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.
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.
We prove the universality of the generalized QDDN1N2 (quadratic dynamical decoupling) pulse sequence for near-optimal suppression of general single-qubit decoherence. Earlier work showed numerically that this dynamical decoupling sequence, which consists of an inner Uhrig DD (UDD) and outer UDD sequence using N1 and N2 pulses, respectively, can eliminate decoherence to O(TN) using O(N2) unequally spaced ``ideal'' (zero-width) pulses, where T is the total evolution time and N=N1=N2. A proof of the universality of QDD has been given for even N1. Here we give a general universality proof of QDD for arbitrary N1 and N2. As in earlier proofs, our result holds for arbitrary bounded environments. Furthermore, we explore the single-axis (polarization) error suppression abilities of the inner and outer UDD sequences. We analyze both the single-axis QDD performance and how the overall performance of QDD depends on the single-axis errors. We identify various performance effects related to the parities and relative magnitudes of N1 and N2. We prove that using QDDN1N2 decoherence can always be eliminated to O(TminN1,N2).
Two topics in the field of quantum information processing, optimized dynamical error suppression and quantum algorithms, are considered here. ? The computational errors induced by the surrounding environment is one of the main obstacles in building a quantum computer. Engineering powerful techniques to combat errors in quantum devices is highly demanding. In the first part of this thesis, I focus on one quantum error correction technique, dynamical decoupling (DD), introduced in Chapter 1. Chapter 2 is dedicated to nested UDD (NUDD), a highly efficient decoupling scheme that utilizes the decoupling characteristics of UDD by multi-layer nesting. UDD (1-layer NUDD) is an optimal DD method for eliminating single-qubit general dephasing, and QDD (2-layer NUDD) is a near-optimal DD method for eliminating one qubit general decoherence. I present a rigorous analytical proof of the performance and universality of QDD/NUDD, and obtain an explicit formula for the decoupling order of each error type, which elucidates the relationship between the error type and characteristics of NUDD. From the explicit formula, a NUDD scheme can be designed accordingly such that optimal efficiency of NUDD is achieved. Moreover, the highly efficient error cancellation mechanism is revealed by the analysis. The proof of QDD has been published in [31], and the proof of NUDD will be submitted for publication shortly. ? Chapter 3 is devoted to the Adiabatic Quantum Computation (AQC). In this work (published in [44]), a general time-optimal strategy, which in principle can optimize any quantum adiabatic algorithm for which the gap is known or can be estimated, is formulated. In addition, I present a natural differential-geometric framework for AQC.
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.
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 Nth order using a sequence of N 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 analyze numerically the performance of the near-optimal quadratic dynamical decoupling (QDD) single-qubit decoherence errors suppression method [J. West et al., Phys. Rev. Lett. 104, 130501 (2010)]. The QDD sequence is formed by nesting two optimal Uhrig dynamical decoupling sequences for two orthogonal axes, comprising N1 and N2 pulses, respectively. Varying these numbers, we study the decoherence suppression properties of QDD directly by isolating the errors associated with each system basis operator present in the system-bath interaction Hamiltonian. Each individual error scales with the lowest order of the Dyson series, therefore immediately yielding the order of decoherence suppression. We show that the error suppression properties of QDD are dependent upon the parities of N1 and N2, and near-optimal performance is achieved for general single-qubit interactions when N1=N2.