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.

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.

On-demand electrical control of spin qubits.

Once called a 'classically non-describable two-valuedness' by Pauli, the electron spin forms a qubit that is naturally robust to electric fluctuations. Paradoxically, a common control strategy is the integration of micromagnets to enhance the coupling between spins and electric fields, which, in turn, hampers noise immunity and adds architectural complexity. Here we exploit a switchable interaction between spins and orbital motion of electrons in silicon quantum dots, without a micromagnet. The weak effects of relativistic spin-orbit interaction in silicon are enhanced, leading to a speed up in Rabi frequency by a factor of up to 650 by controlling the energy quantization of electrons in the nanostructure. Fast electrical control is demonstrated in multiple devices and electronic configurations. Using the electrical drive, we achieve a coherence time T2,Hahn ≈ 50 µs, fast single-qubit gates with Tπ/2 = 3 ns and gate fidelities of 99.93%, probed by randomized benchmarking. High-performance all-electrical control improves the prospects for scalable silicon quantum computing.

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

Nuclear spins were among the first physical platforms to be considered for quantum information processing1,2, because of their exceptional quantum coherence3 and atomic-scale footprint. However, their full potential for quantum computing has not yet been realized, owing to the lack of methods with which to link nuclear qubits within a scalable device combined with multi-qubit operations with sufficient fidelity to sustain fault-tolerant quantum computation. Here we demonstrate universal quantum logic operations using a pair of ion-implanted 31P donor nuclei in a silicon nanoelectronic device. A nuclear two-qubit controlled-Z gate is obtained by imparting a geometric phase to a shared electron spin4, and used to prepare entangled Bell states with fidelities up to 94.2(2.7)%. The quantum operations are precisely characterized using gate set tomography (GST)5, yielding one-qubit average gate fidelities up to 99.95(2)%, two-qubit average gate fidelity of 99.37(11)% and two-qubit preparation/measurement fidelities of 98.95(4)%. These three metrics indicate that nuclear spins in silicon are approaching the performance demanded in fault-tolerant quantum processors6. We then demonstrate entanglement between the two nuclei and the shared electron by producing a Greenberger-Horne-Zeilinger three-qubit state with 92.5(1.0)% fidelity. Because electron spin qubits in semiconductors can be further coupled to other electrons7-9 or physically shuttled across different locations10,11, these results establish a viable route for scalable quantum information processing using donor nuclear and electron spins.

Bell-state tomography in a silicon many-electron artificial molecule.

An error-corrected quantum processor will require millions of qubits, accentuating the advantage of nanoscale devices with small footprints, such as silicon quantum dots. However, as for every device with nanoscale dimensions, disorder at the atomic level is detrimental to quantum dot uniformity. Here we investigate two spin qubits confined in a silicon double quantum dot artificial molecule. Each quantum dot has a robust shell structure and, when operated at an occupancy of 5 or 13 electrons, has single spin-[Formula: see text] valence electron in its p- or d-orbital, respectively. These higher electron occupancies screen static electric fields arising from atomic-level disorder. The larger multielectron wavefunctions also enable significant overlap between neighbouring qubit electrons, while making space for an interstitial exchange-gate electrode. We implement a universal gate set using the magnetic field gradient of a micromagnet for electrically driven single qubit gates, and a gate-voltage-controlled inter-dot barrier to perform two-qubit gates by pulsed exchange coupling. We use this gate set to demonstrate a Bell state preparation between multielectron qubits with fidelity 90.3%, confirmed by two-qubit state tomography using spin parity measurements.

Materials challenges and opportunities for quantum computing hardware.

Quantum computing hardware technologies have advanced during the past two decades, with the goal of building systems that can solve problems that are intractable on classical computers. The ability to realize large-scale systems depends on major advances in materials science, materials engineering, and new fabrication techniques. We identify key materials challenges that currently limit progress in five quantum computing hardware platforms, propose how to tackle these problems, and discuss some new areas for exploration. Addressing these materials challenges will require scientists and engineers to work together to create new, interdisciplinary approaches beyond the current boundaries of the quantum computing field.

Exchange Coupling in a Linear Chain of Three Quantum-Dot Spin Qubits in Silicon.

