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. ...

Superconducting circuits are promising candidates for constructing quantum bits (qubits) in a quantum computer; single-qubit operations are now routine 1,2 , and several examples 3,4,5,6,7,8,9 of two qubit interactions and gates having been demonstrated. These experiments show that two nearby qubits can be readily coupled with local interactions. Performing gates between an arbitrary pair of distant qubits is highly desirable for any quantum computer architecture, but has not yet been demonstrated. An efficient way to achieve this goal is to couple the qubits to a quantum bus, which distributes quantum information among the qubits. Here we show the implementation of such a quantum bus, using microwave photons confined in a transmission line cavity, to couple two superconducting qubits on opposite sides of a chip. The interaction is mediated by the exchange of virtual rather than real photons, avoiding cavity induced loss. Using fast control of the qubits to switch the coupling effectively on and off, we demonstrate coherent transfer of quantum states between the qubits. The cavity is also used to perform multiplexed control and measurement of the qubit states. This approach can be expanded to more than two qubits, and is an attractive architecture for quantum information processing on a chip.There are several physical systems in which one could realize a quantum bus. A particular example is trapped ions 10,11 in which a variety of quantum operations and algorithms have been performed using the quantized motion of the ions (phonons) as the bus. Photons are another natural candidate as a carrier of quantum information 12,13 , because they are highly coherent and can mediate interactions between distant objects. To create a photon bus, it is helpful to utilize the increased interaction strength provided by the techniques of cavity quantum electrodynamics, where an atom is coupled to a single cavity mode. In the strong coupling limit 14 the interaction is coherent, permitting the transfer of quantum information between the atom and the photon. Entanglement between atoms has been demonstrated with Rydberg atom cavity QED 15,16,17 . Circuit QED 18 is a realization of the physics of cavity QED with superconducting qubits coupled to a microwave cavity on a chip. Previous circuit QED experiments with single qubits have achieved 19 the strong coupling limit and have demonstrated 20 the transfer of quantum information from qubit to photon. Here we perform a circuit QED experiment with two qubits strongly coupled to a cavity, and demonstrate a coherent, non-local coupling between the qubits via this bus.Operations with multiple superconducting qubits have been performed and are a subject of current research. The first solid-state quantum gate has been demonstrated with charge qubits 3 . For flux qubits, two-qubit coupling 5 and a controllable coupling mechanism have been realized 7,8,9 . Two phase qubits have also been successfully coupled 4 and the entanglement between them has been observed 6 . All of these interactions h...

Quantum computers, which harness the superposition and entanglement of physical states, could outperform their classical counterparts in solving problems with technological impact-such as factoring large numbers and searching databases. A quantum processor executes algorithms by applying a programmable sequence of gates to an initialized register of qubits, which coherently evolves into a final state containing the result of the computation. Building a quantum processor is challenging because of the need to meet simultaneously requirements that are in conflict: state preparation, long coherence times, universal gate operations and qubit readout. Processors based on a few qubits have been demonstrated using nuclear magnetic resonance, cold ion trap and optical systems, but a solid-state realization has remained an outstanding challenge. Here we demonstrate a two-qubit superconducting processor and the implementation of the Grover search and Deutsch-Jozsa quantum algorithms. We use a two-qubit interaction, tunable in strength by two orders of magnitude on nanosecond timescales, which is mediated by a cavity bus in a circuit quantum electrodynamics architecture. This interaction allows the generation of highly entangled states with concurrence up to 94 per cent. Although this processor constitutes an important step in quantum computing with integrated circuits, continuing efforts to increase qubit coherence times, gate performance and register size will be required to fulfil the promise of a scalable technology.

