A driven Kerr oscillator with two-fold degeneracies for qubit protection.

We present the experimental finding of multiple simultaneous two-fold degeneracies in the spectrum of a Kerr oscillator subjected to a squeezing drive. This squeezing drive resulting from a three-wave mixing process, in combination with the Kerr interaction, creates an effective static two-well potential in the phase space rotating at half the frequency of the sinusoidal drive generating the squeezing. Remarkably, these degeneracies can be turned on-and-off on demand, as well as their number by simply adjusting the frequency of the squeezing drive. We find that when the detuning Δ between the frequency of the oscillator and the second subharmonic of the drive equals an even multiple of the Kerr coefficient K, [Formula: see text], the oscillator displays [Formula: see text] exact, parity-protected, spectral degeneracies, insensitive to the drive amplitude. These degeneracies can be explained by the unusual destructive interference of tunnel paths in the classically forbidden region of the double well static effective potential that models our experiment. Exploiting this interference, we measure a peaked enhancement of the incoherent well-switching lifetime, thus creating a protected cat qubit in the ground state manifold of our oscillator. Our results illustrate the relationship between degeneracies and noise protection in a driven quantum system.

Entanglement of bosonic modes through an engineered exchange interaction.

Quantum computation presents a powerful new paradigm for information processing. A robust universal quantum computer can be realized with any well controlled quantum system, but a successful platform will ultimately require the combination of highly coherent, error-correctable quantum elements with at least one entangling operation between them1,2. Quantum information stored in a continuous-variable system-for example, a harmonic oscillator-can take advantage of hardware-efficient quantum error correction protocols that encode information in the large available Hilbert space of each element3-5. However, such encoded states typically have no controllable direct couplings, making deterministic entangling operations between them particularly challenging. Here we develop an efficient implementation of the exponential-SWAP operation6 and present its experimental realization between bosonic qubits stored in two superconducting microwave cavities. This engineered operation is analogous to the exchange interaction between discrete spin systems, but acts within any encoded subspace of the continuous-variable modes. Based on a control rotation, the operation produces a coherent superposition of identity and SWAP operations between arbitrary states of two harmonic oscillator modes and can be used to enact a deterministic entangling gate within quantum error correction codes. These results provide a valuable building block for universal quantum computation using bosonic modes.

Coexistence of Nonequilibrium Density and Equilibrium Energy Distribution of Quasiparticles in a Superconducting Qubit.

The density of quasiparticles typically observed in superconducting qubits exceeds the value expected in equilibrium by many orders of magnitude. Can this out-of-equilibrium quasiparticle density still possess an energy distribution in equilibrium with the phonon bath? Here, we answer this question affirmatively by measuring the thermal activation of charge-parity switching in a transmon qubit with a difference in superconducting gap on the two sides of the Josephson junction. We then demonstrate how the gap asymmetry of the device can be exploited to manipulate its parity.

Erasure Detection of a Dual-Rail Qubit Encoded in a Double-Post Superconducting Cavity.

Qubits with predominantly erasure errors present distinctive advantages for quantum error correction (QEC) and fault-tolerant quantum computing. Logical qubits based on dual-rail encoding that exploit erasure detection have been recently proposed in superconducting circuit architectures, with either coupled transmons or cavities. Here, we implement a dual-rail qubit encoded in a compact, double-post superconducting cavity. Using an auxiliary transmon, we perform erasure detection on the dual-rail subspace. We characterize the behavior of the code space by a novel method to perform joint-Wigner tomography. This is based on modifying the cross-Kerr interaction between the cavity modes and the transmon. We measure an erasure rate of 3.981±0.003 (ms)^{-1} and a residual, postselected dephasing error rate up to 0.17 (ms)^{-1} within the code space. This strong hierarchy of error rates, together with the compact and hardware-efficient nature of this novel architecture, holds promise in realizing QEC schemes with enhanced thresholds and improved scaling.

