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.

Classical autoencoders are neural networks that can learn efficient low dimensional representations of data in higher dimensional space. The task of an autoencoder is, given an input x, is to map x to a lower dimensional point y such that x can likely be recovered from y. The structure of the underlying autoencoder network can be chosen to represent the data on a smaller dimension, effectively compressing the input. Inspired by this idea, we introduce the model of a quantum autoencoder to perform similar tasks on quantum data. The quantum autoencoder is trained to compress a particular dataset of quantum states, where a classical compression algorithm cannot be employed. The parameters of the quantum autoencoder are trained using classical optimization algorithms. We show an example of a simple programmable circuit that can be trained as an efficient autoencoder. We apply our model in the context of quantum simulation to compress ground states of the Hubbard model and molecular Hamiltonians. * Corresponding author: aspuru@chemistry.harvard.edu Figure 1. a) A graphical representation of a 6-bit autoencoder with a 3-bit latent space. The map E encodes a 6-bit input (red dots) into a 3-bit intermediate state (yellow dots), after which the decoder D attempts to reconstruct the input bits at the output (green dots). b) Circuit implementation of a 6-3-6 quantum autoencoder. without exponentially costly classical memory, for instance, in dimension reduction of quantum data. A related work proposing a quantum autoencoder model establishes a formal connection between classical and quantum feedforward neural networks where a particular setting of parameters in the quantum network reduces to a classical neural network exactly [7]. In this work, we provide a simpler model which we believe more easily captures the essence of an autoencoder, and apply it to ground states of the Hubbard model and molecular Hamiltonians.

Physically motivated quantum algorithms for specific near-term quantum hardware will likely be the next frontier in quantum information science. Here, we show how many of the features of neural networks for machine learning can naturally be mapped into the quantum optical domain by introducing the quantum optical neural network (QONN). Through numerical simulation and analysis we train the QONN to perform a range of quantum information processing tasks, including newly developed protocols for quantum optical state compression, reinforcement learning, and blackbox quantum simulation. We consistently demonstrate our system can generalize from only a small set of training data onto states for which it has not been trained. Our results indicate QONNs are a powerful design tool for quantum optical systems and, leveraging advances in integrated quantum photonics, a promising architecture for next generation quantum processors.

Quantum number-path entanglement is a resource for super-sensitive quantum metrology and in particular provides for sub-shotnoise or even Heisenberg-limited sensitivity. However, such numberpath entanglement has thought to have been resource intensive to create in the first place -typically requiring either very strong nonlinearities, or nondeterministic preparation schemes with feedforward, which are difficult to implement. Very recently, arising from the study of quantum random walks with multi-photon walkers, as well as the study of the computational complexity of passive linear optical interferometers fed with single-photon inputs, it has been shown that such passive linear optical devices generate a superexponentially large amount of number-path entanglement. A logical question to ask is whether this entanglement may be exploited for quantum metrology. We answer that question here in the affirmative by showing that a simple, passive, linear-optical interferometer -fed with only uncorrelated, single-photon inputs, coupled with simple, single-mode, disjoint photodetection -is capable of significantly beating the shotnoise limit. Our result implies a pathway forward to practical quantum metrology with readily available technology.Ever since the early work of Yurke & Yuen it has been understood that quantum number-path entanglement is a resource for super-sensitive quantum metrology, allowing for sensors that beat the shotnoise limit [1,2] [7], protein concentration measurements [8], and microscopy [9,10]. This line of work culminated in the analysis of the bosonic NOON state ((|N, 0 + |0, N )/ √ 2, where N is the total number of photons), which was shown to be optimal for local phase estimation with a fixed, finite number of photons, and in fact allows one to hit the Heisenberg limit and the Quantum Cramér-Rao Bound [11][12][13][14].Let us consider the NOON state as an example, where for this state in a two-mode interferometer we have the condition of all N particles in the first mode (and none in the second mode) superimposed with all N particles in the second mode (and none in the first mode). While such a state is known to be optimal for sensing, its generation is also known to be highly problematic and resource intensive. There are two routes to preparing high-NOON states: the first is to deploy very strong optical nonlinearities [15,16], and the second is to prepare them using measurement and feed-forward [17][18][19]. In many ways * motesk@gmail.com † dr.rohde@gmail.com; URL: http://www.peterrohde.org then NOON-state generators have had much in common with all-optical quantum computers and therefore are just as difficult to build [20]. In addition to the complicated state preparation, typically a complicated measurement scheme, such as parity measurement at each output port, also had to be deployed [21].Recently two independent lines of research, the study of quantum random walks with multi-photon walkers in passive linear-optical interferometers [22][23][24], as well as the quantum complexity analysis o...