We investigate the long-time dynamics of the Sine-Gordon (SG) model under a class of perturbations whose quantum field theoretic analog — via bosonization — corresponds to the massive Schwinger model describing 1+1D relativistic QED of Dirac fermions. Classical SG solutions offer critical insight into non-perturbative effects in this quantum theory, but capturing their long-time behavior poses significant numerical challenges. To address this, we extend a coarse-graining method to spacetime using a dual-mesh construction based on the Minkowski-metric. We first validate the approach against the well-studied variant of the SG model describing magnetic fluxon dynamics in Josephson transmission lines (JTLs), where analytical and numerical benchmarks exist. We then apply the method to the Schwinger-inspired SG model and uncover long-lived bound states — “Schwinger atoms” — in which a soliton is trapped by a fixed central charge. In certain regimes, the system exhibits limit cycles that give rise to positronium-like states of oppositely charged solitons, while in others such formation is suppressed. Accessing such long-time solutions requires a rigorous implementation of outgoing boundary conditions on a finite computational domain that provide radiative dissipation to allow relaxation toward states that exist only in an infinite domain. Here we provide such a construction. Our results also suggest the possibility of analog quantum simulation of relativistic quantum field theories with JTLs. These results demonstrate the utility of spatio-temporal coarse-graining methodology for probing non-perturbative structure formation in non-linear field theories.

We introduce an end-to-end optimization strategy for quantum machine learning that directly targets performance under finite measurement resources, where learning objectives are defined directly at the level of task performance. The method is applied on a Bayesian quantum metrology task since it provides a natural testbed with known fundamental limits and scaling with system size. The sampling-aware hybrid algorithm achieves a single-shot risk within 1 dB of the -20 dB Bayesian limit using 32 qubits. We extend the Bayesian framework from parameter estimation to global function inference, where the task is to infer a target function of the sensor input drawn from an arbitrary prior, and we demonstrate a clear computational-sensing advantage for direct functional inference over indirect reconstruction. We relate the corresponding Bayesian risk to the Capacity metric and argue that the Resolvable Expressive Capacity provides a natural measure of the space of functions accessible in a single shot. The resulting eigentask analysis identifies noise-robust feature combinations that yield compact estimators with improved accuracy and reduced optimization cost in resource-limited or real-time on-device settings.

Superconducting circuits comprising Josephson junctions have spurred significant research activity due to their promise to realize scalable quantum computers. Effective Hamiltonians for these systems have traditionally been derived assuming the junction connects superconducting islands of infinite size. We derive a quantized Hamiltonian for a Josephson junction between finite-sized islands and predict measurable corrections to the qubit frequency and charge susceptibility to test the theory.

We derive a mesoscopic theory of the Josephson junction from nonrelativistic scalar electrodynamics. Our theory reproduces the Josephson relations with the canonical current phase relation acquiring a weak second harmonic term, and it improves the standard lumped-element descriptions employed in circuit quantum electrodynamics by providing spatial resolution of the superconducting order parameter and electromagnetic field. By providing an ab initio derivation of the charge qubit Hamiltonian that relates traditionally free qubit parameters to geometric and material properties, we progress toward the quantum engineering of superconducting circuits at the subnanometer scale.

Real-world experiments on quantum systems are always performed with measurement apparatus whose interaction times with the systems are finite. This restricts the observable quantum states to the space of time-coarse-grained density matrices {ρ(t)}, providing motivation for the time-coarse-graining (TCG) approach that solves for ρ(t) without ever referring to the unobservable ρ(t) of infinite time resolution. Phenomenologically, this implies that coherent transitions far outside the bandwidth would be filtered out, leaving only their effective impacts on the “slow” dynamics resolvable by the finite time resolution of the measurements. Therefore, the TCG framework provides rigorous justification for many existing effective Hamiltonian methods in the literature that aim at capturing the unitary part of the long-time dynamics. However, since time-coarse-graining is fundamentally irreversible, the resulting effective model allows for secular loss of information and dissipation of energy in general, which cannot be captured by any uni- tary effective models and has to be treated with explicit time-coarse-graining. Such incoherent effects are particularly prominent in driven nonlinear quantum systems where exchange of information and energy with the drive gives rise to incoherent effective dynamics at all finite time resolutions in general. While rigorous in principle, existing TCG methods in the literature can be applied only to simple systems by one iteratively solving superoperator equations at low orders in the coupling strengths. The complexity of such methods prevents systematic study of the time-coarse-grained dynamics and limits analytical results to the IR (low-resolution) limit in most cases. We address these limitations in this paper, presenting a systematic time-coarse-graining method that overcomes the complexities and restrictions of current techniques, offer- ing a comprehensive and accurate modeling framework for driven nonlinear quantum systems. We derive closed-form formulas as well as diagrammatic representations for both unitary and nonunitary contribu- tions, in the form of an effective Hamiltonian and nonunitary dissipators at arbitrary order in the coupling strengths, and complement this with a computer-algebra software package. We demonstrate the effective- ness of the method using several typical models of driven nonlinear systems in superconducting circuits, and show that it generalizes and improves on existing methods by providing more accurate results and explaining phenomena that have not been accounted for.

