To push gate performance to levels beyond the thresholds for quantum error correction, it is important to characterize the error sources occurring on quantum gates. However, the characterization of non-Markovian error poses a challenge to current quantum process tomography techniques. Fast Bayesian Tomography (FBT) is a self-consistent gate set tomography protocol that can be bootstrapped from earlier characterization knowledge and be updated in real-time with arbitrary gate sequences. Here we demonstrate how FBT allows for the characterization of key non-Markovian error processes. We introduce two experimental protocols for FBT to diagnose the non-Markovian behavior of two-qubit systems on silicon quantum dots. To increase the efficiency and scalability of the experiment-analysis loop, we develop an online FBT software stack. To reduce experiment cost and analysis time, we also introduce a native readout method and warm boot strategy. Our results demonstrate that FBT is a useful tool for probing non-Markovian errors that can be detrimental to the ultimate realization of fault-tolerant operation on quantum computing.

Commercially viable quantum computing will require large numbers of high-quality qubits. Therefore, leveraging the extensive capabilities of conventional CMOS technology is expected to accelerate the development of a fault-tolerant quantum computer. Here we report on the highest single-qubit control fidelity of 99.9% for any Si/SiO2 spin qubit fabricated in a 300 mm wafer process. This result represents an important milestone towards a large-scale silicon quantum processor on a single chip.

Silicon spin qubits promise to leverage the extraordinary progress in silicon nanoelectronic device fabrication over the past half century to deliver large-scale quantum processors. Despite the scalability advantage of using silicon technology, realising a quantum computer with the millions of qubits required to run some of the most demanding quantum algorithms poses several outstanding challenges, including how to control many qubits simultaneously. Recently, compact 3D microwave dielectric resonators were proposed as a way to deliver the magnetic fields for spin qubit control across an entire quantum chip using only a single microwave source. Although spin resonance of individual electrons in the globally applied microwave field was demonstrated, the spins were controlled incoherently. Here we report coherent Rabi oscillations of single electron spin qubits in a planar SiMOS quantum dot device using a global magnetic field generated off-chip. The observation of coherent qubit control driven by a dielectric resonator establishes a credible pathway to achieving large-scale control in a spin-based quantum computer.

Quantum computing, communications, sensing, and simulations are radically transformative technologies, with great potential to impact industries and economies. Worldwide, national governments, industries, and universities are moving to create a new class of workforce—the Quantum Engineers. Demand for such engineers is predicted to be in the tens of thousands within a five-year timescale, far exceeding the rate at which the world’s universities can produce Ph.D. graduates in the discipline. How best to train this next generation of engineers is currently a matter of debate. Quantum mechanics—long a pillar of traditional physics undergraduate degrees—must now be merged with traditional engineering offerings. This article discusses the history, development, and the first year of operation of the world’s first undergraduate degree in quantum engineering to be grown out of an engineering curriculum. The main purpose of this article is to inform the wider discussion, now being held by many institutions worldwide, on how best to formally educate the Quantum Engineer.

Benchmarking and characterising quantum states and logic gates is essential in the development of devices for quantum computing. We introduce a Bayesian approach to self-consistent process tomography, called fast Bayesian tomography (FBT), and experimentally demonstrate its performance in characterising a two-qubit gate set on a silicon-based spin qubit device. FBT is built on an adaptive self-consistent linearisation that is robust to model approximation errors. Our method offers several advantages over other self-consistent tomographic methods. Most notably, FBT can leverage prior information from randomised benchmarking (or other characterisation measurements), and can be performed in real time, providing continuously updated estimates of full process matrices while data is acquired.

Quantum computing based on spins in the solid state allows for densely-packed arrays of quantum bits. While high-fidelity operation of single qubits has been demonstrated with individual control pulses, the operation of large-scale quantum processors requires a shift in paradigm towards global control solutions. Here we report the experimental implementation of a new type of qubit protocol - the SMART (Sinusoidally Modulated, Always Rotating and Tailored) protocol. As with a dressed qubit, we resonantly drive a two-level system with a continuous microwave field, but here we add a tailored modulation to the dressing field to achieve increased robustness to detuning noise and microwave amplitude fluctuations. We implement this new protocol to control a single spin confined in a silicon quantum dot and confirm the optimal modulation conditions predicted from theory. Universal control of a single qubit is demonstrated using modulated Stark shift control via the local gate electrodes. We measure an extended coherence time of $2$ ms and an average Clifford gate fidelity $>99$ $\%$ despite the relatively long qubit gate times ($>15$ $\unicode[serif]{x03BC}$s, $20$ times longer than a conventional square pulse gate), constituting a significant improvement over a conventional spin qubit and a dressed qubit. This work shows that future scalable spin qubit arrays could be operated using global microwave control and local gate addressability, while maintaining robustness to relevant experimental inhomogeneities.

