# Publications

## Calibration of the cross-resonance two-qubit gate between directly-coupled transmons

Phys. Rev. Applied 12 (0) 064013-064013

AD Patterson, J Rahamim, T Tsunoda, P Spring, S Jebari, K Ratter, M Mergenthaler, G Tancredi, B Vlastakis, M Esposito, PJ Leek

Quantum computation requires the precise control of the evolution of a quantum system, typically through application of discrete quantum logic gates on a set of qubits. Here, we use the cross-resonance interaction to implement a gate between two superconducting transmon qubits with a direct static dispersive coupling. We demonstrate a practical calibration procedure for the optimization of the gate, combining continuous and repeated-gate Hamiltonian tomography with step-wise reduction of dominant two-qubit coherent errors through mapping to microwave control parameters. We show experimentally that this procedure can enable a $\hat{ZX}_{-\pi/2}$ gate with a fidelity $F=97.0(7)\%$, measured with interleaved randomized benchmarking. We show this in a architecture with out-of-plane control and readout that is readily extensible to larger scale quantum circuits.

## Simultaneous Bistability of a Qubit and Resonator in Circuit Quantum Electrodynamics

Physical Review Letters American Physical Society (APS) 118 (0) 040402

TK Mavrogordatos, G Tancredi, M Elliott, MJ Peterer, A Patterson, J Rahamim, PJ Leek, E Ginossar, MH Szymańska

## Measurements of the Correlation Function of a Microwave Frequency Single Photon Source

ArXiv (0)

D Bozyigit, C Lang, L Steffen, JM Fink, M Baur, R Bianchetti, PJ Leek, S Filipp, MPD Silva, A Blais, A Wallraff

At optical frequencies the radiation produced by a source, such as a laser, a black body or a single photon source, is frequently characterized by analyzing the temporal correlations of emitted photons using single photon counters. At microwave frequencies, however, there are no efficient single photon counters yet. Instead, well developed linear amplifiers allow for efficient measurement of the amplitude of an electromagnetic field. Here, we demonstrate how the properties of a microwave single photon source can be characterized using correlation measurements of the emitted radiation with such detectors. We also demonstrate the cooling of a thermal field stored in a cavity, an effect which we detect using a cross-correlation measurement of the radiation emitted at the two ends of the cavity.

## Quantum-to-Classical Transition in Cavity Quantum Electrodynamics

ArXiv (0)

JM Fink, L Steffen, P Studer, LS Bishop, M Baur, R Bianchetti, D Bozyigit, C Lang, S Filipp, PJ Leek, A Wallraff

The quantum properties of electromagnetic, mechanical or other harmonic oscillators can be revealed by investigating their strong coherent coupling to a single quantum two level system in an approach known as cavity quantum electrodynamics (QED). At temperatures much lower than the characteristic energy level spacing the observation of vacuum Rabi oscillations or mode splittings with one or a few quanta asserts the quantum nature of the oscillator. Here, we study how the classical response of a cavity QED system emerges from the quantum one when its thermal occupation -- or effective temperature -- is raised gradually over 5 orders of magnitude. In this way we explore in detail the continuous quantum-to-classical crossover and demonstrate how to extract effective cavity field temperatures from both spectroscopic and time-resolved vacuum Rabi measurements.

## Dynamics of dispersive single qubit read-out in circuit quantum electrodynamics

ArXiv (0)

R Bianchetti, S Filipp, M Baur, JM Fink, M Göppl, PJ Leek, L Steffen, A Blais, A Wallraff

The quantum state of a superconducting qubit nonresonantly coupled to a transmission line resonator can be determined by measuring the quadrature amplitudes of an electromagnetic field transmitted through the resonator. We present experiments in which we analyze in detail the dynamics of the transmitted field as a function of the measurement frequency for both weak continuous and pulsed measurements. We find excellent agreement between our data and calculations based on a set of Bloch-type differential equations for the cavity field derived from the dispersive Jaynes-Cummings Hamiltonian including dissipation. We show that the measured system response can be used to construct a measurement operator from which the qubit population can be inferred accurately. Such a measurement operator can be used in tomographic methods to reconstruct single and multiqubit states in ensemble-averaged measurements.

## Dressed Collective Qubit States and the Tavis-Cummings Model in Circuit QED

ArXiv (0)

JM Fink, R Bianchetti, M Baur, M Goeppl, L Steffen, S Filipp, PJ Leek, A Blais, A Wallraff

We present an ideal realization of the Tavis-Cummings model in the absence of atom number and coupling fluctuations by embedding a discrete number of fully controllable superconducting qubits at fixed positions into a transmission line resonator. Measuring the vacuum Rabi mode splitting with one, two and three qubits strongly coupled to the cavity field, we explore both bright and dark dressed collective multi-qubit states and observe the discrete square root of N scaling of the collective dipole coupling strength. Our experiments demonstrate a novel approach to explore collective states, such as the W-state, in a fully globally and locally controllable quantum system. Our scalable approach is interesting for solid-state quantum information processing and for fundamental multi-atom quantum optics experiments with fixed atom numbers.