The accurate modeling of mode hybridization and calculation of radiative relaxation rates have been crucial to the design and optimization of superconducting quantum devices. In this work, we introduce a spectral theory for the electrohydrodynamics of superconductors that enables the extraction of the relaxation rates of excitations in a general three-dimensional distribution of superconducting bodies. Our approach addresses the long-standing problem of formulating a modal description of open systems that is both efficient and allows second quantization of the radiative hybridized fields. This is achieved through the implementation of finite but transparent boundaries through which radiation can propagate into and out of the computational domain. The resulting spectral problem is defined within a coarse-grained for- mulation of the electrohydrodynamical equations that is suitable for the analysis of the nonequilibrium dynamics of multiscale superconducting quantum systems.

Tackling output sampling noise due to finite shots of quantum measurement is an unavoidable challenge when extracting information in machine learning with physical systems. A technique called Eigentask Learning was developed recently as a framework for learning with infinite input training data in the presence of output sam- pling noise. In the work of Eigentask Learning, numerical evidence was presented that extracting low-noise contributions of features can practically improve performance for machine learning tasks, displaying robustness to overfitting and increasing generalization accuracy. However, it remains unsolved to quantitatively character- ize generalization errors in situations where the training dataset is finite, while output sampling noise still exists. In this study, we use methodologies from statistical mechanics to calculate the training and generalization errors of a generic quantum machine learning system when the input training dataset and output measurement sam- pling shots are both finite. Our analytical findings, supported by numerical validation, offer solid justification that Eigentask Learning provides optimal learning in the sense of minimizing generalization errors.

Model order reduction encompasses mathematical techniques aimed at reducing the complexity of mathematical models in simulations while retaining the essential characteristics and behaviors of the original model. This is particularly useful in the context of large-scale dynamical systems, which can be computationally expensive to analyze and simulate. Here, we present a model order reduction technique to reduce the time complexity of open quantum systems, grounded in the principle of measurement-adapted coarse-graining. This method, governed by a coarse-graining time scale τ and the spectral band center ω0, organizes corrections to the lowestorder model which aligns with the RWA Hamiltonian in certain limits, and rigorously justifies the resulting effective quantum master equation (EQME). The focus on calculating to a high degree of accuracy only what can be resolved by the measurement introduces a principled regularization procedure to address singularities and generates low-stiffness models suitable for efficient long-time integration. Furthermore, the availability of the analytical form of the EQME parameters significantly boosts the interpretive capabilities of the method. As a demonstration, we derive the fourth-order EQME for a challenging problem related to the dynamics of a superconducting qubit under high-power dispersive readout in the presence of a continuum of dissipative waveguide modes. This derivation shows that the lowest-order terms align with previous results, while higher-order corrections suggest new phenomena.

