Single-shot read-out of an individual electron spin in a quantum dot

Gate-defined quantum point contacts in a germanium quantum well

We report an experimental study of quantum point contacts defined in a high-quality strained germanium quantum well with layered electric gates. At a zero magnetic field, we observed quantized conductance plateaus in units of 2e2/h. Bias-spectroscopy measurements reveal that the energy spacing between successive one-dimensional subbands ranges from 1.5 to 5 meV as a consequence of the small effective mass of the holes and the narrow gate constrictions. At finite magnetic fields perpendicular to the device plane, the edges of the conductance plateaus get split due to the Zeeman effect and Landé g factors were estimated to be ∼6.6 for the holes in the germanium quantum well. We demonstrate that all quantum point contacts in the same device have comparable performances, indicating a reliable and reproducible device fabrication process. Thus, our work lays a foundation for investigating multiple forefronts of physics in germanium-based quantum devices that require quantum point contacts as building blocks.

High-Fidelity Optical Readout of a Nuclear-Spin Qubit in Silicon Carbide

Quantum state readout is a key requirement for a successful qubit platform. In this work, we demonstrate a high-fidelity quantum state readout of a V2 center nuclear spin based on a repetitive readout technique. We demonstrate up to 99.5% readout fidelity and 99% for state preparation. Using this efficient readout, we initialize the nuclear spin by measurement and demonstrate its Rabi and Ramsey nutation. Finally, we use the nuclear spin as a long-lived memory for quantum sensing application of a weakly coupled diatomic nuclear-spin bath.

Probing single electrons across 300-mm spin qubit wafers

Building a fault-tolerant quantum computer will require vast numbers of physical qubits. For qubit technologies based on solid-state electronic devices1-3, integrating millions of qubits in a single processor will require device fabrication to reach a scale comparable to that of the modern complementary metal-oxide-semiconductor (CMOS) industry. Equally important, the scale of cryogenic device testing must keep pace to enable efficient device screening and to improve statistical metrics such as qubit yield and voltage variation. Spin qubits1,4,5 based on electrons in Si have shown impressive control fidelities6-9 but have historically been challenged by yield and process variation10-12. Here we present a testing process using a cryogenic 300-mm wafer prober13 to collect high-volume data on the performance of hundreds of industry-manufactured spin qubit devices at 1.6 K. This testing method provides fast feedback to enable optimization of the CMOS-compatible fabrication process, leading to high yield and low process variation. Using this system, we automate measurements of the operating point of spin qubits and investigate the transitions of single electrons across full wafers. We analyse the random variation in single-electron operating voltages and find that the optimized fabrication process leads to low levels of disorder at the 300-mm scale. Together, these results demonstrate the advances that can be achieved through the application of CMOS-industry techniques to the fabrication and measurement of spin qubit devices.

Is Wave Function Collapse Necessary? Explaining Quantum Nondemolition Measurement of a Spin Qubit within Linear Evolution

The measurement problem dates back to the dawn of quantum mechanics. Here, we measure a quantum dot electron spin qubit through off-resonant coupling with a highly redundant ancilla, consisting of thousands of nuclear spins. Large redundancy allows for single-shot measurement with high fidelity ≈99.85%. Repeated measurements enable heralded initialization of the qubit and backaction-free detection of electron spin quantum jumps, attributed to burstlike fluctuations in a thermally populated phonon bath. Based on these results we argue that the measurement, linking quantum states to classical observables, can be made without any "wave function collapse" in agreement with the Quantum Darwinism concept.

Cross-correlations between currents and tunnel magnetoresistance in interacting double quantum dot-Majorana wire system

We theoretically investigate the spin and charge transport properties of a double quantum dot coupled to distinct edges of the nanowire hosting Majorana zero-energy modes. The focus is on the analysis of the currents flowing through the left and right junctions and their cross-correlations. We show that the system reveals very different transport properties depending on the detuning protocol of the quantum dot energy levels. For the symmetric detuning, the current dependencies reveal only two maxima associated with resonant tunneling, and currents in the left and right arms of the system reveal weak positive cross-correlations. On the other hand, for antisymmetric detuning, the flow of electrons into drains is maximized and strongly correlated in one bias voltage direction, while for the opposite bias direction a spin blockade is predicted. Furthermore, we observe a suppression of the current cross-correlations at a highly symmetric detuning point, indicating the involvement of the Majorana zero-energy modes in the transport processes. To gain insight into the role of the spin polarization of the Majorana edge states, we analyze the spin-dependent transport characteristics by considering the relationship between the spin canting angle, which describes the coupling of the Majorana modes to the spin of the quantum dots, and the magnetic configurations of the ferromagnetic drains. Moreover, we examine the non-local zero bias anomaly in the differential conductance, detailed analysis of which revealed a specific operational mode of the device that can facilitate the identification of the Majorana presence in the quantum dot-Majorana wire system. Finally, we also consider the transport properties in different magnetic configurations of the system and discuss the behavior of the associated tunnel magnetoresistance.

