Precision tomography of a three-qubit donor quantum processor in silicon

Cross-architecture tuning of silicon and SiGe-based quantum devices using machine learning

The potential of Si and SiGe-based devices for the scaling of quantum circuits is tainted by device variability. Each device needs to be tuned to operation conditions and each device realisation requires a different tuning protocol. We demonstrate that it is possible to automate the tuning of a 4-gate Si FinFET, a 5-gate GeSi nanowire and a 7-gate Ge/SiGe heterostructure double quantum dot device from scratch with the same algorithm. We achieve tuning times of 30, 10, and 92 min, respectively. The algorithm also provides insight into the parameter space landscape for each of these devices, allowing for the characterization of the regions where double quantum dot regimes are found. These results show that overarching solutions for the tuning of quantum devices are enabled by machine learning.

Valley-Free Silicon Fins Caused by Shear Strain

Electron spins confined in silicon quantum dots are promising candidates for large-scale quantum computers. However, the degeneracy of the conduction band of bulk silicon introduces additional levels dangerously close to the window of computational energies, where the quantum information can leak. The energy of the valley states-typically 0.1 meV-depends on hardly controllable atomistic disorder and still constitutes a fundamental limit to the scalability of these architectures. In this work, we introduce designs of complementary metal-oxide-semiconductor (CMOS)-compatible silicon fin field-effect transistors that enhance the energy gap to noncomputational states by more than one order of magnitude. Our devices comprise realistic silicon-germanium nanostructures with a large shear strain, where troublesome valley degrees of freedom are completely removed. The energy of noncomputational states is therefore not affected by unavoidable atomistic disorder and can further be tuned in situ by applied electric fields. Our design ideas are directly applicable to a variety of setups and will offer a blueprint toward silicon-based large-scale quantum processors.

Machine Learning-Assisted Precision Manufacturing of Atom Qubits in Silicon

Donor-based qubits in silicon, manufactured using scanning tunneling microscope (STM) lithography, provide a promising route to realizing full-scale quantum computing architectures. This is due to the precision of donor placement, long coherence times, and scalability of the silicon material platform. The properties of multiatom quantum dot qubits, however, depend on the exact number and location of the donor atoms within the quantum dots. In this work, we develop machine learning techniques that allow accurate and real-time prediction of the donor number at the qubit site during STM patterning. Machine learning image recognition is used to determine the probability distribution of donor numbers at the qubit site directly from STM images during device manufacturing. Models in excess of 90% accuracy are found to be consistently achieved by mitigating overfitting through reduced model complexity, image preprocessing, data augmentation, and examination of the intermediate layers of the convolutional neural networks. The results presented in this paper constitute an important milestone in automating the manufacture of atom-based qubits for computation and sensing applications.

High-fidelity initialization and control of electron and nuclear spins in a four-qubit register

Single electron spins bound to multi-phosphorus nuclear spin registers in silicon have demonstrated fast (0.8 ns) two-qubitSWAP gates and long spin relaxation times (~30 s). In these spin registers, when the donors are ionized, the nuclear spins remain weakly coupled to their environment, allowing exceptionally long coherence times. When the electron is present, the hyperfine interaction allows coupling of the spin and charge degrees of freedom for fast qubit operation and control. Here we demonstrate the use of the hyperfine interaction to enact electric dipole spin resonance to realize high-fidelity ( F = 10 0 - 6 + 0 %) initialization of all the nuclear spins within a four-qubit nuclear spin register. By controllably initializing the nuclear spins to⇓ ⇓ ⇓ , we achieve single-electron qubit gate fidelities of F = 99.78 ± 0.07% (Clifford gate fidelities of 99.58 ± 0.14%), above the fault-tolerant threshold for the surface code with a coherence time ofT 2 * = 12 μ s .

Benchmarking of X-Ray Fluorescence Microscopy with Ion Beam Implanted Samples Showing Detection Sensitivity of Hundreds of Atoms

Single impurities in insulators are now often used for quantum sensors and single photon sources, while nanoscale semiconductor doping features are being constructed for electrical contacts in quantum technology devices, implying that new methods for sensitive, non-destructive imaging of single- or few-atom structures are needed. X-ray fluorescence (XRF) can provide nanoscale imaging with chemical specificity, and features comprising as few as 100 000 atoms have been detected without any need for specialized or destructive sample preparation. Presently, the ultimate limits of sensitivity of XRF are unknown - here, gallium dopants in silicon are investigated using a high brilliance, synchrotron source collimated to a small spot. It is demonstrated that with a single-pixel integration time of 1 s, the sensitivity is sufficient to identify a single isolated feature of only 3000 Ga impurities (a mass of just 350 zg). With increased integration (25 s), 650 impurities can be detected. The results are quantified using a calibration sample consisting of precisely controlled numbers of implanted atoms in nanometer-sized structures. The results show that such features can now be mapped quantitatively when calibration samples are used, and suggest that, in the near future, planned upgrades to XRF facilities might achieve single-atom sensitivity.

