Over the past few years, machine learning has emerged as a powerful computational tool to tackle complex problems in a broad range of scientific disciplines. In particular, artificial neural networks have been successfully used to mitigate the exponential complexity often encountered in quantum many-body physics, the study of properties of quantum systems built from a large number of interacting particles. In this article, we review some applications of neural networks in condensed matter physics and quantum information, with particular emphasis on hands-on tutorials serving as a quick start for a newcomer to the field. The prerequisites of this tutorial are basic probability theory and calculus, linear algebra, basic notions of neural networks, statistical physics, and quantum mechanics. The reader is introduced to supervised machine learning with convolutional neural networks to learn a phase transition, unsupervised learning with restricted Boltzmann machines to perform quantum tomography, and the variational Monte Carlo method with recurrent neural networks for approximating the ground state of a many-body Hamiltonian. For each algorithm, we briefly review the key ingredients and their corresponding neural-network implementation, and show numerical experiments for a system of interacting Rydberg atoms in two dimensions.

Atomic interference experiments can probe the gravitational redshift via the internal energy splitting of atoms and thus give direct access to test the universality of the coupling between matter-energy and gravity at different spacetime points. By including possible violations of the equivalence principle in a fully quantized treatment of all atomic degrees of freedom, we characterize how the sensitivity to gravitational redshift violations arises in atomic clocks and atom interferometers, as well as their underlying limitations. Specifically, we show that: (i) Contributions beyond linear order to trapping potentials lead to such a sensitivity of trapped atomic clocks. (ii) Bragg-type interferometers, even with a superposition of internal states, with state-independent, linear interaction potentials are at first insensitive to gravitational redshift tests. However, modified configurations, for example by relaunching the atoms, can mimic such tests under certain conditions and may constitute a competitive alternative. (iii) Guided atom interferometers are comparable to atomic clocks. (iv) Internal transitions lead to state-dependent interaction potentials through which light-pulse atom interferometers can become sensitive to gravitational redshift violations.

Quantum simulations of chemistry in first quantization offer some important advantages over approaches in second quantization including faster convergence to the continuum limit and the opportunity for practical simulations outside of the Born-Oppenheimer approximation. However, since all prior work on quantum simulation of chemistry in first quantization has been limited to asymptotic analysis, it has been impossible to directly compare the resources required for these approaches to those required for the more commonly studied algorithms in second quantization. Here, we compile, optimize, and analyze the finite resources required to implement two first quantized quantum algorithms for chemistry from Babbush et al. [Npj Quantum Inf. 5, 92 (2019)] that realize block encodings for the qubitization and interaction-picture frameworks of Low et al. [Quantum 3, 163 (2019), arXiv:1805.00675 (2018)]. The two algorithms we study enable simulation with gate complexities of $\tilde{\mathcal{O}}({\eta }^{8/3}{N}^{1/3}t+{\eta }^{4/3}{N}^{2/3}t)$ and $\tilde{\mathcal{O}}({\eta }^{8/3}{N}^{1/3}t)$ where $\eta$ is the number of electrons, $N$ is the number of plane-wave basis functions, and $t$ is the duration of time evolution ($t$ is linearly inverse to target precision when the goal is to estimate energies). In addition to providing the first explicit circuits and constant factors for any first quantized simulation, and then introducing improvements, which reduce circuit complexity by about a thousandfold over naive implementations for modest sized systems, we also describe new algorithms that asymptotically achieve the same scaling in a real-space representation. Finally, we assess the resources required to simulate various molecules and materials and conclude that the qubitized algorithm will often be more practical than the interaction-picture algorithm. We demonstrate that our qubitization algorithm often requires much less surface-code space-time volume for simulating millions of plane waves than the best second quantized algorithms require for simulating hundreds of Gaussian orbitals.

Tensor-network methods are routinely used in approximating various equilibrium and nonequilibrium scenarios, with the algorithms requiring a small bond dimension at low enough time or inverse temperature. These approaches so far lacked a rigorous mathematical justification, since existing approximations to thermal states and time evolution demand a bond dimension growing with system size. To address this problem, we construct projected entangled-pair operators that approximate, for all local observables, (i) their thermal expectation values and (ii) their Heisenberg time evolution. The bond dimension required does not depend on system size, but only on the temperature or time. We also show how these can be used to approximate thermal correlation functions and expectation values in quantum quenches.

