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arXiv:quant-ph/9603031 · Physical Review A

Error prevention scheme with four particles

Lev Vaidman A, Lior Goldenberg, Stephen J. Wiesner

Published Sep 1, 1996• 115 citations
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Abstract

It is shown that a simplified version of the error correction code recently suggested by Shor [Phys. Rev. A 52, R2493 (1995)] exhibits manifestation of the quantum Zeno effect. Thus, under certain conditions, protection of an unknown quantum state is achieved. Error prevention procedures based on four-particle and two-particle encoding are proposed and it is argued that they have feasible practical implementations.

DOI
10.1103/physreva.54.r1745

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It is shown that a simplified version of the error correction code recently suggested by Shor [Phys. Rev. A 52, R2493 (1995)] exhibits manifestation of the quantum Zeno effect. Thus, under certain conditions, protection of an unknown quantum state is achieved. Error prevention procedures based on four-particle and two-particle encoding are proposed and it is argued that they have feasible practical implementations.
In order to reduce errors, error correction codes need to be implemented fast. They can correct the errors corresponding to the first few orders in the Taylor expansion of the time evolution operator corresponding to the Hamiltonian of the interaction with the environment. If implemented fast enough, the zeroth order error predominates and the dominant effect is of error prevention by measurement (Zeno effect) rather than correction. In this ``Zeno regime,'' codes with less redundancy are sufficient for protection. We describe such a simple scheme, which uses two ``noiseless'' qubits to protect a large number, nnn, of information qubits from noise from the environment. The ``noisless qubits'' can be realized by treating them as logical qubits to be encoded by one of the previously introduced encoding schemes.
We show that multidimensional Zeno effect allows one to efficiently protect quantum systems from coherence loss by a method similar to classical coding. Contrary to the conventional approach, our method is applicable to arbitrary error-induced Hamiltonians and general quantum systems. We propose algorithms for finding the coherence-protecting codes along with a method of practical encoding by non-holonomic control. Two new single-error–correcting codes are constructed that surpass the bounds valid for classical quantum codes.
A scheme is presented for protecting one-qubit quantum information against decoherence due to a general environment and local exchange interactions. The scheme operates essentially by distributing information over two pairs of qubits and through error-prevention procedures. In the scheme, quantum information is encoded through a decoherence-free subspace for collective phase errors and exchange errors affecting the qubits in pairs; leakage out of the encoding space due to amplitude damping is reduced by quantum Zeno effect. In addition, how to construct decoherence-free states for n-qubit information against phase and exchange errors is discussed.
Within quantum information, many methods have been proposed to avoid or correct the deleterious effects of the environment on a system of interest. In this work, expanding on our earlier paper (Paz-Silva et al 2012 Phys. Rev. Lett. 108 080501), we evaluate the applicability of the quantum Zeno effect as one such method. Using the algebraic structure of stabilizer quantum error correction codes as a unifying framework, two open-loop protocols are described which involve frequent non-projective (i.e. weak) measurement of either the full stabilizer group or a minimal generating set thereof. The effectiveness of the protocols is measured by the distance between the final state under the protocol and the final state of an idealized evolution in which system and environment do not interact. Rigorous bounds on this metric are derived which demonstrate that, under certain assumptions, a Zeno effect may be realized with arbitrarily weak measurements, and that this effect can protect an arbitrary, unknown encoded state against the environment arbitrarily well.
Within quantum information, many methods have been proposed to avoid or correct the deleterious effects of the environment on a system of interest. In this work, expanding on our earlier paper (Paz-Silva et al 2012 Phys. Rev. Lett. 108 080501), we evaluate the applicability of the quantum Zeno effect as one such method. Using the algebraic structure of stabilizer quantum error correction codes as a unifying framework, two open-loop protocols are described which involve frequent non-projective (i.e. weak) measurement of either the full stabilizer group or a minimal generating set thereof. The effectiveness of the protocols is measured by the distance between the final state under the protocol and the final state of an idealized evolution in which system and environment do not interact. Rigorous bounds on this metric are derived which demonstrate that, under certain assumptions, a Zeno effect may be realized with arbitrarily weak measurements, and that this effect can protect an arbitrary, unknown encoded state against the environment arbitrarily well.