Search

One of the main postulates of quantum mechanics is that measurements destroy quantum coherence (wave function collapse). Recently it was discovered that in a many-body system dilute local measurements still preserve some coherence across the entire system. As the measurement density is increased, a phase transition occurs that is characterized by the disentanglement of different parts of the system. Unfortunately, this transition is impossible to observe experimentally for macroscopic systems because it requires an exponentially costly full tomography of the many-body wave function or a comparison with the simulation on an oracle classical computer. In this work we report the discovery of another measurement-induced phase transition that can be observed experimentally if quantum dynamics can be reversed. On one side of this phase transition the quantum information encoded in some part of the Hilbert space is fully recovered after the time inversion. On the other side, all quantum information is corrupted. This transition also manifests itself as the change in the behavior of the probability to observe the same measurement outcome in the process that consists of identical blocks repeated many times. In each block the unitary evolution is followed by the measurement. On one side of the transition the probability decreases exponentially with the number of repetitions, on the other it tends to a constant as the number of repetitions is increased. We confirm the existence of the proposed phase transition through numerical simulations of realistic quantum circuits and analytical calculations using an effective random-matrix theory model., Comment: 18 pages, 10 figures; v2: minor improvements

One of the most challenging problems in the computational study of localization in quantum manybody systems is to capture the effects of rare events, which requires sampling over exponentially many disorder realizations. We implement an efficient procedure on a quantum processor, leveraging quantum parallelism, to efficiently sample over all disorder realizations. We observe localization without disorder in quantum many-body dynamics in one and two dimensions: perturbations do not diffuse even though both the generator of evolution and the initial states are fully translationally invariant. The disorder strength as well as its density can be readily tuned using the initial state. Furthermore, we demonstrate the versatility of our platform by measuring Renyi entropies. Our method could also be extended to higher moments of the physical observables and disorder learning.

With intensive Monte Carlo simulations and finite size scaling we undertake a percolation analysis of Wegner's three-dimensional $\mathbb{Z}_2$ lattice gauge model in equilibrium. We confirm that the loops threading excited plaquettes percolate at the thermal critical point $T_{\rm c}$ and we show that their critical exponents coincide with the ones of the loop representation of the dual 3D Ising model. We then construct Fortuin-Kasteleyn (FK) clusters using a random-cluster representation and find that they also percolate at $T_c$ and moreover give access to all thermal critical exponents. The Binder cumulants of the percolation order parameter of both loops and FK clusters demonstrate a pseudo first order transition. This study sheds light on the critical properties of quantum error correction and lattice gauge theories., Comment: 19 pages, 21 figures

Quantum error correction provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. In this work, we present two surface code memories operating below this threshold: a distance-7 code and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of $\Lambda$ = 2.14 $\pm$ 0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% $\pm$ 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit's lifetime by a factor of 2.4 $\pm$ 0.3. We maintain below-threshold performance when decoding in real time, achieving an average decoder latency of 63 $\mu$s at distance-5 up to a million cycles, with a cycle time of 1.1 $\mu$s. To probe the limits of our error-correction performance, we run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or 3 $\times$ 10$^9$ cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms., Comment: 10 pages, 4 figures, Supplementary Information

Understanding how interacting particles approach thermal equilibrium is a major challenge of quantum simulators. Unlocking the full potential of such systems toward this goal requires flexible initial state preparation, precise time evolution, and extensive probes for final state characterization. We present a quantum simulator comprising 69 superconducting qubits which supports both universal quantum gates and high-fidelity analog evolution, with performance beyond the reach of classical simulation in cross-entropy benchmarking experiments. Emulating a two-dimensional (2D) XY quantum magnet, we leverage a wide range of measurement techniques to study quantum states after ramps from an antiferromagnetic initial state. We observe signatures of the classical Kosterlitz-Thouless phase transition, as well as strong deviations from Kibble-Zurek scaling predictions attributed to the interplay between quantum and classical coarsening of the correlated domains. This interpretation is corroborated by injecting variable energy density into the initial state, which enables studying the effects of the eigenstate thermalization hypothesis (ETH) in targeted parts of the eigenspectrum. Finally, we digitally prepare the system in pairwise-entangled dimer states and image the transport of energy and vorticity during thermalization. These results establish the efficacy of superconducting analog-digital quantum processors for preparing states across many-body spectra and unveiling their thermalization dynamics.

