Fuelled by increasing computer power and algorithmic advances, machine learning techniques have become powerful tools for finding patterns in data. Quantum systems produce atypical patterns that classical systems are thought not to produce efficiently, so it is reasonable to postulate that quantum computers may outperform classical computers on machine learning tasks. The field of quantum machine learning explores how to devise and implement quantum software that could enable machine learning that is faster than that of classical computers. Recent work has produced quantum algorithms that could act as the building blocks of machine learning programs, but the hardware and software challenges are still considerable.

Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang [LC17a]. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while typically only use a constant number of ancilla qubits.We show that singular value transformation leads to novel algorithms. We give an efficient solution to a "non-commutative" measurement problem used for efficient ground-state-preparation of certain local Hamiltonians, and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression."Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, and quickly derive the following algorithms: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms.In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.

With rapid recent advances in quantum technology, we are close to the threshold of quantum devices whose computational powers can exceed those of classical supercomputers. Here, we show that a quantum computer can be used to elucidate reaction mechanisms in complex chemical systems, using the open problem of biological nitrogen fixation in nitrogenase as an example. We discuss how quantum computers can augment classical computer simulations used to probe these reaction mechanisms, to significantly increase their accuracy and enable hitherto intractable simulations. Our resource estimates show that, even when taking into account the substantial overhead of quantum error correction, and the need to compile into discrete gate sets, the necessary computations can be performed in reasonable time on small quantum computers. Our results demonstrate that quantum computers will be able to tackle important problems in chemistry without requiring exorbitant resources.quantum computing | quantum algorithms | reaction mechanisms C hemical reaction mechanisms are networks of molecular structures representing short-or long-lived intermediates connected by transition structures. The relative energies of all stable structures determine the relative thermodynamical stability. Differences of the energies of local minima and connecting transition structures determine the rates of interconversion, i.e., the chemical kinetics of the process. As they enter exponential expressions, very accurate energy differences are required for the reliable evaluation of the rate constants. At its core, the detailed understanding and prediction of complex reaction mechanisms then requires highly accurate electronic structure methods. However, the electron correlation problem remains, despite decades of progress (1), one of the most vexing problems in quantum chemistry. Although approximate approaches, such as density functional theory (DFT) (2), are very popular, their accuracy is often too low for quantitative predictions (see, e.g., refs. 3 and 4); this holds particularly true for molecules with many energetically close-lying orbitals. For such problems on classical computers, much less than a hundred strongly correlated electrons are already out of reach for systematically improvable ab initio methods that could achieve the required accuracy.The apparent intractability of accurate simulations for such quantum systems led Richard Feynmann to propose quantum computers. The promise of exponential speedups for quantum simulation on quantum computers was first investigated by Lloyd (5) and Zalka (6) and was directly applied to quantum chemistry by Lidar, Aspuru-Guzik, and others (7-11). Quantum chemistry simulation has remained an active area within quantum algorithm development, with ever more sophisticated methods being used to reduce the costs of quantum chemistry simulation (12)(13)(14)(15)(16)(17)(18)(19)(20).The promise of exponential speedups for the electronic structure problem has led many to suspect that quantum computers will one day revolu...