Session Y47: Superconducting Qubits: Kerr Oscillators, Cats, and More

Quantum tunneling and level crossings in the squeeze-driven Kerr oscillator

The quasienergy spectrum of a squeeze-driven Kerr oscillator was recently measured with a SNAIL transmon superconducting circuit and showed good agreement with the low-level energy spectrum of the oscillator's corresponding static effective Hamiltonian. In this presentation, we provide a detailed analysis of the spectrum and the dynamics of the effective model up to high energies, which should soon be within experimental reach. The spectrum exhibits real (avoided) level crossings for specific values of the Hamiltonian parameters, which can then be chosen to suppress (enhance) quantum tunneling. The parameter values for the crossings can be obtained from a semiclassical approach and can also be identified directly from the dynamics. Our analysis of quantum tunneling is done with the effective flux of the Husimi volume of the evolved states between different regions of the phase space. Both initial coherent states and quench dynamics are considered. We argue that the level crossings and their consequences on the dynamics are typical to any quantum system with one degree of freedom, whose density of states presents a local logarithmic divergence and a local step discontinuity.   

Quantum chaos and quantum phase transitions in Kerr parametric oscillators

Transmon qubits are the predominant element in circuit-based quantum information processing, such as existing quantum computers. But more than qubits, they are multilevel nonlinear oscillators that can be used to investigate fundamental physics questions. We show that on the same Kerr parametric oscillator implemented with driven SNAIL transmon superconducting circuits, one can either simulate excited state quantum phase transitions (ESQPTs) or study quantum chaos. ESQPTs take place in the regime where Kerr-cat qubits are generated, while chaos sets in when the nonlinearities and drive become strong. As the parameters are changed from one limit to the other, perturbative expansions used to derive static effective models cease to capture all the relevant physics of the original driven system.   

Driving superconducting qubits into chaos

Kerr parametric oscillators can operate as Kerr-cat qubits, which offer advantages towards the encoding and manipulation of error-protected quantum information. Recently, they have been implemented with the SNAIL transmon superconducting circuit, which combines Kerr nonlinearity and a squeezing drive. In this presentation, we show that when the nonlinearities and the drive are strong, the qubit melts away due to the onset of chaos. By exploring various ranges of experimentally accessible parameters, we provide an equation for the crossing line between regularity and chaos. If on the one hand chaos puts limits on the Kerr-cat qubit, on the other hand it opens up a new direction of research for superconducting circuits.   

Pulsed dynamics and itinerant cat state generation in a Kerr parametric oscillator

Kerr parametric oscillators (KPOs) have been proposed as a platform for alternative implementations of quantum computing [1] and for quantum simulations [2]. These goals can only be reached using a scalable KPO architecture with high internal quality factors and low pure dephasing. An experiment that can be performed using a single low-loss KPO is the generation of itinerant cat states [3].

Here we present a planar KPO design consisting of a large island shunted to the ground by a single SQUID, with an internal quality factor exceeding 10⁵. We investigate the dynamics of the KPO under pulsed excitations at the fundamental and around double the fundamental frequency, both by monitoring the averaged field emitted by the KPO and by performing Wigner tomography. Using a carefully shaped parametric pulse, we generate itinerant cat states.   

Generation of an entangled cat state from a Bell state in two interacting Kerr Parametric Oscillators

Kerr Parametric Oscillators (KPOs) are parametrically driven weakly nonlinear oscillators. These are promising quantum systems for generation and control of cat states, which can be useful for various applications, such as quantum error correction.

However, scaling with cat states is a non-trivial task because of complexity in entangled state generation compared to that in two-level system encoding.

In this talk, we present our recent experimental progress on a relatively simple method on generation of an entangled cat state from a Bell state in planar KPOs.

We first generate a Bell state by applying a parametric drive to one of KPOs. Subsequently, we create an entangled cat state by applying an adiabatic two-photon pump to each KPO. To characterize the generated entangled cat state, we perform two-mode Wigner tomography via joint measurement of ancilla transmons coupled to each KPO.  

The simplicity of our scheme and planar geometry will be beneficial for scaling.   

Experimental gate operation of entangled cat states in two Kerr parametric oscillators

 A Kerr parametric oscillator (KPO) is parametrically driven weakly nonlinear oscillator and is one of the promising devices for quantum computing, quantum simulation and error correction because cat states can be generated and stabilized easily in this system [1]. We present our experimental efforts towards two-qubit gate operations for cat states in two-dimensional superconducting KPOs. The two KPOs are capacitively coupled, and the detuning between them is much larger than the coupling strength. The gate operation is performed by applying a parametric drive to one of the KPOs [2]. A tunable coupler is not required as our KPO supports three-wave mixing by itself. We create entangled cat states by using this gate operation. We will present experimental results on two-cat-qubit gate operations. This works will be the first step towards practical scaling for continuous variable quantum computation. 

