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Fortschritte der Physik

A Possible Strategy to Defeat Decoherence in Quantum Computation: The Role of Symmetries, Dynamical Algebras, and All That

Mario Rasetti, Paolo Zanardi A

Published May 1, 2000• 5 citations
Publisher

Abstract

The question is addressed whether in the frame of Error Avoiding Codes one can devise strategies whereby a universal set of gates can be constructed allowing for arbitrary quantum computation [in its full extent, from input system preparation, through computation implementation, to output measurement] to be completely performed within the code.

DOI
10.1002/(sici)1521-3978(200005)48:5/7<559::aid-prop559>3.0.co;2-x

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The question is addressed whether in the frame of Error Avoiding Codes one can devise strategies whereby a universal set of gates can be constructed allowing for arbitrary quantum computation [in its full extent, from input system preparation, through computation implementation, to output measurement] to be completely performed within the code.
Let H\mathcal{H}H be the state space of a quantum computer coupled with the environment by a set of error operators spanning a Lie algebra L.\mathcal{L}.L. Suppose L\mathcal{L}L admits an error-avoiding quantum code, i.e., a subspace $\mathcal{C}\ensuremath{\subset}\mathcal{H}$ annihilated by L.\mathcal{L}.L. We show that a universal set of gates over C\mathcal{C}C is obtained by any generic pair of L\mathcal{L}L-invariant gates. Such gates---if not available from the outset---can be obtained by resorting to a symmetrization with respect to the group generated by L.\mathcal{L}.L. Any computation can then be performed completely within the coding decoherence-free subspace.
The notion of symmetry is shown to be at the heart of all error correction/avoidance strategies for preserving quantum coherence of an open quantum system S e.g., a quantum computer. The existence of a non-trivial group of symmetries of the dynamical algebra of S provides state-space sectors immune to decoherence. Such noiseless sectors, that can be viewed as a noncommutative version of the pointer basis, are shown to support universal quantum computation and to be robust against perturbations. When the required symmetry is not present one can generate it artificially resorting to active symmetrization procedures.
In this thesis we describe methods for avoiding the detrimental effects of decoherence while at the same time still allowing for computation of the quantum information. The philosophy of the method discussed in the first part of this thesis is to use a symmetry of the decoherence mechanism to find robust encodings of the quantum information. Stability, control, and methods for using decoherence-free information in a quantum computer are presented with a specific emphasis on decoherence due to a collective coupling between the system and its environment. Universal quantum computation on such collective decoherence decoherence-free encodings is demonstrated. Rigorous definitions of control and the use of encoded universality in quantum computers are addressed. Explicit gate constructions for encoded universality on ion trap and exchange based quantum computers are given. In the second part of the thesis we examine physical systems with error correcting properties. We examine systems that can store quantum information in their ground state such that decoherence processes are prohibited via energetics. We present the theory of supercoherent systems whose ground states are quantum error detecting codes and describe a spin ladder whose ground state has both the error detecting and correcting properties. We conclude by discussing naturally fault-tolerant quantum computation.
A constructive scheme has been devised to enable mapping of any quantum computation into a spintronic circuit in which the computation is encoded in a basis that is, in principle, immune to quantum decoherence. The scheme is implemented by an algorithm that utilizes multiple physical spins to encode each logical bit in such a way that collective errors affecting all the physical spins do not disturb the logical bit. The scheme is expected to be of use to experimenters working on spintronic implementations of quantum logic. Spintronic computing devices use quantum-mechanical spins (typically, electron spins) to encode logical bits. Bits thus encoded (denoted qubits) are potentially susceptible to errors caused by noise and decoherence. The traditional model of quantum computation is based partly on the assumption that each qubit is implemented by use of a single two-state quantum system, such as an electron or other spin-1.2 particle. It can be surprisingly difficult to achieve certain gate operations . most notably, those of arbitrary 1-qubit gates . in spintronic hardware according to this model. However, ironically, certain 2-qubit interactions (in particular, spin-spin exchange interactions) can be achieved relatively easily in spintronic hardware. Therefore, it would be fortunate if it were possible to implement any 1-qubit gate by use of a spin-spin exchange interaction. While such a direct representation is not possible, it is possible to achieve an arbitrary 1-qubit gate indirectly by means of a sequence of four spin-spin exchange interactions, which could be implemented by use of four exchange gates. Accordingly, the present scheme provides for mapping any 1-qubit gate in the logical basis into an equivalent sequence of at most four spin-spin exchange interactions in the physical (encoded) basis. The complexity of the mathematical derivation of the scheme from basic quantum principles precludes a description within this article; it must suffice to report that the derivation provides explicit constructions for finding the exchange couplings in the physical basis needed to implement any arbitrary 1-qubit gate. These constructions lead to spintronic encodings of quantum logic that are more efficient than those of a previously published scheme that utilizes a universal but fixed set of gates.
Scalable quantum computation in realistic devices requires that precise control can be implemented efficiently in the presence of decoherence and operational errors. We propose a general constructive procedure for designing robust unitary gates on an open quantum system without encoding or measurement overhead. Our results allow for a low-level error correction strategy solely based on Hamiltonian engineering using realistic bounded-strength controls and may substantially reduce implementation requirements for fault-tolerant quantum computing architectures.
The dynamical-algebraic structure underlying all the schemes for quantum information stabilization is argued to be fully contained in the reducibility of the operator algebra describing the interaction with the environment of the coding quantum system. This property amounts to the existence of a nontrivial group of symmetries for the global dynamics. We provide a unified framework that allows us to build systematically additional classes of error correcting codes and noiseless subsystems. It is shown that by using symmetrization strategies one can artificially produce noiseless subsystems supporting universal quantum computation.