Nagaoka ferromagnetism observed in a quantum dot plaquette.

Engineered, highly controllable quantum systems are promising simulators of emergent physics beyond the simulation capabilities of classical computers1. An important problem in many-body physics is itinerant magnetism, which originates purely from long-range interactions of free electrons and whose existence in real systems has been debated for decades2,3. Here we use a quantum simulator consisting of a four-electron-site square plaquette of quantum dots4 to demonstrate Nagaoka ferromagnetism5. This form of itinerant magnetism has been rigorously studied theoretically6-9 but has remained unattainable in experiments. We load the plaquette with three electrons and demonstrate the predicted emergence of spontaneous ferromagnetic correlations through pairwise measurements of spin. We find that the ferromagnetic ground state is remarkably robust to engineered disorder in the on-site potentials and we can induce a transition to the low-spin state by changing the plaquette topology to an open chain. This demonstration of Nagaoka ferromagnetism highlights that quantum simulators can be used to study physical phenomena that have not yet been observed in any experimental system. The work also constitutes an important step towards large-scale quantum dot simulators of correlated electron systems.

Universal quantum logic in hot silicon qubits.

Quantum computation requires many qubits that can be coherently controlled and coupled to each other1. Qubits that are defined using lithographic techniques have been suggested to enable the development of scalable quantum systems because they can be implemented using semiconductor fabrication technology2-5. However, leading solid-state approaches function only at temperatures below 100 millikelvin, where cooling power is extremely limited, and this severely affects the prospects of practical quantum computation. Recent studies of electron spins in silicon have made progress towards a platform that can be operated at higher temperatures by demonstrating long spin lifetimes6, gate-based spin readout7 and coherent single-spin control8. However, a high-temperature two-qubit logic gate has not yet been demonstrated. Here we show that silicon quantum dots can have sufficient thermal robustness to enable the execution of a universal gate set at temperatures greater than one kelvin. We obtain single-qubit control via electron spin resonance and readout using Pauli spin blockade. In addition, we show individual coherent control of two qubits and measure single-qubit fidelities of up to 99.3 per cent. We demonstrate the tunability of the exchange interaction between the two spins from 0.5 to 18 megahertz and use it to execute coherent two-qubit controlled rotations. The demonstration of 'hot' and universal quantum logic in a semiconductor platform paves the way for quantum integrated circuits that host both the quantum hardware and its control circuitry on the same chip, providing a scalable approach towards practical quantum information processing.

A programmable two-qubit quantum processor in silicon.

Now that it is possible to achieve measurement and control fidelities for individual quantum bits (qubits) above the threshold for fault tolerance, attention is moving towards the difficult task of scaling up the number of physical qubits to the large numbers that are needed for fault-tolerant quantum computing. In this context, quantum-dot-based spin qubits could have substantial advantages over other types of qubit owing to their potential for all-electrical operation and ability to be integrated at high density onto an industrial platform. Initialization, readout and single- and two-qubit gates have been demonstrated in various quantum-dot-based qubit representations. However, as seen with small-scale demonstrations of quantum computers using other types of qubit, combining these elements leads to challenges related to qubit crosstalk, state leakage, calibration and control hardware. Here we overcome these challenges by using carefully designed control techniques to demonstrate a programmable two-qubit quantum processor in a silicon device that can perform the Deutsch-Josza algorithm and the Grover search algorithm-canonical examples of quantum algorithms that outperform their classical analogues. We characterize the entanglement in our processor by using quantum-state tomography of Bell states, measuring state fidelities of 85-89 per cent and concurrences of 73-82 per cent. These results pave the way for larger-scale quantum computers that use spins confined to quantum dots.

Quantum simulation of a Fermi-Hubbard model using a semiconductor quantum dot array.

