A Quantum Computer Based on Donor-Cluster Arrays in Silicon

TL;DR

This work tackles the scalability challenge of phosphorus-donor silicon qubits by proposing a donor-cluster array where each logical qubit is encoded in a cluster of $N$ donors. Intra-cluster operations rely on NMR for single-qubit gates and ESR for multi-qubit CZ-type gates mediated by a bound electron, while inter-cluster gates exploit exchange-coupled electrons to enable conditional ESR-based entangling operations across neighboring clusters; a micromagnet creates a field gradient to mitigate frequency crowding. The authors provide analytical and numerical analyses of crosstalk and decoherence, deriving parameter regimes that yield intra-cluster fidelities around $99 ightarrow 99.5\\%$ and inter-cluster fidelities around $98\%$, and they discuss how to implement QEC codes (notably XZZX toric codes) on the resulting local all-to-all connected topology. The proposed framework offers a CMOS-compatible, randomness-tolerant pathway toward large-scale, fault-tolerant silicon quantum processors, with explicit design targets for exchange coupling, hyperfine diversity, and readout capabilities.

Abstract

Significant advances in silicon spin qubits highlight the potential of silicon quantum dots for scalable quantum computing, given their compatibility with industrial fabrication and long coherence times. In particular, phosphorus (P)-doped spin qubits possess excellent coherence and have demonstrated high-fidelity two-qubit gates exceeding 99.9%. However, scaling P-donor systems is challenging due to crosstalk caused by the uniformity of individual P donors and the low tolerance for imprecise atomic placement. Stochastic placement can lead to multiple donors located within a small region (diameter <3 nm), forming a so-called donor cluster. Notably, in cluster-based systems, high-fidelity multi-qubit quantum gates and all-to-all connectivity have recently been demonstrated experimentally on nuclear spin qubits. In this work, we propose a scalable cluster-array architecture for nuclear spin qubits and a corresponding control protocol. We analyze crosstalk-induced errors, a major error source, during primitive operations under various parameters, showing that they can be suppressed through device design and control optimization. We evaluate the fidelities of intra- and inter-cluster multi-qubit gates between nuclear spins, confirming the feasibility of our architecture and establishing design requirements and parameter targets. The local all-to-all connectivity within clusters provides unique flexibility for quantum error correction. Our scalable scheme provides a path toward large-scale spin-based quantum processors.

Figures (10)

