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SiPS ... design and implementation - IEEE Workshop on Signal Processing Systems

Application of Weighted Chebyshev Approximation in Pulse Design for Quantum Gates

Qi Ding, Alan V. Oppenheim, Petros T. Boufounos, Simon Gustavsson, Jeffrey A. Grover, Thomas A. Baran, William D. Oliver

Published Nov 4, 2024• 0 citations
Publisher

Abstract

Chebyshev approximation problems are fundamen-tal in approximation theory and are integral to various engi-neering applications. In this study, we suggest that weighted Chebyshev approximation (WCA) can be effectively applied to designing quantum gates. Specifically, we examine a class of two-qubit gates in superconducting qubits known as controlled-phase (CPHASE) gates, which leverage interactions between higher levels controlled by baseband flux pulses. We frame the design of CPHASE gates as a pulse design problem and propose Chebyshev pulses, generated through WCA, as an alternative solution. We utilize an illustrative example to demonstrate our process of designing a Chebyshev pulse with WCA, then contrast it with a benchmark Slepian pulse for comparison. We find that Chebyshev pulses can facilitate faster CPHASE gates while ensuring leakage errors remain below the desired threshold.

DOI
10.1109/sips62058.2024.00024

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Chebyshev approximation problems are fundamen-tal in approximation theory and are integral to various engi-neering applications. In this study, we suggest that weighted Chebyshev approximation (WCA) can be effectively applied to designing quantum gates. Specifically, we examine a class of two-qubit gates in superconducting qubits known as controlled-phase (CPHASE) gates, which leverage interactions between higher levels controlled by baseband flux pulses. We frame the design of CPHASE gates as a pulse design problem and propose Chebyshev pulses, generated through WCA, as an alternative solution. We utilize an illustrative example to demonstrate our process of designing a Chebyshev pulse with WCA, then contrast it with a benchmark Slepian pulse for comparison. We find that Chebyshev pulses can facilitate faster CPHASE gates while ensuring leakage errors remain below the desired threshold.
Despite progress towards achieving low error rates with superconducting qubits, error-prone two-qubit gates remain a bottleneck for realizing large-scale quantum computers. Therefore, a systematic framework to design high-fidelity gates becomes imperative. One type of two-qubit gate in superconducting qubits is the controlled-phase (CPHASE) gate, which utilizes a conditional interaction between higher energy levels of the qubits controlled by a baseband flux pulse on one of the qubits or a tunable coupler. In this work, we study an adiabatic implementation of CPHASE gates and formulate the design of the control trajectory for the gate as a pulse-design problem. We show in simulation that the Chebyshev-based trajectory can, in certain cases, enable gates with gate infidelity lower by an average of 23.3% when compared to the widely used Slepian-based trajectory.
Abstract An integrated programmable controlled-phase (CPHASE) gate has been proposed for quantum information processing applications. This gate can introduce arbitrary phase difference to the target qubit in the case of the control qubit being in the state of |1〉. As the desired phase difference can be utilized after fabrication, unlike the conventional controlled-phase gates that provide hard-coded phase shift, such an integrated gate is expected to pave the way for more versatile operations of current integrated optical circuits as well as possible new applications.
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.
We present theoretical schemes for performing high-fidelity one- and two-qubit pulsed gates for a superconducting flux qubit. The 'IBM qubit' consists of three Josephson junctions, three loops and a superconducting transmission line. Assuming a fixed inductive qubit–qubit coupling, we show that the effective qubit–qubit interaction is tunable by changing the applied fluxes, and can be made negligible, allowing one to perform high-fidelity single qubit gates. Our schemes are tailored to alleviate errors due to 1/ f noise; we find gates with only 1% loss of fidelity due to this source, for pulse times in the range of 20–30 ns for one-qubit gates ( Z rotations, Hadamard) and 60 ns for a two-qubit gate (controlled- Z ). Our relaxation and dephasing time estimates indicate a comparable loss of fidelity from this source. The control of leakage plays an important role in the design of our shaped pulses, preventing shorter pulse times. However, we have found that imprecision in the control of the quantum phase plays a major role in the limitation of the fidelity of our gates.
Fast quantum gates are crucial not only for the contemporary era of noisy intermediate-scale quantum devices but also for the prospective development of practical fault-tolerant quantum computing. Leakage errors, which arise from data qubits jumping beyond the confines of the computational subspace, are the main challenges in realizing nonadiabatically driven, fast gates. In this work, we propose and illustrate the usefulness of reinforcement learning (RL) to generate fast two-qubit gates in practical multilevel superconducting qubits. In particular, we show that the RL controller offers great effectiveness in finding piecewise constant gate-pulse sequences that act on two transmon data qubits coupled by a tunable coupler to generate a controlled-ZZZ (cz) gate with a gate time of 10 ns and an error rate of approximately $4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$. Using a gradient-based method to solve the same optimization problem often does not achieve high fidelity for such fast gates. However, we show that using the gate pulses discovered by RL as an ansatz for the gradient-based controller can substantially enhance fidelity compared to using RL alone. While for a 10-ns pulse, this improvement is marginal, the combined RL + gradient approach decreases the gate errors below ${10}^{\ensuremath{-}4}$ for a gate of length 20 ns.
Superconducting transmon qubits are of great interest for quantum computing and quantum simulation. A key component of quantum chemistry simulation algorithms is breaking up the evolution into small steps, which naturally leads to the need for nonmaximally entangling, arbitrary CPHASE gates. Here we design such microwave-based gates using an analytically solvable approach leading to smooth, simple pulses. We use the local invariants of the evolution operator in SU(4) to develop a method of constructing pulse protocols, which allows for the continuous tuning of the phase. We find CPHASE fidelities of more than 0.999 and gate times as low as 100 ns.