If noisy-intermediate-scale-quantum-era quantum computers are to perform useful tasks, they will need to employ powerful error mitigation techniques. Quasiprobability methods can permit perfect error compensation at the cost of additional circuit executions, provided that the nature of the error model is fully understood and sufficiently local both spatially and temporally. Unfortunately, these conditions are challenging to satisfy. Here we present a method by which the proper compensation strategy can instead be learned ab initio. Our training process uses multiple variants of the primary circuit where all non-Clifford gates are substituted with gates that are efficient to simulate classically. The process yields a configuration that is near optimal versus noise in the real system with its non-Clifford gate set. Having presented a range of learning strategies, we demonstrate the power of the technique both with real quantum hardware (IBM devices) and exactly emulated imperfect quantum computers. The systems suffer a range of noise severities and types, including spatially and temporally correlated variants. In all cases the protocol successfully adapts to the noise and mitigates it to a high degree.

Several quantum many-body models in one dimension possess exact solutions via the Bethe ansatz method, which has been highly successful for understanding their behavior. Nevertheless, there remain physical properties of such models for which analytic results are unavailable and which are also not well described by approximate numerical methods. Preparing Bethe ansatz eigenstates directly on a quantum computer would allow straightforward extraction of these quantities via measurement. We present a quantum algorithm for preparing Bethe ansatz eigenstates of the spin-1/2 $XXZ$ spin chain that correspond to real-valued solutions of the Bethe equations. The algorithm is polynomial in the number of $T$ gates and the circuit depth, with modest constant prefactors. Although the algorithm is probabilistic, with a success rate that decreases with increasing eigenstate energy, we employ amplitude amplification to boost the success probability. The resource requirements for our approach are lower than for other state-of-the-art quantum simulation algorithms for small error-corrected devices and thus may offer an alternative and computationally less demanding demonstration of quantum advantage for physically relevant problems.

Unlike classical systems, a measurement performed on a quantum system always alters its state. In this work, the impacts of two diagnostic schemes to determine the performance of quantum Otto heat engines are compared: In one scheme, the energy of the engine’s working substance is measured after each stroke (repeated measurements), and in the other one, the energies after each stroke are recorded in one or two pointer states and measured only after the completion of a prescribed number of cycles (repeated contacts). A single pointer state suffices if one is only interested in either work or heat. For joint work and heat diagnostics, two pointers are needed. These schemes are applied to Otto engines, whose working substance consists of a two-level system. Depending on the engine protocol, the duration of a single cycle may be infinite or finite. Because in the repeated contact scheme, the number of measurements is drastically reduced compared to the repeated measurement scheme, the quantum coherence after and during the contact diagnostics is much better maintained than for repeated measurements that destroy any coherence at the end of each stroke. We demonstrate that maximum power, reliability, and efficiency of the engine in the presence of repeated contacts typically outperform these figures of merit of repeated measurements. Due to the improved coherence persistence, heat engines with a finite cycle duration require a larger number of cycles to reach a periodically asymptotic state. Overall, our results document the importance of taking into account the particular nature of diagnostic tools for monitoring and testing purposes but also for feedback control, both in theory and experiment.

Quantum compiling, where a parameterized quantum circuit is trained to learn a target unitary, is an important primitive for quantum computing that can be used as a subroutine to obtain optimal circuits or as a tomographic tool to study the dynamics of an experimental system. While much attention has been paid to quantum compiling on discrete-variable hardware, less has been paid to compiling in the continuous-variable paradigm. Here we motivate several, closely related, short-depth continuous-variable algorithms for quantum compilation. We analyze the trainability of our proposed cost functions and numerically demonstrate our algorithms by learning arbitrary Gaussian operations and Kerr nonlinearities. We further make connections between this framework and quantum learning theory in the continuous-variable setting by deriving no-free-lunch theorems. These generalization bounds demonstrate a linear resource reduction for learning Gaussian unitaries using entangled coherent-Fock states and an exponential resource reduction for learning arbitrary unitaries using two-mode-squeezed states.

