Observation of Josephson Harmonics in Tunnel Junctions

  1. Dennis Willsch,
  2. Dennis Rieger,
  3. Patrick Winkel,
  4. Madita Willsch,
  5. Christian Dickel,
  6. Jonas Krause,
  7. Yoichi Ando,
  8. Raphaël Lescanne,
  9. Zaki Leghtas,
  10. Nicholas T. Bronn,
  11. Pratiti Deb,
  12. Olivia Lanes,
  13. Zlatko K. Minev,
  14. Benedikt Dennig,
  15. Simon Geisert,
  16. Simon Günzler,
  17. Sören Ihssen,
  18. Patrick Paluch,
  19. Thomas Reisinger,
  20. Roudy Hanna,
  21. Jin Hee Bae,
  22. Peter Schüffelgen,
  23. Detlev Grützmacher,
  24. Luiza Buimaga-Iarinca,
  25. Cristian Morari,
  26. Wolfgang Wernsdorfer,
  27. David P. DiVincenzo,
  28. Kristel Michielsen,
  29. Gianluigi Catelani,
  30. and Ioan M. Pop
An accurate understanding of the Josephson effect is the keystone of quantum information processing with superconducting hardware. Here we show that the celebrated sinφ current-phase
relation (CφR) of Josephson junctions (JJs) fails to fully describe the energy spectra of transmon artificial atoms across various samples and laboratories. While the microscopic theory of JJs contains higher harmonics in the CφR, these have generally been assumed to give insignificant corrections for tunnel JJs, due to the low transparency of the conduction channels. However, this assumption might not be justified given the disordered nature of the commonly used AlOx tunnel barriers. Indeed, a mesoscopic model of tunneling through an inhomogeneous AlOx barrier predicts contributions from higher Josephson harmonics of several %. By including these in the transmon Hamiltonian, we obtain orders of magnitude better agreement between the computed and measured energy spectra. The measurement of Josephson harmonics in the CφR of standard tunnel junctions prompts a reevaluation of current models for superconducting hardware and it offers a highly sensitive probe towards optimizing tunnel barrier uniformity.

Active resonator depletion with short microwave pulses

  1. Sashank Kaushik Sridhar,
  2. and David P. DiVincenzo
We propose a physical model to explain the phenomenon of photon depletion in superconducting microwave resonators in the dispersive regime, coupled to Josephson junction qubits, via
short microwave pulses. We discuss the conditions for matching the amplitude and phase of the pulse optimally within the framework of the model, allowing for significant reductions in reset times after measurement of the qubits. We consider how to deal with pulses and transient dynamics within the input-output formalism, along with a reassessment of the underlying assumptions for a wide-band pulse.

Circuit quantization with time-dependent magnetic fields for realistic geometries

  1. Roman-Pascal Riwar,
  2. and David P. DiVincenzo
Quantum circuit theory has become a powerful and indispensable tool to predict the dynamics of superconducting circuits. Surprisingly however, the question of how to properly account
for a time-dependent driving via external magnetic fields has hardly been addressed so far. Here, we derive a general recipe to construct a low-energy Hamiltonian, taking as input only the circuit geometry and the solution of the external magnetic fields. A gauge fixing procedure for the scalar and vector potentials is given which assures that time-varying magnetic fluxes make contributions only to the potential function in the Schrödinger equation. Our proposed procedure is valid for continuum geometries and thus significantly generalizes previous efforts, which were based on discrete circuits. We study some implications of our results for the concrete example of a parallel-plate SQUID circuit. We show that if we insist on representing the response of this SQUID with individual, discrete capacitances associated with each individual Josephson junction, this is only possible if we permit the individual capacitance values to be negative, time-dependent or even momentarily singular. Finally, we provide some experimentally testable predictions, such as a strong enhancement of the qubit relaxation rates arising from the effective negative capacitances, and the emergence of a Berry phase due to time dependence of these capacitances.

Transmon platform for quantum computing challenged by chaotic fluctuations

  1. Christoph Berke,
  2. Evangelos Varvelis,
  3. Simon Trebst,
  4. Alexander Altland,
  5. and David P. DiVincenzo
From the perspective of many body physics, the transmon qubit architectures currently developed for quantum computing are systems of coupled nonlinear quantum resonators. A significant
amount of intentional frequency detuning (disorder) is required to protect individual qubit states against the destabilizing effects of nonlinear resonator coupling. Here we investigate the stability of this variant of a many-body localized (MBL) phase for system parameters relevant to current quantum processors of two different types, those using untunable qubits (IBM type) and those using tunable qubits (Delft/Google type). Applying three independent diagnostics of localization theory – a Kullback-Leibler analysis of spectral statistics, statistics of many-body wave functions (inverse participation ratios), and a Walsh transform of the many-body spectrum – we find that these computing platforms are dangerously close to a phase of uncontrollable chaotic fluctuations.

Hardware-Encoding Grid States in a Non-Reciprocal Superconducting Circuit

  1. Martin Rymarz,
  2. Stefano Bosco,
  3. Alessandro Ciani,
  4. and David P. DiVincenzo
We present a circuit design composed of a non-reciprocal device and Josephson junctions whose ground space is doubly degenerate and the ground states are approximate codewords of the
Gottesman-Kitaev-Preskill (GKP) code. We determine the low-energy dynamics of the circuit by working out the equivalence of this system to the problem of a single electron confined in a two-dimensional plane and under the effect of strong magnetic field and of a periodic potential. We find that the circuit is naturally protected against the common noise channels in superconducting circuits, such as charge and flux noise, implying that it can be used for passive quantum error correction. We also propose realistic design parameters for an experimental realization and we describe possible protocols to perform logical one- and two-qubit gates, state preparation and readout.