Continuous quantum phase transitions are widely assumed and frequently observed in various systems of quantum particles or spins. Their characteristic trait involves scaling laws governing a second-order, gradual suppression of the order parameter as the quantum critical point is approached. The localization of Cooper pairs in disordered superconductors and the resulting breakdown of superconductivity have long stood as a prototypical example. Here, we show a departure from this paradigm, showcasing that amorphous superconducting films of indium oxide undergo a distinctive, discontinuous first-order quantum phase transition tuned by disorder. Through systematic measurements of the plasmon spectrum in superconducting microwave resonators, we provide evidence for a marked jump of both the zero-temperature superfluid stiffness and the transition temperature at the critical disorder. This discontinuous transition sheds light on the previously overlooked role of repulsive interactions between Cooper pairs and the subsequent competition between superconductivity and insulating Cooper-pair glass. Furthermore, our investigation shows that the critical temperature of the films no longer relates to the pairing amplitude but aligns with the superfluid stiffness, consistent with the pseudogap regime of preformed Cooper pairs. Our findings raise fundamental new questions into the role of disorder in quantum phase transitions and carry implications for superinductances in quantum circuits., Comment: Updated text

By considering the quench dynamics of two-dimensional frustrated Ising models through numerical simulations, we investigate the dynamical critical behavior on the multicritical Nishimori point (NP). We calculate several dynamical critical exponents, namely, the relaxation exponent $z_{\rm c}$, the autocorrelation exponent $\lambda_{\rm c}$, and the persistence exponent $\theta_{\rm c}$, after a quench from the high temperature phase to the NP. We confirm their universality with respect to the lattice geometry and bond distribution. For a quench from a power-law correlated initial state to the NP, the aging dynamics are much slower. We also look up the issue of multifractality during the critical dynamics by investigating different moments of the spatial correlation function. We observe a single growth law for all the length scales extracted from different moments, indicating that the equilibrium multifractality at the NP does not affect the dynamics., Comment: 28 pages, 9 figures

Simulation of the single-particle tunneling problem by means of the Suzuki-Trotter approximation (STA) is analyzed. Considered is a particle hopping across a chain of sites in presence of a smooth position-dependent potential profile with several local minima that arrange a tunneling problem between the localized states in different minima. The STA error is found to manifest itself in three ways: i) perturbative energy shifts, ii) nonperturbartive renormalization of the tunneling rates, and iii) perturbative leakage of the total probability to other states. Generally, the first type of error is the most essential, as detuning of the tunneling resonance has to be compared with exponentially small tunneling rates. In absence of detuning (e.g. if the resonance is protected by symmetry), STA leads to exponential enhancement of the tunneling rates. The last type of error classifies the overall defect in the wave function and delineates the region of sufficiently weak distortion of the wave function due to STA. The conducted analysis confirms the naive criteria of applicability $\max\{T,P\}\ll\delta t^{-1}$ (with $T,P$ being the typical scales of kinetic and potential terms, respectively), while also revealing the structure of error and its behavior with system parameters. Analysis of the case of large Trotter step is also performed, with the main result being the reconstruction of low-energy spectrum due to coupling between states with energy difference close to $2\pi/\delta t$. The connection of the obtained results with rigorous upper error bounds on the STA error is discussed, with particular emphasis on why these rigorous bounds are not always saturated. We also point out that the proposed problem can be directly implemented on existing quantum devices [arXiv:2012.00921]. In particular, we give a detailed description of an experimental design that demonstrates the described physics., Comment: 38 pages, 12 figures. Changes: fixed several typos, replaced certain figures for better readability

