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Experiments on disordered alloys suggest that spin glasses can be brought into low-energy states faster by annealing quantum fluctuations than by conventional thermal annealing. Due to the importance of spin glasses as a paradigmatic computational testbed, reproducing this phenomenon in a programmable system has remained a central challenge in quantum optimization. Here we achieve this goal by realizing quantum critical spin-glass dynamics on thousands of qubits with a superconducting quantum annealer. We first demonstrate quantitative agreement between quantum annealing and time-evolution of the Schr\"odinger equation in small spin glasses. We then measure dynamics in 3D spin glasses on thousands of qubits, where simulation of many-body quantum dynamics is intractable. We extract critical exponents that clearly distinguish quantum annealing from the slower stochastic dynamics of analogous Monte Carlo algorithms, providing both theoretical and experimental support for a scaling advantage in reducing energy as a function of annealing time.

We report on measurements of flux and charge noise in an rf-SQUID flux qubit using macroscopic resonant tunneling (MRT). We measure rates of incoherent tunneling from the lowest energy state in the initial well to the ground and first excited states in the target well. The result of the measurement consists of two peaks. The first peak corresponds to tunneling to the ground state of the target well, and is dominated by flux noise. The second peak is due to tunneling to the excited state and is wider due to an intrawell relaxation process dominated by charge noise. We develop a theoretical model that allows us to extract information about flux and charge noise within one experimental setup. The model agrees very well with experimental data over a wide dynamic range and provides parameters that characterize charge and flux noise., 11 pages, 5 figures

The promise of quantum computing lies in harnessing programmable quantum devices for practical applications such as efficient simulation of quantum materials and condensed matter systems. One important task is the simulation of geometrically frustrated magnets in which topological phenomena can emerge from competition between quantum and thermal fluctuations. Here we report on experimental observations of equilibration in such simulations, measured on up to 1440 qubits with microsecond resolution. By initializing the system in a state with topological obstruction, we observe quantum annealing (QA) equilibration timescales in excess of one microsecond. Measurements indicate a dynamical advantage in the quantum simulation compared with spatially local update dynamics of path-integral Monte Carlo (PIMC). The advantage increases with both system size and inverse temperature, exceeding a million-fold speedup over an efficient CPU implementation. PIMC is a leading classical method for such simulations, and a scaling advantage of this type was recently shown to be impossible in certain restricted settings. This is therefore an important piece of experimental evidence that PIMC does not simulate QA dynamics even for sign-problem-free Hamiltonians, and that near-term quantum devices can be used to accelerate computational tasks of practical relevance., Experimental demonstration of quantum speedup that scales with the system size is the goal of near-term quantum computing. Here, the authors demonstrate such scaling advantage for a D-Wave quantum annealer over analogous classical algorithms in simulations of frustrated quantum magnets.

Quantum simulation has emerged as a valuable arena for demonstrating and understanding the capabilities of near-term quantum computers. Quantum annealing has been used successfully in simulating a range of open quantum systems, both at equilibrium and out of equilibrium. However, in all previous experiments, annealing has been too slow to simulate a closed quantum system coherently, due to the onset of thermal effects from the environment. Here we demonstrate coherent evolution through a quantum phase transition in the paradigmatic setting of the 1D transverse-field Ising chain, using up to 2000 superconducting flux qubits in a programmable quantum annealer. In large systems we observe the quantum Kibble-Zurek mechanism with theoretically predicted kink statistics, as well as characteristic positive kink-kink correlations, independent of system temperature. In small chains, excitation statistics validate the picture of a Landau-Zener transition at a minimum gap. In both cases, results are in quantitative agreement with analytical solutions to the closed-system quantum model. For slower anneals we observe anti-Kibble-Zurek scaling in a crossover to the open quantum regime. These experiments demonstrate that large-scale quantum annealers can be operated coherently, paving the way to exploiting coherent dynamics in quantum optimization, machine learning, and simulation tasks.

