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Multidimensional cluster states are a key resource for robust quantum communication, measurement-based quantum computing and quantum metrology. Here, we present a device capable of emitting large-scale entangled microwave photonic states in a two dimensional ladder structure. The device consists of a pair of coupled superconducting transmon qubits which are each tuneably coupled to a common output waveguide. This architecture permits entanglement between each transmon and a deterministically emitted photonic qubit. By interleaving two-qubit gates with controlled photon emission, we generate 2 x n grids of time- and frequency-multiplexed cluster states of itinerant microwave photons. We measure a signature of localizable entanglement across up to 20 photonic qubits. We expect the device architecture to be capable of generating a wide range of other tensor network states such as tree graph states, repeater states or the ground state of the toric code, and to be readily scalable to generate larger and higher dimensional states., Comment: 21 pages, 19 figures

Despite the high measurement fidelity that can now be reached, the dispersive qubit readout of circuit quantum electrodynamics is plagued by a loss of its quantum nondemolition character and a decrease in fidelity with increased measurement strength. In this work we elucidate the nature of this dynamical process, which we refer to as transmon ionization. We develop a comprehensive framework which provides a unified physical picture of the origin of transmon ionization. This framework consists of three complementary levels of descriptions: a fully quantized transmon-resonator model, a semiclassical model where the resonator is treated as a classical drive on the transmon, and a fully classical model. Crucially, all three approaches preserve the full cosine potential of the transmon, and lead to similar predictions. This framework identifies the multiphoton resonances responsible for transmon ionization. It also allows us to efficiently compute numerical estimates of the photon number threshold for ionization, which are in remarkable agreement with recent experimental results. The set of tools developed within this work are both conceptually and computationally simple, and we expect them to become an integral part of the theoretical support of all circuit QED experiments.

We introduce a quantum-inspired approximation algorithm for MaxCut based on low-depth Clifford circuits. We start by showing that the solution unitaries found by the adaptive quantum approximation optimization algorithm (ADAPT-QAOA) for the MaxCut problem on weighted fully connected graphs are (almost) Clifford circuits. Motivated by this observation, we devise an approximation algorithm for MaxCut, \emph{ADAPT-Clifford}, that searches through the Clifford manifold by combining a minimal set of generating elements of the Clifford group. Our algorithm finds an approximate solution of MaxCut on an $N$-vertex graph by building a depth $O(N)$ Clifford circuit. The algorithm has runtime complexity $O(N^2)$ and $O(N^3)$ for sparse and dense graphs, respectively, and space complexity $O(N^2)$, with improved solution quality achieved at the expense of more demanding runtimes. We implement ADAPT-Clifford and characterize its performance on graphs with positive and signed weights. The case of signed weights is illustrated with the paradigmatic Sherrington-Kirkpatrick model, for which our algorithm finds solutions with ground-state mean energy density corresponding to $\sim94\%$ of the Parisi value in the thermodynamic limit. The case of positive weights is investigated by comparing the cut found by ADAPT-Clifford with the cut found with the Goemans-Williamson (GW) algorithm. For both sparse and dense instances we provide copious evidence that, up to hundreds of nodes, ADAPT-Clifford finds cuts of lower energy than GW., Comment: Revised version, several typos corrected, extended appendices

We study the competing effects of collective generalized measurements and interaction-induced scrambling in the dynamics of an ensemble of spin-1/2 particles at the level of quantum trajectories. This setup can be considered as analogous to the one leading to measurement-induced transitions in quantum circuits. We show that the interplay between collective unitary dynamics and measurements leads to three regimes of the average Quantum Fisher Information (QFI), which is a witness of multipartite entanglement, as a function of the monitoring strength. While both weak and strong measurements lead to extensive QFI density (i.e., individual quantum trajectories yield states displaying Heisenberg scaling), an intermediate regime of classical-like states emerges for all system sizes where the measurement effectively competes with the scrambling dynamics and precludes the development of quantum correlations, leading to sub-Heisenberg-limited states. We characterize these regimes and the crossovers between them using numerical and analytical tools, and discuss the connections between our findings, entanglement phases in monitored many-body systems, and the quantum-to-classical transition., Comment: 16 pages, 7 figures. Typos fixed, section IV. B added in v3 - v4 fixes minor formatting issues. Accepted for publication in Quantum