Qudit-Based Spectroscopy for Measurement and Control of Nuclear-Spin Qubits in Silicon Carbide

Nuclear spins with hyperfine coupling to single electron spins are highly valuable quantum bits. Here we probe and characterize the particularly rich nuclear-spin environment around single silicon vacancy color centers (V2) in 4H-SiC. By using the electron spin-3/2 qudit as a four level sensor, we identify several sets of ^{29}Si and ^{13}C nuclear spins through their hyperfine interaction. We extract the major components of their hyperfine coupling via optical detected nuclear magnetic resonance, and assign them to shells in the crystal via the density function theory simulations. We utilize the ground-state level anticrossing of the electron spin for dynamic nuclear polarization and achieve a nuclear-spin polarization of up to 98±6%. We show that this scheme can be used to detect the nuclear magnetic resonance signal of individual spins and demonstrate their coherent control. Our work provides a detailed set of parameters and first steps for future use of SiC as a multiqubit memory and quantum computing platform.

High-fidelity spin qubit operation and algorithmic initialization above 1 K

The encoding of qubits in semiconductor spin carriers has been recognized as a promising approach to a commercial quantum computer that can be lithographically produced and integrated at scale1-10. However, the operation of the large number of qubits required for advantageous quantum applications11-13 will produce a thermal load exceeding the available cooling power of cryostats at millikelvin temperatures. As the scale-up accelerates, it becomes imperative to establish fault-tolerant operation above 1 K, at which the cooling power is orders of magnitude higher14-18. Here we tune up and operate spin qubits in silicon above 1 K, with fidelities in the range required for fault-tolerant operations at these temperatures19-21. We design an algorithmic initialization protocol to prepare a pure two-qubit state even when the thermal energy is substantially above the qubit energies and incorporate radiofrequency readout to achieve fidelities up to 99.34% for both readout and initialization. We also demonstrate single-qubit Clifford gate fidelities up to 99.85% and a two-qubit gate fidelity of 98.92%. These advances overcome the fundamental limitation that the thermal energy must be well below the qubit energies for the high-fidelity operation to be possible, surmounting a main obstacle in the pathway to scalable and fault-tolerant quantum computation.

Navigating the 16-dimensional Hilbert space of a high-spin donor qudit with electric and magnetic fields

Efficient scaling and flexible control are key aspects of useful quantum computing hardware. Spins in semiconductors combine quantum information processing with electrons, holes or nuclei, control with electric or magnetic fields, and scalable coupling via exchange or dipole interaction. However, accessing large Hilbert space dimensions has remained challenging, due to the short-distance nature of the interactions. Here, we present an atom-based semiconductor platform where a 16-dimensional Hilbert space is built by the combined electron-nuclear states of a single antimony donor in silicon. We demonstrate the ability to navigate this large Hilbert space using both electric and magnetic fields, with gate fidelity exceeding 99.8% on the nuclear spin, and unveil fine details of the system Hamiltonian and its susceptibility to control and noise fields. These results establish high-spin donors as a rich platform for practical quantum information and to explore quantum foundations.

Accelerated Adiabatic Passage of a Single Electron Spin Qubit in Quantum Dots

Adiabatic processes can keep the quantum system in its instantaneous eigenstate, which is robust to noises and dissipation. However, it is limited by sufficiently slow evolution. Here, we experimentally demonstrate the transitionless quantum driving (TLQD) of the shortcuts to adiabaticity in gate-defined semiconductor quantum dots (QDs) to greatly accelerate the conventional adiabatic passage for the first time. For a given efficiency of quantum state transfer, the acceleration can be more than twofold. The dynamic properties also prove that the TLQD can guarantee fast and high-fidelity quantum state transfer. In order to compensate for the diabatic errors caused by dephasing noises, the modified TLQD is proposed and demonstrated in experiment by enlarging the width of the counterdiabatic drivings. The benchmarking shows that the state transfer fidelity of 97.8% can be achieved. This work will greatly promote researches and applications about quantum simulations and adiabatic quantum computation based on the gate-defined QDs.

Universality and Multiplication of Gigahertz-Operated Silicon Pumps with Parts Per Million-Level Uncertainty

The universality of physical phenomena is a pivotal concept underlying quantum standards. In this context, the realization of a quantum current standard using silicon single-electron pumps necessitates the verification of the equivalence across multiple devices. Herein, we experimentally investigate the universality of pumped currents from two different silicon single-electron devices which are placed inside the cryogen-free dilution refrigerator whose temperature (mixing chamber plate) was ∼150 mK under the operation of the pump devices. By direct comparison using an ultrastable current amplifier as a galvanometer, we confirm that two pumped currents are consistent with ∼1 ppm uncertainty. Furthermore, we realize quantum-current multiplication with a similar uncertainty by adding the currents of two different gigahertz (GHz)-operated silicon pumps, whose generated currents are confirmed to be identical. These results pave the way for realizing a quantum current standard in the nanoampere range and a quantum metrology triangle experiment using silicon pump devices.