Quantum gates between spin qubits can be implemented leveraging the natural Heisenberg exchange interaction between two electrons in contact with each other. This interaction is controllable by electrically tailoring the overlap between electronic wave functions in quantum dot systems, as long as they occupy neighboring dots. An alternative route is the exploration of superexchange-the coupling between remote spins mediated by a third idle electron that bridges the distance between quantum dots. We experimentally demonstrate direct exchange coupling and provide evidence for second neighbor mediated superexchange in a linear array of three single-electron spin qubits in silicon, inferred from the electron spin resonance frequency spectra. We confirm theoretically, through atomistic modeling, that the device geometry only allows for sizable direct exchange coupling for neighboring dots, while next-nearest neighbor coupling cannot stem from the vanishingly small tail of the electronic wave function of the remote dots, and is only possible if mediated.

Conditional quantum operation of two exchange-coupled single-donor spin qubits in a MOS-compatible silicon device.

Silicon nanoelectronic devices can host single-qubit quantum logic operations with fidelity better than 99.9%. For the spins of an electron bound to a single-donor atom, introduced in the silicon by ion implantation, the quantum information can be stored for nearly 1 second. However, manufacturing a scalable quantum processor with this method is considered challenging, because of the exponential sensitivity of the exchange interaction that mediates the coupling between the qubits. Here we demonstrate the conditional, coherent control of an electron spin qubit in an exchange-coupled pair of 31P donors implanted in silicon. The coupling strength, J = 32.06 ± 0.06 MHz, is measured spectroscopically with high precision. Since the coupling is weaker than the electron-nuclear hyperfine coupling A ≈ 90 MHz which detunes the two electrons, a native two-qubit controlled-rotation gate can be obtained via a simple electron spin resonance pulse. This scheme is insensitive to the precise value of J, which makes it suitable for the scale-up of donor-based quantum computers in silicon that exploit the metal-oxide-semiconductor fabrication protocols commonly used in the classical electronics industry.

Controllable freezing of the nuclear spin bath in a single-atom spin qubit.

The quantum coherence and gate fidelity of electron spin qubits in semiconductors are often limited by nuclear spin fluctuations. Enrichment of spin-zero isotopes in silicon markedly improves the dephasing time [Formula: see text], which, unexpectedly, can extend two orders of magnitude beyond theoretical expectations. Using a single-atom 31P qubit in enriched 28Si, we show that the abnormally long [Formula: see text] is due to the freezing of the dynamics of the residual 29Si nuclei, caused by the electron-nuclear hyperfine interaction. Inserting a waiting period when the electron is controllably removed unfreezes the nuclear dynamics and restores the ergodic [Formula: see text] value. Our conclusions are supported by a nearly parameter-free modeling of the 29Si nuclear spin dynamics, which reveals the degree of backaction provided by the electron spin. This study clarifies the limits of ergodic assumptions in nuclear bath dynamics and provides previously unidentified strategies for maximizing coherence and gate fidelity of spin qubits in semiconductors.

Coherent electrical control of a single high-spin nucleus in silicon.

Nuclear spins are highly coherent quantum objects. In large ensembles, their control and detection via magnetic resonance is widely exploited, for example, in chemistry, medicine, materials science and mining. Nuclear spins also featured in early proposals for solid-state quantum computers1 and demonstrations of quantum search2 and factoring3 algorithms. Scaling up such concepts requires controlling individual nuclei, which can be detected when coupled to an electron4-6. However, the need to address the nuclei via oscillating magnetic fields complicates their integration in multi-spin nanoscale devices, because the field cannot be localized or screened. Control via electric fields would resolve this problem, but previous methods7-9 relied on transducing electric signals into magnetic fields via the electron-nuclear hyperfine interaction, which severely affects nuclear coherence. Here we demonstrate the coherent quantum control of a single 123Sb (spin-7/2) nucleus using localized electric fields produced within a silicon nanoelectronic device. The method exploits an idea proposed in 196110 but not previously realized experimentally with a single nucleus. Our results are quantitatively supported by a microscopic theoretical model that reveals how the purely electrical modulation of the nuclear electric quadrupole interaction results in coherent nuclear spin transitions that are uniquely addressable owing to lattice strain. The spin dephasing time, 0.1 seconds, is orders of magnitude longer than those obtained by methods that require a coupled electron spin to achieve electrical driving. These results show that high-spin quadrupolar nuclei could be deployed as chaotic models, strain sensors and hybrid spin-mechanical quantum systems using all-electrical controls. Integrating electrically controllable nuclei with quantum dots11,12 could pave the way to scalable, nuclear- and electron-spin-based quantum computers in silicon that operate without the need for oscillating magnetic fields.

A silicon quantum-dot-coupled nuclear spin qubit.