Quantum annealing (QA) is a heuristic algorithm for finding low-energy configurations of a system, with applications in optimization, machine learning, and quantum simulation. Up to now, all implementations of QA have been limited to qubits coupled via a single degree of freedom. This gives rise to a stoquastic Hamiltonian that has no sign problem in quantum Monte Carlo (QMC) simulations. In this paper, we report implementation and measurements of two superconducting flux qubits coupled via two canonically conjugate degrees of freedom (charge and flux) to achieve a nonstoquastic Hamiltonian. Such coupling can enhance performance of QA processors, extend the range of quantum simulations. We perform microwave spectroscopy to extract circuit parameters and show that the charge coupling manifests itself as a YY interaction in the computational basis. We observe destructive interference in quantum coherent oscillations between the computational basis states of the two-qubit system. Finally, we show that the extracted Hamiltonian is nonstoquastic over a wide range of parameters., Comment: 15 pages, 12 figures

The celebrated work of Berezinskii, Kosterlitz and Thouless in the 1970s revealed exotic phases of matter governed by topological properties of low-dimensional materials such as thin films of superfluids and superconductors. Key to this phenomenon is the appearance and interaction of vortices and antivortices in an angular degree of freedom---typified by the classical XY model---due to thermal fluctuations. In the 2D Ising model this angular degree of freedom is absent in the classical case, but with the addition of a transverse field it can emerge from the interplay between frustration and quantum fluctuations. Consequently a Kosterlitz-Thouless (KT) phase transition has been predicted in the quantum system by theory and simulation. Here we demonstrate a large-scale quantum simulation of this phenomenon in a network of 1,800 in situ programmable superconducting flux qubits arranged in a fully-frustrated square-octagonal lattice. Essential to the critical behavior, we observe the emergence of a complex order parameter with continuous rotational symmetry, and the onset of quasi-long-range order as the system approaches a critical temperature. We use a simple but previously undemonstrated approach to statistical estimation with an annealing-based quantum processor, performing Monte Carlo sampling in a chain of reverse quantum annealing protocols. Observations are consistent with classical simulations across a range of Hamiltonian parameters. We anticipate that our approach of using a quantum processor as a programmable magnetic lattice will find widespread use in the simulation and development of exotic materials., 17 pages, 15 figures

Perturbative anticrossings have long been identified as a potential computational bottleneck for quantum annealing. This bottleneck can appear, for example, when a uniform transverse driver Hamiltonian is applied to each qubit. Previous theoretical research sought to alleviate such anticrossings by adjusting the transverse driver Hamiltonians on individual qubits according to a perturbative approximation. Here we apply this principle to a physical implementation of quantum annealing in a D-Wave 2000Q system. We use samples from the quantum annealing hardware and per-qubit anneal offsets to produce nonuniform driver Hamiltonians. On small instances with severe perturbative anticrossings, our algorithm yields an increase in minimum eigengaps, ground state success probabilities, and escape rates from metastable valleys. We also demonstrate that the same approach can mitigate biased sampling of degenerate ground states., 9 pages, 11 figures

We have developed a quantum annealing processor, based on an array of tunable coupled rf-SQUID flux qubits, fabricated in a superconducting integrated circuit process. Implementing this type of processor at a scale of 512 qubits and 1472 programmable interqubit couplers and operating at ~ 20 mK has required attention to a number of considerations that one may ignore at the smaller scale of a few dozen or so devices. Here, we discuss some of these considerations, and the delicate balance necessary for the construction of a practical processor that respects the demanding physical requirements imposed by a quantum algorithm. In particular, we will review some of the design tradeoffs at play in the floor planning of the physical layout, driven by the desire to have an algorithmically useful set of interqubit couplers, and the simultaneous need to embed programmable control circuitry into the processor fabric. In this context, we have developed a new ultralow-power embedded superconducting digital-to-analog flux converter (DAC) used to program the processor with zero static power dissipation, optimized to achieve maximum flux storage density per unit area. The 512 single-stage, 3520 two-stage, and 512 three-stage flux DACs are controlled with an XYZ addressing scheme requiring 56 wires. Our estimate of on-chip dissipated energy for worst-case reprogramming of the whole processor is ~ 65 fJ. Several chips based on this architecture have been fabricated and operated successfully at our facility, as well as two outside facilities (see, for example, the recent reporting by Jones).