## Measurement of Autler-Townes and Mollow transitions in a strongly driven superconducting qubit

ArXiv (0)

M Baur, S Filipp, R Bianchetti, JM Fink, M Göppl, L Steffen, PJ Leek, A Blais, A Wallraff

We present spectroscopic measurements of the Autler-Townes doublet and the sidebands of the Mollow triplet in a driven superconducting qubit. The ground to first excited state transition of the qubit is strongly pumped while the resulting dressed qubit spectrum is probed with a weak tone. The corresponding transitions are detected using dispersive read-out of the qubit coupled off-resonantly to a microwave transmission line resonator. The observed frequencies of the Autler-Townes and Mollow spectral lines are in good agreement with a dispersive Jaynes-Cummings model taking into account higher excited qubit states and dispersive level shifts due to off-resonant drives.

## Modelling Enclosures for Large-Scale Superconducting Quantum Circuits

ArXiv (0)

PA Spring, T Tsunoda, B Vlastakis, PJ Leek

Superconducting quantum circuits are typically housed in conducting enclosures in order to control their electromagnetic environment. As devices grow in physical size, the electromagnetic modes of the enclosure come down in frequency and can introduce unwanted long-range cross-talk between distant elements of the enclosed circuit. Incorporating arrays of inductive shunts such as through-substrate vias or machined pillars can suppress these effects by raising these mode frequencies. Here, we derive simple, accurate models for the modes of enclosures that incorporate such inductive-shunt arrays. We use these models to predict that cavity-mediated inter-qubit couplings and drive-line cross-talk are exponentially suppressed with distance for arbitrarily large quantum circuits housed in such enclosures, indicating the promise of this approach for quantum computing. We find good agreement with a finite-element simulation of an example device containing more than 400 qubits.

## Excitation and detection of propagating spin waves at the single magnon level

ArXiv (0)

Ferro- and ferrimagnets play host to small-signal, microwave-frequency magnetic excitations called spin waves, the quanta of which are known as magnons. Over the last decade, the field of spin-wave dynamics has contributed much to our understanding of fundamental magnetism. To date, experiments have focussed overwhelmingly on the study of room-temperature systems within classical limits. Here we demonstrate, for the first time, the excitation and detection of propagating spin waves at the single magnon level. Our results allow us to project that coupling of propagating spin-wave excitations to quantum circuits is achievable, enabling fundamental quantum-level studies of magnon systems and potentially opening doors to novel hybrid quantum measurement and information processing devices.

## Realization of a Carbon-Nanotube-Based Superconducting Qubit

ArXiv (0)

M Mergenthaler, A Nersisyan, A Patterson, M Esposito, A Baumgartner, C Schönenberger, GAD Briggs, EA Laird, PJ Leek

Hybrid circuit quantum electrodynamics (QED) involves the study of coherent quantum physics in solid state systems via their interactions with superconducting microwave circuits. Here we present an implementation of a hybrid superconducting qubit that employs a carbon nanotube as a Josephson junction. We realize the junction by contacting a carbon nanotube with a superconducting Pd/Al bi-layer, and implement voltage tunability of the qubit frequency using a local electrostatic gate. We demonstrate strong dispersive coupling to a coplanar waveguide resonator via observation of a resonator frequency shift dependent on applied gate voltage. We extract qubit parameters from spectroscopy using dispersive readout and find qubit relaxation and coherence times in the range of $10-200~\rm{ns}$.

## Efficient Hamiltonian programming in qubit arrays with nearest-neighbour couplings

ArXiv (0)

T Tsunoda, G Bhole, SA Jones, JA Jones, PJ Leek

We consider the problem of selectively controlling couplings in a practical quantum processor with always-on interactions that are diagonal in the computational basis, using sequences of local NOT gates. This methodology is well-known in NMR implementations, but previous approaches do not scale efficiently for the general fully-connected Hamiltonian, where the complexity of finding time-optimal solutions makes them only practical up to a few tens of qubits. Given the rapid growth in the number of qubits in cutting-edge quantum processors, it is of interest to investigate the applicability of this control scheme to much larger scale systems with realistic restrictions on connectivity. Here we present an efficient scheme to find near time-optimal solutions that can be applied to engineered qubit arrays with local connectivity for any number of qubits, indicating the potential for practical quantum computing in such systems.

## Critical slowing down in the bistable regime of circuit quantum electrodynamics

ArXiv (0)

P Brookes, G Tancredi, AD Patterson, J Rahamim, M Esposito, PJ Leek, E Ginossar, MH Szymanska

We investigate the dynamics of the bistable regime of the generalized Jaynes-Cummings Hamiltonian (GJC), realised by a circuit quantum electrodynamics (cQED) system consisting of a transmon qubit coupled to a microwave cavity. In this regime we observe critical slowing down in the approach to the steady state. By measuring the response of the cavity to a step function drive pulse we characterize this slowing down as a function of driving frequency and power. We find that the critical slowing down saturates as the driving power is increased. We compare these results with the predictions of analytical and numerical calculations both with and without the Duffing approximation. We find that the Duffing approximation incorrectly predicts that the critical slowing down timescale increases exponentially with the drive, whereas the GJC model accurately predicts the saturation seen in our data, suggesting a different process of quantum activation.