Closed-loop optimization for high-fidelity controlled-Z gates in superconducting qubits

Semiconductor spin qubits demonstrated single-qubit gates with fidelities up to benchmarked in the single-qubit subspace. However, tomographic characterizations reveal non-negligible crosstalk errors in a larger space. Additionally, it was long thought that the two-qubit gate performance is limited by charge noise, which couples to the qubits via the exchange interaction. Here, we show that coherent error sources such as a limited bandwidth of the control signals, diabaticity errors, microwave crosstalk, and non-linear transfer functions can equally limit the fidelity. We report a simple theoretical framework for pulse optimization that relates erroneous dynamics to spectral concentration problems and allows for the reuse of existing signal shaping methods on a larger set of gate operations. We apply this framework to common gate operations for spin qubits and show that simple pulse shaping techniques can significantly improve the performance of these gate operations in the presence of such coherent error sources. The methods presented in the paper were used to demonstrate two-qubit gate fidelities with in Xue et al (2022 Nature 601 343). We also find that single and two-qubit gates can be optimized using the same pulse shape. We use analytic derivations and numerical simulations to arrive at predicted gate fidelities greater than 99.9% with duration less than, where is the difference in qubit frequencies.

<sec>Spin qubits in silicon-based semiconductor quantum dots have become one of the prominent candidates for realizing fault-tolerant quantum computing due to their long coherence time, good controllability, and compatibility with modern advanced integrated circuit manufacturing processes. In recent years, due to the remarkable advances in silicon-based materials, the structure of quantum dot and its fabrication process, and qubit manipulation technology, the great progress of high-fidelity state preparation and readout, single- and two-qubit gates of spin qubits for silicon based semiconductor quantum computation has been achieved. The control fidelities for single- and two-qubit gates all exceed 99%—fault tolerance threshold required by the surface code known for its exceptionally high tolerance to errors. In this paper, we briefly introduce the basic concepts of silicon-based semiconductor quantum dots, discuss the state-of-art technologies used to improve the fidelities of single- and two-qubit gates, and finally highlight the research directions that should be focused on.</sec><sec>This paper is organized as follows. Firstly, we introduce three major types of quantum dot (QD) devices fabricated on different silicon-based substrates, including Si/SiGe heterojunction and Si/SiO<sub>2</sub>. The spin degree of electron or nucleus hosted in QD can be encoded into spin qubits. Electron spin qubits can be thermally initialized to ground state by using an electron reservoirs, and can be read out by spin-charge conversion mechanism: energy-selective readout (Elzerman readout) with reservoirs or Pauli spin blockade (PSB) without reservoirs. Additionally, high fidelity single-shot readout has been demonstrated by using radio-frequency gate reflectometry combined with the PSB, which has unique advantages in large-scale qubit array. To coherently control the spin qubits, electron dipole resonance (ESR) or electron dipole spin resonance (EDSR) for electron and nuclear magnetic resonance (NMR) for nucleus are introduced. With the help of isotope purification greatly improving the dephasing time of qubit and fast single-qubit manipulation based on EDSR, fidelity above 99.9% can be reached. For the two-qubit gates based on exchange interaction between electron spins, the strength of interaction <inline-formula><tex-math id="M1">\begin{document}$ J $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M1.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M1.png"/></alternatives></inline-formula> combined with Zeeman energy difference <inline-formula><tex-math id="M2">\begin{document}$ \Delta {E}_{z} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M2.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M2.png"/></alternatives></inline-formula> determines the energy levels of system, which lead to the different two-qubit gates, such as controlled-Z (CZ), controlled-Rotation (CROT), and the square root of the SWAP gate (<inline-formula><tex-math id="M3">\begin{document}$ \sqrt{\rm{S}\rm{W}\rm{A}\rm{P}} $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M3.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="23-20221900_M3.png"/></alternatives></inline-formula>) gates. In order to improve the fidelity of two-qubit gates, a series of key technologies is used experimentally, they being isotope purification, symmetry operation, careful Hamiltonian engineering, and gate set tomography. Fidelity of two-qubit gates exceeding 99% has been demonstrated for electron spin qubits in Si/SiGe quantum dots and nuclear spin qubits in donors. These advances have pushed the silicon-based spin qubit platform to become a major stepping stone towards fault-tolerant quantum computation. Finally, we discuss the future study of spin qubits, that is, how to effectively expand the number of qubits, and many other problems to be explored and solved.</sec>

