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Microwave photons are important carriers of quantum information in many promising platforms for quantum computing. They can be routinely generated, controlled, and teleported in experiments, indicating a variety of applications in quantum technology. However, observation of quantum statistical properties of microwave photons remains demanding: The energy of several microwave photons is considerably smaller than the thermal fluctuation of any room-temperature detector, while amplification necessarily induces noise. Here, we present a measurement technique with a nanobolometer that directly measures the photon statistics at millikelvin and overcomes this trade-off. We apply our method to thermal states generated by a blackbody radiator operating in the regime of circuit quantum electrodynamics. We demonstrate the photon number resolvedness of the nanobolometer, and reveal the n(n+1)-scaling law of the photon number variance as indicated by the Bose--Einstein distribution. By engineering the coherent and incoherent proportions of the input field, we observe the transition between super-Poissonian and Poissonian statistics of the microwave photons from the bolometric second-order correlation measurement. This technique is poised to serve in fundamental tests of quantum mechanics with microwave photons and function as a scalable readout solution for a quantum information processor.

Compact inductor-capacitor (LC) resonators, in contrast to coplanar waveguide (CPW) resonators, have a simple lumped-element circuit representation but usually call for sophisticated finite-element method (FEM) simulations for an accurate modelling. Here we present an simple analytical model for a family of coplanar LC resonators where the electrical properties are directly obtained from the circuit geometry with a satisfying accuracy. Our experimental results on $10$ high-internal-quality-factor resonators ($Q_{\rm i}\gtrsim 2\times 10^{5}$), with frequency ranging roughly from $300\,{\rm MHz}$ to $1\,{\rm GHz}$, show an excellent consistency with both the derived analytical model and detailed FEM simulations. These results showcase the ability to design sub-gigahertz resonators with less than $2\%$ deviation in the resonance frequency, which has immediate applications, for example, in the implementation of ultrasensitive cryogenic detectors. The achieved compact resonator size of the order of a square millimeter indicates a feasible way to integrate hundreds of microwave resonators on a single chip for realizing photonic lattices.

Simple, precise, and robust control is demanded for operating a large quantum information processor. However, existing routes to high-fidelity quantum control rely heavily on arbitrary waveform generators that are difficult to scale up. Here, we show that arbitrary control of a quantum system can be achieved by simply turning on and off the control fields in a proper sequence. The switching instances can be designed by conventional quantum optimal control algorithms, while the required computational resources for matrix exponential can be substantially reduced. We demonstrate the flexibility and robustness of the resulting control protocol, and apply it to superconducting quantum circuits for illustration. We expect this proposal to be readily achievable with current semiconductor and superconductor technologies, which offers a significant step towards scalable quantum computing.

Understanding the non-deterministic behavior of deterministic nonlinear systems has been an implicit dream since Lorenz named it the "butterfly effect". A prominent example is the hysteresis and bistability of the Duffing oscillator, which in the classical description is attributed to the coexistence of two steady states in a double-well potential. However, this interpretation fails in the quantum-mechanical perspective, where a single unique steady state is allowed in the whole parameter space. Here, we measure the non-equilibrium dynamics of a superconducting Duffing oscillator and reconcile the classical and quantum descriptions in a unified picture of quantum metastability. We demonstrate that the two classically regarded steady states are in fact metastable states. They have a remarkably long lifetime in the classical hysteresis regime but must eventually relax into a single unique steady state allowed by quantum mechanics. By engineering the lifetime of the metastable states sufficiently large, we observe a first-order dissipative phase transition, which mimics a sudden change of the mean field in a 11-site Bose-Hubbard lattice. We also reveal the two distinct phases of the transition by quantum state tomography, namely a coherent-state phase and a squeezed-state phase separated by a critical point. Our results reveal a smooth quantum state evolution behind a sudden dissipative phase transition, and they form an essential step towards understanding hysteresis and instability in non-equilibrium systems.

We describe a unified quantum approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. We also generalize the method to a chain of resonators with time delays, and reveal several transport properties similar to a photonic crystal. It is shown that both the quantum and classical analyses provide consistent results, and they together reveal different decay and decoherence mechanisms in a general microwave resonator. These results form a solid basis for understanding the scattering spectrums of networks of microwave resonators, and pave the way for applying superconducting microwave resonators in complex circuits.

We describe a unified classical approach for analyzing the scattering coefficients of superconducting microwave resonators with a variety of geometries. To fill the gap between experiment and theory, we also consider the influences of small circuit asymmetry and the finite length of the feedlines, and describe a procedure to correct them in typical measurement results. We show that, similar to the transmission coefficient of a hanger-type resonator, the reflection coefficient of a necklace- or bridge-type resonator does also contain a reference point which can be used to characterize the electrical properties of a microwave resonator in a single step. Our results provide a comprehensive understanding of superconducting microwave resonators from the design concepts to the characterization details.

