In recent years, hybrid semiconductor–superconductor qubits have emerged as a compelling alternative to more traditional qubit platforms, owing to their unique combination of material properties and device-level tunability. A key feature of these systems is their gate-tunable Josephson coupling, which enables the realization of superconducting qubit architectures with full electric field control and which offers a promising route toward scalable and low-crosstalk quantum processors. One of the central motivations behind this approach is the possibility of unifying the advantages of superconducting and spin-based qubits. This approach to creating hybrid qubits can be achieved, for example, by encoding quantum information in the spin degree of freedom of a quasiparticle occupying an Andreev bound state, thereby combining long coherence times with fast and flexible control. Recent progress has been further accelerated by the bottom-up engineering of Andreev bound states into coupled quantum dot arrays, leading to the realization of novel device architectures such as minimal Kitaev chains hosting (poor man’s) Majorana states, precursors of topologically protected Majorana zero modes, and, very recently, to the first coherent control of a Majorana parity qubit. In parallel, alternative strategies have focused on the design of Hamiltonian-protected hybrid qubits, including Cooper pair parity qubits and fluxon parity qubits, which aim to enhance robustness against local noise and decoherence. These approaches leverage the interplay between superconducting phase dynamics and the discrete charge or flux degrees of freedom in hybrid junctions. In this article, we review recent theoretical and experimental advances in the development of semiconductor–superconductor hybrid qubits. We provide a comprehensive overview of the underlying physical mechanisms, device implementations, and emerging architectures, with particular emphasis on their potential to enable (topologically) protected quantum information processing. While many of these qubit designs remain at the proof-of-concept stage, the rapid pace of progress suggests that practical demonstrations may be within reach in the near future.
The NVFP4 lower-precision format, supported in hardware by NVIDIA Blackwell GPUs, promises to allow, for the first time, end-to-end fully-quantized pre-training of massive models such as LLMs. Yet, existing quantized training methods still sacrifice some of the representation capacity of this format in favor of more accurate unbiased quantized gradient estimation by stochastic rounding (SR), losing noticeable accuracy relative to standard FP16 and FP8 training. In this paper, improve the state of the art for quantized training in NVFP4 via a novel unbiased quantization routine for micro-scaled formats, called MS-EDEN, that has more than 2x lower quantization error than SR. We integrate it into a novel fully-NVFP4 quantization scheme for linear layers, called Quartet II. We show analytically that Quartet II achieves consistently better gradient estimation across all major matrix multiplications, both on the forward and on the backward passes. In addition, our proposal synergizes well with recent training improvements aimed specifically at NVFP4. We further validate Quartet II on end-to-end LLM training with up to 1.9B parameters on 38B tokens. We provide kernels for execution on NVIDIA Blackwell GPUs with up to 4.2x speedup over BF16. Our code is available at https://github.com/IST-DASLab/Quartet-II .
With probability 1, we assess the average behaviour of various arithmetic functions at the values of degree d polynomials f that are ordered by height. This allows us to establish averaged versions of the Bateman-Horn conjecture, the polynomial Chowla conjecture and to address a basic question about the integral Hasse principle for norm form equations. Moreover, we are able to quantify the error term in the asymptotics and the size of the exceptional set of f, both with arbitrary logarithmic power savings.
Our work establishes a three-term asymptotic expansion of the ground state energy of a dilute gas of spin 1/2 fermions with repulsive short-range interactions, validating a formula predicted by Huang and Yang in 1957. The formula is universal in the sense that it holds for a large class of interaction potentials and depends on those only via their scattering length. We have recently proved an upper bound on the ground state energy of the desired form, and the present work completes the program by proving the matching lower bound.
Kinetically constrained models were originally introduced to capture slow relaxation in glassy systems, where dynamics are hindered by local constraints instead of energy barriers. Their quantum counterparts have recently drawn attention for exhibiting highly degenerate eigenstates at zero energy—known as zero modes—stemming from chiral symmetry. Yet, the structure and implications of these zero modes remain poorly understood. In this work, we focus on the properties of the zero mode subspace in quantum kinetically constrained models with a U(1) particle-conservation symmetry. We use the U(1) East, which lacks inversion symmetry, and the inversion-symmetric U(1) East-West models to illustrate our two main results. First, we observe that the simultaneous presence of constraints and chiral symmetry generally leads to a parametric increase in the number of zero modes due to the fragmentation of the many-body Hilbert space into disconnected sectors. Second, we generalize the concept of compact localized states from single-particle physics and introduce the notion of collective bound states, a special kind of nonergodic eigenstates that are robust to enlarging the system size. We formulate sufficient criteria for their existence, arguing that the degenerate zero mode subspace plays a central role, and demonstrate bound states in both example models and in a two-dimensional model, the U(1) North-East, and in the pair-flip model, a system without particle conservation. Our results motivate a systematic study of bound states and their relation to ergodicity breaking, transport, and other properties of quantum kinetically constrained models.
