The next few years will be exciting as prototype universal quantum processors emerge, enabling the implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation and which have the potential to significantly expand the breadth of applications for which quantum computers have an established advantage. A leading candidate is Farhi et al.'s quantum approximate optimization algorithm, which alternates between applying a cost function based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the quantum alternating operator ansatz, is the consideration of general parameterized families of unitaries rather than only those corresponding to the time evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach, in the spirit of the quantum approximate optimization algorithm, to a wide variety of approximate optimization, exact optimization, and sampling problems. In addition to introducing the quantum alternating operator ansatz, we lay out design criteria for mixing operators, detail mappings for eight problems, and provide a compendium with brief descriptions of mappings for a diverse array of problems.
Many NP-hard problems can be seen as the task of finding a ground state of a disordered highly connected Ising spin glass. If solutions are sought by means of quantum annealing, it is often necessary to represent those graphs in the annealer's hardware by means of the graph-minor embedding technique, generating a final Hamiltonian consisting of coupled chains of ferromagnetically bound spins, whose binding energy is a free parameter. In order to investigate the effect of embedding on problems of interest, the fully connected Sherrington-Kirkpatrick model with random AE1 couplings is programmed on the D-Wave Two TM annealer using up to 270 qubits interacting on a Chimera-type graph. We present the best embedding prescriptions for encoding the Sherrington-Kirkpatrick problem in the Chimera graph. The results indicate that the optimal choice of embedding parameters could be associated with the emergence of the spin-glass phase of the embedded problem, whose presence was previously uncertain. This optimal parameter setting allows the performance of the quantum annealer to compete with (and potentially outperform, in the absence of analog control errors) optimized simulated annealing algorithms.
Physical annealing systems provide heuristic approaches to solving combinatorial optimization problems. Here, we benchmark two types of annealing machines—a quantum annealer built by D-Wave Systems and measurement-feedback coherent Ising machines (CIMs) based on optical parametric oscillators—on two problem classes, the Sherrington-Kirkpatrick (SK) model and MAX-CUT. The D-Wave quantum annealer outperforms the CIMs on MAX-CUT on cubic graphs. On denser problems, however, we observe an exponential penalty for the quantum annealer [exp(–αDWN2)] relative to CIMs [exp(–αCIMN)] for fixed anneal times, both on the SK model and on 50% edge density MAX-CUT. This leads to a several orders of magnitude time-to-solution difference for instances with over 50 vertices. An optimal–annealing time analysis is also consistent with a substantial projected performance difference. The difference in performance between the sparsely connected D-Wave machine and the fully-connected CIMs provides strong experimental support for efforts to increase the connectivity of quantum annealers.
We report on a case study in programming an early quantum annealer to attack optimization problems related to operational planning. While a number of studies have looked at the performance of quantum annealers on problems native to their architecture, and others have examined performance of select problems stemming from an application area, ours is one of the first studies of a quantum annealer's performance on parametrized families of hard problems from a practical domain. We explore two different general mappings of planning problems to quadratic unconstrained binary optimization (QUBO) problems, and apply them to two parametrized families of planning problems, navigation-type and scheduling-type. We also examine two more compact, but problemtype specific, mappings to QUBO, one for the navigation-type planning problems and one for the scheduling-type planning problems. We study embedding properties and parameter setting, and examine their effect on the efficiency with which the quantum annealer solves these problems. From these results we derive insights useful for the programming and design of future quantum annealers: problem choice, the mapping used, the properties of the embedding, and the annealing profile all matter, each significantly affecting the performance.