Besides noticeable challenges in implementing low-error single- and two-qubit quantum gates in superconducting quantum processors, the readout technique and analysis are a key factor in determining the efficiency and performance of quantum processors. Being able to efficiently implement quantum algorithms involving entangling gates and asses their output is mandatory for quantum utility. In a transmon-based 5-qubit superconducting quantum processor, we compared the performance of quantum circuits involving an increasing level of complexity, from single-qubit circuits to maximally entangled Bell circuits. This comparison highlighted the importance of the readout analysis and helped us optimize the protocol for more advanced quantum algorithms. Here, we report the results obtained from the analysis of the outputs of quantum circuits using two readout paradigms, referred to as “multiplied readout probabilities” and “conditional readout probabilities.” The first method is suitable for single-qubit circuits, while the second is essential for accurately interpreting the outputs of circuits involving two-qubit gates.

Two-qubit gates are important components of quantum computing. However, unwanted interactions between qubits (so-called parasitic gates) can be particularly problematic and degrade the performance of quantum applications. In this work, we present two software methods to mitigate parasitic two-qubit gate errors. The first approach is built upon the Cartan’s KAK decomposition and keeps the original unitary decomposition for the error-free native two-qubit gate. It counteracts a parasitic two-qubit gate by only applying single-qubit rotations and therefore has no two-qubit gate overhead. We show the optimal choice of single-qubit mitigation gates. The second approach applies a numerical optimisation algorithm to re-compile a target unitary into the error-parasitic two-qubit gate plus single-qubit gates. We demonstrate these approaches on the CPhase-parasitic iSWAP-like gates. The KAK-based approach helps decrease unitary infidelity by a factor of 3 compared to the noisy implementation without error mitigation. When arbitrary single-qubit rotations are allowed, recompilation could completely mitigate the effect of parasitic errors but may require more native gates than the KAK-based approach. We also compare their average gate fidelity under realistic noise models, including relaxation and depolarising errors. Numerical results suggest that different approaches are advantageous in different error regimes, providing error mitigation guidance for near-term quantum computers.

The development of noisy intermediate-scale quantum (NISQ) devices has extended the scope of executable quantum circuits with high-fidelity single- and two-qubit gates. Equipping NISQ devices with three-qubit gates will enable the realization of more complex quantum algorithms and efficient quantum error correction protocols with reduced circuit depth. Several three-qubit gates have been implemented for superconducting qubits, but their use in gate synthesis has been limited due to their low fidelity. Here, using fixed-frequency superconducting qubits, we demonstrate a high-fidelity iToffoli gate based on two-qubit interactions, the so-called cross-resonance effect. As with the Toffoli gate, this three-qubit gate can be used to perform universal quantum computation. The iToffoli gate is implemented by simultaneously applying microwave pulses to a linear chain of three qubits, revealing a process fidelity as high as 98.26(2)%. Moreover, we numerically show that our gate scheme can produce additional three-qubit gates which provide more efficient gate synthesis than the Toffoli and iToffoli gates. Our work not only brings a high-fidelity iToffoli gate to current superconducting quantum processors but also opens a pathway for developing multi-qubit gates based on two-qubit interactions.

We propose a scalable neutral atom quantum computer with an on-demand interaction through a selective two-qubit gate operation. Atoms are trapped by a lattice of near field Fresnel diffraction lights so that each trap captures a single atom. One-qubit gate operation is implemented by a gate control laser beam which is applied to an individual atom. Two-qubit gate operation between an arbitrary pair of atoms is implemented by sending these atoms to a state-dependent optical lattice and making them collide so that a particular two-qubit state acquires a dynamical phase. We give numerical evaluations corresponding to these processes, from which we estimate the upper bound of a two-qubit gate operation time and corresponding gate fidelity. Our proposal is feasible within currently available technology developed in cold atom gas, MEMS, nanolithography, and various areas in optics.

