Search

Scalable quantum computing can become a reality with error correction, provided coherent qubits can be constructed in large arrays. The key premise is that physical errors can remain both small and sufficiently uncorrelated as devices scale, so that logical error rates can be exponentially suppressed. However, energetic impacts from cosmic rays and latent radioactivity violate both of these assumptions. An impinging particle ionizes the substrate, radiating high energy phonons that induce a burst of quasiparticles, destroying qubit coherence throughout the device. High-energy radiation has been identified as a source of error in pilot superconducting quantum devices, but lacking a measurement technique able to resolve a single event in detail, the effect on large scale algorithms and error correction in particular remains an open question. Elucidating the physics involved requires operating large numbers of qubits at the same rapid timescales as in error correction, exposing the event's evolution in time and spread in space. Here, we directly observe high-energy rays impacting a large-scale quantum processor. We introduce a rapid space and time-multiplexed measurement method and identify large bursts of quasiparticles that simultaneously and severely limit the energy coherence of all qubits, causing chip-wide failure. We track the events from their initial localised impact to high error rates across the chip. Our results provide direct insights into the scale and dynamics of these damaging error bursts in large-scale devices, and highlight the necessity of mitigation to enable quantum computing to scale.

Quantum many-body systems display rich phase structure in their low-temperature equilibrium states1. However, much of nature is not in thermal equilibrium. Remarkably, it was recently predicted that out-of-equilibrium systems can exhibit novel dynamical phases2–8 that may otherwise be forbidden by equilibrium thermodynamics, a paradigmatic example being the discrete time crystal (DTC)7,9–15. Concretely, dynamical phases can be defined in periodically driven many-body-localized (MBL) systems via the concept of eigenstate order7,16,17. In eigenstate-ordered MBL phases, the entire many-body spectrum exhibits quantum correlations and long-range order, with characteristic signatures in late-time dynamics from all initial states. It is, however, challenging to experimentally distinguish such stable phases from transient phenomena, or from regimes in which the dynamics of a few select states can mask typical behaviour. Here we implement tunable controlled-phase (CPHASE) gates on an array of superconducting qubits to experimentally observe an MBL-DTC and demonstrate its characteristic spatiotemporal response for generic initial states7,9,10. Our work employs a time-reversal protocol to quantify the impact of external decoherence, and leverages quantum typicality to circumvent the exponential cost of densely sampling the eigenspectrum. Furthermore, we locate the phase transition out of the DTC with an experimental finite-size analysis. These results establish a scalable approach to studying non-equilibrium phases of matter on quantum processors., A study establishes a scalable approach to engineer and characterize a many-body-localized discrete time crystal phase on a superconducting quantum processor.

Interactions in quantum systems can spread initially localized quantum information into the exponentially many degrees of freedom of the entire system. Understanding this process, known as quantum scrambling, is key to resolving several open questions in physics. Here, by measuring the time-dependent evolution and fluctuation of out-of-time-order correlators, we experimentally investigate the dynamics of quantum scrambling on a 53-qubit quantum processor. We engineer quantum circuits that distinguish operator spreading and operator entanglement and experimentally observe their respective signatures. We show that whereas operator spreading is captured by an efficient classical model, operator entanglement in idealized circuits requires exponentially scaled computational resources to simulate. These results open the path to studying complex and practically relevant physical observables with near-term quantum processors.

Realizing the potential of quantum computing requires sufficiently low logical error rates1. Many applications call for error rates as low as 10−15 (refs. 2–9), but state-of-the-art quantum platforms typically have physical error rates near 10−3 (refs. 10–14). Quantum error correction15–17 promises to bridge this divide by distributing quantum logical information across many physical qubits in such a way that errors can be detected and corrected. Errors on the encoded logical qubit state can be exponentially suppressed as the number of physical qubits grows, provided that the physical error rates are below a certain threshold and stable over the course of a computation. Here we implement one-dimensional repetition codes embedded in a two-dimensional grid of superconducting qubits that demonstrate exponential suppression of bit-flip or phase-flip errors, reducing logical error per round more than 100-fold when increasing the number of qubits from 5 to 21. Crucially, this error suppression is stable over 50 rounds of error correction. We also introduce a method for analysing error correlations with high precision, allowing us to characterize error locality while performing quantum error correction. Finally, we perform error detection with a small logical qubit using the 2D surface code on the same device18,19 and show that the results from both one- and two-dimensional codes agree with numerical simulations that use a simple depolarizing error model. These experimental demonstrations provide a foundation for building a scalable fault-tolerant quantum computer with superconducting qubits., Repetition codes running many cycles of quantum error correction achieve exponential suppression of errors with increasing numbers of qubits.