Time-dependent drives play a crucial role in quantum computing efforts with circuit quantum electrodynamics. They enable single-qubit control, entangling logical operations, as well as qubit readout. However, their presence can lead to deleterious effects such as large ac-Stark shifts and unwanted qubit transitions ultimately reflected into reduced control or readout fidelities. Qubit cloaking was introduced in Lled\'o, Dassonneville, et al. [C. Lled\'o, R. Dassonneville, A. Moulinas et al., Nat. Commun. \textbf{14}, 6313 (2023)] to temporarily decouple the qubit from the coherent photon population of a driven cavity, allowing for the application of arbitrary displacements to the cavity field while avoiding the deleterious effects on the qubit. For qubit readout, cloaking permits to prearm the cavity with an, in principle, arbitrarily large number of photons, in anticipation to the qubit-state-dependent evolution of the cavity field, allowing for improved readout strategies. Here we take a closer look at two of them. First, arm-and-release readout, introduced together with qubit cloaking, where after arming the cavity the cloaking mechanism is released and the cavity field evolves under the application of a constant drive amplitude. Second, an arm-and-longitudinal readout scheme, where the cavity drive amplitude is slowly modulated after the release. We show that the two schemes complement each other, offering an improvement over the standard dispersive readout for any values of the dispersive interaction and cavity decay rate, as well as any target measurement integration time. Our results provide a recommendation for improving qubit readout without changes to the standard circuit QED architecture., Comment: Revised version with typos corrected, comments welcome

We revisit well-known protocols in quantum metrology using collective spins and propose a unifying picture for optimal state preparation based on a semiclassical description in phase space. We show how this framework allows for quantitative predictions of the timescales required to prepare various metrologically useful states, and that these predictions remain accurate even for moderate system sizes, surprisingly far from the classical limit. Furthermore, this framework allows us to build a geometric picture that relates optimal (exponentially fast) entangled probe preparation to the existence of separatrices connecting saddle points in phase space. We illustrate our results with the paradigmatic examples of the two-axis counter-twisting and twisting-and-turning Hamiltonians, where we provide analytical expressions for all the relevant optimal time scales. Finally, we propose a generalization of these models to include $p$-body collective interaction (or $p$-order twisting), beyond the usual case of $p=2$. Using our geometric framework, we prove a no-go theorem for the local optimality of these models for $p>2$., Comment: 15 pages, 6 figures, 9 pages appendix

In this work we revisit the problem of equilibration in isolated many-body interacting quantum systems. We pay particular attention to quantum chaotic Hamiltonians, and rather than focusing on the properties of the asymptotic states and how they adhere to the predictions of the Eigenstate Thermalization Hypothesis, we focus on the equilibration process itself, i.e., \emph{the road to equilibrium}. Along the road to equilibrium the diagonal ensembles obey an emergent form of the second law of thermodynamics and we provide an information theoretic proof of this fact. With this proof at hand we show that the road to equilibrium is nothing but a hierarchy in time of diagonal ensembles. Furthermore, introducing the notions of statistical complexity and the entropy-complexity plane, we investigate the uniqueness of the road to equilibrium in a generic many-body system by comparing its trajectories in the entropy-complexity plane to those generated by a random Hamiltonian. Finally by treating the random Hamiltonian as a perturbation we analyzed the stability of entropy-complexity trajectories associated with the road to equilibrium for a chaotic Hamiltonian and different types of initial states., Comment: revised version, 14 pages, 4 figures, short appendix

We show the emergence of Floquet time crystal (FTC) phases in the Floquet dynamics of periodically driven $p$-spin models, which describe a collection of spin-1/2 particles with all-to-all $p$-body interactions. Given the mean-field nature of these models, we treat the problem exactly in the thermodynamic limit and show that, for a given $p$, these systems can host various robust time-crystalline responses with period $nT$, where $T$ is the period of the drive and $n$ an integer between 2 and $p$. In particular, the case of four-body interactions ($p=4$) gives rise to both a usual period-doubling crystal, and also a novel period-quadrupling phase. We develop a comprehensive framework to predict robust subharmonic response in classical area-preserving maps, and use this as a basis to predict the occurrence and characterize the stability of the resulting mean-field FTC phases in the quantum regime. Our analysis reveals that the robustness of the time-crystal behavior is reduced as their period increases, and establishes a connection between the emergence of time crystals, described by eigenstate ordering and robust subharmonic response, and the phenomenology of excited state and dynamical quantum phase transitions. Finally, for the models hosting two or more coexisting time crystal phases, we define protocols where the periodic subharmonic response of the system can be varied in time via the non-periodic modulation of an external control parameter., Comment: 20 pages, 8 figures, closer in form to the published version

We study the behavior of errors in the quantum simulation of spin systems with long-range multi-body interactions resulting from the Trotter-Suzuki decomposition of the time-evolution operator. We identify a regime where the Floquet operator underlying the Trotter decomposition undergoes sharp changes even for small variations in the simulation step size. This results in a time evolution operator that is very different from the dynamics generated by the targeted Hamiltonian, which leads to a proliferation of errors in the quantum simulation. These regions of sharp change in the Floquet operator, referred to as structural instability regions, appear typically at intermediate Trotter step sizes and in the weakly-interacting regime, and are thus complementary to recently revealed quantum chaotic regimes of the Trotterized evolution (Sieberer et al., npj Quantum Information 5, 1 (2019)). We characterize these structural instability regimes in $p$-spin models, transverse-field Ising models with all-to-all $p$-body interactions, and analytically predict their occurrence based on unitary perturbation theory. We further show that the effective Hamiltonian associated with the Trotter decomposition of the unitary time-evolution operator, when the Trotter-step size is chosen to be in the structural instability region, is very different from the target Hamiltonian, which explains the large errors that can occur in the simulation in the regions of instability. These results have implications for the reliability of near-term gate-based quantum simulators, and reveal an important interplay between errors and the physical properties of the system being simulated., Comment: 20 pages, 4 figures