Static Effective Hamiltonian of a Rapidly Driven Nonlinear System.

We present a recursive formula for the computation of the static effective Hamiltonian of a system under a fast-oscillating drive. Our analytical result is well-suited to symbolic calculations performed by a computer and can be implemented to arbitrary order, thus overcoming limitations of existing time-dependent perturbation methods and allowing computations that were impossible before. We also provide a simple diagrammatic tool for calculation and treat illustrative examples. By construction, our method applies directly to both quantum and classical systems; the difference is left to a low-level subroutine. This sheds light on the relationship between seemingly disconnected independently developed methods in the literature and has direct applications in quantum engineering.

Proposal for Heralded Generation and Detection of Entangled Microwave-Optical-Photon Pairs.

Quantum state transfer between microwave and optical frequencies is essential for connecting superconducting quantum circuits to optical systems and extending microwave quantum networks over long distances. However, establishing such a quantum interface is extremely challenging because the standard direct quantum transduction requires both high coupling efficiency and small added noise. We propose an entanglement-based scheme-generating microwave-optical entanglement and using it to transfer quantum states via quantum teleportation-which can bypass the stringent requirements in direct quantum transduction and is robust against loss errors. In addition, we propose and analyze a counterintuitive design-suppress the added noise by placing the device at a higher temperature environment-which can improve both the device quality factor and power handling capability. We systematically analyze the generation and verification of entangled microwave-optical-photon pairs. The parameter for entanglement verification favors the regime of cooperativity mismatch and can tolerate certain thermal noises. Our scheme is feasible given the latest advances on electro-optomechanics, and can be generalized to various physical systems.

Coherent suppression of electromagnetic dissipation due to superconducting quasiparticles.

Owing to the low-loss propagation of electromagnetic signals in superconductors, Josephson junctions constitute ideal building blocks for quantum memories, amplifiers, detectors and high-speed processing units, operating over a wide band of microwave frequencies. Nevertheless, although transport in superconducting wires is perfectly lossless for direct current, transport of radio-frequency signals can be dissipative in the presence of quasiparticle excitations above the superconducting gap. Moreover, the exact mechanism of this dissipation in Josephson junctions has never been fully resolved experimentally. In particular, Josephson's key theoretical prediction that quasiparticle dissipation should vanish in transport through a junction when the phase difference across the junction is π (ref. 2) has never been observed. This subtle effect can be understood as resulting from the destructive interference of two separate dissipative channels involving electron-like and hole-like quasiparticles. Here we report the experimental observation of this quantum coherent suppression of quasiparticle dissipation across a Josephson junction. As the average phase bias across the junction is swept through π, we measure an increase of more than one order of magnitude in the energy relaxation time of a superconducting artificial atom. This striking suppression of dissipation, despite the presence of lossy quasiparticle excitations above the superconducting gap, provides a powerful tool for minimizing decoherence in quantum electronic systems and could be directly exploited in quantum information experiments with superconducting quantum bits.

Degeneracy-Preserving Quantum Nondemolition Measurement of Parity-Type Observables for Cat Qubits.

A central requirement for any quantum error correction scheme is the ability to perform quantum nondemolition measurements of an error syndrome, corresponding to a special symmetry property of the encoding scheme. It is in particular important that such a measurement does not introduce extra error mechanisms, not included in the error model of the correction scheme. In this Letter, we ensure such a robustness by designing an interaction with a measurement device that preserves the degeneracy of the measured observable. More precisely, we propose a scheme to perform continuous and quantum nondemolition measurement of photon-number parity in a microwave cavity. This corresponds to the error syndrome in a class of error correcting codes called the cat codes, which have recently proven to be efficient and versatile for quantum information processing. In our design, we exploit the strongly nonlinear Hamiltonian of a high-impedance Josephson circuit, coupling a high-Q cavity storage cavity mode to a low-Q readout one. By driving the readout resonator at its resonance, the phase of the reflected or transmitted signal carries directly exploitable information on parity-type observables for encoded cat qubits of the high-Q mode.

