Experimental Bayesian Calibration of Trapped-Ion Entangling Operations
Lukas Gerster, Fernando Martínez-García, Pavel Hrmo, Martin W. van Mourik, Benjamin Wilhelm, Davide Vodola, Markus Müller, R. Blatt, Philipp Schindler, Thomas Monz
The performance of quantum gate operations is experimentally determined by how correctly operational parameters can be determined and set, and how stable these parameters can be maintained.In addition, gates acting on different sets of qubits require unique sets of control parameters.Thus, an efficient multidimensional parameter estimation procedure is crucial to calibrate even medium-sized quantum processors.Here, we develop and characterize an efficient calibration protocol to automatically estimate and adjust experimental parameters of the widely used two-qubit Mølmer-Sørensen entangling gate operation in a trapped-ion quantum information processor.The protocol exploits Bayesian parameter estimation methods that include a stopping criterion based on a desired gate infidelity.We experimentally demonstrate a tune-up procedure that leads to a residual median gate infidelity due to miscalibration of 1.3(1) × 10 -3 , requiring 1200 ± 500 experimental cycles, while completing the entire gate calibration procedure in less than one minute, which provides a significant speedup over commonly used manual tune-up routines.This approach is applicable to other quantum information processor architectures with known or sufficiently characterized theoretical models.
The performance of quantum gate operations is experimentally determined by how correctly operational parameters can be determined and set, and how stable these parameters can be maintained.In addition, gates acting on different sets of qubits require unique sets of control parameters.Thus, an efficient multidimensional parameter estimation procedure is crucial to calibrate even medium-sized quantum processors.Here, we develop and characterize an efficient calibration protocol to automatically estimate and adjust experimental parameters of the widely used two-qubit Mølmer-Sørensen entangling gate operation in a trapped-ion quantum information processor.The protocol exploits Bayesian parameter estimation methods that include a stopping criterion based on a desired gate infidelity.We experimentally demonstrate a tune-up procedure that leads to a residual median gate infidelity due to miscalibration of 1.3(1) × 10 -3 , requiring 1200 ± 500 experimental cycles, while completing the entire gate calibration procedure in less than one minute, which provides a significant speedup over commonly used manual tune-up routines.This approach is applicable to other quantum information processor architectures with known or sufficiently characterized theoretical models.
Achieving precise control over quantum systems has groundbreaking ap-plications, such as simulating complex quantum systems, or implementing quantum algorithms with an improved efficiency over classical algorithms. Quantum computing has been under development for the last decades, re-sulting in the so-called Noisy Intermediate-Scale Quantum (NISQ) devices with about 50 qubits. While these devices are built with the purpose of achieving practical advantages over classical processors, they are vulnerable to noise sources. In order to account for error sources, one needs to im-plement Quantum Error Correction (QEC) techniques. These techniques protect quantum information by encoding a logical qubit into several phys-ical ones. For the implementation of NISQ devices and QEC techniques, it is vital to keep improving the quantum operations and states involved. Here we develop calibration protocols based on the application of Bayesian inference. We show their advantages compared to the more traditional frequentist approaches, such as their ability to maximise the information gained with each measurement, or their straightforward automation. In our first line of research, we develop a Bayesian protocol for the correction of unwanted phases appearing in the experimental preparation of Steane code states. This protocol requires 13 times less measurements than the frequentist approach. In our second line of research, we develop a proto-col for locking a laser to the qubit transition frequency in a trapped-ion architecture. This calibration is vital, since the correct implementation of single-qubit gates depends on it. In our final line of research, we study the Mølmer-Sørensen entangling gate, used in trapped-ion systems. First we introduce the calibration parameters, and derive a semianalytical model of the effects of one of them, for which there was no previous analytical understanding. Finally, we develop a Bayesian protocol for the calibration of these parameters, which was successfully implemented and tested in an experimental trapped-ion quantum processor.