Ultrahigh-fidelity spatial mode quantum gates in high-dimensional space by diffractive deep neural networks

While the spatial mode of photons is widely used in quantum cryptography, its potential for quantum computation remains largely unexplored. Here, we showcase the use of the multi-dimensional spatial mode of photons to construct a series of high-dimensional quantum gates, achieved through the use of diffractive deep neural networks (D2NNs). Notably, our gates demonstrate high fidelity of up to 99.6(2)%, as characterized by quantum process tomography. Our experimental implementation of these gates involves a programmable array of phase layers in a compact and scalable device, capable of performing complex operations or even quantum circuits. We also demonstrate the efficacy of the D2NN gates by successfully implementing the Deutsch algorithm and propose an intelligent deployment protocol that involves self-configuration and self-optimization. Moreover, we conduct a comparative analysis of the D2NN gate's performance to the wave-front matching approach. Overall, our work opens a door for designing specific quantum gates using deep learning, with the potential for reliable execution of quantum computation.

Single-Electron Occupation in Quantum Dot Arrays at Selectable Plunger Gate Voltage

The small footprint of semiconductor qubits is favorable for scalable quantum computing. However, their size also makes them sensitive to their local environment and variations in the gate structure. Currently, each device requires tailored gate voltages to confine a single charge per quantum dot, clearly challenging scalability. Here, we tune these gate voltages and equalize them solely through the temporary application of stress voltages. In a double quantum dot, we reach a stable (1,1) charge state at identical and predetermined plunger gate voltage and for various interdot couplings. Applying our findings, we tune a 2 × 2 quadruple quantum dot such that the (1,1,1,1) charge state is reached when all plunger gates are set to 1 V. The ability to define required gate voltages may relax requirements on control electronics and operations for spin qubit devices, providing means to advance quantum hardware.

Atomic Engineering of Molecular Qubits for High-Speed, High-Fidelity Single Qubit Gates

Universal quantum computing requires fast single- and two-qubit gates with individual qubit addressability to minimize decoherence errors during processor operation. Electron spin qubits using individual phosphorus donor atoms in silicon have demonstrated long coherence times with high fidelities, providing an attractive platform for scalable quantum computing. While individual qubit addressability has been demonstrated by controlling the hyperfine interaction between the electron and nuclear wave function in a global magnetic field, the small hyperfine Stark coefficient of 0.34 MHz/MV m-1 achieved to date has limited the speed of single quantum gates to ∼42 μs to avoid rotating neighboring qubits due to power broadening from the antenna. The use of molecular 2P qubits with more than one donor atom has not only demonstrated fast (0.8 ns) two-qubit SWAP gates and long spin relaxation times of ∼30 s but provides an alternate way to achieve high selectivity of the qubit resonance frequency. Here, we show in two different devices that by placing the donors with comparable interatomic spacings (∼0.8 nm) but along different crystallographic axes, either the [110] or [310] orientations using STM lithography, we can engineer the hyperfine Stark shift from 1 MHz/MV m-1 to 11.2 MHz/MV m-1, respectively, a factor of 10 difference. NEMO atomistic calculations show that larger hyperfine Stark coefficients of up to ∼70 MHz/MV m-1 can be achieved within 2P molecules by placing the donors ≥5 nm apart. When combined with Gaussian pulse shaping, we show that fast single qubit gates with 2π rotation times of 10 ns and ∼99% fidelity single qubit operations are feasible without affecting neighboring qubits. By increasing the single qubit gate time to ∼550 ns, two orders of magnitude faster than previously measured, our simulations confirm that >99.99% single qubit control fidelities are achievable.

The complexity of NISQ

The recent proliferation of NISQ devices has made it imperative to understand their power. In this work, we define and study the complexity class NISQ, which encapsulates problems that can be efficiently solved by a classical computer with access to noisy quantum circuits. We establish super-polynomial separations in the complexity among classical computation, NISQ, and fault-tolerant quantum computation to solve some problems based on modifications of Simon's problems. We then consider the power of NISQ for three well-studied problems. For unstructured search, we prove that NISQ cannot achieve a Grover-like quadratic speedup over classical computers. For the Bernstein-Vazirani problem, we show that NISQ only needs a number of queries logarithmic in what is required for classical computers. Finally, for a quantum state learning problem, we prove that NISQ is exponentially weaker than classical computers with access to noiseless constant-depth quantum circuits.