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

This paper studies a model with multifractal wave functions and fractal mini-bands which allows for analytical derivation of its phase diagram

A promising approach to study condensed-matter systems is to simulate them on an engineered quantum platform1–4. However, the accuracy needed to outperform classical methods has not been achieved so far. Here, using 18 superconducting qubits, we provide an experimental blueprint for an accurate condensed-matter simulator and demonstrate how to investigate 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 measure the energy eigenvalues of this wire with an error of approximately 0.01 rad, whereas typical energy scales are of the order of 1 rad. Insight into the fidelity of this algorithm is gained by highlighting the robust properties of a Fourier transform, including the ability to resolve eigenenergies with a statistical uncertainty of 10−4 rad. We also synthesize magnetic flux and disordered local potentials, which are two key tenets of a condensed-matter system. When sweeping the magnetic flux we observe avoided level crossings in the spectrum, providing a detailed fingerprint of the spatial distribution of local disorder. By combining these methods we reconstruct electronic properties of the eigenstates, observing persistent currents and a strong suppression of conductance with added disorder. Our work describes an accurate method for quantum simulation5,6 and paves the way to study new quantum materials with superconducting qubits. As a blueprint for high-precision quantum simulation, an 18-qubit algorithm that consists of more than 1,400 two-qubit gates is demonstrated, and reconstructs the energy eigenvalues of the simulated one-dimensional wire to a precision of 1 per cent.

Disorder has different profound effects on superconducting thin films. For a large variety of materials, increasing disorder reduces electronic screening which enhances electron-electron repulsion. These fermionic effects lead to a mechanism described by Finkelstein: when disorder combined to electron-electron interactions increases, there is a global decrease of the superconducting energy gap $\Delta$ and of the critical temperature $T_c$, the ratio $\Delta$/$k_BT_c$ remaining roughly constant. In addition, in most films an emergent granularity develops with increasing disorder and results in the formation of inhomogeneous superconducting puddles. These gap inhomogeneities are usually accompanied by the development of bosonic features: a pseudogap develops above the critical temperature $T_c$ and the energy gap $\Delta$ starts decoupling from $T_c$. Thus the mechanism(s) driving the appearance of these gap inhomogeneities could result from a complicated interplay between fermionic and bosonic effects. By studying the local electronic properties of a NbN film with scanning tunneling spectroscopy (STS) we show that the inhomogeneous spatial distribution of $\Delta$ is locally strongly correlated to a large depletion in the local density of states (LDOS) around the Fermi level, associated to the Altshuler-Aronov effect induced by strong electronic interactions. By modelling quantitatively the measured LDOS suppression, we show that the latter can be interpreted as local variations of the film resistivity. This local change in resistivity leads to a local variation of $\Delta$ through a local Finkelstein mechanism. Our analysis furnishes a purely fermionic scenario explaining quantitatively the emergent superconducting inhomogeneities, while the precise origin of the latter remained unclear up to now., Comment: 11 pages, 4 figures

A major issue for the implementation of large scale superconducting quantum circuits is the interaction with interfacial two-level system defects (TLS) that leads to qubit relaxation and impedes qubit operation in certain frequency ranges that also drift in time. Another major challenge comes from non-equilibrium quasiparticles (QPs) that result in qubit dephasing and relaxation. In this work we show that such QPs can also serve as a source of TLS. Using spectral and temporal mapping of TLS-induced fluctuations in frequency tunable resonators, we identify a subset of the general TLS population that are highly coherent TLS with a low reconfiguration temperature $\sim$ 300 mK, and a non-uniform density of states. These properties can be understood if these TLS are formed by QPs trapped in shallow subgap states formed by spatial fluctutations of the superconducting order parameter $\Delta$. Magnetic field measurements of one such TLS reveals a link to superconductivity. Our results imply that trapped QPs can induce qubit relaxation., Comment: 7 pages, 4 figures, plus 18 pages, 13 figures supplemental

We propose a novel platform for the study of quantum phase transitions in one dimension (1D QPT). The system consists of a specially designed chain of asymmetric SQUIDs; each SQUID contains several Josephson junctions with one junction shared between the nearest-neighbor SQUIDs.We develop the theoretical description of the low energy part of the spectrum. In particular, we show that the system exhibits the quantum phase transition of Ising type. In the vicinity of the transition the low energy excitations of the system can be described by Majorana fermions. This allow us to compute the matrix elements of the physical perturbations in the low energy sector. In the microwave experiments with this system, we explored the phase boundaries between the ordered and disordered phases and the critical behavior of the system's low-energy modes close to the transition. Due to the flexible chain design and control of the parameters of individual Josephson junctions, future experiments will be able to address the effects of non-integrability and disorder on the 1D QPT., Comment: 12 pages, 7 figures

