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Many quantum machine learning algorithms use a quantum linear system solver (QLS) as a subroutine. HHL type QLS algorithms achieve exponential speedup over classical algorithms for sparse matrices, however for dense matrices the speed-up is less profound, In this paper, Wossnig et al. describe a new QLS algorithm using the quantum singular value estimation. When applied to a dense matrix with spectral norm bounded by a constant, the runtime of this proposed algorithm is bounded by O(κ^2 √n.polylog(n)/e), which is a quadratic improvement over HHL based QLS algorithms. In comparison, classical (non-quantum) linear system solvers typically require time O(n^3) for dense matrices.

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Quantum computers can be used to address molecular structure, materials science and condensed matter physics problems, which currently stretch the limits of existing high-performance computing resources. Finding exact numerical solutions to these interacting fermion problems has exponential cost, while Monte Carlo methods are plagued by the fermionic sign problem. In Quantum Computational Chemistry solutions, the Variational Quantum Eigensolver (VQE) algorithm offers a hybrid classical-quantum, and thus low quantum circuit depth, alternative to the Phase Estimation algorithm used to measure the ground-state energy of a molecular Hamiltonian. In VQE the quantum computer is used to prepare variational trial states that depend on a set of parameters. Then, the expectation value of the energy is estimated and used by a classical optimizer to generate a new set of improved parameters. The advantage of VQE over classical simulation methods is that in VQE one can prepare trial states that are not amenable to efficient classical numerics. In this paper, Kandala et al. demonstrate the experimental results for determining the ground state energy for molecules of increasing size, up to BeH2 using the VQE algorithm.

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In this article Boixo et al. (Google/UCSB) present their proposed quantum supremacy experiment. Their proposal uses the task of sampling from the output distributions of (pseudo-) random quantum circuits, which, classically, requires a direct numerical simulation of the circuit, with computational cost exponential in the number of qubits.. They estimate that in such experiment, quantum supremacy can be achieved in the near-term with approximately 50 superconducting qubits. Furthermore they introduce cross entropy as a practical test of quantum supremacy.

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Most near-term quantum-computational chemistry experiments have so-far been implemented by applying the Variational Quantum Eigensolver (VQE) classical-quantum hybrid algorithm as an alternative to Quantum Phase Estimation (QPE). This is due to the fact that QPE requires many orders of magnitude more quantum gates than is feasible with typical coherence times of current and near-term quantum-processors. As an alternative, in this paper, Paesani et al. report experimental results of a recently proposed adaptive Bayesian approach to quantum phase estimation and use it to simulate molecular energies on a Silicon quantum photonic device. The approach is verified to be well suited for NISQ quantum-processors by investigating its superior robustness to noise and decoherence compared to the iterative phase estimation algorithm. There results shows a promising route to unlock the power of QPE much sooner than previously believed possible.

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This essay by The Economist journalist Jason Palmer provides an introductory overview of the state of development of different quantum-technologies, their potential use-cases and current investments and patent applications per country

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Quantum sensing and metrology may benefit from a spatially distributed network architecture employing entangled states and measurements to enhance precision. Given the challenges faced in the creation and manipulation of entangled states, a complete understanding of when entanglement is (and is not) critical to optimizing estimation precision is importance. In this paper, Proctor et al. introduce a general model for a network of quantum sensors, and use this model to determine whether precision enhancement can be achieved in a range of practical applications.

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Several physical platforms are aiming to realize a fully programmable, coherent and scalable quantum annealing device. In this paper, Glaetzle et al. show that combining a quantum simulation toolbox for Rydberg atoms with the Lechner-Hauke-Zoller (LHZ) architecture allows one to build a prototype for a coherent adiabatic quantum computer with all-to-all Ising interactions. 

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Networks of coupled optical parametric oscillators (OPOs) are an alternative physical system for solving Ising type problems. Theoretical/numerical investigations have shown that in principle quantum effects (like entanglement between delay-coupled pulses) can play meaningful roles in such systems. In this paper, McMahon et al. (and an earlier paper of Inagaki et al.), show that this type of architecture is relatively scalable and can be used to solve max cut problems accurately, although in the current prototype devices the quantum features are 'washed out' by high round-trip losses (typically 10 dB), to the point that a purely semi-classical description of the system is sufficient to explain all the observed experimental results. The next step would be to realize this architecture in a system where the quantum nature is not lost.

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In this paper, Venturelli et al. present a quantum annealing solver for the renowned job-shop scheduling problem (JSP). They formulate the problem as a time-indexed quadratic unconstrained binary optimization problem, several pre-processing and graph embedding strategies are employed to compile optimally parametrized families of the JSP for scheduling instances of up to six jobs and six machines on the D-Wave Systems Vesuvius (DW2) processor. Problem simplifications and partitioning algorithms are discussed and the results from the processor are compared against state-of-the-art global-optimum solvers.

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In this paper, Puri et al. propose an alternative to the typical quantum annealing architecture with a scalable network of all-to-all connected, two-photon driven Kerr-nonlinear resonators. Each of these resonators encode an Ising spin in a robust degenerate subspace formed by two coherent states of opposite phases. A fully-connected optimization problem is mapped onto local fields driving the resonators, which are themselves connected by local four-body interactions. They describe an adiabatic annealing protocol in this system and analyze its performance in the presence of photon loss. Numerical simulations indicate substantial resilience to this noise channel, making it a promising platform for implementing a large scale quantum Ising machine.

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A recommendation engine uses the past purchases or ratings to construct a (partial) preference matrix, which is used to provide personalized recommendations to individual users. In this paper, Kerenidis et al. present a quantum recommendation system which updates the partial preference matrix each time data comes in and can, based on such matrix, provides a recommendation in time polylog to the matrix-dimension, exponentially faster than classical recommendation systems.

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Mean-variance portfolio optimization problems are traditionally solved as continuous-variable problems. However, for assets that can only be traded in large lots, or for asset managers who are constrained to trading large blocks of assets, solving the continuous problem yields an approximation. The discrete problem, is expected to provide better results, but is non-convex due to the fragmented nature of the domain, and is therefore much harder to solve. In this paper, Rosenberg et al. attempt to solve a discrete multi-period portfolio optimisation problem using D-Wave Systems' quantum annealer. They derive a formulation of the problem, discuss several possible integer encoding schemes, and present numerical examples that show high success rates. They also present insight into how results may be improved using suitable software enhancements, and why current quantum annealing technology limits the size of problem that can be successfully solved today. The formulation presented is specifically designed to be scalable, with the expectation that as quantum annealing technology improves, larger problems will be solvable using the same techniques. 

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