Super-compact universal quantum logic gates with inverse-designed elements

Integrated quantum photonic circuit is a promising platform for the realization of quantum information processing in the future. To achieve the large-scale quantum photonic circuits, the applied quantum logic gates should be as small as possible for the high-density integration on chips. Here, we report the implementation of super-compact universal quantum logic gates on silicon chips by the method of inverse design. In particular, the fabricated controlled-NOT gate and Hadamard gate are both nearly a vacuum wavelength, being the smallest optical quantum gates reported up to now. We further design the quantum circuit by cascading these fundamental gates to perform arbitrary quantum processing, where the corresponding size is about several orders smaller than that of previous quantum photonic circuits. Our study paves the way for the realization of large-scale quantum photonic chips with integrated sources and can have important applications in the field of quantum information processes.

Jellybean Quantum Dots in Silicon for Qubit Coupling and On-Chip Quantum Chemistry

The small size and excellent integrability of silicon metal-oxide-semiconductor (SiMOS) quantum dot spin qubits make them an attractive system for mass-manufacturable, scaled-up quantum processors. Furthermore, classical control electronics can be integrated on-chip, in-between the qubits, if an architecture with sparse arrays of qubits is chosen. In such an architecture qubits are either transported across the chip via shuttling or coupled via mediating quantum systems over short-to-intermediate distances. This paper investigates the charge and spin characteristics of an elongated quantum dot-a so-called jellybean quantum dot-for the prospects of acting as a qubit-qubit coupler. Charge transport, charge sensing, and magneto-spectroscopy measurements are performed on a SiMOS quantum dot device at mK temperature and compared to Hartree-Fock multi-electron simulations. At low electron occupancies where disorder effects and strong electron-electron interaction dominate over the electrostatic confinement potential, the data reveals the formation of three coupled dots, akin to a tunable, artificial molecule. One dot is formed centrally under the gate and two are formed at the edges. At high electron occupancies, these dots merge into one large dot with well-defined spin states, verifying that jellybean dots have the potential to be used as qubit couplers in future quantum computing architectures.

Isotopically Enriched Layers for Quantum Computers Formed by 28Si Implantation and Layer Exchange

²⁸Si enrichment is crucial for production of group IV semiconductor-based quantum computers. Cryogenically cooled, monocrystalline ²⁸Si is a spin-free, vacuum-like environment where qubits are protected from sources of decoherence that cause loss of quantum information. Currently, ²⁸Si enrichment techniques rely on deposition of centrifuged SiF₄ gas, the source of which is not widely available, or bespoke ion implantation methods. Previously, conventional ion implantation into ^naturalSi substrates has produced heavily oxidized ²⁸Si layers. Here we report on a novel enrichment process involving ion implantation of ²⁸Si into Al films deposited on native-oxide free Si substrates followed by layer exchange crystallization. We measured continuous, oxygen-free epitaxial ²⁸Si enriched to 99.7%. Increases in isotopic enrichment are possible, and improvements in crystal quality, aluminum content, and thickness uniformity are required before the process can be considered viable. TRIDYN models, used to model 30 keV ²⁸Si implants into Al to understand the observed post-implant layers and to investigate the implanted layer exchange process window over different energy and vacuum conditions, showed that the implanted layer exchange process is insensitive to implantation energy and would increase in efficiency with oxygen concentrations in the implanter end-station by reducing sputtering. Required implant fluences are an order of magnitude lower than those required for enrichment by direct ²⁸Si implants into Si and can be chosen to control the final thickness of the enriched layer. We show that implanted layer exchange could potentially produce quantum grade ²⁸Si using conventional semiconductor foundry equipment within production-worthy time scales.

Electrical Control of Uniformity in Quantum Dot Devices

Highly uniform quantum systems are essential for the practical implementation of scalable quantum processors. While quantum dot spin qubits based on semiconductor technology are a promising platform for large-scale quantum computing, their small size makes them particularly sensitive to their local environment. Here, we present a method to electrically obtain a high degree of uniformity in the intrinsic potential landscape using hysteretic shifts of the gate voltage characteristics. We demonstrate the tuning of pinch-off voltages in quantum dot devices over hundreds of millivolts that then remain stable at least for hours. Applying our method, we homogenize the pinch-off voltages of the plunger gates in a linear array for four quantum dots, reducing the spread in pinch-off voltages by one order of magnitude. This work provides a new tool for the tuning of quantum dot devices and offers new perspectives for the implementation of scalable spin qubit arrays.

Reducing charge noise in quantum dots by using thin silicon quantum wells

Charge noise in the host semiconductor degrades the performance of spin-qubits and poses an obstacle to control large quantum processors. However, it is challenging to engineer the heterogeneous material stack of gate-defined quantum dots to improve charge noise systematically. Here, we address the semiconductor-dielectric interface and the buried quantum well of a ²⁸Si/SiGe heterostructure and show the connection between charge noise, measured locally in quantum dots, and global disorder in the host semiconductor, measured with macroscopic Hall bars. In 5 nm thick ²⁸Si quantum wells, we find that improvements in the scattering properties and uniformity of the two-dimensional electron gas over a 100 mm wafer correspond to a significant reduction in charge noise, with a minimum value of 0.29 ± 0.02 μeV/Hz^½ at 1 Hz averaged over several quantum dots. We extrapolate the measured charge noise to simulated dephasing times to CZ-gate fidelities that improve nearly one order of magnitude. These results point to a clean and quiet crystalline environment for integrating long-lived and high-fidelity spin qubits into a larger system.

