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The resonator-induced phase (RIP) gate is a multi-qubit entangling gate that allows a high degree of flexibility in qubit frequencies, making it attractive for quantum operations in large-scale architectures. We experimentally realize the RIP gate with four superconducting qubits in a three-dimensional (3D) circuit-quantum electrodynamics architecture, demonstrating high-fidelity controlled-Z (CZ) gates between all possible pairs of qubits from two different 4-qubit devices in pair subspaces. These qubits are arranged within a wide range of frequency detunings, up to as large as 1.8 GHz. We further show a dynamical multi-qubit refocusing scheme in order to isolate out 2-qubit interactions, and combine them to generate a four-qubit Greenberger-Horne-Zeilinger state., Comment: 5 pages, 4 figures in main text with 9 pages 4 figures in the supplemental material

In quantum information science, the phase of a wavefunction plays an important role in encoding information. While most experiments in this field rely on dynamic effects to manipulate this information, an alternative approach is to use geometric phase, which has been argued to have potential fault tolerance. We demonstrate the controlled accumulation of a geometric phase, Berry's phase, in a superconducting qubit, manipulating the qubit geometrically using microwave radiation, and observing the accumulated phase in an interference experiment. We find excellent agreement with Berry's predictions, and also observe a geometry dependent contribution to dephasing., 5 pages, 4 figures, version with high resolution figures available at http://qudev.ethz.ch/content/science/PubsPapers.html

Electromagnetic signals in circuits consist of discrete photons, though conventional voltage sources can only generate classical fields with a coherent superposition of many different photon numbers. While these classical signals can control and measure bits in a quantum computer (qubits), single photons can carry quantum information, enabling non-local quantum interactions, an important resource for scalable quantum computing. Here, we demonstrate an on-chip single photon source in a circuit quantum electrodynamics (QED) architecture, with a microwave transmission line cavity that collects the spontaneous emission of a single superconducting qubit with high efficiency. The photon source is triggered by a qubit rotation, as a photon is generated only when the qubit is excited. Tomography of both qubit and fluorescence photon shows that arbitrary qubit states can be mapped onto the photon state, demonstrating an ability to convert a stationary qubit into a flying qubit. Both the average power and voltage of the photon source are characterized to verify performance of the system. This single photon source is an important addition to a rapidly growing toolbox for quantum optics on a chip., 6 pages, 5 figures, hires version at http://www.eng.yale.edu/rslab/papers/single_photon_hires.pdf

Randomized benchmarking is a widely used experimental technique to characterize the average error of quantum operations. Benchmarking procedures that scale to enable characterization of $n$-qubit circuits rely on efficient procedures for manipulating those circuits and, as such, have been limited to subgroups of the Clifford group. However, universal quantum computers require additional, non-Clifford gates to approximate arbitrary unitary transformations. We define a scalable randomized benchmarking procedure over $n$-qubit unitary matrices that correspond to protected non-Clifford gates for a class of stabilizer codes. We present efficient methods for representing and composing group elements, sampling them uniformly, and synthesizing corresponding $\mathrm{poly}(n)$-sized circuits. The procedure provides experimental access to two independent parameters that together characterize the average gate fidelity of a group element., 5+4 pages, 1 figure

Continuous measurements play a pivotal role in the study of dynamical open quantum systems. `Dyne' detections are among the most widespread and efficient measurement schemes, and give rise to quantum diffusion of the conditioned state. In this work we study under what conditions the detector dependency of the conditional state of a quantum system subject to diffusive monitoring can be demonstrated experimentally, in the sense of ruling our any detector-independent pure-state dynamical model for the system. We consider an arbitrary number L of environments to which the system is coupled, and an arbitrary number K of different types of dyne detections. We prove that non-trivial necessary conditions for such a demonstration can be determined efficiently by semi-definite programming. To determine sufficient conditions, different physical environmental couplings and Hamiltonians for a qubit, and different sets of diffusive monitorings are scrutinized. We compare the threshold efficiencies that are sufficient in the various cases, as well as cases previously considered in the literature, to suggest the most feasible experimental options., 11 pages, 5 figures, 1 table

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., The physical realization of a quantum computer requires built-in error-correcting codes that compensate the disruption of quantum information arising from noise. Here, the authors demonstrate a quantum error detection scheme for arbitrary single-qubit errors on a four superconducting qubit lattice.