[1] Grimm, A., Frattini, N.E., Puri, S. et al. Stabilization and operation of a Kerr-cat qubit. Nature 584, 205–209 (2020)

[2] Hiroomi Chono, Two-qubit gate using conditional driving for highly detuned Kerr-nonlinear parametric oscillators, Phys. Rev. Research 4, 043054 (2022)   

Quantum supervised machine learning with Kerr-nonlinear oscillators

We propose a strategy for obtaining the expressivity in quantum machine learning without the data reuploading, which repeatedly encode into the quantum state. Using Kerr-nonlinear bosonic cavities, we can make full use of a large Hilbert space even if we have a single bosonic mode and it brings highly expressive quantum machine learning. Our numerical simulations show that the expressibility of our method with only one mode is much higher than that of the conventional method with six qubits. Our results pave the way towards a resource efficient algorithm of quantum supervised machine learning.   

Quantum state tomography for Kerr parametric oscillators with reflection measurement

Kerr parametric oscillators (KPOs) implemented in the circuit QED architecture can work as qubits. The readout of KPOs is required in their applications to quantum annealing and universal quantum computation. In this presentation, we show a scheme of state tomography for KPOs with reflection measurement. A previous paper showed that the reflection coefficient depends on the state of the KPO [1]. However, it is nontrivial if tomography of a qubit encoded into a KPO can be performed in a practical way mitigating the decoherence during the measurement, and how accurate it is. We present that the reflection coefficient has a one-to-one correspondence with a diagonal element of the density matrix of the qubit when a probe frequency is properly chosen, and an additional single-photon-drive is utilized. Our scheme offers a novel method to readout the qubit along an axis of the Bloch sphere, and therefore single-qubit gates followed by the reflection measurement can constitute state tomography [2]. 

[1] S. Masuda, et al., NJP 23, 093023 (2021).

[2] Y. Suzuki, et al., Phys. Rev. Applied 20, 034031 (2023).   

Efficient and versatile tomography of bosonic quantum states (Part I)

The use of bosonic superconducting circuits for quantum information processing is promising due to their hardware efficiency and error-correction capabilities. In this setting, conventional methods for characterizing bosonic states, such as Wigner tomography, require many measurements and are not feasible for scaling up to multimode systems. Here, we present a Quantum Reservoir Processing (QRP) approach that uses ergodic dynamics on the bosonic systems with operations as simple as displacements. By identifying the action of the dynamics as a linear map, we perform state estimation from a few measured simple observables. We show that our approach allows for robust reconstruction of an arbitrary input state of dimension D with high fidelity from D²-1 measured photon number distribution, which is not only fewer than the measurements required for standard Wigner tomography but also simpler to execute. We provide systematic and thorough error analysis to show how coherent and decoherent errors affect the observables, and therefore, the state reconstruction. Our protocol offers an efficient, robust, and versatile tool for state tomography in bosonic superconducting circuits.   

Efficient and versatile tomography of bosonic quantum states (Part II)

Quantum reservoir processing (QRP) is an alternative approach to conventional Wigner tomography for bosonic superconducting circuits. The estimation of an arbitrary state of dimension D only requires D^2-1 measurements of the photon number distribution, which are then inverted with a linear map to reconstruct the density matrix.

Here we present the experimental data where we show that, for the same number of measurements, QRP reconstructs states with higher fidelity than Wigner tomography. We also show how QRP fidelity scales more favorably both under coherent and incoherent errors.

This project may contribute a new tool, particularly useful in the presence of incoherent errors, for efficient state estimation and reconstruction for quantum information processing with bosonic modes.

  

Universal control of a bosonic mode via drive-activated native cubic interactions

Bosonic modes provide a hardware-efficient alternative to qubit-based quantum information processing. However, achieving universal control on bosons requires access to a nonlinearity, or to resourceful non-Gaussian quantum states like cubic phase states. Superconducting microwave circuits offer such strong nonlinearities but face other challenges, like parasitic state distortion due to the Kerr effect and shorter coherence times.

In this talk, we will demonstrate how these difficulties can be overcome. We harness the 3rd order non-linearity of a SNAIL (Superconducting Nonlinear Asymmetric Inductive eLement) dipole terminated resonator through simultaneous flux and charge pumping to obtain the desired cubic state, 45 times faster than decoherence. In parallel, we minimize the 4th order Kerr effect by adjusting the flux DC bias. Achieving this required meticulous pulse calibration and circuit modeling. We will delve into the details of these processes and discuss how our simulation efforts shed light on the primary causes of infidelity in our current experimental setup.   