Universal logic with encoded spin qubits in silicon

Quantum computation features known examples of hardware acceleration for certain problems, but is challenging to realize because of its susceptibility to small errors from noise or imperfect control. The principles of fault tolerance may enable computational acceleration with imperfect hardware, but they place strict requirements on the character and correlation of errors1. For many qubit technologies2-21, some challenges to achieving fault tolerance can be traced to correlated errors arising from the need to control qubits by injecting microwave energy matching qubit resonances. Here we demonstrate an alternative approach to quantum computation that uses energy-degenerate encoded qubit states controlled by nearest-neighbour contact interactions that partially swap the spin states of electrons with those of their neighbours. Calibrated sequences of such partial swaps, implemented using only voltage pulses, allow universal quantum control while bypassing microwave-associated correlated error sources1,22-28. We use an array of six 28Si/SiGe quantum dots, built using a platform that is capable of extending in two dimensions following processes used in conventional microelectronics29. We quantify the operational fidelity of universal control of two encoded qubits using interleaved randomized benchmarking30, finding a fidelity of 96.3% ± 0.7% for encoded controlled NOT operations and 99.3% ± 0.5% for encoded SWAP. The quantum coherence offered by enriched silicon5-9,16,18,20,22,27,29,31-37, the all-electrical and low-crosstalk-control of partial swap operations1,22-28 and the configurable insensitivity of our encoding to certain error sources28,33,34,38 all combine to offer a strong pathway towards scalable fault tolerance and computational advantage.

Generalized Toffoli Gate Decomposition Using Ququints: Towards Realizing Grover's Algorithm with Qudits

Qubits, which are the quantum counterparts of classical bits, are used as basic information units for quantum information processing, whereas underlying physical information carriers, e.g., (artificial) atoms or ions, admit encoding of more complex multilevel states-qudits. Recently, significant attention has been paid to the idea of using qudit encoding as a way for further scaling quantum processors. In this work, we present an efficient decomposition of the generalized Toffoli gate on five-level quantum systems-so-called ququints-that use ququints' space as the space of two qubits with a joint ancillary state. The basic two-qubit operation we use is a version of the controlled-phase gate. The proposed N-qubit Toffoli gate decomposition has O(N) asymptotic depth and does not use ancillary qubits. We then apply our results for Grover's algorithm, where we indicate on the sizable advantage of using the qudit-based approach with the proposed decomposition in comparison to the standard qubit case. We expect that our results are applicable for quantum processors based on various physical platforms, such as trapped ions, neutral atoms, protonic systems, superconducting circuits, and others.

An electrically driven single-atom "flip-flop" qubit

The spins of atoms and atom-like systems are among the most coherent objects in which to store quantum information. However, the need to address them using oscillating magnetic fields hinders their integration with quantum electronic devices. Here, we circumvent this hurdle by operating a single-atom "flip-flop" qubit in silicon, where quantum information is encoded in the electron-nuclear states of a phosphorus donor. The qubit is controlled using local electric fields at microwave frequencies, produced within a metal-oxide-semiconductor device. The electrical drive is mediated by the modulation of the electron-nuclear hyperfine coupling, a method that can be extended to many other atomic and molecular systems and to the hyperpolarization of nuclear spin ensembles. These results pave the way to the construction of solid-state quantum processors where dense arrays of atoms can be controlled using only local electric fields.

The Use of Exchange Coupled Atom Qubits as Atomic-Scale Magnetic Field Sensors

Phosphorus atoms in silicon offer a rich quantum computing platform where both nuclear and electron spins can be used to store and process quantum information. While individual control of electron and nuclear spins has been demonstrated, the interplay between them during qubit operations has been largely unexplored. This study investigates the use of exchange-based operation between donor bound electron spins to probe the local magnetic fields experienced by the qubits with exquisite precision at the atomic scale. To achieve this, coherent exchange oscillations are performed between two electron spin qubits, where the left and right qubits are hosted by three and two phosphorus donors, respectively. The frequency spectrum of exchange oscillations shows quantized changes in the local magnetic fields at the qubit sites, corresponding to the different hyperfine coupling between the electron and each of the qubit-hosting nuclear spins. This ability to sense the hyperfine fields of individual nuclear spins using the exchange interaction constitutes a unique metrology technique, which reveals the exact crystallographic arrangements of the phosphorus atoms in the silicon crystal for each qubit. The detailed knowledge obtained of the local magnetic environment can then be used to engineer hyperfine fields in multi-donor qubits for high-fidelity two-qubit gates.

