The entropy accumulation theorem—and its subsequent generalized version—is a powerful tool in the security analysis of many device-dependent and device-independent cryptography protocols. However, it has the drawback that the finite-size bounds it yields are not necessarily optimal; furthermore, it relies on the construction of an affine min-tradeoff function, which can often be challenging to construct optimally in practice. In this work, we address both of these challenges simultaneously by deriving a new entropy accumulation bound. Our bound yields significantly better finite-size performance and can be computed as an intuitively interpretable convex optimization, without any specification of affine min-tradeoff functions. Furthermore, it can be applied directly at the level of Rényi entropies if desired, yielding fully-Rényi security proofs. Our proof techniques are based on elaborating on a connection between entropy accumulation and the frameworks of quantum probability estimation or $f$-weighted Rényi entropies; in the process, we obtain some new results with respect to those frameworks as well. In particular, those findings imply that our bounds apply to prepare-and-measure protocols without the virtual tomography procedures or repetition-rate restrictions previously required for entropy accumulation.

The realization of large-scale quantum computers requires not only quantum error correction but also fault-tolerant (FT) operations to handle errors that propagate into harmful errors. Recently, flag-based protocols have been introduced that use ancillary qubits to flag harmful errors. However, there is no clear recipe for finding a FT quantum circuit with flag-based protocols, especially when we consider hardware constraints, such as the qubit connectivity and available gate set. This work presents a novel approach to automatically discover compact and hardware-adapted FT quantum circuits to make significant progress towards scalable FT quantum computing. We employ reinforcement learning (RL) as an enabling tool, leveraging a fast, parallelized stabilizer quantum circuit simulator and a nontrivial reward function specifically adapted to the problem. We show that, in the task of FT logical state preparation, RL discovers not only circuits with fewer gates and ancillary qubits than published results but also novel circuits without and with hardware constraints of up to distance-5 codes with 25 physical qubits, and they can be implemented directly in experiments. Furthermore, RL allows for straightforward exploration of different qubit connectivities and the use of transfer learning to accelerate the discovery. More generally, our work sets the framework towards the use of RL or other machine learning techniques for FT quantum circuit discovery with hardware constraints to make real progress towards the realization of large-scale quantum computers, addressing tasks beyond state preparation, including magic state preparation, logical gate synthesis, and syndrome measurement.

Collective modes in multilayer graphene, such as plasmons and phonons, exhibit sensitivity to displacement fields and interlayer coupling, distinguishing them from their counterparts in single-layer graphene. Here, we engineer collective modes in charge-transfer heterostructures composed of multilayer graphene and $\alpha \text{−}{\mathrm{RuCl}}_3$. In heterostructures with a single $\alpha \text{−}{\mathrm{RuCl}}_3$ interface, the charge transfer generates displacement fields up to 7 V/nm at the interface between $\alpha \text{−}{\mathrm{RuCl}}_3$ and the adjacent graphene layer—the highest value achieved through charge-transfer methods. As a result of the broken inversion symmetry, we discover enhanced nonlinear optical response and modified phonon selection rules. Conversely, we find that multilayer graphene sandwiched between two $\alpha \text{−}{\mathrm{RuCl}}_3$ flakes causes displacement fields to cancel. There, we achieve carrier densities as high as $8\times {10}^{13}{\mathrm{cm}}^{-2}$ in multilayer graphene and restore the phonon selection rules to their unperturbed state. Meanwhile, we demonstrate that plasmonic properties derive from the depletion of multiple valence bands. As a result of the quasilinear band dispersion, these “Dirac multiband plasmons” are relatively unaffected by displacement fields. On the other hand, the inverted heterostructure sequence—two multilayer graphene sheets encapsulating $\alpha \text{−}{\mathrm{RuCl}}_3$—activates significant alteration of the plasmons via interlayer plasmon-plasmon coupling. Hence, multilayer graphene and $\alpha \text{−}{\mathrm{RuCl}}_3$ heterostructures offer a gate-free platform for engineering collective modes derived from inversion symmetry and interlayer coupling.