Photoionisation detection of a single Er3+ ion with sub-100-ns time resolution

Efficient detection of single optical centres in solids is essential for quantum information processing, sensing and single-photon generation applications. In this work, we use radio-frequency (RF) reflectometry to electrically detect the photoionisation induced by a single Er3+ ion in Si. The high bandwidth and sensitivity of the RF reflectometry provide sub-100-ns time resolution for the photoionisation detection. With this technique, the optically excited state lifetime of a single Er3+ ion in a Si nano-transistor is measured for the first time to be [Formula: see text]s. Our results demonstrate an efficient approach for detecting a charge state change induced by Er excitation and relaxation. This approach could be used for fast readout of other single optical centres in solids and is attractive for large-scale integrated optical quantum systems thanks to the multi-channel RF reflectometry demonstrated with frequency multiplexing techniques.

Heterogeneous integration of single InAs/InP quantum dots with the SOI chip using direct bonding

Quantum information processing with photons in small-footprint and highly integrated silicon-based photonic chips requires incorporating non-classical light sources. In this respect, self-assembled III-V semiconductor quantum dots (QDs) are an attractive solution, however, they must be combined with the silicon platform. Here, by utilizing the large-area direct bonding technique, we demonstrate the hybridization of InP and SOI chips, which allows for coupling single photons to the SOI chip interior, offering cost-effective scalability in setting up a multi-source environment for quantum photonic chips. We fabricate devices consisting of self-assembled InAs QDs embedded in the tapered InP waveguide (WG) positioned over the SOI-defined Si WG. Focusing on devices generating light in the telecom C-band compatible with the low-loss optical fiber networks, we demonstrate the light coupling between InP and SOI platforms by observing photons outcoupled at the InP-made circular Bragg grating outcoupler fabricated at the end of an 80 µm-long Si WG, and at the cleaved edge of the Si WG. Finally, for a device with suppressed multi-photon generation events exhibiting 80% single photon generation purity, we measure the photon number outcoupled at the cleaved facet of the Si WG. We estimate the directional on-chip photon coupling between the source and the Si WG to 5.1%.

Engineering colloidal semiconductor nanocrystals for quantum information processing

Quantum information processing-which relies on spin defects or single-photon emission-has shown quantum advantage in proof-of-principle experiments including microscopic imaging of electromagnetic fields, strain and temperature in applications ranging from battery research to neuroscience. However, critical gaps remain on the path to wider applications, including a need for improved functionalization, deterministic placement, size homogeneity and greater programmability of multifunctional properties. Colloidal semiconductor nanocrystals can close these gaps in numerous application areas, following years of rapid advances in synthesis and functionalization. In this Review, we specifically focus on three key topics: optical interfaces to long-lived spin states, deterministic placement and delivery for sensing beyond the standard quantum limit, and extensions to multifunctional colloidal quantum circuits.

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.

Molecular nanomagnets: a viable path toward quantum information processing?

Molecular nanomagnets (MNMs), molecules containing interacting spins, have been a playground for quantum mechanics. They are characterized by many accessible low-energy levels that can be exploited to store and process quantum information. This naturally opens the possibility of using them as qudits, thus enlarging the tools of quantum logic with respect to qubit-based architectures. These additional degrees of freedom recently prompted the proposal for encoding qubits with embedded quantum error correction (QEC) in single molecules. QEC is the holy grail of quantum computing and this qudit approach could circumvent the large overhead of physical qubits typical of standard multi-qubit codes. Another important strength of the molecular approach is the extremely high degree of control achieved in preparing complex supramolecular structures where individual qudits are linked preserving their individual properties and coherence. This is particularly relevant for building quantum simulators, controllable systems able to mimic the dynamics of other quantum objects. The use of MNMs for quantum information processing is a rapidly evolving field which still requires to be fully experimentally explored. The key issues to be settled are related to scaling up the number of qudits/qubits and their individual addressing. Several promising possibilities are being intensively explored, ranging from the use of single-molecule transistors or superconducting devices to optical readout techniques. Moreover, new tools from chemistry could be also at hand, like the chiral-induced spin selectivity. In this paper, we will review the present status of this interdisciplinary research field, discuss the open challenges and envisioned solution paths which could finally unleash the very large potential of molecular spins for quantum technologies.

Enhanced Spin Polarization from Biaxially Strained Colloidal Quantum Dots

Electron and hole spin polarization is crucial for quantum dots to be used in spin lasers and quantum information processing. However, the degree of spin polarization in II-VI and III-V semiconductor quantum dots is low because of the degenerated valence band. Here, we increase the light and heavy hole degeneracy by introducing biaxial strain into CdSe-based quantum dots, enabling the degree of spin polarization to be increased from 20% to 50% under photoexcitation. The optical gain threshold measurement further reveals that the increase in polarization helps to reduce the gain threshold.

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.

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.

Excited state spectroscopy and spin splitting in single layer MoS2 quantum dots

Semiconducting transition metal dichalcogenides (TMDCs) are very promising materials for quantum dots and spin-qubit implementation. Reliable operation of spin qubits requires the knowledge of the Landé g-factor, which can be measured by exploiting the discrete energy spectrum on a quantum dot. However, the quantum dots realized in TMDCs are yet to reach the required control and quality for reliable measurement of excited state spectroscopy and the g-factor, particularly in atomically thin layers. Quantum dot sizes reported in TMDCs so far are not small enough to observe discrete energy levels on them. Here, we report on electron transport through discrete energy levels of quantum dots in a single layer MoS2 isolated from its environment using a dual gate geometry. The quantum dot energy levels are separated by a few (5-6) meV such that the ground state and the first excited state transitions are clearly visible, thanks to the low contact resistance of ∼700 Ω and relatively low gate voltages. This well-resolved energy separation allowed us to accurately measure the ground state g-factor of ∼5 in MoS2 quantum dots. We observed a spin-filling sequence in our quantum dots under a perpendicular magnetic field. Such a system offers an excellent testbed to measure the key parameters for evaluation and implementation of spin-valley qubits in TMDCs, thus accelerating the development of quantum systems in two-dimensional semiconducting TMDCs.