Understanding universal aspects of quantum dynamics is an unresolved problem in statistical mechanics. In particular, the spin dynamics of the 1D Heisenberg model were conjectured to belong to the Kardar-Parisi-Zhang (KPZ) universality class based on the scaling of the infinite-temperature spin-spin correlation function. In a chain of 46 superconducting qubits, we study the probability distribution, $P(\mathcal{M})$, of the magnetization transferred across the chain's center. The first two moments of $P(\mathcal{M})$ show superdiffusive behavior, a hallmark of KPZ universality. However, the third and fourth moments rule out the KPZ conjecture and allow for evaluating other theories. Our results highlight the importance of studying higher moments in determining dynamic universality classes and provide key insights into universal behavior in quantum systems.

The $\pm J$ Ising model is a simple frustrated spin model, where the exchange couplings independently take the discrete value $-J$ with probability $p$ and $+J$ with probability $1-p$. It is especially appealing due to its connection to quantum error correcting codes. Here, we investigate the nonequilibrium critical behavior of the two-dimensional $\pm J$ Ising model, after a quench from different initial conditions to a critical point $T_c(p)$ on the paramagnetic-ferromagnetic (PF) transition line, especially, above, below and at the multicritical Nishimori point (NP). The dynamical critical exponent $z_c$ seems to exhibit non-universal behavior for quenches above and below the NP, which is identified as a pre-asymptotic feature due to the repulsive fixed point at the NP. Whereas, for a quench directly to the NP, the dynamics reaches the asymptotic regime with $z_c \simeq 6.02(6)$. We also consider the geometrical spin clusters (of like spin signs) during the critical dynamics. Each universality class on the PF line is uniquely characterized by the stochastic Loewner evolution (SLE) with corresponding parameter $\kappa$. Moreover, for the critical quenches from the paramagnetic phase, the model, irrespective of the frustration, exhibits an emergent critical percolation topology at the large length scales., Comment: 29 pages, 11 figures

Indistinguishability of particles is a fundamental principle of quantum mechanics. For all elementary and quasiparticles observed to date - including fermions, bosons, and Abelian anyons - this principle guarantees that the braiding of identical particles leaves the system unchanged. However, in two spatial dimensions, an intriguing possibility exists: braiding of non-Abelian anyons causes rotations in a space of topologically degenerate wavefunctions. Hence, it can change the observables of the system without violating the principle of indistinguishability. Despite the well developed mathematical description of non-Abelian anyons and numerous theoretical proposals, the experimental observation of their exchange statistics has remained elusive for decades. Controllable many-body quantum states generated on quantum processors offer another path for exploring these fundamental phenomena. While efforts on conventional solid-state platforms typically involve Hamiltonian dynamics of quasi-particles, superconducting quantum processors allow for directly manipulating the many-body wavefunction via unitary gates. Building on predictions that stabilizer codes can host projective non-Abelian Ising anyons, we implement a generalized stabilizer code and unitary protocol to create and braid them. This allows us to experimentally verify the fusion rules of the anyons and braid them to realize their statistics. We then study the prospect of employing the anyons for quantum computation and utilize braiding to create an entangled state of anyons encoding three logical qubits. Our work provides new insights about non-Abelian braiding and - through the future inclusion of error correction to achieve topological protection - could open a path toward fault-tolerant quantum computing.