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.

A fault-tolerant quantum processor may be configured using stationary qubits interacting only with their nearest neighbours, but at the cost of significant overheads in physical qubits per logical qubit. Such overheads could be reduced by coherently transporting qubits across the chip, allowing connectivity beyond immediate neighbours. Here we demonstrate high-fidelity coherent transport of an electron spin qubit between quantum dots in isotopically-enriched silicon. We observe qubit precession in the inter-site tunnelling regime and assess the impact of qubit transport using Ramsey interferometry and quantum state tomography techniques. We report a polarization transfer fidelity of 99.97% and an average coherent transfer fidelity of 99.4%. Our results provide key elements for high-fidelity, on-chip quantum information distribution, as long envisaged, reinforcing the scaling prospects of silicon-based spin qubits.

Quantum computers are expected to outperform conventional computers in several important applications, from molecular simulation to search algorithms, once they can be scaled up to large numbers—typically millions—of quantum bits (qubits)1–3. For most solid-state qubit technologies—for example, those using superconducting circuits or semiconductor spins—scaling poses a considerable challenge because every additional qubit increases the heat generated, whereas the cooling power of dilution refrigerators is severely limited at their operating temperature (less than 100 millikelvin)4–6. Here we demonstrate the operation of a scalable silicon quantum processor unit cell comprising two qubits confined to quantum dots at about 1.5 kelvin. We achieve this by isolating the quantum dots from the electron reservoir, and then initializing and reading the qubits solely via tunnelling of electrons between the two quantum dots7–9. We coherently control the qubits using electrically driven spin resonance10,11 in isotopically enriched silicon12 28Si, attaining single-qubit gate fidelities of 98.6 per cent and a coherence time of 2 microseconds during ‘hot’ operation, comparable to those of spin qubits in natural silicon at millikelvin temperatures13–16. Furthermore, we show that the unit cell can be operated at magnetic fields as low as 0.1 tesla, corresponding to a qubit control frequency of 3.5 gigahertz, where the qubit energy is well below the thermal energy. The unit cell constitutes the core building block of a full-scale silicon quantum computer and satisfies layout constraints required by error-correction architectures8,17. Our work indicates that a spin-based quantum computer could be operated at increased temperatures in a simple pumped 4He system (which provides cooling power orders of magnitude higher than that of dilution refrigerators), thus potentially enabling the integration of classical control electronics with the qubit array18,19. A scalable silicon quantum processor unit cell made of two qubits confined to quantum dots operates at about 1.5 K, achieving 98.6% single-qubit gate fidelities and a 2 μs coherence time.

Once the periodic properties of elements were unveiled, chemical behaviour could be understood in terms of the valence of atoms. Ideally, this rationale would extend to quantum dots, and quantum computation could be performed by merely controlling the outer-shell electrons of dot-based qubits. Imperfections in semiconductor materials disrupt this analogy, so real devices seldom display a systematic many-electron arrangement. We demonstrate here an electrostatically confined quantum dot that reveals a well defined shell structure. We observe four shells (31 electrons) with multiplicities given by spin and valley degrees of freedom. Various fillings containing a single valence electron—namely 1, 5, 13 and 25 electrons—are found to be potential qubits. An integrated micromagnet allows us to perform electrically-driven spin resonance (EDSR), leading to faster Rabi rotations and higher fidelity single qubit gates at higher shell states. We investigate the impact of orbital excitations on single qubits as a function of the dot deformation and exploit it for faster qubit control.