Room-temperature electric field control of spin filtering efficiency for enhanced modulation of optical spin polarization in a defect-functional 0D-2D hybrid nanostructure

In order to accomplish spin-based photoelectric information processing, it is necessary to modulate electron spin polarization in III-V semiconductor quantum dots (QDs) using an electric field. However, there is a principal limitation to the spin polarization degree and its control range, as the electron spin polarization is rapidly lost during injection into the QDs at room temperature (RT). Here, electric field control of optical spin polarization in the range of 15-40% is demonstrated at RT using InAs QDs tunnel-coupled with a defect-functional GaNAs quantum well (QW) spin filter. This compares with an electric field control of 1-4% for InAs QDs tunnel-coupled with an InGaAs QW. Transient polarization in the range of 30-60% is also obtained in the ultrafast time domain of less than 100 ps, the degree of polarization depending on the electric field. The enhanced polarization control is achieved by tuning the amplified spin polarization of electrons tunnel-injected from the GaNAs QW into QDs via the electric-field-dependent spin-filtering efficiency of GaNAs. These findings will provide a new way to extensively modulate the electron spin polarization in opto-semiconductors, by electric-field-induced on/off switching of spin amplification.

Single-Gate In-Transistor Readout of Current Superposition and Collapse Utilizing Quantum Tunneling and Ferroelectric Switching

In nanostructure assemblies, the superposition of current paths forms microscopic electric circuits, and different circuit networks produce varying results, particularly when utilized as transistor channels for computing applications. However, the intricate nature of assembly networks and the winding paths of commensurate currents hinder standard circuit modeling. Inspired by the quantum collapse of superposition states for information decoding in quantum circuits, the implementation of analogous current path collapse to facilitate the detection of microscopic circuits by modifying their network topology is explored. Here, the superposition and collapse of current paths in gate-all-around polysilicon nanosheet arrays are demonstrated to enrich the computational resources within transistors by engineering the channel length and quantity. Switching the ferroelectric polarization of Hf0.5 Zr0.5 O2 gate dielectric, which drives these transistors out-of-equilibrium, decodes the output polymorphism through circuit topological modifications. Furthermore, a protocol for the single-electron readout of ferroelectric polarization is presented with tailoring the channel coherence. The introduction of lateral path superposition results into intriguing metal-to-insulator transitions due to transient behavior of ferroelectric switching. This ability to adjust the current networks within transistors and their interaction with ferroelectric polarization in polycrystalline nanostructures lays the groundwork for generating diverse current characteristics as potential physical databases for optimization-based computing.

Cavity-enhanced single-shot readout of a quantum dot spin within 3 nanoseconds

Rapid, high-fidelity single-shot readout of quantum states is a ubiquitous requirement in quantum information technologies. For emitters with a spin-preserving optical transition, spin readout can be achieved by driving the transition with a laser and detecting the emitted photons. The speed and fidelity of this approach is typically limited by low photon collection rates and measurement back-action. Here we use an open microcavity to enhance the optical readout signal from a semiconductor quantum dot spin state, largely overcoming these limitations. We achieve single-shot readout of an electron spin in only 3 nanoseconds with a fidelity of (95.2 ± 0.7)%, and observe quantum jumps using repeated single-shot measurements. Owing to the speed of our readout, errors resulting from measurement-induced back-action have minimal impact. Our work reduces the spin readout-time well below both the achievable spin relaxation and dephasing times in semiconductor quantum dots, opening up new possibilities for their use in quantum technologies.

Single-electron spin resonance detection by microwave photon counting

Electron spin resonance spectroscopy is the method of choice for characterizing paramagnetic impurities, with applications ranging from chemistry to quantum computing^1,2, but it gives access only to ensemble-averaged quantities owing to its limited signal-to-noise ratio. Single-electron spin sensitivity has, however, been reached using spin-dependent photoluminescence^3-5, transport measurements^6-9 and scanning-probe techniques^10-12. These methods are system-specific or sensitive only in a small detection volume^13,14, so that practical single-spin detection remains an open challenge. Here, we demonstrate single-electron magnetic resonance by spin fluorescence detection¹⁵, using a microwave photon counter at millikelvin temperatures¹⁶. We detect individual paramagnetic erbium ions in a scheelite crystal coupled to a high-quality-factor planar superconducting resonator to enhance their radiative decay rate¹⁷, with a signal-to-noise ratio of 1.9 in one second integration time. The fluorescence signal shows anti-bunching, proving that it comes from individual emitters. Coherence times up to 3 ms are measured, limited by the spin radiative lifetime. The method has the potential to be applied to arbitrary paramagnetic species with long enough non-radiative relaxation times, and allows single-spin detection in a volume as large as the resonator magnetic mode volume (approximately 10 μm³ in the present experiment), orders of magnitude larger than other single-spin detection techniques. As such, it may find applications in magnetic resonance and quantum computing.