We introduce kicked $p$-spin models describing a family of transverse Ising-like models for an ensemble of spin-$1/2$ particles with all-to-all $p$-body interaction terms occurring periodically in time as delta-kicks. This is the natural generalization of the well-studied quantum kicked top ($p$=2)[Haake, Ku\'s, and Scharf, Z. Phys. B 65, 381 (1987)]. We fully characterize the classical nonlinear dynamics of these models, including the transition to global Hamiltonian chaos. The classical analysis allows us to build a classification for this family of models, distinguishing between $p=2$ and $p>2$, and between models with odd and even $p$'s. Quantum chaos in these models is characterized in both kinematic and dynamic signatures. For the latter we show numerically that the growth rate of the out-of-time-order correlator is dictated by the classical Lyapunov exponent. Finally, we argue that the classification of these models constructed in the classical system applies to the quantum system as well., Comment: 14 pages, 6 figures, and appendices. Close to the published version

We study the application of a new method for simulating nonlinear dynamics of many-body spin systems using quantum measurement and feedback [Mu\~noz-Arias et al., Phys. Rev. Lett. 124, 110503 (2020)] to a broad class of many-body models known as $p$-spin Hamiltonians, which describe Ising-like models on a completely connected graph with $p$-body interactions. The method simulates the desired mean field dynamics in the thermodynamic limit by combining nonprojective measurements of a component of the collective spin with a global rotation conditioned on the measurement outcome. We apply this protocol to simulate the dynamics of the $p$-spin Hamiltonians and demonstrate how different aspects of criticality in the mean-field regime are readily accessible with our protocol. We study applications including properties of dynamical phase transitions and the emergence of spontaneous symmetry breaking in the adiabatic dynamics of the collective spin for different values of the parameter $p$. We also demonstrate how this method can be employed to study the quantum-to-classical transition in the dynamics continuously as a function of system size., Comment: 16 pages, 7 figures

We study a method to simulate quantum many-body dynamics of spin ensembles using measurement-based feedback. By performing a weak collective measurement on a large ensemble of two-level quantum systems and applying global rotations conditioned on the measurement outcome, one can simulate the dynamics of a mean-field quantum kicked top, a standard paradigm of quantum chaos. We analytically show that there exists a regime in which individual quantum trajectories adequately recover the classical limit, and show the transition between noisy quantum dynamics to full deterministic chaos described by classical Lyapunov exponents. We also analyze the effects of decoherence, and show that the proposed scheme represents a robust method to explore the emergence of chaos from complex quantum dynamics in a realistic experimental platform based on an atom-light interface., Comment: 6 pages, 4 figures and supplementary material

The interband dynamics of a two-band Bose-Hubbard model is studied with strongly correlated bosons forming single-site double occupancies referred to as doublons. Our model for resonant doublon interband coupling exhibits interesting dynamical features such as quantum Zeno effect, the generation of states such as a two-band Bell-like state and an upper-band Mott-like state. The evolution of the asymptotic state is controlled here by the effective opening of one or both of the two bands, which models decay channels., Comment: 6 pages, 4 figures. Comments are welcome

A many-body Wannier-Stark system coupled to an effective reservoir is studied within a non-Hermitian approach in the presence of a periodic driving. We show how the interplay of dissipation and driving dynamically induces a subspace of states which are very robust against dissipation. We numerically probe the structure of these asymptotic states and their robustness to imperfections in the initial-state preparation and to the size of the system. Moreover, the asymptotic states are found to be strongly entangled making them interesting for further applications., Comment: 8 pages, 5 figures! Comments are welcome!

An unit filling Bose-Hubbard Hamilonian embedded in strong Stark field is studied in the off-resonant regime inhibiting single and many-particle first order tunneling resonances. We investigate the occurrence of coherent dipole wave-like propagation along of an optical lattice by means of an effective Hamiltonian accounting for second order tunneling processes. It is shown that dipole wavefunction evolution in the short-time limit is ballistic and that finite size effects induce dynamical self-interference patterns known as quantum carperts. We also present the effects of the border right after the first reflection showing that the wavefunction diffuses normally with the variance changing linearly in time. This work extends the rich physical phenomenology of the tilted one dimensional lattice systems in a scenario of many interacting quantum particles, the so-called many-body Wannier-Stark system., Comment: 8 pages, 8 figures. Comments are welcome