Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging and complex landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chemistry, such as the electronic structure of molecules. In the past two decades, significant advances have been made in developing algorithms and physical hardware for quantum computing, heralding a revolution in simulation of quantum systems. This Review provides an overview of the algorithms and results that are relevant for quantum chemistry. The intended audience is both quantum chemists who seek to learn more about quantum computing and quantum computing researchers who would like to explore applications in quantum chemistry.

The promise of quantum neural nets, which utilize quantum effects to model complex data sets, has made their development an aspirational goal for quantum machine learning and quantum computing in general. Here we provide new methods of training quantum Boltzmann machines, which are a class of recurrent quantum neural network. Our work generalizes existing methods and provides new approaches for training quantum neural networks that compare favorably to existing methods. We further demonstrate that quantum Boltzmann machines enable a form of quantum state tomography that not only estimates a state but provides a prescription for generating copies of the reconstructed state. Classical Boltzmann machines are incapable of this. Finally we compare small non-stoquastic quantum Boltzmann machines to traditional Boltzmann machines for generative tasks and observe evidence that quantum models outperform their classical counterparts.Introduction-The Boltzmann machine is a widely used type of recurrent neural net that, unlike the feed forward neural nets used in many applications, is capable of generating new examples of the training data [1]. This makes it an excellent model to use in cases where data is missing. We focus on Boltzmann machines because, of all neural net models, the Boltzmann machine is perhaps the most natural one for physicists. It models the input data as if it came from an Ising model in thermal equilibrium. The goal of training is then to find the Ising model that is most likely to reproduce the input data which is known as a training set.The close analogy between this model and physics has made it a natural fit for quantum computing and quantum annealing. A number of proposals have been put forward for accelerating Boltzmann machines in current generation quantum annealers [2-4] and quantum computers [5], the latter showing polynomial speedups relative to classical training [6]. While these methods showed that quantum technologies can train Boltzmann machines more accurately and at lower cost than classical methods, the question of whether transitioning from an Ising model to a quantum model for the data would provide substantial improvements.This question is addressed in [7], wherein a new method for training Boltzmann machines is provided that uses transverse Ising models in thermal equilibrium to model the data. While such models are trainable and can outperform classical Boltzmann machines, the training procedure proposed therein suffers two drawbacks. First, it is unable to learn quantum terms from classical data. Second, the transverse Ising models considered are widely believed to be simulatable using quantum Monte-Carlo methods. This means that such models are arguably not quantum and as such the benchmarks they give do not necessarily apply to manifestly quantum models. Here we rectify these issues by giving new training methods that do not suffer these drawbacks and illustrate their performance for models that are manifestly quantum.The first, and arguably most important, task when approach...

We study the glued-trees problem of Childs, et. al. [1] in the adiabatic model of quantum computing and provide an annealing schedule to solve an oracular problem exponentially faster than classically possible. The Hamiltonians involved in the quantum annealing do not suffer from the socalled sign problem. Unlike the typical scenario, our schedule is efficient even though the minimum energy gap of the Hamiltonians is exponentially small in the problem size. We discuss generalizations based on initial-state randomization to avoid some slowdowns in adiabatic quantum computing due to small gaps.PACS numbers: 03.67. Ac, 03.67.Lx, 42.50.Lc Quantum annealing is a powerful heuristic to solve problems in optimization [2,3]. In quantum computing, the method consists of preparing a low-energy or ground state |ψ of a quantum system such that, after a simple measurement, the optimal solution is obtained with large probability. |ψ is prepared by following a particular annealing schedule, with a parametrized Hamiltonian path subject to initial and final conditions. A ground state of the initial Hamiltonian is then transformed to |ψ by varying the parameter adiabatically. In contrast to more general quantum adiabatic state transformations, the Hamiltonians along the path in quantum annealing are termed stoquastic and do not suffer from the so-called numerical sign problem [4]: for a specified basis, the offdiagonal Hamiltonian-matrix entries are nonpositive [5]. This property is useful for classical simulations [3].A sufficient condition for convergence of the quantum method is given by the quantum adiabatic approximation. It asserts that, if the rate of change of the Hamiltonian scales with the energy gap ∆ between their two lowest-energy states, |ψ can be prepared with controlled accuracy [6,7]. Such an approximation may also be necessary [8]. However, it could result in undesired overheads if ∆ is small but transitions between the lowestenergy states are forbidden due to selection rules, or if transitions between lowest-energy states can be exploited to prepare |ψ . The latter case corresponds to the annealing schedule in this Letter. It turns out that the relevant energy gap for the adiabatic approximation in these cases is not ∆ and can be much bigger.Because of the properties of the Hamiltonians, the annealing can also be simulated using probabilistic classical methods such as quantum Monte-Carlo (QMC) [9]. The goal in QMC is to sample according to the distribution of the ground state, i.e. with probabilities coming from amplitudes squared. While we lack of necessary conditions that guarantee convergence, the power of QMC is widely recognized [3,9,10]. In fact, if the Hamiltonians satisfy an additional frustration-free property, efficient QMC simulations for quantum annealing exist [11,12]. This places a doubt on whether a quantum-computer simulation of general quantum annealing processes can ever be done using substantially less resources than QMC or any other classical simulation.Towards answering this question, we...