In recent years qubit designs such as transmons approached the fidelities of up to 0.999. However, even these devices are still insufficient for realizing quantum error correction requiring better than 0.9999 fidelity. Topologically protected superconducting qubits are arguably most prospective for building a realistic quantum computer as they are intrinsically protected from noise and leakage errors that occur in transmons. We propose a topologically protected qubit design based on a $\pi$-periodic Josephson element and a universal set of gates: protected Clifford group and highly robust (with infidelity $\sim 10^{-4}$) non-discrete holonomic phase gate. The qubit is controlled via charge($Q$) and flux($\Phi$)-biases. The holonomic gate is realized by quickly, but adiabatically, going along a particular closed path in the two-dimensional $\{\Phi,Q\}$-space -- a path where computational states are always degenerate, but Berry curvature is localized inside the path. This gate is robust against currently achievable noise levels. This qubit architecture allows building a realistic scalable superconducting quantum computer with leakage and noise-induced errors as low as $10^{-4}$, which allows performing realistic error correction codes with currently available fabrication techniques., Comment: 13 pages, 4 figures

Strongly disordered pseudogapped superconductors are expected to display arbitrarily high values of kinetic inductance close to the superconductor-insulator transition (SIT), which make them attractive for the implementation of large dissipationless inductance. We develop the theory of the collective modes in these superconductors and discuss associated dissipation at microwave frequencies. We obtain the collective mode spectra dependence on the disorder level and conclude that collective modes become a relevant source of dissipation and noise in the outer proximity of the SIT.

10 pags., 9 figs., 1 tab., We report the results of the numerical study of the nondissipative quantum Josephson junction chain with the focus on the statistics of many-body wave functions and local energy spectra. The disorder in this chain is due to the random offset charges. This chain is one of the simplest physical systems to study many-body localization. We show that the system may exhibit three distinct regimes: insulating, characterized by the full localization of many-body wave functions, a fully delocalized (metallic) one characterized by the wave functions that take all the available phase volume, and the intermediate regime in which the volume taken by the wave function scales as a nontrivial power of the full Hilbert-space volume. In the intermediate nonergodic regime the Thouless conductance (generalized to the many-body problem) does not change as a function of the chain length indicating a failure of the conventional single-parameter scaling theory of localization transition. The local spectra in this regime display the fractal structure in the energy space which is related with the fractal structure of wave functions in the Hilbert space. A simple theory of fractality of local spectra is proposed, and a scaling relationship between fractal dimensions in the Hilbert and energy spaces is suggested and numerically tested., This work was supported by ARO Grant No. W911NF- 13-1-0431 and Russian Science Foundation Grant No. 14-42- 00044. M.P. acknowledges support from Juan de la Cierva Grant No. IJCI-2015-23260, MINECO/FEDER Project No. FIS2015-70856-P, CAM PRICYT Project No. QUITEMAD+ S2013/ICE-2801, and Proyecto de la Fundacion Seneca Grant No. 19907/GERM/15.

The concept of non-ergodicity in quantum many body systems can be discussed in the context of the wave functions of the many body system or as a property of the dynamical observables, such as time-dependent spin correlators. In the former approach the non-ergodic delocalized states is defined as the one in which the wave functions occupy a volume that scales as a non-trivial power of the full phase space. In this work we study the simplest spin glass model and find that in the delocalized non-ergodic regime the spin-spin correlators decay with the characteristic time that scales as non-trivial power of the full Hilbert space volume. The long time limit of this correlator also scales as a power of the full Hilbert space volume. We identify this phase with the glass phase whilst the many body localized phase corresponds to a 'hyperglass' in which dynamics is practically absent. We discuss the implications of these finding to quantum information problems., Changes with respect to the first version (Dec. 2018): titles is slightly changed, abstract is extended, discussion of previous papers on similar subject is considerably augmented

In this Letter we introduce a chopped Josephson junction array as a tractable short-ranged frustrated model of a glass without quenched disorder. We study its configurational entropy and conclude that there is no true thermodynamic phase transition at finite temperature.