We present a method for mitigating measurement errors on quantum computing platforms that does not form the full assignment matrix, or its inverse, and works in a subspace defined by the noisy input bit strings. This method accommodates both uncorrelated and correlated errors and allows for the computation of accurate error bounds. Additionally, we detail a matrix-free preconditioned iterative-solution method that converges in $\mathcal{O}(1)$ steps that is performant and uses orders of magnitude less memory than direct factorization. We demonstrate the validity of our method and mitigate errors in a few seconds on numbers of qubits that would otherwise be impractical.

Quantum simulators have made a remarkable progress towards exploring the dynamics of many-body systems, many of which offer a formidable challenge to both theoretical and numerical methods. While state-of-the-art quantum simulators are, in principle, able to simulate quantum dynamics well outside the domain of classical computers, they are noisy and limited in the variability of the initial state of the dynamics and the observables that can be measured. Despite these limitations, here we show that such a quantum simulator can be used to in effect solve for the dynamics of a many-body system. We develop an efficient numerical technique that facilitates classical simulations in regimes not accessible to exact calculations or other established numerical techniques. The method is based on approximations that are well suited to describe localized one-dimensional Fermi-Hubbard systems. Since this new method does not have an error estimate and the approximations do not hold in general, we use a neutral-atom Fermi-Hubbard quantum simulator with $L_{\exp }\simeq 290$ lattice sites to benchmark its performance in terms of accuracy and convergence for evolution times up to $700$ tunneling times. We then use these approximations in order to derive a simple prediction of the behavior of interacting Bloch oscillations for spin-imbalanced Fermi-Hubbard systems, which we show to be in quantitative agreement with experimental results. Finally, we demonstrate that the convergence of our method is the slowest when the entanglement depth developed in the many-body system we consider is neither too small nor too large. This represents a promising regime for near-term applications of quantum simulators.

Quantum computers promise tremendous impact across applications—and have shown great strides in hardware engineering—but remain notoriously error prone. Careful design of low-level controls has been shown to compensate for the processes that induce hardware errors, leveraging techniques from optimal and robust control. However, these techniques rely heavily on the availability of highly accurate and detailed physical models, which generally achieve only sufficient representative fidelity for the most simple operations and generic noise modes. In this work, we use deep reinforcement learning to design a universal set of error-robust quantum logic gates in runtime on a superconducting quantum computer, without requiring knowledge of a specific Hamiltonian model of the system, its controls, or its underlying error processes. We experimentally demonstrate that a fully autonomous deep-reinforcement-learning agent can design single qubit gates up to $3\times$ faster than default DRAG operations without additional leakage error, and exhibiting robustness against calibration drifts over weeks. We then show that $ZX(-\pi /2)$ operations implemented using the cross-resonance interaction can outperform hardware default gates by over $2\times$ and equivalently exhibit superior calibration-free performance up to 25 days post optimization. We benchmark the performance of deep-reinforcement-learning-derived gates against other black-box optimization techniques, showing that deep reinforcement learning can achieve comparable or marginally superior performance, even with limited hardware access.

Bell’s theorem is typically understood as the proof that quantum theory is incompatible with local-hidden-variable models. More generally, we can see the violation of a Bell inequality as witnessing the impossibility of explaining quantum correlations with classical causal models. The violation of a Bell inequality, however, does not exclude classical models where some level of measurement dependence is allowed, that is, the choice made by observers can be correlated with the source generating the systems to be measured. Here, we show that the level of measurement dependence can be quantitatively upper bounded if we arrange the Bell test within a network. Furthermore, we also prove that these results can be adapted in order to derive nonlinear Bell inequalities for a large class of causal networks and to identify quantumly realizable correlations that violate them.