Thermodynamic uncertainty relations in mesoscopic devices

We investigate the thermodynamic uncertainty relations (TURs) in mesoscopic devices for all universal symmetry classes of Wigner-Dyson and Dirac (chiral). The observables of interest include the TUR (MS), which is defined in terms of the ratio between the mean noise and mean conductance, as well as a new TUR (R) proposed in this article, which is based on the ensemble mean of the noise-to-conductance ratio. A detailed study is made on the quantum interference corrections associated with the TURs. We also analyze the influence of orbital and sublattice/chiral degrees of freedom for the validity of the observables in these chaotic mesoscopic billiards. Our investigation is based on the concatenation between the Landauer-Büttiker theory, the Mahaux-Wendeinmüller theory, and the TURs. We simulate the universal mesoscopic chaotic quantum dots using the random-matrix theory and compare our numerical results with the pertinent experimental data. The results were obtained for a different number of channels and tunneling rates that vary from the opaque to the ideal regime and, in all cases, demonstrate a clear phenomenological distinction between the TURs. In particular, the opaque regime engenders remarkable differences between the observables, even in the semiclassical regime, which characterizes a clear violation of the central limit theorem. Furthermore, we show that the phenomenology of the quantum interference corrections is strikingly robust, surprisingly exhibiting an order of magnitude greater than the supposedly leading semiclassical term for the TUR (R).

Coherence Characteristics of a GaAs Single Heavy-Hole Spin Qubit Using a Modified Single-Shot Latching Readout Technique

We present an experimental study of the coherence properties of a single heavy-hole spin qubit formed in one quantum dot of a gated GaAs/AlGaAs double quantum dot device. We use a modified spin-readout latching technique in which the second quantum dot serves both as an auxiliary element for a fast spin-dependent readout within a 200 ns time window and as a register for storing the spin-state information. To manipulate the single-spin qubit, we apply sequences of microwave bursts of various amplitudes and durations to make Rabi, Ramsey, Hahn-echo, and CPMG measurements. As a result of the qubit manipulation protocols combined with the latching spin readout, we determine and discuss the achieved qubit coherence times: T1, TRabi, T2*, and T2CPMG vs. microwave excitation amplitude, detuning, and additional relevant parameters.

ESR-STM on diamagnetic molecule: C60 on graphene

Electron Spin Resonance-Scanning Tunneling Microscopy (ESR-STM) of C₆₀ radical ion on graphene is a first demonstration of ESR-STM on diamagnetic molecules. ESR-STM signal at g_average=2.0±0.1 was measured in accordance with macroscopic ESR of C₆₀ radical ion. The ESR-STM signal was bias voltage dependent, as it reflects the charge state of the molecule. The signal appears in the bias voltage which enables the ionization of the lowest unoccupied molecular orbital (LUMO) - creation of radical anion, and the highest occupied molecular orbital (HOMO) - creation of a radical cation of the C₆₀ molecule when it deposited on graphene. In parallel, ESR-STM signal at g_average=1.7±0.1 was ascribed to Tungsten oxide (WO₃) at the tip apex. In several experiments, triplet spectrum was observed, and we ascribed their origin to zero-field splitting of doubly ionized C₁₂₀O^-2 dimer, as argued in previous ESR experiments of C₆₀ samples. Second possibility is hyperfine coupling with two ¹³C nuclei. In addition, we further validate the interference mechanism previously suggested for ESR-STM noise spectroscopy. The ability of ESR-STM to observe ESR of diamagnetic molecules in parallel with observing their electronic structure, provides a general single molecule identification technique.

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.

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.

Variable-Barrier Quantum Coulomb Blockade Effect in Nanoscale Transistors

Current-voltage characteristics of a quantum dot in double-barrier configuration, as formed in the nanoscale channel of silicon transistors, were analyzed both experimentally and theoretically. Single electron transistors (SET) made in a SOI-FET configuration using silicon quantum dot as well as phosphorus donor quantum dots were experimentally investigated. These devices exhibited a quantum Coulomb blockade phenomenon along with a detectable effect of variable tunnel barriers. To replicate the experimental results, we developed a generalized formalism for the tunnel-barrier dependent quantum Coulomb blockade by modifying the rate-equation approach. We qualitatively replicate the experimental results with numerical calculation using this formalism for two and three energy levels participated in the tunneling transport. The new formalism supports the features of most of the small-scaled SET devices.

Entropy Measurement of a Strongly Coupled Quantum Dot

The spin 1/2 entropy of electrons trapped in a quantum dot has previously been measured with great accuracy, but the protocol used for that measurement is valid only within a restrictive set of conditions. Here, we demonstrate a novel entropy measurement protocol that is universal for arbitrary mesoscopic circuits and apply this new approach to measure the entropy of a quantum dot hybridized with a reservoir. The experimental results match closely to numerical renormalization group (NRG) calculations for small and intermediate coupling. For the largest couplings investigated in this Letter, NRG calculations predict a suppression of spin entropy at the charge transition due to the formation of a Kondo singlet, but that suppression is not observed in the experiment.