With improved gate calibrations reducing unitary errors, we achieve a benchmarked single-qubit gate fidelity of 99.95% with superconducting qubits in a circuit quantum electrodynamics system. We present a method for distinguishing between unitary and non-unitary errors in quantum gates by interleaving repetitions of a target gate within a randomized benchmarking sequence. The benchmarking fidelity decays quadratically with the number of interleaved gates for unitary errors but linearly for non-unitary, allowing us to separate systematic coherent errors from decoherent effects. With this protocol we show that the fidelity of the gates is not limited by unitary errors, but by another drive-activated source of decoherence such as amplitude fluctuations., 5 pages, 4 figures

Physical implementations of qubits can be extremely sensitive to environmental coupling, which can result in decoherence. While efforts are made for protection, coupling to the environment is necessary to measure and manipulate the state of the qubit. As such, the goal of having long qubit energy relaxation times is in competition with that of achieving high-fidelity qubit control and measurement. Here we propose a method that integrates filtering techniques for preserving superconducting qubit lifetimes together with the dispersive coupling of the qubit to a microwave resonator for control and measurement. The result is a compact circuit that protects qubits from spontaneous loss to the environment, while also retaining the ability to perform fast, high-fidelity readout. Importantly, we show the device operates in a regime that is attainable with current experimental parameters and provide a specific example for superconducting qubits in circuit quantum electrodynamics., 9 pages, 6 figures, 1 table

The resonator-induced phase gate is a multi-qubit controlled-phase gate for fixed-frequency superconducting qubits. Through off-resonant driving of a bus resonator, statically coupled qubits acquire a state-dependent phase. However, photon loss leads to dephasing during the gate, and any residual entanglement between the resonator and qubits after the gate leads to decoherence. Here we consider how to shape the drive pulse to minimize these unwanted effects. First, we review how the gate's entangling and dephasing rates depend on the system parameters and validate closed-form solutions against direct numerical solution of a master equation. Next, we propose spline pulse shapes that reduce residual qubit-bus entanglement, are robust to imprecise knowledge of the resonator shift, and can be shortened by using higher-degree polynomials. Finally, we present a procedure that optimizes over the subspace of pulses that leave the resonator unpopulated. This finds shaped drive pulses that further reduce the gate duration. Assuming realistic parameters, we exhibit shaped pulses that have the potential to realize ~212 ns spline pulse gates and ~120 ns optimized gates with ~6e-4 average gate infidelity. These examples do not represent fundamental limits of the gate and in principle even shorter gates may be achievable., 13 pages, 10 figures

The ability to perform fast, high-fidelity readout of quantum bits (qubits) is essential to the goal of building a quantum computer. However, coupling a fast measurement channel to a superconducting qubit typically also speeds up its relaxation via spontaneous emission. Here we use impedance engineering to design a filter by which photons may easily leave the resonator at the cavity frequency but not at the qubit frequency. We implement this broadband filter in both an on-chip and off-chip configuration., Comment: 5 pages, 3 figures

Decoherence of superconducting transmon qubits is purported to be consistent with surface loss from two-level systems on the substrate surface. Here, we present a study of surface loss in transmon devices, explicitly designed to have varying sensitivities to different surface loss contributors. Our experiments also encompass two particular different sapphire substrates, which reveal the onset of a yet unknown additional loss mechanism outside of surface loss for one of the substrates. Tests across different wafers and devices demonstrate substantial variation, and we emphasize the importance of testing large numbers of devices for disentangling different sources of decoherence., Comment: 4 pages, 3 figures

Using a circuit QED device, we demonstrate a simple qubit measurement pulse shape that yields fast ring-up and ring-down of the readout resonator regardless of the qubit state. The pulse differs from a square pulse only by the inclusion of additional constant-amplitude segments designed to effect a rapid transition from one steady-state population to another. Using a Ramsey experiment performed shortly after the measurement pulse to quantify the residual population, we find that compared to a square pulse followed by a delay, this pulse shape reduces the timescale for cavity ring-down by more than twice the cavity time constant. At low drive powers, this performance is achieved using pulse parameters calculated from a linear cavity model; at higher powers, empirical optimization of the pulse parameters leads to similar performance.