Recent investigations have unveiled exotic quantum phases that elude characterization by simple bipartite correlation functions. In these phases, long-range entanglement arising from tripartite correlations plays a central role. Consequently, the study of multipartite correlations has become a focal point in modern physics. Here, conditional mutual information (CMI) is one of the most well-established information-theoretic measures, adept at encapsulating the essence of various exotic phases, including topologically ordered ones. Within the realm of quantum many-body physics, it has been a long-sought goal to establish a quantum analog to the Hammersley-Clifford theorem that bridges the two concepts of the Gibbs state and the Markov network. This theorem posits that the correlation length of CMI remains short-range across all thermal equilibrium quantum phases. In this work, we demonstrate that CMI exhibits exponential decay with respect to distance, with its correlation length increasing polynomially with respect to the inverse temperature. While this clustering theorem has previously been believed to hold for high temperatures devoid of thermal phase transitions, it has remained elusive at low temperatures, where genuine long-range entanglement is corroborated to exist by the quantum topological order. Our findings unveil that, even at low temperatures, a broad class of tripartite entanglement cannot manifest in the long-range regime. To achieve the proof, we establish a comprehensive formalism for analyzing the locality of effective Hamiltonians on subsystems, commonly known as the “entanglement Hamiltonian” or “Hamiltonian of mean force.” As one outcome of our analyses, we enhance the prior clustering theorem concerning bipartite entanglement. In essence, we investigate genuine bipartite entanglement that extends beyond the limitations of the positive-partial-transpose class.

Optical frequency combs have emerged as a cornerstone for a wide range of areas, including spectroscopy, ranging, optical clocks, time and frequency transfer, waveform synthesis, and communications. However, quantum-mechanical fluctuations of the optical carrier impose fundamental performance limits on the precision of classical optical frequency combs, particularly in their use for interferometry and spectroscopy. Entanglement, as a quintessential quantum resource, allows for surpassing the fundamental limits of classical systems. Here, we introduce entanglement into the realm of optical frequency combs, formulating entangled dual-comb spectroscopy (EDCS) that surmounts the fundamental limits of classical DCS. EDCS capitalizes on tailored entangled structures across the frequency comb, enabling simultaneous detection of all comb lines below the standard quantum limit of classical DCS. Applying EDCS in gas detection, we achieve a 2.6-dB enhancement in signal-to-noise ratio and a 1.7-fold reduction in integration time over classical DCS, rendering EDCS particularly suited for dynamic chemical and biological sensing, where fast, precise measurements subject to power constraints are required. EDCS opens a new avenue for exploiting quantum frequency combs, underscoring their prospects in a plethora of applications in precision metrology, spectroscopy, and timekeeping.

Preparation of high-fidelity logical magic states has remained as a necessary but daunting step towards building a large-scale fault-tolerant quantum computer. One approach is to fault-tolerantly prepare a magic state in one code and then switch to another, a method known as code switching. We experimentally demonstrate this protocol on an ion-trap quantum processor, yielding a logical magic state encoded in an error-correcting code with state-of-the-art logical fidelity. Our experiment is based on the first demonstration of code switching between color codes, from the fifteen-qubit quantum Reed-Muller code to the seven-qubit Steane code. We prepare an encoded magic state in the Steane code with 82.58% probability, with an infidelity of at most $5.1(2.7)\times {10}^{-4}$. The reported infidelity is lower than the leading infidelity of the physical operations utilized in the protocol by a factor of at least 2.7, indicating the quantum processor is below the pseudothreshold. Furthermore, we create two copies of the magic state in the same quantum processor and perform a logical Bell basis measurement for a sample-efficient certification of the encoded magic state. The high-fidelity magic state can be combined with the already-demonstrated fault-tolerant Clifford gates, state preparation, and measurement of the 2D color code, completing a universal set of fault-tolerant computational primitives with logical error rates equal or better than the physical two-qubit error rate.

