Milestones Toward Majorana-Based Quantum Computing

We introduce a scheme for preparation, manipulation, and read out of Majorana zero modes in semiconducting wires with mesoscopic superconducting islands. Our approach synthesizes recent advances in materials growth with tools commonly used in quantum-dot experiments, including gate control of tunnel barriers and Coulomb effects, charge sensing, and charge pumping. We outline a sequence of milestones interpolating between zero-mode detection and quantum computing that includes (1) detection of fusion rules for non-Abelian anyons using either proximal charge sensors or pumped current, (2) validation of a prototype topological qubit, and (3) demonstration of non-Abelian statistics by braiding in a branched geometry. The first two milestones require only a single wire with two islands, and additionally enable sensitive measurements of the system’s excitation gap, quasiparticle poisoning rates, residual Majorana zero-mode splittings, and topological-qubit coherence times. These pre-braiding experiments can be adapted to other manipulation and read out schemes as well.

Popular Summary

The quest for solid-state devices harboring exotic excitations known as Majorana zero modes is under way in laboratories across the world. Majorana modes promise to reveal new facets of quantum mechanics that can be harnessed for fault-tolerant “topological” quantum information processing—potentially leading to scalable quantum computing hardware whose capabilities far exceed those of classical systems. Given numerous observations supporting the onset of Majorana modes in setups engineered from well-understood components, Majorana control and eventual applications seem within reach. Here, we develop a possible road map based on a series of milestone experiments, achievable in relatively simple devices, that exposes foundational properties of Majorana modes directly relevant for quantum computation.

Our approach builds on widely studied hybrid nanowire-superconductor Majorana platforms. We introduce a new all-electrical control and readout scheme for such devices that leverages tools that have long been successfully deployed in quantum-dot and spin-qubit studies. Utilizing these capabilities, we elucidate precise protocols for (i) detecting the nontrivial “fusion rules” that quantify the fate of two initially well-separated Majorana modes brought together, (ii) validating the topological protection of quantum information encoded by a prototype Majorana-based qubit, and (iii) braiding the positions of Majorana modes to reveal the highly exotic form of quantum exchange statistics that they underpin. Notably, the first two milestones require single-nanowire setups already available in the laboratory and can also be adapted to other platforms as a natural precursor to braiding.

Successful demonstration of these experiments would verify the basic tenets underlying topological quantum information schemes and potentially establish Majorana-based qubits as viable components for quantum computing hardware.

Figure 1

Semiconducting wire coated with a superconducting island and bulk SC that are bridged by a gate-tunable “valve.” The valve controls the carrier density in the barrier region and thereby modulates the ratio of Josephson energy $E_J$ to the island charging energy $E_C$. In addition, a back gate at voltage $V_g$ tunes the charge on the island. (a) When the valve is open ($E_J\gg E_C$) an applied magnetic field $\mathbf{B}$ drives the wire into a topological superconducting state hosting Majorana zero modes ${\gamma }_{1,2}$. The system thus supports degenerate ground states with even and odd fermion parity. (b) Closing the valve ($E_J\ll E_C$) restores charging energy and converts these parity eigenstates into nondegenerate states with island charges $Q_o$ and $Q_e$, as shown on the right. The gate-controlled “parity-to-charge conversion” illustrated here is central to the manipulation and read out schemes developed in this paper.

Figure 2

Energy levels versus back gate voltage $V_g$ for the single-island setup in Fig. 1 with the gate-tunable valve (a) fully open, (b) partially transmitting, and (c) fully closed. Red and blue curves correspond to states with even and odd electron number. For the open configuration in (a), Josephson energy $E_J$ dominates charging energy $E_C$. The island thus hosts a pair of Majorana zero modes that encode an approximate twofold topological ground-state degeneracy. Residual charging energy splits the degeneracy, but only by an amount $\mathrm{\Delta }E$ exponentially small in $E_J/E_C$. Closing the valve effectively ramps up Coulomb effects, which dominate in the fully closed configuration of (c). There, the eigenstates are generally nondegenerate (except with fine-tuning) and carry well-defined island charges $\dots Q-1,Q,Q+1\dots$, as labeled on the right. Crucially, the topologically degenerate ground states smoothly evolve into nondegenerate charge eigenstates as the valve closes—signifying parity-to-charge conversion. The red and blue dots illustrate this phenomenon at one particular gate voltage indicated by the vertical dashed line.

