Transport of spin qubits with donor chains under realistic experimental conditions

The ability to transport quantum information across some distance can facilitate the design and operation of a quantum processor. One-dimensional spin chains provide a compact platform to realize scalable spin transport for a solid-state quantum computer. Here, we model odd-sized donor chains in silicon under a range of experimental nonidealities, including variability of donor position within the chain. We show that the tolerance against donor placement inaccuracies is greatly improved by operating the spin chain in a mode where the electrons are confined at the Si-${\mathrm{SiO}}_2$ interface. We then estimate the required time scales and exchange couplings, and the level of noise that can be tolerated to achieve high-fidelity transport. We also propose a protocol to calibrate and initialize the chain, thereby providing a complete guideline for realizing a functional donor chain and utilizing it for spin transport.

Figure 1

Schematic of a spin chain (gray) where the spins at the edges $({\mathrm{S}}_{c\left(1\right)}$ and ${\mathrm{S}}_{c\left(N\right)})$ are exchange coupled to source and target qubits $({\mathrm{S}}_s$ and ${\mathrm{S}}_t)$. When ${\mathrm{\Delta }}_c\gg h{\gamma }_e{B}_0$, the chain forms an extended qubit, allowing spin transport from the source to the target. The magnitude of the effective coupling between the chain and source qubits, $J_s$, approaches $J_{s\text{−}c\left(1\right)}/\sqrt{N}$ for large $N$ [24].

Figure 2

(a), (b) Schematics of electron orbital wave functions in a three-donor chain operated in the (a) bulklike and (b) interface modes. The electrons and ${}^{31}\mathrm{P}$ nuclei are represented in red and black, respectively. (c) Targeted interdonor separation ${\mu }_d$ to obtain at least 90% yield, as a function of donor placement error ${\sigma }_d$. Chains are simulated for the bulklike and interface modes for $N=3$, 5, and 7, with $B_0=1$ T. The donors in the chain are assumed to be placed along the [100] crystallographic axis. Pulling electrons to the Si-${\mathrm{SiO}}_2$ interface with a chain gate increases the robustness of the chain to donor placement inaccuracies.

Figure 3

Schematic of the system used for transport. A source donor is exchange coupled to a donor chain, which is in turn coupled to a target donor. The target donor is then linked to the remainder of the quantum processor. A single-electron transistor (SET) tunnel coupled to the source donor serves to initialize and measure the state of the source electron spin qubit.

Figure 4

ESR spectra of a donor electron coupled to a chain qubit, operated in the interface mode. The spectra are shown for the cases where measurement of only (a) the donor electron or (b) the chain qubit is possible. (a) and (b) are used to calibrate $J_s$ and $J_t$, respectively.

Figure 5

(a) Pulsing scheme for the adiabatic spin transport protocol. (b) Sketch of the time evolution of the eigenenergies $(y$ axis not to scale). The approximate eigenstates are labeled at the start and end of the protocol. The $|\uparrow \rangle$ and $|\downarrow \rangle$ states are transported via the two labeled adiabatic passages. They belong to two independent blocks of the Hamiltonian as grouped by the shaded boxes.

Figure 6

(a) Protocol for initializing the chain and target qubits into the near-singlet $|{\phi }_{S_{ct}}\rangle$ state of chain and target qubits. (b) Initialization error as a function of the product of $T_{\mathrm{SP}}$ and $J_{\max }$, for three values of $\mathrm{\Delta }{B}_z/J_{\max }$. The fidelities are approximately given by Eq. (8).

Figure 7

Error of the transport protocol as a function of $J_{\max }{T}_{\mathrm{AP}}/h$, assuming $\mathrm{\Delta }{B}_z=0$ and $J_{\min }=0$. The envelope of the transport error in the adiabatic regime is given by Eq. (9) (dashed line). The protocol is highly robust to errors in $J_{\max }$ and $T_{\mathrm{AP}}$.

Figure 8

The transport error due to (a) $\mathrm{\Delta }{B}_z$ normalized to $J_{\max }$, and (b) limited tunability $(J_{\min }/J_{\max })$ of exchange couplings. These are obtained from numerical simulations of the adiabatic protocol for $J_{\max }{T}_{\mathrm{AP}}/h\approx {10}^1,{10}^2$, and ${10}^3$. The cutoff value of $\mathrm{\Delta }{B}_z/J_{\max }$ such that $\mathrm{\Delta }{B}_z$ does not affect the transport fidelity is given in Eq. (11). The dashed line in (b) is the error due to “spin leakage” as described by Eq. (12).

Figure 9

Calculations of error due to noise added to (a) ${\epsilon }_s,{\epsilon }_c$, and ${\epsilon }_t$ and (b) $J_s$ and $J_t$. The errors are plotted as a function of the product of the power spectral density of the white noise added $(S_{\delta {\epsilon }_i}$ and $S_{\delta J})$ and $T_{\mathrm{AP}}$. We assume $\mathrm{\Delta }{B}_z/h=100$ MHz. The solid lines are obtained from numerical simulations of the adiabatic protocol for $J_{\max }{T}_{\mathrm{AP}}/h\approx {10}^1,{10}^2$, and ${10}^3$. The dashed purple lines correspond to the error due to the loss of adiabaticity for $J_{\max }{T}_{\mathrm{AP}}/h\approx {10}^3$. The dashed magenta lines are fits of the total error due to the noise given by Eqs. (14) and (15). The dashed gray line in panel (a) shows the error due to dephasing.

Figure 10

Tight-binding calculation of the exchange coupling as a function of donor separation, for the bulklike (blue) and interface (green) modes of operation modes. Insets: schematics of the electron orbital wave functions for the two operation modes, with the electrons and ${}^{31}\mathrm{P}$ nuclei represented in red and black, respectively.

Figure 11

(a) Electron spin resonance (ESR) frequencies for a three-donor chain operated in the bulklike mode, with $B_0=1$ T and $J_c/h=1$ THz. (b) ESR spectrum of a donor coupled to the edge of an $N=3$ donor chain. (c) ESR spectrum of an $N=3$ chain qubit coupled to a single donor, i.e., assuming that the chain, not the donor, is being measured.

Figure 12

Bloch sphere in the adiabatic frame at the (a) start and (b) end of the singlet initialization protocol. The red dot represents the instantaneous state. The initial state of the system is oriented along the $z$ axis. In the adiabatic limit, the “circle of precession” (dashed red circle) around the eigenstate $|{\mathrm{\Phi }}_{1_{\mathrm{SP}}}\left(t\right)\rangle$ (blue arrow) follows the evolution of $|{\mathrm{\Phi }}_{1_{\mathrm{SP}}}\left(t\right)\rangle$ and does not change in diameter.