Pauli blocking in Fermi liquids imposes strong phase-space constraints on quasiparticle lifetimes, leading to a well-known quadratic-in-temperature decay rate of quasiparticle modes at low temperatures. In two-dimensional systems, however, even longer-lived modes are predicted (dubbed “odd-parity” modes) that involve a collective deformation of the Fermi distribution. Here, we present an efficient method to evaluate the full spectrum of relaxational eigenmodes of a Fermi liquid within kinetic theory. We employ this method to study the experimentally relevant case of a Fermi liquid with screened Coulomb interactions and map out the decay rates of quasiparticle modes beyond the asymptotic low-temperature limit up to the Fermi temperature, thus covering the entire temperature range of typical experiments. We confirm the existence of anomalously long-lived odd-parity modes and provide a comprehensive classification and detailed analysis of the relaxation spectrum. In particular, we find that (i) the odd-parity effect in the decay rates extends to temperatures as large as $T=0.15T_F$, (ii) there is only a small number of long-lived odd-parity modes, with an infinite number of remaining modes that show standard Fermi-liquid scaling, and (iii) the ratio between the odd- and even-parity lifetimes is tunable with the Coulomb interaction strength, in addition to temperature, which reflects a difference in the microscopic relaxation mechanism of the modes. Our findings provide a comprehensive description of the nonequilibrium relaxation behavior of two-dimensional electron gases and bridge a significant gap in our understanding of these systems.

Tensor networks provide a natural language for noninvertible symmetries in general Hamiltonian lattice models. We use ZX-diagrams, which are tensor network presentations of quantum circuits, to define a noninvertible operator implementing the Wegner duality in $3+1\mathrm{D}$ lattice ${\mathbb{Z}}_2$ gauge theory. The noninvertible algebra, which mixes with lattice translations, can be efficiently computed using ZX-calculus. We further deform the ${\mathbb{Z}}_2$ gauge theory while preserving the duality and find a model with nine exactly degenerate ground states on a torus, consistent with the Lieb-Schultz-Mattis-type constraint imposed by the symmetry. Finally, we provide a ZX-diagram presentation of the noninvertible duality operators (including noninvertible parity and reflection symmetries) of generalized Ising models based on graphs, encompassing the $1+1\mathrm{D}$ Ising model, the three-spin Ising model, the Ashkin-Teller model, and the $2+1\mathrm{D}$ plaquette Ising model. The mixing (or lack thereof) with spatial symmetries is understood from a unifying perspective based on graph theory.

Beginning with high-$T_c$ cuprate materials, it has been observed that many superconductors exhibit so-called “Homes scaling,” in which the zero-temperature superfluid density ${\rho }_{s0}$ is proportional to the product of the normal-state dc conductivity and the superconducting transition temperature ${\sigma }_{\mathrm{dc}}{T}_c$. For conventional, $s$-wave superconductors, such scaling has been shown to be a natural consequence of elastic-scattering disorder, not only in the extreme dirty limit, but across a broad range of scattering parameters. Here we show that when an analogous calculation is carried out for elastic scattering in $d$-wave superconductors, a stark contrast emerges, with ${\rho }_{s0}\propto {({\sigma }_{\mathrm{dc}}{T}_c)}^2$ in the dirty limit, in apparent violation of Homes scaling. Within a simple approximate Migdal-Eliashberg treatment of inelastic scattering, we show how the observed Homes scaling is recovered. The normal-state behavior of near-optimally-doped cuprates is dominated by inelastic scattering, but significant deviations from Homes scaling occur for disorder-dominated cuprate systems, such as underdoped ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{6.333}$ and overdoped ${\mathrm{La}}_{2-x}{\mathrm{Sr}}_x{\mathrm{CuO}}_4$, and in very clean materials with little inelastic scattering, such as ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$. We present a revised analysis where both axes of the original Homes scaling plot are normalized by the Drude plasma weight ${\omega }_{p,D}^2$ and show that a new universal scaling emerges, in which the superfluid fractions of dirty $s$-wave and dirty $d$-wave superconductors coalesce to a single point at which normal-state scattering is occurring at the Planckian bound. The combined result is a new tool for classifying superconductors in terms of order parameter symmetry, as well as scattering strength and character. Although our model starts from a Fermi-liquid assumption, it describes underdoped cuprates surprisingly well.