Microscopic Origin of the Effective Spin-Spin Interaction in a Semiconductor Quantum Dot Ensemble

We present a microscopic model for a singly charged quantum dot (QD) ensemble to reveal the origin of the long-range effective interaction between the electron spins in the QDs. Wilson's numerical renormalization group (NRG) is used to calculate the magnitude and the spatial dependency of the effective spin-spin interaction mediated by the growth-induced wetting layer. Surprisingly, we found an antiferromagnetic Heisenberg coupling for very short inter-QD distances that is caused by the significant particle-hole asymmetry of the wetting layer band at very low filling. Using the NRG results obtained from realistic parameters as input for a semiclassical simulation for a large QD ensemble, we demonstrate that the experimentally reported phase shifts in the coherent spin dynamics between single- and two-color laser pumping can be reproduced by our model, solving a long-standing open problem of the microscopic origin of the inter-QD electron spin-spin interaction.

A single hole spin with enhanced coherence in natural silicon

Semiconductor spin qubits based on spin-orbit states are responsive to electric field excitations, allowing for practical, fast and potentially scalable qubit control. Spin electric susceptibility, however, renders these qubits generally vulnerable to electrical noise, which limits their coherence time. Here we report on a spin-orbit qubit consisting of a single hole electrostatically confined in a natural silicon metal-oxide-semiconductor device. By varying the magnetic field orientation, we reveal the existence of operation sweet spots where the impact of charge noise is minimized while preserving an efficient electric-dipole spin control. We correspondingly observe an extension of the Hahn-echo coherence time up to 88 μs, exceeding by an order of magnitude existing values reported for hole spin qubits, and approaching the state-of-the-art for electron spin qubits with synthetic spin-orbit coupling in isotopically purified silicon. Our finding enhances the prospects of silicon-based hole spin qubits for scalable quantum information processing.

A shuttling-based two-qubit logic gate for linking distant silicon quantum processors

Control of entanglement between qubits at distant quantum processors using a two-qubit gate is an essential function of a scalable, modular implementation of quantum computation. Among the many qubit platforms, spin qubits in silicon quantum dots are promising for large-scale integration along with their nanofabrication capability. However, linking distant silicon quantum processors is challenging as two-qubit gates in spin qubits typically utilize short-range exchange coupling, which is only effective between nearest-neighbor quantum dots. Here we demonstrate a two-qubit gate between spin qubits via coherent spin shuttling, a key technology for linking distant silicon quantum processors. Coherent shuttling of a spin qubit enables efficient switching of the exchange coupling with an on/off ratio exceeding 1000, while preserving the spin coherence by 99.6% for the single shuttling between neighboring dots. With this shuttling-mode exchange control, we demonstrate a two-qubit controlled-phase gate with a fidelity of 93%, assessed via randomized benchmarking. Combination of our technique and a phase coherent shuttling of a qubit across a large quantum dot array will provide feasible path toward a quantum link between distant silicon quantum processors, a key requirement for large-scale quantum computation.

A coherent quantum link between distant quantum processors is desirable for scaling up of quantum computation. Noiri et al. demonstrate a strategy to link distant quantum processors in silicon, by implementing a shuttling-based two-qubit gate between spin qubits in a Si/SiGe triple quantum dot.

Ramped measurement technique for robust high-fidelity spin qubit readout

State preparation and measurement of single-electron spin qubits typically rely on spin-to-charge conversion where a spin-dependent charge transition of the electron is detected by a coupled charge sensor. For high-fidelity, fast readout, this process requires that the qubit energy is much larger than the temperature of the system limiting the temperature range for measurements. Here, we demonstrate an initialization and measurement technique that involves voltage ramps rather than static voltages allowing us to achieve state-to-charge readout fidelities above 99% for qubit energies almost half that required by traditional methods. This previously unidentified measurement technique is highly relevant for achieving high-fidelity electron spin readout at higher temperature operation and offers a number of pragmatic benefits compared to traditional energy-selective readout such as real-time dynamic feedback and minimal alignment procedures.

Quantum error correction with silicon spin qubits

Future large-scale quantum computers will rely on quantum error correction (QEC) to protect the fragile quantum information during computation1,2. Among the possible candidate platforms for realizing quantum computing devices, the compatibility with mature nanofabrication technologies of silicon-based spin qubits offers promise to overcome the challenges in scaling up device sizes from the prototypes of today to large-scale computers3-5. Recent advances in silicon-based qubits have enabled the implementations of high-quality one-qubit and two-qubit systems6-8. However, the demonstration of QEC, which requires three or more coupled qubits1, and involves a three-qubit gate9-11 or measurement-based feedback, remains an open challenge. Here we demonstrate a three-qubit phase-correcting code in silicon, in which an encoded three-qubit state is protected against any phase-flip error on one of the three qubits. The correction to this encoded state is performed by a three-qubit conditional rotation, which we implement by an efficient single-step resonantly driven iToffoli gate. As expected, the error correction mitigates the errors owing to one-qubit phase-flip, as well as the intrinsic dephasing mainly owing to quasi-static phase noise. These results show successful implementation of QEC and the potential of a silicon-based platform for large-scale quantum computing.