• Figure 1: Scalable cluster-based spin qubit scheme in $^{28}$Si. Donor clusters are arranged into a two-dimensional array. Each donor cluster contains multiple P donors and one electron bound to them. Top gates (TGs) above the clusters are used to adjust the energy detuning between clusters. To achieve tunability of exchange interactions between neighboring electrons, the inter-cluster TGs is used to adjust tunneling. In a state-of-the-art device, the exchange interaction is tuned by adjusting detuning between clusters. Additionally, single-lead quantum dots (SLQDs) are incorporated between clusters for readout and loading of electrons.
• Figure 1: Scalable cluster-based spin qubit scheme in $^{28}$Si. (a) Donor clusters are arranged into a two-dimensional array. Each donor cluster contains multiple P donors and one electron bound to them. To achieve tunability of exchange interactions between neighboring electrons, the inter-cluster units are quantum dot (QDs) to induce tunable superexchange interactions. Top gates (TGs) above clusters are used to adjusting energy detuning between clusters. The inter-cluster units could be quantum dots (QDs) or TG to adjust tunneling between clusters. Additionally, single-lead quantum dots are incorporated between clusters for readout and loading of electrons.
• Figure 2: Addressability in the scalable cluster-based qubit scheme. (a) The clusters are arranged linearly with a longitudinal magnetic field gradient $\Delta B$ generated by the micromagnet. The spatial distribution of the magnetic field gradient ensures frequency discrimination between qubits in distinct donor clusters. (b) Nuclear spin qubits within a cluster are coupled to a single confined electron via hyperfine interactions. (c) The nuclear spin qubits, coupled via effective exchange interaction between electron spins, belong to adjacent 2P and 3P clusters, respectively. (d) Schematic diagram of electron spin energy levels. The spin-up and spin-down states of the electron are split under the magnetic field, with the corresponding splitting magnitude determining the ESR driving frequency. When the electron is confined to a 2P cluster, its spin energy levels undergo further splitting due to hyperfine interactions with the two donors. When the electron spin couples with the neighboring electron spin, its energy levels further split according to the nuclear spin states of the adjacent cluster. The electron in the right cluster is assumed to be initialized in the spin-down state. For simplicity, the energy levels corresponding to the spin-up state of the right electron are not included.
• Figure 2: Scalable cluster-based spin qubit scheme in $^{28}$Si. Donor clusters are arranged into a one-dimensional chain. Each donor cluster contains 2-4 P donors and one electron bound to them. To achieve tunability of exchange interactions between neighboring electrons, the inter-cluster units can be implemented either as top gates (TGs) for adjusting tunneling or as quantum dot (QDs) to induce tunable superexchange interactions. Additionally, single-lead quantum dots (SLQDs) are incorporated between clusters for readout and loading of electrons. The control gates for the clusters and the SLQDs are arranged on opposite sides (upper and lower) of the cluster chain. Moreover, a micromagnet is positioned adjacent to the cluster chain to generate magnetic field gradients, which mitigate the frequency crowding issue. (b) The infidelity $e_{\rm{Intra-CNOT}}$ of the intra-cluster CNOT-type gate is plotted as a function of the residual exchange interaction $J_{\rm{off}}$ and the hyperfine interaction $A_{0}$. (c) The infidelity $e_{\rm{Inter-Toffoli}}$ of the inter-cluster Toffoli-type gate is plotted as a function of the activated exchange interaction $J_{\rm{on}}$ and the hyperfine interaction $A_{0}$.
• Figure 3: Schematic of qubit initialization, readout, and ESR-based CZ gate. (a) Circuit for the control protocol includes qubit initialization, operation, and readout. Detailed sequences for the initialization and readout of a specific nuclear spin in each cluster are shown. Taking $\mathrm{N_{\rm{2L}}}$$\mid\Downarrow\rangle$ initialization as an example, the electron is initially prepared to $\mid\downarrow\rangle$, then all ESR $\pi$ pulses involving $\mathrm{N_{\rm{2L}}}$ are applied to flip the corresponding electron spin; followed by the NMR $\pi$ pulse conditional on the electron spin being in $\mid\downarrow\rangle$. The three steps above constitute $\mathrm{N_{\rm{2L}}}$$\mid\Downarrow\rangle$ initialization procedure. The final nuclear spin readout, also implemented by a bound electron, constitutes a quantum non-demolition (QND) measurement. This property enables repeated measurements to suppress SPAM errors. (b) The ESR frequency of the electron in the left cluster is shown as a function of the exchange interaction strength $J$. Here, a 2P-1P pair is taken as an example. Near $J=0$, the frequencies splits into four groups depending on the nuclear spin states in the left cluster: $\mid\Downarrow_{\rm{1L}}\Downarrow_{\rm{2L}}\rangle$ (red), $\mid\Uparrow_{\rm{1L}}\Downarrow_{\rm{2L}}\rangle$ (orange), $\mid\Downarrow_{\rm{1L}}\Uparrow_{\rm{2L}}\rangle$ (green), and $\mid\Uparrow_{\rm{1L}}\Uparrow_{\rm{2L}}\rangle$ (blue). The solid curves correspond to the electron spin in the right cluster being $\mid\downarrow_{\rm{R}}\rangle$, while the dashed curves represent the electron spin being $\mid\uparrow_{\rm{R}}\rangle$. The splitting between frequencies with the same color and line style depends on the nuclear spin state on the right. (c) Intra-cluster CZ operation based on a conditional 2$\uppi$ ESR rotation with $J=J_{\rm{off}}$ [(c) in Figure (b)]. (d) Inter-cluster CZ operation based on a conditional 2$\uppi$ ESR rotation with $J=J_{\rm{on}}$ [(d) in Figure (b)]. (e) The crosstalk error during 2$\uppi$ ESR rotations is plotted as a function of detuning between driving frequency and the off-resonant frequency. The insets show the time evolution of the electron spin population under a large detuning (45 MHz) and a small detuning (0.25 MHz). Here, the solid curves correspond to the target nuclear spin state, while the dashed curves represent a non-target nuclear spin state.