We present a method for detecting electromagnetic (EM) modes that couple to a superconducting qubit in a circuit-QED architecture. Based on measurement-induced dephasing, this technique allows the measurement of modes that have a high quality factor (Q) and may be difficult to detect through standard transmission and reflection measurements at the device ports. In this scheme the qubit itself acts as a sensitive phase meter, revealing modes that couple to it through measurements of its coherence time. Such modes are indistinguishable from EM modes that do not couple to the qubit using a vector network analyzer. Moreover, this technique provides useful characterization parameters including the quality factor and the coupling strength of the unwanted resonances. We demonstrate the method for detecting both high-Q coupling resonators in planar devices as well as spurious modes produced by a 3D cavity., Comment: 4 pages, 2 figures; updated to published version

Quantum codes excel at correcting local noise but fail to correct leakage faults that excite qubits to states outside the computational space. Aliferis and Terhal have shown that an accuracy threshold exists for leakage faults using gadgets called leakage reduction units (LRUs). However, these gadgets reduce the accuracy threshold and can increase overhead and experimental complexity, and these costs have not been thoroughly understood. Our work explores a variety of techniques for leakage-resilient, fault-tolerant error correction in the context of topological codes. Our contributions are threefold. First, we develop a leakage model that differs in critical details from earlier models. Second, we use Monte-Carlo simulations to survey several syndrome extraction circuits. Third, given the capability to perform three-outcome measurements, we present a dramatically improved syndrome processing algorithm. Our simulation results show that simple circuits with one extra CNOT per qubit and no additional ancillas reduce the accuracy threshold by less than a factor of 4 when leakage and depolarizing noise rates are comparable. This becomes a factor of 2 when the decoder uses 3-outcome measurements. Finally, when the physical error rate is less than 2 x 10^-4, placing LRUs after every gate may achieve the lowest logical error rates of all of the circuits we considered. We expect the closely related planar and rotated codes to exhibit the same accuracy thresholds and that the ideas may generalize naturally to other topological codes., 20 pages, 12 figures

We demonstrate the ability to control spontaneous emission from a superconducting qubit coupled to a cavity. The time domain profile of the emitted photon is shaped into a symmetric truncated exponential. The experiment is enabled by a qubit coupled to a cavity, with a coupling strength that can be tuned in tens of nanoseconds while maintaining a constant dressed state emission frequency. Symmetrization of the photonic wave packet will enable use of photons as flying qubits for transferring the quantum state between atoms in distant cavities.

We analyze the Purcell relaxation rate of a superconducting qubit coupled to a resonator, which is coupled to a transmission line and pumped by an external microwave drive. Considering the typical regime of the qubit measurement, we focus on the case when the qubit frequency is significantly detuned from the resonator frequency. Surprisingly, the Purcell rate decreases when the strength of the microwave drive is increased. This suppression becomes significant in the nonlinear regime. In the presence of the microwave drive, the loss of photons to the transmission line also causes excitation of the qubit; however, the excitation rate is typically much smaller than the relaxation rate. Our analysis also applies to a more general case of a two-level quantum system coupled to a cavity., Published version

High-fidelity measurements are important for the physical implementation of quantum information protocols. Current methods for classifying measurement trajectories in superconducting qubit systems produce fidelities that are systematically lower than those predicted by experimental parameters. Here, we place current classification methods within the framework of machine learning algorithms and improve on them by investigating more sophisticated ML approaches. We find that non-linear algorithms and clustering methods produce significantly higher assignment fidelities that help close the gap to the fidelity achievable under ideal noise conditions. Clustering methods group trajectories into natural subsets within the data, which allows for the diagnosis of specific systematic errors. We find large clusters in the data associated with relaxation processes and show these are the main source of discrepancy between our experimental and achievable fidelities. These error diagnosis techniques help provide a concrete path forward to improve qubit measurements., Comment: 8 pages, 6 figures, submitted version