We present an analytic enumeration of the metastable states, $N_s$, in a periodic long-range Josephson array frustrated by a transverse field. We find that the configurational entropy, $S_{conf} \equiv \ln N_s$, is extensive and scales with frustration, confirming that the non-random system is glassy. We also find that $S_{conf}$ is different from that of its disordered analogue, despite that fact that the two models share the same dynamical equations.

We discuss a general formalism that allows study of transitions over barriers in spin glasses with long-range interactions that contain a large but finite number $N$ of spins. We apply this formalism to the Sherrington-Kirkpatrick model with finite $N$ and derive equations for the dynamical order parameters which allow ``instanton'' solutions describing transitions over the barriers separating metastable states. Specifically, we study these equations for a glass state that was obtained in a slow cooling process ending a little below ${T}_{c}$ and show that these equations allow ``instanton'' solutions which erase the response of the glass to the perturbations applied during the slow cooling process. The corresponding action of these solutions gives the energy of the barriers, we find that it scales as ${\ensuremath{\tau}}^{6}$ where $\ensuremath{\tau}$ is the reduced temperature.

We have observed the effect of the Aharonov-Casher (AC) interference on the spectrum of a superconducting system containing a symmetric Cooper pair box (CPB) and a large inductance. By varying the charge $n_{g}$ induced on the CPB island, we observed oscillations of the device spectrum with the period $\Delta n_{g}=2e$. These oscillations are attributed to the charge-controlled AC interference between the fluxon tunneling processes in the CPB Josephson junctions. Total suppression of the tunneling (complete destructive interference) has been observed for the charge $n_{g}=e(2n+1)$. The CPB in this regime represents the $4\pi$-periodic Josephson element, which can be used for the development of the parity-protected superconducting qubits., Comment: 6 pages, 3 figures

International audience; We extend the Keldysh technique to enable the computation of out-of-time order correlators such as 〈O(t)Õ(0)O(t)Õ(0)〉 . We show that the behavior of these correlators is described by equations that display initially an exponential instability which is followed by a linear propagation of the decoherence between two initially identically copies of the quantum many body systems with interactions. At large times the decoherence propagation (quantum butterfly effect) is described by a diffusion equation with non-linear dissipation known in the theory of combustion waves. The solution of this equation is a propagating non-linear wave moving with constant velocity despite the diffusive character of the underlying dynamics.Our general conclusions are illustrated by the detailed computations for the specific models describing the electrons interacting with bosonic degrees of freedom (phonons, two-level-systems etc.) or with each other.

We combine numerical diagonalization with a semi-analytical calculations to prove the existence of the intermediate non-ergodic but delocalized phase in the Anderson model on disordered hierarchical lattices. We suggest a new generalized population dynamics that is able to detect the violation of ergodicity of the delocalized states within the Abou-Chakra, Anderson and Thouless recursive scheme. This result is supplemented by statistics of random wave functions extracted from exact diagonalization of the Anderson model on ensemble of disordered Random Regular Graphs (RRG) of N sites with the connectivity K=2. By extrapolation of the results of both approaches to N->infinity we obtain the fractal dimensions D_{1}(W) and D_{2}(W) as well as the population dynamic exponent D(W) with the accuracy sufficient to claim that they are non-trivial in the broad interval of disorder strength W_{E}10^{5} reveals a singularity in D_{1,2}(W)-dependencies which provides a clear evidence for the first order transition between the two delocalized phases on RRG at W_{E}\approx 10.0. We discuss the implications of these results for quantum and classical non-integrable and many-body systems., Comment: 4 pages paper with 5 figures + Supplementary Material with 5 figures

The search for Majorana bound states in solid-state physics has been limited to materials that display a gap in their bulk spectrum. We show that such unpaired states appear in certain quasi-one-dimensional Josephson-junction arrays with gapless bulk excitations. The bulk modes mediate a coupling between Majorana bound states via the Ruderman-Kittel-Yosida-Kasuya mechanism. As a consequence, the lowest energy doublet acquires a finite energy difference. For a realistic set of parameters this energy splitting remains much smaller than the energy of the bulk eigenstates even for short chains of length L∼10.

