Quantum computers can be used to address molecular structure, materials science and condensed matter physics problems, which currently stretch the limits of existing high-performance computing resources [1]. Finding exact numerical solutions to these interacting fermion problems has exponential cost, while Monte Carlo methods are plagued by the fermionic sign problem. These limitations of classical computational methods have made even few-atom molecular structures problems of practical interest for medium-sized quantum computers.Yet, thus far experimental implementations have been restricted to molecules involving only Period I elements [2][3][4][5][6][7][8]. Here, we demonstrate the experimental optimization of up to six-qubit Hamiltonian problems with over a hundred Pauli terms, determining the ground state energy for molecules of increasing size, up to BeH 2 . This is enabled by a hardware-efficient variational quantum eigensolver with trial states specifically tailored to the available interactions in our quantum processor, combined with a compact encoding of fermionic Hamiltonians [9] and a robust stochastic optimization routine [10]. We further demonstrate the flexibility of our approach by applying the technique to a problem of quantum magnetism [11]. Across all studied problems, we find agreement between experiment and numerical simulations with a noisy model of the device. These results help elucidate the requirements for scaling the method to larger systems, and aim at bridging the gap between problems at the forefront of high-performance computing and their implementation on quantum hardware.The fundamental goal of addressing molecular structure problems is to solve for the ground state energy of many-body interacting fermionic Hamiltonians. Solving this problem on a quantum computer relies on a mapping between fermionic and qubit operators [12]. This restates it as a specific instance of a local Hamiltonian problem on a set of qubits. Given a k-local Hamiltonian H, composed of terms that act on at most k qubits, the solution to the local Hamiltonian problem amounts to finding its * These authors contributed equally to this work.smallest eigenvalue E G ,To date, no efficient algorithm is known that can solve this problem in full generality. For k ≥ 2 the problem is known to be QMA-complete [13]. However, it is expected that physical systems have Hamiltonians that do not constitute hard instances of this problem, and can be solved efficiently on a quantum computer, while remaining hard to solve classically. Following Feynman's idea for quantum simulation, a quantum algorithm for the ground state problem of interacting fermions was proposed in [14] and [15]. The approach relies on a good initial state that has a large overlap with the ground state and then solves the problem using the quantum phase estimation algorithm (PEA) [16]. While PEA has been demonstrated to achieve extremely accurate energy estimates for quantum chemistry [2, 3, 5, 8], it applies stringent requirements on quantum coherence.An a...

Machine learning and quantum computing are two technologies each with the potential for altering how computation is performed to address previously untenable problems. Kernel methods for machine learning are ubiquitous for pattern recognition, with support vector machines (SVMs) being the most well-known method for classification problems. However, there are limitations to the successful solution to such problems when the feature space becomes large, and the kernel functions become computationally expensive to estimate. A core element to computational speed-ups afforded 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 novel methods on a superconducting processor. Both methods represent the feature space of a classification problem by a quantum state, taking advantage of the large dimensionality of quantum Hilbert space to obtain an enhanced solution. One method, the quantum variational classifier builds on [1, 2] and operates through using a variational quantum circuit to classify a training set in direct analogy to conventional SVMs. In the second, a quantum kernel estimator, we estimate the kernel function and optimize the classifier directly. The two methods present a new class of tools for exploring the applications of noisy intermediate scale quantum computers [3] to machine learning.The intersection between machine learning and quantum computing has been dubbed quantum machine learning, and has attracted considerable attention in recent years [4][5][6]. This has led to a number of recently proposed quantum algorithms [1,2,[7][8][9]. Here, we present a quantum algorithm that has the potential to run on near-term quantum devices. A natural class of algorithms for such noisy devices are short-depth circuits, which are amenable to error-mitigation techniques that reduce the effect of decoherence [10,11]. There are convincing arguments that indicate that even very sim- ple circuits are hard to simulate by classical computational means [12,13]. The algorithm we propose takes on the original problem of supervised learning: the construction of a classifier. For this problem, we are given data from a training set T and a test set S of a subset Ω ⊂ R d . Both are assumed to be labeled by a map m : T ∪ S → {+1, −1} unknown to the algorithm. The training algorithm only receives the labels of the training data T . The goal is to infer an approximate map on the test setm : S → {+1, −1} such that it agrees with high probability with the true map m( s) =m( s) on the test data s ∈ S. For such a learning task to be meaningful it is assumed that there is a correlation between the labels given for training and the true map. A classical approach to constructing an approximate labeling function uses socalled support vector machines (SVMs) [14]. The data gets mapped non-linearly to a high dimensional space, the feature space, where a hyperplane is constructed to separate the labeled samples. ...