Interacting fermions on a lattice can develop strong quantum correlations, which are the cause of the classical intractability of many exotic phases of matter. Current efforts are directed towards the control of artificial quantum systems that can be made to emulate the underlying Fermi-Hubbard models. Electrostatically confined conduction-band electrons define interacting quantum coherent spin and charge degrees of freedom that allow all-electrical initialization of low-entropy states and readily adhere to the Fermi-Hubbard Hamiltonian. Until now, however, the substantial electrostatic disorder of the solid state has meant that only a few attempts at emulating Fermi-Hubbard physics on solid-state platforms have been made. Here we show that for gate-defined quantum dots this disorder can be suppressed in a controlled manner. Using a semi-automated and scalable set of experimental tools, we homogeneously and independently set up the electron filling and nearest-neighbour tunnel coupling in a semiconductor quantum dot array so as to simulate a Fermi-Hubbard system. With this set-up, we realize a detailed characterization of the collective Coulomb blockade transition, which is the finite-size analogue of the interaction-driven Mott metal-to-insulator transition. As automation and device fabrication of semiconductor quantum dots continue to improve, the ideas presented here will enable the investigation of the physics of ever more complex many-body states using quantum dots.

Tunable Coupling and Isolation of Single Electrons in Silicon Metal-Oxide-Semiconductor Quantum Dots.

Extremely long coherence times, excellent single-qubit gate fidelities, and two-qubit logic have been demonstrated with silicon metal-oxide-semiconductor spin qubits, making it one of the leading platforms for quantum information processing. Despite this, a long-standing challenge in this system has been the demonstration of tunable tunnel coupling between single electrons. Here we overcome this hurdle with gate-defined quantum dots and show couplings that can be tuned on and off for quantum operations. We use charge sensing to discriminate between the (2,0) and (1,1) charge states of a double quantum dot and show excellent charge sensitivity. We demonstrate tunable coupling up to 13 GHz, obtained by fitting charge polarization lines, and tunable tunnel rates down to <1 Hz, deduced from the random telegraph signal. The demonstration of tunable coupling between single electrons in a silicon metal-oxide-semiconductor device provides significant scope for high-fidelity two-qubit logic toward quantum information processing with standard manufacturing.

Spin Lifetime and Charge Noise in Hot Silicon Quantum Dot Qubits.

We investigate the magnetic field and temperature dependence of the single-electron spin lifetime in silicon quantum dots and find a lifetime of 2.8 ms at a temperature of 1.1 K. We develop a model based on spin-valley mixing and find that Johnson noise and two-phonon processes limit relaxation at low and high temperature, respectively. We also investigate the effect of temperature on charge noise and find a linear dependence up to 4 K. These results contribute to the understanding of relaxation in silicon quantum dots and are promising for qubit operation at elevated temperatures.

Current-Phase Relation of Ballistic Graphene Josephson Junctions.

The current-phase relation (CPR) of a Josephson junction (JJ) determines how the supercurrent evolves with the superconducting phase difference across the junction. Knowledge of the CPR is essential in order to understand the response of a JJ to various external parameters. Despite the rising interest in ultraclean encapsulated graphene JJs, the CPR of such junctions remains unknown. Here, we use a fully gate-tunable graphene superconducting quantum intereference device (SQUID) to determine the CPR of ballistic graphene JJs. Each of the two JJs in the SQUID is made with graphene encapsulated in hexagonal boron nitride. By independently controlling the critical current of the JJs, we can operate the SQUID either in a symmetric or asymmetric configuration. The highly asymmetric SQUID allows us to phase-bias one of the JJs and thereby directly obtain its CPR. The CPR is found to be skewed, deviating significantly from a sinusoidal form. The skewness can be tuned with the gate voltage and oscillates in antiphase with Fabry-Pérot resistance oscillations of the ballistic graphene cavity. We compare our experiments with tight-binding calculations that include realistic graphene-superconductor interfaces and find a good qualitative agreement.

Side Gate Tunable Josephson Junctions at the LaAlO3/SrTiO3 Interface.

Novel physical phenomena arising at the interface of complex oxide heterostructures offer exciting opportunities for the development of future electronic devices. Using the prototypical LaAlO3/SrTiO3 interface as a model system, we employ a single-step lithographic process to realize gate-tunable Josephson junctions through a combination of lateral confinement and local side gating. The action of the side gates is found to be comparable to that of a local back gate, constituting a robust and efficient way to control the properties of the interface at the nanoscale. We demonstrate that the side gates enable reliable tuning of both the normal-state resistance and the critical (Josephson) current of the constrictions. The conductance and Josephson current show mesoscopic fluctuations as a function of the applied side gate voltage, and the analysis of their amplitude enables the extraction of the phase coherence and thermal lengths. Finally, we realize a superconducting quantum interference device in which the critical currents of each of the constriction-type Josephson junctions can be controlled independently via the side gates.