Both simulated quantum annealing and physical quantum annealing have shown the emergence of "heavy tails" in their performance as optimizers: The total time needed to solve a set of random input instances is dominated by a small number of very hard instances. Classical simulated annealing, in contrast, does not show such heavy tails. Here we explore the origin of these heavy tails, which appear for inputs with high local degeneracy---large isoenergetic clusters of states in Hamming space. This category includes the low-precision Chimera-structured problems studied in recent benchmarking work comparing the D-Wave Two quantum annealing processor with simulated annealing. On similar inputs designed to suppress local degeneracy, performance of a quantum annealing processor on hard instances improves by orders of magnitude at the 512-qubit scale, while classical performance remains relatively unchanged. Simulations indicate that perturbative crossings are the primary factor contributing to these heavy tails, while sensitivity to Hamiltonian misspecification error plays a less significant role in this particular setting., 14 pages. Corrected annealing schedule and dependent simulations

We have studied different quantum dynamics of superconducting nano-circuits with Josephson junctions. A dc SQUID, when it is strongly decoupled from the environment, demonstrates two-level and multilevel dynamics. We have realized a two qubits coupled circuit based on a dc SQUID in parallel with an asymmetric Cooper pair transistor (ACPT). The ACPT behaves as a charge qubit. Its asymmetry produces a strong tunable coupling with the dc SQUID which is used to realize entangled states between the two qubits and new read-out of the charge qubit based on adiabatic quantum transfer. We have measured the current---phase relations of different rhombi chains in the presence or absence of quantum fluctuations which confirm theoretical predictions.

To understand the origins of synchronous and asynchronous phase slippages observed in an array of apertures connecting two reservoirs of superfluid 4He, we have investigated the role of thermal fluctuations in the critical velocity and the possible effects of having an array rather than a single aperture through several model simulations. The results are compared with recent experiments carried out near the superfluid transition temperature with an array of apertures as well as those carried out at low temperatures with a single aperture.

Propagation of acoustic waves in liquid media containing many air-filled bubbles is studied using a self-consistent approach. It is shown that under proper conditions, multiple scattering leads to a peculiar phase transition in acoustic propagation. When the phase transition occurs, not only the acoustic waves are confined in the neighborhood of the transmitting source, but a previously unsuspected collective behavior of the air bubbles appears, responsible for effective cancellation of the propagating wave. A novel phase diagram method is used to depict the phase transition.

This sequential paper describes a comparison between the simple closed solution proposed in a previous paper [, J. Acoust. Soc. Am. 101, 1945–1952 (1997)] with the exact solution for scattering of acoustic waves by gas-filled prolate spheroids at low frequencies. The analysis and comparison show that the low-frequency acoustic scattering by prolate spheroid gas bubbles has resonance features, and can be rather accurately described by the proposed simple formula. It is shown that the results from the simple analytic solution agree remarkably well with those obtained by the exact solution.

This paper presents a new approach for calculating sound scattering by slender scatterers. The method is based on the assumption that the sound scattering function can be approximated by superimposing the scattered waves from a series of small adjacent cylindrical elements, with each of these elements scattering sound as though it were part of an infinite straight cylinder of the same radius. Using the Kirchhoff integral theorem, the bistatic scattering function is derived rigorously within the accuracy of the assumption. To validate the approach, the solution is applied to sound scattering by fluid prolate spheroids, for which the exact solution is known. Numerical comparison with the exact solution reveals that the new approach is very versatile; it can predict scattering patterns remarkably comparable to that obtained by the exact solution for a wide range of incident angles. For incidence within approximately ±20° of the broadside direction, the scattering patterns predicted by the new method coincide uniformly with the exact results. Agreement in the absolute scattering amplitude slowly deteriorates as the incidence is tilted away from the broadside aspect. However, the method consistently produces scattering patterns highly correlated to the results from the exact solution; this is potentially useful for target identification based on pattern recognition techniques. The results also show that the new method works better on the whole as the elongation of the target increases. Furthermore, calculation of cross-correlation coefficients indicates that the present method works slightly better for rigid bodies than for weak fluid bodies. The low-frequency scattering by gas-filled spheroids is also considered. It is shown that near resonance the scattering is almost isotropic, in agreement with the prediction of previous simple analytic solution.