99.92%-Fidelity cnot Gates in Solids by Noise Filtering

Inevitable interactions with the reservoir largely degrade the performance of entangling gates, which hinders practical quantum computation from coming into existence. Here, we experimentally demonstrate a 99.920(7)%-fidelity controlled-not gate by suppressing the complicated noise in a solid-state spin system at room temperature. We found that the fidelity limited at 99% in previous works results from considering only static classical noise, and, thus, in this work, a complete noise model is constructed by also considering the time dependence and the quantum nature of the spin bath. All noises in the model are dynamically corrected by an exquisitely designed shaped pulse, giving the resulting error below 10^{-4}. The residual gate error is mainly originated from the longitudinal relaxation and the waveform distortion that can both be further reduced technically. Our noise-resistant method is universal and will benefit other solid-state spin systems.

Colloquium: Advances in automation of quantum dot devices control

Arrays of quantum dots (QDs) are a promising candidate system to realize scalable, coupled qubit systems and serve as a fundamental building block for quantum computers. In such semiconductor quantum systems, devices now have tens of individual electrostatic and dynamical voltages that must be carefully set to localize the system into the single-electron regime and to realize good qubit operational performance. The mapping of requisite QD locations and charges to gate voltages presents a challenging classical control problem. With an increasing number of QD qubits, the relevant parameter space grows sufficiently to make heuristic control unfeasible. In recent years, there has been considerable effort to automate device control that combines script-based algorithms with machine learning (ML) techniques. In this Colloquium, a comprehensive overview of the recent progress in the automation of QD device control is presented, with a particular emphasis on silicon- and GaAs-based QDs formed in two-dimensional electron gases. Combining physics-based modeling with modern numerical optimization and ML has proven effective in yielding efficient, scalable control. Further integration of theoretical, computational, and experimental efforts with computer science and ML holds vast potential in advancing semiconductor and other platforms for quantum computing.

Robust electron spin qubit control in a nanowire double quantum dot

Robust and efficient manipulation of electron spin qubits in quantum dots is of great significance for the reliable realization of quantum computers and execution of quantum algorithms. In this paper, we study the robust control on a singlet-triplet qubit based on inverse engineering, one technique of shortcuts to adiabaticity (STA), in a nanowire double quantum dot in the presence of magnetic field and strong spin-orbit coupling. The optimization of STA with respect to the systematic errors, contributed from the control field and the perturbative interaction, is explored. Moreover, we also apply optimal control techniques combining with STA, referred to as robust inverse optimization, to design optimal control fields and optimal operation time. This article is part of the theme issue 'Shortcuts to adiabaticity: theoretical, experimental and interdisciplinary perspectives'.

Computational Insights into Electronic Excitations, Spin-Orbit Coupling Effects, and Spin Decoherence in Cr(IV)-Based Molecular Qubits

The great success of point defects and dopants in semiconductors for quantum information processing has invigorated a search for molecules with analogous properties. Flexibility and tunability of desired properties in a large chemical space have great advantages over solid-state systems. The properties analogous to point defects were demonstrated in the Cr(IV)-based molecular family, Cr(IV)(aryl)₄, where the electronic spin states were optically initialized, read out, and controlled. Despite this kick-start, there is still a large room for enhancing properties crucial for molecular qubits. Here, we provide computational insights into key properties of the Cr(IV)-based molecules aimed at assisting the chemical design of efficient molecular qubits. Using the multireference ab initio methods, we investigate the electronic states of Cr(IV)(aryl)₄ molecules with slightly different ligands, showing that the zero-phonon line energies agree with the experiment and that the excited spin-triplet and spin-singlet states are highly sensitive to small chemical perturbations. By adding spin-orbit interaction, we find that the sign of the uniaxial zero-field splitting (ZFS) parameter is negative for all considered molecules and discuss optically induced spin initialization via non-radiative intersystem crossing. We quantify (super)hyperfine coupling to the ⁵³Cr nuclear spin and to the ¹³C and ¹H nuclear spins, and we discuss electron spin decoherence. We show that the splitting or broadening of the electronic spin sub-levels due to superhyperfine interaction with ¹H nuclear spins decreases by an order of magnitude when the molecules have a substantial transverse ZFS parameter.