Figure 3

Geometries utilized for the proposed milestone experiments. (a)–(d) Two-island setup with gate-tunable valves that electrically tune the Josephson-to-charging-energy ratios for each pair of adjacent superconducting regions. Various interesting regimes are possible depending on the valve configurations: When cut off from the outer bulk superconductors, the islands can (a) form independent double quantum dots or (b) effectively combine into a single dot. Restoring the connection to the outer superconductors quenches charging energy and generates topological superconductivity on the islands. Either two or four Majorana zero modes (denoted by $\times$’s) appear depending on the middle-valve configuration as shown in (c) and (d). (e) Trijunction geometry that enables braiding through similar valve manipulations.

Figure 4

Alternative two-island setup involving trivial superconducting wires forming gate-tunable Josephson junctions with conventional bulk superconductors. This shunt-junction configuration is expected to more efficiently quench the island charging energy in the open-valve configuration shown here. Indeed, the vertical wires can support $N\gg 1$ channels (unlike the horizontal wire hosting Majorana modes), thus enhancing the maximum Josephson energy $E_J$.

Figure 5

Correspondence between non-Abelian statistics and nontrivial fusion rules for Ising anyons $\sigma$ that obey $\sigma \times \sigma =I+\psi$. Top: Each of the four Ising anyons shown binds a Majorana zero mode. With fixed total parity the system then admits two degenerate ground states ${\psi }_{i=1,2}$. Adiabatically braiding a pair of anyons sends ${\psi }_i\to U_{ij}{\psi }_j$, where $U_{ij}$ denotes the braid matrix. Bottom: Alternatively fusing the pair of anyons hybridizes the associated Majorana modes and splits the twofold ground-state degeneracy. The lower and upper states correspond to the Ising anyons coalescing into the identity $(I)$ and fermion $(\psi )$ fusion channels. While fusion and braiding are intimately linked, establishing the former experimentally is expected to be much simpler; see Figs. 6 and 7.

Figure 6

Protocol to detect the Ising-anyon fusion rule $\sigma \times \sigma =I+\psi$ via charge sensing. The device contains two superconducting islands (red) Josephson coupled to outer trivial bulk superconductors (blue); intervening gate-tunable valves control the Josephson-to-charging-energy ratios. The system is initialized into a unique ground state by opening the middle valve while closing the outer valves. In the control experiment (right path), the middle valve closes before opening the outer valves, thus nucleating out of the vacuum two pairs of Ising anyons binding Majorana modes ${\gamma }_{1,2}$ and ${\gamma }_{3,4}$, as indicated by dashed lines. Reclosing the outer valves fuses these same pairs back into the vacuum, corresponding to the $I$ fusion channel. Charge read out on the islands is correspondingly deterministic. Instead, closing the middle valve after opening the outer valves (left path) first nucleates Ising anyons binding ${\gamma }_{1,4}$ and later ${\gamma }_{2,3}$. Closing the outer valves now fuses the anyons in a nontrivial way that accesses both the $I$ and $\psi$ fusion channels. This leads to an equal-amplitude superposition of states in the measurement basis and thus probabilistic charge read out.

Figure 7

(a) Nontrivial cycle that selectively pumps charge $2e$ whenever Ising anyons fuse into the $\psi$ channel, yielding a quantized zero-bias dc current [Eq. (22)] passing between the outer grounded superconductors. The sequence initially follows the left path from Fig. 6. Once the anyons nontrivially fuse into a superposition of $I$ and $\psi$ channels (bottom configuration), the cycle probabilistically pumps charge by pursuing the manipulations sketched in Fig. 8 and described in the text. Opening the middle valve then restores the islands to their original state, whereupon the process repeats. (b) Control cycle that initially follows the right path from Fig. 6 but is otherwise identical to (a). This alternate sequence blocks the $\psi$ fusion channel and hence eliminates the current. The two cycles sketched here reveal the nontrivial Ising-anyon fusion rules via a macroscopic signal.