The discovery of a new, $e^{+}{e}^{-}$ reaction channel in deuteron-deuteron (DD) fusion at very low energies might have major implications for understanding primordial and stellar nucleosynthesis, where electron-positron reaction channels are typically not considered. It could also enable research on metal hydride fusion, potentially paving the way for the design and construction of next-generation fusion energy sources. Following the first experimental indications of electron emission, we present here an extensive experimental study confirming emission of high-energy electrons from DD reaction at very low energy. A simultaneous use of Si charged particle detectors of different thicknesses and large-volume NaI(Tl) and HPGe detectors has allowed the determination of the branching ratios between emitted protons, neutrons, and $e^{+}{e}^{-}$ pairs for deuteron energies down to 5 keV. The high-energy positrons could be unambiguously detected by their bremsstrahlung spectra and annihilation radiation, supported by the geant4 Monte Carlo simulations. The theoretical calculations, based on a destructive interference between the threshold resonance and the known broad resonance in ${}^4\mathrm{He}$, agree very well with experimentally observed increase of branching ratios for lowering projectile energies. The partial width of the threshold resonance for the internal $e^{+}{e}^{-}$ pair creation should be at least 10 times larger than that of the proton channel.

The ability to distribute heralded entanglement between distant matter nodes is a primitive for the implementation of large-scale quantum networks. Some of the most crucial requirements for future applications include high heralding rates at telecom wavelengths, multiplexed operation, and on-demand retrieval of stored excitations for synchronization of separate quantum links. Despite tremendous progress in various physical systems, the demonstration of telecom-heralded entanglement between quantum nodes featuring both multiplexed operation and on-demand retrieval remains elusive. In this work, we combine narrow band parametric photon-pair sources and solid-state quantum memories based on rare-earth doped crystals to demonstrate telecom-heralded entanglement between spatially separated spin-wave quantum memories with fully adjustable recall time and temporal multiplexing of 15 modes. In a first experiment, the storage in the spin state is conditioned on the entanglement heralding. We take advantage of the control over readout pulse phase to achieve feedforward conditional phase shifts on the stored photons depending on which heralding detector clicked. We exploit this effect to double the entanglement heralding rate for a given quantum state up to $510\text{counts}/\mathrm{s}$, with an associated detection rate of $0.32\text{counts}/\mathrm{s}$ and measured positive concurrence by up to 6 standard deviations. In a second experiment, we simulate the communication time of a long-distance link by implementing an unconditional storage scheme with a dead time of $100\mathrm{\mu }\mathrm{s}$. We take advantage of temporal multiplexing to increase the entanglement rates by a factor of 15 with respect to single mode storage, reaching a value of $22\text{counts}/\mathrm{s}$ per heralding detector. These results establish our architecture as a prime candidate for the implementation of scalable high-rate quantum network links.

Electrons trapped on the surface of cryogenic substrates (liquid helium, solid neon, or hydrogen) are an emerging platform for quantum information processing made attractive by the inherent purity of the electron environment, the scalability of trapping devices, and the predicted long lifetime of electron spin states. Here we demonstrate the spatial control and detection of single electrons above the surface of liquid helium at temperatures above 1 K. A superconducting coplanar waveguide resonator is used to read out the charge state of an electron trap defined by gate electrodes beneath the helium surface. Dispersive frequency shifts are observed as the trap is loaded with electrons, from several tens down to single electrons. These frequency shifts are in good agreement with our theoretical model that treats each electron as a classical oscillator coupled to the cavity field. This sensitive charge readout scheme can aid efforts to develop large-scale quantum processors that require the high cooling powers available in cryostats operating above 1 K.