In order to treat all-to-all-connected quadratic binary optimization problems (QUBOs) with hardware quantum annealers, an embedding of the original problem is required due to the sparsity of the topology of the hardware. The embedding of fully connected graphs—typically found in industrial applications—incurs a quadratic space overhead and thus a significant overhead in the time to solution. Here, we investigate this embedding penalty of established planar embedding schemes such as square-lattice embedding, embedding on a chimera lattice, and the Lechner-Hauke-Zoller scheme, using simulated quantum annealing on classical hardware. Large-scale quantum Monte Carlo simulation suggests a polynomial time-to-solution overhead. Our results demonstrate that standard analog quantum annealing hardware is at a disadvantage in comparison to classical digital annealers, as well as gate-model quantum annealers, and could also serve as a benchmark for improvements of the standard quantum annealing protocol.

Quantum classification and hypothesis testing (state and channel discrimination) are two tightly related subjects, the main difference being that the former is data driven: how to assign to quantum states $\rho (x)$ the corresponding class $c$ (or hypothesis) is learnt from examples during training, where $x$ can be either tunable experimental parameters or classical data “embedded” into quantum states. Does the model generalize? This is the main question in any data-driven strategy, namely the ability to predict the correct class even of previously unseen states. Here we establish a link between quantum classification and quantum information theory, by showing that the accuracy and generalization capability of quantum classifiers depend on the (Rényi) mutual information $I(C:Q)$ and $I_2(X:Q)$ between the quantum state space $Q$ and the classical parameter space $X$ or class space $C$. Based on the above characterization, we then show how different properties of $Q$ affect classification accuracy and generalization, such as the dimension of the Hilbert space, the amount of noise, and the amount of neglected information from $X$ via, e.g., pooling layers. Moreover, we introduce a quantum version of the information bottleneck principle that allows us to explore the various trade-offs between accuracy and generalization. Finally, in order to check our theoretical predictions, we study the classification of the quantum phases of an Ising spin chain, and we propose the variational quantum information bottleneck method to optimize quantum embeddings of classical data to favor generalization.

An arbitrary operator corresponding to a physical observable cannot be measured in a single measurement on currently available quantum hardware. To obtain the expectation value of the observable, one needs to partition its operator into measurable fragments. However, the observable and its fragments generally do not share any eigenstates and thus the number of measurements needed to obtain the expectation value of the observable can grow rapidly even when the wave function prepared is close to an eigenstate of the observable. We provide a unified Lie-algebraic framework for developing efficient measurement schemes for quantum observables. It is based on two elements: (1) embedding the observable operator in a Lie algebra and (2) transforming Lie-algebra elements into those of a Cartan subalgebra (CSA) using unitary operators. The CSA plays the central role because all its elements are mutually commutative and thus can be measured simultaneously. We illustrate the framework on measuring expectation values of Hamiltonians appearing in the variational quantum eigensolver approach to quantum chemistry. The CSA approach puts many recently proposed methods for the measurement optimization within a single framework and allows one not only to reduce the number of measurable fragments but also the total number of measurements.

Time evolution of quantum many-body systems typically leads to a state with maximal entanglement allowed by symmetries. Two distinct routes to impede entanglement growth are inducing localization via spatial disorder, or subjecting the system to nonunitary evolution, e.g., via projective measurements. Here we employ the idea of spacetime rotation of a circuit to explore the relation between systems that fall into these two classes. In particular, by spacetime rotating unitary Floquet circuits that display a localization transition, we construct nonunitary circuits that display a rich variety of entanglement scaling and phase transitions. One outcome of our approach is a nonunitary circuit for free fermions in one dimension that exhibits an entanglement transition from logarithmic scaling to volume-law scaling. This transition is accompanied by a “purification transition” analogous to that seen in hybrid projective-unitary circuits. We follow a similar strategy to construct a nonunitary two-dimensional (2D) Clifford circuit that shows a transition from area to volume-law entanglement scaling. Similarly, we spacetime rotate a 1D spin chain that hosts many-body localization to obtain a nonunitary circuit that exhibits an entanglement transition. Finally, we introduce an unconventional correlator and argue that if a unitary circuit hosts a many-body localization transition then the correlator is expected to be singular in its nonunitary counterpart as well.

