High-fidelity control of superconducting qubits requires carefully shaped microwave pulses that account for multiple error channels. In this work, we present a pedagogical introduction to pulse-shaping techniques for transmon qubits, aiming to provide a unified, accessible framework that integrates physical intuition for pulse design, analytical understanding of gate-level descriptions, and practical considerations of hardware. This article further aims to serve as a guide for students and early researchers entering superconducting quantum computing. We begin by examining simple pulse envelopes and their spectral properties, highlighting how finite bandwidth leads to leakage outside the computational subspace. These observations motivate the introduction of the derivative removal by adiabatic gate (DRAG) technique, which uses a quadrature component proportional to the pulse's time derivative to suppress off-resonant excitations. We analyze the single-qubit case using the Magnus expansion, which provides a clear understanding of the order-by-order introduction of error channels. We discuss the practical hardware realities of control pulse generation, focusing on arbitrary waveform generators (AWG), local oscillators (LO), and IQ mixing. Common imperfections are discussed in terms of their impact on the effective pulse shape and qubit Hamiltonian. Finally, we extend the discussion to two-qubit operations, focusing on the cross-resonance gate and the emergence of effective interactions.
Recent achievements in the physical size of quantum computers based on superconducting qubits foresee reaching the next milestones of the exponentially growing number of qubits. The promises in the scalability present opportunities in integrated control electronics operating at 4K stage in the dilution refrigerator. In addition to the number of physical qubits, however, commercialization necessitates a further increase in the number of consecutive high-fidelity gate operations within a coherence time of
∼100μs. It leads to the stringent requirement on the duration of qubit-driving microwave pulse
(i.e., <20ns). The microwave pulses should only drive
∣0⟩↔∣1⟩for single qubit gate operations while avoiding leakage to
∣1⟩↔∣2⟩which is the major factor that degrades the gate fidelity. The frequency for
∣1⟩↔∣2⟩,ω12, is typically lower than the frequency for
∣0⟩↔∣1⟩,ω01, by
2π×(160MHz−to−360MHz). The frequency difference
(ω01−ω12), called anharmonicity, is an individual qubit characteristic and becomes the margin for spectral management. But, a short driving pulse even with Gaussian, sine, or raised-cosine shapes eventually widens the spectral width. It would range a few hundred MHz and result in a considerable spectral leakage at
ω12. To mitigate this spectral leakage, the derivative removal by adiabatic gate (DRAG) has been adopted in the pulse shaping [1], [2]. As shown in Fig. 34.4.1, the first-order DRAG pulse shaping is realized by adding two parts as \begin{equation*}\mathsf{O}(\mathrm{t})=\mathsf{S}_{\mathsf{M}}(\mathsf{t})\cdot\mathsf{cos}(\omega_{01}\mathsf{t}+\varphi)+\mathsf{q}_{\mathsf{scale}}\mathsf{S}_{\mathsf{M}}(\mathsf{t})^{\prime}\cdot\mathsf{sin}(\omega_{01}\mathsf{t}+\varphi)\end{equation*} where
SM(t)is the main pulse shape and
qscaleis a coefficient inversely proportional to the anharmonicity. This first-order DRAG makes an asymmetrical spectrum that lowers the left frequency side and raises the right frequency side. Since the
ω12is always on the left side of
ω01, the DRAG can effectively minimize the leakage.
Improving the speed and fidelity of quantum logic gates is essential to reach quantum advantage with future quantum computers. However, fast logic gates lead to increased leakage errors in superconducting quantum processors based on qubits with low anharmonicity, such as transmons. To reduce leakage errors, we propose and experimentally demonstrate two new analytical methods, Fourier ansatz spectrum tuning derivative removal by adiabatic gate (FAST DRAG) and higher-derivative (HD) DRAG, both of which enable shaping single-qubit control pulses in the frequency domain to achieve stronger suppression of leakage transitions compared to previously demonstrated pulse shapes. Using the new methods to suppress the e f transition of a transmon qubit with an anharmonicity of − 212 MHz, we implement R X ( π / 2 ) gates achieving a leakage error below 3.0 × 10 − 5 down to a gate duration of 6.25 ns without the need for iterative closed-loop optimization. The obtained leakage error represents a 20-fold reduction in leakage compared to a conventional cosine DRAG pulse. Employing the FAST DRAG method, we further achieve an error per gate of ( 1.56 ± 0.07 ) × 10 − 4 at a 7.9-ns gate duration, outperforming conventional pulse shapes both in terms of error and gate speed. Furthermore, we study error-amplifying measurements for the characterization of temporal microwave control-pulse distortions, and demonstrate that non-Markovian coherent errors caused by such distortions may be a significant source of error for sub-10-ns single-qubit gates unless corrected using predistortion. Published by the American Physical Society 2024
For superconducting qubits, microwave pulses drive rotations around the Bloch sphere. The phase of these drives can be used to generate zero-duration arbitrary virtual
Z gates, which, combined with two ${X}_{\ensuremath{\pi}/2}$ gates, can generate any SU(2) gate. Here we show how to best utilize these virtual
Z gates to both improve algorithms and correct pulse errors. We perform randomized benchmarking using a Clifford set of Hadamard and
Z gates and show that the error per Clifford is reduced versus a set consisting of standard finite-duration
X and
Y gates.
