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A key challenge in quantum computing is speeding up measurement and initialization. Here, we experimentally demonstrate a dispersive measurement method for superconducting qubits that simultaneously measures the qubit and returns the readout resonator to its initial state. The approach is based on universal analytical pulses and requires knowledge of the qubit and resonator parameters, but needs no direct optimization of the pulse shape, even when accounting for the nonlinearity of the system. Moreover, the method generalizes to measuring an arbitrary number of modes and states. For the qubit readout, we can drive the resonator to $\sim 10^2$ photons and back to $\sim 10^{-3}$ photons in less than $3 \kappa^{-1}$, while still achieving a $T_1$-limited assignment error below 1\%. We also present universal pulse shapes and experimental results for qutrit readout., Comment: 8 pages, 6 figures

The discovery of topological order has revolutionized the understanding of quantum matter in modern physics and provided the theoretical foundation for many quantum error correcting codes. Realizing topologically ordered states has proven to be extremely challenging in both condensed matter and synthetic quantum systems. Here, we prepare the ground state of the toric code Hamiltonian using an efficient quantum circuit on a superconducting quantum processor. We measure a topological entanglement entropy near the expected value of $\ln2$, and simulate anyon interferometry to extract the braiding statistics of the emergent excitations. Furthermore, we investigate key aspects of the surface code, including logical state injection and the decay of the non-local order parameter. Our results demonstrate the potential for quantum processors to provide key insights into topological quantum matter and quantum error correction., Comment: 6 pages 4 figures, plus supplementary materials

Quantum computing can become scalable through error correction, but logical error rates only decrease with system size when physical errors are sufficiently uncorrelated. During computation, unused high energy levels of the qubits can become excited, creating leakage states that are long-lived and mobile. Particularly for superconducting transmon qubits, this leakage opens a path to errors that are correlated in space and time. Here, we report a reset protocol that returns a qubit to the ground state from all relevant higher level states. We test its performance with the bit-flip stabilizer code, a simplified version of the surface code for quantum error correction. We investigate the accumulation and dynamics of leakage during error correction. Using this protocol, we find lower rates of logical errors and an improved scaling and stability of error suppression with increasing qubit number. This demonstration provides a key step on the path towards scalable quantum computing.

A promising approach to study condensed-matter systems is to simulate them on an engineered quantum platform. However, achieving the accuracy needed to outperform classical methods has been an outstanding challenge. Here, using eighteen superconducting qubits, we provide an experimental blueprint for an accurate condensed-matter simulator and demonstrate how to probe fundamental electronic properties. We benchmark the underlying method by reconstructing the single-particle band-structure of a one-dimensional wire. We demonstrate nearly complete mitigation of decoherence and readout errors and arrive at an accuracy in measuring energy eigenvalues of this wire with an error of ~0.01 rad, whereas typical energy scales are of order 1 rad. Insight into this unprecedented algorithm fidelity is gained by highlighting robust properties of a Fourier transform, including the ability to resolve eigenenergies with a statistical uncertainty of 1e-4 rad. Furthermore, we synthesize magnetic flux and disordered local potentials, two key tenets of a condensed-matter system. When sweeping the magnetic flux, we observe avoided level crossings in the spectrum, a detailed fingerprint of the spatial distribution of local disorder. Combining these methods, we reconstruct electronic properties of the eigenstates where we observe persistent currents and a strong suppression of conductance with added disorder. Our work describes an accurate method for quantum simulation and paves the way to study novel quantum materials with superconducting qubits.

Quantum algorithms offer a dramatic speedup for computational problems in machine learning, material science, and chemistry. However, any near-term realizations of these algorithms will need to be heavily optimized to fit within the finite resources offered by existing noisy quantum hardware. Here, taking advantage of the strong adjustable coupling of gmon qubits, we demonstrate a continuous two-qubit gate set that can provide a 3x reduction in circuit depth as compared to a standard decomposition. We implement two gate families: an iSWAP-like gate to attain an arbitrary swap angle, $\theta$, and a CPHASE gate that generates an arbitrary conditional phase, $\phi$. Using one of each of these gates, we can perform an arbitrary two-qubit gate within the excitation-preserving subspace allowing for a complete implementation of the so-called Fermionic Simulation, or fSim, gate set. We benchmark the fidelity of the iSWAP-like and CPHASE gate families as well as 525 other fSim gates spread evenly across the entire fSim($\theta$, $\phi$) parameter space achieving purity-limited average two-qubit Pauli error of $3.8 \times 10^{-3}$ per fSim gate., Comment: 20 pages, 17 figures

