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arXiv:0803.1427 · New Journal of Physics

Exact results on dynamical decoupling by π pulses in quantum information processes

Götz Silvester Uhrig

Published Aug 15, 2008• 169 citations
arXivHTMLPDF

Abstract

The aim of dynamical decoupling consists in the suppression of decoherence by appropriate coherent control of a quantum register. Effectively, the interaction with the environment is reduced. In particular, a sequence of π pulses is considered. Here we present exact results on the suppression of the coupling of a quantum bit to its environment by optimized sequences of π pulses. The effect of various cut-offs of the spectral density of the environment is investigated. As a result we show that the harder the cut-off is, the better an optimized pulse sequence can deal with it. For cut-offs which are neither completely hard nor very soft we advocate iterated optimized sequences.

DOI
10.1088/1367-2630/10/8/083024

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The aim of dynamical decoupling consists in the suppression of decoherence by appropriate coherent control of a quantum register. Effectively, the interaction with the environment is reduced. In particular, a sequence of π pulses is considered. Here we present exact results on the suppression of the coupling of a quantum bit to its environment by optimized sequences of π pulses. The effect of various cut-offs of the spectral density of the environment is investigated. As a result we show that the harder the cut-off is, the better an optimized pulse sequence can deal with it. For cut-offs which are neither completely hard nor very soft we advocate iterated optimized sequences.
Reliable long-time storage of arbitrary quantum states is a key element of quantum information processing. In order to dynamically decouple a spin or quantum bit from a dephasing environment by non-instantaneous pulses, we introduce an optimized sequence of N control π pulses that are realistic in the sense that they have a finite duration and a finite amplitude. We show that optimized dynamical decoupling is still applicable and that higher-order decoupling can be reached if shaped pulses are implemented. The sequence suppresses decoherence up to the order , with T being the total duration of the sequence and τ mx the maximum length of the pulses. The exponent depends on the shape of the pulse. Based on existing experiments, a concrete setup for the verification of the properties of the advocated sequence is proposed.
We study the time evolution of a qubit linearly coupled with a quantum environment under a sequence of short π pulses.Our attention is focused on the case where qubit-environment interactions induce the decoherence with population decay.We assume that the environment consists of a set of bosonic excitations.The time evolution of the reduced density matrix for the qubit is calculated in the presence of periodic short π pulses.We confirm that the decoherence is suppressed if the pulse interval is shorter than the correlation time for qubit-environment interactions.
The pulse control of decoherence in a qubit interacting with a quantum environment is studied with focus on a general case where decoherence is induced by both pure dephasing and population decay. To observe how the decoherence is suppressed by periodic pi pulses, we present a simple method to calculate the time evolution of a qubit under arbitrary pulse sequences consisting of bit-flips and/or phase-flips. We examine the effectiveness of the two typical sequences: bb sequence consisting of only bit-flips, and bp sequence consisting of both bit- and phase-flips. It is shown that the effectiveness of the pulse sequences depends on a relative strength of the two decoherence processes especially when a pulse interval is slightly shorter than qubit-environment correlation times. In the short-interval limit, however, the bp sequence is always more effective than, or at least as effective as, the bb sequence.
Aperiodic dynamical decoupling (DD) sequences of π pulses are of great interest to decoherence control and have been recently extended from single-qubit to two-qubit systems. If the environmental noise power spectrum is made available, then one may further optimize aperiodic DD sequences to reach higher efficiency of decoherence suppression than known universal schemes. This possibility is investigated in this work for the protection of two-qubit states, using an exactly solvable pure dephasing model including both local and nonlocal noise. The performance of optimized DD sequences in protecting two-qubit states is compared with that achieved by nested Uhrig's DD (nested-UDD) sequences, for several different types of noise spectrum. Except for the cases with noise spectrum decaying slowly in the high-frequency regime, optimized DD sequences with tens of control pulses can perform orders of magnitude better than that of nested-UDD. A two-qubit system with highly unbalanced local noise is also examined to shed more light on a recent experiment. Possible experiments that may be motivated by this work are discussed.
Decoherence is one of the most important obstacles that must be overcome in quantum information processing. It depends on the qubit-environment coupling strength, but also on the spectral composition of the noise generated by the environment. If the spectral density is known, fighting the effect of decoherence can be made more effective. Applying sequences of inversion pulses to the qubit system, we developed a method for dynamical decoupling noise spectroscopy. We generate effective filter functions that probe the environmental spectral density without requiring assumptions about its shape. Comparing different pulse sequences, we recover the complete spectral density function and distinguish different contributions to the overall decoherence.