We develop a semi-quantitative theory of electron pairing and resulting superconductivity in bulk "poor conductors" in which Fermi energy $E_F$ is located in the region of localized states not so far from the Anderson mobility edge $E_c$. We review the existing theories and experimental data and argue that a large class of disordered films is described by this model. Our theoretical analysis is based on the analytical treatment of pairing correlations, described in the basis of the exact single-particle eigenstates of the 3D Anderson model, which we combine with numerical data on eigenfunction correlations. Fractal nature of critical wavefunction's correlations is shown to be crucial for the physics of these systems. We identify three distinct phases: 'critical' superconductive state formed at $E_F=E_c$, superconducting state with a strong pseudogap, realized due to pairing of weakly localized electrons and insulating state realized at $E_F$ still deeper inside localized band. The 'critical' superconducting phase is characterized by the enhancement of the transition temperature with respect to BCS result, by the inhomogeneous spatial distribution of superconductive order parameter and local density of states. The major new feature of the pseudo-gaped state is the presence of two independent energy scales: superconducting gap $\Delta$, that is due to many-body correlations and a new "pseudogap" energy scale $\Delta_P$ which characterizes typical binding energy of localized electron pairs and leads to the insulating behavior of the resistivity as a function of temperature above superconductive $T_c$. Two gap nature of the "pseudo-gaped superconductor" is shown to lead to a number of unusual physical properties., Comment: 110 pages, 39 figures. The revised version corrects a number of typos, adds references and discussion of recent results

We analyze the critical behavior of the superfluid density ${\ensuremath{\rho}}_{s}$ in strongly disordered superconductors near the superconductor-insulator transition and compare it with the behavior of the spectral gap $\mathrm{\ensuremath{\Delta}}$ for collective excitations. We show that, in contrast to conventional superconductors, the superconductors with preformed pairs display an unusual scaling relation ${\ensuremath{\rho}}_{s}\ensuremath{\propto}{\mathrm{\ensuremath{\Delta}}}^{2}$ close to the superconductor-insulator transition. This relation has been reported in very recent experiments.

We use a tunneling model for two-level systems in insulators that takes into account the interaction between them and a slow power-law dependence of their density of states. We show that the predictions of this model are in a perfect agreement with the recent studies of the noise in high quality superconducting resonators. The predictions also agree with the temperature dependence of the TLS dephasing rates observed in phase qubits. Both observations are impossible to explain in the framework of the standard tunneling model of TLS. We discuss the origin of the universal dimensionless parameter that controls the interaction between TLS in glasses and show that it is consistent with the assumptions of the model.

Strictly speaking, the laws of the conventional statistical physics, based on the equipartition postulate [Gibbs J W (1902) Elementary Principles in Statistical Mechanics, developed with especial reference to the rational foundation of thermodynamics] and ergodicity hypothesis [Boltzmann L (1964) Lectures on Gas Theory], apply only in the presence of a heat bath. Until recently this restriction was believed to be not important for real physical systems because a weak coupling to the bath was assumed to be sufficient. However, this belief was not examined seriously until recently when the progress in both quantum gases and solid-state coherent quantum devices allowed one to study the systems with dramatically reduced coupling to the bath. To describe such systems properly one should revisit the very foundations of statistical mechanics. We examine this general problem for the case of the Josephson junction chain that can be implemented in the laboratory and show that it displays a novel high-temperature nonergodic phase with finite resistance. With further increase of the temperature the system undergoes a transition to the fully localized state characterized by infinite resistance and exponentially long relaxation.

We review recent results on a new class of Josephson arrays which have nontrivial topology and exhibit novel quantum states at low temperatures. One of these states is characterized by long-range order in a two-Cooper-pair condensate and by a discrete topological order parameter. The second state is insulating and can be considered as being the result of an evolution of the former state due to Bose-condensation of usual superconductive vortices with a flux quantum Φ0. The quantum phase transition between these two states is controlled by variation of the external magnetic field. Both the superconductive and insulating states are characterized by the presence of 2K-degenerate ground states, with K being the number of topologically different cycles existing in the plane of the array. This degeneracy is “protected” from the external perturbations (and noise) by the topological order parameter and spectral gap. We show that under ideal conditions the low-order effect of the external perturbations on this dege...

All physical implementations of quantum bits (qubits), carrying the information and computation in a putative quantum computer, have to meet the conflicting requirements of environmental decoupling while remaining manipulable through designed external signals. Proposals based on quantum optics naturally emphasize the aspect of optimal isolation, while those following the solid state route exploit the variability and scalability of modern nanoscale fabrication techniques. Recently, various designs using superconducting structures have been successfully tested for quantum coherent operation, however, the ultimate goal of reaching coherent evolution over thousands of elementary operations remains a formidable task. Protecting qubits from decoherence by exploiting topological stability, a qualitatively new proposal due to Kitaev, holds the promise for long decoherence times, but its practical physical implementation has remained unclear so far. Here, we show how strongly correlated systems developing an isolated two-fold degenerate quantum dimer liquid groundstate can be used in the construction of topologically stable qubits and discuss their implementation using Josephson junction arrays., 6 pages, 4 figures