The interplay of interactions and strong disorder can lead to an exotic quantum many-body localized (MBL) phase. Beyond the absence of transport, the MBL phase has distinctive signatures, such as slow dephasing and logarithmic entanglement growth; they commonly result in slow and subtle modification of the dynamics, making their measurement challenging. Here, we experimentally characterize these properties of the MBL phase in a system of coupled superconducting qubits. By implementing phase sensitive techniques, we map out the structure of local integrals of motion in the MBL phase. Tomographic reconstruction of single and two qubit density matrices allowed us to determine the spatial and temporal entanglement growth between the localized sites. In addition, we study the preservation of entanglement in the MBL phase. The interferometric protocols implemented here measure affirmative correlations and allow us to exclude artifacts due to the imperfect isolation of the system. By measuring elusive MBL quantities, our work highlights the advantages of phase sensitive measurements in studying novel phases of matter., Comment: 5+28 pages, 5+22 figures, updated version

We demonstrate diabatic two-qubit gates with Pauli error rates down to $4.3(2)\cdot 10^{-3}$ in as fast as 18 ns using frequency-tunable superconducting qubits. This is achieved by synchronizing the entangling parameters with minima in the leakage channel. The synchronization shows a landscape in gate parameter space that agrees with model predictions and facilitates robust tune-up. We test both iSWAP-like and CPHASE gates with cross-entropy benchmarking. The presented approach can be extended to multibody operations as well., Comment: Main text: 6 pages, 4 figures. Supplementary: 2 pages, 2 figures

For space observatories, the glitches caused by high energy phonons created by the interaction of cosmic ray particles with the detector substrate lead to dead time during observation. Mitigating the impact of cosmic rays is therefore an important requirement for detectors to be used in future space missions. In order to investigate possible solutions, we carry out a systematic study by testing four large arrays of Microwave Kinetic Inductance Detectors (MKIDs), each consisting of $\sim$960 pixels and fabricated on monolithic 55 mm $\times$ 55 mm $\times$ 0.35 mm Si substrates. We compare the response to cosmic ray interactions in our laboratory for different detector arrays: A standard array with only the MKID array as reference; an array with a low $T_c$ superconducting film as phonon absorber on the opposite side of the substrate; and arrays with MKIDs on membranes. The idea is that the low $T_c$ layer down-converts the phonon energy to values below the pair breaking threshold of the MKIDs, and the membranes isolate the sensitive part of the MKIDs from phonons created in the substrate. We find that the dead time can be reduced up to a factor of 40 when compared to the reference array. Simulations show that the dead time can be reduced to below 1 % for the tested detector arrays when operated in a spacecraft in an L2 or a similar far-Earth orbit. The technique described here is also applicable and important for large superconducting qubit arrays for future quantum computers., Comment: 5 pages, 4 figures; accepted to APL

Superconducting qubits are an attractive platform for quantum computing since they have demonstrated high-fidelity quantum gates and extensibility to modest system sizes. Nonetheless, an outstanding challenge is stabilizing their energy-relaxation times, which can fluctuate unpredictably in frequency and time. Here, we use qubits as spectral and temporal probes of individual two-level-system defects to provide direct evidence that they are responsible for the largest fluctuations. This research lays the foundation for stabilizing qubit performance through calibration, design, and fabrication., Comment: 7 main pages, 3 main figures, 5 supplemental pages, 5 supplemental figures

We develop a high speed on-chip flux measurement using a capacitively shunted SQUID as an embedded cryogenic transducer and apply this technique to the qualification of a near-term scalable printed circuit board (PCB) package for frequency tunable superconducting qubits. The transducer is a flux tunable LC resonator where applied flux changes the resonant frequency. We apply a microwave tone to probe this frequency and use a time-domain homodyne measurement to extract the reflected phase as a function of flux applied to the SQUID. The transducer response bandwidth is 2.6 GHz with a maximum gain of $\rm 1200^\circ/\Phi_0$ allowing us to study the settling amplitude to better than 0.1%. We use this technique to characterize on-chip bias line routing and a variety of PCB based packages and demonstrate that step response settling can vary by orders of magnitude in both settling time and amplitude depending on if normal or superconducting materials are used. By plating copper PCBs in aluminum we measure a step response consistent with the packaging used for existing high-fidelity qubits.

