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Journal of Physics B Atomic Molecular and Optical Physics

Filter-design perspective applied to dynamical decoupling of a multi-qubit system

Zhi-Kun Su, Shaoji Jiang

Published Jan 9, 2012• 5 citations
Publisher

Abstract

We employ the filter-design perspective and derive the filter functions according to nested Uhrig dynamical decoupling (NUDD) and symmetric dynamical decoupling (SDD) in the pure-dephasing spin-boson model with N qubits. The performances of NUDD and SDD are discussed in detail for a two-qubit system. The analysis shows that (i) SDD outperforms NUDD for the bath with a soft cutoff while NUDD approaches SDD as the cutoff becomes harder; (ii) if the qubits are coupled to a common reservoir, SDD helps to protect the decoherence-free subspace while NUDD destroys it; (iii) when the imperfect control pulses with finite width are considered, NUDD is affected in both the high-fidelity regime and coherence time regime while SDD is affected in the coherence time regime only.

DOI
10.1088/0953-4075/45/2/025502

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Bulletin of the American Physical Society•Feb 27, 2012•1 citation
We employ the filter-design perspective and derive the filter functions according to nested Uhrig dynamical decoupling (NUDD) and symmetric dynamical decoupling (SDD) in the pure-dephasing spin-boson model with N qubits. The performances of NUDD and SDD are discussed in detail for a two-qubit system. The analysis shows that (i) SDD outperforms NUDD for the bath with a soft cutoff while NUDD approaches SDD as the cutoff becomes harder; (ii) if the qubits are coupled to a common reservoir, SDD helps to protect the decoherence-free subspace while NUDD destroys it; (iii) when the imperfect control pulses with finite width are considered, NUDD is affected in both the high-fidelity regime and coherence time regime while SDD is affected in the coherence time regime only.
We employ the filter-design perspective and derive the filter functions according to nested Uhrig dynamical decoupling (NUDD) and Symmetric dynamical decoupling (SDD) in the pure-dephasing spin-boson model with N qubits. The performances of NUDD and SDD are discussed in detail for a two-qubit system. The analysis shows that (i) SDD outperforms NUDD for the bath with a soft cutoff while NUDD approaches SDD as the cutoff becomes harder; (ii) if the qubits are coupled to a common reservoir, SDD helps to protect the decoherence-free subspace while NUDD destroys it; (iii) when the the imperfect control pulses with finite width are considered, NUDD is affected in both the high-fidelity regime and coherence time regime while SDD is affected in the coherence time regime only.
Based on a theorem we establish on dynamical decoupling of time-dependent systems, we present a scheme of nested Uhrig dynamical decoupling (NUDD) to protect multiqubit systems in generic quantum baths to arbitrary decoupling orders, using only single-qubit operations. The number of control pulses in NUDD increases polynomially with the decoupling order. For general multilevel systems, this scheme can preserve a set of unitary Hermitian system operators which mutually either commute or anticommute, and hence all operators in the Lie algebra generated from this set of operators, generating an effective symmetry group for the system up to a given order of precision. NUDD can be implemented with pulses of finite amplitude, up to an error in the second order of the pulse durations.
We study the suppression of qubit dephasing through Uhrig dynamical decoupling (UDD) in nontrivial environments modeled within the spin-boson formalism. In particular, we address the case of (i) a qubit coupled to a bosonic bath with power-law spectral density, and (ii) a qubit coupled to a single harmonic oscillator that dissipates energy into a bosonic bath, which embodies an example of a structured bath for the qubit. We then model the influence of random time jitter in the UDD protocol by sorting pulse-application times from Gaussian distributions centered at appropriate values dictated by the optimal protocol. In case (i) we find that, when few pulses are applied and a sharp cutoff is considered, longer coherence times and robust UDD performances (against random timing errors) are achieved for a super-Ohmic bath. On the other hand, when an exponential cutoff is considered, a super-Ohmic bath is undesirable. In case (ii) the best scenario is obtained for an overdamped harmonic motion. Our study provides relevant information for the implementation of optimized schemes for the protection of quantum states from decoherence.
Dynamical decoupling (DD) emerges as a promising tool to suppress decoherence of quantum systems by eliminating the system-environment coupling. We investigate its power in combating pure dephasing of a qubit under random classical noise. Interestingly, the decoherence rate can be viewed as the interplay between noise spectrum and a controllable filter function. More specifically, the filter function is determined by the DD sequence we employed. Here we take Uhrig DD (UDD) as a benchmark sequence to illustrate the filtering process. We find that UDD sequence works as a perfect dynamical filter for low-frequency noise. For spectrum with hard cutoff, UDD increases the performance order by order with linearly increased number of pulses. However, for noise spectrum with soft cutoff, the required pulse number grows exponentially.
We study the dynamical decoupling of multiqubit states from environment. For a system of mmm qubits, the nested Uhrig dynamical decoupling (NUDD) sequence can efficiently suppress generic decoherence induced by the system-environment interaction to order NNN using (N+1)2m{(N+1)}^{2m}(N+1)2m pulses. We prove that the NUDD sequence is universal, i.e., it can restore the coherence of an mmm-qubit quantum system independent of the details of the system-environment interaction. We also construct a general mapping between dynamical decoupling problems and discrete quantum walks in certain functional spaces.
The Uhrig dynamical decoupling sequence achieves high-order decoupling of a single system qubit from its dephasing bath through the use of bang-bang Pauli pulses at appropriately timed intervals. High-order decoupling of single and multiple qubit systems from baths causing both dephasing and relaxation can also be achieved through the nested application of Uhrig sequences, again using single-qubit Pauli pulses. For the three-qubit decoherence free subsystem (DFS) and related subsystem encodings, Pauli pulses are not naturally available operations; instead, exchange interactions provide all required encoded operations. Here we demonstrate that exchange interactions alone can achieve high-order decoupling against general noise in the three-qubit DFS. We present decoupling sequences for a three-qubit DFS coupled to classical and quantum baths and evaluate the performance of the sequences through numerical simulations.