Second-Harmonic Coherent Driving of a Spin Qubit in a Si/SiGe Quantum Dot.

We demonstrate coherent driving of a single electron spin using second-harmonic excitation in a Si/SiGe quantum dot. Our estimates suggest that the anharmonic dot confining potential combined with a gradient in the transverse magnetic field dominates the second-harmonic response. As expected, the Rabi frequency depends quadratically on the driving amplitude, and the periodicity with respect to the phase of the drive is twice that of the fundamental harmonic. The maximum Rabi frequency observed for the second harmonic is just a factor of 2 lower than that achieved for the first harmonic when driving at the same power. Combined with the lower demands on microwave circuitry when operating at half the qubit frequency, these observations indicate that second-harmonic driving can be a useful technique for future quantum computation architectures.

Nuclear spin effects in semiconductor quantum dots.

The interaction of an electronic spin with its nuclear environment, an issue known as the central spin problem, has been the subject of considerable attention due to its relevance for spin-based quantum computation using semiconductor quantum dots. Independent control of the nuclear spin bath using nuclear magnetic resonance techniques and dynamic nuclear polarization using the central spin itself offer unique possibilities for manipulating the nuclear bath with significant consequences for the coherence and controlled manipulation of the central spin. Here we review some of the recent optical and transport experiments that have explored this central spin problem using semiconductor quantum dots. We focus on the interaction between 10(4)-10(6) nuclear spins and a spin of a single electron or valence-band hole. We also review the experimental techniques as well as the key theoretical ideas and the implications for quantum information science.

Spin-relaxation anisotropy in a GaAs quantum dot.

We report that the electron spin-relaxation time T_{1} in a GaAs quantum dot with a spin-1/2 ground state has a 180° periodicity in the orientation of the in-plane magnetic field. This periodicity has been predicted for circular dots as being due to the interplay of Rashba and Dresselhaus spin orbit contributions. Different from this prediction, we find that the extrema in the T_{1} do not occur when the magnetic field is along the [110] and [11[over ¯]0] crystallographic directions. This deviation is attributed to an elliptical dot confining potential. The T_{1} varies by more than 1 order of magnitude when rotating a 3 T field, reaching about 80 ms for the optimal angle. We infer from the data that in our device the signs of the Rashba and Dresselhaus constants are opposite.

Resolving spin-orbit- and hyperfine-mediated electric dipole spin resonance in a quantum dot.

We investigate the electric manipulation of a single-electron spin in a single gate-defined quantum dot. We observe that so-far neglected differences between the hyperfine- and spin-orbit-mediated electric dipole spin resonance conditions have important consequences at high magnetic fields. In experiments using adiabatic rapid passage to invert the electron spin, we observe an unusually wide and asymmetric response as a function of the magnetic field. Simulations support the interpretation of the line shape in terms of four different resonance conditions. These findings may lead to isotope-selective control of dynamic nuclear polarization in quantum dots.

Steady-state entanglement in the nuclear spin dynamics of a double quantum dot.

We propose a scheme for the deterministic generation of steady-state entanglement between the two nuclear spin ensembles in an electrically defined double quantum dot. Because of quantum interference in the collective coupling to the electronic degrees of freedom, the nuclear system is actively driven into a two-mode squeezedlike target state. The entanglement buildup is accompanied by a self-polarization of the nuclear spins towards large Overhauser field gradients. Moreover, the feedback between the electronic and nuclear dynamics leads to multistability and criticality in the steady-state solutions.

Simultaneous spin-charge relaxation in double quantum dots.

We investigate phonon-induced spin and charge relaxation mediated by spin-orbit and hyperfine interactions for a single electron confined within a double quantum dot. A simple toy model incorporating both direct decay to the ground state of the double dot and indirect decay via an intermediate excited state yields an electron spin relaxation rate that varies nonmonotonically with the detuning between the dots. We confirm this model with experiments performed on a GaAs double dot, demonstrating that the relaxation rate exhibits the expected detuning dependence and can be electrically tuned over several orders of magnitude. Our analysis suggests that spin-orbit mediated relaxation via phonons serves as the dominant mechanism through which the double-dot electron spin-flip rate varies with detuning.

Graphene at high bias: cracking, layer by layer sublimation, and fusing.

