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arXiv:0808.1640 · Physical Review A

Decoherence-free quantum information in Markovian systems

Manas K. Patra, Peter G. Brooke

Published Jul 29, 2008• 9 citations
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Abstract

Decoherence in Markovian systems can result indirectly from the action of a system Hamiltonian which is usually fixed and unavoidable. Here, we show that in general in Markovian systems, because of the system Hamiltonian, quantum information decoheres. We give conditions for the system Hamiltonian that must be satisfied if coherence is to be preserved. Finally, we show how to construct robust subspaces for quantum-information processing.

DOI
10.1103/physreva.78.010308

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Decoherence in Markovian systems can result indirectly from the action of a system Hamiltonian which is usually fixed and unavoidable. Here, we show that in general in Markovian systems, because of the system Hamiltonian, quantum information decoheres. We give conditions for the system Hamiltonian that must be satisfied if coherence is to be preserved. Finally, we show how to construct robust subspaces for quantum-information processing.
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.
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.
The thesis is contributed to the study of the decoherence dynamics of dissipative qubit systems. We reveal the profound impact of the formation of a bound state between the qubit and its local environment on the decoherence dynamics of qubit system in non-Markovian regime. We find that the decoherence can be suppressed greatly under the non-Markovian dynamics when the bound state is formed. Consequently, the entanglement can be partially preserved in the steady state. A criterion on this decoherence/disentanglement suppression is given explicitly. It can give a useful guideline on the decoherence control in experiment. We also find that the entanglement can be stably distributed among all components and particularly it always satisfies an identity. Our unified treatment includes the previous results in the literatures as special cases. The result reveals the profound nature of the entanglement and should have significant implications for quantum information processing.
Decoherence free subspaces are a steady-state structure of the open quantum system which preserves quantum coherence between the states lying with in it and thus has found a variety of applications throughout quantum information science and technology. In this paper we study the computational complexity of deciding whether an open quantum system admits a decoherence free subspace or not. More specifically we study this problem with in the context of Markovian open quantum systems, governed by the time-independent Lindblad master equation. Along the way we introduce the kkk-Local Lindbladian problem, which captures the difficulty of computing purity decay rates under Lindbladian dynamics. We show that both problems are hard for the complexity class Quantum Merlin Arthur (QMA) when the locality k≥5k \geq 5k≥5, with the first under perfect completeness and the second being complete for QMA. Our hardness construction generalizes Kitaev's clock Hamiltonian construction to the open quantum system setting by encoding the execution of a quantum circuit into the steady subspace of a Lindbladian containing both pure and mixed history states. This subspace is then mixed depending on the output of the encoded circuit. Our results suggest that deciding whether a generic Markovian open quantum system admits a decoherence free subspace is intractable even for quantum computation.