The ground-state energy of small molecules is determined efficiently using six qubits of a superconducting quantum processor. Quantum simulation is currently the most promising application of quantum computers. However, only a few quantum simulations of very small systems have been performed experimentally. Here, researchers from IBM present quantum simulations of larger systems using a variational quantum eigenvalue solver (or eigensolver), a previously suggested method for quantum optimization. They perform quantum chemical calculations of LiH and BeH2 and an energy minimization procedure on a four-qubit Heisenberg model. Their application of the variational quantum eigensolver is hardware-efficient, which means that it is optimized on the given architecture. Noise is a big problem in this implementation, but quantum error correction could eventually help this experimental set-up to yield a quantum simulation of chemically interesting systems on a quantum computer. Quantum computers can be used to address electronic-structure problems and problems in materials science and condensed matter physics that can be formulated as interacting fermionic problems, problems which stretch the limits of existing high-performance computers1. Finding exact solutions to such problems numerically has a computational cost that scales exponentially with the size of the system, and Monte Carlo methods are unsuitable owing to the fermionic sign problem. These limitations of classical computational methods have made solving even few-atom electronic-structure problems interesting for implementation using medium-sized quantum computers. Yet experimental implementations have so far been restricted to molecules involving only hydrogen and helium2,3,4,5,6,7,8. Here we demonstrate the experimental optimization of Hamiltonian problems with up to six qubits and more than one hundred Pauli terms, determining the ground-state energy for molecules of increasing size, up to BeH2. We achieve this result by using a variational quantum eigenvalue solver (eigensolver) with efficiently prepared trial states that are tailored specifically to the interactions that are available in our quantum processor, combined with a compact encoding of fermionic Hamiltonians9 and a robust stochastic optimization routine10. We demonstrate the flexibility of our approach by applying it to a problem of quantum magnetism, an antiferromagnetic Heisenberg model in an external magnetic field. In all cases, we find agreement between our experiments and numerical simulations using a model of the device with noise. Our results help to elucidate the requirements for scaling the method to larger systems and for bridging the gap between key problems in high-performance computing and their implementation on quantum hardware.

Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets

Machine learning and quantum computing are two technologies that each have the potential to alter how computation is performed to address previously untenable problems. Kernel methods for machine learning are ubiquitous in pattern recognition, with support vector machines (SVMs) being the best known method for classification problems. However, there are limitations to the successful solution to such classification problems when the feature space becomes large, and the kernel functions become computationally expensive to estimate. A core element in the computational speed-ups enabled by quantum algorithms is the exploitation of an exponentially large quantum state space through controllable entanglement and interference. Here we propose and experimentally implement two quantum algorithms on a superconducting processor. A key component in both methods is the use of the quantum state space as feature space. The use of a quantum-enhanced feature space that is only efficiently accessible on a quantum computer provides a possible path to quantum advantage. The algorithms solve a problem of supervised learning: the construction of a classifier. One method, the quantum variational classifier, uses a variational quantum circuit1,2 to classify the data in a way similar to the method of conventional SVMs. The other method, a quantum kernel estimator, estimates the kernel function on the quantum computer and optimizes a classical SVM. The two methods provide tools for exploring the applications of noisy intermediate-scale quantum computers3 to machine learning.

Supervised learning with quantum-enhanced feature spaces.

A quantum simulator is a type of quantum computer that controls the interactions between quantum bits (or qubits) in a way that can be mapped to certain quantum many-body problems. As it becomes possible to exert more control over larger numbers of qubits, such simulators will be able to tackle a wider range of problems, such as materials design and molecular modelling, with the ultimate limit being a universal quantum computer that can solve general classes of hard problems. Here we use a quantum simulator composed of up to 53 qubits to study non-equilibrium dynamics in the transverse-field Ising model with long-range interactions. We observe a dynamical phase transition after a sudden change of the Hamiltonian, in a regime in which conventional statistical mechanics does not apply. The qubits are represented by the spins of trapped ions, which can be prepared in various initial pure states. We apply a global long-range Ising interaction with controllable strength and range, and measure each individual qubit with an efficiency of nearly 99 per cent. Such high efficiency means that arbitrary many-body correlations between qubits can be measured in a single shot, enabling the dynamical phase transition to be probed directly and revealing computationally intractable features that rely on the long-range interactions and high connectivity between qubits.

