Fault-Tolerant Quantum Dynamical Decoupling

Dynamical decoupling pulse sequences have been used to extend coherence times in quantum systems ever since the discovery of the spin-echo effect. Here we introduce a method of recursively concatenated dynamical decoupling pulses, designed to overcome both decoherence and operational errors. This is important for coherent control of quantum systems such as quantum computers. For bounded-strength, non-Markovian environments, such as for the spin-bath that arises in electron- and nuclear-spin based solid-state quantum computer proposals, we show that it is strictly advantageous to use concatenated, as opposed to standard periodic dynamical decoupling pulse sequences. Namely, the concatenated scheme is both fault-tolerant and super-polynomially more efficient, at equal cost. We derive a condition on the pulse noise level below which concatenated is guaranteed to reduce decoherence.Comment: 5 pages, 4 color eps figures. v3: Minor changes. To appear in Phys. Rev. Let

Optimally combining dynamical decoupling and quantum error correction

We show how dynamical decoupling (DD) and quantum error correction (QEC) can be optimally combined in the setting of fault tolerant quantum computing. To this end we identify the optimal generator set of DD sequences designed to protect quantum information encoded into stabilizer subspace or subsystem codes. This generator set, comprising the stabilizers and logical operators of the code, minimizes a natural cost function associated with the length of DD sequences. We prove that with the optimal generator set the restrictive local-bath assumption used in earlier work on hybrid DD-QEC schemes, can be significantly relaxed, thus bringing hybrid DD-QEC schemes, and their potentially considerable advantages, closer to realization.Comment: 6 pages, 1 figur

Long-time Low-latency Quantum Memory by Dynamical Decoupling

Quantum memory is a central component for quantum information processing devices, and will be required to provide high-fidelity storage of arbitrary states, long storage times and small access latencies. Despite growing interest in applying physical-layer error-suppression strategies to boost fidelities, it has not previously been possible to meet such competing demands with a single approach. Here we use an experimentally validated theoretical framework to identify periodic repetition of a high-order dynamical decoupling sequence as a systematic strategy to meet these challenges. We provide analytic bounds-validated by numerical calculations-on the characteristics of the relevant control sequences and show that a "stroboscopic saturation" of coherence, or coherence plateau, can be engineered, even in the presence of experimental imperfection. This permits high-fidelity storage for times that can be exceptionally long, meaning that our device-independent results should prove instrumental in producing practically useful quantum technologies.Comment: abstract and authors list fixe

Efficient Coherent Control by Optimized Sequences of Pulses of Finite Duration

Reliable long-time storage of arbitrary quantum states is a key element for quantum information processing. In order to dynamically decouple a spin or quantum bit from a dephasing environment, we introduce an optimized sequence of $N$ control pulses of finite durations \tau\pp and finite amplitudes. The properties of this sequence of length $T$ stem from a mathematically rigorous derivation. Corrections occur only in order $T^{N+1}$ and \tau\pp^3 without mixed terms such as T^N\tau\pp or T^N\tau\pp^2. Based on existing experiments, a concrete setup for the verification of the properties of the advocated realistic sequence is proposed.Comment: 8 pages, 1 figur

Reducing sequencing complexity in dynamical quantum error suppression by Walsh modulation

We study dynamical error suppression from the perspective of reducing sequencing complexity, in order to facilitate efficient semi-autonomous quantum-coherent systems. With this aim, we focus on digital sequences where all interpulse time periods are integer multiples of a minimum clock period and compatibility with simple digital classical control circuitry is intrinsic, using so-called em Walsh functions as a general mathematical framework. The Walsh functions are an orthonormal set of basis functions which may be associated directly with the control propagator for a digital modulation scheme, and dynamical decoupling (DD) sequences can be derived from the locations of digital transitions therein. We characterize the suite of the resulting Walsh dynamical decoupling (WDD) sequences, and identify the number of periodic square-wave (Rademacher) functions required to generate a Walsh function as the key determinant of the error-suppressing features of the relevant WDD sequence. WDD forms a unifying theoretical framework as it includes a large variety of well-known and novel DD sequences, providing significant flexibility and performance benefits relative to basic quasi-periodic design. We also show how Walsh modulation may be employed for the protection of certain nontrivial logic gates, providing an implementation of a dynamically corrected gate. Based on these insights we identify Walsh modulation as a digital-efficient approach for physical-layer error suppression.Comment: 15 pages, 3 figure

Partitioned trace distances

New quantum distance is introduced as a half-sum of several singular values of difference between two density operators. This is, up to factor, the metric induced by so-called Ky Fan norm. The partitioned trace distances enjoy similar properties to the standard trace distance, including the unitary invariance, the strong convexity and the close relations to the classical distances. The partitioned distances cannot increase under quantum operations of certain kind including bistochastic maps. All the basic properties are re-formulated as majorization relations. Possible applications to quantum information processing are briefly discussed.Comment: 8 pages, no figures. Significant changes are made. New section on majorization is added. Theorem 4.1 is extended. The bibliography is enlarged

Control of electron spin decoherence caused by electron-nuclear spin dynamics in a quantum dot

Control of electron spin decoherence in contact with a mesoscopic bath of many interacting nuclear spins in an InAs quantum dot is studied by solving the coupled quantum dynamics. The nuclear spin bath, because of its bifurcated evolution predicated on the electron spin up or down state, measures the which-state information of the electron spin and hence diminishes its coherence. The many-body dynamics of nuclear spin bath is solved with a pair-correlation approximation. In the relevant timescale, nuclear pair-wise flip-flops, as elementary excitations in the mesoscopic bath, can be mapped into the precession of non-interacting pseudo-spins. Such mapping provides a geometrical picture for understanding the decoherence and for devising control schemes. A close examination of nuclear bath dynamics reveals a wealth of phenomena and new possibilities of controlling the electron spin decoherence. For example, when the electron spin is flipped by a $\pi$-pulse at $\tau$, its coherence will partially recover at $\sqrt{2}\tau$ as a consequence of quantum disentanglement from the mesoscopic bath. In contrast to the re-focusing of inhomogeneously broadened phases by conventional spin-echoes, the disentanglement is realized through shepherding quantum evolution of the bath state via control of the quantum object. A concatenated construction of pulse sequences can eliminate the decoherence with arbitrary accuracy, with the nuclear-nuclear spin interaction strength acting as the controlling small parameter

Frozen and Invariant Quantum Discord under Local Dephasing Noise

In this chapter, we intend to explore and review some remarkable dynamical properties of quantum discord under various different open quantum system models. Specifically, our discussion will include several concepts connected to the phenomena of time invariant and frozen quantum discord. Furthermore, we will elaborate on the relation of these two phenomena to the non-Markovian features of the open system dynamics and to the usage of dynamical decoupling protocols.Comment: 29 pages, 8 figure