On-chip microwave source of coherent states with in-situ control of the photon numebr distribution

Populating superconducting cavities with specifically designed coherent states is a fundamental step in many quantum error correction and quantum comunication protocols [1-2]. In the microwave regime, the possibility to design and engineer superconducting circuits behaving like artificial atoms supports the realization of quantum optics protocols, including microwave photons generation [3]. Here, we propose and theoretically investigate a new design that allows on-chip generation of microwave coherent states with on-demand control on the photon number distribution. The scheme is based on driving a superconducting circuit acting as a photon source and controlling the steady-state photons injected in a target resonator via an external magnetic flux. We will discuss the experimental feasibility of the proposed design and its possible implementation for driving multiple superconducting cavities.

[1] Cai, Weizhou, et al. "Bosonic quantum error correction codes in superconducting quantum circuits." Fundamental Research 1.1 (2021): 50-67.

 

[2] Wang, W., et al. "Converting quasiclassical states into arbitrary fock state superpositions in a superconducting circuit." Physical Review Letters 118.22 (2017): 223604.

 

[3] You, J. Q., et al. "Persistent single-photon production by tunable on-chip micromaser with a superconducting quantum circuit." Physical Review B 75.10 (2007): 104516.   

Ancilla-fault tolerant control of linear quantum memories with dynamic dispersive coupling

Superconducting resonators offer a rich playground for continuous-variable quantum simulations, computation, and error correction. In isolation, these resonators are linear and only suffer a single correctible error (photon loss), making them potential quantum memories. There has been much progress towards controlling these memories by dispersively coupling them to ancillary superconducting qubits, but this introduces non-linearity and dephasing to the memories, and control fidelity is often limited by the ancilla qubit's decoherence. We introduce a novel architecture that breaks this trade-off, by inserting a quasi-linear coupler between the memories and the qubit. This 'Linear INductive Coupler' (LINC), a novel three-wave mixing element that is linear when undriven, ensures that the inherited non-linearity and dephasing of the memories are both much smaller than their linewidths. Gaussian control of the memories, like beamsplitting and squeezing, is directly enabled by parametrically driving the LINC. When non-Gaussian control or tomography is required, the LINC can activate a dynamic dispersive coupling between the memory and the ancilla qubit, similar to other recent proposals [1]. We further show that full parametric control of the memory-qubit interaction enables first-order tolerance to both decay and low-frequency dephasing of the qubit, achieving universal control limited only by the memories' long lifetimes.   

Nonlinearity-engineered Threaded SQUID: A Novel Nonlinear Device for Bosonic Quantum Error Correction

Bosonic error correction codes, such as cat codes, are promising candidates for quantum information processing. However, the stabilization and control of these codes require complicated high-order nonlinear Hamiltonians, which are challenging to implement in practice. The realization of effective high-order Hamiltonians usually demands strong drives at carefully selected frequencies, which also induce unavoidable residual Hamiltonians and decoherence effects. To get around, we design a novel multi-loop Josephson interference device that selectively produces odd-order nonlinear Hamiltonians when subjected to specific magnetic drives. Different from the anharmonic threaded SQUID (ATS) with two single-junction branches, this new device is expected to eliminate the residual first-order Hamiltonian and suppress static Kerr nonlinearity while retaining the third-order and higher ones by introducing a multi-junction branch. The device that we propose can be used for the implementation of bias-preserving CNOT gates and other gate operations on Kerr-cat qubits. In this talk, we will present the theoretical part related to this novel device.   

Analysis of DC-biased Josephson junctions for cat qubit stabilisation

Current implementations of qubits continue to exhibit too frequent errors to be scaled into useful quantum machines. An emerging approach is to encode quantum information in the two metastable states of an oscillator exchanging pairs of photons with its environment, a mechanism shown to provide protection against bit flips. In this talk, we propose a theoretical study of a novel circuit design for the implementation of this so called dissipative cat qubit. The proposed circuit replaces the perturbative effect of a microwave pump to mediate the two-photon exchange, by the natural periodic evolution of a DC voltage biased Josephson junction. Our design is predicted to showcase a two-photon exchange rate larger than that of the microwave pump-based cat qubit implementation while dynamically averaging the usually resonant parasitic terms such as Kerr and cross Kerr. In addition to addressing qubit stabilization, we propose to use injection locking to prevent long-term drifts of the cat qubit phase associated to DC voltage noise.