As quantum systems expand in size and complexity, manual qubit characterization and gate optimization will be a non-scalable and time-consuming venture. Physical qubits have to be carefully calibrated because quantum processors are very sensitive to the external environment, with control hardware parameters slowly drifting during operation, affecting gate fidelity. Currently, existing calibration techniques require complex and lengthy measurements to independently control the different parameters of each gate and are unscalable to large quantum systems. Therefore, fully automated protocols with the desired functionalities are required to speed up the calibration process. This paper aims to propose single-qubit calibration of superconducting qubits under continuous weak measurements from a real physical experimental settings point of view. We propose a real-time optimal estimator of qubit states, which utilizes weak measurements and Bayesian optimization to find the optimal control pulses for gate design. Our numerical results demonstrate a significant reduction in the calibration process, obtaining a high gate fidelity. Using the proposed estimator we estimated the qubit state with and without measurement noise and the estimation error between the qubit state and the estimator state is less than 0.02. With this setup, we drive an approximated pi pulse with final fidelity of 0.9928. This shows that our proposed strategy is robust against the presence of measurement and environmental noise and can also be applicable for the calibration of many other quantum computation technologies.
A major challenge for the realization of useful universal quantum computers is achieving high fidelity two-qubit entangling gate operations. However, calibration errors can affect the quantum gate operations and limit their fidelity. To reduce such errors it is desirable to have an analytical understanding and quantitative predictions of the effects that miscalibrations of gate parameters have on the gate performance. In this work, we study a systematic perturbative expansion in miscalibrated parameters of the M\o{}lmer-S\o{}rensen entangling gate, which is widely used in trapped-ion quantum processors. Our analytical treatment particularly focuses on systematic center-line detuning miscalibrations. Via a unitary Magnus expansion, we compute the gate evolution operator, which allows us to obtain relevant key properties such as relative phases, electronic populations, quantum state purity and fidelities. These quantities, subsequently, are used to assess the performance of the gate using the fidelity of entangled states as performance metric. We verify the predictions from our model by benchmarking them against measurements in a trapped-ion quantum processor. The method and the results presented here can help design and calibrate high-fidelity gate operations of large-scale quantum computers.
The calibration of high-quality two-qubit entangling gates is an essential component in engineering large-scale, fault-tolerant quantum computers. However, many standard calibration techniques are based on randomized circuits that are only quadratically sensitive to calibration errors. As a result, these approaches are inefficient, requiring many experimental shots to achieve acceptable performance. In this work, we demonstrate that robust phase estimation can enable high-precision, Heisenberg-limited estimates of coherent errors in multi-qubit gates. Equipped with an efficient estimator, the calibration problem may be reduced to a simple optimization loop that minimizes the estimated coherent error. We experimentally demonstrate our calibration protocols by improving the operation of a two-qubit controlled-Z gate on a superconducting processor, and we validate the improved performance with gate set tomography. Our methods are applicable to gates in other quantum hardware platforms such as ion traps and neutral atoms, and on other multi-qubit gates, such as CNOT or iSWAP.
The fidelity of a quantum gate is sensitive to small deviations in the physical control parameters. Unfortunately, it is generally difficult to exactly model the implemented Hamiltonian for a set of user-defined parameters, necessitating on-device calibration. Here, we present an active learning framework based on Bayesian optimization with a Gaussian Process surrogate to find the optimal parameter set. We validate the technique through numerical calibration of the laser amplitude and frequencies that implement the trapped-ion Mølmer Sørensen gate. We show that a Gaussian process can model the Hamiltonian dynamics. The addition of active learning accelerates the discovery of the optimal parameter set with speed and final fidelity dependent on the quantum projection noise of the data. These results establish the utility of active learning and surrogate models for quantum calibration and control.
Continuously parameterized two-qubit gates are a key feature of state-of-the-art trapped-ion quantum processors as they have favorable error scalings and show distinct improvements in circuit performance over more restricted maximally entangling gatesets. In this work, we provide a comprehensive and pedagogical discussion on how to practically implement these continuously parameterized Mølmer Sørensen gates on the Quantum Scientific Computing Open User Testbed (QSCOUT), a low-level trappedion processor. To generate the arbitrary entangling angles, θ, we simply scale the amplitude of light used to generate the entanglement. However, doing so requires careful consideration of amplifier saturation as well as the variable light shifts that result. As such, we describe a method to calibrate and cancel the dominant fourth-order effects, followed by a dynamic virtual phase advance during the gate to cancel any residual light shifts, and find a linear scaling between θ and the residual light shift. Once, we have considered and calibrated these effects, we demonstrate performance improvement with decreasing θ. Finally, we describe nuances of hardware control to transform the XX-type interaction of the arbitraryangle Mølmer-Sørensen gate into a phase-agnostic and crosstalk-mitigating ZZ interaction.