Noise Threshold and Resource Cost of Fault-Tolerant Quantum Computing with Majorana Fermions in Hybrid Systems

Fault-tolerant quantum computing in systems composed of both Majorana fermions and topologically unprotected quantum systems, e.g., superconducting circuits or quantum dots, is studied in this Letter. Errors caused by topologically unprotected quantum systems need to be corrected with error-correction schemes, for instance, the surface code. We find that the error-correction performance of such a hybrid topological quantum computer is not superior to a normal quantum computer unless the topological charge of Majorana fermions is insusceptible to noise. If errors changing the topological charge are rare, the fault-tolerance threshold is much higher than the threshold of a normal quantum computer and a surface-code logical qubit could be encoded in only tens of topological qubits instead of about 1,000 normal qubits.

Figure 1

(a) A hybrid platform of implementing the surface code of MF qubits. Measurement TUQSs and PP TUQSs are coupled to one-dimensional semiconducting wires. MFs are created and moved on the wire network by adjusting gate voltages [39]. Measurement and PP operations are performed by coupling MFs to corresponding TUQSs. See Ref. [41] for details of implementing the surface code on such a platform. (b) The surface code of MF qubits. Each blue square denotes a qubit encoded in four MFs $\{s,x,y,z\}$: the crossed circle denotes $s$, and solid circles, respectively, denote $x$, $y$, and $z$ in a clockwise direction. Each red (green) octagon corresponds to an $X$ ($Z$) stabilizer, which is the product of Pauli operators ${\sigma }^x$ (${\sigma }^z$) of four surrounding qubits; i.e., the stabilizer equals the product of the fermion parity of eight MFs on the octagon and fermion parities of two qubits (marked with dashed lines) [41]. Measuring stabilizers needs ancillary MFs, which are not shown in this figure.

Figure 2

(a) The measurement of eight-MF fermion parity $c_0{c}_1\dots c_7$. (b) An effective PP with the partial-error detection. (c) An effective PP with the full-error detection. (d) An effective PP using an entangled state. In (b)–(d), $c_0{c}_1{c}_2{c}_3$ is measured. MFs without a number are ancillaries. Each purple square denotes a PP, each red pipe with outgoing arrows denotes the creation of a pair of MFs $a$ and $b$ in the eigenstate of $iab$, and each green pipe with incoming arrows denotes the measurement of $iab$. See Ref. [41] for details.

Figure 3

(a) Thresholds of the MF-based fault-tolerant quantum computing. On each curve, from left to right, the ratios $p_F/p_B$ of marks are 0, ${10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$, ${10}^{-1}$, 1, $\infty$, respectively; i.e., empty marks on the vertical (horizontal) axis correspond to $p_F=0$ ($p_B=0$). Assisted by detecting qubit-charge errors, the threshold is higher than only detecting Pauli errors. When $p_F\ll p_B$, the threshold can be improved by using distilled PPs, where the distillation is only performed for one round. (b) Number of qubits for encoding a logical qubit to achieve the logical-qubit error rate $p_L$ (per round of stabilizer measurements). The fidelity of PPs is $\sim 99.9\%$, i.e., $p_B=0.02\%$ when $p_F/p_B=1$ and $p_B=0.1\%$ when $p_F/p_B={10}^{-1}$, ${10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$. For normal qubits, stabilizer measurements are implemented using controlled- not gates with the fidelity 99.9% [61]. From top to bottom, the solid lines, respectively, correspond to $p_F/p_B=1$, ${10}^{-1}$, and the dashed lines, respectively, correspond to $p_F/p_B={10}^{-2}$, ${10}^{-3}$, ${10}^{-4}$. Data are obtained numerically. See Ref. [41] for more data and details.