We propose a tunable coupler consisting of N fixed-frequency qubits, which can tune and even amplify the effective interaction between two superconducting quantum circuits. The tuning range of the interaction is proportional to N, with a minimum value of zero and a maximum that can exceed the physical coupling rates between the coupler and the circuits. The effective coupling rate is determined by the collective magnetic quantum number of the qubit ensemble, which takes only discrete values and is free from collective decay and decoherence. Using single-photon pi-pulses, the coupling rate can be switched between arbitrary choices of the initial and final values within the dynamic range in a single step without going through intermediate values. A cascade of the couplers for amplifying small interactions or weak signals is also discussed. These results should not only stimulate interest in exploring the collective effects in quantum information processing, but also enable development of applications in tuning and amplifying the interactions in a general cavity-QED system.

We have fabricated and studied a system of two tunable and coupled nonlinear superconducting resonators. The nonlinearity is introduced by galvanically coupled dc-SQUIDs. We simulate the system response by means of a circuit model, which includes an additional signal path introduced by the electromagnetic environment. Furthermore, we present two methods allowing us to experimentally determine the nonlinearity. First, we fit the measured frequency and flux dependence of the transmission data to simulations based on the equivalent circuit model. Second, we fit the power dependence of the transmission data to a model that is predicted by the nonlinear equation of motion describing the system. Our results show that we are able to tune the nonlinearity of the resonators by almost two orders of magnitude via an external coil and two on-chip antennas. The studied system represents the basic building block for larger systems, allowing for quantum simulations of bosonic many-body systems with a larger number of lattice sites.

Designing a high-quality control is crucial for reliable quantum computation. Among the existing approaches, closed-loop leaning control is an effective choice. Its efficiency depends on the learning algorithm employed, thus deserving algorithmic comparisons for its practical applications. Here, we assess three representative learning algorithms, including GRadient Ascent Pulse Engineering (GRAPE), improved Nelder-Mead (NMplus) and Differential Evolution (DE), by searching for high-quality control pulses to prepare the Bell state. We first implement each algorithm experimentally in a nuclear magnetic resonance system and then conduct a numerical study considering the impact of some possible significant experimental uncertainties. The experiments report the successful preparation of the high-fidelity target state with different convergence speeds by the three algorithms, and these results coincide with the numerical simulations when potential uncertainties are negligible. However, under certain significant uncertainties, these algorithms possess distinct performance with respect to their resulting precision and efficiency. This study provides insight to aid in the practical application of different closed-loop learning algorithms in realistic physical scenarios., Comment: 9 pages, 3 figures

Quantum Fourier transform (QFT) is a key ingredient of many quantum algorithms where a considerable amount of ancilla qubits and gates are often needed to form a Hilbert space large enough for high-precision results. Qubit recycling reduces the number of ancilla qubits to one but imposes the requirement of repeated measurements and feedforward within the coherence time of the qubits. Moreover, recycling only applies to certain cases where QFT can be carried out in a semi-classical way. Here, we report a novel approach based on two harmonic resonators which form a high-dimensional Hilbert space for the realization of QFT. By employing the all-resonant and perfect state-transfer methods, we develop a protocol that transfers an unknown multi-qubit state to one resonator. QFT is performed by the free evolution of the two resonators with a cross-Kerr interaction. Then, the fully-quantum result can be localized in the second resonator by a projective measurement. Qualitative analysis shows that a 2^10-dimensional QFT can be realized in current superconducting quantum circuits which paves the way for implementing various quantum algorithms in the noisy intermediate-scale quantum (NISQ) era.

A common way to manipulate a quantum system, for example spins or artificial atoms, is to use properly tailored control pulses. In order to accomplish quantum information tasks before coherence is lost, it is crucial to implement the control in the shortest possible time. Here we report the near time-optimal preparation of a Bell state with fidelity higher than $99\%$ in an NMR experiment, which is feasible by combining the synergistic capabilities of modelling and experiments operating in tandem. The pulses preparing the Bell state are found by experiments that are recursively assisted with a gradient-based optimization algorithm working with a model. Thus, we explore the interplay between model-based numerical optimal design and experimental-based learning control. Utilizing the balanced synergism between the dual approaches should have broad applications for accelerating the search for optimal quantum controls.

