Decoherence in quantum computers is formulated within the Semigroup approach. The error generators are identified with the generators of a Lie algebra. This allows for a comprehensive description which includes as a special case the frequently assumed spin-boson model. A generic condition is presented for error-less quantum computation: decoherence-free subspaces are spanned by those states which are annihilated by all the generators. It is shown that these subspaces are stable to perturbations and moreover, that universal quantum computation is possible within them.

Adiabatic quantum computing (AQC) started as an approach to solving optimization problems, and has evolved into an important universal alternative to the standard circuit model of quantum computing, with deep connections to both classical and quantum complexity theory and condensed matter physics. In this review we give an account of most of the major theoretical developments in the field, while focusing on the closedsystem setting. The review is organized around a series of topics that are essential to an understanding of the underlying principles of AQC, its algorithmic accomplishments and limitations, and its scope in the more general setting of computational complexity theory. We present several variants of the adiabatic theorem, the cornerstone of AQC, and we give examples of explicit AQC algorithms that exhibit a quantum speedup. We give an overview of several proofs of the universality of AQC and related Hamiltonian quantum complexity theory. We finally devote considerable space to Stoquastic AQC, the setting of most AQC work to date, where we discuss obstructions to success and their possible resolutions.

Quantum technology is maturing to the point where quantum devices, such as quantum communication systems, quantum random number generators and quantum simulators, may be built with capabilities exceeding classical computers. A quantum annealer, in particular, solves hard optimisation problems by evolving a known initial configuration at non-zero temperature towards the ground state of a Hamiltonian encoding a given problem. Here, we present results from experiments on a 108 qubit D-Wave One device based on superconducting flux qubits. The strong correlations between the device and a simulated quantum annealer, in contrast with weak correlations between the device and classical annealing or classical spin dynamics, demonstrate that the device performs quantum annealing. We find additional evidence for quantum annealing in the form of small-gap avoided level crossings characterizing the hard problems. To assess the computational power of the device we compare it to optimised classical algorithms.Annealing a material by slow cooling is an ancient technique to improve the properties of glasses, metals and steel that has been used for more than seven millennia [1]. Mimicking this process in computer simulations is the idea behind simulated annealing as an optimisation method [2], which views the cost function of an optimisation problem as the energy of a physical system. Its configurations are sampled in a Monte Carlo simulation using the Metropolis algorithm [3], escaping from local minima by thermal fluctuations to find lower energy configurations. The goal is to find the global energy minimum (or at least a close approximation) by slowly lowering the temperature and thus obtain the solution to the optimisation problem.The phenomenon of quantum tunneling suggests that it can be more efficient to explore the state space quantum mechanically in a quantum annealer [4][5][6]. In simulated quantum annealing [7,8], one makes use of this effect by adding quantum fluctuations, which are slowly reduced while keeping the temperature constant and positive -ultimately ending up in a low energy configuration of the optimisation problem. Simulated quantum annealing, using a quantum Monte Carlo algorithm, has been observed to be more efficient than thermal annealing for certain spin glass models [8], although the opposite has been observed for k-satisfiability problems [9]. Further speedup may be expected in physical quantum annealing, either as an experimental technique for annealing a quantum spin glass [10], or -and this is what we will focus on here -as a computational technique in a programmable quantum device.In this work we report on computer simulations and experimental tests on a D-Wave One device [11] in order to address central open questions about quantum annealers: is the device actually a quantum annealer, i.e., do the quantum effects observed on 8 [11,12] and 16 qubits [13] persist when scaling problems up to more than 100 qubits, or do short coherence times turn the device into a classical, thermal annealer? Which ...

The development of small-scale quantum devices raises the question of how to fairly assess and detect quantum speedup. Here, we show how to define and measure quantum speedup and how to avoid pitfalls that might mask or fake such a speedup. We illustrate our discussion with data from tests run on a D-Wave Two device with up to 503 qubits. By using random spin glass instances as a benchmark, we found no evidence of quantum speedup when the entire data set is considered and obtained inconclusive results when comparing subsets of instances on an instance-by-instance basis. Our results do not rule out the possibility of speedup for other classes of problems and illustrate the subtle nature of the quantum speedup question.

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 electronand 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. Introduction.-In spite of considerable recent progress, coherent control and quantum information processing (QIP) is still plagued by the problems associated with controllability of quantum systems under realistic conditions. The two main obstacles in any experimental realization of QIP are (i) faulty controls, i.e., control parameters which are limited in range and precision, and (ii) decoherence-errors due to inevitable system-bath interactions. Nuclear magnetic resonance (NMR) has been a particularly fertile arena for the development of many methods to overcome such problems, starting with the discovery of the spin-echo effect, and followed by methods such as refocusing, and composite pulse sequences [1]. Closely related to the spin-echo effect and refocusing is the method of dynamical decoupling (DD) pulses introduced into QIP in order to overcome decoherence-errors [2,3]. In standard DD one uses a periodic sequence of fast and strong symmetrizing pulses to reduce the undesired parts of the system-bath interaction Hamiltonian H SB , causing decoherence. Since DD requires no encoding overhead, no measurements, and no feedback, it is an economical alternative to the method of quantum error correcting codes (QECC) [e.g., [4,5,6], and references therein], in the non-Markovian regime [7].Here we introduce concatenated DD (CDD) pulse sequences, which have a recursive temporal structure. We show both numerically and analytically that CDD pulse sequences have two important advantages over standard, periodic DD (PDD): (i) Significant fault-tolerance to both random and systematic pulse-control errors (see Ref.[8] for a related study), (ii) CDD is significantly more efficient at decoupling than PDD, when compared at equal switching times and pulse numbers. These advantages simplify the requirements of DD (fast-paced strong pulses) in general, and bring it closer to utility in QIP as a feedback-free error correction scheme.The noisy quantum control problem.-The problem of