Complex integrated circuits require multiple wiring layers. In complementary metal-oxide-semiconductor (CMOS) processing, these layers are robustly separated by amorphous dielectrics. These dielectrics would dominate energy loss in superconducting integrated circuits. Here we demonstrate a procedure that capitalizes on the structural benefits of inter-layer dielectrics during fabrication and mitigates the added loss. We separate and support multiple wiring layers throughout fabrication using SiO$_2$ scaffolding, then remove it post-fabrication. This technique is compatible with foundry level processing and the can be generalized to make many different forms of low-loss multi-layer wiring. We use this technique to create freestanding aluminum vacuum gap crossovers (airbridges). We characterize the added capacitive loss of these airbridges by connecting ground planes over microwave frequency $\lambda/4$ coplanar waveguide resonators and measuring resonator loss. We measure a low power resonator loss of $\sim 3.9 \times 10^{-8}$ per bridge, which is 100 times lower than dielectric supported bridges. We further characterize these airbridges as crossovers, control line jumpers, and as part of a coupling network in gmon and fuxmon qubits. We measure qubit characteristic lifetimes ($T_1$'s) in excess of 30 $\mu$s in gmon devices.

Statistical mechanics is founded on the assumption that a system can reach thermal equilibrium, regardless of the starting state. Interactions between particles facilitate thermalization, but, can interacting systems always equilibrate regardless of parameter values\,? The energy spectrum of a system can answer this question and reveal the nature of the underlying phases. However, most experimental techniques only indirectly probe the many-body energy spectrum. Using a chain of nine superconducting qubits, we implement a novel technique for directly resolving the energy levels of interacting photons. We benchmark this method by capturing the intricate energy spectrum predicted for 2D electrons in a magnetic field, the Hofstadter butterfly. By increasing disorder, the spatial extent of energy eigenstates at the edge of the energy band shrink, suggesting the formation of a mobility edge. At strong disorder, the energy levels cease to repel one another and their statistics approaches a Poisson distribution - the hallmark of transition from the thermalized to the many-body localized phase. Our work introduces a new many-body spectroscopy technique to study quantum phases of matter.

Fundamental questions in chemistry and physics may never be answered due to the exponential complexity of the underlying quantum phenomena. A desire to overcome this challenge has sparked a new industry of quantum technologies with the promise that engineered quantum systems can address these hard problems. A key step towards demonstrating such a system will be performing a computation beyond the capabilities of any classical computer, achieving so-called quantum supremacy. Here, using 9 superconducting qubits, we demonstrate an immediate path towards quantum supremacy. By individually tuning the qubit parameters, we are able to generate thousands of unique Hamiltonian evolutions and probe the output probabilities. The measured probabilities obey a universal distribution, consistent with uniformly sampling the full Hilbert-space. As the number of qubits in the algorithm is varied, the system continues to explore the exponentially growing number of states. Combining these large datasets with techniques from machine learning allows us to construct a model which accurately predicts the measured probabilities. We demonstrate an application of these algorithms by systematically increasing the disorder and observing a transition from delocalized states to localized states. By extending these results to a system of 50 qubits, we hope to address scientific questions that are beyond the capabilities of any classical computer.

We present a fabrication process for fully superconducting interconnects compatible with superconducting qubit technology. These interconnects allow for the 3D integration of quantum circuits without introducing lossy amorphous dielectrics. They are composed of indium bumps several microns tall separated from an aluminum base layer by titanium nitride which serves as a diffusion barrier. We measure the whole structure to be superconducting (transition temperature of 1.1$\,$K), limited by the aluminum. These interconnects have an average critical current of 26.8$\,$mA, and mechanical shear and thermal cycle testing indicate that these devices are mechanically robust. Our process provides a method that reliably yields superconducting interconnects suitable for use with superconducting qubits.