The introduction of an optical resonator can enable efficient and precise interaction between a photon and a solid-state emitter. It facilitates the study of strong light-matter interaction, polaritonic physics and presents a powerful interface for quantum communication and computing. A pivotal aspect in the progress of light-matter interaction with solid-state systems is the challenge of combining the requirements of cryogenic temperature and high mechanical stability against vibrations while maintaining sufficient degrees of freedom for in situ tunability. Here, we present a fiber-based open Fabry-Pérot cavity in a closed-cycle cryostat exhibiting ultrahigh mechanical stability while providing wide-range tunability in all three spatial directions. We characterize the setup and demonstrate the operation with the root-mean-square cavity-length fluctuation of less than 90 pm at temperature of 6.5 K and integration bandwidth of 100 kHz. Finally, we benchmark the cavity performance by demonstrating the strong-coupling formation of exciton polaritons in monolayer ${\mathrm{W}\mathrm{Se}}_2$ with a cooperativity of 1.6. This set of results manifests the open cavity in a closed-cycle cryostat as a versatile and powerful platform for low-temperature cavity QED experiments.

Achieving high-precision detection of time-dependent signals within noise is a ubiquitous issue in physics and a critical task in metrology. Lock-in amplifiers are detectors that can extract alternating signals within extreme noise via a known carrier frequency. Here we present a protocol for achieving an entanglement-enhanced lock-in amplifier via use of many-body multipulse quantum interferometry. The many-body quantum lock-in amplifier is implemented by application of a periodic multi-$\pi$-pulse sequence during the interrogation. Our analytic results show that, by our choosing suitable input states and readout operations, the frequency and amplitude of an unknown alternating field can be simultaneously extracted via population measurements. The lock-in point can be determined via the symmetry of the signal during a single interrogation time or the time-averaged signals for multiple interrogation times. We find that the measurement signal at the lock-in point is independent of the interrogation time. In particular, if we input spin cat states and apply interaction-based readout operations, the measurement precisions for frequency and amplitude can both approach the Heisenberg limit. Moreover, our many-body quantum amplifier is also robust with regard to extreme stochastic noise. Our study paves a new way for measuring time-dependent signals with many-body quantum systems, and provides a feasible method for achieving Heisenberg-limited detection of alternating signals.

We argue that an excess in entanglement between the visible and hidden units in a quantum neural network can hinder learning. In particular, we show that quantum neural networks that satisfy a volume law in the entanglement entropy will give rise to models that are not suitable for learning with high probability. Using arguments from quantum thermodynamics, we then show that this volume law is typical and that there exists a barren plateau in the optimization landscape due to entanglement. More precisely, we show that for any bounded objective function on the visible layers, the Lipshitz constants of the expectation value of that objective function will scale inversely with the dimension of the hidden subsystem with high probability. We show how this can cause both gradient-descent and gradient-free methods to fail. We note that similar problems can happen with quantum Boltzmann machines, although stronger assumptions on the coupling between the hidden and/or visible subspaces are necessary. We highlight how pretraining such generative models may provide a way to navigate these barren plateaus.

Bosonic qubits are a promising route to building fault-tolerant quantum computers on a variety of physical platforms. Studying the performance of bosonic qubits under realistic gates and measurements is challenging with existing analytical and numerical tools. We present a novel formalism for simulating classes of states that can be represented as linear combinations of Gaussian functions in phase space. This formalism allows us to analyze and simulate a wide class of non-Gaussian states, transformations, and measurements. We demonstrate how useful classes of bosonic qubits—Gottesman-Kitaev-Preskill (GKP), cat, and Fock states—can be simulated using this formalism, opening the door to investigating the behavior of bosonic qubits under Gaussian channels and measurements, non-Gaussian transformations such as those achieved via gate teleportation, and important non-Gaussian measurements such as threshold and photon-number detection. Our formalism enables simulating these situations with levels of accuracy that are not feasible with existing methods. Finally, we use a method informed by our formalism to simulate circuits critical to the study of fault-tolerant quantum computing with bosonic qubits but beyond the reach of existing techniques. Specifically, we examine how finite-energy GKP states transform under realistic qubit phase gates; interface with a continuous-variable cluster state; and transform under non-Clifford t gate teleportation using magic states. We implement our simulation method as a part of the open-source Strawberry Fields python library.