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 don't 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 quasi-probability distribution.From the time quantum computation generated wide spread interest, the strongest objection to its viability was the sensitivity to errors and noise. In an early paper, William Unruh [1] found that the coupling to the environment sets an ultimate time and size limit for any quantum computation. This initially curbed the hopes that the full advantage of quantum computing could be harnessed, since it set limits on the scalability of any algorithm. This problem was, at least in theory, remedied with the advent of quantum error correction [2][3][4]. It was proven that if both the decoherence and the imprecision of gates could be reduced below a finite threshold value, then quantum computation could be performed indefinitely [5,6]. Although it is the ultimate goal to reach this threshold in an experiment that is scalable to larger sizes, the overhead that is needed to implement a fully fault-tolerant gate set with current codes [7] seems prohibitively large [8,9]. In turn, it is expected that in the near term the progress in quantum experiments will lead to devices with dynamics, which are beyond what can be simulated with a conventional computer. This leads to the question: what computational tasks could be accomplished with only limited, or no error correction?The suggestions of near-term applications in such quantum devices mostly center around quantum simulations with short-depth circuit [10][11][12] and approximate optimization algorithms [13]. Furthermore, certain problems in material simulation may be tackled by hybrid quantum-classical algorithms [14]. In most such applications, the task can be abstracted to applying a short-depth quantum circuits to some simple initial state and then estimating the expectation value of some observable after the circuit has been applied. This estimation must be accurate enough to achieve a simulation precision comparable or exceeding that of classical algorithms. Yet, although the quantum system evolves coherently for the most part of the short-depth circuit, the effects of decoherence already become apparent...

Quantum computation, a completely different paradigm of computing, benefits from theoretically proven speed-ups for certain problems and opens up the possibility of exactly studying the properties of quantum systems [1]. Yet, because of the inherent 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 a significant overhead of physical qubits, in order to realize fault-tolerance via quantum error correction [2, 3]. However, recent theoretical work [4, 5] has shown that the accuracy of computation based off expectation values of quantum observables can be enhanced through an extrapolation of results from a collection of varying noisy experiments. 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 on canonical single-and two-qubit experiments and then extend its application to the variational optimization [6][7][8] of Hamiltonians for quantum chemistry and magnetism [9]. We effectively demonstrate that the suppression of incoherent errors helps unearth otherwise inaccessible accuracies to the variational solutions using our noisy processor. These results demonstrate that error mitigation techniques will be critical to significantly enhance the capabilities of nearterm quantum computing hardware.Quantum computation can be extended indefinitely if decoherence and inaccuracies in the implementation of gates can be brought below an error-correction threshold [2, 3]. However, the resource requirements for a fullyfault tolerant architecture lie beyond the scope of nearterm quantum hardware [10]. In the absence of quantum error correction, the dominant sources of noise in current hardware are unitary gate errors and decoherence, both of which set a limit on the size of the computation that can be carried out. In this context, hybrid-quantum algorithms [7, 8, 11] with short-depth quantum circuits have been designed to perform computations within the available coherence window, while also demonstrating some robustness to coherent unitary errors [9, 12]. However, even when restricting to short depth circuits, the effect of decoherence already becomes evident for small experiments [9]. The recently proposed zero-noise extrapolation method [4, 5, 13] presents a route to mitigating incoherent errors and significantly improving the accuracy of the computation. It is important to note that, unlike quantum error-correction this technique does not allow for an indefinite extension of the computation time, and only provides corrections to expectation values, without correcting for the full statistical behavior. However, since it does not require any additional quantum resources, the technique is extremely well suited for practical implementations with near-term hardware.We shall first briefly describe the proposal of [4] and discuss important...

Universal fault-tolerant quantum computers will require error-free execution of long sequences of quantum gate operations, which is expected to involve millions of physical qubits. Before the full power of such machines will be available, near-term quantum devices will provide several hundred qubits and limited error correction. Still, there is a realistic prospect to run useful algorithms within the limited circuit depth of such devices. Particularly promising are optimization algorithms that follow a hybrid approach: the aim is to steer a highly entangled state on a quantum system to a target state that minimizes a cost function via variation of some gate parameters. This variational approach can be used both for classical optimization problems as well as for problems in quantum chemistry. The challenge is to converge to the target state given the limited coherence time and connectivity of the qubits. In this context, the quantum volume as a metric to compare the power of near-term quantum devices is discussed.With focus on chemistry applications, a general description of variational algorithms is provided and the mapping from fermions to qubits is explained. Coupledcluster and heuristic trial wave-functions are considered for efficiently finding molecular ground states. Furthermore, simple error-mitigation schemes are introduced that could improve the accuracy of determining ground-state energies. Advancing these techniques may lead to near-term demonstrations of useful quantum computation with systems containing several hundred qubits.PACS numbers: quantum computation, quantum chemistry, quantum algorithms