Practical quantum computing will require error rates that are well below what is achievable with physical qubits. Quantum error correction offers a path to algorithmically-relevant error rates by encoding logical qubits within many physical qubits, where increasing the number of physical qubits enhances protection against physical errors. However, introducing more qubits also increases the number of error sources, so the density of errors must be sufficiently low in order for logical performance to improve with increasing code size. Here, we report the measurement of logical qubit performance scaling across multiple code sizes, and demonstrate that our system of superconducting qubits has sufficient performance to overcome the additional errors from increasing qubit number. We find our distance-5 surface code logical qubit modestly outperforms an ensemble of distance-3 logical qubits on average, both in terms of logical error probability over 25 cycles and logical error per cycle ($2.914\%\pm 0.016\%$ compared to $3.028\%\pm 0.023\%$). To investigate damaging, low-probability error sources, we run a distance-25 repetition code and observe a $1.7\times10^{-6}$ logical error per round floor set by a single high-energy event ($1.6\times10^{-7}$ when excluding this event). We are able to accurately model our experiment, and from this model we can extract error budgets that highlight the biggest challenges for future systems. These results mark the first experimental demonstration where quantum error correction begins to improve performance with increasing qubit number, illuminating the path to reaching the logical error rates required for computation., Comment: Main text: 6 pages, 4 figures. v2: Update author list, references, Fig. S12, Table IV

Systems of correlated particles appear in many fields of science and represent some of the most intractable puzzles in nature. The computational challenge in these systems arises when interactions become comparable to other energy scales, which makes the state of each particle depend on all other particles. The lack of general solutions for the 3-body problem and acceptable theory for strongly correlated electrons shows that our understanding of correlated systems fades when the particle number or the interaction strength increases. One of the hallmarks of interacting systems is the formation of multi-particle bound states. In a ring of 24 superconducting qubits, we develop a high fidelity parameterizable fSim gate that we use to implement the periodic quantum circuit of the spin-1/2 XXZ model, an archetypal model of interaction. By placing microwave photons in adjacent qubit sites, we study the propagation of these excitations and observe their bound nature for up to 5 photons. We devise a phase sensitive method for constructing the few-body spectrum of the bound states and extract their pseudo-charge by introducing a synthetic flux. By introducing interactions between the ring and additional qubits, we observe an unexpected resilience of the bound states to integrability breaking. This finding goes against the common wisdom that bound states in non-integrable systems are unstable when their energies overlap with the continuum spectrum. Our work provides experimental evidence for bound states of interacting photons and discovers their stability beyond the integrability limit., Comment: 7 pages + 15 pages supplements

Inherent symmetry of a quantum system may protect its otherwise fragile states. Leveraging such protection requires testing its robustness against uncontrolled environmental interactions. Using 47 superconducting qubits, we implement the one-dimensional kicked Ising model which exhibits non-local Majorana edge modes (MEMs) with $\mathbb{Z}_2$ parity symmetry. Remarkably, we find that any multi-qubit Pauli operator overlapping with the MEMs exhibits a uniform late-time decay rate comparable to single-qubit relaxation rates, irrespective of its size or composition. This characteristic allows us to accurately reconstruct the exponentially localized spatial profiles of the MEMs. Furthermore, the MEMs are found to be resilient against certain symmetry-breaking noise owing to a prethermalization mechanism. Our work elucidates the complex interplay between noise and symmetry-protected edge modes in a solid-state environment.

We propose a quantum algorithm for inferring the molecular nuclear spin Hamiltonian from time-resolved measurements of spin-spin correlators, which can be obtained via nuclear magnetic resonance (NMR). We focus on learning the anisotropic dipolar term of the Hamiltonian, which generates dynamics that are challenging-to-classically-simulate in some contexts. We demonstrate the ability to directly estimate the Jacobian and Hessian of the corresponding learning problem on a quantum computer, allowing us to learn the Hamiltonian parameters. We develop algorithms for performing this computation on both noisy near-term and future fault-tolerant quantum computers. We argue that the former is promising as an early beyond-classical quantum application since it only requires evolution of a local spin Hamiltonian. We investigate the example of a protein (ubiquitin) confined in a membrane as a benchmark of our method. We isolate small spin clusters, demonstrate the convergence of our learning algorithm on one such example, and then investigate the learnability of these clusters as we cross the ergodic to non-ergodic phase transition by suppressing the dipolar interaction. We see a clear correspondence between a drop in the multifractal dimension measured across many-body eigenstates of these clusters, and a transition in the structure of the Hessian of the learning cost-function (from degenerate to learnable). Our hope is that such quantum computations might enable the interpretation and development of new NMR techniques for analyzing molecular structure., Comment: 20 pages, 6 page appendix with some FT and NISQ algorithm details, 7 figures

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.