Probabilistic computing with p-bits is emerging as a computational paradigm for machine learning and for facing combinatorial optimization problems (COPs) with the so-called probabilistic Ising machines (PIMs). From a hardware point of view, the key elements that characterize a PIM are the random number generation, the nonlinearity, the network of coupled probabilistic bits, and the energy-minimization algorithm. Regarding the energy-minimization algorithm in this work we show that PIMs using the simulated quantum annealing (SQA) schedule exhibit better performance as compared to simulated annealing and parallel tempering in solving a number of COPs, such as maximum satisfiability problems, the planted Ising problem, and the traveling salesman problem. Additionally, we design and simulate the architecture of a fully connected CMOS-based PIM that is able to run the SQA algorithm having a spin-update time of 8 ns with a power consumption of 0.22 mW. Our results also show that SQA increases the reliability and the scalability of PIMs by compensating for device variability at an algorithmic level enabling the development of their implementation combining CMOS with different technologies such as spintronics. This work shows that the characteristics of the SQA are hardware agnostic and can be applied in the codesign of any hybrid analog-digital Ising machine implementation. Our results open a promising direction for the implementation of a new generation of reliable and scalable PIMs.

The XENONnT experiment has achieved an exceptionally low ${}^{222}\mathrm{Rn}$ activity concentration within its inner 5.9 tonne liquid xenon detector of $(0.90\pm 0.02\mathrm{stat}\pm 0.07\mathrm{syst})\mathrm{\mu }\mathrm{Bq}{\mathrm{kg}}^{-1}$, equivalent to about 430 ${}^{222}\mathrm{Rn}$ atoms per tonne of xenon. This was achieved by active online radon removal via cryogenic distillation after stringent material selection. The achieved ${}^{222}\mathrm{Rn}$ activity concentration is 5 times lower than that in other currently operational multitonne liquid xenon detectors engaged in dark matter searches. This breakthrough enables the pursuit of various rare event searches that lie beyond the confines of the standard model of particle physics, with world-leading sensitivity. The ultralow ${}^{222}\mathrm{Rn}$ levels have diminished the radon-induced background rate in the detector to a point where it is for the first time comparable to the solar neutrino-induced background, which is poised to become the primary irreducible background in liquid xenon-based detectors.

The optical properties of atomically thin semiconductors are dominated by excitons, tightly bound electron-hole pairs, which give rise to particularly rich and remarkable physics. Despite their importance, the microscopic formation mechanisms of excitons remain very poorly understood due to the complex interplay of concurrent phenomena occurring on an ultrafast timescale. Here, we investigate the exciton formation processes in 2D materials based on transition-metal dichalcogenide (TMD) monolayers using a technique based on the control of excitation light polarization. It allows us to distinguish between the two competing models of exciton formation: geminate and bimolecular formation. The geminate process is the direct formation of the exciton from the initially photogenerated electron-hole pair before the loss of correlation between them, whereas the bimolecular process corresponds to the random binding of free-electron-hole pairs from the initially photogenerated plasma. These processes control the exciton formation time. Our findings reveal that the luminescence intensity is higher by up to 40% for circularly polarized excitation compared to linearly polarized excitation for laser energy above the free-carrier gap. We show that this spin-dependent exciton emission is a fingerprint of the bimolecular formation process. Importantly, we observe that exciton linear polarization (valley coherence) persists even for laser excitation energies exceeding the gap. We demonstrate that it is the result of a fraction of excitons formed by a geminate process. This shows that two formation processes coexist for excitation energies above the gap, where both mechanisms operate concurrently. Similar results obtained on the two most emblematic materials of the TMD semiconductor family, ${\mathrm{WSe}}_2$ and ${\mathrm{MoS}}_2$ monolayers, confirm this dual formation mechanism. These findings overturn the prevailing simplistic view of purely geminate or bimolecular exciton formation and provide crucial insights into exciton physics and means to preserve spin or valley coherence. Our methodology can be applied to a broader range of semiconductor for investigating the intricate interplay between nonresonant excitation conditions, Coulomb interactions, electron-phonon couplings, excitonic dynamics, and quantum coherence.

