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A bstract We compute conductivities of strongly-interacting and non-uniform charge densities dual to inhomogeneous anti-de Sitter-black hole spacetimes. Backreacting bulk scalars with periodic boundary profiles, we construct generalizations of ReissnerNordström-AdS that interpolate between those used in two previous studies — one that reports power-law scaling for the boundary optical conductivity and one that does not. We find no evidence for power-law scaling of the conductivity, thereby corroborating the previous negative result that gravitational crystals are insufficient to generate the power-law mid-infrared conductivity observed in cuprate superconductors.

Quantum error correction 1 , 2 , 3 – 4 provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, in which 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. Here we present two below-threshold surface code memories on our newest generation of superconducting processors, Willow: 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 Λ  = 2.14 ± 0.02 when increasing the code distance by 2, culminating in a 101-qubit distance-7 code with 0.143% ± 0.003 per cent error per cycle of error correction. This logical memory is also beyond breakeven, exceeding the lifetime of its best physical qubit by a factor of 2.4 ± 0.3. Our system maintains below-threshold performance when decoding in real time, achieving an average decoder latency of 63 microseconds at distance 5 up to a million cycles, with a cycle time of 1.1 microseconds. We also 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 × 10 9 cycles. Our results indicate device performance that, if scaled, could realize the operational requirements of large-scale fault-tolerant quantum algorithms. Two below-threshold surface code memories on superconducting processors markedly reduce logical error rates, achieving high efficiency and real-time decoding, indicating potential for practical large-scale fault-tolerant quantum algorithms.

We present the full charge and energy diffusion constants for the Einstein-Maxwell dilaton (EMD) action for Lifshitz spacetime characterized by a dynamical critical exponent \\(z\\). Therein we compute the fully renormalized static thermodynamic potential explicitly, which confirms the forms of all thermodynamic quantities including the Bekenstein-Hawking entropy and Smarr-like relationship. Our exact computation demonstrates a modification to the Lifshitz Ward identity for the EMD theory. For transport, we target our analysis at finite chemical potential and include axion fields to generate momentum dissipation. While our exact results corroborate anticipated bounds, we are able to demonstrate that the diffusivities are governed by the engineering dimension of the diffusion coefficient, \\([D]=2-z\\). Consequently, a \\(\\)-function defined as the derivative of the trace of the diffusion matrix with respect to the effective lattice spacing changes sign precisely at \\(z=2\\). At \\(z=2\\), the diffusion equation exhibits perfect scale invariance and the corresponding diffusion constant is the pure number \\(1/d_s\\) for both the charge and energy sectors, where \\(d_s\\) is the number of spatial dimensions. Further, we find that as \\(zınfty\\), the charge diffusion constant vanishes, indicating charge localization. Deviation from universal decoupled transport obtains when either the chemical potential or momentum dissipation are large relative to temperature, an echo of strong thermoelectric interactions.

The electromagnetic couplings among resonators and transmission lines are discussed. A single resonator coupled to an N-port microwave network is formulated. The equation of motion of the resonator and the input-output relations of the network are obtained. Methods of extracting the couplings from electromagnetic solutions are also discussed.

The precise engineering of electromagnetic couplings is paramount for constructing scalable and highfidelity superconducting quantum processors. While essential for orchestrating qubit operations, these couplings also present significant design challenges, including the mitigation of crosstalk and the management of environmental decoherence. A clear and unified theoretical framework is therefore crucial for the design, simulation, and analysis of these complex quantum circuits. This paper presents a comprehensive theoretical treatment of the fundamental electromagnetic coupling mechanisms in superconducting devices. Starting from first principles, we formulate the equations of motion and derive the input-output relations for canonical systems, including a single resonator coupled to a multi-port microwave network, interacting resonators, and coupled transmission lines. We review rigorous definitions for key parameters such as the energy decay rate (appa) and the dimensionless coupling coefficient () and connect these formalisms to practical methods of parameter extraction from electromagnetic simulations. This work provides a rigorous and pedagogical foundation for understanding and modeling linear electromagnetic interactions, serving as a vital resource for the development of advanced superconducting quantum hardware.

Electromagnetic finite element method (FEM) implementations using traditional basis functions struggle to accurately represent field behavior near singular features such as conducting wedges. To combat this, specialized singular basis functions have been introduced to directly model the singular fields in these regions, leading to substantially improved performance. While these efforts have been pursued extensively in 2D, few functions have been developed for 3D elements. In this work, we develop basis functions for this in tetrahedra. Unlike prior functions, these basis functions are additive, meaning they are included alongside the standard vector basis functions to achieve more robust performance. Further, these functions are designed to be adaptable to tetrahedra touching several unique singular features by using combinations of basis functions singular with respect to each node and edge in the element, making them applicable to highly complex geometries. Higher-order interpolatory versions of the basis functions for modeling singular behavior with greater accuracy are also provided. These basis functions lead to substantial improvements in accuracy relative to the standard basis functions, and allow otherwise expensive simulations to be performed at far lower costs. As an application example, we perform simulations to extract critical quantities for designing superconducting qubits that significantly depend on the behavior of singular fields. In Ansys HFSS, this took 21.27 hours and a peak memory usage of 6.23 TB with 800 processors available, while using our singular basis functions achieved comparable results in 196 seconds while using 27.24 GB of memory and only 16 processors. Due to these benefits, our singular basis functions could be applied to enable design optimization of electromagnetic geometries with dominantly singular behavior, such as superconducting qubits.