We present a theoretical study of an electronic quantum refrigerator based on four quantum dots arranged in a square configuration, in contact with as many thermal reservoirs. We show that the system implements the minimal mechanism for acting as a self-contained quantum refrigerator, by demonstrating heat extraction from the coldest reservoir and the cooling of the nearby quantum-dot.PACS numbers: 03.65. Yz, 73.63.Kv, The increasing interest in quantum thermal machines has its roots in the need to understand the relations between thermodynamics and quantum mechanics [1,2]. The progress in this field may as well have important applications in the control of heat transport in nano-devices . In a series of recent works [4-6] the fundamental limits to the dimensions of a quantum refrigerator have been found. It has been further demonstrated that these machines could still attain Carnot-efficiency  thus launching the call for the implementation of the smallest possible quantum refrigerator. Refs.[4-6] considered selfcontained thermal machines defined as those that perform a cycle without the supply of external work, their action being grounded on the steady-state heat transfer from thermal reservoirs at different temperatures. The major difficulty in the realization [7,8] of self-contained refrigerators (SCRs) is the engineering of the crucial three-body interaction enabling the coherent transition between a doubly excited state in contact with a hot (H) and cold (C) reservoir, and a singly-excited state coupled to an intermediate (or "room" -R) temperature bath. We get around this problem by proposing an experimentally feasible implementation of a minimal SCR with semiconducting quantum dots (QDs) operating in the Coulomb blockade regime. We are thus able to establish a connection between the general theory of quantum machines and the heat transport in nanoelectronics .QDs contacted by leads were proposed as ideal systems for achieving high thermopower  or anomalous thermal effects . Here we study a four-QD planar array (hereafter named a "quadridot" for simplicity) coupled to independent electron reservoirs as shown in Fig. 1; with proper (but realistic) tuning of the parameters, we will show that the quadridot acts as a SCR which pumps energy from the high temperature reservoir H and the low temperature reservoir C to the intermediate temperature reservoirs R 1 , R 2 . Furthermore we will analyze the conditions under which the quadridot is able to cool the dot QD 2 which is directly connected to the bath C, at an effective temperature that is lower than the one it would have had in the absence of the other reservoirs. This will lead us to introduce an operative definition of the local effective temperature depending on the measurement FIG. 1: [Color online]The quadridot. The four quantum dots QD1, QD2, QD3, and QD4 are weakly coupled to the reservoirs R1, C, R2, and H, respectively, which are all grounded and maintained at temperatures TH > (TR 1 = TR 2 = TR) > TC. Tunneling is allowed onl...
We investigate alternative annealing schedules on the current generation of quantum annealing hardware (the D-Wave 2000Q), which includes the use of forward and reverse annealing with an intermediate pause. This work provides new insights into the inner workings of these devices (and quantum devices in general), particular into how thermal effects govern the system dynamics. We show that a pause mid-way through the anneal can cause a dramatic change in the output distribution, and we provide evidence suggesting thermalization is indeed occurring during such a pause. We demonstrate that upon pausing the system in a narrow region shortly after the minimum gap, the probability of successfully finding the ground state of the problem Hamiltonian can be increased over an order of magnitude. We relate this effect to relaxation (i.e. thermalization) after diabatic and thermal excitations that occur in the region near to the minimum gap. For a set of large-scale problems of up to 500 qubits, we demonstrate that the distribution returned from the annealer very closely matches a (classical) Boltzmann distribution of the problem Hamiltonian, albeit one with a temperature at least 1.5 times higher than the (effective) temperature of the annealer. Moreover, we show that larger problems are more likely to thermalize to a classical Boltzmann distribution. arXiv:1810.05881v2 [quant-ph]
We investigate a hybrid quantum-classical solution method to the mean-variance portfolio optimization problems. Starting from real financial data statistics and following the principles of the Modern Portfolio Theory, we generate parametrized samples of portfolio optimization problems that can be related to quadratic binary optimization forms programmable in the analog D-Wave Quantum Annealer 2000Q TM . The instances are also solvable by an industry-established Genetic Algorithm approach, which we use as a classical benchmark. We investigate several options to run the quantum computation optimally, ultimately discovering that the best results in terms of expected time-to-solution as a function of number of variables for the hardest instances set are obtained by seeding the quantum annealer with a solution candidate found by a greedy local search and then performing a reverse annealing protocol. The optimized reverse annealing protocol is found to be more than 100 times faster than the corresponding forward quantum annealing on average.
Abstract. To run quantum algorithms on emerging gate-model quantum hardware, quantum circuits must be compiled to take into account constraints on the hardware. For near-term hardware, with only limited means to mitigate decoherence, it is critical to minimize the duration of the circuit. We investigate the application of temporal planners to the problem of compiling quantum circuits to newly emerging quantum hardware. While our approach is general, we focus on compiling to superconducting hardware architectures with nearest neighbor constraints. Our initial experiments focus on compiling Quantum Alternating Operator Ansatz (QAOA) circuits whose high number of commuting gates allow great flexibility in the order in which the gates can be applied. That freedom makes it more challenging to find optimal compilations but also means there is a greater potential win from more optimized compilation than for less flexible circuits. We map this quantum circuit compilation problem to a temporal planning problem, and generated a test suite of compilation problems for QAOA circuits of various sizes to a realistic hardware architecture. We report compilation results from several state-of-the-art temporal planners on this test set. This early empirical evaluation demonstrates that temporal planning is a viable approach to quantum circuit compilation.
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