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arXiv:0911.1676 · Physical Review A

Universal dynamical decoupling: Two-qubit states and beyond

Musawwadah Mukhtar, Thuan Beng Saw, Wee Tee Soh, Jiangbin Gong

Published Jan 29, 2010• 34 citations
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Abstract

Uhrig's dynamical decoupling pulse sequence has emerged as a universal and highly promising approach to decoherence suppression. So far, both the theoretical and experimental studies have examined single-qubit decoherence only. This work extends Uhrig's universal dynamical decoupling from one-qubit to two-qubit systems and even to general multilevel quantum systems. In particular, we show that by designing appropriate control Hamiltonians for a two-qubit or a multilevel system, Uhrig's pulse sequence can also preserve a generalized quantum coherence measure to the order of $1+O({T}^{N+1})$ with only $N$ pulses. Our results lead to a very useful scheme for efficiently locking two-qubit entangled states. Future important applications of Uhrig's pulse sequence in preserving the quantum coherence of multilevel quantum systems can also be anticipated.

DOI
10.1103/physreva.81.012331

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Uhrig's dynamical decoupling pulse sequence has emerged as a universal and highly promising approach to decoherence suppression. So far, both the theoretical and experimental studies have examined single-qubit decoherence only. This work extends Uhrig's universal dynamical decoupling from one-qubit to two-qubit systems and even to general multilevel quantum systems. In particular, we show that by designing appropriate control Hamiltonians for a two-qubit or a multilevel system, Uhrig's pulse sequence can also preserve a generalized quantum coherence measure to the order of 1+O(TN+1)1+O({T}^{N+1})1+O(TN+1) with only NNN pulses. Our results lead to a very useful scheme for efficiently locking two-qubit entangled states. Future important applications of Uhrig's pulse sequence in preserving the quantum coherence of multilevel quantum systems can also be anticipated.
Future quantum technologies rely heavily on good protection of quantum entanglement against environment-induced decoherence. A recent study showed that an extension of Uhrig's dynamical decoupling (UDD) sequence can (in theory) lock an arbitrary but known two-qubit entangled state to the NNNth order using a sequence of NNN control pulses [Mukhtar et al., Phys. Rev. A 81, 012331 (2010)]. By nesting three layers of explicitly constructed UDD sequences, here we first consider the protection of unknown two-qubit states as superposition of two known basis states, without making assumptions of the system-environment coupling. It is found that the obtained decoherence suppression can be highly sensitive to the ordering of the three UDD layers and can be remarkably effective with the correct ordering. The detailed theoretical results are useful for general understanding of the nature of controlled quantum dynamics under nested UDD. As an extension of our three-layer UDD, it is finally pointed out that a completely unknown two-qubit state can be protected by nesting four layers of UDD sequences. This work indicates that when UDD is applicable (e.g., when the environment has a sharp frequency cutoff and when control pulses can be taken as instantaneous pulses), dynamical decoupling using nested UDD sequences is a powerful approach for entanglement protection.
Dynamical decoupling (DD) eliminates qubit-bath coupling by applying a sequence of instantaneous pulses. While qubit-bath couplings generally lead to qubit relaxation and dephasing, qubit-qubit couplings are often used to manipulate or control quantum states. We investigate the available control operations in two DD schemes, named periodic DD (PDD) and Uhrig DD (UDD), to see whether universal quantum computation can be realized in these decoupled systems. We find that universal control is possible using Heisenberg interaction in both periodically decoupled system and Uhrig decoupled system, and the available control operations under two kinds of DD sequences obey the same commutation relation. In the UDD case, we also derive a rough bound for control errors.
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.
Stroboscopic spin flips have already been shown to prolong the coherence times of quantum systems under noisy environments. Uhrig's dynamical decoupling scheme provides an optimal sequence for a quantum system interacting with a dephasing bath. Several experimental demonstrations have already verified the efficiency of such dynamical decoupling schemes in preserving single-qubit coherences. In this work we describe the experimental study of Uhrig's dynamical decoupling in preserving two-qubit entangled states using an ensemble of spin-1/2 nuclear pairs in solution state. We find that the performance of odd-order Uhrig sequences in preserving entanglement is superior to both even-order Uhrig sequences and periodic spin-flip sequences. We also find that there exists an optimal order of the Uhrig sequence in which a singlet state can be stored at high correlation for about 30 seconds.
The optimal N-pulse dynamical decoupling discovered by Uhrig for a spin-boson model [Phys. Rev. Lett. 98, 100504 (2007)10.1103/PhysRevLett.98.100504] is proved to be universal in suppressing to O(T;{N+1}) the pure dephasing or the longitudinal relaxation of a qubit (or spin 1/2) coupled to a generic bath in a short-time evolution of duration T. For suppressing the longitudinal relaxation, a Uhrig pi-pulse sequence can be generalized to be a superposition of the ideal Uhrig pi-pulse sequence as the core and an arbitrarily shaped pulse sequence satisfying certain symmetry requirements. The generalized Uhrig dynamical decoupling offers the possibility of manipulating the qubit while simultaneously combating the longitudinal relaxation.
Dynamical decoupling (DD) sequences were invented to eliminate the direct coupling between a qubit and its environment. We further investigate the possibility of decoupling the indirect qubit–qubit interaction induced by a common environment, and successfully find simplified solutions that preserve the bipartite quantum states to arbitrary order. Through analysing the exact dynamics of the controlled two-qubit density matrix, we have proven that applying independent Uhrig DD (UDD) on each qubit will effectively eliminate both the qubit–environment and indirect qubit–qubit couplings to the same order as in the single-qubit case, only if orders of the two UDD sequences have different parities. More specifically, UDD( n ) on one qubit with UDD( m ) on another are able to produce min ( n , m )th-order suppression while n + m is odd. Our results can be used to reduce the pulse number in relevant experiments for protecting bipartite quantum states, or dynamically manipulating the indirect interaction within a certain quantum gate and quantum bus.