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arXiv:1310.8363 · Physical Review A

Improving frequency selection of driven pulses using derivative-based transition suppression

Felix Motzoi, Frank K. Wilhelm

Published Dec 13, 2013• 104 citations
arXivHTMLPDF

Abstract

Many techniques in quantum control rely on frequency separation as a means for suppressing unwanted couplings. In its simplest form, the mechanism relies on the low bandwidth of control pulses of long duration. Here we perform a higher-order quantum-mechanical treatment that allows for higher precision and shorter times. In particular, we identify three kinds of off-resonant effects: (i) simultaneous unwanted driven couplings (e.g., due to drive crosstalk), (ii) additional (initially undriven) transitions such as those in an infinite ladder system, and (iii) sideband frequencies of the driving wave form such as we find in corrections to the rotating-wave approximation. With a framework that is applicable to all three cases, in addition to the known adiabatic error responsible for a shift of the energy levels we typically see in the spectroscopy of such systems, we derive error terms in a controlled expansion corresponding to higher-order adiabatic effects and diabatic excitations. We show, by also expanding the driving wave form in a basis of different order derivatives of a trial function (typically a Gaussian) these different error terms can be corrected for in a systematic way, hence strongly improving quantum control of systems with dense spectra.

DOI
10.1103/physreva.88.062318

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Many techniques in quantum control rely on frequency separation as a means for suppressing unwanted couplings. In its simplest form, the mechanism relies on the low bandwidth of control pulses of long duration. Here we perform a higher-order quantum-mechanical treatment that allows for higher precision and shorter times. In particular, we identify three kinds of off-resonant effects: (i) simultaneous unwanted driven couplings (e.g., due to drive crosstalk), (ii) additional (initially undriven) transitions such as those in an infinite ladder system, and (iii) sideband frequencies of the driving wave form such as we find in corrections to the rotating-wave approximation. With a framework that is applicable to all three cases, in addition to the known adiabatic error responsible for a shift of the energy levels we typically see in the spectroscopy of such systems, we derive error terms in a controlled expansion corresponding to higher-order adiabatic effects and diabatic excitations. We show, by also expanding the driving wave form in a basis of different order derivatives of a trial function (typically a Gaussian) these different error terms can be corrected for in a systematic way, hence strongly improving quantum control of systems with dense spectra.
The task of controlling a quantum system under time and bandwidth limitations is made difficult by unwanted excitations of spectrally neighboring energy levels. In this article we review the Derivative Removal by Adiabatic Gate (DRAG) framework. DRAG is a multi-transition variant of counterdiabatic driving, where multiple low-lying gapped states in an adiabatic evolution can be avoided simultaneously, greatly reducing operation times compared to the adiabatic limit. In its essence, the method corresponds to a convergent version of the superadiabatic expansion where multiple counterdiabaticity conditions can be met simultaneously. When transitions are strongly crowded, the system of equations can instead be favorably solved by an average Hamiltonian (Magnus) expansion, suggesting the use of additional sideband control. We give some examples of common systems where DRAG and variants thereof can be applied to improve performance.
Three-level systems appear in various quantum information processing platforms.In several control schemes, the excited level serves as an auxiliary state for implementing gate operations between the lower qubit states.However, extra excited levels give rise to unwanted transitions that cause leakage and other errors, degrading the gate fidelity.We focus on a coherent-population-trapping scheme for gates and design protocols that reduce the effects of the unwanted off-resonant couplings and improve the gate performance up to several orders of magnitude.For a particular setup of unwanted couplings, we find an exact solution, which leads to error-free gate operations via only a static detuning modification.In the general case, we improve gate operations by adding corrective modulations to the pulses, thereby generalizing the derivative removal by adiabatic gates protocol to systems.Our techniques enable fast and high-fidelity gates and apply to a wide range of optically driven platforms, such as quantum dots, color centers, and trapped ions.