Josephson junctions form the essential non-linearity for almost all superconducting qubits. The junction is formed when two superconducting electrodes come within $\sim$1 nm of each other. Although the capacitance of these electrodes is a small fraction of the total qubit capacitance, the nearby electric fields are more concentrated in dielectric surfaces and can contribute substantially to the total dissipation. We have developed a technique to experimentally investigate the effect of these electrodes on the quality of superconducting devices. We use $\lambda$/4 coplanar waveguide resonators to emulate lumped qubit capacitors. We add a variable number of these electrodes to the capacitive end of these resonators and measure how the additional loss scales with number of electrodes. We then reduce this loss with fabrication techniques that limit the amount of lossy dielectrics. We then apply these techniques to the fabrication of Xmon qubits on a silicon substrate to improve their energy relaxation times by a factor of 5.

An ideal preamplifier for qubit measurement must not only provide high gain and near quantum-limited noise performance, but also isolate the delicate quantum circuit from noisy downstream measurement stages while producing negligible backaction. Here we use a Superconducting Low-inductance Undulatory Galvanometer (SLUG) microwave amplifier to read out a superconducting transmon qubit, and we characterize both reverse isolation and measurement backaction of the SLUG. For appropriate dc bias, the SLUG achieves reverse isolation that is better than that of a commercial cryogenic isolator. Moreover, SLUG backaction is dominated by thermal emission from dissipative elements in the device. When the SLUG is operated in pulsed mode, it is possible to characterize the transmon qubit using a measurement chain that is free from cryogenic isolators or circulators with no measurable degradation of qubit performance., Comment: 5 pages, 4 figures

By analyzing the dissipative dynamics of a tunable gap flux qubit, we extract both sides of its two-sided environmental flux noise spectral density over a range of frequencies around $2k_BT/h \approx 1\,\rm{GHz}$, allowing for the observation of a classical-quantum crossover. Below the crossover point, the symmetric noise component follows a $1/f$ power law that matches the magnitude of the $1/f$ noise near $1\,{\rm{Hz}}$. The antisymmetric component displays a 1/T dependence below $100\,\rm{mK}$, providing dynamical evidence for a paramagnetic environment. Extrapolating the two-sided spectrum predicts the linewidth and reorganization energy of incoherent resonant tunneling between flux qubit wells., Comment: paper + supplement

Surface distributions of two level system (TLS) defects and magnetic vortices are limiting dissipation sources in superconducting quantum circuits. Arrays of flux-trapping holes are commonly used to eliminate loss due to magnetic vortices, but may increase dielectric TLS loss. We find that dielectric TLS loss increases by approximately 25% for resonators with a hole array beginning 2 $\mu \text{m}$ from the resonator edge, while the dielectric loss added by holes further away was below measurement sensitivity. Other forms of loss were not affected by the holes. Additionally, we estimate the loss due to residual magnetic effects to be $9\times 10^{-10} /\mu\text{T} $ for resonators patterned with flux-traps and operated in magnetic fields up to $5$ $\mu\text{T}$. This is orders of magnitude below the total loss of the best superconducting coplanar waveguide resonators., Comment: 7 Pages, 4 Figures

Many superconducting qubit systems use the dispersive interaction between the qubit and a coupled harmonic resonator to perform quantum state measurement. Previous works have found that such measurements can induce state transitions in the qubit if the number of photons in the resonator is too high. We investigate these transitions and find that they can push the qubit out of the two-level subspace, and that they show resonant behavior as a function of photon number. We develop a theory for these observations based on level crossings within the Jaynes-Cummings ladder, with transitions mediated by terms in the Hamiltonian that are typically ignored by the rotating wave approximation. We find that the most important of these terms comes from an unexpected broken symmetry in the qubit potential. We confirm the theory by measuring the photon occupation of the resonator when transitions occur while varying the detuning between the qubit and resonator.

The intriguing many-body phases of quantum matter arise from the interplay of particle interactions, spatial symmetries, and external fields. Generating these phases in an engineered system could provide deeper insight into their nature and the potential for harnessing their unique properties. However, concurrently bringing together the main ingredients for realizing many-body phenomena in a single experimental platform is a major challenge. Using superconducting qubits, we simultaneously realize synthetic magnetic fields and strong particle interactions, which are among the essential elements for studying quantum magnetism and fractional quantum Hall (FQH) phenomena. The artificial magnetic fields are synthesized by sinusoidally modulating the qubit couplings. In a closed loop formed by the three qubits, we observe the directional circulation of photons, a signature of broken time-reversal symmetry. We demonstrate strong interactions via the creation of photon-vacancies, or "holes", which circulate in the opposite direction. The combination of these key elements results in chiral groundstate currents, the first direct measurement of persistent currents in low-lying eigenstates of strongly interacting bosons. The observation of chiral currents at such a small scale is interesting and suggests that the rich many-body physics could survive to smaller scales. We also motivate the feasibility of creating FQH states with near future superconducting technologies. Our work introduces an experimental platform for engineering quantum phases of strongly interacting photons and highlight a path toward realization of bosonic FQH states., Comment: in Nature Physics (2016)