Efforts to develop useful quantum computers have been blocked primarily by environmental noise. Quantum annealing is a scheme of quantum computation that is predicted to be more robust against noise, because despite the thermal environment mixing the system's state in the energy basis, the system partially retains coherence in the computational basis, and hence is able to establish well-defined eigenstates. Here we examine the environment's effect on quantum annealing using 16 qubits of a superconducting quantum processor. For a problem instance with an isolated small-gap anticrossing between the lowest two energy levels, we experimentally demonstrate that, even with annealing times eight orders of magnitude longer than the predicted single-qubit decoherence time, the probabilities of performing a successful computation are similar to those expected for a fully coherent system. Moreover, for the problem studied, we show that quantum annealing can take advantage of a thermal environment to achieve a speedup factor of up to 1,000 over a closed system.

We describe a quantum tunneling spectroscopy technique that requires only low-bandwidth control. The method involves coupling a probe qubit to the system under study to create a localized probe state. The energy of the probe state is then scanned with respect to the unperturbed energy levels of the probed system. Incoherent tunneling transitions that flip the state of the probe qubit occur when the energy bias of the probe is close to an eigenenergy of the probed system. Monitoring these transitions allows the reconstruction of the probed system eigenspectrum. We demonstrate this method on an rf SQUID flux qubit.

A superconducting flux qubit is inductively coupled to a Superconducting QUantum Interference Device (SQUID) magnetometer, capacitively shunted to form a 1.294-GHz resonator. The qubit-state-dependent resonator frequency is weakly probed with a microwave signal and detected with a Microstrip SQUID Amplifier. At a mean resonator occupation $\bar{n}$ = 1.5 photons, the readout visibility is increased by a factor of 4.5 over that using a cryogenic semiconductor amplifier. As $\bar{n}$ is increased from 0.008 to 0.1, no reduction in $T_1$ is observed, potentially enabling continuous monitoring of the qubit state., Comment: 5 pages, 5 figures

4; International audience; We investigate a quadratic-quartic anharmonic oscillator formed by a potential well between two potential barriers. We realize this novel potential shape with a superconducting circuit comprised of a loop interrupted by two Josephson junctions (dc SQUID), with near-zero current bias and flux bias near half a flux quantum. We investigate escape out of the central well, which can occur via tunneling through either of the two barriers, and find good agreement with a generalized double-path macroscopic quantum tunneling theory. We also demonstrate that this system exhibits an ''optimal line'' in current and flux bias space along which the oscillator, which can be operated as a phase qubit, is insensitive to decoherence due to low-frequency current fluctuations.

This article presents our procedure to measure the quantum state of a dc SQUID within a few nanoseconds, using an adiabatic dc flux pulse. Detection of the ground state is governed by standard macroscopic quantum theory (MQT), with a small correction due to residual noise in the bias current. In the two level limit, where the SQUID constitutes a phase qubit, an observed contrast of 0.54 indicates a significant loss in contrast compared to the MQT prediction. It is attributed to spurious depolarization (loss of excited state occupancy) during the leading edge of the adiabatic flux measurement pulse. We give a simple phenomenological relaxation model which is able to predict the observed contrast of multilevel Rabi oscillations for various microwave amplitudes., 10 pages, 8 figures

We have realized a tunable coupling over a large frequency range between an asymmetric Cooper pair transistor (charge qubit) and a dc SQUID (phase qubit). Our circuit enables the independent manipulation of the quantum states of each qubit as well as their entanglement. The measurements of the charge qubit's quantum states is performed by resonant read-out via the measurement of the quantum states of the SQUID. The measured coupling strength is in agreement with an analytic theory including a capacitive and a tunable Josephson coupling between the two qubits., Comment: 5 pages