High-fidelity control of superconducting qubits requires carefully shaped microwave pulses to avoid several different kinds of error at once. This article is a pedagogical bridging text aimed at upper-level undergraduate and early graduate students who have completed an introductory quantum mechanics course and a first course in quantum computing or quantum information, but who have not yet encountered the physical implementation of qubit gates. We integrate physical intuition for pulse design, analytical gate-level descriptions, and practical hardware considerations into a single, derivation-driven narrative, with explicit learning objectives. We begin with simple pulse envelopes and their spectral properties, showing how finite bandwidth produces leakage outside the computational subspace. This motivates the derivative removal by adiabatic gate (DRAG) technique, which we derive explicitly using the Magnus expansion, obtaining a clear, order-by-order account of which physical error channel appears at which order and why DRAG's cancellation is necessarily incomplete. We discuss the practical hardware realities of control pulse generation, focusing on arbitrary waveform generators (AWG), local oscillators (LO), and IQ mixing. Finally, we extend the discussion to two-qubit operation via the cross-resonance gate, and interpret how driving the control qubit at the target qubit's transition frequency necessarily produces several unwanted interaction terms alongside the desired one, and how successive generations of pulse-engineering strategies have been designed to suppress them.
An off-resonant error for a driven quantum system refers to interactions due to the input drives having nonzero spectral overlap with unwanted system transitions. For the cross-resonance gate, this includes leakage as well as off-diagonal computational interactions that lead to bit-flip error on the control qubit. In this paper, we quantify an off-resonant error, with more focus on the less studied off-diagonal control interactions, for a direct controlled-not (cnot) gate implementation. Our results are based on a numerical simulation of the dynamics while we demonstrate the connection to time-dependent Schrieffer-Wolff and Magnus perturbation theories. We present two methods for suppressing such error terms. First, pulse parameters need to be optimized so off-resonant transition frequencies coincide with the local minima due to the pulse spectrum sidebands. Second, we show the advantage of a YYY-DRAG pulse on the control qubit in mitigating the off-resonant error. Depending on qubit-qubit detuning, the proposed methods can improve the average off-resonant error from approximately ${10}^{\ensuremath{-}3}$ closer to the ${10}^{\ensuremath{-}4}$ level for a direct cnot calibration.
High-fidelity quantum control relies on accurate models of driven dynamics. We examine this re- quirement for single-qubit gates in superconducting transmons by comparing control pulses derived from the standard Duffing approximation and from a Hamiltonian constructed by diagonalizing the transmon eigenbasis. Using the same correction-pulse construction for both models, we show that correction fields derived from the Duffing approximation can substantially reduce the gate error pre- dicted by that model while remaining less effective when combined with an independently calibrated baseline pulse in the diagonalized-transmon model. In the fast-gate regime, such transferred correc- tions can even fail to improve over the uncorrected diagonalized-transmon baseline. We show that small model-dependent differences in both the energy spectrum and the representation of the drive operator can compound during driven evolution, resulting in different predicted error generators and correction pulses. A mismatch in the accumulated AC Stark phase provides one illustrative di- agnostic of this dynamical model dependence. We further demonstrate that the model Hamiltonian informs the choice of control framework: Omitting relevant leakage pathways or higher-order error channels can lead to an overly restricted correction strategy. Including these channels motivates an extended correction framework that improves the gate performance using the same physical control resources.
Quantum information and quantum computation are linked by a common mathematical and physical framework of quantum mechanics. The manipulation of the predicted dynamics and its optimization is known as quantum control. Many techniques, originating in the study of nuclear magnetic resonance, have found common usage in methods for processing quantum information and steering physical systems into desired states. This thesis expands on these techniques, with careful attention to the regime where competing effects in the dynamics are present, and no semi-classical picture exists where one effect dominates over the others. That is, the transition between the diabatic and adiabatic error regimes is examined, with the use of such techniques as time-dependent diagonalization, interaction frames, average-Hamiltonian expansion, and numerical optimization with multiple time-dependences. The results are applied specifically to superconducting systems, but are general and improve on existing methods with regard to selectivity and crosstalk problems, filtering of modulation of resonance between qubits, leakage to non-compuational states, multi-photon virtual transitions, and the strong driving limit.