Two-dimensional (2D) crystals proved revolutionary soon after graphene was discovered in 2004. However, 2D amorphous materials down to a single layer of atoms only became accessible in 2020, and they remain largely unexplored. In particular, the thermophysical properties of amorphous materials are of great interest upon transition from 3D to 2D. Here, we probe and simulate thermal transport in monolayer amorphous carbon (MAC). An ultralow cross-plane thermal conductivity ($\kappa$) is measured for van der Waals stacked multilayers, which is comparable to that of randomly stacked graphene despite the extra disorder in MAC. This result reveals the predominant role of the weak interlayer interactions in 2D materials. Meanwhile, an unexpectedly high in-plane $\kappa$ is obtained for freestanding monolayers, which is a few times higher than what is predicted by conventional wisdom for 3D amorphous carbon with a similar ${\mathrm{sp}}^2$ fraction. This observation is primarily attributed to the dimensionality-induced reduction of anharmonicity and the unique low-frequency out-of-plane vibrational modes in MAC. Amorphous materials at the 2D limit open up new avenues for understanding and manipulating heat at the atomic scale.

Antiferromagnets that break both space-time reversal and translation-spin-rotation symmetries were recently predicted [L.-D. Yuan, Z. Wang, J.-W. Luo, E. I. Rashba, and A. Zunger, Phys. Rev. B 102, 014422 (2020)] to possess splitting between the otherwise spin-degenerate energy bands even without the relativistic spin-orbit coupling (SOC). Here, we point out that such nonrelativistic spin splitting (NRSS)—in particular, “spin splitting type 4” (SST-4) symmetry-broken antiferromagnets—can be divided into subgroups having distinct patterns of spin splitting and spin textures, depending on additional auxiliary symmetries of spin interconversion and polarity. These SST-4 subgroups include the $\alpha$-type (no spin-interconverting symmetry) having spin splitting at the Brillouin zone center, as well as the $\beta$ subgroup in which a rotation symmetry is applied and determines the alternating spin texture and the $\gamma$ subgroup having exclusively reflection spin-interconverting symmetry. Unlike ferrimagnets, the $\alpha$-type compounds are shown to have tiny net magnetization at finite temperature and thus avoid the adverse effect of the stray field. The $\alpha$ and $\beta$ subgroups can be either polar or nonpolar, whereas the $\gamma$ subgroup is polar only, providing a basis for possible switching by external fields. The combination of NRSS-enabling and auxiliary symmetries is used here as a filter for identifying previously synthesized compounds as specific prototypes. Their characteristic splitting and spin polarization are calculated by density functional theory to the benefit of potential future experimental testing. Interesting results are as follows: (i) SOC-independent NRSS can exceed the magnitude of the SOC-induced Rashba and Dresselhaus spin splitting in semiconductors. (ii) Examples of predicted $\alpha$-type insulating compounds include ${\mathrm{BiCrO}}_3$ (nonpolar) and ${\mathrm{Mn}}_2{\mathrm{ScSbO}}_6$ (polar), the latter having spin splitting of 158 meV and 160 meV in the valence and conduction bands, respectively. (iii) The $\beta$-type (${\text{Cu}}_2{\text{Y}}_2{\text{O}}_5$ and ${\text{FeF}}_2$) and $\gamma$-type compounds (${\text{Mn}}_4{\text{Nb}}_2{\text{O}}_9$ and ${\text{FeScO}}_3$) are distinguished both by their auxiliary symmetries and polarity. The spin textures of $\gamma$-type compounds are mirror reflected with spin degeneracy of the wave vectors on that mirror. These observations will likely broaden the experimental playing field of NRSS physics significantly.