Harnessing many-body spin environment for long coherence storage and high-fidelity single-shot qubit readout

There is a growing interest in hybrid solid-state quantum systems where nuclear spins, interfaced to the electron spin qubit, are used as quantum memory or qubit register. These approaches require long nuclear spin coherence, which until now seemed impossible owing to the disruptive effect of the electron spin. Here we study InGaAs semiconductor quantum dots, demonstrating millisecond-long collective nuclear spin coherence even under inhomogeneous coupling to the electron central spin. We show that the underlying decoherence mechanism is spectral diffusion induced by a fluctuating electron spin. These results provide new understanding of the many-body coherence in central spin systems, required for development of electron-nuclear spin qubits. As a demonstration, we implement a conditional gate that encodes electron spin state onto collective nuclear spin coherence, and use it for a single-shot readout of the electron spin qubit with >99% fidelity.

Benchmarking Noise and Dephasing in Emerging Electrical Materials for Quantum Technologies

As quantum technologies develop, a specific class of electrically conducting materials is rapidly gaining interest because they not only form the core quantum-enabled elements in superconducting qubits, semiconductor nanostructures, or sensing devices, but also the peripheral circuitry. The phase coherence of the electronic wave function in these emerging materials will be crucial when incorporated in the quantum architecture. The loss of phase memory, or dephasing, occurs when a quantum system interacts with the fluctuations in the local electromagnetic environment, which manifests in "noise" in the electrical conductivity. Hence, characterizing these materials and devices therefrom, for quantum applications, requires evaluation of both dephasing and noise, although there are very few materials where these properties are investigated simultaneously. Here, the available data on magnetotransport and low-frequency fluctuations in electrical conductivity are reviewed to benchmark the dephasing and noise. The focus is on new materials that are of direct interest to quantum technologies. The physical processes causing dephasing and noise in these systems are elaborated, the impact of both intrinsic and extrinsic parameters from materials synthesis and devices realization are evaluated, and it is hoped that a clearer pathway to design and characterize both material and devices for quantum applications is thus provided.

Two-qubit silicon quantum processor with operation fidelity exceeding 99

Silicon spin qubits satisfy the necessary criteria for quantum information processing. However, a demonstration of high-fidelity state preparation and readout combined with high-fidelity single- and two-qubit gates, all of which must be present for quantum error correction, has been lacking. We use a two-qubit Si/SiGe quantum processor to demonstrate state preparation and readout with fidelity greater than 97%, combined with both single- and two-qubit control fidelities exceeding 99%. The operation of the quantum processor is quantitatively characterized using gate set tomography and randomized benchmarking. Our results highlight the potential of silicon spin qubits to become a dominant technology in the development of intermediate-scale quantum processors.

A Robust Protocol for Entropy Measurement in Mesoscopic Circuits

Previous measurements utilizing Maxwell relations to measure change in entropy, S, demonstrated remarkable accuracy in measuring the spin-1/2 entropy of electrons in a weakly coupled quantum dot. However, these previous measurements relied upon prior knowledge of the charge transition lineshape. This had the benefit of making the quantitative determination of entropy independent of scale factors in the measurement itself but at the cost of limiting the applicability of the approach to simple systems. To measure the entropy of more exotic mesoscopic systems, a more flexible analysis technique may be employed; however, doing so requires a precise calibration of the measurement. Here, we give details on the necessary improvements made to the original experimental approach and highlight some of the common challenges (along with strategies to overcome them) that other groups may face when attempting this type of measurement.

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 processing^1,2, 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 electrons^7-9 or physically shuttled across different locations^10,11, these results establish a viable route for scalable quantum information processing using donor nuclear and electron spins.

Fast universal quantum gate above the fault-tolerance threshold in silicon

Fault-tolerant quantum computers that can solve hard problems rely on quantum error correction¹. One of the most promising error correction codes is the surface code², which requires universal gate fidelities exceeding an error correction threshold of 99 per cent³. Among the many qubit platforms, only superconducting circuits⁴, trapped ions⁵ and nitrogen-vacancy centres in diamond⁶ have delivered this requirement. Electron spin qubits in silicon^7-15 are particularly promising for a large-scale quantum computer owing to their nanofabrication capability, but the two-qubit gate fidelity has been limited to 98 per cent owing to the slow operation¹⁶. Here we demonstrate a two-qubit gate fidelity of 99.5 per cent, along with single-qubit gate fidelities of 99.8 per cent, in silicon spin qubits by fast electrical control using a micromagnet-induced gradient field and a tunable two-qubit coupling. We identify the qubit rotation speed and coupling strength where we robustly achieve high-fidelity gates. We realize Deutsch-Jozsa and Grover search algorithms with high success rates using our universal gate set. Our results demonstrate universal gate fidelity beyond the fault-tolerance threshold and may enable scalable silicon quantum computers.