Although linear quantum amplification has proven essential to the processing of weak quantum signals, extracting higher-order quantum features such as correlations in principle demands nonlinear operations. However, nonlinear processing of quantum signals is often associated with non-idealities and excess noise, and absent a general framework to harness nonlinearity, such regimes are typically avoided. Here we present a framework to uncover general quantum signal processing principles of a broad class of bosonic quantum nonlinear processors (QNPs), inspired by a remarkably analogous paradigm in nature: the processing of environmental stimuli by nonlinear, noisy neural ensembles, to enable perception. Using a quantum-coherent description of a QNP monitoring a quantum signal source, we show that quantum nonlinearity can be harnessed to calculate higher-order features of an incident quantum signal, concentrating them into linearly-measurable observables, a transduction not possible using linear amplifiers. Secondly, QNPs provide coherent nonlinear control over quantum fluctuations including their own added noise, enabling noise suppression in an observable without suppressing transduced information, a paradigm that bears striking similarities to optimal neural codings that allow perception even under highly stochastic neural dynamics. Unlike the neural case, we show that QNP-engineered noise distributions can exhibit non-classical correlations, providing a new means to harness resources such as entanglement. Finally, we show that even simple QNPs in realistic measurement chains can provide enhancements of signal-to-noise ratio for practical tasks such as quantum state discrimination. Our work provides pathways to utilize nonlinear quantum systems as general computation devices, and enables a new paradigm for nonlinear quantum information processing.

The practical implementation of many quantum algorithms known today is limited by the coherence time of the executing quantum hardware and quantum sampling noise. Here we present a machine learning algorithm, NISQRC, for qubit-based quantum systems that enables inference on temporal data over durations unconstrained by decoherence. NISQRC leverages mid-circuit measurements and deterministic reset operations to reduce circuit executions, while still maintaining an appropriate length persistent temporal memory in the quantum system, confirmed through the proposed Volterra Series analysis. This enables NISQRC to overcome not only limitations imposed by finite coherence, but also information scrambling in monitored circuits and sampling noise, problems that persist even in hypothetical fault-tolerant quantum computers that have yet to be realized. To validate our approach, we consider the channel equalization task to recover test signal symbols that are subject to a distorting channel. Through simulations and experiments on a 7-qubit quantum processor we demonstrate that NISQRC can recover arbitrarily long test signals, not limited by coherence time.

We develop and demonstrate a trainable temporal postprocessor (TPP) harnessing a simple but versatile machine learning algorithm to provide optimal processing of quantum measurement data subject to arbitrary noise processes for the readout of an arbitrary number of quantum states. We demonstrate the TPP on the essential task of qubit state readout, which has historically relied on temporal processing via matched filters in spite of their applicability for only specific noise conditions. Our results show that the TPP can reliably outperform standard filtering approaches under complex readout conditions, such as high-power readout. Using simulations of quantum measurement noise sources, we show that this advantage relies on the TPP’s ability to learn optimal linear filters that account for general quantum noise correlations in data, such as those due to quantum jumps, or correlated noise added by a phase-preserving quantum amplifier. Furthermore, we derive an exact analytic form for the optimal TPP weights: this positions the TPP as a linearly scaling generalization of matched filtering, valid for an arbitrary number of states under the most general readout noise conditions, all while preserving a training complexity that is essentially negligible in comparison with that of training neural networks for processing temporal quantum measurement data. The TPP can be autonomously and reliably trained on measurement data and requires only linear operations, making it ideal for field-programmable gate array implementations in circuit QED for real-time processing of measurement data from general quantum systems.

By applying a hydrodynamic representation of nonrelativistic scalar electrodynamics to the superconducting order parameter, we predict a negative (attractive) pressure between planar superconducting bodies. For con- ventional superconductors with London penetration depth λL ≈ 100 nm, the pressure reaches tens of N/mm2 at angstrom separations. The resulting surface energies are in better agreement with experimental values than those predicted by the Hartree-Fock theory, and the emergent electric-field screening length is comparable to that of the Thomas-Fermi theory. The model circumvents the bulk limitations of the Bardeen-Cooper-Schrieffer and Ginzburg-Landau theories to the analysis of superconducting quantum devices.

The expressive capacity of physical systems employed for learning is limited by the unavoidable presence of noise in their extracted outputs. Though present in physical systems across both the classical and quantum regimes, the precise impact of noise on learning remains poorly understood. Focusing on supervised learning, we present a mathematical framework for evaluating the resolvable expressive capacity (REC) of general physical systems under finite sampling noise and provide a methodology for extracting its extrema, the eigentasks. Eigentasks are a native set of functions that a given physical system can approximate with minimal error. We show that the REC of a quantum system is limited by the fundamental theory of quantum measurement and obtain a tight upper bound for the REC of any finitely sampled physical system. We then provide empirical evidence that extracting low-noise eigentasks can lead to improved performance for machine learning tasks such as classification, displaying robustness to overfitting. We present analyses suggesting that correlations in the measured quantum system enhance learning capacity by reducing noise in eigentasks. The applicability of these results in practice is demonstrated with experiments on superconducting quantum processors. Our findings have broad implications for quantum machine learning and sensing applications.