We present a method to optimize qubit control parameters during error detection which is compatible with large-scale qubit arrays. We demonstrate our method to optimize single or two-qubit gates in parallel on a nine-qubit system. Additionally, we show how parameter drift can be compensated for during computation by inserting a frequency drift and using our method to remove it. We remove both drift on a single qubit and independent drifts on all qubits simultaneously. We believe this method will be useful in keeping error rates low on all physical qubits throughout the course of a computation. Our method is O(1) scalable to systems of arbitrary size, providing a path towards controlling the large numbers of qubits needed for a fault-tolerant quantum computer, Comment: 8 pages with supplemental, 7 figures

Statistical mechanics is founded on the assumption that all accessible configurations of a system are equally likely. This requires dynamics that explore all states over time, known as ergodic dynamics. In isolated quantum systems, however, the occurrence of ergodic behavior has remained an outstanding question. Here, we demonstrate ergodic dynamics in a small quantum system consisting of only three superconducting qubits. The qubits undergo a sequence of rotations and interactions and we measure the evolution of the density matrix. Maps of the entanglement entropy show that the full system can act like a reservoir for individual qubits, increasing their entropy through entanglement. Surprisingly, these maps bear a strong resemblance to the phase space dynamics in the classical limit; classically chaotic motion coincides with higher entanglement entropy. We further show that in regions of high entropy the full multi-qubit system undergoes ergodic dynamics. Our work illustrates how controllable quantum systems can investigate fundamental questions in non-equilibrium thermodynamics.

We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future., Comment: 13 pages, 7 figures. This revision is to correct an error in the coefficients of identity in Table 1

A major challenge in quantum computing is to solve general problems with limited physical hardware. Here, we implement digitized adiabatic quantum computing, combining the generality of the adiabatic algorithm with the universality of the digital approach, using a superconducting circuit with nine qubits. We probe the adiabatic evolutions, and quantify the success of the algorithm for random spin problems. We find that the system can approximate the solutions to both frustrated Ising problems and problems with more complex interactions, with a performance that is comparable. The presented approach is compatible with small-scale systems as well as future error-corrected quantum computers., Comment: Main text: 7 pages, 5 figures. Supplementary: 12 pages, 9 figures

Leakage errors occur when a quantum system leaves the two-level qubit subspace. Reducing these errors is critically important for quantum error correction to be viable. To quantify leakage errors, we use randomized benchmarking in conjunction with measurement of the leakage population. We characterize single qubit gates in a superconducting qubit, and by refining our use of Derivative Reduction by Adiabatic Gate (DRAG) pulse shaping along with detuning of the pulses, we obtain gate errors consistently below $10^{-3}$ and leakage rates at the $10^{-5}$ level. With the control optimized, we find that a significant portion of the remaining leakage is due to incoherent heating of the qubit., Comment: 10 pages, 10 figures including supplement; fixed typos in metadata

Weak measurement has provided new insight into the nature of quantum measurement, by demonstrating the ability to extract average state information without fully projecting the system. For single qubit measurements, this partial projection has been demonstrated with violations of the Leggett-Garg inequality. Here we investigate the effects of weak measurement on a maximally entangled Bell state through application of the Hybrid Bell-Leggett-Garg inequality (BLGI) on a linear chain of four transmon qubits. By correlating the results of weak ancilla measurements with subsequent projective readout, we achieve a violation of the BLGI with 27 standard deviations of certainty.

Josephson parametric amplifiers have become a critical tool in superconducting device physics due to their high gain and quantum-limited noise. Traveling wave parametric amplifiers (TWPAs) promise similar noise performance while allowing for significant increases in both bandwidth and dynamic range. We present a TWPA device based on an LC-ladder transmission line of Josephson junctions and parallel plate capacitors using low-loss amorphous silicon dielectric. Crucially, we have inserted $\lambda/4$ resonators at regular intervals along the transmission line in order to maintain the phase matching condition between pump, signal, and idler and increase gain. We achieve an average gain of 12\,dB across a 4\,GHz span, along with an average saturation power of -92\,dBm with noise approaching the quantum limit.