Figure 8

Manipulations used to pump charge in the cycle sketched in Fig. 7. Each panel displays the energies versus gate voltage $V_{g,L/R}$ on the left or right islands. The sequence begins from the bottom configuration of Fig. 7, wherein the anyons fuse to yield a superposition of charge states $|Q_L,Q_R⟩$ ($I$ channel) and $|Q_L-1,Q_R+1⟩$ ($\psi$ channel). Panel (a) denotes these charge states with circles and also indicates the initial gate voltages $V_{L/R}$. Next, one opens the outer valves, leading to the energies in (b), and then sweeps the gate voltages to $V_{L/R}^{\prime }$. Closing the outer valves again and subsequently restoring the initial gate voltages as shown in (c) completes the desired pumping: $|Q_L,Q_R⟩$ evolves trivially while the excited state $|Q_L-1,Q_R+1⟩$ changes to $|Q_L+1,Q_R-1⟩$. The latter evolution indicates charge $2e$ flowing between the outer superconductors when the fusion channel $\psi$ appears.

Figure 9

(a) Topological qubit and (b) reference nontopological “Andreev qubit” that we seek to sharply distinguish. For the former, the qubit is stored in the topologically degenerate ground states encoded by Majorana zero modes; in the latter, the occupation numbers for “accidental” near-zero-energy Andreev levels specify the qubit states. Because of the very different sensitivity to local noise sources, we argue that the topological qubit’s coherence times and oscillation frequency exhibit nontrivial scaling relations that are generically absent in the Andreev qubit and can be used to identify topological protection in a modest pre-braiding experiment.

Figure 10

Protocol for measuring the topological qubit’s dephasing time $T_2$ and oscillation frequency ${\omega }_0$. The left side illustrates the valve manipulations while the right shows the corresponding qubit orientation on the Bloch sphere. After initialization and a $\pi /2$ pulse, the qubit unitarily evolves for time $t$. During this evolution, the relative phase between $|0⟩$ and $|1⟩$ changes by an amount $\delta \varphi (t)$ set by ${\omega }_0$ and time-dependent noise in the splitting of the qubit states. Crucially, the latter two factors are intimately related for the topological qubit because they are constrained by the same exponential suppression. After a second $\pi /2$ pulse, the probability of measuring the $|1⟩$ state is $P(t)=|\beta {|}^2=\frac{1}{2}[1+\cos \delta \varphi (t)]$. Noise averaging gives Eq. (29), which simultaneously probes both $T_2$ and ${\omega }_0$. The nontrivial connection between uniform quantities and fluctuations for the topological qubit implies the unusual scaling relation between ${\omega }_0$ and $1/T_2$ in Eq. (31) that signifies topological protection. A similar scaling relation links $T_1$ and $T_2$; see Eq. (33).

Figure 11

(a) Elementary braid of Majorana zero modes in a trijunction hosting a single topological qubit. Modulating the three valves adjacent to the vertical island controls both charging effects and the coupling between neighboring topological superconductors. Steps (1)–(3) use this flexibility to braid the inner Majorana modes ${\gamma }_2$ and ${\gamma }_3$, thus nontrivially rotating the topological qubit via the non-Abelian braid matrix in Eq. (35). (b) Experimental protocols for verifying “rigid” qubit rotation induced by braiding. Beginning from the ground state of decoupled, charging-energy-dominated islands, the topological qubit is initialized into the $|0⟩$ state by opening the outer valves. A single braid (upper path) or double braid (lower path) rotates the topological qubit, and then charging energy is restored for read out. Under the single braid the qubit rotates into an equal superposition of $|0⟩$ and $|1⟩$ leading to maximally uncertain measurement outcome; the double braid flips the qubit to $|1⟩$ and returns a unique measurement value.

Figure 12

Potential $V(\phi )$ from Eq. (b3), normalized by $E_J$ defined in Eq. (b4), at three different transmissions $T$. At $T\ll 1$, the potential recovers the usual cosine form familiar from standard SIS junctions, while larger $T$ stiffens the potential compared to this case. Such stiffening yields relatively minor qualitative effects on the low-lying level structure and charge screening for the junction, even at perfect transmission where $T=1$.