Strongly correlated quantum systems give rise to many exotic physical phenomena, including high-temperature superconductivity. Simulating these systems on quantum computers may avoid the prohibitively high computational cost incurred in classical approaches. However, systematic errors and decoherence effects presented in current quantum devices make it difficult to achieve this. Here, we simulate the dynamics of the one-dimensional Fermi-Hubbard model using 16 qubits on a digital superconducting quantum processor. We observe separations in the spreading velocities of charge and spin densities in the highly excited regime, a regime that is beyond the conventional quasiparticle picture. To minimize systematic errors, we introduce an accurate gate calibration procedure that is fast enough to capture temporal drifts of the gate parameters. We also employ a sequence of error-mitigation techniques to reduce decoherence effects and residual systematic errors. These procedures allow us to simulate the time evolution of the model faithfully despite having over 600 two-qubit gates in our circuits. Our experiment charts a path to practical quantum simulation of strongly correlated phenomena using available quantum devices., Comment: 20 pages, 15 figures

As the search continues for useful applications of noisy intermediate scale quantum devices, variational simulations of fermionic systems remain one of the most promising directions. Here, we perform a series of quantum simulations of chemistry the largest of which involved a dozen qubits, 78 two-qubit gates, and 114 one-qubit gates. We model the binding energy of ${\rm H}_6$, ${\rm H}_8$, ${\rm H}_{10}$ and ${\rm H}_{12}$ chains as well as the isomerization of diazene. We also demonstrate error-mitigation strategies based on $N$-representability which dramatically improve the effective fidelity of our experiments. Our parameterized ansatz circuits realize the Givens rotation approach to non-interacting fermion evolution, which we variationally optimize to prepare the Hartree-Fock wavefunction. This ubiquitous algorithmic primitive corresponds to a rotation of the orbital basis and is required by many proposals for correlated simulations of molecules and Hubbard models. Because non-interacting fermion evolutions are classically tractable to simulate, yet still generate highly entangled states over the computational basis, we use these experiments to benchmark the performance of our hardware while establishing a foundation for scaling up more complex correlated quantum simulations of chemistry., Comment: updated link to experiment code, new version containing expanded data sets and corrected figure label

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

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., Comment: 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

We extend the Keldysh technique to enable the computation of out-of-time order correlators. 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 formulate the generalized 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.

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

The origin of continuous energy spectra in large disordered interacting quantum systems is one of the key unsolved problems in quantum physics. While 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 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 two different 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 spins-1/2 with anisotropic exchange interactions and random transverse fields. Our results present the first evidence for a well-defined quantum phase transition between domains of discrete and continous many-body spectra in a class of random spin models. Because this model also describes the physics of the superconductor-insulator transition in disordered superconductors like InO and similar materials, our results imply the appearance of novel insulating phases in the vicinity of this transition., Comment: 18 pages, 7 figures, original version of the paper accepted for publication (in slightly revised form) in Nature Communications

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.

We develop a technique to compute the high-frequency asymptotics of spin correlators in weakly interacting disordered spin systems. We show that the dynamical spin correlator decreases exponentially at high frequencies, $< SS>_{\omega}\sim\exp(-\tau_{*}\omega)$ and compute the characteristic time $\tau^{*}$ of this dependence. In a typical random configuration, some fraction of spins form strongly coupled pairs, which behave as two-level systems. Their switching dynamics is driven by the high-frequency noise from the surrounding spins, resulting in low-frequency $1/f$ noise in the magnetic susceptibility and other physical quantities. We discuss application of these results to the problem of susceptibility and flux noise in superconducting circuits at mK temperatures.