Graphene and few-layer graphene at high bias expose a wealth of phenomena due to the high temperatures reached. With in situ transmission electron microscopy, we observe directly how the current modifies the structure, and vice versa. In some samples, cracks propagate from the edges of the flakes, leading to the formation of narrow constrictions or to nanometer spaced gaps after breakdown. In other samples, we find layer-by-layer evaporation of few-layer graphene, which could be exploited for the controlled production of single layer graphene from multilayered samples. Surprisingly, we even find that two pieces of graphene that overlap can heal out at high bias and form one continuous sheet. These findings open up new avenues to structure graphene for specific device applications.

Single-shot measurement of triplet-singlet relaxation in a Si/SiGe double quantum dot.

We investigate the lifetime of two-electron spin states in a few-electron Si/SiGe double dot. At the transition between the (1,1) and (0,2) charge occupations, Pauli spin blockade provides a readout mechanism for the spin state. We use the statistics of repeated single-shot measurements to extract the lifetimes of multiple states simultaneously. When the magnetic field is zero, we find that all three triplet states have equal lifetimes, as expected, and this time is ~10 ms. When the field is nonzero, the T(0) lifetime is unchanged, whereas the T- lifetime increases monotonically with the field, reaching 3 sec at 1 T.

Driven coherent oscillations of a single electron spin in a quantum dot.

The ability to control the quantum state of a single electron spin in a quantum dot is at the heart of recent developments towards a scalable spin-based quantum computer. In combination with the recently demonstrated controlled exchange gate between two neighbouring spins, driven coherent single spin rotations would permit universal quantum operations. Here, we report the experimental realization of single electron spin rotations in a double quantum dot. First, we apply a continuous-wave oscillating magnetic field, generated on-chip, and observe electron spin resonance in spin-dependent transport measurements through the two dots. Next, we coherently control the quantum state of the electron spin by applying short bursts of the oscillating magnetic field and observe about eight oscillations of the spin state (so-called Rabi oscillations) during a microsecond burst. These results demonstrate the feasibility of operating single-electron spins in a quantum dot as quantum bits.

Generating entanglement and squeezed states of nuclear spins in quantum dots.

We present a scheme for achieving coherent spin squeezing of nuclear spin states in semiconductor quantum dots. The nuclear polarization dependence of the electron spin resonance generates a unitary evolution that drives nuclear spins into a collective entangled state. The polarization dependence of the resonance generates an area-preserving, twisting dynamics that squeezes and stretches the nuclear spin Wigner distribution without the need for nuclear spin flips. Our estimates of squeezing times indicate that the entanglement threshold can be reached in current experiments.

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

Spin is a fundamental property of all elementary particles. Classically it can be viewed as a tiny magnetic moment, but a measurement of an electron spin along the direction of an external magnetic field can have only two outcomes: parallel or anti-parallel to the field. This discreteness reflects the quantum mechanical nature of spin. Ensembles of many spins have found diverse applications ranging from magnetic resonance imaging to magneto-electronic devices, while individual spins are considered as carriers for quantum information. Read-out of single spin states has been achieved using optical techniques, and is within reach of magnetic resonance force microscopy. However, electrical read-out of single spins has so far remained elusive. Here we demonstrate electrical single-shot measurement of the state of an individual electron spin in a semiconductor quantum dot. We use spin-to-charge conversion of a single electron confined in the dot, and detect the single-electron charge using a quantum point contact; the spin measurement visibility is approximately 65%. Furthermore, we observe very long single-spin energy relaxation times (up to approximately 0.85 ms at a magnetic field of 8 T), which are encouraging for the use of electron spins as carriers of quantum information.

Strong spin-photon coupling in silicon.

Long coherence times of single spins in silicon quantum dots make these systems highly attractive for quantum computation, but how to scale up spin qubit systems remains an open question. As a first step to address this issue, we demonstrate the strong coupling of a single electron spin and a single microwave photon. The electron spin is trapped in a silicon double quantum dot, and the microwave photon is stored in an on-chip high-impedance superconducting resonator. The electric field component of the cavity photon couples directly to the charge dipole of the electron in the double dot, and indirectly to the electron spin, through a strong local magnetic field gradient from a nearby micromagnet. Our results provide a route to realizing large networks of quantum dot-based spin qubit registers.