Single nuclear spins in the solid state are a potential future platform for quantum computing1-3, because they possess long coherence times4-6 and offer excellent controllability7. Measurements can be performed via localized electrons, such as those in single atom dopants8,9 or crystal defects10-12. However, establishing long-range interactions between multiple dopants or defects is challenging13,14. Conversely, in lithographically defined quantum dots, tunable interdot electron tunnelling allows direct coupling of electron spin-based qubits in neighbouring dots15-20. Moreover, the compatibility with semiconductor fabrication techniques21 may allow for scaling to large numbers of qubits in the future. Unfortunately, hyperfine interactions are typically too weak to address single nuclei. Here we show that for electrons in silicon metal-oxide-semiconductor quantum dots the hyperfine interaction is sufficient to initialize, read out and control single 29Si nuclear spins. This approach combines the long coherence times of nuclear spins with the flexibility and scalability of quantum dot systems. We demonstrate high-fidelity projective readout and control of the nuclear spin qubit, as well as entanglement between the nuclear and electron spins. Crucially, we find that both the nuclear spin and electron spin retain their coherence while moving the electron between quantum dots. Hence we envision long-range nuclear-nuclear entanglement via electron shuttling3. Our results establish nuclear spins in quantum dots as a powerful new resource for quantum processing.

A quantum-dot spin qubit with coherence limited by charge noise and fidelity higher than 99.9.

The isolation of qubits from noise sources, such as surrounding nuclear spins and spin-electric susceptibility 1-4 , has enabled extensions of quantum coherence times in recent pivotal advances towards the concrete implementation of spin-based quantum computation. In fact, the possibility of achieving enhanced quantum coherence has been substantially doubted for nanostructures due to the characteristic high degree of background charge fluctuations 5-7 . Still, a sizeable spin-electric coupling will be needed in realistic multiple-qubit systems to address single-spin and spin-spin manipulations 8-10 . Here, we realize a single-electron spin qubit with an isotopically enriched phase coherence time (20 µs) 11,12 and fast electrical control speed (up to 30 MHz) mediated by extrinsic spin-electric coupling. Using rapid spin rotations, we reveal that the free-evolution dephasing is caused by charge noise-rather than conventional magnetic noise-as highlighted by a 1/f spectrum extended over seven decades of frequency. The qubit exhibits superior performance with single-qubit gate fidelities exceeding 99.9% on average, offering a promising route to large-scale spin-qubit systems with fault-tolerant controllability.

Multiple-Quantum Transitions and Charge-Induced Decoherence of Donor Nuclear Spins in Silicon.

We study single- and multiquantum transitions of the nuclear spins of an ensemble of ionized arsenic donors in silicon and find quadrupolar effects on the coherence times, which we link to fluctuating electrical field gradients present after the application of light and bias voltage pulses. To determine the coherence times of superpositions of all orders in the 4-dimensional Hilbert space, we use a phase-cycling technique and find that, when electrical effects were allowed to decay, these times scale as expected for a fieldlike decoherence mechanism such as the interaction with surrounding ^{29}Si nuclear spins.

A dressed spin qubit in silicon.

Coherent dressing of a quantum two-level system provides access to a new quantum system with improved properties-a different and easily tunable level splitting, faster control and longer coherence times. In our work we investigate the properties of the dressed, donor-bound electron spin in silicon, and assess its potential as a quantum bit in scalable architectures. The two dressed spin-polariton levels constitute a quantum bit that can be coherently driven with an oscillating magnetic field, an oscillating electric field, frequency modulation of the driving field or a simple detuning pulse. We measure coherence times of and , one order of magnitude longer than those of the undressed spin. Furthermore, the use of the dressed states enables coherent coupling of the solid-state spins to electric fields and mechanical oscillations.

Broadband, large-area microwave antenna for optically detected magnetic resonance of nitrogen-vacancy centers in diamond.

We report on a microwave planar ring antenna specifically designed for optically detected magnetic resonance (ODMR) of nitrogen-vacancy (NV) centers in diamond. It has the resonance frequency at around 2.87 GHz with the bandwidth of 400 MHz, ensuring that ODMR can be observed under external magnetic fields up to 100 G without the need of adjustment of the resonance frequency. It is also spatially uniform within the 1-mm-diameter center hole, enabling the magnetic-field imaging in the wide spatial range. These features facilitate the experiments on quantum sensing and imaging using NV centers at room temperature.

Bell's inequality violation with spins in silicon.