We present methods and results of shot-by-shot correlation of noisy measurements to extract entangled state and process tomography in a superconducting qubit architecture. We show that averaging continuous values, rather than counting discrete thresholded values, is a valid tomographic strategy and is in fact the better choice in the low signal-to-noise regime. We show that the effort to measure $N$-body correlations from individual measurements scales exponentially with $N$, but with sufficient signal-to-noise the approach remains viable for few-body correlations. We provide a new protocol to optimally account for the transient behavior of pulsed measurements. Despite single-shot measurement fidelity that is less than perfect, we demonstrate appropriate processing to extract and verify entangled states and processes., 7 pages, 5 figures; updated references

In this paper, we analyze the performance of randomized benchmarking protocols on gate sets under a variety of realistic error models that include systematic rotations, amplitude damping, leakage to higher levels, and 1/f noise. We find that, in almost all cases, benchmarking provides better than a factor-of-two estimate of average error rate, suggesting that randomized benchmarking protocols are a valuable tool for verification and validation of quantum operations. In addition, we derive new models for fidelity decay curves under certain types of non-Markovian noise models such as 1/f and leakage errors. We also show that, provided the standard error of the fidelity measurements is small, only a small number of trials are required for high confidence estimation of gate errors., 13 pages, 11 figures

We introduce a new entangling gate between two fixed-frequency qubits statically coupled via a microwave resonator bus which combines the following desirable qualities: all-microwave control, appreciable qubit separation for reduction of crosstalk and leakage errors, and the ability to function as a two-qubit conditional-phase gate. A fixed, always-on interaction is explicitly designed between higher energy (non-computational) states of two transmon qubits, and then a conditional-phase gate is `activated' on the otherwise unperturbed qubit subspace via a microwave drive. We implement this microwave-activated conditional-phase gate with a fidelity from quantum process tomography of 87%., 5 figures

Quantum process tomography is a necessary tool for verifying quantum gates and diagnosing faults in architectures and gate design. We show that the standard approach of process tomography is grossly inaccurate in the case where the states and measurement operators used to interrogate the system are generated by gates that have some systematic error, a situation all but unavoidable in any practical setting. These errors in tomography can not be fully corrected through oversampling or by performing a larger set of experiments. We present an alternative method for tomography to reconstruct an entire library of gates in a self-consistent manner. The essential ingredient is to define a likelihood function that assumes nothing about the gates used for preparation and measurement. In order to make the resulting optimization tractable we linearize about the target, a reasonable approximation when benchmarking a quantum computer as opposed to probing a black-box function., 10 pages, 6 figures

We demonstrate enhanced relaxation and dephasing times of transmon qubits, up to ~ 60 \mu s by fabricating the interdigitated shunting capacitors using titanium nitride (TiN). Compared to lift-off aluminum deposited simultaneously with the Josephson junction, this represents as much as a six-fold improvement and provides evidence that previous planar transmon coherence times are limited by surface losses from two-level system (TLS) defects residing at or near interfaces. Concurrently, we observe an anomalous temperature dependent frequency shift of TiN resonators which is inconsistent with the predicted TLS model., Comment: 3 figures

Quantum error correction (QEC) is an essential step towards realising scalable quantum computers. Theoretically, it is possible to achieve arbitrarily long protection of quantum information from corruption due to decoherence or imperfect controls, so long as the error rate is below a threshold value. The two-dimensional surface code (SC) is a fault-tolerant error correction protocol} that has garnered considerable attention for actual physical implementations, due to relatively high error thresholds ~1%, and restriction to planar lattices with nearest-neighbour interactions. Here we show a necessary element for SC error correction: high-fidelity parity detection of two code qubits via measurement of a third syndrome qubit. The experiment is performed on a sub-section of the SC lattice with three superconducting transmon qubits, in which two independent outer code qubits are joined to a central syndrome qubit via two linking bus resonators. With all-microwave high-fidelity single- and two-qubit nearest-neighbour entangling gates, we demonstrate entanglement distributed across the entire sub-section by generating a three-qubit Greenberger-Horne-Zeilinger (GHZ) state with fidelity ~94%. Then, via high-fidelity measurement of the syndrome qubit, we deterministically entangle the otherwise un-coupled outer code qubits, in either an even or odd parity Bell state, conditioned on the syndrome state. Finally, to fully characterize this parity readout, we develop a new measurement tomography protocol to obtain a fidelity metric (90% and 91%). Our results reveal a straightforward path for expanding superconducting circuits towards larger networks for the SC and eventually a primitive logical qubit implementation., Comment: 9 pages, 4 main figures, 3 extended data figures

