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.

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.

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.

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.

Entanglement in a solid-state spin ensemble.

Entanglement is the quintessential quantum phenomenon. It is a necessary ingredient in most emerging quantum technologies, including quantum repeaters, quantum information processing and the strongest forms of quantum cryptography. Spin ensembles, such as those used in liquid-state nuclear magnetic resonance, have been important for the development of quantum control methods. However, these demonstrations contain no entanglement and ultimately constitute classical simulations of quantum algorithms. Here we report the on-demand generation of entanglement between an ensemble of electron and nuclear spins in isotopically engineered, phosphorus-doped silicon. We combined high-field (3.4 T), low-temperature (2.9 K) electron spin resonance with hyperpolarization of the (31)P nuclear spin to obtain an initial state of sufficient purity to create a non-classical, inseparable state. The state was verified using density matrix tomography based on geometric phase gates, and had a fidelity of 98% relative to the ideal state at this field and temperature. The entanglement operation was performed simultaneously, with high fidelity, on 10(10) spin pairs; this fulfils one of the essential requirements for a silicon-based quantum information processor.

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.

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.

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.

High-sensitivity magnetometry based on quantum beats in diamond nitrogen-vacancy centers.

We demonstrate an absolute magnetometer based on quantum beats in the ground state of nitrogen-vacancy centers in diamond. We show that, by eliminating the dependence of spin evolution on the zero-field splitting D, the magnetometer is immune to temperature fluctuation and strain inhomogeneity. We apply this technique to measure low-frequency magnetic field noise by using a single nitrogen-vacancy center located within 500 nm of the surface of an isotopically pure (99.99% 12C) diamond. The photon-shot-noise limited sensitivity achieves 38 nT/sqrt[Hz] for 4.45 s acquisition time, a factor of sqrt[2] better than the implementation which uses only two spin levels. For long acquisition times (>10 s), we realize up to a factor of 15 improvement in magnetic sensitivity, which demonstrates the robustness of our technique against thermal drifts. Applying our technique to nitrogen-vacancy center ensembles, we eliminate dephasing from longitudinal strain inhomogeneity, resulting in a factor of 2.3 improvement in sensitivity.

Optical and spin coherence properties of nitrogen-vacancy centers placed in a 100 nm thick isotopically purified diamond layer.

We have studied optical and spin properties of near-surface nitrogen-vacancy (NV) centers incorporated during chemical vapor phase growth of isotopically purified (12)C single-crystal diamond layers. The spectral diffusion-limited line width of zero-phonon luminescence from the NV centers is 1.2 ± 0.5 GHz, a considerable improvement over that of NV centers formed by ion implantation and annealing. Enhanced spin dephasing times (T(2)* ≈ 90 µs, T(2) ≈ 1.7 ms) due to the reduction of (13)C nuclear spins persist even for NV centers placed within 100 nm of the surface.

Thermal stability and surface passivation of Ge nanowires coated by epitaxial SiGe shells.

Epitaxial growth of a highly strained, coherent SiGe alloy shell around a Ge nanowire core is investigated as a method to achieve surface passivation and carrier confinement, important in realizing nanowire devices. The high photoluminescence intensity observed from the core-shell nanowires with spectral features similar to that of bulk Ge indicates effective surface passivation. Thermal stability of these core-shell heterostructures has been systematically investigated, with a method demonstrated to avoid misfit strain relaxation during postgrowth annealing.

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.

Electron spin coherence exceeding seconds in high-purity silicon.

Silicon is one of the most promising semiconductor materials for spin-based information processing devices. Its advanced fabrication technology facilitates the transition from individual devices to large-scale processors, and the availability of a (28)Si form with no magnetic nuclei overcomes a primary source of spin decoherence in many other materials. Nevertheless, the coherence lifetimes of electron spins in the solid state have typically remained several orders of magnitude lower than that achieved in isolated high-vacuum systems such as trapped ions. Here we examine electron spin coherence of donors in pure (28)Si material (residual (29)Si concentration <50 ppm) with donor densities of 10(14)-10(15) cm(-3). We elucidate three mechanisms for spin decoherence, active at different temperatures, and extract a coherence lifetime T(2) up to 2 s. In this regime, we find the electron spin is sensitive to interactions with other donor electron spins separated by ~200 nm. A magnetic field gradient suppresses such interactions, producing an extrapolated electron spin T(2) of 10 s at 1.8 K. These coherence lifetimes are without peer in the solid state and comparable to high-vacuum qubits, making electron spins of donors in silicon ideal components of quantum computers, or quantum memories for systems such as superconducting qubits.

Coherent storage of photoexcited triplet states using 29Si nuclear spins in silicon.

Pulsed electron paramagnetic resonance spectroscopy of the photoexcited, metastable triplet state of the oxygen-vacancy center in silicon reveals that the lifetime of the m(s)=±1 sublevels differs significantly from that of the m(s)=0 state. We exploit this significant difference in decay rates to the ground singlet state to achieve nearly ~100% electron-spin polarization within the triplet. We further demonstrate the transfer of a coherent state of the triplet electron spin to, and from, a hyperfine-coupled, nearest-neighbor (29)Si nuclear spin. We measure the coherence time of the (29)Si nuclear spin employed in this operation and find it to be unaffected by the presence of the triplet electron spin and equal to the bulk value measured by nuclear magnetic resonance.

Control of optical bandgap energy and optical absorption coefficient by geometric parameters in sub-10 nm silicon-nanodisc array structure.

A sub-10 nm, high-density, periodic silicon-nanodisc (Si-ND) array has been fabricated using a new top-down process, which involves a 2D array bio-template etching mask made of Listeria-Dps with a 4.5 nm diameter iron oxide core and damage-free neutral-beam etching (Si-ND diameter: 6.4 nm). An Si-ND array with an SiO(2) matrix demonstrated more controllable optical bandgap energy due to the fine tunability of the Si-ND thickness and diameter. Unlike the case of shrinking Si-ND thickness, the case of shrinking Si-ND diameter simultaneously increased the optical absorption coefficient and the optical bandgap energy. The optical absorption coefficient became higher due to the decrease in the center-to-center distance of NDs to enhance wavefunction coupling. This means that our 6 nm diameter Si-ND structure can satisfy the strict requirements of optical bandgap energy control and high absorption coefficient for achieving realistic Si quantum dot solar cells.

Damage-free top-down processes for fabricating two-dimensional arrays of 7 nm GaAs nanodiscs using bio-templates and neutral beam etching.

The first damage-free top-down fabrication processes for a two-dimensional array of 7 nm GaAs nanodiscs was developed by using ferritin (a protein which includes a 7 nm diameter iron core) bio-templates and neutral beam etching. The photoluminescence of GaAs etched with a neutral beam clearly revealed that the processes could accomplish defect-free etching for GaAs. In the bio-template process, to remove the ferritin protein shell without thermal damage to the GaAs, we firstly developed an oxygen-radical treatment method with a low temperature of 280 °C. Then, the neutral beam etched the defect-free nanodisc structure of the GaAs using the iron core as an etching mask. As a result, a two-dimensional array of GaAs quantum dots with a diameter of ∼ 7 nm, a height of ∼ 10 nm, a high taper angle of 88° and a quantum dot density of more than 7 × 10(11) cm(-2) was successfully fabricated without causing any damage to the GaAs.

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.

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.

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.