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arXiv:2101.11592 · Physical review. B./Physical review. B

Fast high-fidelity single-qubit gates for flip-flop qubits in silicon

F. A. Calderon-Vargas, Edwin Barnes, Sophia E. Economou

Published Oct 10, 2022• 10 citations
arXivHTMLPDF

Abstract

Flip-flop qubits, encoded in the antiparallel spin states of electron and nuclear spins in phosphorus donors in silicon, are promising for quantum processors due to their long coherence times, long-range two-qubit coupling, and CMOS compatibility. However, the large Hilbert space complicates understanding and controlling the dynamics. In this work, the authors use sophisticated techniques to derive effective flip-flop qubit Hamiltonians and design optimal control pulses that implement high-fidelity single-qubit gates that are much faster and more robust than previous proposals.

DOI
10.1103/physrevb.106.165302

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Flip-flop qubits, encoded in the antiparallel spin states of electron and nuclear spins in phosphorus donors in silicon, are promising for quantum processors due to their long coherence times, long-range two-qubit coupling, and CMOS compatibility. However, the large Hilbert space complicates understanding and controlling the dynamics. In this work, the authors use sophisticated techniques to derive effective flip-flop qubit Hamiltonians and design optimal control pulses that implement high-fidelity single-qubit gates that are much faster and more robust than previous proposals.
A high degree of controllability and long coherence time make the nuclear spin of a phosphorus donor in isotopically purified silicon a promising candidate for a quantum bit. However, long-distance two-qubit coupling and fast, robust gates remain outstanding challenges for these systems. Here, following recent proposals for long-distance coupling via dipole-dipole interactions, we present a simple method to implement fast, high-fidelity arbitrary single- and two-qubit gates in the absence of charge noise. Moreover, we provide a method to make the single-qubit gates robust to moderate levels of charge noise to well within an error bound of ${10}^{\ensuremath{-}3}$.
Practical quantum computers require a large network of highly coherent qubits, interconnected in a design robust against errors. Donor spins in silicon provide state-of-the-art coherence and quantum gate fidelities, in a platform adapted from industrial semiconductor processing. Here we present a scalable design for a silicon quantum processor that does not require precise donor placement and leaves ample space for the routing of interconnects and readout devices. We introduce the flip-flop qubit, a combination of the electron-nuclear spin states of a phosphorus donor that can be controlled by microwave electric fields. Two-qubit gates exploit a second-order electric dipole-dipole interaction, allowing selective coupling beyond the nearest-neighbor, at separations of hundreds of nanometers, while microwave resonators can extend the entanglement to macroscopic distances. We predict gate fidelities within fault-tolerance thresholds using realistic noise models. This design provides a realizable blueprint for scalable spin-based quantum computers in silicon.Quantum computers will require a large network of coherent qubits, connected in a noise-resilient way. Tosi et al. present a design for a quantum processor based on electron-nuclear spins in silicon, with electrical control and coupling schemes that simplify qubit fabrication and operation.
Abstract Impurities hosted in semiconducting solid matrices represent an extensively studied platform for quantum computing applications. In this scenario, the so-called flip-flop qubit emerges as a convenient choice for scalable implementations in silicon. Flip-flop qubits are realized implanting phosphorous donor in isotopically purified silicon, and encoding the logical states in the donor nuclear spin and in its bound electron. Electrically modulating the hyperfine interaction by applying a vertical electric field causes an Electron Dipole Spin Resonance (EDSR) transition between the states with antiparallel spins {∣↓⇑⟩,∣↑⇓⟩}\{|\downarrow \Uparrow \rangle ,|\uparrow \Downarrow \rangle \}{∣↓⇑⟩,∣↑⇓⟩} {|↓⇑〉,|↑⇓〉} , that are chosen as the logical states. When two qubits are considered, the dipole-dipole interaction is exploited to establish long-range coupling between them. A universal set of quantum gates for flip-flop qubits is here proposed and the effect of a realistic 1/f noise on the gate fidelity is investigated for the single qubit Rz(−π2)R_{z}(-\frac{\pi }{2})Rz​(−2π​) Rz(−π2) and Hadamard gate and for the two-qubit iSWAP\sqrt{\mathit{iSWAP}}iSWAP​ iSWAP gate.
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.
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\sqrt{iSWAP}iSWAP​ and iSWAPiSWAPiSWAP. We then optimize the control parameters of physically realizable, electrically driven RzR_zRz​ 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.
Abstract Single electron spins bound to multi-phosphorus nuclear spin registers in silicon have demonstrated fast (0.8 ns) two-qubit SWAP\sqrt{\mathrm{SWAP}}SWAP​ SWAP gates and long spin relaxation times (~30 s). In these spin registers, when the donors are ionized, the nuclear spins remain weakly coupled to their environment, allowing exceptionally long coherence times. When the electron is present, the hyperfine interaction allows coupling of the spin and charge degrees of freedom for fast qubit operation and control. Here we demonstrate the use of the hyperfine interaction to enact electric dipole spin resonance to realize high-fidelity ( F=100−6+0F=10{0}_{-6}^{+0}F=100−6+0​ F = 10 0 − 6 + 0 %) initialization of all the nuclear spins within a four-qubit nuclear spin register. By controllably initializing the nuclear spins to ∣⇓⇓⇓⟩\left\vert \Downarrow \Downarrow \Downarrow \right\rangle∣⇓⇓⇓⟩ ⇓ ⇓ ⇓ , we achieve single-electron qubit gate fidelities of F = 99.78 ± 0.07% (Clifford gate fidelities of 99.58 ± 0.14%), above the fault-tolerant threshold for the surface code with a coherence time of $${T}_{2}^{\,* }=12\,\upmu {{{\rm{s}}}}$$ T 2 * = 12 μ s .