High-fidelity numerical methods that model the physical layout of a device are essential for the design of many technologies. For methods that characterize electromagnetic effects, these numerical methods are referred to as computational electromagnetics (CEM) methods. Although the CEM research field is mature, emerging applications can still stress the capabilities of the techniques in use today. The design of superconducting circuit quantum devices falls in this category due to the unconventional material properties and important features of the devices covering nanometer to centimeter scales. Such multiscale devices can stress the fundamental properties of CEM tools which can lead to an increase in simulation times, a loss in accuracy, or even cause no solution to be reliably found. While these challenges are being investigated by CEM researchers, knowledge about them is limited in the broader community of users of these CEM tools. This review is meant to serve as a practical introduction to the fundamental aspects of the major CEM techniques that a researcher may need to choose between to model a device, as well as provide insight into what steps they may take to alleviate some of their challenges. Our focus is on highlighting the main concepts without rigorously deriving all the details, which can be found in many textbooks and articles. After covering the fundamentals, we discuss more advanced topics related to the challenges of modeling multiscale devices with specific examples from superconducting circuit quantum devices. We conclude with a discussion on future research directions that will be valuable for improving the ability to successfully design increasingly more sophisticated superconducting circuit quantum devices. Although our focus and examples are taken from this area, researchers from other fields will still benefit from the details discussed here.

Lattice gauge theories (LGTs) can be employed to understand a wide range of phenomena, from elementary particle scattering in high-energy physics to effective descriptions of many-body interactions in materials. Studying dynamical properties of emergent phases can be challenging as it requires solving many-body problems that are generally beyond perturbative limits. Here, we investigate the dynamics of local excitations in a \\(Z_2\\) LGT using a two-dimensional lattice of superconducting qubits. We first construct a simple variational circuit which prepares low-energy states that have a large overlap with the ground state; then we create charge excitations with local gates and simulate their quantum dynamics via a discretized time evolution. As the electric field coupling constant is increased, our measurements show signatures of transitioning from deconfined to confined dynamics. For confined excitations, the electric field induces a tension in the string connecting them. Our method allows us to experimentally image string dynamics in a (2+1)D LGT from which we uncover two distinct regimes inside the confining phase: for weak confinement the string fluctuates strongly in the transverse direction, while for strong confinement transverse fluctuations are effectively frozen. In addition, we demonstrate a resonance condition at which dynamical string breaking is facilitated. Our LGT implementation on a quantum processor presents a novel set of techniques for investigating emergent excitations and string dynamics.

Quantum error correction is essential for bridging the gap between the error rates of physical devices and the extremely low logical error rates required for quantum algorithms. Recent error-correction demonstrations on superconducting processors have focused primarily on the surface code, which offers a high error threshold but poses limitations for logical operations. In contrast, the color code enables much more efficient logic, although it requires more complex stabilizer measurements and decoding techniques. Measuring these stabilizers in planar architectures such as superconducting qubits is challenging, and so far, realizations of color codes have not addressed performance scaling with code size on any platform. Here, we present a comprehensive demonstration of the color code on a superconducting processor, achieving logical error suppression and performing logical operations. Scaling the code distance from three to five suppresses logical errors by a factor of \\(_3/5\\) = 1.56(4). Simulations indicate this performance is below the threshold of the color code, and furthermore that the color code may be more efficient than the surface code with modest device improvements. Using logical randomized benchmarking, we find that transversal Clifford gates add an error of only 0.0027(3), which is substantially less than the error of an idling error correction cycle. We inject magic states, a key resource for universal computation, achieving fidelities exceeding 99% with post-selection (retaining about 75% of the data). Finally, we successfully teleport logical states between distance-three color codes using lattice surgery, with teleported state fidelities between 86.5(1)% and 90.7(1)%. This work establishes the color code as a compelling research direction to realize fault-tolerant quantum computation on superconducting processors in the near future.

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 \\(\\) = 2.14 \\(\\) 0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% \\(\\) 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 \\(\\) 0.3. We maintain below-threshold performance when decoding in real time, achieving an average decoder latency of 63 \\(\\)s at distance-5 up to a million cycles, with a cycle time of 1.1 \\(\\)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 \\(\\) 10\\(^9\\) cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms.