Donor qubits in bulk doped silicon have many competitive advantages for quantum computation in the solid state: not only do they offer a fast way to scalability, but they also show some of the longest coherence times found in any quantum computation proposal. We determine the densities of entangling gates in randomly doped silicon comprising two different dopant species. First, we define conditions and plot maps of the relative locations of the dopants necessary for them to form exchange interaction-mediated entangling gates. Second, using nearest neighbor Poisson point process theory, we calculate the doping densities necessary for maximal densities of single and dual-species gates. Third, using the moving-average cluster expansion technique, we make predictions for a proof of principle experiment demonstrating the control of the far-from-equilibrium magnetization dynamics of one species by the orbital excitation of another. We find agreement of our results with a Monte Carlo simulation that handles multiple donor structures and scales optimally with the number of dopants. The simulator can also extract donor structures not captured by our Poisson point process theory. The combined approaches to density optimization in random distributions presented here may be useful for other condensed matter systems as well as applications outside physics.
Donor qubits in bulk doped silicon have many competitive advantages for quantum computation in the solid state: not only do they offer a fast way to scalability, but they also show some of the longest coherence times found in any quantum computation proposal. We determine the densities of entangling gates in randomly doped silicon comprising two different dopant species. First, we define conditions and plot maps of the relative locations of the dopants necessary for them to form exchange interaction-mediated entangling gates. Second, using nearest neighbor Poisson point process theory, we calculate the doping densities necessary for maximal densities of single and dual-species gates. Third, using the moving-average cluster expansion technique, we make predictions for a proof of principle experiment demonstrating the control of the far-from-equilibrium magnetization dynamics of one species by the orbital excitation of another. We find agreement of our results with a Monte Carlo simulation that handles multiple donor structures and scales optimally with the number of dopants. The simulator can also extract donor structures not captured by our Poisson point process theory. The combined approaches to density optimization in random distributions presented here may be useful for other condensed matter systems as well as applications outside physics.
Spin qubits of donors in silicon show some of the longest coherence times recorded and hold the promise of scalability with seamless integration of quantum computing into semiconductor fabrication. However, current entangling gate implementations rely on highly challenging precision donor placement and substantial gate tuning, rendering scaling challenging. In this work, we show how to implement placement-insensitive ultrafast entangling gates in deep and shallow donors in silicon by relying on the Rydberg blockade induced by strong interactions between low lying orbital excited states. We calculate multivalley interactions for several donor species using the Finite Element Method, and show that Van der Waals interactions between donors, calculated here for the first time, are important even for low-lying excited states. We show that blockade entangling gate operation is possible within the short lifetime of donor excited states with over 99% fidelity for the creation of a Bell state in the presence of decoherence. This gate substantially relaxes the requirements on single donor placement while allowing for billions of entangling gate operations within the single-qubit lifetime, thus paving the way for large scale quantum computation in silicon.
Several solid-state quantum information processor proposals are based on silicon [16-18, 20, 52], including the original architecture conceived by Kane in 1998 [22], in which the quantum information was stored in the spin states of 31P donors. The interest raised by silicon in the field of quantum information is twofold. On the one hand, it represents the main semiconductor used for large- scale fabrication. On the other hand, it owns interesting physical properties suitable for preserving andmanipulating quantum states, such as long coherence times and relatively low disturbance of nuclear spins, which are further enhanced by employing purified 28Si. The employment of individual donors in silicon covers all the aspects of quantum information encoding and processing, like quantumbit (qubit) storage, implementation of quantum logic gates, coherent transfer of quantum states, and readout schemes. This chapter is devoted to the physics of few dopants in nanometric silicon devices and their applications to quantum information processing purposes. In the first part, an overview of the physics of themost common donors and acceptors in silicon is presented. Next, the quantum information key concepts and their implementation in donor-based architectures are discussed. Finally, decoherence effects and the concept of measurement of quantum states are presented for the case of electron states bound to donors in silicon. Between the sixties and the early eighties of the twentieth century, all the theoretical and experimental aspects related to the doping of bulk silicon have been investigated and explained. A silicon crystal consists of a diamond lattice constituted by two interleaved cubic lattices (face-centered cubic, FCC), the second of which has the origin in the center of the tetrahedron given by the origin (0,0,0) and the centers of the faces 100. Each Si atom has four valence electrons, which create a covalent bond with the shared electrons of four other Si atoms. The substitution of silicon atoms with atoms of a group V element (indicated as donors, typically P, As, and Sb in silicon) generates an n-doped silicon crystal (n-Si). Each donor introduces an eccess electron in the crystal, and it provides new energy levels in the band gap, close to the conduction band edge. Similarly, substituting silicon atoms with atoms of a group III element (an acceptor, typically B in silicon) generates a p-doped silicon crystal (p-Si). Each acceptor introduces an electron hole in the crystal, and it provides new energy levels in the band gap, close to the valence band. The diffusion of impurities in semiconductors alters the conduction properties from an insulating to a metallic regime. Depending on the different doping concentration, the wavefunction of the electron (hole) states introduced by each donor (acceptor) may overlap negligibly or substantially with those of neighboring sites. Such a transition from low doping to high doping as a function of the average distance between neighboring sites is described in terms of an Anderson-Mott transition and produces additional impurity bands (Hubbard bands) below the conduction band edge at sufficiently high concentration [1, 34]. The transport is governed by mechanisms based on localized states at low density, while it is based on delocalized states at high density. Four conventional regimes of impurity concentration are defined [50]. The dilute concentration of impurities holds for n < 1 · 1016 cm−3. There, the problem of the donor is that of a hydrogen atom with a scaled Rydberg and radius. Between the densities n of 1 ·1016 cm−3 and 2 · 1017 cm−3 (rNN = 11.9 nm), respectively, the regime is called semidilute, and it is characterized by formation of pairs. Above 2 · 1017 cm−3 and below the metallic behavior, which occurs at 3.7 · 1018 cm−3, the regime is called intermediate, and the formation of random clusters leads to effects generally accounted by the Hubbard band formation. Above n > 3.7 · 1018 cm−3, silicon is treated as a metal. In this section the physics of the first three regimes of donor concentration and the Anderson-Mott transition are described. The Anderson-Mott transition has been observed down to microscopic scale by employing arrays of few deterministically implanted As ions (see chapter 5) in Si transistors [39]. Analogous arguments and treatment can be given to acceptor concentration regimes.