We report the observation of quantum interference in superfluid $^{4}\mathrm{He}$. The interferometer, an analog of a dc-superconducting quantum interference device (SQUID), employs a recently reported phenomenon wherein superfluid $^{4}\mathrm{He}$ exhibits Josephson frequency oscillations in an array of submicron apertures. An interference pattern is generated by reorienting the loop of the superfluid ``SQUID'' with respect to the Earth's rotation vector, thereby varying the rotation flux in the loop. The experiment is performed at $2\phantom{\rule{0.2em}{0ex}}\mathrm{K}$, a temperature 2000 times higher than previously achieved with superfluid $^{3}\mathrm{He}$. We find that the interference exists not only when the aperture array current-phase relation is a sinusoidal function characteristic of the Josephson effect, but also at lower temperatures where it is linear and oscillations occur by phase slips. The modest requirements for the interferometer ($2\phantom{\rule{0.2em}{0ex}}\mathrm{K}$ cryogenics and fabrication of apertures at the level of $100\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$) and its potential resolution suggest that, when engineering challenges such as vibration isolation are met, superfluid $^{4}\mathrm{He}$ interferometers could become important scientific probes.

Studies of superfluid 4He weak‐links require calibration constants which permit the determination of the pressure and temperature differences which drive Josephson oscillations. We describe a technique for calibrating 4He weak‐link cells in which a heater is used to induce fountain pressures detected by the deflection of a diaphragm. The technique determines the diaphragm spring constant, the inner cell volume, and the thermal conductance of the inner cells walls. This information is used to convert the measured deflection of the diaphragm into the total chemical potential difference across the weak link.

Research and development of superfluid weak links has been hindered by the absence of a source of dc chemical potential, similar to a simple battery or voltage source for analogous superconducting devices. We describe here a method for generating a dc chemical potential difference, Δμ across a weak link array in superfluid 4He. The presence of a Δμ forces quantum oscillations at a Josephson frequency, selectable by the adjustment of input power to a heater. We discuss a case in which the frequency locks onto a resonance feature where it exhibits remarkable stability, and amplitude magnification by a factor of 40.

We have investigated the dynamics of superfluid phase slippage in an array of apertures. The magnitude of the dissipative phase slips shows that they occur simultaneously in all the apertures when the temperature is around 10 mK below the superfluid transition, and subsequently lose their simultaneity as the temperature is lowered. We find that when periodic synchronous phase slippage occurs, the synchronicity exists from the very first phase slip, and therefore is not due to mode locking of interacting oscillators. When the system is allowed to relax freely from a given initial energy, the total number of phase slips that occur and the energy left in the system after the last phase slip depends reproducibly on the initial energy. We find the energy remaining after the final phase slip is a periodic function of the initial system energy. This dependence directly reveals the discrete and dissipative nature of the phase slips and is a powerful diagnostic for investigation of synchronicity in the array. When the array slips synchronously, this periodic energy function is a sharp sawtooth. As the temperature is lowered and the degree of synchronicity drops, the peak of this sawtooth becomes rounded, suggesting a broadening of the time interval over which the array slips. The underlying mechanism for the higher temperature synchronous behavior and the following loss of synchronicity at lower temperatures is not yet understood. We discuss the implications of our measurements and pose several questions that need to be resolved by a theory explaining the synchronous behavior in this quantum system. An understanding of the array phase slip process is essential to the optimization of superfluid `dc-SQUID' gyroscopes and interferometers., Comment: 10 pages, 4 figures

We find that a temperature differential can drive superfluid oscillations in 4He. The oscillations are excited by a heater which causes a time dependent temperature differential across an array of 70 nm apertures. By measuring the oscillation frequency and simultaneously determining both temperature and pressure differentials we prove the validity of the most general form of the Josephson frequency relation. These observations were made near saturated vapor pressure, within a few mK of the superfluid transition temperature.