Simulating quantum physics with a device which itself is quantum mechanical, a notion Richard Feynman originated, would be an unparallelled computational resource. However, the universal quantum simulation of fermionic systems is daunting due to their particle statistics, and Feynman left as an open question whether it could be done, because of the need for non-local control. Here, we implement fermionic interactions with digital techniques in a superconducting circuit. Focusing on the Hubbard model, we perform time evolution with constant interactions as well as a dynamic phase transition with up to four fermionic modes encoded in four qubits. The implemented digital approach is universal and allows for the efficient simulation of fermions in arbitrary spatial dimensions. We use in excess of 300 single-qubit and two-qubit gates, and reach global fidelities which are limited by gate errors. This demonstration highlights the feasibility of the digital approach and opens a viable route towards analog-digital quantum simulation of interacting fermions and bosons in large-scale solid state systems., Comment: Main text: 5 pages, 5 figures. Supplementary: 7 pages, 6 figures

Quantum computing becomes viable when a quantum state can be preserved from environmentally-induced error. If quantum bits (qubits) are sufficiently reliable, errors are sparse and quantum error correction (QEC) is capable of identifying and correcting them. Adding more qubits improves the preservation by guaranteeing increasingly larger clusters of errors will not cause logical failure - a key requirement for large-scale systems. Using QEC to extend the qubit lifetime remains one of the outstanding experimental challenges in quantum computing. Here, we report the protection of classical states from environmental bit-flip errors and demonstrate the suppression of these errors with increasing system size. We use a linear array of nine qubits, which is a natural precursor of the two-dimensional surface code QEC scheme, and track errors as they occur by repeatedly performing projective quantum non-demolition (QND) parity measurements. Relative to a single physical qubit, we reduce the failure rate in retrieving an input state by a factor of 2.7 for five qubits and a factor of 8.5 for nine qubits after eight cycles. Additionally, we tomographically verify preservation of the non-classical Greenberger-Horne-Zeilinger (GHZ) state. The successful suppression of environmentally-induced errors strongly motivates further research into the many exciting challenges associated with building a large-scale superconducting quantum computer., Comment: Main text 5 pages, 4 figures. Supplemental 25 pages, 31 figures

A precise measurement of dephasing over a range of timescales is critical for improving quantum gates beyond the error correction threshold. We present a metrological tool, based on randomized benchmarking, capable of greatly increasing the precision of Ramsey and spin echo sequences by the repeated but incoherent addition of phase noise. We find our SQUID-based qubit is not limited by $1/f$ flux noise at short timescales, but instead observe a telegraph noise mechanism that is not amenable to study with standard measurement techniques., Comment: 11 pages, 7 figures

Many superconducting qubits are highly sensitive to dielectric loss, making the fabrication of coherent quantum circuits challenging. To elucidate this issue, we characterize the interfaces and surfaces of superconducting coplanar waveguide resonators and study the associated microwave loss. We show that contamination induced by traditional qubit lift-off processing is particularly detrimental to quality factors without proper substrate cleaning, while roughness plays at most a small role. Aggressive surface treatment is shown to damage the crystalline substrate and degrade resonator quality. We also introduce methods to characterize and remove ultra-thin resist residue, providing a way to quantify and minimize remnant sources of loss on device surfaces.

The discovery of topological phases in condensed matter systems has changed the modern conception of phases of matter. The global nature of topological ordering makes these phases robust and hence promising for applications. However, the non-locality of this ordering makes direct experimental studies an outstanding challenge, even in the simplest model topological systems, and interactions among the constituent particles adds to this challenge. Here we demonstrate a novel dynamical method to explore topological phases in both interacting and non-interacting systems, by employing the exquisite control afforded by state-of-the-art superconducting quantum circuits. We utilize this method to experimentally explore the well-known Haldane model of topological phase transitions by directly measuring the topological invariants of the system. We construct the topological phase diagram of this model and visualize the microscopic evolution of states across the phase transition, tasks whose experimental realizations have remained elusive. Furthermore, we developed a new qubit architecture that allows simultaneous control over every term in a two-qubit Hamiltonian, with which we extend our studies to an interacting Hamiltonian and discover the emergence of an interaction-induced topological phase. Our implementation, involving the measurement of both global and local textures of quantum systems, is close to the original idea of quantum simulation as envisioned by R. Feynman, where a controllable quantum system is used to investigate otherwise inaccessible quantum phenomena. This approach demonstrates the potential of superconducting qubits for quantum simulation and establishes a powerful platform for the study of topological phases in quantum systems.