Quantum communication is needed to build powerful quantum computers and establish reliable quantum networks. At its basis lies the ability to generate and distribute entanglement to separate quantum systems, which can be used to run remote quantum operations on them or teleport quantum states from one system to another with the help of classical channels. To this end, it is useful to harness the resource of continuous-variable (CV) entanglement, since it can be efficiently and unconditionally produced by squeezing light in a nonlinear medium and can be easily manipulated, distributed, and measured using standard components. While various aspects of CV-based quantum communication have been successfully demonstrated in the optical domain, some key capabilities, such as entanglement swapping, have been lacking in the microwave domain. Here, we demonstrate three key elements of CV-based microwave quantum communication: (i) a Josephson mixer operating as a nondegenerate two-mode entangler with maximum measured logarithmic negativity $E_N=1.5$, (ii) a quantum teleportation apparatus, capable of teleporting vacuum and coherent states with a maximum fidelity of 73%, which exceeds the 50% classical limit and is mainly limited by intermediate losses in the setup, and (iii) an entanglement-swapping system which generates entanglement between two remote noninteracting modes via entanglement-swapping operations applied to input vacuum and coherent states with maximum measured logarithmic negativity $E_N=0.53$. Such hardware-efficient CV entanglement building blocks that are based on nondegenerate Josephson mixers could enable wide-ranging applications in modular quantum computation, quantum cryptography, and quantum communication.

Understanding the physics of the two-dimensional Hubbard model is widely believed to be a key step in achieving a full understanding of high-$T_{\mathrm{c}}$ cuprate superconductors. In recent years, progress has been made by large-scale numerical simulations at finite doping and, on the other hand, by microscopic theories able to capture the physics of individual charge carriers. In this work, we study single pairs of dopants in a cylindrical system using the density-matrix renormalization group algorithm. We identify two coexisting charge configurations that couple to the spin environment in different ways: a tightly bound configuration featuring (next-)nearest-neighbor pairs and a stripelike configuration of dopants on opposite sides of the cylinder, accompanied by a spin domain wall. Thus, we establish that the interplay between stripe order and uniform pairing, central to the models’ phases at finite doping, has its origin at the single-pair level. By interpolating between the Hubbard and the related $t\text{−}J$ model, we are able to quantitatively understand discrepancies in the pairing properties of the two models through the three-site hopping term usually omitted from the $t\text{−}J$ Hamiltonian. This term is closely related to a next-nearest-neighbor tunneling $t^{\prime }$, which we observe to upset the balance between the competing stripe and pair states on the two-dopant level.

We initiate the study of state complexity for continuous-variable quantum systems. Concretely, we consider a setup with bosonic modes and auxiliary qubits, where available operations include Gaussian one- and two-mode operations and single- and two-qubit operations as well as qubit-controlled phase-space displacements. We define the (approximate) complexity of a bosonic state by the minimum size of a circuit that prepares an approximation to the state in trace distance. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state $|{\mathsf{G}\mathsf{K}\mathsf{P}}_{\kappa ,\mathrm{\Delta }}⟩$. Here, ${\kappa }^{-2}$ is the variance of the envelope, and ${\mathrm{\Delta }}^2$ is the variance of the individual peaks. We show that the circuit accepts with constant probability and—conditioned on acceptance—the output state is polynomially close in $(\kappa ,\mathrm{\Delta })$ to the state $|{\mathsf{G}\mathsf{K}\mathsf{P}}_{\kappa ,\mathrm{\Delta }}⟩$. The size of our circuit is linear in $(\log 1/\kappa ,\log 1/\mathrm{\Delta })$. To our knowledge, this is the first protocol for GKP-state preparation with fidelity guarantees for the prepared state. We also show converse bounds, establishing that the linear circuit-size dependence of our construction is optimal. This fully characterizes the complexity of GKP states.