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Physical Review A

S-matrix approach to the construction of decoherence-free subspaces

Moshe Shapiro, Paul W. Brumer

Published Nov 18, 2002• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.66.052308

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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.
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.
Decoherence-free subspaces protect quantum information from the effects of noise that is correlated across the physical qubits used to implement them. Given the ability to impose suitable Hamiltonians upon such a multi-qubit system, one can also implement a set of logical gates which enables universal computation on this information without compromising this protection. Real physical systems, however, seldom come with the correct Hamiltonians built-in, let alone the ability to turn them off and on at will. In the course of our development of quantum information processing devices based on liquid-state NMR, we have found the task of operating on quantum information encoded in decoherence-free subspaces rather more challenging than is commonly assumed. This contribution presents an overview of these challenges and the methods we have developed for overcoming them in practice. These methods promise to be broadly applicable to many of the physical systems proposed for the implementation of quantum information processing devices.
Introduction In the previous chapter we saw how quantum information can be protected by active means. A combination of encoding and syndrome measurements, followed by corrective operations conditioned on the syndrome measurement outcomes, led to a method by which quantum information could be error corrected. In this chapter we will see how quantum information can be protected passively . Rather than attempting to fix errors, we will hide the quantum information from the environment, so that it never becomes corrupted. This “quiet corner” of the system's Hilbert space is called a decoherence-free subspace (DFS) or noiseless subsystem (NS). The main mechanism that will allow us to play this trick is symmetry . Every student of physics knows that symmetries lead to conservation laws. In our case the conserved quantity will turn out to be quantum information. Of course such symmetries must be exceedingly rare, or else protected quantum information would be all around us. Thus we will be forced to make some very strong assumptions, and ask the environment to “cooperate.” While we will definitely allow the environment to couple to our system, we will only allow it to do so in a certain symmetric manner. We will later discuss how relaxing this “cooperation requirement” still allows quantum information to be passively protected for a short time. In Chapter 4 we will see how, by using dynamical decoupling techniques, it becomes possible to force the environment to cooperate, i.e., to dynamically generate symmetries leading to a DFS or NS.
Decoherence-Free Subsystems (DFS) are a powerful means of protecting quantum information against noise with known symmetry properties. Although Hamiltonians theoretically exist that can implement a universal set of logic gates on DFS encoded qubits without ever leaving the protected subsystem, the natural Hamiltonians that are available in specific implementations do not necessarily have this property. Here we describe some of the principles that can be used in such cases to operate on encoded qubits without losing the protection offered by the DFS. In particular, we show how dynamical decoupling can be used to control decoherence during the unavoidable excursions outside of the DFS. By means of cumulant expansions, we show how the fidelity of quantum gates implemented by this method on a simple two-physical-qubit DFS depends on the correlation time of the noise responsible for decoherence. We further show by means of numerical simulations how our previously introduced "strongly modulating pulses" for NMR quantum information processing can permit high-fidelity operations on multiple DFS encoded qubits in practice, provided that the rate at which the system can be modulated is fast compared to the correlation time of the noise. The principles thereby illustrated are expected to be broadly applicable to many implementations of quantum information processors based on DFS encoded qubits.
We consider the input-output process with low-QQQ cavities confining atoms for quantum-information processing, based on a general expression for reflection coefficient of the single photon. Focusing on the qubit encoding in decoherence-free subspace (DFS), we propose entanglement generation, universal quantum computing, and multiqubit controlled phase flip, using individual photon input and output from the cavities. Because of the DFS and no requirement for high-QQQ cavities, our schemes are not only immune to dephasing but also feasible with currently available technology.