We implement a complete randomized benchmarking protocol on a system of two superconducting qubits. The protocol consists of randomizing over gates in the Clifford group, which experimentally are generated via an improved two-qubit cross-resonance gate implementation and single-qubit unitaries. From this we extract an optimal average error per Clifford of 0.0936. We also perform an interleaved experiment, alternating our optimal two-qubit gate with random two-qubit Clifford gates, to obtain a two-qubit gate error of 0.0653. We compare these values with a two-qubit gate error of ~0.12 obtained from quantum process tomography, which is likely limited by state preparation and measurement errors., 4 figures plus supplementary material

We report a system where fixed interactions between non-computational levels make bright the otherwise forbidden two-photon 00 --> 11 transition. The system is formed by hand selection and assembly of two discrete component transmon-style superconducting qubits inside a rectangular microwave cavity. The application of a monochromatic drive tuned to this transition induces two-photon Rabi-like oscillations between the ground and doubly-excited states via the Bell basis. The system therefore allows all-microwave two-qubit universal control with the same techniques and hardware required for single qubit control. We report Ramsey-like and spin echo sequences with the generated Bell states, and measure a two-qubit gate fidelity of 90% (unconstrained) and 86% (maximum likelihood estimator)., 5 pages, 4 figures. V2: add supplemental material about the Schrieffer-Wolff transformation

We observe measurement-induced qubit state mixing in a transmon qubit dispersively coupled to a planar readout cavity. Our results indicate that dephasing noise at the qubit-readout detuning frequency is up-converted by readout photons to cause spurious qubit state transitions, thus limiting the nondemolition character of the readout. Furthermore, we use the qubit transition rate as a tool to extract an equivalent flux noise spectral density at f ~ 1 GHz and find agreement with values extrapolated from a $1/f^\alpha$ fit to the measured flux noise spectral density below 1 Hz., Comment: 5 pages, 4 figures. Final journal version

We study bifurcation measurement of a multi-level superconducting qubit using a nonlinear resonator biased in the straddling regime, where the resonator frequency sits between two qubit transition frequencies. We find that high-fidelity bifurcation measurements are possible because of the enhanced qubit-state-dependent pull of the resonator frequency, the behavior of qubit-induced nonlinearities and the reduced Purcell decay rate of the qubit that can be realized in this regime. Numerical simulations find up to a threefold improvement in qubit readout fidelity when operating in, rather than outside of, the straddling regime. High-fidelity measurements can be obtained at much smaller qubit-resonator couplings than current typical experimental realizations, reducing spectral crowding and potentially simplifying the implementation of multi-qubit devices., 9 pages, 6 figures

We provide a general method to find the Hamiltonian of a linear circuit in the presence of a nonlinearity. Focussing on the case of a Josephson junction embedded in a transmission-line resonator, we solve for the normal modes of the system by taking into account exactly the effect of the quadratic (i.e. inductive) part of the Josephson potential. The nonlinearity is then found to lead to self and cross-Kerr effect, as well as beam-splitter type interactions between modes. By adjusting the parameters of the circuit, the Kerr coefficient K can be made to reach values that are weak (K < \kappa), strong (K > \kappa) or even very strong (K >> \kappa) with respect to the photon-loss rate \kappa. In the latter case, the resonator+junction circuit corresponds to an in-line version of the transmon. By replacing the single junction by a SQUID, the Kerr coefficient can be tuned in-situ, allowing for example the fast generation of Schr\"odinger cat states of microwave light. Finally, we explore the maximal strength of qubit-resonator coupling that can be reached in this setting., Comment: 15 pages, 12 figures; Published version with minor changes and corrections

