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arXiv:quant-ph/0304018

Realistic Decoherence Free Subspaces

K. M. Fonseca-Romero, S. G. Mokarzel, Marcelo Terra Cunha, Maria Carolina Nemes

Published Apr 2, 2003• 1 citation
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Abstract

Decoherence free subspaces (DFS) is a theoretical tool towards experimental implementation of quantum information storage and processing. However, they represent an experimental challenge, since conditions for their existence are very stringent. This work explores the situation in which a system of $N$ oscillators coupled to a bath of harmonic oscillators is close to satisfy the conditions for the existence of DFS. We show, in the Born-Markov limit and for small deviations from separability and degeneracy conditions, that there are {\emph{weak decoherence subspaces}} which resemble the original notion of DFS.

DOI
10.48550/arxiv.quant-ph/0304018

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Decoherence free subspaces (DFS) is a theoretical tool towards experimental implementation of quantum information storage and processing. However, they represent an experimental challenge, since conditions for their existence are very stringent. This work explores the situation in which a system of NNN oscillators coupled to a bath of harmonic oscillators is close to satisfy the conditions for the existence of DFS. We show, in the Born-Markov limit and for small deviations from separability and degeneracy conditions, that there are {\emph{weak decoherence subspaces}} which resemble the original notion of DFS.
We introduce a generalized theory of decoherence-free subspaces and subsystems (DFSs), which do not require accurate initialization. We derive a set of conditions for the existence of DFSs within this generalized framework. By relaxing the initialization requirement we show that a DFS can tolerate arbitrarily large preparation errors. This has potentially significant implications for experiments involving DFSs, in particular for the experimental implementation, over DFSs, of the large class of quantum algorithms which can function with arbitrary input states.
We consider N identical oscillators coupled to a single environment and show that the conditions for the existence of decoherence free subspaces are degeneracy of the oscillator frequencies and separability of the coupling with the environment. A formal exact equation for the evolution in the case of two oscillators is found and the decoherence free subspace is explicitly determined. A full analytical solution for any initial condition and general parameters (frequencies and dissipation constants) is given in the markovian approximation and zero temperature. We find that slight relaxation of degeneracy and separability conditions leads to the appearence of two components in the dynamical evolution with very different decoherence times. The ratio between the characteristic time of the weak and strong decoherent components is given by τWD/τSD≈k/δkτ_{WD}/τ_{SD}\approx k/δkτWD​/τSD​≈k/δk, where δkδkδk is a measure of the nonseparability of the coupling to the environment and kkk is the mean decay constant of the oscillators.
Analyzing the S matrices associated with a host of relaxation processes allows one, using extra ``chaperon'' states, to construct a decoherence-free subspace (DFS) that is immune to the effects of relaxation. The method does not require knowledge of the system-bath Hamiltonian, which is rarely known. Thus, a DFS can be constructed directly from experimentally determined relaxation and dephasing rates of the relevant decoherence processes. Although the method requires resetting the coefficients of the chaperon states at various times, this resetting does not require determining the coefficients of the states belonging to the DFS that one uses for control, computations, or information storage.
Complex quantum systems, such as a quantum computer, will always be coupled in some way to the environment. This can cause what's called decoherence, a destructive process by which information is lost from the system into the environment. In this bachelor thesis paper, we discuss decoherence-free subspaces within networks of coupled quantum harmonic oscillators (or QHOs). We investigate where such noiseless subspaces (or NSs) occur most frequently in an ensemble of Erdos-Renyi networks, for which we do not yet consider the influence of the bath. We then proceed by adding the bath into the equation, using some of the theory of open quantum systems. Specifically, we derive the Lindblad master equation and show its form for the case of our networks. Consequently, we simulate the behavior of the moments of the position operators for the graphs with 3 nodes, both by means of the full Lindblad equation, and by first tracing out those moments to obtain their differential equations. We compare those two results to each other, and also look back to the situation before adding the bath to see if the NSs are still present. From the results of the simulations, we can conclude several things. Firstly, we see that for ensembles with probability of connection p very close to either 0 or 1, both to number of noiseless modes and the probability of finding at least one is largest. This is credited to their relatively high degrees of symmetry. Secondly, in the results of the density matrix and moment simulations, we see that, indeed, the noiseless modes are preserved when considering the influence of the bath. Furthermore, we can conclude that simulation of the density matrix for the case of coupled QHOs in a network is in many cases not stable; the cutoff at a finite level s needed to simulate an otherwise infinite- dimensional operator leads to non-positivity of the density matrix. Therefore, it is best to simulate the moments from their respective differential equations, as opposed to the full Lindblad master equation. Finally, the differential equations for the moments and their solutions show that there are indeed noiseless clusters for eigenmodes perpendicular to the center of mass, as predicted.
Coherence in an open quantum system is degraded through its interaction with a bath. This decoherence can be avoided by restricting the dynamics of the system to special decoherence-free subspaces. These subspaces are usually constructed under the assumption of spatially symmetric system-bath coupling. Here we show that decoherence-free subspaces may appear without spatial symmetry. Instead, we consider a model of system-bath interactions in which to first order only multiple-qubit coupling to the bath is present, with single-qubit system-bath coupling absent. We derive necessary and sufficient conditions for the appearance of decoherence-free states in this model, and give a number of examples. In a sequel paper we show how to perform universal and fault tolerant quantum computation on the decoherence-free subspaces considered in this paper.
A decoherence-free subspace (DFS) isolates quantum information from deleterious environmental interactions. We give explicit sequences of strong and fast ["bang-bang" (BB)] pulses that create the conditions allowing for the existence of DFSs that support scalable, universal quantum computation. One such example is the creation of the conditions for collective decoherence, wherein all system particles are coupled in an identical manner to their environment. The BB pulses needed for this are generated using only the Heisenberg exchange interaction. In conjunction with previous results, this shows that Heisenberg exchange is by itself an enabler of universal fault-tolerant quantum computation on DFSs.