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A critical question for the field of quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of state-of-the-art classical computers, achieving so-called quantum supremacy. We study the task of sampling from the output distributions of (pseudo-)random quantum circuits, a natural task for benchmarking quantum computers. Crucially, sampling this distribution classically requires a direct numerical simulation of the circuit, with computational cost exponential in the number of qubits. This requirement is typical of chaotic systems. We extend previous results in computational complexity to argue more formally that this sampling task must take exponential time in a classical computer. We study the convergence to the chaotic regime using extensive supercomputer simulations, modeling circuits with up to 42 qubits -the largest quantum circuits simulated to date for a computational task that approaches quantum supremacy. We argue that while chaotic states are extremely sensitive to errors, quantum supremacy can be achieved in the near-term with approximately fifty superconducting qubits. We introduce cross entropy as a useful benchmark of quantum circuits which approximates the circuit fidelity. We show that the cross entropy can be efficiently measured when circuit simulations are available. Beyond the classically tractable regime, the cross entropy can be extrapolated and compared with theoretical estimates of circuit fidelity to define a practical quantum supremacy test. arXiv:1608.00263v3 [quant-ph]

Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2-design characteristic of random circuits, and that solutions to this problem must be studied.

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.

We develop generalized bounds for quantum single-parameter estimation problems for which the coupling to the parameter is described by intrinsic multisystem interactions. For a Hamiltonian with k-system parameter-sensitive terms, the quantum limit scales as 1/Nk, where N is the number of systems. These quantum limits remain valid when the Hamiltonian is augmented by any parameter-independent interaction among the systems and when adaptive measurements via parameter-independent coupling to ancillas are allowed.

Quantum annealing (QA) has been proposed as a quantum enhanced optimization heuristic exploiting tunneling. Here, we demonstrate how finite range tunneling can provide considerable computational advantage. For a crafted problem designed to have tall and narrow energy barriers separating local minima, the D-Wave 2X quantum annealer achieves significant runtime advantages relative to Simulated Annealing (SA). For instances with 945 variables, this results in a time-to-99%success-probability that is ∼ 10 8 times faster than SA running on a single processor core. We also compared physical QA with Quantum Monte Carlo (QMC), an algorithm that emulates quantum tunneling on classical processors. We observe a substantial constant overhead against physical QA: D-Wave 2X again runs up to ∼ 10 8 times faster than an optimized implementation of QMC on a single core. We note that there exist heuristic classical algorithms that can solve most instances of Chimera structured problems in a timescale comparable to the D-Wave 2X. However, we believe that such solvers will become ineffective for the next generation of annealers currently being designed. To investigate whether finite range tunneling will also confer an advantage for problems of practical interest, we conduct numerical studies on binary optimization problems that cannot yet be represented on quantum hardware. For random instances of the number partitioning problem, we find numerically that QMC, as well as other algorithms designed to simulate QA, scale better than SA. We discuss the implications of these findings for the design of next generation quantum annealers.

Abstract. We develop from first principles Markovian master equations suited for studying the time evolution of a system evolving adiabatically while coupled weakly to a thermal bath. We derive two sets of equations in the adiabatic limit, one using the rotating wave (secular) approximation that results in a master equation in Lindblad form, the other without the rotating wave approximation but not in Lindblad form. The two equations make markedly different predictions depending on whether or not the Lamb shift is included. Our analysis keeps track of the various time and energy scales associated with the various approximations we make, and thus allows for a systematic inclusion of higher order corrections, in particular beyond the adiabatic limit. We use our formalism to study the evolution of an Ising spin chain in a transverse field and coupled to a thermal bosonic bath, for which we identify four distinct evolution phases. While we do not expect this to be a generic feature, in one of these phases dissipation acts to increase the fidelity of the system state relative to the adiabatic ground state.