Quantum computing on magnetic racetracks with flying domain wall qubits

Domain walls (DWs) on magnetic racetracks are at the core of the field of spintronics, providing a basic element for classical information processing. Here, we show that mobile DWs also provide a blueprint for large-scale quantum computers. Remarkably, these DW qubits showcase exceptional versatility, serving not only as stationary qubits, but also performing the role of solid-state flying qubits that can be shuttled in an ultrafast way. We estimate that the DW qubits are long-lived because they can be operated at sweet spots to reduce potential noise sources. Single-qubit gates are implemented by moving the DW, and two-qubit entangling gates exploit naturally emerging interactions between different DWs. These gates, sufficient for universal quantum computing, are fully compatible with current state-of-the-art experiments on racetrack memories. Further, we discuss possible strategies for qubit readout and initialization, paving the way toward future quantum computers based on mobile topological textures on magnetic racetracks.

Figure 1

Sketch of two DW qubits on two parallel ferrimagnetic racetracks. Spins stand for the uncompensated magnetic moments of the sublattices. The spin texture of the right racetrack has a well-defined positive chirality and is in state $|⥀\rangle$, whereas the texture within the left one has negative chirality and is in state $|⥁\rangle$. Single-qubit gates are implemented by shuttling the domain wall and controlling its velocity $v\left(t\right)$. Two-qubit entangling gates between different racetracks take advantage of intertrack exchange and dipolar interactions.

Figure 2

The DW qubit. (a) The effective potential energy $V\left(\mathrm{\Phi }\right)$. Two states with opposite chirality, which span the Hilbert space of the DW qubit, are localized at two minima of the potential. They are hybridized by the tunneling $t_g$ through the barrier $V_0$ that is controlled by $B_y$ and $K_y$. The detuning $\varepsilon$ of the two minima is proportional to $B_x$. (b) Energy dispersion of the Hamiltonian (3) against $\varepsilon$. A finite value of $B_x$ favors states with a well-defined chirality. (c) The physical qubit splitting $\tilde{\mathrm{\Omega }}$ as a function of the dimensionless magnetic field $b_x$ and $b_y$ with zero shuttling velocity. The red star marks the qubit operational point used for estimations. Here, we used the parameters given in Table 1.

Figure 3

DW qubit performance. (a) Single- and two-qubit gate time as a function of the shuttling velocity $v$. (b) Qubit relaxation $T_1$ and dephasing $T_2$ time as function of $v$ at different temperatures $T$. (c) The relaxation rate $T_1^{-1}$ of a stationary qubit as a function of dimensionless magnetic fields $b_x,b_y$. (d) The dephasing rate $T_2^{-1}$ of a stationary qubit as a function of dimensionless magnetic fields $b_x,b_y$. The red star marks the working point of the qubit used in our estimations.

Figure 4

Construction of the computational space of DW qubits. (a). The potential energy of the collective coordinate $\mathrm{\Phi }$. The dashed gray curve shows the case of zero magnetic field $b_y=0$, see Eq. (B1). The green curve corresponds to a finite magnetic field $b_y=0.15$. We take $b_x=0$ for both cases. Because of $b_y$, the tunneling barrier at $\mathrm{\Phi }=\pi /2+2\pi \mathbb{Z}$ is suppressed, whereas the barrier at $-\pi /2+2\pi \mathbb{Z}$ is increased. (b) A detuning $\varepsilon$ due to a finite $b_x$, where we set $b_x=0.025$.

Figure 5

Two-qubit interaction Hamiltonian. A top view of two domain walls sitting on two parallel magnetic racetracks with a distance $D\left(t\right)$ from each other. One domain wall is moving with a velocity $v\left(t\right)$. A two-qubit gate can be achieved as this domain wall passes by the other one.

Figure 6

Coherence time of DW qubits. (a) $T_2$ as a function of dimensionless magnetic field $b_x$ and shuttling velocity $v$. (b) $T_1$ as a function of $b_x$ and $v$. (c) The dephasing rate $T_2^{-1}$ as a function of $b_x$ and $b_y$. (d) The relaxation rate $T_1^{-1}$ as a function of $b_x$ and $b_y$. We fixed $b_y=0.15$ in (a) and (b). In (c) and (d), we fixed the velocity to be $v=10\text{m/s}$.

Figure 7

Qubit readout via a paramagnetic dot. When the dot is comparable with the domain-wall size, electrons with similar directions would tunnel to the dot, leading to the formation of a ferromagnetic domain. When we interpret the magnetization direction of the domain in the right (left) hemisphere as positive (negative) chirality state, a 75%-reliable measurement of the chirality is obtained.