Bell's theorem proves the existence of entangled quantum states with no classical counterpart. An experimental violation of Bell's inequality demands simultaneously high fidelities in the preparation, manipulation and measurement of multipartite quantum entangled states, and provides a single-number benchmark for the performance of devices that use such states for quantum computing. We demonstrate a Bell/ Clauser-Horne-Shimony-Holt inequality violation with Bell signals up to 2.70(9), using the electron and the nuclear spins of a single phosphorus atom embedded in a silicon nanoelectronic device. Two-qubit state tomography reveals that our prepared states match the target maximally entangled Bell states with >96% fidelity. These experiments demonstrate complete control of the two-qubit Hilbert space of a phosphorus atom and highlight the important function of the nuclear qubit to expand the computational basis and maximize the readout fidelity.

Electrically controlling single-spin qubits in a continuous microwave field.

Large-scale quantum computers must be built upon quantum bits that are both highly coherent and locally controllable. We demonstrate the quantum control of the electron and the nuclear spin of a single (31)P atom in silicon, using a continuous microwave magnetic field together with nanoscale electrostatic gates. The qubits are tuned into resonance with the microwave field by a local change in electric field, which induces a Stark shift of the qubit energies. This method, known as A-gate control, preserves the excellent coherence times and gate fidelities of isolated spins, and can be extended to arbitrarily many qubits without requiring multiple microwave sources.

Interaction of Strain and Nuclear Spins in Silicon: Quadrupolar Effects on Ionized Donors.

The nuclear spins of ionized donors in silicon have become an interesting quantum resource due to their very long coherence times. Their perfect isolation, however, comes at a price, since the absence of the donor electron makes the nuclear spin difficult to control. We demonstrate that the quadrupolar interaction allows us to effectively tune the nuclear magnetic resonance of ionized arsenic donors in silicon via strain and determine the two nonzero elements of the S tensor linking strain and electric field gradients in this material to S(11)=1.5×10(22) V/m2 and S(44)=6×10(22) V/m2. We find a stronger benefit of dynamical decoupling on the coherence properties of transitions subject to first-order quadrupole shifts than on those subject to only second-order shifts and discuss applications of quadrupole physics including mechanical driving of magnetic resonance, cooling of mechanical resonators, and strain-mediated spin coupling.

Submillisecond hyperpolarization of nuclear spins in silicon.

In this Letter, we devise a fast and effective nuclear spin hyperpolarization scheme, which is, in principle, magnetic field independent. We use this scheme to experimentally demonstrate polarizations of up to 66% for phosphorus donor nuclear spins in bulk silicon, which are created within less than 100 µs in a magnetic field of 0.35 T at a temperature of 5 K. The polarization scheme is based on a spin-dependent recombination process via weakly coupled spin pairs, for which the recombination time constant strongly depends on the relative orientation of the two spins. We further use this scheme to measure the nuclear spin relaxation time and find a value of ∼100 ms under illumination, in good agreement with the value calculated for nuclear spin flips induced by repeated ionization and deionization processes.

Storing quantum information for 30 seconds in a nanoelectronic device.

The spin of an electron or a nucleus in a semiconductor naturally implements the unit of quantum information--the qubit. In addition, because semiconductors are currently used in the electronics industry, developing qubits in semiconductors would be a promising route to realize scalable quantum information devices. The solid-state environment, however, may provide deleterious interactions between the qubit and the nuclear spins of surrounding atoms, or charge and spin fluctuations arising from defects in oxides and interfaces. For materials such as silicon, enrichment of the spin-zero (28)Si isotope drastically reduces spin-bath decoherence. Experiments on bulk spin ensembles in (28)Si crystals have indeed demonstrated extraordinary coherence times. However, it remained unclear whether these would persist at the single-spin level, in gated nanostructures near amorphous interfaces. Here, we present the coherent operation of individual (31)P electron and nuclear spin qubits in a top-gated nanostructure, fabricated on an isotopically engineered (28)Si substrate. The (31)P nuclear spin sets the new benchmark coherence time (>30 s with Carr-Purcell-Meiboom-Gill (CPMG) sequence) of any single qubit in the solid state and reaches >99.99% control fidelity. The electron spin CPMG coherence time exceeds 0.5 s, and detailed noise spectroscopy indicates that--contrary to widespread belief--it is not limited by the proximity to an interface. Instead, decoherence is probably dominated by thermal and magnetic noise external to the device, and is thus amenable to further improvement.