Coherent Two-Dimensional and Broadband Electronic Spectroscopies

Over the past few decades, coherent broadband spectroscopy has been widely used to improve our understanding of ultrafast processes (e.g., photoinduced electron transfer, proton transfer, and proton-coupled electron transfer reactions) at femtosecond resolution. The advances in femtosecond laser technology along with the development of nonlinear multidimensional spectroscopy enabled further insights into ultrafast energy transfer and carrier relaxation processes in complex biological and material systems. New discoveries and interpretations have led to improved design principles for optimizing the photophysical properties of various artificial systems. In this review, we first provide a detailed theoretical framework of both coherent broadband and two-dimensional electronic spectroscopy (2DES). We then discuss a selection of experimental approaches and considerations of 2DES along with best practices for data processing and analysis. Finally, we review several examples where coherent broadband and 2DES were employed to reveal mechanisms of photoinitiated ultrafast processes in molecular, biological, and material systems. We end the review with a brief perspective on the future of the experimental techniques themselves and their potential to answer an even greater range of scientific questions.

Time-Resolved Photoionization Detection of a Single Er3+ Ion in Silicon

The detection of charge trap ionization induced by resonant excitation enables spectroscopy on single Er³⁺ ions in silicon nanotransistors. In this work, a time-resolved detection method is developed to investigate the resonant excitation and relaxation of a single Er³⁺ ion in silicon. The time-resolved detection is based on a long-lived current signal with a tunable reset and allows the measurement under stronger and shorter resonant excitation in comparison to time-averaged detection. Specifically, the short-pulse study gives an upper bound of 23.7 μs on the decay time of the ⁴I_13/2 state of the Er³⁺ ion. The fast decay and the tunable reset allow faster repetition of the single-ion detection, which is attractive for implementing this method in large-scale quantum systems of single optical centers. The findings on the detection mechanism and dynamics also provide an important basis for applying this technique to detect other single optical centers in solids.

Harnessing the Quantum Behavior of Spins on Surfaces

The desire to control and measure individual quantum systems such as atoms and ions in a vacuum has led to significant scientific and engineering developments in the past decades that form the basis of today's quantum information science. Single atoms and molecules on surfaces, on the other hand, are heavily investigated by physicists, chemists, and material scientists in search of novel electronic and magnetic functionalities. These two paths crossed in 2015 when it was first clearly demonstrated that individual spins on a surface can be coherently controlled and read out in an all-electrical fashion. The enabling technique is a combination of scanning tunneling microscopy (STM) and electron spin resonance, which offers unprecedented coherent controllability at the Angstrom length scale. This review aims to illustrate the essential ingredients that allow the quantum operations of single spins on surfaces. Three domains of applications of surface spins, namely quantum sensing, quantum control, and quantum simulation, are discussed with physical principles explained and examples presented. Enabled by the atomically-precise fabrication capability of STM, single spins on surfaces might one day lead to the realization of quantum nanodevices and artificial quantum materials at the atomic scale.

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.

Quantum-coherent nanoscience

For the past three decades nanoscience has widely affected many areas in physics, chemistry and engineering, and has led to numerous fundamental discoveries, as well as applications and products. Concurrently, quantum science and technology has developed into a cross-disciplinary research endeavour connecting these same areas and holds burgeoning commercial promise. Although quantum physics dictates the behaviour of nanoscale objects, quantum coherence, which is central to quantum information, communication and sensing, has not played an explicit role in much of nanoscience. This Review describes fundamental principles and practical applications of quantum coherence in nanoscale systems, a research area we call quantum-coherent nanoscience. We structure this Review according to specific degrees of freedom that can be quantum-coherently controlled in a given nanoscale system, such as charge, spin, mechanical motion and photons. We review the current state of the art and focus on outstanding challenges and opportunities unlocked by the merging of nanoscience and coherent quantum operations.

An Operation Guide of Si-MOS Quantum Dots for Spin Qubits

In the last 20 years, silicon quantum dots have received considerable attention from academic and industrial communities for research on readout, manipulation, storage, near-neighbor and long-range coupling of spin qubits. In this paper, we introduce how to realize a single spin qubit from Si-MOS quantum dots. First, we introduce the structure of a typical Si-MOS quantum dot and the experimental setup. Then, we show the basic properties of the quantum dot, including charge stability diagram, orbital state, valley state, lever arm, electron temperature, tunneling rate and spin lifetime. After that, we introduce the two most commonly used methods for spin-to-charge conversion, i.e., Elzerman readout and Pauli spin blockade readout. Finally, we discuss the details of how to find the resonance frequency of spin qubits and show the result of coherent manipulation, i.e., Rabi oscillation. The above processes constitute an operation guide for helping the followers enter the field of spin qubits in Si-MOS quantum dots.

Quantum tomography of an entangled three-qubit state in silicon

Quantum entanglement is a fundamental property of coherent quantum states and an essential resource for quantum computing¹. In large-scale quantum systems, the error accumulation requires concepts for quantum error correction. A first step toward error correction is the creation of genuinely multipartite entanglement, which has served as a performance benchmark for quantum computing platforms such as superconducting circuits^2,3, trapped ions⁴ and nitrogen-vacancy centres in diamond⁵. Among the candidates for large-scale quantum computing devices, silicon-based spin qubits offer an outstanding nanofabrication capability for scaling-up. Recent studies demonstrated improved coherence times^6-8, high-fidelity all-electrical control^9-13, high-temperature operation^14,15 and quantum entanglement of two spin qubits^9,11,12. Here we generated a three-qubit Greenberger-Horne-Zeilinger state using a low-disorder, fully controllable array of three spin qubits in silicon. We performed quantum state tomography¹⁶ and obtained a state fidelity of 88.0%. The measurements witness a genuine Greenberger-Horne-Zeilinger class quantum entanglement that cannot be separated into any biseparable state. Our results showcase the potential of silicon-based spin qubit platforms for multiqubit quantum algorithms.