Holonomic Quantum Control with Continuous Variable Systems.

Universal computation of a quantum system consisting of superpositions of well-separated coherent states of multiple harmonic oscillators can be achieved by three families of adiabatic holonomic gates. The first gate consists of moving a coherent state around a closed path in phase space, resulting in a relative Berry phase between that state and the other states. The second gate consists of "colliding" two coherent states of the same oscillator, resulting in coherent population transfer between them. The third gate is an effective controlled-phase gate on coherent states of two different oscillators. Such gates should be realizable via reservoir engineering of systems that support tunable nonlinearities, such as trapped ions and circuit QED.

Asymmetric frequency conversion in nonlinear systems driven by a biharmonic pump.

A novel mechanism of asymmetric frequency conversion is investigated in nonlinear dispersive devices driven parametrically with a biharmonic pump. When the relative phase between the first and second harmonics combined in a two-tone pump is appropriately tuned, nonreciprocal frequency conversion, either upward or downward, can occur. Full directionality and efficiency of the conversion process is possible, provided that the distribution of pump power over the harmonics is set correctly. While this asymmetric conversion effect is generic, we describe its practical realization in a model system consisting of a current-biased, resistively shunted Josephson junction. Here, the multiharmonic Josephson oscillations, generated internally from the static current bias, provide the pump drive.

Inelastic microwave photon scattering off a quantum impurity in a Josephson-junction array.

Quantum fluctuations in an anharmonic superconducting circuit enable frequency conversion of individual incoming photons. This effect, linear in the photon beam intensity, leads to ramifications for the standard input-output circuit theory. We consider an extreme case of anharmonicity in which photons scatter off a small set of weak links within a Josephson junction array. We show that this quantum impurity displays Kondo physics and evaluate the elastic and inelastic photon scattering cross sections. These cross sections reveal many-body properties of the Kondo problem that are hard to access in its traditional fermionic version.

Three-wave mixing with three incoming waves: signal-idler coherent attenuation and gain enhancement in a parametric amplifier.

We demonstrate the time-reversed process of nondegenerate three-wave parametric amplification from three distinct sources in the fully nonlinear regime using a Josephson amplifier. In the reverse process, coherent attenuation, signal and idler beams destructively interfere in the presence of a pump to generate additional pump photons. This effect is observed through the symmetric phase-dependent amplification and attenuation of the signal and idler beams and, in the depleted pump regime, through the phase-dependent modulation of the amplifier gain, directly probing the enhancement of the pump. Results are found to be in good agreement with theory.

Hardware-efficient autonomous quantum memory protection.

We propose to encode a quantum bit of information in a superposition of coherent states of an oscillator, with four different phases. Our encoding in a single cavity mode, together with a protection protocol, significantly reduces the error rate due to photon loss. This protection is ensured by an efficient quantum error correction scheme employing the nonlinearity provided by a single physical qubit coupled to the cavity. We describe in detail how to implement these operations in a circuit quantum electrodynamics system. This proposal directly addresses the task of building a hardware-efficient quantum memory and can lead to important shortcuts in quantum computing architectures.

Spectral kissing and its dynamical consequences in the squeeze-driven Kerr oscillator.

Transmon qubits are the predominant element in circuit-based quantum information processing, such as existing quantum computers, due to their controllability and ease of engineering implementation. But more than qubits, transmons are multilevel nonlinear oscillators that can be used to investigate fundamental physics questions. Here, they are explored as simulators of excited state quantum phase transitions (ESQPTs), which are generalizations of quantum phase transitions to excited states. We show that the spectral kissing (coalescence of pairs of energy levels) experimentally observed in the effective Hamiltonian of a driven SNAIL-transmon is an ESQPT precursor. We explore the dynamical consequences of the ESQPT, which include the exponential growth of out-of-time-ordered correlators, followed by periodic revivals, and the slow evolution of the survival probability due to localization. These signatures of ESQPT are within reach for current superconducting circuits platforms and are of interest to experiments with cold atoms and ion traps.