In bulk materials superconductivity is remarkably robust with respect to non-magnetic disorder. In the two-dimensional limit however, the quantum condensate suffers from the effects produced by disorder and electron correlations which both tend to destroy superconductivity . The recent discovery of superconductivity in single atomic layers of Pb, the striped incommensurate (SIC) and $\sqrt{7}\times\sqrt{3}$ Pb/Si(111), opened an unique opportunity to probe the influence of well-identified structural disorder on two-dimensional superconductivity at the atomic and mesoscopic scale. In these two ultimate condensates we reveal how the superconducting spectra loose their conventional character, by mapping the local tunneling density of states. We report variations of the spectral properties even at scales significantly shorter than the coherence length. Furthermore, fine structural differences between the two monolayers, such as their atomic density, lead to very different superconducting behaviour. The denser SIC remains globally robust to disorder, as are thicker Pb films, whereas in the slighly more diluted $\sqrt{7}\times\sqrt{3}$ system superconductivity is strongly fragilized. A consequence of this weakness is revealed at monoatomic steps of $\sqrt{7}\times\sqrt{3}$, which disrupt superconductivity at the atomic scale. This effect witnesses that each individual step edge is a Josephson barrier. At a mesoscopic scale the weakly linked superconducting atomic terraces of $\sqrt{7}\times\sqrt{3}$ form a native network of Josephson junctions. We anticipate the Pb/Si(111) system to offer the unique opportunity to tune the superconducting coupling between adjacent terraces, paving a new way of designing atomic scale quantum devices compatible with silicon technology., submitted to Nature Physics

We present a content-addressable memory whose central component is a superconducting crossbar array with 2N elements connected by N2 junctions. Because multiple pathways exist between any two elements, this storage device is tolerant to physical defects in the interconnections. Furthermore, each pattern of N bits is stored non-locally in the N2 junctions, so information access and retrieval are tolerant to input errors. This superconducting memory should exhibit picosecond single-bit acquisition times with negligible energy dissipation during switching and multiple non-destructive read-outs of the stored data.

The superconducting phase qubit combines Josephson junctions into superconducting loops and defines one of the promising solid state device implementations for quantum computing. Here, we propose two hardware realizations for superconducting phase qubits, where the first is based on 2 φ -periodic s-wave–d-wave–s-wave Josephson junctions, while the second proposal involves five Josephson junctions arranged in a loop which is frustrated by a π-junction, a Josephson junction with a ground state characterized by a π -phase shift across.

We demonstrate evidence of coherent magnetic flux tunneling through superconducting nanowires patterned in a thin highly disordered NbN film. The phenomenon is revealed as a superposition of flux states in a fully metallic superconducting loop with the nanowire acting as an effective tunnel barrier for the magnetic flux, and reproducibly observed in different wires. The flux superposition achieved in the fully metallic NbN rings proves the universality of the phenomenon previously reported for InOx. We perform microwave spectroscopy and study the tunneling amplitude as a function of the wire width, compare the experimental results with theories, and estimate the parameters for existing theoretical models., Comment: 5 pages, 3 figures. Supplementary material added

We have studied the low-energy excitations in a minimalistic protected Josephson circuit which contains two basic elements (rhombi) characterized by the $\pi$ periodicity of the Josephson energy. Novel design of these elements, which reduces their sensitivity to the offset charge fluctuations, has been employed. We have observed that the life time $T_{1}$ of the first excited state of this quantum circuit in the protected regime is increased up to $70\mu s$, a factor of $\sim100$ longer than that in the unprotected state. The quality factor $\omega_{01}T_{1}$ of this qubit exceeds $10^{6}$ . Our results are in agreement with theoretical expectations; they demonstrate the feasibility of symmetry protection in the rhombi-based qubits fabricated with existing technology., Comment: 5 pages, 4 figures, Supplementary Materials

A theory of the frequency dependence of the interplane conductivity of a strongly anisotropic superconductor is presented. The form of the conductivity is shown to be a sensitive probe of the strength of quantum and thermal fluctuations of the phase of the superconducting order parameter. The temperature dependence of the superfluid stiffness and of the form of the absorbtion at frequencies of the order of twice the superconducting gap is shown to depend on the interplay between superconducting pairing, phase coherence, and the mechanism by which electrons are scattered. Measurements of the c-axis conductivity of high-${T}_{c}$ superconductors are interpreted in terms of the theory.

