We experimentally study the performance of a programmable quantum annealing processor, the D-Wave One (DW1) with up to 108 qubits, on maximum SAT problem with 2 variables per clause (MAX 2-SAT) problems. We consider ensembles of random problems characterized by a fixed clause density, an external parameter which we tune through its critical value in our experiments. We demonstrate that the DW1 is sensitive to the critical value of the clause density. The DW1 results are verified and compared with akmaxsat, an exact, state-of-the-art algorithm. We study the relative performance of the two solvers and how they correlate in terms of problem hardness. We find that the DW1 performance scales more favorably with problem size and that problem hardness correlation is essentially non-existent. We discuss the relevance and limitations of such a comparison.Keywords: adiabatic quantum computation, quantum annealing, computational complexity IntroductionAdiabatic quantum computation (AQC) is a model of solving computational problems, in particular hard optimization problems, by evolving a closed system in the ground state manifold Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. P f ad, represents the solution to the computational problem. AQC has been proven to be polynomially equivalent to standard, closed-system, circuit model QC [3][4][5][6][7], but so far it is unclear whether this equivalence extends to the open system, non-zero New J. Phys. 16 (2014) 045006 S Santra et al 2 New J. Phys. 16 (2014) 045006 S Santra et al 3 5 In more detail, the McGeoch and Wang (MW) study [35], working with the DW2, used a 439 qubit subgraph of Chimera and considered three problems: (1) Chimera-structured QUBO instances (this is actually an ensemble of uniform samples from the Ising model on Chimera with ∈ − J h , { 1,1} ij j[36]), (2) Weighted Max 2-SAT, (3) the Quadratic Assignment Problem. Their main conclusion is that in their experiments the DW2 (together with a software layer called Blackbox) outperformed the software against which it was tested. In the case of problem (1), the DW2 is reported as outperforming its nearest rival (CPLEX), amongst those tried, by a factor of 3600. The times recorded by MW were for the first point that CPLEX found the optimal solution, and not the time at which it proved it optimality. However, several researchers have reported classical implementations for all three problems which outperform the DW2 and in particular the MW benchmarks [36,37]. Our ensemble of MAX 2-SAT problems differs from the weighted MAX 2-SAT problems considered by MW, since we used uniform weights with the additional constraint of a fixed clause density. In addition, unlike MWʼs case (2), our MAX 2-SAT problem ensemble inherits the native Chimera graph structure along with the connectivity contraints of the processor by design, which eliminates the ...

Dynamical decoupling is a coherent control technique where the intrinsic and extrinsic couplings of a quantum system are effectively averaged out by application of specially designed driving fields (refocusing pulse sequences). This entails pumping energy into the system, which can be especially dangerous when it has sharp spectral features like a cavity mode close to resonance. In this work we show that such an effect can be avoided with properly constructed refocusing sequences. To this end we construct the average Hamiltonian expansion for the system evolution operator associated with a single ``soft'' pi-pulse. To second order in the pulse duration, we characterize a symmetric pulse shape by three parameters, two of which can be turned to zero by shaping. We express the effective Hamiltonians for several pulse sequences in terms of these parameters, and use the results to analyze the structure of error operators for controlled Jaynes-Cummings Hamiltonian. When errors are cancelled to second order, numerical simulations show excellent qubit fidelity with strongly-suppressed oscillator heating.Comment: 9pages, 5eps figure

We analyze numerically the performance of the near-optimal quadratic dynamical decoupling (QDD) singlequbit 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.

We utilize genetic algorithms aided by simulated annealing to find optimal dynamical decoupling (DD) sequences for a single-qubit system subjected to a general decoherence model under a variety of control pulse conditions. We focus on the case of sequences with equal pulse-intervals and perform the optimization with respect to pulse type and order. In this manner we obtain robust DD sequences, first in the limit of ideal pulses, then when including pulse imperfections such as finite pulse duration and qubit rotation (flip-angle) errors. Although our optimization is numerical, we identify a deterministic structure that underlies the top-performing sequences. We use this structure to devise DD sequences which outperform previously designed concatenated DD (CDD) and quadratic DD (QDD) sequences in the presence of pulse errors. We explain our findings using time-dependent perturbation theory and provide a detailed scaling analysis of the optimal sequences. arXiv:1210.5538v2 [quant-ph] 7 Aug 2013 1 Our choice of units is arbitrary but is meant to be commensurate with electron spin qubits in, e.g., quantum dots.

We introduce high-order dynamical decoupling strategies for open system adiabatic quantum computation. Our numerical results demonstrate that a judicious choice of high-order dynamical decoupling method, in conjunction with an encoding which allows computation to proceed alongside decoupling, can dramatically enhance the fidelity of adiabatic quantum computation in spite of decoherence.Introduction.-In adiabatic quantum computation (AQC) a problem is solved by evolving in the ground state manifold of an adiabatic Hamiltonian H ad (t), with t ∈ [0, T ] [1, 2]. The ground state of the beginning Hamiltonian H B = H ad (0) is assumed to be easy to prepare, while the final one, the ground state of the problem Hamiltonian H P = H ad (T ), represents the solution to the computational problem. AQC has been shown to be computationally equivalent to the standard circuit model of QC [3][4][5][6][7], and is being pursued experimentally using superconducting flux qubits [8] and nuclear magnetic resonance [9,10]. However, in spite of evidence of intrinsic robustness [11][12][13][14][15][16] and proposals to protect AQC against decoherence [17,18], AQC still lacks a complete theory of fault-tolerance, unlike the circuit model of QC [19][20][21][22][23][24]. In fact, even identifying an acceptable notion of fault-tolerant AQC (FTAQC) is an open problem. In the circuit model a fault-tolerant simulation allows one to generate the output of a given ideal circuit, to arbitrary accuracy, using the faulty components of another circuit [25]. A similar definition for AQC would presumably involve the simulation of an ideal adiabatic evolution using a faulty one, but if "faulty" is simply taken to mean "non-adiabatic", then one can just use the equivalence proof [3][4][5][6][7] and circuit-model fault-tolerance [19][20][21][22][23][24] to argue that the problem is already solved. However, this argument misses the point since to qualify as AQC, at least the computation should remain adiabatic, i.e., the defining feature of AQC-the adiabatic preparation of the ground state of H P -should be preserved. On the other hand, it seems too restrictive to require the techniques used to address decoherence and noise to be adiabatic as well. We thus propose the following characterization of FTAQC: 'Given a closed-system AQC specified by H B and H P , and > 0, a fault tolerant opensystem simulation will use adiabatic evolution between two faulty, encoded HamiltoniansH B andH P derived from H B and H P , so that the final system-only state of the simulation is efficiently decodable to a state that is -close (in fidelity) to the ground state of H P . In addition, the simulation may involve any other faulty non-adiabatic error-correction, suppression, or avoidance operations.'

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We consider the effect of broadband decoherence on the performance of refocusing sequences, having in mind applications of dynamical decoupling in concatenation with quantum error correcting codes as the first stage of coherence protection. Specifically, we construct cumulant expansions of effective decoherence operators for a qubit driven by a pulse of a generic symmetric shape, and for several sequences of π-and π/2-pulses. While, in general, the performance of soft pulses in decoupling sequences in the presence of Markovian decoherence is worse than that of the ideal δ-pulses, it can be substantially improved by shaping.