We use quantum process tomography to characterize a full universal set of all-microwave gates on two superconducting single-frequency single-junction transmon qubits. All extracted gate fidelities, including those for Clifford group generators, single-qubit pi/4 and pi/8 rotations, and a two-qubit controlled-NOT, exceed 95% (98%), without (with) accounting for state preparation and measurement errors. Furthermore, we introduce a process map representation in the Pauli basis which is visually efficient and informative. This high-fidelity gate set serves as another critical building block towards scalable architectures of superconducting qubits for error correction schemes., Comment: 4 figures, 2 tables plus supplementary material

We report a superconducting artificial atom with a coherence time of ${T}_{2}^{*}=92$ $\ensuremath{\mu}$s and energy relaxation time ${T}_{1}=70$ $\ensuremath{\mu}$s. The system consists of a single Josephson junction transmon qubit on a sapphire substrate 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 promise as a robust and easily produced artificial quantum system whose intrinsic coherence properties are sufficient to allow tests of quantum error correction.

We provide an efficient method for computing the maximum-likelihood mixed quantum state (with density matrix $\ensuremath{\rho}$) given a set of measurement outcomes in a complete orthonormal operator basis subject to Gaussian noise. Our method works by first changing basis yielding a candidate density matrix $\ensuremath{\mu}$ which may have nonphysical (negative) eigenvalues, and then finding the nearest physical state under the 2-norm. Our algorithm takes at worst $O({d}^{4})$ for the basis change plus $O({d}^{3})$ for finding $\ensuremath{\rho}$ where $d$ is the dimension of the quantum state. In the special case where the measurement basis is strings of Pauli operators, the basis change takes only $O({d}^{3})$ as well. The workhorse of the algorithm is a new linear-time method for finding the closest probability distribution (in Euclidean distance) to a set of real numbers summing to one.

We present a tunable coupling qubit (TCQ), which has independent and fast control over the qubit energy and the coupling strength to a superconducting microwave cavity in a circuit quantum electrodynamics architecture.

Cavity and circuit QED study light-matter interaction at its most fundamental level. Yet, this interaction is most often neglected when considering the coupling of this system with an environment. In this paper, we show how this simplification, which leads to the standard quantum optics master equation, is at the root of unphysical effects. Including qubit relaxation and dephasing, and cavity relaxation, we derive a master equation that takes into account the qubit-resonator coupling. Special attention is given to the ultrastrong coupling regime, where the failure of the quantum optical master equation is manifest. In this situation, our model predicts an asymmetry in the vacuum Rabi splitting that could be used to probe dephasing noise at unexplored frequencies. We also show how fluctuations in the qubit frequency can cause sideband transitions, squeezing, and Casimir-like photon generation., Comment: 16 pages, 6 figures

Dynamical quantum jumps were initially conceived by Bohr as objective events associated with the emission of a light quantum by an atom. Since the early 1990s they have come to be understood as being associated rather with the detection of a photon by a measurement device, and that different detection schemes result in different types of jumps (or diffusion). Here we propose experimental tests to rigorously prove the detector-dependence of the stochastic evolution of an individual atom. The tests involve no special preparation of the atom or field, and the required efficiency can be as low as \eta ~58%., Comment: 4+ pages, 1 figure. Accepted for publication in Physical Review Letters

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%.

