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Journal of Physics A Mathematical and Theoretical

Immunizing quantum information against decoherence via decoherence-free environments

Murat Çetinbaş, Gary W. Leach

Published Feb 24, 2011• 4 citations
Publisher

Abstract

We construct decoherence-free environments (DFEs) within the Hamiltonian formalism and derive an exact equation of motion for a quantum subsystem interacting with a DFE. We discuss the feasibility of DFEs as a means to protect the quantum information in subsystem degrees of freedom. By studying a number of examples we show how to avoid entanglement-induced decoherence as well as decoherence induced by incoherence dynamics in a quantum subsystem coupled to a DFE.

DOI
10.1088/1751-8113/44/13/135301

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We construct decoherence-free environments (DFEs) within the Hamiltonian formalism and derive an exact equation of motion for a quantum subsystem interacting with a DFE. We discuss the feasibility of DFEs as a means to protect the quantum information in subsystem degrees of freedom. By studying a number of examples we show how to avoid entanglement-induced decoherence as well as decoherence induced by incoherence dynamics in a quantum subsystem coupled to a DFE.
We present a theoretical discussion of various aspects of the dynamics of information in quantum systems of qubits. We begin by reviewing ideas about entanglement and information in systems of qubits, different models for realistic qubits coupled to an external environment, how this causes the loss of information stored in qubits, and some practical designs for qubits that have been created. Then we study how the state of many-body systems can be decomposed in terms of its different subsystems and derived a hierarchy of equations describing the motion of these different parts. We show that in a qubit system, the central objects in this hierarchy are correlators between different components of different qubits. Systems where the environment coupled to a central qubit is a "spin bath'' of environmental qubits are then considered in detail and we find that information lost by the central qubit is transferred via a cascade to higher and higher order correlations with the environment. We discuses this process in the realistic example of an "Fe₈'' magnetic molecule qubit. Finally we discuss simple models of the dynamics of many entangled qubits.
Decoherence-free subsystems (DFSs) are a powerful means of protecting quantum information against noise with known symmetry properties. Although Hamiltonians that can implement a universal set of logic gates on DFS encoded qubits without ever leaving the protected subsystem theoretically exist, 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 DFSs. 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.
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 analyze decoherence-free (DF) quantum information in the presence of an arbitrary non-nearest-neighbor bath-induced system Hamiltonian using a Markovian master equation. We show that the most appropriate encoding for NNN qubits is probably contained within the $\ensuremath{\sim}\frac{2}{9}N$ excitation subspace. We give a time scale over which one would expect to apply other methods to correct for the system Hamiltonian. In order to remain applicable to experiment, we then focus on small systems, and present examples of DF quantum information for three and four qubits. We give an encoding for four qubits that, while quantum information remains in the two-excitation subspace, protects against an arbitrary bath-induced system Hamiltonian. Although our results are general to any system of qubits that satisfies our assumptions, throughout the paper we use dipole-coupled qubits as an example physical system.
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.