We have studied the microwave response of a single Cooper-pair transistor (CPT) coupled to a lumped-element microwave resonator. The resonance frequency of this circuit, $f_{r}$, was measured as a function of the charge $n_{g}$ induced on the CPT island by the gate electrode, and the phase difference across the CPT, $\phi_{B}$, which was controlled by the magnetic flux in the superconducting loop containing the CPT. The observed $f_{r}(n_{g},\phi_{B})$ dependences reflect the variations of the CPT Josephson inductance with $n_{g}$ and $\phi_{B}$ as well as the CPT excitation when the microwaves induce transitions between different quantum states of the CPT. The results are in excellent agreement with our simulations based on the numerical diagonalization of the circuit Hamiltonian. This agreement over the whole range of $n_{g}$ and $\phi_{B}$ is unexpected, because the relevant energies vary widely, from 0.1K to 3K. The observed strong dependence $f_{r}(n_{g},\phi_{B})$ near the resonance excitation of the CPT provides a tool for sensitive charge measurements., Comment: 10 pages, 6 figures

We develop an analytical theory for quantum phase transitions driven by disorder in magnets and superconductors. We study these transitions with a cavity approximation which becomes exact on a Bethe lattice with large branching number. We find two different disordered phases, characterized by very different relaxation rates, which both exhibit strong inhomogeneities typical of glassy physics., Comment: 4 pages, 1 figure

For successful realization of a quantum computer, its building blocks (qubits) should be simultaneously scalable and sufficiently protected from environmental noise. Recently, a novel approach to the protection of superconducting qubits has been proposed. The idea is to prevent errors at the "hardware" level, by building a fault-free (topologically protected) logical qubit from "faulty" physical qubits with properly engineered interactions between them. It has been predicted that the decoupling of a protected logical qubit from local noises would grow exponentially with the number of physical qubits. Here we report on the proof-of-concept experiments with a prototype device which consists of twelve physical qubits made of nanoscale Josephson junctions. We observed that due to properly tuned quantum fluctuations, this qubit is protected against magnetic flux variations well beyond linear order, in agreement with theoretical predictions. These results demonstrate the feasibility of topologically protected superconducting qubits., Comment: 25 pages, 5 figures

We introduce a toy model that allows us to study the physical properties of a spin impurity coupled to the electrons in the superconducting island. We show that when the coupling of the spin is of the order of the superconducting gap two almost degenerate subgap states are formed. By computing the Berry phase that is associated with the superconducting phase rotations in this model, we prove that these subgap states are characterized by a different charge and demonstrate that the switching between these states has the same effect as quasiparticle poisoning (unpoisoning) of the island. We also show that an impurity coupled to both the island and the lead generates Josepshon current fluctuations., Comment: 5 pages, 1 figure

We analyze the data and discuss their implications for the microscopic origin of the low frequency flux noise in superconducting circuits. We argue that this noise is produced by spins at the superconductor insulator boundary whose dynamics is due to RKKY interaction. We show that this mechanism explains size independence of the noise, different frequency dependences of the spectra reported in large and small SQUIDs and gives the correct intensity for realistic parameters., Comment: 4 pages, no figures

We study electronic transport in Anderson insulators with strong Coulomb interactions in dimensions d>=2. Close to the metal insulator transition where the single particle localization length is much larger than interparticle-distance, the interactions lead to a strongly correlated quantum glass phase. Even though single particle excitations are localized and the system is insulating, there are collective electronic modes which remain delocalized down to parametrically small energies. These collective excitations serve as a continuous bath which can provide the activation energy for variable range hopping transport. This circumvents the energy conservation problem arising when only discrete particle-hole excitations are present. In contrast to the weak and material-dependent phonon-assisted hopping mechanism, the activation by an electronic bath leads to a nearly universal prefactor e^2/h of the Efros-Shklovskii conductance, as is observed in many recent experiments.