Single-electron spin qubits employ magnetic fields on the order of 1 Tesla or above to enable quantum state readout via spin-dependent-tunnelling. This requires demanding microwave engineering for coherent spin resonance control, which limits the prospects for large scale multi-qubit systems. Alternatively, singlet-triplet readout enables high-fidelity spin-state measurements in much lower magnetic fields, without the need for reservoirs. Here, we demonstrate low-field operation of metal-oxide-silicon quantum dot qubits by combining coherent single-spin control with high-fidelity, single-shot, Pauli-spin-blockade-based ST readout. We discover that the qubits decohere faster at low magnetic fields with T_2^ Rabi =18.6 μs and T_2^* =1.4 μs at 150 mT. Their coherence is limited by spin flips of residual 29 Si nuclei in the isotopically enriched 28 Si host material, which occur more frequently at lower fields. Our finding indicates that new trade-offs will be required to ensure the frequency stabilization of spin qubits, and highlights the importance of isotopic enrichment of device substrates for the realization of a scalable silicon-based quantum processor.

Implementing high-fidelity quantum control and reducing the effect of the coupling between a quantum system and its environment is a major challenge in developing quantum information technologies. Here, we show that there exists a geometrical structure hidden within the time-dependent Schr\"odinger equation that provides a simple way to view the entire solution space of pulses that suppress noise errors in a system's evolution. In this framework, any single-qubit gate that is robust against quasistatic noise to first order corresponds to a closed three-dimensional space curve, where the driving fields that implement the robust gate can be read off from the curvature and torsion of the space curve. Gates that are robust to second order are in one-to-one correspondence with closed curves whose projections onto three mutually orthogonal planes each enclose a vanishing net area. We use this formalism to derive new examples of dynamically corrected gates generated from smooth pulses. We also show how it can be employed to analyze the noise-cancellation properties of pulses generated from numerical algorithms such as GRAPE. A similar geometrical framework exists for quantum systems of arbitrary Hilbert space dimension.

Universal quantum computation will require qubit technology based on a scalable platform1, together with quantum error correction protocols that place strict limits on the maximum infidelities for one- and two-qubit gate operations2,3. Although various qubit systems have shown high fidelities at the one-qubit level4-10, the only solid-state qubits manufactured using standard lithographic techniques that have demonstrated two-qubit fidelities near the fault-tolerance threshold6 have been in superconductor systems. Silicon-based quantum dot qubits are also amenable to large-scale fabrication and can achieve high single-qubit gate fidelities (exceeding 99.9 per cent) using isotopically enriched silicon11,12. Two-qubit gates have now been demonstrated in a number of systems13-15, but as yet an accurate assessment of their fidelities using Clifford-based randomized benchmarking, which uses sequences of randomly chosen gates to measure the error, has not been achieved. Here, for qubits encoded on the electron spin states of gate-defined quantum dots, we demonstrate Bell state tomography with fidelities ranging from 80 to 89 per cent, and two-qubit randomized benchmarking with an average Clifford gate fidelity of 94.7 per cent and an average controlled-rotation fidelity of 98 per cent. These fidelities are found to be limited by the relatively long gate times used here compared with the decoherence times of the qubits. Silicon qubit designs employing fast gate operations with high Rabi frequencies16,17, together with advanced pulsing techniques18, should therefore enable much higher fidelities in the near future.

Spin qubits created from gate-defined silicon metal–oxide–semiconductor quantum dots are a promising architecture for quantum computation. The high single qubit fidelities possible in these systems, combined with quantum error correcting codes, could potentially offer a route to fault-tolerant quantum computing. To achieve fault tolerance, however, gate error rates must be reduced to below a certain threshold and, in general, correlated errors must be removed. Here we show that pulse engineering techniques can be used to reduce the average Clifford gate error rates for silicon quantum dot spin qubits down to 0.043%. This represents a factor of three improvement over state-of-the-art silicon quantum dot devices and extends the randomized benchmarking coherence time to 9.4 ms. By including tomographically complete measurements in our randomized benchmarking, we infer a higher-order feature of the noise called the unitarity, which measures the coherence of noise. This, in turn, allows us to theoretically predict that average gate error rates as low as 0.026% may be achievable with further pulse improvements. These spin qubit fidelities are ultimately limited by incoherent noise, which we attribute to charge noise from the silicon device structure or the environment. Pulse engineering techniques can be used to reduce the average Clifford gate error rates for silicon quantum dot spin qubits down to 0.043%, a factor of three improvement over state-of-the-art silicon devices.

