Pareto-optimality of Majoranas in hybrid platforms

TL;DR

This work addresses the optimization of Majorana qubits in two hybrid platforms—proximitized nanowires and quantum dot chains—by casting the problem as a multi-objective optimization between the topological gap $E_\text{gap}$ and the localization length $\xi$. Using perturbative and transfer-matrix analyses, the authors derive effective Hamiltonians for the QD chain up to fourth order and map the NW and QD systems onto Pareto fronts to identify optimal trade-offs. They find that, while both platforms can achieve similar quality in some regimes, QD chains can realize regimes with $\xi$ below 100 nm and finite $E_\text{gap}$, thanks to tunable disorder profiles and long-range couplings, whereas NWs are more vulnerable to disorder and require longer devices. The results imply that near-term quantum computing architectures could benefit from tuned QD chains, though achieving optimal performance requires iterative tuning as chain length increases and long-range effects emerge.

Abstract

To observe Majorana bound states, and especially to use them as a qubit, requires careful optimization of competing quality metrics. We systematically compare Majorana quality in proximitized semiconductor nanowires and quantum dot chains. Using multi-objective optimization, we analyze the fundamental trade-offs between topological gap and localization length, two key metrics that determine MBS coherence and operational fidelity. We demonstrate that these quantities cannot be simultaneously optimized in realistic models, creating Pareto frontiers that define the achievable parameter space. Our results show that QD chains achieve both comparable quality as nanowires and a regime with a much shorter localization length, making them particularly promising for near-term quantum computing applications where device length and disorder are limiting factors.

Figures (8)

• Figure 1: Illustration of the MOO problem and the Pareto front. The filled regions represent all the possible combinations of $E_\text{gap}$ and $\xi$ for a given platform, and the solid lines represent the Pareto front. (a) A single platform's Pareto front with two solutions with an optimal error rate for two choices of $(T_1, L_1)$ and $(T_2, L_2)$. The dashed lines indicate the quadrant where all solutions are strictly better for any choice of $L$ and $T$. (b) The Pareto fronts of three different platforms.
• Figure 2: Majorana quality in the topological phase of the NW model given by Eq. \ref{['eq:HNW']}. (a) Topological gap $E_\text{gap}$ as a function of the chemical potential $\mu$ and the magnetic field $B$. (b) Localization length $\xi$ as a function of the chemical potential $\mu$ and the magnetic field $B$. The red line is the Pareto front, and its end points---the orange and blue dots are the parameters that optimize $\xi$ or $E_\text{gap}$, respectively. (c) Band structure of the two extreme cases given by the orange and blue dots in panels (a) and (b). We use the parameters listed in Table \ref{['tab:params']} as $\text{InSbAs}$ with $m_\text{eff}/m_e=0.0162$, $g_\text{sm}=6.8$, $\alpha=20$ nm meV, and $\Delta=0.2$ meV.
• Figure 3: (a) Schematic of the quantum dot chain with $N$ quantum dots (blue) connected by $N-1$ ABS (orange) via hopping with amplitude $t$. Quantum dots have chemical potential $\mu_D$ and ABS have chemical potential $\mu_A$. (b) Schematic of the effective Hamiltonian for the quantum dot chain with position-dependent chemical potential, nearest-neighbor CAR and ECT, and long-range couplings (purple lines).
• Figure 4: Majorana quality in the topological phase of the QD chain given by Eq. \ref{['eq:h_dot_abs']} in the weak-coupling limit. (a) Topological gap $E_\text{gap}$ as a function of the coupling strength $t$ and the rescaled and recentered chemical potential $\tilde{\mu}$ such that $\mu_D = (\tilde{\mu} + \delta\mu^{(2)})t^2$. (b) Localization length $\xi$ as a function of the coupling strength $t$ and the rescaled and recentered chemical potential $\tilde{\mu}$. Red region in panels (a) and (b) show the Pareto front for the QD chain pymoo. The blue and orange dots indicate the parameters that optimize $\xi$ or $E_\text{gap}$, respectively. The purple line indicates the two-site sweet spot condition $\mu_D = E_Z$.
• Figure 5: Comparison of the Pareto fronts between the QD chain (solid lines) and the NW model (dashed lines) for three different material parameter configurations listed in Table \ref{['tab:params']}. Both models include renormalization effects as described in Appendix \ref{['app:renormalization']}. The distribution parameters of the Pareto front and their cut-offs are discussed in detail in Appendix \ref{['app:distribution']}.