Z gates can correct unitary rotation errors for weakly anharmonic qubits as an alternative to pulse-shaping techniques such as derivative removal by adiabatic gate (DRAG). We investigate leakage and show that a combination of DRAG pulse shaping to minimize leakage and
Z gates to correct rotation errors realizes a 13.3 ns ${X}_{\ensuremath{\pi}/2}$ gate characterized by low error $[1.95(3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}]$ and low leakage $[3.1(6)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}]$. Ultimately leakage is limited by the finite temperature of the qubit, but this limit is two orders of magnitude smaller than pulse errors due to decoherence.
Simple analytic pulse-shaping techniques are of great practical utility in quantum control, with prime examples being the DRAG method for suppressing leakage in superconducting microwave gates and the transitionless-driving approach to shortcuts-to-adiabaticity. Standard versions of these methods require two orthogonal control channels, with the second channel effectively breaking time-reversal symmetry. This appears to rule out their use in settings with only a single real-valued control field, such as the kind of baseband flux control that is common in many superconducting circuit architectures. We show here that a simple analytic pulse-shaping technique derived via a Magnus expansion is effective even with just a single baseband control channel. We demonstrate its efficacy by simulating a two-qubit gate between transmons realized with a tunable coupler and baseband flux pulses. Our corrections dramatically reduce non-adiabatic leakage caused by ramping the coupler: for realistic device parameters, leakage in a fast iSWAP gate is suppressed by up to three orders of magnitude. Our approach is general, goes beyond simply suppressing unwanted spectral weight at leakage transitions, and can be applied to a variety of platforms.
A fundamental challenge in quantum computing is to increase the number of operations within the qubit coherence time. While this can be achieved by decreasing the gate duration, the use of shorter signals increases their bandwidth and can cause leakage into energetically separated states. A common method to suppress leakage for short pulses is the Derivative Removal by Adiabatic Gate (DRAG) method, which, however, relies on IQ modulation of radio-frequency (RF) signals, thus cannot be applied to the baseband signals, e.g., for semiconductor spin qubits. This paper proposes a novel technique, Delayed Leakage Reduction (DLR), that suppresses leakage at targeted frequencies even for baseband control by using time-delayed repetitions of the control signal to enable rapid, high-fidelity operations. We apply DLR on the adiabatic CZ gate between two spin qubits and achieve fidelities exceeding 99.9% within 9.4 ns for a resonance frequency difference of only 100 MHz. Towards the experimental realization of the proposed control method, we also assess the impact on the fidelity of the sampling rate, noise, and signal quantization of the electronic hardware generating the control pulse, thus setting the minimum hardware requirements for any experimental demonstration.
Single-qubit gates on superconducting quantum processors are typically implemented using microwave pulses applied through dedicated control lines. However, these microwave pulses may also drive other qubits due to crosstalk arising from capacitive coupling and wavefunction overlap in systems with closely spaced transition frequencies. Crosstalk and frequency crowding increase errors during simultaneous single-qubit operations relative to isolated gates, thus forming a major bottleneck for scaling superconducting quantum processors. In this work, we combine model-based qubit frequency optimization with pulse shaping to demonstrate crosstalk error mitigation in single-qubit gates on a 49-qubit superconducting quantum processor. We introduce and experimentally verify an analytical model of simultaneous single-qubit gate error caused by microwave crosstalk that depends on a given pulse shape. By employing a model-based optimization strategy of qubit frequencies, we minimize the crosstalk-induced error across the processor and achieve a mean simultaneous single-qubit gate fidelity of 99.96% for a 16-ns gate duration, approaching the mean individual gate fidelity. To further reduce the simultaneous error and required qubit frequency bandwidth on high-crosstalk qubit pairs, we introduce a crosstalk transition suppression (CTS) pulse shaping technique that minimizes the spectral energy around transitions inducing leakage and crosstalk errors. Finally, we combine CTS with model-based frequency optimization across the device and experimentally show a systematic reduction in the required qubit frequency bandwidth for high-fidelity simultaneous gates, supported by simulations of systems with up to 1000 qubits. By alleviating constraints on qubit frequency bandwidth for parallel single-qubit operations, this work represents an important step for scaling towards larger quantum processors.