We compute the current voltage characteristic of a chain of identical Josephson circuits characterized by a large ratio of Josephson to charging energy that are envisioned as the implementation of topologically protected qubits. We show that in the limit of small coupling to the environment it exhibits a non-monotonous behavior with a maximum voltage followed by a parametrically large region where $V\propto 1/I$. We argue that its experimental measurement provides a direct probe of the amplitude of the quantum transitions in constituting Josephson circuits and thus allows their full characterization., Comment: 12 pages, 4 figures

We show that trapped ions can be used to simulate a highly symmetrical Hamiltonian with eingenstates naturally protected against local sources of decoherence. This Hamiltonian involves long range coupling between particles and provides a more efficient protection than nearest neighbor models discussed in previous works. Our results open the perspective of experimentally realizing in controlled atomic systems, complex entangled states with decoherence times up to nine orders of magnitude longer than isolated quantum systems., Comment: 4 pages

We analyze data on the critical current and normal state resistance noise in Josephson junctions and argue that the noise in the critical current is due to a mechanism that is absent in the normal state. We estimate the noise produced by conventional Two Level Systems (TLSs) in the insulating barrier and find that it agrees both in magnitude and in temperature dependence with the resistance fluctuations in the normal state but it is not sufficient to explain the critical current noise observed in large superconducting contacts. We propose a novel microscopic mechanism for the noise in the superconducting state in which the noise is due to electron tunneling between weak Kondo states at subgap energies. We argue that the noise produced by this mechanism gives temperature, area dependence and intensity that agree with the data., Comment: 4 pages, 1 figure

The noise in physical qubits is fundamentally asymmetric: in most devices, phase errors are much more probable than bit flips. We propose a quantum error correcting code which takes advantage of this asymmetry and shows good performance at a relatively small cost in redundancy, requiring less than a doubling of the number of physical qubits for error correction.

We discuss the origin of decoherence in Josephson junction qubits. We find that two level systems in the surrounding insulator cannot be the dominant source of noise in small qubits. We argue that electron traps in the Josephson barrier with large Coulomb repulsion would give noise that agrees both in magnitude and in temperature dependence with experimental data., Comment: 4 pages, no figures

We develop a theory of a pseudogap state appearing near the superconductor-insulator transition in strongly disordered metals with attractive interaction. We show that such an interaction combined with the fractal nature of the single particle wave functions near the mobility edge leads to an anomalously large single particle gap in the superconducting state near SI transition that persists and even increases in the insulating state long after the superconductivity is destroyed. We give analytic expressions for the value of the pseudogap in terms of the inverse participation ratio of the corresponding localization problem., Comment: 4 pages, 2 figures

We present a theory of the subgap tails of the density of states in a diffusive superconductor containing magnetic impurities. We show that the subgap tails have two contributions: one arising from mesoscopic gap fluctuations, previously discussed by Lamacraft and Simons, and the other associated to the long-wave fluctuations of the concentration of magnetic impurities. We study the latter both in small superconducting grains and in bulk systems [$d=1,2,3$], and establish the dimensionless parameter that controls which of the two contributions dominates the subgap tails. We observe that these contributions are related to each other by dimensional reduction. We apply the theory to estimate the effects of a weak concentration of magnetic impurities [$\approx 1 {\rm p.p.m}$] on the phase coherence of Josephson qubits. We find that at these typical concentrations, magnetic impurities are relevant for the dephasing in large qubits, designed around a $10 {\rm \mu m}$ scale, where they limit the quality factor to be $Q<10^4-10^5$., Comment: 13 pages, 1 figure

We propose a new method to measure the coherence time of superconducting phase qubits based on the analysis of the magnetic-field dependent dc nonlinear Andreev current across a high-resistance tunnel contact between the qubit and a dirty metal wire and derive a quantitative relation between the subgap I-V characteristic and the internal correlation function of the qubit., Comment: LaTeX 2.09, 11 pages, 2 eps-figures; special LaTeX style file included. Contribution to the Proceedings of "Electron Transport In Mesoscopic Systems", LT22 Satellite Conference in Goteborg, Sweden, 12-15 August 1999. To be published in a special issue of J. Low Temp. Phys. The corresponding Extended Abstract (2 pages) was posted previously at cond-mat/9907317