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. ...

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...

We report a superconducting artificial atom with an observed quantum coherence time of T * 2 =95µs and energy relaxation time T1=70µs. The system consists of a single Josephson junction transmon qubit embedded in an otherwise empty copper waveguide cavity whose lowest eigenmode is dispersively coupled to the qubit transition. We attribute the factor of four increase in the coherence quality factor relative to previous reports to device modifications aimed at reducing qubit dephasing from residual cavity photons. This simple device holds great promise as a robust and easily produced artificial quantum system whose intrinsic coherence properties are sufficient to allow tests of quantum error correction. PACS numbers: 03.67.Ac, 42.50.Pq, 85.25.-j Superconducting quantum circuits are a leading candidate technology for large scale quantum computing. They have been used to show a violation of a Bell-type inequality [1]; implement a simple two-qubit gate favorable for scaling [2]; generate three-qubit entanglement [3]; perform a routine relevant to error correction [4];and very recently to demonstrate a universal set of quantum gates with fidelities greater than 95% [5]. Most of these devices employ small angle-evaporated Josephson junctions as their critical non-linear circuit components. Devices designs appear to be consistent with the basic requirements for quantum error correction (QEC) and fault tolerance [6]. However, the construction and operation of much larger systems capable of meaningful tests of such procedures will require individual qubits and junctions with a very high degree of coherence. Current estimates for threshold error rates -and the cumulative nature of errors originating from control, measurement, and decoherence -make likely the need for quantum lifetimes at least 10 3 times longer than gate and measurement times [7], corresponding to 20 to 200µs for typical systems.To this end, improvements in qubit lifetimes have continued for the past decade, spurred largely by clever methods of decoupling noise and loss mechanisms from the qubit transition and thus realizing Hamiltonians more closely resembling their idealized versions. Recently, Paik, et al. made a breakthrough advance [8] by embedding a transmon qubit [9, 10] in a superconducting waveguide cavity. Dubbed three-dimensional circuit QED (3D cQED), this system produced significantly enhanced qubit lifetimes of T 1 =25-60µs and T * 2 =10-20µs, corresponding to quality factors for dissipation and decoherence of Q 1 ≈1.8×10 6 and Q 2 ≈7×10 5 , respectively.These results lead to two important questions. First, are similar coherence properties observable using other fabrication processes, facilities, and measurement setups? Second, what is the origin of the dephasing process suppressing T * 2 well below the no-pure-dephasing limit of 2T 1 ? Is it intrinsic to the junctions or to this qubit ar-chitecture? The weight and urgency of these questions are increased by implications on scaling potential: if the results are reproducible and decoherence tim...

The ability to detect and deal with errors when manipulating quantum systems is a fundamental requirement for fault-tolerant quantum computing. Unlike classical bits that are subject to only digital bit-flip errors, quantum bits are susceptible to a much larger spectrum of errors, for which any complete quantum error-correcting code must account. Whilst classical bit-flip detection can be realized via a linear array of qubits, a general fault-tolerant quantum error-correcting code requires extending into a higher-dimensional lattice. Here we present a quantum error detection protocol on a two-by-two planar lattice of superconducting qubits. The protocol detects an arbitrary quantum error on an encoded two-qubit entangled state via quantum non-demolition parity measurements on another pair of error syndrome qubits. This result represents a building block towards larger lattices amenable to fault-tolerant quantum error correction architectures such as the surface code.

We demonstrate an all-microwave two-qubit gate on superconducting qubits which are fixed in frequency at optimal bias points. The gate requires no additional subcircuitry and is tunable via the amplitude of microwave irradiation on one qubit at the transition frequency of the other. We use the gate to generate entangled states with a maximal extracted concurrence of 0.88, and quantum process tomography reveals a gate fidelity of 81%.