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Implementing a Universal Gate Set on a Logical Qubit Encoded in an Oscillator
Reinier W. Heeres, Philip Reinhold, Nissim Ofek, Luigi Frunzio, Liang Jiang, Michel H. Devoret, Robert J. Schoelkopf
TL;DR
The paper addresses how to manipulate an encoded qubit in an oscillator despite residual system interactions and the need for precise whole-system control. It uses GRAPE with a characterized oscillator-transmon Hamiltonian and experimentally constrained pulses, achieving high-fidelity operations while improving preparation speed.
Problem
Encoded-qubit manipulation requires precise control of the full system, whose residual interactions must be accounted for when designing logical operations.
Method
GRAPE optimizes time-dependent oscillator and transmon drives using a characterized Hamiltonian while incorporating amplitude, bandwidth, frequency, and truncation constraints.
Results
The cooling protocol reduces the transmon population to approximately 1% and shortens the inter-experimental delay from approximately 18 ms to below 1 ms.
Takeaways & Limitations
Numerical control of a coupled oscillator-transmon system supports high-fidelity operations on a logical qubit encoded in an oscillator.
Abstract
from arXiv · showhide
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 (99%, 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.
I. SYSTEM HAMILTONIAN
The appendix specifies the full time-dependent system Hamiltonian by separating oscillator, transmon, interaction, and drive contributions, with parameters measured experimentally.
- System Hamiltonian: The appendix presents the full system Hamiltonian to the precision with which the system was characterized.
- System Hamiltonian: The Hamiltonian is decomposed into oscillator, transmon, interaction, and time-dependent driving terms.
- System Hamiltonian: Decoherence effects are simulated with a Markovian Lindblad master equation.
- System Hamiltonian: The measured system parameters are reported in Table SI.
II. GRAPE IMPLEMENTATION
GRAPE optimizes simultaneous oscillator and transmon drives for state transfers while incorporating experimental constraints on pulse amplitude, bandwidth, frequency, and oscillator truncation.
- GRAPE implementation: GRAPE maximizes coherent average fidelity over simultaneous oscillator and transmon controls to prepare desired joint-system operations.
- GRAPE implementation: The dispersive shift and its second-order correction are calibrated from transmon spectroscopy under several oscillator displacements.
- GRAPE implementation: The optimization uses efficient gradient calculations and Quasi-Newton methods to handle 2200 pulse parameters.
- GRAPE implementation: Penalty terms constrain pulse amplitude and bandwidth, while Fourier reparametrization enforces minimum and maximum allowed frequencies.
- GRAPE implementation: Solutions are optimized across several photon-number truncations so relevant dynamics remain consistent within the finite simulated oscillator space.
- GRAPE implementation: Optimized waveforms include complex time-domain drives and their Fourier spectra for transmon and oscillator controls.
III. MEASUREMENT SETUP
The measurement setup digitally generates, frequency-converts, amplifies, filters, and demodulates control and readout signals for the oscillator-transmon system.
- Measurement setup: An FPGA controller generates three I/Q waveform pairs using 500 Msample/s DACs, which are upconverted through I/Q mixers.
IV. SYSTEM PREPARATION
Measurement-based feedback cools the storage resonator and transmon before experiments, with Q-switching used when residual photons remain.
- System preparation: Feedback cooling initializes the storage resonator and transmon by checking the oscillator state and ending with the transmon in its ground state.
- System preparation: Q-switching drives couple the storage mode to the short-lived readout mode when the oscillator is not empty.
- System preparation: Figure S4 summarizes the preparation protocol, oscillator and transmon lifetimes, and spectroscopy after cooling.
- System preparation: The cooling protocol reduces the transmon population to approximately 1%.
- System preparation: The protocol decreases the inter-experimental delay from approximately 18 ms to below 1 ms.
V. EMPIRICAL TUNING
The control waveforms were empirically refined using randomized benchmarking and frequency-domain corrections to compensate for non-flat experimental transfer functions.
- The delay optimization was evaluated through the randomized benchmarking decay constant.
- Randomized benchmarking was used to fine-tune the resulting pulse waveforms.
- Frequency-domain correction applied a linear amplitude weighting with coefficient b and delay parameter τ before inverse transforming the waves.
VI. ADDITIONAL DATA
Additional measurements examined process tomography and the encoded qubit’s lifetime, including periodic fidelity revivals caused by Kerr-induced basis-state rephasing.
- Process tomography results for the encoded operations are provided in the supplementary data.
- The resonator’s Kerr nonlinearity of −3.7 kHz scrambles encoded basis states during free evolution.
- Periodic process-fidelity revivals occur because the encoded basis states rephase during waiting.The revival periodicity is not 1/K because of an intentional frequency detuning.