Quantum gates for Majoranas zero modes in topological superconductors in one-dimensional geometry

We propose and analyze a physical system capable of performing topological quantum computation with Majorana zero modes (MZMs) in a one-dimensional topological superconductor (1DTS). One of the leading methods to realize quantum gates in a 1DTS is to use T junctions, which allows one to maneuver MZMs in such a manner as to achieve braiding. In this paper, we propose a scheme for implementing quantum logical gates in a purely one-dimensional geometry without T junctions, instead replacing it with an auxiliary qubit. This has the additional benefit of introducing a non-Clifford gate, corresponding to one- and two-logical-qubit $Z$ rotations. We first design a topologically protected logical $Z$ gate based entirely on local interactions within the 1DTS. Using an auxiliary qubit coupled to the topological superconductors, we extend the $Z$ gate to non-Clifford single- and multiqubit arbitrary rotations with partial topological protection. Finally, to perform universal quantum computing, we introduce a scheme for performing arbitrary unitary rotations, albeit without topological protection. We develop a formalism based on unitary braids which creates transitions between different topological phases of the 1DTS system. The unitary formalism can be simply converted to an equivalent adiabatic scheme, which we numerically simulate, and we show that high-fidelity operations should be possible with reasonable parameters.

Figure 1

An example of a controllable one-dimensional topological superconductor (1DTS) system that is considered in this paper. A superconducting strip is placed on a semiconductor wire, which has regions in the topological phase ($T$ phase) and a normal phase ($N$ phase). Keyboard gates locally adjust the chemical potential, which changes the phase of the fermions in the superconducting strip. Junction gates are used to break the strips into regions, which at the end points have Majorana zero modes (MZMs), encoding the logical states. A coupler, which consists of a qubit coupled to two local sites, allows for braiding operations to be performed (see Sec. 4). Operations to control the local site are present to perform universal qubit gates (see Sec. 5). See Refs. [42] for an experimentally implemented scheme with a similar configuration.

Figure 2

(a) The normal phase $N$; (b) the topological phase $T$.

Figure 3

Visual notation in which rows are topological quantum states characterized by a particular pairing and transitions between rows are braids. In this example we can see the effect of $U_{(1,l)(2,l)}{|001\rangle }_T={|0\rangle }_N\otimes {|01\rangle }_T$.

Figure 4

A step-by-step procedure of the converting $T$-phase state into the $N$-phase state by conversion operator $U_c$.

Figure 5

Performing a double-braid operation (63) by (a) reversing the sign of the coupling sequentially in a chain and (b) moving a domain through a region in which the coupling parameter is reversed. The process of moving is controlled by adjusting the potential ${\mu }_j$ on particular sites indicated using circles. The level of the potential is indicated by vertical position of the curved line. The topological domains are indicated by the shaded regions and are spread over four sites indicating their current positions.

Figure 6

A process of moving the domain by a single site to the right. The extension operator $U_j^{\left(e\right)}$ extends the domain by one site, and then the retraction operator $U_{j-1}^{\left(r\right)}$ retracts the domain by one site from left to right. The overall effect is the moving operator $U_{(j-1)\left(j\right)}^{\left(m\right)}=U_{j-1}^{\left(r\right)}{U}_j^{\left(e\right)}$, which moves the entire domain by one site.

Figure 7

The Hamiltonian terms associated with the four-nanowire system, where each nanowire has a length of $M$ sites. There are two topological domains present in the system indicated by the shaded regions and stored on the two leftmost nanowires. The sites are represented by circles. The vertical position of the curved line represents the level of ${\mu }_j$ potential on each site. Between the regions we indicate breakable couplings by the open or closed links, for $s_1$ and $s_2$, respectively.

Figure 8

(a) Quantum circuit implementing a $Z_L$ rotation by an arbitrary angle $\varphi$ by acting on a coupler which controls the nanowire state $|\psi \rangle$ through controlled-$Z_L$ operation. (b) Extension to multiple logical qubits constructed using the same principles by adding additional controlled-$Z_L$ operations.

Figure 9

Moving the topological domains (shaded regions spread over sites) through the three-spin coupling (three connected circles). Shown are the corresponding sequences for the (a) adiabatic and (b) braiding formulations. In both cases the logical $Z$ gate is controlled by the coupler spin. The curved horizontal line indicates the level of ${\mu }_j$ for each site.

Figure 10

Adiabatic scheme to entangle two topological qubits. Each chain is of length $M=2$, and at the end of the sequence a $Z_1{Z}_2$ Hamiltonian is implemented. Steps 1, 5, and 6 represent the unitary operations applied to the coupler. Steps 2–4 and 6–8 represent moving the topological regions between the chains. Open circles denote fermion sites, the circle with a center dot is the coupler, the switch denotes a region with suppressed $t_j$ and ${\mathrm{\Delta }}_j$ such that the tunneling between the sites is decreased, the shaded ovals denote the topological regions, and the curved line gives the value of ${\mu }_j$ for the site.

Figure 11

(a) The adiabatic protocol and (b) the sequence of local braids or a unitary protocol, both implementing the $U_{\mathit{u}}^T\left(\varphi \right)$ operation.

Figure 12

Energy (upper graph) and the spin expectation value (lower graph) of the state plots for the $U_Z\left(\pi \right)$ gate.

Figure 13

Energy and the spin expectation value of the state plots for the $U_{{\mathbf{u}}_{\mathbf{x}}}^T\left(\pi \right)$ gate.

Figure 14

Fidelity plots vs adiabatic time constant obtained after numerically simulating the entire protocol for all three gates $U_Z^{12}\left(\varphi \right)$, $U_Z\left(\varphi \right)$, and $U_{\mathbf{u}}^T\left(\pi \right)$ with $\mathbf{u}={\mathbf{u}}_{\mathbf{x}}$. Fidelity is calculated for a series of angles $\varphi$ ranging over 16 evenly distributed angles $\varphi$ from the unit circle.