Universal quantum computation on decoherence-free subspaces and subsystems (DFSs) is examined with particular emphasis on using only physically relevant interactions. A necessary and sufficient condition for the existence of decoherence-free (noiseless) subsystems in the Markovian regime is derived here for the first time. A stabilizer formalism for DFSs is then developed which allows for the explicit understanding of these in their dual role as quantum error correcting codes. Conditions for the existence of Hamiltonians whose induced evolution always preserves a DFS are derived within this stabilizer formalism. Two possible collective decoherence mechanisms arising from permutation symmetries of the system-bath coupling are examined within this framework. It is shown that in both cases universal quantum computation which always preserves the DFS (natural fault-tolerant computation) can be performed using only two-body interactions. This is in marked contrast to standard error correcting codes, where all known constructions using one or two-body interactions must leave the codespace during the on-time of the fault-tolerant gates. A further consequence of our universality construction is that a single exchange Hamiltonian can be used to perform universal quantum computation on an encoded space whose asymptotic coding efficiency is unity. The exchange Hamiltonian, which is naturally present in many quantum systems, is thus asymptotically universal.

We develop a general theory of the relation between quantum phase transitions (QPTs) characterized by nonanalyticities in the energy and bipartite entanglement. We derive a functional relation between the matrix elements of two-particle reduced density matrices and the eigenvalues of general two-body Hamiltonians of d-level systems. The ground state energy eigenvalue and its derivatives, whose non-analyticity characterizes a QPT, are directly tied to bipartite entanglement measures. We show that first-order QPTs are signalled by density matrix elements themselves and second-order QPTs by the first derivative of density matrix elements. Our general conclusions are illustrated via several quantum spin models.PACS numbers: 03.65. Ud,75.10.Pq Recently, a great deal of effort has been devoted to the understanding of the connections between quantum information [1] and the theory of quantum critical phenomena [2]. A key novel observation is that quantum entanglement can play an important role in a quantum phase transition (QPT) [3,4,5,6,7,8,9,10,11,12,13,14,15,16]. In particular, for a number of spin systems, it has been shown that QPTs are signalled by a critical behavior of bipartite entanglement as measured, for instance, in terms of the concurrence [17]. For the case of second-order QPTs (2QPTs), the critical point was found to be associated to a singularity in the derivative of the ground state concurrence, as first illustrated, for the transverse field Ising chain, in Ref. [3], and generalized in Refs. [4,5,6] (see Refs. [7,8,9,10,11] for an analysis in terms of other entanglement measures). In the case of first-order QPTs (1QPTs), discontinuities in the ground state concurrence were shown to detect the QPT [12,13,14]. The studies conducted to date are based on the analysis of particular many-body models. Hence the general connection between bipartite entanglement and QPTs is not yet well understood. The aim of this work is to discuss, in a general framework, how bipartite entanglement can be related to a QPT characterized by nonanalyticities in the energy.Expectation values and the reduced density matrix.-The most general Hamiltonian of non-identical particles, up to two-body interactions, readswhere {|α i } is a basis for the Hilbert space, α, β, γ, δ ∈ {0, 1, ..., d − 1}, and i, j enumerate N "qudits" (dlevel systems).Let E = ψ|H|ψ be the energy in a non-degenerate eigenstate |ψ of the Hamiltonian. The two-spin reduced density operatorρ ij is given byρ ij = m m|ψ ψ|m , with m running over all the d N −2 orthonormal basis vectors, excluding qudits i and j.ρ ij has a d 2 × d 2 matrix representation ρ ij , with elements ρ with U(ij) denoting a d 2 × d 2 matrix whose elements are, where N i is the number of qudits that qudit i interacts with, and δ j βδ is the Kronecker symbol on qudit j. Clearly, Eq. (2) holds not only for the Hamiltonian operator but for any observable. Indeed, it turns out that the expectation value (or eigenvalue, for an eigenstate) of any two-qudit observable in an arbitrary state |ψ is a ...

A general scheme to perform universal, fault-tolerant quantum computation within decoherence-free subspaces (DFSs) is presented. At most two-qubit interactions are required, and the system remains within the DFS throughout the entire implementation of a quantum gate. We show explicitly how to perform universal computation on clusters of the four-qubit DFS encoding one logical qubit each under spatially symmetric (collective) decoherence. Our results have immediate relevance to quantum computer implementations in which quantum logic is implemented through exchange interactions, such as the recently proposed spin-spin coupled quantum dot arrays and donor-atom arrays.