Fixed-frequency transmon quantum computers (QCs) have advanced in coherence times, addressability, and gate fidelities. Unfortunately, these devices are restricted by the number of on-chip qubits, capping processing power and slowing progress toward fault-tolerance. Although emerging transmon devices feature over 100 qubits, building QCs large enough for meaningful demonstrations of quantum advantage requires overcoming many design challenges. For example, today’s transmon qubits suffer from significant variation due to limited precision in fabrication. As a result, barring significant improvements in current fabrication techniques, scaling QCs by building ever larger individual chips with more qubits is hampered by device variation. Severe device variation that degrades QC performance is referred to as a defect. Here, we focus on a specific defect known as a frequency collision. When transmon frequencies collide, their difference falls within a range that limits two-qubit gate fidelity. Frequency collisions occur with greater probability on larger QCs, causing collision-free yields to decline as the number of on-chip qubits increases. As a solution, we propose exploiting the higher yields associated with smaller QCs by integrating quantum chiplets within quantum multi-chip modules (MCMs). Yield, gate performance, and application-based analysis show the feasibility of QC scaling through modularity. Our results demonstrate that chiplet architectures, relative to monolithic designs, benefit from average yield improvements ranging from 9.6 – 92.6 × for ≲5 qubit machines. In addition, our simulations explore the design space of chiplet systems and discover configurations that demonstrate average two-qubit gate infidelity reductions that are at best 0.815 × their monolithic counterpart. Finally, we observe that carefully-selected modular systems achieve fidelity improvements on a range of benchmark circuits.

In trapped-ion quantum computers, two-qubit entangling gates are generated by applying spin-dependent force which uses phonons to mediate interaction between the internal states of the ions. To maintain high-fidelity two-qubit gates under fluctuating experimental parameters, robust pulse-design methods are applied to remove the residual spin-motion entanglement in the presence of motional mode-frequency drifts. Here we propose an improved pulse-design method that also guarantees the robustness of the two-qubit rotation angle against uniform mode-frequency drifts by concatenating pulses with opposite sensitivity of the angle to mode-frequency drifts. We experimentally verify significantly improved robustness of the rotation angle against uniform mode-frequency drifts, as well as observe an improvement in gate fidelity from 97.84(10)% to 98.11(11)%, compared to a single frequency-modulated pulse. © 2023 American Physical Society.

Fault-tolerant quantum computers that can solve hard problems rely on quantum error correction1. One of the most promising error correction codes is the surface code2, which requires universal gate fidelities exceeding an error correction threshold of 99 per cent3. Among the many qubit platforms, only superconducting circuits4, trapped ions5 and nitrogen-vacancy centres in diamond6 have delivered this requirement. Electron spin qubits in silicon7-15 are particularly promising for a large-scale quantum computer owing to their nanofabrication capability, but the two-qubit gate fidelity has been limited to 98 per cent owing to the slow operation16. Here we demonstrate a two-qubit gate fidelity of 99.5 per cent, along with single-qubit gate fidelities of 99.8 per cent, in silicon spin qubits by fast electrical control using a micromagnet-induced gradient field and a tunable two-qubit coupling. We identify the qubit rotation speed and coupling strength where we robustly achieve high-fidelity gates. We realize Deutsch-Jozsa and Grover search algorithms with high success rates using our universal gate set. Our results demonstrate universal gate fidelity beyond the fault-tolerance threshold and may enable scalable silicon quantum computers.

Rydberg quantum gate serving as an indispensable computing unit for neutral-atom quantum computation, has attracted intense research efforts for the last decade. However, the state-of-the-art experiments have not reached the high gate fidelity as predicted by most theories due to the unexpected large loss remaining in Rydberg and intermediate states. In this paper, we report our findings in constructing a native two-qubit controlled-NOT gate based on pulse optimization. We focus on the method of commonly-used two-photon Rydberg excitation with smooth Gaussian-shaped pulses which is straightforward for experimental demonstration. By utilizing optimized pulse shapes the scheme reveals a remarkable reduction in the decays from Rydberg and intermediate states, as well as a high-tolerance to the residual thermal motion of atoms. We extract a conservative lower bound for the gate fidelity after taking into account the experimental imperfections. Our results not only reduce the gap between experimental and theoretical prediction because of the optimal control, but also facilitate the connectivity of distant atomic qubits in a larger atom array by reducing the requirement of strong blockade, which is promising for developing multiqubit quantum computation in large-scale atomic arrays.