Preserving coherence long enough to perform meaningful calculations is one of the major challenges on the pathway to large scale quantum computer implementations. Noise coupled from the environment is the main contributing factor to decoherence but can be mitigated via engineering design and control solutions. However, this is only possible after acquiring a thorough understanding of the dominant noise sources and their spectrum. In this paper, we employ a silicon quantum dot spin qubit as a metrological device to study the noise environment experienced by the qubit. We compare the sensitivity of this qubit to electrical noise with that of an implanted phosphorus donor in silicon qubit in the same environment and measurement set-up. Our results show that, as expected, a quantum dot spin qubit is more sensitive to electrical noise than a donor spin qubit due to the larger Stark shift, and the noise spectroscopy data shows pronounced charge noise contributions at intermediate frequencies (2-20 kHz).

Valence band holes confined in silicon quantum dots are attracting significant attention for use as spin qubits. However, experimental studies of single-hole spins have been hindered by challenges in fabrication and stability of devices capable of confining a single hole. To fully utilize hole spins as qubits, it is crucial to have a detailed understanding of the spin and orbital states. Here we show a planar silicon metal-oxide-semiconductor-based quantum dot device and demonstrate operation down to the last hole. Magneto-spectroscopy studies show magic number shell filling consistent with the Fock–Darwin states of a circular two-dimensional quantum dot, with the spin filling sequence of the first six holes consistent with Hund’s rule. Next, we use pulse-bias spectroscopy to determine that the orbital spectrum is heavily influenced by the strong hole–hole interactions. These results provide a path towards scalable silicon hole-spin qubits.

The performance requirements for fault-tolerant quantum computing are very stringent. Qubits must be manipulated, coupled, and measured with error rates well below 1%. For semiconductor implementations, silicon quantum dot spin qubits have demonstrated average single-qubit Clifford gate error rates that approach this threshold, notably with error rates of 0.14% in isotopically enriched $^{28}$Si/SiGe devices. This gate performance, together with high-fidelity two-qubit gates and measurements, is only known to meet the threshold for fault-tolerant quantum computing in some architectures when assuming that the noise is incoherent, and still lower error rates are needed to reduce overhead. Here we experimentally show that pulse engineering techniques, widely used in magnetic resonance, improve average Clifford gate error rates for silicon quantum dot spin qubits to 0.043%,a factor of 3 improvement on previous best results for silicon quantum dot devices. By including tomographically complete measurements in randomised benchmarking, we infer a higher-order feature of the noise called the unitarity, which measures the coherence of noise. This in turn allows us to theoretically predict that average gate error rates as low as 0.026% may be achievable with further pulse improvements. These fidelities are ultimately limited by Markovian noise, which we attribute to charge noise emanating from the silicon device structure itself, or the environment.

We identify the presence of monatomic steps at the Si/SiGe or Si/SiO2 interface as a dominant source of variations in the dephasing time of silicon (Si) quantum dot (QD) spin qubits. First, using atomistic tight-binding calculations we show that the g-factors and their Stark shifts undergo variations due to these steps. We compare our theoretical predictions with experiments on QDs at a Si/SiO2 interface, in which we observe significant differences in Stark shifts between QDs in two different samples. We also experimentally observe variations in the g-factors of one-electron and three-electron spin qubits realized in three neighboring QDs on the same sample, at a level consistent with our calculations. The dephasing times of these qubits also vary, most likely due to their varying sensitivity to charge noise, resulting from different interface conditions. More importantly, from our calculations we show that by employing the anisotropic nature of the spin-orbit interaction (SOI) in a Si QD, we can minimize and control these variations. Ultimately, we predict that the dephasing times of the Si QD spin qubits will be anisotropic and can be improved by at least an order of magnitude, by aligning the external dc magnetic field towards specific crystal directions, given other decoherence mechanisms do not dominate over charge noise.

Silicon quantum dot spin qubits provide a promising platform for large-scale quantum computation because of their compatibility with conventional CMOS manufacturing and the long coherence times accessible using ^28Si enriched material. A scalable error-corrected quantum processor, however, will require control of many qubits in parallel, while performing error detection across the constituent qubits. Spin resonance techniques are a convenient path to parallel two-axis control, while Pauli spin blockade can be used to realize local parity measurements for error detection. Despite this, silicon qubit implementations have so far focused on either single-spin resonance control, or control and measurement via voltage-pulse detuning in the two-spin singlet-triplet basis, but not both simultaneously. Here, we demonstrate an integrated device platform incorporating a silicon metal-oxide-semiconductor double quantum dot that is capable of single-spin addressing and control via electron spin resonance, combined with high-fidelity spin readout in the singlet-triplet basis.