Coherent multimode instabilities are responsible for several phenomena of recent interest in semiconductor lasers, such as the generation of frequency combs and ultrashort pulses. These techonologies have proven disruptive in optical telecommunications and spectroscopy applications. While the standard Maxwell-Bloch equations (MBEs) encompass such complex lasing phenomena, their integration is computationally expensive and offers limited analytical insight. In this paper, we demonstrate an efficient spectral approach to the simulation of multimode instabilities via a quantitative analysis of the instability of single-frequency lasing in ring lasers, referred to as the Lorenz-Haken (LH) instability or the RNGH instability in distinct parameter regimes. Our approach, referred to as CFTD, uses generally non-Hermitian Constant Flux modes to obtain projected Time Domain equations. CFTD provides excellent agreement with finite-difference integration of the MBEs across a wide range of parameters in regimes of non-stationary inversion, including frequency comb formation and spatiotemporal chaos. We also develop a modal linear stability analysis using CFTD to efficiently predict multimode instabilities in lasers. The combination of numerical accuracy, speedup, and semi-analytic insight across a variety of dynamical regimes make the CFTD approach ideal to analyze multimode instabilities in lasers, especially in more complex geometries or coupled laser arrays.

The expressive capacity for learning with quantum systems is fundamentally limited by the quantum sampling noise incurred during measurement. While studies suggest that noise limits the resolvable capacity of quantum systems, its precise impact on learning remains an open question. We develop a framework for quantifying the expressive capacity of qubit-based systems from finite numbers of projective measurements, and calculate a tight bound on the expressive capacity and the corresponding accuracy limit that we compare to experiments on superconducting quantum processors. We uncover the native function set a finitely-sampled quantum system can approximate, called eigentasks. We then demonstrate how low-noise eigentasks improve performance for tasks such as classification in a way that is robust to noise and overfitting. We also present experimental and numerical analyses suggesting that entanglement enhances learning capacity by reducing noise in eigentasks. Our results are broadly relevant to quantum machine learning and sensing applications

Modeling the behavior of superconducting electronic circuits containing Josephson junctions is crucial for the design of superconducting information processors and devices. In this paper, we introduce DEC-QED, a computational approach for modeling the electrodynamics of superconducting electronic circuits containing Josephson junctions in arbitrary three-dimensional electromagnetic environments. DEC-QED captures the nonlinear response and induced currents in BCS superconductors and accurately captures phenomena such as the Meissner effect, flux quantization, and Josephson effects. Using a spatial coarse-graining formulation based on discrete exterior calculus (DEC), DEC-QED can accurately simulate transient and long-time dynamics in superconductors. The expression of the entire electrodynamic problem in terms of the gauge-invariant flux field and charges makes the resulting classical field theory suitable for second quantization.

We study the radiative properties -- the Lamb shift, Purcell decay rate and the spontaneous emission dynamics -- of an artificial atom coupled to a long, multimode cavity formed by an array of Josephson junctions. Introducing a tunable coupling element between the atom and the array, we demonstrate that such a system can exhibit a crossover from a perturbative to non-perturbative regime of light-matter interaction as one strengthens the coupling between the atom and the Josephson junction array (JJA). As a consequence, the concept of spontaneous emission as the occupation of the local atomic site being governed by a single complex-valued exponent breaks down. This breakdown, we show, can be interpreted in terms of formation of hybrid atom-resonator modes with radiative losses that are non-trivially related to the effective coupling between individual modes. We develop a singular function expansion approach for the description of the open quantum system dynamics in such a multimode non-perturbative regime. This modal framework generalizes the normal mode description of quantum fields in a finite volume, incorporating exact radiative losses and incident quantum noise at the delimiting surface. Our results are pertinent to recent experiments with Josephson atoms coupled to high impedance Josephson junction arrays.