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arXiv:1104.5507 · Physical Review Letters

Zeno Effect for Quantum Computation and Control

Gerardo A. Paz-Silva, A. T. Rezakhani, Jason Dominy, Daniel A. Lidar

Published Feb 22, 2012• 119 citations
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Abstract

It is well known that the quantum Zeno effect can protect specific quantum states from decoherence by using projective measurements. Here we combine the theory of weak measurements with stabilizer quantum error correction and detection codes. We derive rigorous performance bounds which demonstrate that the Zeno effect can be used to protect appropriately encoded arbitrary states to arbitrary accuracy while at the same time allowing for universal quantum computation or quantum control.

DOI
10.1103/physrevlett.108.080501

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It is well known that the quantum Zeno effect can protect specific quantum states from decoherence by using projective measurements. Here we combine the theory of weak measurements with stabilizer quantum error correction and detection codes. We derive rigorous performance bounds which demonstrate that the Zeno effect can be used to protect appropriately encoded arbitrary states to arbitrary accuracy while at the same time allowing for universal quantum computation or quantum control.
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.
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.
We show that multidimensional Zeno effect combined with non-holonomic control allows to efficiently protect quantum systems from decoherence by a method similar to classical coding.Contrary to the conventional approach, our method is applicable to arbitrary error-inducing Hamiltonians and general quantum systems.We also propose algorithms of finding encoding that approaches the Hamming upper bound along with methods of practical realizations of the encodings.Two new codes protecting 2 information qubits out of 7 and 4 information qubits out of 9 against a single error with arbitrarily small probability of failure are constructed as an example.
Projective measurements are an essential element of quantum mechanics. In most cases, they cause an irreversible change of the quantum system on which they act. However, measurements can also be used to stabilize quantum states from decay processes, which is known as the quantum Zeno effect (QZE). Here, we demonstrate this effect for the case of a superposition state of a nuclear spin qubit, using an ancilla to perform the measurement. As a result, the quantum state of the qubit is protected against dephasing without relying on an ensemble nature of NMR experiments. We also propose a scheme to protect an arbitrary state by using QZE.
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.