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Observation of a many-body dynamical phase transition with a 53-qubit quantum simulator

Many experimental proposals for noisy intermediate scale quantum devices involve training a parameterized quantum circuit with a classical optimization loop. Such hybrid quantum-classical algorithms are popular for applications in quantum simulation, optimization, and machine learning. Due to its simplicity and hardware efficiency, random circuits are often proposed as initial guesses for exploring the space of quantum states. We show that the exponential dimension of Hilbert space and the gradient estimation complexity make this choice unsuitable for hybrid quantum-classical algorithms run on more than a few qubits. Specifically, we show that for a wide class of reasonable parameterized quantum circuits, the probability that the gradient along any reasonable direction is non-zero to some fixed precision is exponentially small as a function of the number of qubits. We argue that this is related to the 2-design characteristic of random circuits, and that solutions to this problem must be studied. Gradient-based hybrid quantum-classical algorithms are often initialised with random, unstructured guesses. Here, the authors show that this approach will fail in the long run, due to the exponentially-small probability of finding a large enough gradient along any direction.

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Barren Plateaus in Quantum Neural Network Training Landscapes

The aim of this review is to provide quantum engineers with an introductory guide to the central concepts and challenges in the rapidly accelerating field of superconducting quantum circuits. Over the past twenty years, the field has matured from a predominantly basic research endeavor to a one that increasingly explores the engineering of larger-scale superconducting quantum systems. Here, we review several foundational elements—qubit design, noise properties, qubit control, and readout techniques—developed during this period, bridging fundamental concepts in circuit quantum electrodynamics and contemporary, state-of-the-art applications in gate-model quantum computation.

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A quantum engineer's guide to superconducting qubits

Two schemes are presented that mitigate the effect of errors and decoherence in short-depth quantum circuits. The size of the circuits for which these techniques can be applied is limited by the rate at which the errors in the computation are introduced. Near-term applications of early quantum devices, such as quantum simulations, rely on accurate estimates of expectation values to become relevant. Decoherence and gate errors lead to wrong estimates of the expectation values of observables used to evaluate the noisy circuit. The two schemes we discuss are deliberately simple and do not require additional qubit resources, so to be as practically relevant in current experiments as possible. The first method, extrapolation to the zero noise limit, subsequently cancels powers of the noise perturbations by an application of Richardson's deferred approach to the limit. The second method cancels errors by resampling randomized circuits according to a quasiprobability distribution.

Error Mitigation for Short-Depth Quantum Circuits

Quantum computation, a paradigm of computing that is completely different from classical methods, benefits from theoretically proved speed-ups for certain problems and can be used to study the properties of quantum systems1. Yet, because of the inherently fragile nature of the physical computing elements (qubits), achieving quantum advantages over classical computation requires extremely low error rates for qubit operations, as well as substantial physical qubits, to realize fault tolerance via quantum error correction2,3. However, recent theoretical work4,5 has shown that the accuracy of computation (based on expectation values of quantum observables) can be enhanced through an extrapolation of results from a collection of experiments of varying noise. Here we demonstrate this error mitigation protocol on a superconducting quantum processor, enhancing its computational capability, with no additional hardware modifications. We apply the protocol to mitigate errors in canonical single- and two-qubit experiments and then extend its application to the variational optimization6–8 of Hamiltonians for quantum chemistry and magnetism9. We effectively demonstrate that the suppression of incoherent errors helps to achieve an otherwise inaccessible level of accuracy in the variational solutions using our noisy processor. These results demonstrate that error mitigation techniques will enable substantial improvements in the capabilities of near-term quantum computing hardware. The accuracy of computations on noisy, near-term quantum systems can be enhanced by extrapolating results from experiments with various noise levels, without requiring additional hardware modifications.

Error mitigation extends the computational reach of a noisy quantum processor

We introduce quantum versions of the $\chi^2$-divergence, provide a detailed analysis of their properties, and apply them in the investigation of mixing times of quantum Markov processes. An approach similar to the one presented in [1-3] for classical Markov chains is taken to bound the trace-distance from the steady state of a quantum processes. A strict spectral bound to the convergence rate can be given for time-discrete as well as for time-continuous quantum Markov processes. Furthermore the contractive behavior of the $\chi^2$-divergence under the action of a completely positive map is investigated and contrasted to the contraction of the trace norm. In this context we analyse different versions of quantum detailed balance and, finally, give a geometric conductance bound to the convergence rate for unital quantum Markov processes.

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Kristan Temme

Papers

Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets

Supervised learning with quantum-enhanced feature spaces.

Error Mitigation for Short-Depth Quantum Circuits

Error mitigation extends the computational reach of a noisy quantum processor

The $\chi^2$ - divergence and Mixing times of quantum Markov processes