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arXiv:1701.08260 · Physical Review A

Experimental protection of arbitrary states in a two-qubit subspace by nested Uhrig dynamical decoupling

Harpreet Singh, Arvind, Kavita Dorai

Published May 18, 2017• 8 citations
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Abstract

We experimentally demonstrate the efficacy of a three-layer nested Uhrig dynamical decoupling (NUDD) sequence to preserve arbitrary quantum states in a two-dimensional subspace of the four-dimensional two-qubit Hilbert space on a nuclear magnetic resonance quantum information processor. The effect of the state preservation is studied first on four known states, including two product states and two maximally entangled Bell states. Next, to evaluate the preservation capacity of the NUDD scheme, we apply it to eight randomly generated states in the subspace. Although, the preservation of different states varies, the scheme, on the average, performs very well. The complete tomographs of the states at different time points are used to compute fidelity. State fidelities using NUDD protection are compared with those obtained without using any protection. The nested pulse schemes are complex in nature and require careful experimental implementation.

DOI
10.1103/physreva.95.052337

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We experimentally demonstrate the efficacy of a three-layer nested Uhrig dynamical decoupling (NUDD) sequence to preserve arbitrary quantum states in a two-dimensional subspace of the four-dimensional two-qubit Hilbert space on a nuclear magnetic resonance quantum information processor. The effect of the state preservation is studied first on four known states, including two product states and two maximally entangled Bell states. Next, to evaluate the preservation capacity of the NUDD scheme, we apply it to eight randomly generated states in the subspace. Although, the preservation of different states varies, the scheme, on the average, performs very well. The complete tomographs of the states at different time points are used to compute fidelity. State fidelities using NUDD protection are compared with those obtained without using any protection. The nested pulse schemes are complex in nature and require careful experimental implementation.
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.
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.
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.
This thesis deals with the generation, estimation and preservation of novel quantum states of two and three qubits, on an NMR quantum information processor. Using the maximum likelihood ansatz, we have developed a method for state estimation such that the reconstructed density matrix does not have negative eigenvalues and the errors are within the space of valid density operators. Due to interactions with the environment, unwanted changes occur in the system, leading to decoherence. Controlling decoherence is one of the biggest challenges to be overcome to build quantum computers. We have used several experimental strategies to decouple the quantum system from its environment. These strategies are based on how much we know about system-environment interaction and what states we want to preserve. We first consider a case where we are aware of the system state but have no knowledge about its interaction with the environment. We demonstrate the efficacy of the super-Zeno scheme to tackle decoherence in this case. Then we consider a situation where only the subspace is known to which the system state belongs. To address such a situation we used a nested Uhrig dynamical decoupling scheme. Next, we consider situations where we have knowledge of the state of the system as well as its interaction with the environment. In such situations, since the noise model is known, decoupling strategies can be explicitly designed to cancel this noise. Using these decoupling strategies, we can experimentally extend the lifetime of time-invariant discord of two-qubit Bell-diagonal states and are also able to preserve the entanglement of three NMR qubits, to a remarkable extent.
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.