Course 9 Solid state quantum bit circuits

Recent experiments demonstrated that the spin state of individual atoms on surfaces can be quantum-coherently controlled through all-electric electron spin resonance. By constructing interacting arrays of atoms, this results in an atomic-scale qubit platform. However, the static exchange coupling between qubits, limited lifetime, and polarization of the initial state impose significant limits on high-fidelity quantum control. We address this issue using open quantum systems simulation and quantum optimal control theory. We demonstrate the conditions under which high-fidelity operations ( $${\mathcal{F}}\,\gtrsim\, 0.9$$ F ≳ 0.9 ) are feasible in this qubit platform, and show how the Krotov method of quantum optimal control theory adapts to specific noise sources to outperform the conventional Rabi drivings. Finally, we re-examine the experimental setup used in the initial demonstration of this qubit platform and propose optimized experimental designs to maximize gate fidelity in this platform.

Practical Quantum computing hinges on the ability to control large numbers of qubits with high fidelity. Quantum dots define a promising platform due to their compatibility with semiconductor manufacturing. Moreover, high-fidelity operations above 99.9% have been realized with individual qubits, though their performance has been limited to 98.67% when driving two qubits simultaneously. Here we present single-qubit randomized benchmarking in a two-dimensional array of spin qubits, finding native gate fidelities as high as 99.992(1)%. Furthermore, we benchmark single qubit gate performance while simultaneously driving two and four qubits, utilizing a novel benchmarking technique called N-copy randomized benchmarking, designed for simple experimental implementation and accurate simultaneous gate fidelity estimation. We find two- and four-copy randomized benchmarking fidelities of 99.905(8)% and 99.34(4)% respectively, and that next-nearest neighbor pairs are highly robust to cross-talk errors. These characterizations of single-qubit gate quality are crucial for scaling up quantum information technology.

We demonstrate a controlled-Z gate between capacitively coupled fluxonium qubits with transition frequencies 72.3 and 136.3 MHz. The gate is activated by a 61.6-ns-long pulse at a frequency between noncomputational transitions |10⟩−|20⟩ and |11⟩−|21⟩, during which the qubits complete only four and eight Larmor periods, respectively. The measured gate error of (8±1)×10−3 is limited by decoherence in the noncomputational subspace, which will likely improve in the next-generation devices. Although our qubits are about 50 times slower than transmons, the two-qubit gate is faster than microwave-activated gates on transmons, and the gate error is on par with the lowest reported. Architectural advantages of low-frequency fluxoniums include long qubit coherence time, weak hybridization in the computational subspace, suppressed residual ZZ-coupling rate (here 46 kHz), and the absence of either excessive parameter-matching or complex pulse-shaping requirements.6 MoreReceived 4 November 2020Accepted 17 March 2021DOI:https://doi.org/10.1103/PhysRevX.11.021026Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasCapacitanceChargeEntanglement productionJosephson effectQuantum computationQuantum controlQuantum gatesQuantum information with solid state qubitsSuperconductivityQuantum InformationInterdisciplinary PhysicsCondensed Matter, Materials & Applied Physics

High-fidelity single- and multiqubit operations compose the backbone of quantum information processing. This fidelity is based on the ability to couple single- or two-qubit levels in an extremely coherent and precise manner. A necessary condition for coherent quantum evolution is a highly stable local oscillator driving these transitions. Here we study the effect of fast noise, that is, noise at frequencies much higher than the local oscillator linewidth, on the fidelity of one- and two-qubit gates in a trapped-ion system. We analyze and measure the effect of fast noise on single-qubit operations, including resonant $\ensuremath{\pi}$ rotations and off-resonant sideband transitions. We further numerically analyze the effect of fast phase noise on the M\o{}lmer-S\o{}rensen two-qubit gate. We find a unified and simple way to estimate the performance of all of these operations through a single parameter given by the noise power spectral density at the qubit response frequency. While our analysis focuses on phase noise and on trapped-ion systems, it is relevant for other sources of fast noise as well as for other qubit systems in which spinlike qubits are coupled by a common bosonic field. Our analysis can help in guiding the design of quantum hardware platforms and gates, improving their fidelity towards fault-tolerant quantum computing.