We propose a new method to measure the coherence time ofsuperconducting phase qubits based on the analysis of themagnetic-field dependent dc nonlinear Andreev current across ahigh-resistance tunnel contact between the qubit and a dirtymetal wire and derive a quantitative relation between thesubgap I–V characteristic and the internal correlationfunction of the qubit.

We develop a general technique to calculate the probability of transitions over the barriers in spin-glasses in the framework of the dynamical theory. We use Lagrangian formulation of the instanton dynamics in which the transitions are represented by instantons. We derive the full set of the equations that determine the instantons but instead of solving them directly we prove that an instanton process can be mapped into a usual process going back in time which simplifies the problem significantly. We apply this general considerations to a simple example of the spherical Sherrington-Kirkpatrick model and we find the probability of the transition between the metastable states which is in agreement with physical expectations., 18 pages, 2 figures

The temperature dependence of the c axis spectral weight (frequency integral of the interplane conductivity) of high transition temperature (high- T c ) superconductors is shown to be a probe of thermal and quantal fluctuations of the phase of the superconducting order parameter. The behavior of underdoped cuprates is shown to be a natural consequence of superconducting pairing without long-ranged phase coherence. Very underdoped cuprates are found to have strong phase fluctuations, even for temperatures much less than the transition temperature.

We present a Boltzmann equation analysis of the transport properties of a model of electrons with a lifetime that is short everywhere except near the Brillouin-zone diagonals. The assumed lifetime is directly implied by photoemission and $c$-axis transport data. We find quantitative agreement between calculations and ac and dc longitudinal and Hall resistivity, but the predicted longitudinal magnetoresistance disagrees with experiment. A possible microscopic origin of the anomalous lifetime is discussed.

The experimental discovery of a number of 'strange metals' has reopened the question of the low temperature behavior of interacting Fermi systems. Here we provide a subjective overview of some aspects of the resulting theoretical work. It seems to us that from a theoretical standpoint Landau's Fermi-liquid theory has proven to be a remarkably robust description of clean Fermi systems. The only well documented theoretical examples of non-Fermi-liquid behavior are metals subject to gauge interactions or at quantum critical points. The experimental anomalies which prompted the reexamination of Fermi-liquid theory remain in many cases mysterious.

We show that recent measurements of the magnetic field dependence of the magnetization, specific heat and resistivity of overdoped $T_c \sim 17K$ $Tl_{2}Ba_{2}CuO_{6+\delta}$ in the vicinity of the superconducting $H_{c2}$ imply that the vortex viscosity is anomalously small and that the material studied is inhomogeneous with small, a few hundred $\AA$, regions in which the local $T_{c}$ is much higher than the bulk $T_{c}$. The anomalously small vortex viscosity can be derived from a microscopic model in which the quasiparticle lifetime varies dramatically around the Fermi surface, being small everywhere except along the zone diagonal (``cold spot''). We propose experimental tests of our results.

We discuss the phenomenology of superconductivity resulting from the Bose condensation of preformed pairs coexisting with unpaired fermions. We show that this transition is more mean-field-like than the usual Bose condensation, i.e., it is characterized by a relatively small value of the Ginzburg parameter. We consider the Hall effect in the vortex-flow regime and in the fluctuational regime above ${\mathrm{T}}_{\mathrm{c}}$, and show that in this situation it is much less than in the transition driven entirely by Bose condensation but much larger than in usual superconductivity. We analyze the available Hall data and conclude that this phenomenology describes reasonably well the data in the underdoped materials of Y-Ba-Cu-O family but is not an appropriate description of optimally doped materials or underdoped La-Sr-Cu-O.

We predict the universal power law dependence of localization length on magnetic field in the strongly localized regime. This effect is due to the orbital quantum interference. Physically, this dependence shows up in an anomalously large negative magnetoresistance in the hopping regime. The reason for the universality is that the problem of the electron tunneling in a random media belongs to the same universality class as directed polymer problem even in the case of wave functions of random sign. We present numerical simulations which prove this conjecture. We discuss the existing experiments that show anomalously large magnetoresistance. We also discuss the role of localized spins in real materials and the spin polarizing effect of magnetic field., Comment: 21 pages, 16 figures, 52 citations, pdflatex

We present an analytic study of a dynamical instability in a periodic long-range Josephson array frustrated by a weak transverse field. This glass transition is characterized by a diverging relaxation time and a jump in the Edwards-Anderson order parameter; it is {\sl not} accompanied by a coinciding static transition., 4 pages, 2 Postscript figures in a separate file