We demonstrate coherent control and measurement of a superconducting qubit coupled to a superconducting coplanar waveguide resonator with a dynamically tunable qubit-cavity coupling strength. Rabi oscillations are measured for several coupling strengths showing that the qubit transition can be turned off by a factor of more than 1500. We show how the qubit can still be accessed in the off state via fast flux pulses. We perform pulse delay measurements with synchronized fast flux pulses on the device and observe $T_1$ and $T_2$ times of 1.6 and 1.9 $\mu$s, respectively. This work demonstrates how this qubit can be incorporated into quantum computing architectures., Comment: 4 pages, 4 figures

In this Letter we propose a fully scalable randomized benchmarking protocol for quantum information processors. We prove that the protocol provides an efficient and reliable estimate of the average error-rate for a set operations (gates) under a very general noise model that allows for both time and gate-dependent errors. In particular we obtain a sequence of fitting models for the observable fidelity decay as a function of a (convergent) perturbative expansion of the gate errors about the mean error. We illustrate the protocol through numerical examples.

In qubits made from a weakly anharmonic oscillator the leading source of error at short gate times is leakage of population out of the two dimensional Hilbert space that forms the qubit. In this paper we develop a general scheme based on an adiabatic expansion to find pulse shapes that correct this type of error. We find a family of solutions that allows tailoring to what is practical to implement for a specific application. Our result contains and improves the previously developed DRAG technique [F. Motzoi, et. al., Phys. Rev. Lett. 103, 110501 (2009)] and allows a generalization to other non-linear oscillators with more than one leakage transition., Comment: 15 pages, 9 figures, small changes

In this article, we develop a numerical method to find optimal control pulses that accounts for the separation of timescales between the variation of the input control fields and the applied Hamiltonian. In traditional numerical optimization methods, these timescales are treated as being the same. While this approximation has had much success, in applications where the input controls are filtered substantially or mixed with a fast carrier, the resulting optimized pulses have little relation to the applied physical fields. Our technique remains numerically efficient in that the dimension of our search space is only dependent on the variation of the input control fields, while our simulation of the quantum evolution is accurate on the timescale of the fast variation in the applied Hamiltonian., Comment: 10 pages, 6 figures

We describe and expand upon the scalable randomized benchmarking protocol proposed in Phys. Rev. Lett. 106, 180504 (2011) which provides a method for benchmarking quantum gates and estimating the gate-dependence of the noise. The protocol allows the noise to have weak time and gate-dependence, and we provide a sufficient condition for the applicability of the protocol in terms of the average variation of the noise. We discuss how state preparation and measurement errors are taken into account and provide a complete proof of the scalability of the protocol. We establish a connection in special cases between the error rate provided by this protocol and the error strength measured using the diamond norm distance., Comment: 19 pages, no figures, published version

We numerically demonstrate the control of motional degrees of freedom of an ensemble of neutral atoms in an optical lattice with a shallow trapping potential. Taking into account the range of quasimomenta across different Brillouin zones results in an ensemble whose members effectively have inhomogeneous control fields as well as spectrally distinct control Hamiltonians. We present an ensemble-averaged optimal control technique that yields high fidelity control pulses, irrespective of quasimomentum, with average fidelities above 98%. The resulting controls show a broadband spectrum with gate times in the order of several free oscillations to optimize gates with up to 13.2% dispersion in the energies from the band structure. This can be seen as a model system for the prospects of robust quantum control. This result explores the limits of discretizing a continuous ensemble for control theory.

We introduce a new type of superconducting charge qubit that has a $V$-shaped energy spectrum and uses quantum interference to provide independently tunable qubit energy and coherent coupling to a superconducting cavity. Dynamic access to the strong coupling regime is demonstrated by tuning the coupling strength from less than 200 kHz to greater than 40 MHz. This tunable coupling can be used to protect the qubit from cavity-induced relaxation and avoid unwanted qubit-qubit interactions in a multiqubit system.

We employ simultaneous shaping of in-phase and out-of-phase resonant microwave drives to reduce single-qubit gate errors arising from the weak anharmonicity of transmon superconducting artificial atoms. To reduce the effect of higher levels present in the transmon spectrum, we apply Gaussian and derivative-of-Gaussian envelopes to the in-phase and out-of-phase quadratures, respectively, and optimize over their relative amplitude. Using randomized benchmarking, we obtain a minimum average error per gate of $0.007\ifmmode\pm\else\textpm\fi{}0.005$ using 4-ns-wide pulses, which is limited by decoherence. This simple optimization technique works for multiple transmons coupled to a single microwave resonator in a quantum bus architecture.