In non-Hermitian coulped-resonator networks, the eigenvectors of degenerate eigenmodes may become parallel due to the singularity at so-called Exceptional Points (EP). To exploit the parametric sensitivity at EPs, an important problem is, given an arbitrary set of coupled resonators, how to generate a desired EP by properly coupling them together. This paper provides the solution for the case of three resonators. We show that all physically admissible EPs can be realized with either weakly coupled linear networks or strongly coupled circular networks, and the latter type of EPs has not been reported in the literature. Each admissible EP eigenvalue can be realized by two and only two resonator networks, and the formulas for calculating the required coupling constants are provided. The characteristics of these EPs are illustrated by the change of transmission spectra near them, which verify the enhanced sensitivity induced by the singularity of EPs., Comment: 27 pages, 10 figures

By simultaneously coupling multiple two-level artificial atoms to two superconducting resonators, we design a quantum switch that tunes the resonator-resonator coupling strength from zero to a large value proportional to the number of qubits. This process is implemented by engineering the qubits into different subradiant states, where the microwave photons decay from different qubits destructively interfere with each other such that the resonator-resonator coupling strength keeps stable in an open environment. Based on a three-step control scheme, we switch the coupling strength among different values within nanoseconds without changing the transition frequency of the qubits. We also apply the quantum switch to a network of superconducting resonators, and demonstrate its potential applications in quantum simulation and quantum information storage and processing.

Illuminated by the Pulse Width Modulation (PWM) technology in classical control engineering, we propose the PWM approximation which transforms continuous and bang-bang control into each other. This method works by squeezing the approximation error into high-frequency components above a prescribed cutoff frequency $\Omega$, which can be tailored by modulating the switch time of the bang-bang control. Because the quantum dynamics under bang-bang control can be efficiently calculated, PWM allows us to speed up quantum control design particularly in large dimensional systems with small number of control variables. Moreover, PWM paves the way to implement various bang-bang control protocols in the literature with an arbitrary waveform, which is more desirable in laboratory.

This paper focuses on accelerating quantum optimal control design for complex quantum systems. Based on our previous work [{arXiv:1607.04054}], we combine Pulse Width Modulation (PWM) and gradient descent algorithm into solving quantum optimal control problems, which shows distinct improvement of computational efficiency in various cases. To further apply this algorithm to potential experiments, we also propose the smooth realization of the optimized control solution, e.g. using Gaussian pulse train to replace rectangular pulses. Based on the experimental data of the D-Norleucine molecule, we numerically find optimal control functions in $3$-qubit and $6$-qubit systems, and demonstrate its efficiency advantage compared with basic GRAPE algorithm., Comment: Withdraw for the poor style and the unclear messages. A better version has been updated (see arXiv:2209.09869)

This paper proposes a numerical method for solving time-dependent Schrodinger equations with finite spectral bandwidth, which applies to both periodic and non-periodic cases. We introduce the concept of Pulse Width Modulation (PWM), which is broadly used in classical control engineering, to simulate time-dependent quantum dynamics by switching between a finite number of constant Hamiltonians. The switching timings can be programmed to improve the precision and save computational resources. The effectiveness of the PWM method is demonstrated by numerical simulations, which shows that it is faster and more robust than the standard piecewise constant control systems. Moreover, we also propose the realization of PWM method for potential implementation in experimental systems., Comment: Withdraw for the poor style and the unclear messages. A better version has been updated (see arXiv:2209.09869)

Here we consider the speed at which quantum information can be transferred between the nodes of a linear network. Because such nodes are linear oscillators, this speed is also important in the cooling and state preparation of mechanical oscillators, as well as frequency conversion. We show that if there is no restriction on the size of the linear coupling between two oscillators, then there exist control protocols that will swap their respective states with high fidelity within a time much less than a single oscillation period. Standard gradient search methods fail to find these fast protocols. We were able to do so by augmenting standard search methods with a path-tracing technique, demonstrating that this technique has remarkable power to solve time-optimal control problems, as well as confirming the highly challenging nature of these problems. As a further demonstration of the power of path-tracing, first introduced by Moore-Tibbets et al. [Phys. Rev. A 86, 062309 (2012)], we apply it to the generation of entanglement in a linear network., Comment: 6 pages, 3 eps figures

A Mini-Orange spectrometer used for in-beam measurements of internal conversion electrons, which consists of a Si(Li) detector and different sets of SmO$_5$ permanent magnets for filtering and transporting the conversion electrons to the Si(Li) detector, has been developed at China Institute of Atomic Energy. The working principle and configuration of the Mini-Orange spectrometer are described. The performance of the setup is illustrated by measured singles conversion electron spectra using the Mini-Orange spectrometer.