Simple Impedance Response Formulas for the Dispersive Interaction Rates in the Effective Hamiltonians of Low Anharmonicity Superconducting Qubits

  1. Firat Solgun,
  2. David P. DiVincenzo,
  3. and Jay M. Gambetta
For superconducting quantum processors consisting of low anharmonicity qubits such as transmons we give a complete microwave description of the system in the qubit subspace. We assume
that the qubits are dispersively coupled to a distributed microwave structure such that the detunings of the qubits from the internal modes of the microwave structure are stronger than their couplings. We define qubit ports across the terminals of the Josephson junctions and drive ports where transmission lines carrying drive signals reach the chip and we obtain the multiport impedance response of the linear passive part of the system between the ports. We then relate interaction parameters in between qubits and between the qubits and the environment to the entries of this multiport impedance function: in particular we show that the exchange coupling rate J between qubits is related in a simple way to the off-diagonal entry connecting the qubit ports. Similarly we relate couplings of the qubits to voltage drives and lossy environment to the entries connecting the qubits and the drive ports. Our treatment takes into account all the modes (possibly infinite) that might be present in the distributed electromagnetic structure and provides an efficient method for the modeling and analysis of the circuits.

A model study of present-day Hall-effect circulators

  1. Benedikt Placke,
  2. Stefano Bosco,
  3. and David P. DiVincenzo
Stimulated by the recent implementation of a three-port Hall-effect microwave circulator of Mahoney et al. (MEA), we present model studies of the performance of this device. Our calculations
are based on the capacitive-coupling model of Viola and DiVincenzo (VD). Based on conductance data from a typical Hall-bar device obtained from a two-dimensional electron gas (2DEG) in a magnetic field, we numerically solve the coupled field-circuit equations to calculate the expected performance of the circulator, as determined by the S parameters of the device when coupled to 50Ω ports, as a function of frequency and magnetic field. Above magnetic fields of 1.5T, for which a typical 2DEG enters the quantum Hall regime (corresponding to a Landau-level filling fraction ν of 20), the Hall angle θH=tan−1σxy/σxx always remains close to 90∘, and the S parameters are close to the analytic predictions of VD for θH=π/2. As anticipated by VD, MEA find the device to have rather high (kΩ) impedance, and thus to be extremely mismatched to 50Ω, requiring the use of impedance matching. We incorporate the lumped matching circuits of MEA in our modeling and confirm that they can produce excellent circulation, although confined to a very small bandwidth. We predict that this bandwidth is significantly improved by working at lower magnetic field when the Landau index is high, e.g. ν=20, and the impedance mismatch is correspondingly less extreme. Our modeling also confirms the observation of MEA that parasitic port-to-port capacitance can produce very interesting countercirculation effects.

Multi-Qubit Joint Measurements in Circuit QED: Stochastic Master Equation Analysis

  1. Ben Criger,
  2. and David P. DiVincenzo
We derive a family of stochastic master equations describing homodyne measurement of multi-qubit diagonal observables in circuit quantum electrodynamics. Our approach replaces the polaron-like
transformation of previous work, which required a lengthy calculation for the physically interesting case of three qubits and two resonator modes. The technique introduced here makes this calculation straightforward and manifestly correct. Using this technique, we are able to show that registers larger than one qubit evolve under a non-Markovian master equation. We perform numerical simulations of the three-qubit, two-mode case from previous work, obtaining an average post-measurement state fidelity near 94%.

Methodology for bus layout for topological quantum error correcting codes

  1. Martin Wosnitzka,
  2. Fabio L. Pedrocchi,
  3. and David P. DiVincenzo
Most quantum computing architectures can be realized as two-dimensional lattices of qubits that interact with each other. We take transmon qubits and transmission line resonators as
promising candidates for qubits and couplers; we use them as basic building elements of a quantum code. We then propose a simple framework to determine the optimal experimental layout to realize quantum codes. We show that this engineering optimization problem can be reduced to the solution of standard binary linear programs. While solving such programs is a NP-hard problem, we propose a way to find scalable optimal architectures that require solving the linear program for a restricted number of qubits and couplers. We apply our methods to two celebrated quantum codes, namely the surface code and the Fibonacci code.

Multiport Impedance Quantization

  1. Firat Solgun,
  2. and David P. DiVincenzo
With the increase of complexity and coherence of superconducting systems made using the principles of circuit quantum electrodynamics, more accurate methods are needed for the characterization,
analysis and optimization of these quantum processors. Here we introduce a new method of modelling that can be applied to superconducting structures involving multiple Josephson junctions, high-Q superconducting cavities, external ports, and voltage sources. Our technique, an extension of our previous work on single-port structures [1], permits the derivation of system Hamiltonians that are capable of representing every feature of the physical system over a wide frequency band and the computation of T1 times for qubits. We begin with a black box model of the linear and passive part of the system. Its response is given by its multiport impedance function Zsim(w), which can be obtained using a finite-element electormagnetics simulator. The ports of this black box are defined by the terminal pairs of Josephson junctions, voltage sources, and 50 Ohm connectors to high-frequency lines. We fit Zsim(w) to a positive-real (PR) multiport impedance matrix Z(s), a function of the complex Laplace variable s. We then use state-space techniques to synthesize a finite electric circuit admitting exactly the same impedance Z(s) across its ports; the PR property ensures the existence of this finite physical circuit. We compare the performance of state-space algorithms to classical frequency domain methods, justifying their superiority in numerical stability. The Hamiltonian of the multiport model circuit is obtained by using existing lumped element circuit quantization formalisms [2, 3]. Due to the presence of ideal transformers in the model circuit, these quantization methods must be extended, requiring the introduction of an extension of the Kirchhoff voltage and current laws.