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arXiv:1104.5021 · Physical Review A

Universal dynamical decoupling of multiqubit states from environment

Liang Jiang, Adilet Imambekov

Published Dec 28, 2011• 34 citations
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Abstract

We study the dynamical decoupling of multiqubit states from environment. For a system of $m$ qubits, the nested Uhrig dynamical decoupling (NUDD) sequence can efficiently suppress generic decoherence induced by the system-environment interaction to order $N$ using ${(N+1)}^{2m}$ pulses. We prove that the NUDD sequence is universal, i.e., it can restore the coherence of an $m$-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.

DOI
10.1103/physreva.84.060302

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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.
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.
Nested Uhrig dynamical decoupling (NUDD) is a highly efficient quantum error suppression scheme that builds on optimized single axis UDD sequences. We prove the universality of NUDD and analyze its suppression of different error types in the setting of generalized control pulses. We present an explicit lower bound for the decoupling order of each error type, which we relate to the sequence orders of the nested UDD layers. We find that the error suppression capabilities of NUDD are strongly dependent on the parities and relative magnitudes of all nested UDD sequence orders. This allows us to predict the optimal arrangement of sequence orders. We test and confirm our analysis using numerical simulations.
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) 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.
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.