We analyze pairing in two dimensional spin liquids. We argue that interplane pairing enhanced by magnetic correlations is the most plausible explanation of the spin gap phenomenon observed in underdoped cuprates. The details of the pairing theory depend on the in-plane antiferromagnetic correlations. We consider two models: $2p_F$ correlations induced by a strong gauge field interaction and undamped spin waves. We estimate the pairing temperature $T_s$ and the angular dependence of the gap function and discuss physical consequences., Comment: 1 pages, 2 Postscript figures in a separate file

We reformulate the two-channel Kondo model to explicitly remove the unscattered charge degrees of freedom. This procedure permits us to move the non-Fermi-liquid fixed point to infinite coupling where we can apply a perturbative strong-coupling expansion. The fixed-point Hamiltonian involves a three-body Majorana zero mode whose scattering effects give rise to marginal self-energies. The compactified model is the N=3 member of a family of O(N) Kondo models that can be solved by semiclassical methods in the large N limit. For odd N, fermionic ``kink'' fluctuations about the N=\ensuremath{\infty} mean-field theory generate a fermionic N-body bound state which asymptotically decouples at low energies. For N=3, our semiclassical methods fully recover the non-Fermi-liquid physics of the original two-channel model. Using the same methods, we find that the corresponding O(3) Kondo lattice model develops a spin-gap and a gapless band of coherently propagating three-body bound states. Its strong-coupling limit offers a rather interesting realization of a marginal Fermi liquid.

We study T=0 spin-density-wave transitions in two-dimensional Fermi liquids in which the ordering wave vector Q is such that the tangents to the Fermi line at the points connected by Q are parallel (e.g., Q=2${\mathit{p}}_{\mathit{F}}$ in a system with a circular Fermi line) and the Fermi line is not flat. We show that the transition is first order if the ordering wave vector Q is not commensurate with a reciprocal lattice vector G, i.e., Q\ensuremath{\ne}G/2. If Q is close to G/2 the transition is weakly first order and an intermediate scaling regime exists; in this regime the 2${\mathit{p}}_{\mathit{F}}$ susceptibility and observables such as the NMR rates ${\mathit{T}}_{1}$ and ${\mathit{T}}_{2}$ have scaling forms which we determine.

Strongly correlated two dimensional electrons are believed to form a spin liquid in some regimes of density and temperature. As the density is varied, one expects a transition from this spin liquid state to a spin density wave antiferromagnetic state. In this paper we show that it is self-consistent to assume that this transition is second order and, on this assumption, determine the critical behavior of the $2p_F$ susceptibility, the NMR rates $T_1$ and $T_2$ and the uniform susceptibility. We compare our results to data on high $T_c$ materials.

We study a highly disordered network of superconducting granules linked by weak Josephson junctions in magnetic field and develop a mean field theory for this problem. The diamagnetic response to a slow {\it variations} of magnetic field is found to be analogous to the response of a type-II superconductor with extremely strong pinning. We calculate an effective penetration depth $\lambda_g$ and critical current $j_c$ and find that both $\lambda_g^{-1}$ and $j_c$ are non-zero but are strongly suppressed by frustration.

The origin of continuous energy spectra in large disordered interacting quantum systems is one of the key unsolved problems in quantum physics. Although small quantum systems with discrete energy levels are noiseless and stay coherent forever in the absence of any coupling to external world, most large-scale quantum systems are able to produce a thermal bath and excitation decay. This intrinsic decoherence is manifested by a broadening of energy levels, which aquire a finite width. The important question is: what is the driving force and the mechanism of transition(s) between these two types of many-body systems - with and without intrinsic decoherence? Here we address this question via the numerical study of energy-level statistics of a system of interacting spin-1/2 with random transverse fields. We present the first evidence for a well-defined quantum phase transition between domains of discrete and continous many-body spectra in such spin models, implying the appearance of novel insulating phases in the vicinity of the superconductor-insulator transition in InO(x) and similar materials.

In a number of recent experiments with microwave high quality superconducting coplanar waveguide (CPW) resonators an anomalously weak power dependence of the quality factor has been observed. We argue that this observation implies that the monochromatic radiation does not saturate the Two Level Systems (TLS) located at the interface oxide surfaces of the resonator and suggests the importance of their interactions. We estimate the microwave loss due to interacting TLS and show that the interactions between TLS lead to a drift of their energies that result in a much slower, logarithmic dependence of their absorption on the radiation power in agreement with the data.