Celebrated interference experiments have demonstrated the wave nature of light and electrons, quantum interference being the manifestation of wave-particle duality. More recently, double-path interference experiments have also demonstrated the quantum-wave nature of beams of neutrons, atoms and Bose-Einstein condensates. In condensed matter systems, double-path quantum interference is observed in the d.c. superconducting quantum interference device (d.c. SQUID). Here we report a double-path quantum interference experiment involving a liquid: superfluid 3He. Using a geometry analogous to the superconducting d.c. SQUID, we control a quantum phase shift by using the rotation of the Earth, and find the classic interference pattern with periodicity determined by the 3He quantum of circulation.

Multiple scattering of waves leads to many peculiar phenomena such as complete band gaps in periodic structures and wave localization in disordered media. Within a band gap excitations are evanescent; when localized they remain confined in space until dissipated. Here we report acoustic band gap and localization in a 2D system of air-cylinders in water. Exact numerical calculations reveal the unexpected result that localization is relatively independent of the precise location or organization of the scatterers. Localization occurs within a finite region of frequencies, coincident with the complete band gap predicted by a conventional band structure calculation for a periodic lattice of scatterers. Inside the gap or localization regime, a previously uninvestigated stable collective behavior of the cylinders appears., Comment: 10 pages, 4 figures

Acoustic wave propagation in liquid media containing many parallel air-filled cylinders is considered. A self-consistent method is used to compute rigorously the propagation, incorporating all orders of multiple scattering. It is shown that under proper conditions, multiple scattering leads to a peculiar phase transition in acoustic propagation. When the phase transition occurs, a collective behavior of the cylinders appears and the acoustic waves are confined in a region of space in the neighborhood of the transmission source. A novel phase diagram is used to describe such phase transition. Originally submitted on April 6, 99., 5 pages, 5 color figures

A band‐structure calculation for a periodic array of scatterers reveals band gaps—bands of frequency outside of which propagation is relatively unhindered, inside of which waves are evanescent and cannot propagate away from their source. In the case of an array of resonant isotropic 2‐D acoustic scatterers, a direct, exact, self‐consistent technique incorporating all orders of multiple scattering can be used to calculate the scattered field due to a given source. This can be done for the periodic case where the band‐structure calculation is relevant: a region of frequencies is found in which the scattered field is localized around the source, and this region coincides with the calculated band gap. The self‐consistent technique has the advantage that it can also be used for an arbitrary random distribution of scatterers. Surprisingly, the localization region proves relatively independent of the scatterer distribution. Further investigation reveals that as the frequency is brought within the localization re...

Multiple scattering of waves may lead to many peculiar phenomena such as complete band gaps in periodic structures and wave localization in disordered media. Within the gaps excitations are evanescent, when localized waves remain confined in space until dissipated. Here we show that waves are not always localized in 2‐D, but when localized there is a coincidence between the gaps in periodic structures and the ranges of localized states in the case of acoustic propagation through water containing an array of air‐cylinders. We report that inside the gap or localization regime, an interesting collective behavior of the cylinders appears. Exact numerical calculations reveal the unexpected result that localization is relatively independent of the precise location or organization of the scatterers.

Wave localization is the peculiar property of a scattering medium that completely blocks propagation, thus inducing a surprising phase transition, for example in optical or acoustic transparency or electrical conductivity. When localized, waves remain confined in space until dissipated. This property has been explained for randomly disordered scatterers. It is generally believed that disorder is an essential ingredient of localization, and in two dimensions (2D) waves are always localized with any amount of randomness. Here it is shown that waves are not always localized in 2D, but under proper conditions even periodic structures can achieve localization in the case of acoustic propagation through water containing an array of air cylinders. An essential component of localization in this case arises from the natural resonance and an amazing collective behavior of the cylinders. Exact numerical calculations reveal the unexpected result that localization is relatively independent of the precise location or organization of the scatterers. The results provide insight on 2D wave localization and offer an opportunity for further investigation of important questions about localization phenomena. [Work supported by NSC and NCU of ROC.]