We apply the method of compressed sensing (CS) quantum process tomography (QPT) to characterize quantum gates based on superconducting Xmon and phase qubits. Using experimental data for a two-qubit controlled-Z gate, we obtain an estimate for the process matrix $\chi$ with reasonably high fidelity compared to full QPT, but using a significantly reduced set of initial states and measurement configurations. We show that the CS method still works when the amount of used data is so small that the standard QPT would have an underdetermined system of equations. We also apply the CS method to the analysis of the three-qubit Toffoli gate with numerically added noise, and similarly show that the method works well for a substantially reduced set of data. For the CS calculations we use two different bases in which the process matrix $\chi$ is approximately sparse, and show that the resulting estimates of the process matrices match each ther with reasonably high fidelity. For both two-qubit and three-qubit gates, we characterize the quantum process by not only its process matrix and fidelity, but also by the corresponding standard deviation, defined via variation of the state fidelity for different initial states., Comment: 16 pages, 11 figures

One of the key challenges in quantum information is coherently manipulating the quantum state. However, it is an outstanding question whether control can be realized with low error. Only gates from the Clifford group -- containing $\pi$, $\pi/2$, and Hadamard gates -- have been characterized with high accuracy. Here, we show how the Platonic solids enable implementing and characterizing larger gate sets. We find that all gates can be implemented with low error. The results fundamentally imply arbitrary manipulation of the quantum state can be realized with high precision, providing new practical possibilities for designing efficient quantum algorithms., Comment: 8 pages, 4 figures, including supplementary material

Accurate methods of assessing the performance of quantum gates are extremely important. Quantum process tomography and randomized benchmarking are the current favored methods. Quantum process tomography gives detailed information, but significant approximations must be made to reduce this information to a form quantum error correction simulations can use. Randomized benchmarking typically outputs just a single number, the fidelity, giving no information on the structure of errors during the gate. Neither method is optimized to assess gate performance within an error detection circuit, where gates will be actually used in a large-scale quantum computer. Specifically, the important issues of error composition and error propagation lie outside the scope of both methods. We present a fast, simple, and scalable method of obtaining exactly the information required to perform effective quantum error correction from the output of continuously running error detection circuits, enabling accurate prediction of large-scale behavior., Comment: 9 pages, 6 figures, comments and additional reference suggestions welcome

Understanding complex quantum matter presents a central challenge in condensed matter physics. The difficulty lies in the exponential scaling of the Hilbert space with the system size, making solutions intractable for both analytical and conventional numerical methods. As originally envisioned by Richard Feynman, this class of problems can be tackled using controllable quantum simulators. Despite many efforts, building an quantum emulator capable of solving generic quantum problems remains an outstanding challenge, as this involves controlling a large number of quantum elements. Here, employing a multi-element superconducting quantum circuit and manipulating a single microwave photon, we demonstrate that we can simulate the weak localization phenomenon observed in mesoscopic systems. By engineering the control sequence in our emulator circuit, we are also able to reproduce the well-known temperature dependence of weak localization. Furthermore, we can use our circuit to continuously tune the level of disorder, a parameter that is not readily accessible in mesoscopic systems. By demonstrating a high level of control and complexity, our experiment shows the potential for superconducting quantum circuits to realize scalable quantum simulators., Comment: 9 pages, including supplement

We introduce a superconducting qubit architecture that combines high-coherence qubits and tunable qubit-qubit coupling. With the ability to set the coupling to zero, we demonstrate that this architecture is protected from the frequency crowding problems that arise from fixed coupling. More importantly, the coupling can be tuned dynamically with nanosecond resolution, making this architecture a versatile platform with applications ranging from quantum logic gates to quantum simulation. We illustrate the advantages of dynamic coupling by implementing a novel adiabatic controlled-Z gate, at a speed approaching that of single-qubit gates. Integrating coherence and scalable control, our "gmon" architecture is a promising path towards large-scale quantum computation and simulation.