Devitalizing noise-driven instability of entangling logic in silicon devices with bias controls

The quality of quantum bits (qubits) in silicon is highly vulnerable to charge noise that is omnipresent in semiconductor devices and is in principle hard to be suppressed. For a realistically sized quantum dot system based on a silicon-germanium heterostructure whose confinement is manipulated with electrical biases imposed on top electrodes, we computationally explore the noise-robustness of 2-qubit entangling operations with a focus on the controlled-X (CNOT) logic that is essential for designs of gate-based universal quantum logic circuits. With device simulations based on the physics of bulk semiconductors augmented with electronic structure calculations, we not only quantify the degradation in fidelity of single-step CNOT operations with respect to the strength of charge noise, but also discuss a strategy of device engineering that can significantly enhance noise-robustness of CNOT operations with almost no sacrifice of speed compared to the single-step case. Details of device designs and controls that this work presents can establish practical guideline for potential efforts to secure silicon-based quantum processors using an electrode-driven quantum dot platform.

Universal control of a six-qubit quantum processor in silicon

Future quantum computers capable of solving relevant problems will require a large number of qubits that can be operated reliably1. However, the requirements of having a large qubit count and operating with high fidelity are typically conflicting. Spins in semiconductor quantum dots show long-term promise2,3 but demonstrations so far use between one and four qubits and typically optimize the fidelity of either single- or two-qubit operations, or initialization and readout4-11. Here, we increase the number of qubits and simultaneously achieve respectable fidelities for universal operation, state preparation and measurement. We design, fabricate and operate a six-qubit processor with a focus on careful Hamiltonian engineering, on a high level of abstraction to program the quantum circuits, and on efficient background calibration, all of which are essential to achieve high fidelities on this extended system. State preparation combines initialization by measurement and real-time feedback with quantum-non-demolition measurements. These advances will enable testing of increasingly meaningful quantum protocols and constitute a major stepping stone towards large-scale quantum computers.

Surface code for low-density qubit array

Surface code is a promising candidate for the quantum error corrections needed for fault-tolerant quantum computations because it can operate on a two-dimensional grid of qubits. However, the gates and control lines become dense as more and more qubits are integrated, making their design and control difficult. This problem can be alleviated if the surface code can operate on sparse qubit arrays. Here, we give an solution for an array in which qubits are placed on edges as well as on nodes of a two-dimensional grid. The qubits on the edges are divided into two groups: those in one group act as the deputies of data qubits; the others act as deputies of the syndrome qubits. Syndrome outputs are obtained by multiplying the measured values of the syndrome and edge qubits. The procedure for the quantum part is the same as that of the ordinary surface code, making the surface code applicable to sparse qubit arrays.

Interplay of magnetic states and hyperfine fields of iron dimers on MgO(001)

Individual nuclear spin states can have very long lifetimes and could be useful as qubits. Progress in this direction was achieved on MgO/Ag(001) via detection of the hyperfine interaction (HFI) of Fe, Ti and Cu adatoms using scanning tunneling microscopy. Previously, we systematically quantified from first-principles the HFI for the whole series of 3d transition adatoms (Sc-Cu) deposited on various ultra-thin insulators, establishing the trends of the computed HFI with respect to the filling of the magnetic s- and d-orbitals of the adatoms and on the bonding with the substrate. Here we explore the case of dimers by investigating the correlation between the HFI and the magnetic state of free standing Fe dimers, single Fe adatoms and dimers deposited on a bilayer of MgO(001). We find that the magnitude of the HFI can be controlled by switching the magnetic state of the dimers. For short Fe-Fe distances, the antiferromagnetic state enhances the HFI with respect to that of the ferromagnetic state. By increasing the distance between the magnetic atoms, a transition toward the opposite behavior is observed. Furthermore, we demonstrate the ability to substantially modify the HFI by atomic control of the location of the adatoms on the substrate. Our results establish the limits of applicability of the usual hyperfine hamiltonian and we propose an extension based on multiple scattering processes.

Optical demonstration of quantum fault-tolerant threshold

A major challenge in practical quantum computation is the ineludible errors caused by the interaction of quantum systems with their environment. Fault-tolerant schemes, in which logical qubits are encoded by several physical qubits, enable to the output of a higher probability of correct logical qubits under the presence of errors. However, strict requirements to encode qubits and operators render the implementation of a full fault-tolerant computation challenging even for the achievable noisy intermediate-scale quantum technology. Especially the threshold for fault-tolerant computation still lacks experimental verification. Here, based on an all-optical setup, we experimentally demonstrate the existence of the threshold for the fault-tolerant protocol. Four physical qubits are represented as the spatial modes of two entangled photons, which are used to encode two logical qubits. The experimental results clearly show that when the error rate is below the threshold, the probability of correct output in the circuit, formed with fault-tolerant gates, is higher than that in the corresponding non-encoded circuit. In contrast, when the error rate is above the threshold, no advantage is observed in the fault-tolerant implementation. The developed high-accuracy optical system may provide a reliable platform to investigate error propagation in more complex circuits with fault-tolerant gates.

