Fault-tolerant quantum random-number generator certified by Majorana fermions

Braiding of Majorana fermions gives accurate topological quantum operations that are intrinsically robust to noise and imperfection, providing a natural method to realize fault-tolerant quantum information processing. Unfortunately, it is known that braiding of Majorana fermions is not sufficient for the implementation of universal quantum computation. Here we show that topological manipulation of Majorana fermions provides the full set of operations required to generate random numbers by way of quantum mechanics and to certify its genuine randomness through violation of a multipartite Bell inequality. The result opens a perspective to apply Majorana fermions for the robust generation of certified random numbers, which has important applications in cryptography and other related areas.

Figure 1

Plot of the function $f\left(\widehat{L}\right)$ vs violation $\widehat{L}$ of the MABK inequality. The function is calculated through optimization based on the semidefinite programming, with the details shown in Appendix . Inset (a) shows the lower bound of the min-entropy $kf({\mathcal{L}}_m-\epsilon )-{\log }_2\frac{1}{\delta }$ vs the number of trials $k$. Here we assume an observed MABK violation lies within the interval $3.9={\mathcal{L}}_m\le \widehat{L}<{\mathcal{L}}_{\text{max}}=4$ with probability $\delta$. The parameters are chosen as $\delta =0.001$ and ${\epsilon }^{\prime }=0.01$. The bound $kf({\mathcal{L}}_m-\epsilon )$ depends on the input probability distribution $P\left(xyz\right)$ through the parameter $r={\min }_{xyz}P\left(xyz\right)$. The blue square line represents the bound under a uniform distribution [$P\left(xyz\right)=1/4$ for all $(x,y,z)\in \mathcal{S}$], while the red dotted line shows the bound under a biased probability distribution with $P\left(011\right)=P\left(101\right)=P\left(110\right)=\alpha k^{-1/2}$ and $P\left(000\right)=1-3\alpha k^{-1/2}$ with $\alpha =10$. It consumes less randomness to generate a biased distribution for the input bits, so the net amount of randomness, defined as the number of output random bits minus that of the input, becomes positive when $k$ is large (typically $k$ needs to be of the order ${10}^5$). Inset (b) plots the net amount of randomness generated after $k$ trails under a biased distribution of the inputs. The parameters are the same as those in inset (a).

Figure 2

Illustration of the encoding scheme for a logic qubit using Majorana fermions and two single-qubit operations that can be implemented through anyon braiding. Each qubit is encoded by four Majorana fermions. (a) A counterclockwise braiding of Majorana fermions 2 and 3 implements a unitary gate $B_{23}$ on the corresponding qubit. (b) Implementation of the Hadamard gate through composition of anyon braiding. In both (a) and (b), time flows from left to right and $\simeq$ means equal up to an irrelevant overall phase.