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 present an experimental realization of the transmon qubit, an improved superconducting charge qubit derived from the Cooper pair box. We experimentally verify the predicted exponential suppression of sensitivity to 1/f charge noise [J. Koch et al., Phys. Rev. A 76, 042319 (2007)]. This removes the leading source of dephasing in charge qubits, resulting in homogenously broadened transitions with relaxation and dephasing times in the microsecond range. Our systematic characterization of the qubit spectrum, anharmonicity, and charge dispersion shows excellent agreement with theory, rendering the transmon a promising qubit for future steps towards solid-state quantum information processing. PACS numbers: 03.67.Lx, 74.50.+r, Over the last decade, superconducting qubits have gained substantial interest as an attractive option for quantum information processing, cf. Refs. [1,2,3] for recent reviews. Although there already exist different realizations of superconducting qubits [4,5,6,7], all their coherence times are several orders of magnitude too short for large-scale quantum computation. Fortunately, an increase of coherence times from 2 ns in the first superconducting qubit [4] to microsecond times in present experiments [8,9,10,11] has already been shown, giving rise to hope that the remaining gap can be overcome by optimized quantum circuits and better materials. Coherence times can be either limited by dissipation (T 1 ) or dephasing (T * 2 ). Most superconducting qubits have dephasing times much shorter than the limit T * 2 = 2T 1 imposed by dissipation, because they are plagued by the influence of 1/f noise in charge, flux, or critical current. The transmon qubit is an improved design [12] derived from the original charge qubit [13] that renders it immune to its primary source of noise, 1/f charge noise, without making it more susceptible to either flux or critical current noise.The transmon consists of two superconducting islands connected by a Josephson tunnel junction. The tunneling of Cooper pairs between the two islands is governed by two energy scales: the charging energy E C and the Josephson energy E J . The transmon has a Hamiltonian identical to the Cooper pair box (CPB),wheren denotes the number of excess Cooper pairs on one of the islands and n g the offset charge due to the electrostatic environment. Because there are no dc connections to the qubit, n is integer-valued like an angular momentum, and the conjugate variableφ is a compact angle. Despite its basic CPB nature, the transmon is operated in a vastly different param-The primary benefit of this new regime is a suppression of the sensitivity to charge noise, which is exponential in the ratio E J /E C . The qubit spectrum becomes more uniformly spaced in the transmon, but it has been shown in [12] that the anharmonicity in the spectrum only decays as a weak alge-braic function of E J /E C , allowing it to be used as an effective two-level system. One of the reasons for the long coherence times of the design is that the state of the transmon q...

Traditionally, quantum entanglement has played a central role in foundational discussions of quantum mechanics. The measurement of correlations between entangled particles can exhibit results at odds with classical behavior. These discrepancies increase exponentially with the number of entangled particles 1 . When entanglement is extended from just two quantum bits (qubits) to three, the incompatibilities between classical and quantum correlation properties can change from a violation of inequalities 2 involving statistical averages to sign differences in deterministic observations 3 . With the ample confirmation of quantum mechanical predictions by experiments 4-7 , entanglement has evolved from a philosophical conundrum to a key resource for quantum-based technologies, like quantum cryptography and computation 8 . In particular, maximal entanglement of more than two qubits is crucial to the implementation of quantum error correction protocols. While entanglement of up to 3, 5, and 8 qubits has been demonstrated among spins 9 , photons 7 , and ions 10 , respectively, entanglement in engineered solid-state systems has been limited to two qubits [11][12][13][14][15] . Here, we demonstrate three-qubit entanglement in a superconducting circuit, creating Greenberger-HorneZeilinger (GHZ) states with fidelity of 88%, measured with quantum state tomography.Several entanglement witnesses show violation of biseparable bounds by 830 ± 80%. Our entangling sequence realizes the first step of basic quantum error correction, namely the encoding of a logical qubit into a manifold of GHZ-like states using a repetition code. The integration of encoding, decoding and error-correcting steps in a feedback loop will be the next milestone for quantum computing with integrated circuits.With steady improvements in qubit coherence, control, and readout over a decade, superconducting quantum circuits 16 have recently attained two milestones for solidstate two-qubit entanglement. The first is the violation of Bell inequalities without a detection loophole, realized with phase qubits by minimizing cross-talk between high-fidelity individual qubit readouts 14 . Second is the realization of simple quantum algorithms 13 , achieved through improved two-qubit gates and coherence in cir- (inset) to Q4] inside a meandering coplanar waveguide resonator. Local flux-bias lines allow qubit tuning on nanosecond timescales with room-temperature voltages Vi. Microwave pulses at qubit transition frequencies f1, f2, and f3 realize single-qubit x-and y-rotations in 8 ns. Q4 (operational but unused) is biased at its maximal frequency of 12.27 GHz to minimize its interaction with the qubits employed. Pulsed measurement of cavity homodyne voltage VH (at the bare cavity frequency fc = 9.070 GHz) allows joint qubit readout. A detailed schematic of the measurement setup is shown in Supplementary Fig. S2. b, Grey-scale images of cavity transmission and qubit spectroscopy versus local tuning of Q1 show avoided crossings with Q2 (66 MHz splitting), with Q3 (128 ...

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...