We present a superconducting qubit for the circuit quantum electrodynamics architecture that has a tunable coupling strength g. We show that this coupling strength can be tuned from zero to values that are comparable with other superconducting qubits. At g = 0 the qubit is in a decoherence free subspace with respect to spontaneous emission induced by the Purcell effect. Furthermore we show that in the decoherence free subspace the state of the qubit can still be measured by either a dispersive shift on the resonance frequency of the resonator or by a cycling-type measurement., 4 pages, 3 figures

We present a quantum electrical circuit with Josephson junctions formed of two anharmonic oscillators coupled with an interaction $g\gamma_{1}^{2}\gamma_{2}^{2}$ where $\gamma_{1}$ and $\gamma_{2}$ are position-like coordinates. This type of coupling allows the quantum non-demolition measurement of the energy of one oscillator by monitoring the frequency of the second oscillator. Despite the fundamental tradeoff between the coupling strength $g$ and maximum photon storage capacity of the oscillators, it is possible to achieve high fidelity detection of up to 10 photons over time scale of the order of microseconds. We discuss the possibility of observing quantum jumps in the number of photons and related applications., Comment: 5 pages, 3 figures

A single-channel joint readout is used to analyze highly entangled two-qubit states in a circuit quantum electrodynamics architecture. The measurement model for the readout is fully characterized, demonstrating a large sensitivity to two-qubit correlations. We quantify the high degree of entanglement by measuring a violation of the Clauser-Horne-Shimony-Holt inequality with a value of $2.61\ifmmode\pm\else\textpm\fi{}0.04$, without optimizing the preparation of the two-qubit state. In its present form, this joint readout can resolve improvements to the fidelity of two-qubit operations and be extended to three or four qubits.

In dispersive readout schemes, qubit-induced nonlinearity typically limits the measurement fidelity by reducing the signal-to-noise ratio (SNR) when the measurement power is increased. Contrary to seeing the nonlinearity as a problem, here we propose to use it to our advantage in a regime where it can increase the SNR. We show analytically that such a regime exists if the qubit has a many-level structure. We also show how this physics can account for the high-fidelity avalanchelike measurement recently reported by Reed {\it et al.} [arXiv:1004.4323v1]., 4 pages, 5 figures

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. 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 involving statistical averages to sign differences in deterministic observations. With the ample confirmation of quantum mechanical predictions by experiments, entanglement has evolved from a philosophical conundrum to a key resource for quantum-based technologies, like quantum cryptography and computation. 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, photons, and ions, respectively, entanglement in engineered solid-state systems has been limited to two qubits. Here, we demonstrate three-qubit entanglement in a superconducting circuit, creating Greenberger-Horne-Zeilinger (GHZ) states with fidelity of 88%, measured with quantum state tomography. Several entanglement witnesses show violation of bi-separable bounds by 830\pm80%. 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., 7 pages, 4 figures, and Supplementary Information (4 figures).

The biexciton cascade in a quantum dot can be used to generate entangled-photon pairs rapidly and deterministically (on demand). However, due to a large fine-structure splitting between intermediate exciton energy levels, which-path information encoded in the frequencies of emitted photon pairs leads to a small degree of entanglement. Here we show that this information can be efficiently erased by modulating the exciton and biexciton energy levels, giving rise to new decay paths through additional sidebands. The resulting degree of entanglement is substantial, and can be made maximal through spectral filtering, with only a nominal reduction in collection efficiency., Comment: 4 pages, 3 figures; New title, discussion of dephasing added, version published in Phys. Rev. B

Joint measurements of multiple qubits have been shown to open new possibilities for quantum information processing. Here, we present an approach based on homodyne detection to realize such measurements in the dispersive regime of cavity/circuit QED. By changing details of the measurement, the readout can be tuned from extracting only single-qubit to only multi-qubit properties. We obtain a reduced stochastic master equation describing this measurement and its effect on the qubits. As an example, we present results showing parity measurements of two qubits. In this situation, measurement of an initially unentangled state can yield with near unit probability a state of significant concurrence., 4 pages, 4 figures