Universal quantum computation with hybrid spin-Majorana qubits

We theoretically propose a set of universal quantum gates acting on a hybrid qubit formed by coupling a quantum-dot spin qubit and Majorana fermion qubit. First, we consider a quantum dot that is tunnel coupled to two topological superconductors. The effective spin-Majorana exchange facilitates a hybrid cnot gate for which either qubit can be the control or target. The second setup is a modular scalable network of topological superconductors and quantum dots. As a result of the exchange interaction between adjacent spin qubits, a cnot gate is implemented that acts on neighboring Majorana qubits and eliminates the necessity of interqubit braiding. In both setups, the spin-Majorana exchange interaction allows for a phase gate, acting on either the spin or the Majorana qubit, and for a swap or hybrid swap gate which is sufficient for universal quantum computation without projective measurements.

Figure 1

(a) Setup of two TSCs (red bars) furnishing two MFs ($\mathsf{X}$'s) on the left TSC, ${\gamma }_l^{\prime }$ and ${\gamma }_l$, and two MFs on the right TSC, ${\gamma }_r^{\prime }$ and ${\gamma }_r$, in contact with a nontopological SC (gray); the MFs on each TSC can overlap, causing a splitting $\delta$. Between the two TSCs is a quantum dot (blue disk) with two single-electron levels of up, ${\epsilon }_{\uparrow }$, and down, ${\epsilon }_{\downarrow }$, spin. The MFs are coupled to the dot through the tunneling elements $t_{\nu }$ and $t_{\nu }^{\prime }$, where $\nu$ labels the right ($r$) and left ($l$) TSCs. (b) MaSH network of TSCs where a grid of hybrid qubits (red and gray crosses) are long-distance coupled by tunably connecting the spin-1/2 quantum dots, with strength $\mathcal{J}$, via floating gates (solid yellow lines), e.g., hybrid qubit (1) is coupled to hybrid qubit (2). Braiding of MFs utilizes the T junctions of the TSCs on each hybrid qubit, for instance, hybrid qubit (3). Note that the T junctions are isolated units without tunnel connection to each other.

Figure 2

Schematic of the MF qubit formed by a left and right TSC (red bars) tunnel coupled by a quantum dot (blue disk) with spin-1/2 ground state and some of the processes that result from the coupling between spin and MF qubit dictated by ${\mathcal{H}}_T$. The odd parity of the TSCs is indicated by a straight line (white) between the MFs (white crosses), i.e., $f_{\nu }^{†}{f}_{\nu }=1$. The four MF states give rise to two types of MF qubits: (a) the degenerate odd-parity states of the MF qubit (even total system parity) with one fermion on the left TSC (left panel) and one on the right TSC (right panel); (b) the degenerate even-parity states of the MF qubit (odd total system parity) with no fermions on either TSC (left panel) and with one fermion on each TSC (right panel). (c),(d) The virtual processes described by ${\mathcal{H}}_s$; the remaining undepicted processes are similar but take place on the one, three, and four total electron state. (e),(f) The transfer of an electron from one TSC to the other due to ${\mathcal{H}}_o$. (g),(h) The processes determined by ${\mathcal{H}}_e$ that map the system between the two states in the even-parity sector of the MF qubit. The other processes resulting from ${\mathcal{H}}_o$ and ${\mathcal{H}}_e$ are obtained by exchanging the right and left TSCs (or initial and final states) in (c)–(h).

Figure 3

Schematic of the hybrid swap (hswap) gate obtained as follows: apply the hcnot gate using, say, the spin qubit (SQ) as the control and the MF qubit (MQ) as the target qubit; apply the hcnot gate, reversing the roles of the qubits; apply the hcnotgate with the control and target qubits as in the first operation. Starting with the initial state $|x,y\rangle$ such that $x,y\in \{0,1\}$, where we identify 0 (1) with the $\left|\downarrow \right.〉$ ($\left|\uparrow \right.〉$) and $|l\rangle$ ($|r\rangle$) state of the spin and MF qubit, respectively, by applying the pictured gate sequence, one obtains $|x,y\rangle \to {(-1)}^x|x,y\oplus x\rangle \to {(-1)}^y|y,y\oplus x\rangle \to |y,x\rangle$; this results in a coherent swap of states between the spin and MF qubit.

Figure 4

Implementation of the necessary gates for universal quantum computation: (a) Hadamard and $\pi /4$-phase gate as a result of braiding, (b) $\pi /8$-phase gate obtained by coupling the MF qubit and the fixed spin qubit, and (c) cnot gate obtained through an effective coupling of two MF qubits facilitated by a long-range interaction between the corresponding spin qubits.