Microwave characterization of Josephson junction arrays: implementing a low loss superinductance.

We have measured the plasma resonances of an array of Josephson junctions in the regime E(J)>>E(C), up to the ninth harmonic by incorporating it as part of a resonator capacitively coupled to a coplanar waveguide. From the characteristics of the resonances, we infer the successful implementation of a superinductance, an electrical element with a nondissipative impedance greater than the resistance quantum [R(Q)=h/(2e)(2) is approximately equal to 6.5 kΩ] at microwave frequencies. Such an element is crucial for preserving the quantum coherence in circuits exploiting large fluctuations of the superconducting phase. Our results show internal losses less than 20 ppm, self-resonant frequencies greater than 10 GHz, and phase-slip rates less than 1 mHz, enabling direct application of such arrays for quantum information and metrology. Arrays with a loop geometry also demonstrate a new manifestation of flux quantization in a dispersive analog of the Little-Parks effect.

Quantum control of bosonic modes with superconducting circuits.

Bosonic modes have wide applications in various quantum technologies, such as optical photons for quantum communication, magnons in spin ensembles for quantum information storage and mechanical modes for reversible microwave-to-optical quantum transduction. There is emerging interest in utilizing bosonic modes for quantum information processing, with circuit quantum electrodynamics (circuit QED) as one of the leading architectures. Quantum information can be encoded into subspaces of a bosonic superconducting cavity mode with long coherence time. However, standard Gaussian operations (e.g., beam splitting and two-mode squeezing) are insufficient for universal quantum computing. The major challenge is to introduce additional nonlinear control beyond Gaussian operations without adding significant bosonic loss or decoherence. Here we review recent advances in universal control of a single bosonic code with superconducting circuits, including unitary control, quantum feedback control, driven-dissipative control and holonomic dissipative control. Various approaches to entangling different bosonic modes are also discussed.

Implementing a universal gate set on a logical qubit encoded in an oscillator.

A logical qubit is a two-dimensional subspace of a higher dimensional system, chosen such that it is possible to detect and correct the occurrence of certain errors. Manipulation of the encoded information generally requires arbitrary and precise control over the entire system. Whether based on multiple physical qubits or larger dimensional modes such as oscillators, the individual elements in realistic devices will always have residual interactions, which must be accounted for when designing logical operations. Here we demonstrate a holistic control strategy which exploits accurate knowledge of the Hamiltonian to manipulate a coupled oscillator-transmon system. We use this approach to realize high-fidelity (98.5%, inferred), decoherence-limited operations on a logical qubit encoded in a superconducting cavity resonator using four-component cat states. Our results show the power of applying numerical techniques to control linear oscillators and pave the way for utilizing their large Hilbert space as a resource in quantum information processing.A logical qubit is a two-dimensional subspace of a higher dimensional system, whose manipulation requires precise control over the whole system. Here the authors demonstrate a control strategy which exploits precise knowledge of the Hamiltonian to manipulate a coupled oscillator-transmon system.

Fluxonium: single cooper-pair circuit free of charge offsets.

The promise of single Cooper-pair quantum circuits based on tunnel junctions for metrology and quantum information applications is severely limited by the influence of offset charges: random, slowly drifting microscopic charges inherent in many solid-state systems. By shunting a small junction with the Josephson kinetic inductance of a series array of large-capacitance tunnel junctions, thereby ensuring that all superconducting islands are connected to the circuit by at least one large junction, we have realized a new superconducting artificial atom that is totally insensitive to offset charges. Yet its energy levels manifest the anharmonic structure associated with single Cooper-pair effects, a useful component for solid-state quantum computation.