string network with negligible loop
production and small-scale structure will asymptotically behave during the
collapse phase as a radiation fluid. In realistic networks these two effects
are important, making this solution only approximate. We derive new scaling
solutions describing this effect, and test them against high-resolution
numerical simulations. A string network in a contracting universe, together
with the gravitational radiation background it has generated, can significantly
affect the dynamics of the universe both locally and globally. The network can
be an important source of radiation, entropy and inhomogeneity. We discuss the
possible implications of these findings for bouncing and cyclic cosmological
models.Comment: 11 RevTeX 4 pages, 6 figures; version to appear in Phys. Rev.
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
Decoherence control in microwave cavities
We present a scheme able to protect the quantum states of a cavity mode
against the decohering effects of photon loss. The scheme preserves quantum
states with a definite parity, and improves previous proposals for decoherence
control in cavities. It is implemented by sending single atoms, one by one,
through the cavity. The atomic state gets first correlated to the photon number
parity. The wrong parity results in an atom in the upper state. The atom in
this state is then used to inject a photon in the mode via adiabatic transfer,
correcting the field parity. By solving numerically the exact master equation
of the system, we show that the protection of simple quantum states could be
experimentally demonstrated using presently available experimental apparatus.Comment: 13 pages, RevTeX, 8 figure