Preparation and Readout of Multielectron High-Spin States in a Gate-Defined GaAs/AlGaAs Quantum Dot

We report the preparation and readout of multielectron high-spin states, a three-electron quartet, and a four-electron quintet, in a gate-defined GaAs/AlGaAs single quantum dot using spin filtering by quantum Hall edge states coupled to the dot. The readout scheme consists of mapping from multielectron to two-electron spin states and a subsequent two-electron spin readout, thus obviating the need to resolve dense multielectron energy levels. Using this technique, we measure the relaxations of the high-spin states and find them to be an order of magnitude faster than those of low-spin states. Numerical calculations of spin relaxation rates using the exact diagonalization method agree with the experiment. The technique developed here offers a new tool for the study and application of high-spin states in quantum dots.

Robust and fast post-processing of single-shot spin qubit detection events with a neural network

Establishing low-error and fast detection methods for qubit readout is crucial for efficient quantum error correction. Here, we test neural networks to classify a collection of single-shot spin detection events, which are the readout signal of our qubit measurements. This readout signal contains a stochastic peak, for which a Bayesian inference filter including Gaussian noise is theoretically optimal. Hence, we benchmark our neural networks trained by various strategies versus this latter algorithm. Training of the network with 10⁶ experimentally recorded single-shot readout traces does not improve the post-processing performance. A network trained by synthetically generated measurement traces performs similar in terms of the detection error and the post-processing speed compared to the Bayesian inference filter. This neural network turns out to be more robust to fluctuations in the signal offset, length and delay as well as in the signal-to-noise ratio. Notably, we find an increase of 7% in the visibility of the Rabi oscillation when we employ a network trained by synthetic readout traces combined with measured signal noise of our setup. Our contribution thus represents an example of the beneficial role which software and hardware implementation of neural networks may play in scalable spin qubit processor architectures.

An ultra-stable 1.5 T permanent magnet assembly for qubit experiments at cryogenic temperatures

Magnetic fields are a standard tool in the toolbox of every physicist and are required for the characterization of materials, as well as the polarization of spins in nuclear magnetic resonance or electron paramagnetic resonance experiments. Quite often, a static magnetic field of sufficiently large, but fixed, magnitude is suitable for these tasks. Here, we present a permanent magnet assembly that can achieve magnetic field strengths of up to 1.5 T over an air gap length of 7 mm. The assembly is based on a Halbach array of neodymium magnets, with the inclusion of the soft magnetic material Supermendur to boost the magnetic field strength inside the air gap. We present the design, simulation, and characterization of the permanent magnet assembly, measuring an outstanding magnetic field stability with a drift rate of |D| < 2.8 ppb/h. Our measurements demonstrate that this assembly can be used for spin qubit experiments inside a dilution refrigerator, successfully replacing the more expensive and bulky superconducting solenoids.

Single-Shot Readout of a Driven Hybrid Qubit in a GaAs Double Quantum Dot

We report a single-shot-based projective readout of a semiconductor hybrid qubit formed by three electrons in a GaAs double quantum dot. Voltage-controlled adiabatic transitions between the qubit operations and readout conditions allow high-fidelity mapping of quantum states. We show that a large ratio both in relaxation time vs tunneling time (≈50) and singlet-triplet splitting vs thermal energy (≈20) allows energy-selective tunneling-based spin-to-charge conversion with a readout visibility of ≈92.6%. Combined with ac driving, we demonstrate high visibility coherent Rabi and Ramsey oscillations of a hybrid qubit in GaAs. Further, we discuss the generality of the method for use in other materials, including silicon.

Coherent control of a donor-molecule electron spin qubit in silicon

Donor spins in silicon provide a promising material platform for large scale quantum computing. Excellent electron spin coherence times of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{2}^{* }=268$$\end{document}T2*=268 μs with fidelities of 99.9% have been demonstrated for isolated phosphorus donors in isotopically pure ²⁸Si, where donors are local-area-implanted in a nanoscale MOS device. Despite robust single qubit gates, realising two-qubit exchange gates using this technique is challenging due to the statistical nature of the dopant implant and placement process. In parallel a precision scanning probe lithography route has been developed to place single donors and donor molecules on one atomic plane of silicon with high accuracy aligned to heavily phosphorus doped silicon in-plane gates. Recent results using this technique have demonstrated a fast (0.8 ns) two-qubit gate with two P donor molecules placed 13 nm apart in ^natSi. In this paper we demonstrate a single qubit gate with coherent oscillations of the electron spin on a P donor molecule in ^natSi patterned by scanning tunneling microscope (STM) lithography. The electron spin exhibits excellent coherence properties, with a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{2}$$\end{document}T2 decoherence time of 298 ± 30 μs, and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}_{2}^{* }$$\end{document}T2* dephasing time of 295 ± 23 ns.

Multi-donor molecules in Si provide a promising qubit platform, offering advantages over single donor qubits in terms of performance and fabrication. Here, the authors report a single qubit gate and long coherence times in a P donor molecule qubit in natural Si patterned by scanning tunneling microscope hydrogen lithography.