We present a method for optimizing quantum control in experimental systems, using a subset of randomized benchmarking measurements to rapidly infer error. This is demonstrated to improve single- and two-qubit gates, minimize gate bleedthrough, where a gate mechanism can cause errors on subsequent gates, and identify control crosstalk in superconducting qubits. This method is able to correct parameters to where control errors no longer dominate, and is suitable for automated and closed-loop optimization of experimental systems., Comment: 7 pages, 7 figures including supplementary

A quantum computer can solve hard problems - such as prime factoring, database searching, and quantum simulation - at the cost of needing to protect fragile quantum states from error. Quantum error correction provides this protection, by distributing a logical state among many physical qubits via quantum entanglement. Superconductivity is an appealing platform, as it allows for constructing large quantum circuits, and is compatible with microfabrication. For superconducting qubits the surface code is a natural choice for error correction, as it uses only nearest-neighbour coupling and rapidly-cycled entangling gates. The gate fidelity requirements are modest: The per-step fidelity threshold is only about 99%. Here, we demonstrate a universal set of logic gates in a superconducting multi-qubit processor, achieving an average single-qubit gate fidelity of 99.92% and a two-qubit gate fidelity up to 99.4%. This places Josephson quantum computing at the fault-tolerant threshold for surface code error correction. Our quantum processor is a first step towards the surface code, using five qubits arranged in a linear array with nearest-neighbour coupling. As a further demonstration, we construct a five-qubit Greenberger-Horne-Zeilinger (GHZ) state using the complete circuit and full set of gates. The results demonstrate that Josephson quantum computing is a high-fidelity technology, with a clear path to scaling up to large-scale, fault-tolerant quantum circuits., Comment: 15 pages, 13 figures, including supplementary material

Progress in superconducting qubit experiments with greater numbers of qubits or advanced techniques such as feedback requires faster and more accurate state measurement. We have designed a multiplexed measurement system with a bandpass filter that allows fast measurement without increasing environmental damping of the qubits. We use this to demonstrate simultaneous measurement of four qubits on a single superconducting integrated circuit, the fastest of which can be measured to 99.8% accuracy in 140ns. This accuracy and speed is suitable for advanced multi-qubit experiments including surface code error correction., Comment: Five figures

We describe the high fidelity dispersive measurement of a superconducting qubit using a microwave amplifier based on the Superconducting Low-inductance Undulatory Galvanometer (SLUG). The SLUG preamplifier achieves gain of 19 dB and yields a signal-to-noise ratio improvement of 9 dB over a state-of-the-art HEMT amplifier. We demonstrate a separation fidelity of 99% at 700 ns compared to 59% with the HEMT alone. The SLUG displays a large dynamic range, with an input saturation power corresponding to 700 photons in the readout cavity., Comment: 10 pages, 3 figures

We demonstrate a high efficiency deterministic quantum receiver to convert flying qubits to logic qubits. We employ a superconducting resonator, which is driven with a shaped pulse through an adjustable coupler. For the ideal "time reversed" shape, we measure absorption and receiver fidelities at the single microwave photon level of, respectively, 99.41% and 97.4%. These fidelities are comparable with gates and measurement and exceed the deterministic quantum communication and computation fault tolerant thresholds., Comment: Main paper: 5 pages, 4 figures. Supplement: 11 pages, 12 figures. Revised abstract and introduction. Minor changes to Figure 1 and figure captions

Superconducting resonators, used in astronomy and quantum computation, couple strongly to microscopic two-level defects. We monitor the microwave response of superconducting resonators and observe fluctuations in dissipation and resonance frequency. We present a unified model where the observed dissipative and dispersive effects can be explained as originating from a bath of fluctuating two-level systems. From these measurements, we quantify the number and distribution of the defects.

We demonstrate a planar, tunable superconducting qubit with energy relaxation times up to 44 microseconds. This is achieved by using a geometry designed to both minimize radiative loss and reduce coupling to materials-related defects. At these levels of coherence, we find a fine structure in the qubit energy lifetime as a function of frequency, indicating the presence of a sparse population of incoherent, weakly coupled two-level defects. This is supported by a model analysis as well as experimental variations in the geometry. Our `Xmon' qubit combines facile fabrication, straightforward connectivity, fast control, and long coherence, opening a viable route to constructing a chip-based quantum computer., Comment: 10 pages, 9 figures, including supplementary material