Competitive Hydrogen Migration in Silicon Nitride Nanoclusters: Reaction Kinetics Generalized from Supervised Machine Learning

The rate coefficients for 52 hydrogen shift reactions for silicon nitrides containing up to 6 atoms of silicon and nitrogen have been calculated using the G3//B3LYP composite method and statistical thermodynamics. The overall reaction of substituted acyclic and cyclic silylenes to their respective silene and imine species by a 1,2-hydrogen shift reaction was sorted by three different types of H shift reactions using overall reaction thermodynamics: (1) endothermic H shift between N and Si:, (2) endothermic H shift between Si and Si:, and (3) exothermic H shift between Si and Si:. Endothermic H shift reactions between Si atoms have one dominant activation barrier where the exothermic H shift reaction between Si atoms has two barriers and a stable intermediate. The rate-determining step was determined to be from the intermediate to the substituted silene, and then kinetic parameters for the overall reaction were calculated for the two-step pathway. The single event pre-exponential factors, Ã, and activation energies, E_a, for the three different classes of hydrogen shift reactions of silicon nitrides were computed. The hydrogen shift reaction was explored for acyclic and cyclic monofunctional silicon nitrides, and the type of hydrogen shift reaction gives the most significant influence on the kinetic parameters. Using a supervised machine learning approach, the models for predicting the energy barrier of three different hydrogen shift reactions were generalized and suggested based on selected descriptors.

Fault-tolerant operation of a logical qubit in a diamond quantum processor

Solid-state spin qubits is a promising platform for quantum computation and quantum networks^1,2. Recent experiments have demonstrated high-quality control over multi-qubit systems^3–8, elementary quantum algorithms^8–11 and non-fault-tolerant error correction^12–14. Large-scale systems will require using error-corrected logical qubits that are operated fault tolerantly, so that reliable computation becomes possible despite noisy operations^15–18. Overcoming imperfections in this way remains an important outstanding challenge for quantum science^15,19–27. Here, we demonstrate fault-tolerant operations on a logical qubit using spin qubits in diamond. Our approach is based on the five-qubit code with a recently discovered flag protocol that enables fault tolerance using a total of seven qubits^28–30. We encode the logical qubit using a new protocol based on repeated multi-qubit measurements and show that it outperforms non-fault-tolerant encoding schemes. We then fault-tolerantly manipulate the logical qubit through a complete set of single-qubit Clifford gates. Finally, we demonstrate flagged stabilizer measurements with real-time processing of the outcomes. Such measurements are a primitive for fault-tolerant quantum error correction. Although future improvements in fidelity and the number of qubits will be required to suppress logical error rates below the physical error rates, our realization of fault-tolerant protocols on the logical-qubit level is a key step towards quantum information processing based on solid-state spins.

By using a five-qubit error-correcting code with a recently discovered flag protocol, a logical qubit that is operated fault-tolerantly is realized based on solid-state spin qubits in diamond.

Near-Field Generation and Control of Ultrafast, Multipartite Entanglement for Quantum Nanoplasmonic Networks

For a quantum Internet, one needs reliable sources of entangled particles that are compatible with measurement techniques enabling time-dependent, quantum error correction. Ideally, they will be operable at room temperature with a manageable decoherence versus generation time. To accomplish this, we theoretically establish a scalable, plasmonically based archetype that uses quantum dots (QD) as quantum emitters, known for relatively low decoherence rates near room temperature, that are excited using subdiffracted light from a near-field transducer (NFT). NFTs are a developing technology that allow rasterization across arrays of qubits and remarkably generate enough power to strongly drive energy transitions on the nanoscale. This eases the fabrication of QD media, while efficiently controlling picosecond-scale dynamic entanglement of a multiqubit system that approaches maximum fidelity, along with fluctuation between tripartite and bipartite entanglement. Our strategy radically increases the scalability and accessibility of quantum information devices while permitting fault-tolerant quantum computing using time-repetition algorithms.

Quantum logic with spin qubits crossing the surface code threshold

High-fidelity control of quantum bits is paramount for the reliable execution of quantum algorithms and for achieving fault tolerance-the ability to correct errors faster than they occur1. The central requirement for fault tolerance is expressed in terms of an error threshold. Whereas the actual threshold depends on many details, a common target is the approximately 1% error threshold of the well-known surface code2,3. Reaching two-qubit gate fidelities above 99% has been a long-standing major goal for semiconductor spin qubits. These qubits are promising for scaling, as they can leverage advanced semiconductor technology4. Here we report a spin-based quantum processor in silicon with single-qubit and two-qubit gate fidelities, all of which are above 99.5%, extracted from gate-set tomography. The average single-qubit gate fidelities remain above 99% when including crosstalk and idling errors on the neighbouring qubit. Using this high-fidelity gate set, we execute the demanding task of calculating molecular ground-state energies using a variational quantum eigensolver algorithm5. Having surpassed the 99% barrier for the two-qubit gate fidelity, semiconductor qubits are well positioned on the path to fault tolerance and to possible applications in the era of noisy intermediate-scale quantum devices.