Abstract Donor spins in silicon have achieved record values of coherence times and single-qubit gate fidelities. The next stage of development involves demonstrating high-fidelity two-qubit logic gates, where the most natural coupling is the exchange interaction. To aid the efficient design of scalable donor-based quantum processors, we model the two-electron wave function using a full configuration interaction method within a multi-valley effective mass theory. We exploit the high computational efficiency of our code to investigate the exchange interaction, valley population, and electron densities for two phosphorus donors in a wide range of lattice positions, orientations, and as a function of applied electric fields. The outcomes are visualized with interactive images where donor positions can be swept while watching the valley and orbital components evolve accordingly. Our results provide a physically intuitive and quantitatively accurate understanding of the placement and tuning criteria necessary to achieve high-fidelity two-qubit gates with donors in silicon.
Silicon-based donor flip-flop qubits offer a promising path toward scalable, fault-tolerant quantum computing by combining the long coherence times of nuclear spins with fast, fully electrical control and long-range dipole-dipole coupling between qubits. However, realizing high-fidelity entangling operations in this platform remains challenging. The entangling interaction is intrinsically coupled to electron orbital dynamics, which can lead to leakage into non-computational states and unwanted phase accumulation. Furthermore, in multi-qubit architectures, residual dipolar couplings from spectator qubits distort the effective interaction landscape. In this work, we employ a numerical simulation framework, FlipFlopQSim, that models the spin-orbital dynamics of interacting flip-flop qubits to extract effective logical operations from realistic electrical control pulses. Using Makhlin invariants, we map the entangling landscape generated by electrically controlled dipole-dipole interactions and identify operating regions locally equivalent to canonical two-qubit gates, such as iSWAP and iSWAP. We then optimize the control parameters of physically realizable, electrically driven Rz rotations to implement the necessary local corrections and maximize composite gate fidelity. Finally, we scale our analysis to multi-qubit registers with various geometries and connectivity patterns to evaluate spectator-induced distortions. Our results demonstrate that high-fidelity entangling operations cannot be optimized in isolation; rather, they require a co-design approach that simultaneously optimizes pulse control, local phase compensation, and physical device geometry. This work provides a robust numerical framework for assessing the scalability of electrically controlled silicon quantum processors and outlines key design principles for robust multi-qubit gate implementation.
Scalable quantum processors require high-fidelity universal quantum logic operations in a manufacturable physical platform. Donors in silicon provide atomic size, excellent quantum coherence and compatibility with standard semiconductor processing, but no entanglement between donor-bound electron spins has been demonstrated to date. Here we present the experimental demonstration and tomography of universal one- and two-qubit gates in a system of two weakly exchange-coupled electrons, bound to single phosphorus donors introduced in silicon by ion implantation. We observe that the exchange interaction has no effect on the qubit coherence. We quantify the fidelity of the quantum operations using gate set tomography (GST), and we use the universal gate set to create entangled Bell states of the electrons spins, with fidelity 91.3 ± 3.0%, and concurrence 0.87 ± 0.05. These results form the necessary basis for scaling up donor-based quantum computers. Donors spins in silicon are coherent, high-performance qubits, but scale-up has been challenging. Here the authors present the first experimental demonstration of exchange-based, entangling two qubit gates between electrons bound to 31P donors in Si.
Gate control of donor electrons near interfaces is a generic ingredient of donor-based quantum computing. Here, we address the question: how is the phonon-assisted qubit relaxation time T 1 affected as the electron is shuttled between the donor and the interface? We focus on the example of the 'flip-flop qubit' (Tosi et al arXiv:1509.08538v1), defined as a combination of the nuclear and electronic states of a phosphorus donor in silicon, promising fast electrical control and long dephasing times when the electron is halfway between the donor and the interface. We theoretically describe orbital relaxation, flip-flop relaxation, and electron spin relaxation. We estimate that the flip-flop qubit relaxation time can be of the order of 100 μs, 8 orders of magnitude shorter than the value for an on-donor electron in bulk silicon, and a few orders of magnitude shorter (longer) than the predicted inhomogeneous dephasing time (gate times). All three relaxation processes are boosted by (i) the nontrivial valley structure of the electron-phonon interaction, and (ii) the different valley compositions of the involved electronic states.