Scalable designs for quasiparticle-poisoning-protected topological quantum computation with Majorana zero modes

We present designs for scalable quantum computers composed of qubits encoded in aggregates of four or more Majorana zero modes, realized at the ends of topological superconducting wire segments that are assembled into superconducting islands with significant charging energy. Quantum information can be manipulated according to a measurement-only protocol, which is facilitated by tunable couplings between Majorana zero modes and nearby semiconductor quantum dots. Our proposed architecture designs have the following principal virtues: (1) the magnetic field can be aligned in the direction of all of the topological superconducting wires since they are all parallel; (2) topological T junctions are not used, obviating possible difficulties in their fabrication and utilization; (3) quasiparticle poisoning is abated by the charging energy; (4) Clifford operations are executed by a relatively standard measurement: detection of corrections to quantum dot energy, charge, or differential capacitance induced by quantum fluctuations; (5) it is compatible with strategies for producing good approximate magic states.

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Figure 1

An example of a scalable hexon architecture. The minimal building block defining a qubit and an ancilla are one-sided hexons, which are topological Cooper pair boxes containing six MZMs (magnified in the left panel). Note: the illustration is not drawn to scale; in practice, the length $L$ of 1DTS wires is much larger than the coherence length $\xi$ and vertical separation distances between wires are much smaller than $\xi$. The measurement of joint parities of MZMs becomes possible by selective coupling to quantum dots. The latter are defined and controlled by gates as depicted in the magnification in the right panel. Two-MZM measurements within a hexon and four-MZM measurements involving two hexons (with two MZMs from a given hexon) enable Clifford-complete operations on the array of qubits.

Figure 2

Appropriately tuning the gates shown in the magnification of the right of Fig. 1 creates the scenarios depicted in the left and right panels here. Left panel: a device configuration for measuring the two-MZM parity $p_{12}$ (eigenvalue of $i{\gamma }_1{\gamma }_2$). MZMs ${\gamma }_1$ and ${\gamma }_2$ are coupled to a single quantum dot with tunneling amplitudes $t_1$ and $t_2$, respectively. Right panel: a device configuration for measuring the four-MZM parity $p=p_{12}{p}_{34}$, where $p_{jk}$ is the eigenvalue of $i{\gamma }_j{\gamma }_k$. MZMs ${\gamma }_1,{\gamma }_3$ are tunnel coupled to the upper quantum dot, while MZMs ${\gamma }_2$ and ${\gamma }_4$ are tunnel coupled to the lower quantum dot. Both geometries can be modified to measure nonadjacent pairs of MZMs, as demonstrated in Fig. 9.

Figure 3

An example configuration for a joint parity measurement of eight MZMs involving two one-sided hexons and two coherent links (using the same legend as Fig. 2). Four of the MZMs involved in the measurement are associated with coherent links and are used to facilitate the measurement of the other four MZMs, which are associated to the hexons. The resulting measurement can provide a two-qubit entangling operation on the two hexons.

Figure 4

Energy as a function of dimensionless induced charges on the quantum dots for the system shown in the right panel of Fig. 2. Top panel: stability diagram for the decoupled system ($t_j=0$) as a function of the occupation numbers $(n_{f,1},n_{f,2})$ of the double quantum dot system in the ground state. The color scale refers to the ground-state energy, whose precise values away from zero (indicated by white) are unimportant for the current discussion. Bottom panel: the four lowest energies ${\varepsilon }_{\beta }^{\text{tot}}/E_C$ as a function of $n_{g,1}$ for $n_{g,2}=(1+h/{\varepsilon }_C)/2$ with tunneling amplitudes $t_1=0.1E_C$ and $t_{j\ne 1}=0.2E_C$. We use the parameter values $N_{g,a}=0$, ${\varepsilon }_C=10E_C$, $h=E_C/2$, and ${\varepsilon }_M=E_C/2$. For nonvanishing tunneling amplitudes, the quantum dot states $(1,0)$ and $(0,1)$ hybridize. The symmetric combination of the $(1,0)$ and $(0,1)$ states has energy ${\varepsilon }_2^{\text{tot}}$ (shown in red) and the antisymmetric combination has energy ${\varepsilon }_1^{\text{tot}}$ (shown in black). These energies ${\varepsilon }_1^{\text{tot}}$ and ${\varepsilon }_2^{\text{tot}}$ depend on the joint parity $p$ of the four MZMs; the solid curves correspond to even parity $p=1$ and dashed curves to odd parity $p=-1$. As our model only considers two quantum dot levels, the states $(0,0)$ and $(1,1)$ do not hybridize. These states have corresponding parity-independent energies ${\varepsilon }_0^{\text{tot}}$ (shown as the blue dotted-dashed curve) and ${\varepsilon }_3^{\text{tot}}$ (shown as the purple dotted-dashed curve), respectively. From the stability diagram (top panel), we see that the mutual charging energy ${\varepsilon }_M$ increases the range of $n_{g,1}$ and $n_{g,2}$ for which the parity-dependent energy ${\varepsilon }_1^{\text{tot}}$ is the ground state.

Figure 5

Average charge (in units of electron charge) on the upper quantum dot as a function of the dimensionless induced charge $n_{g,1}$ for the system shown in the right panel of Fig. 2. We assume the system is in the ground state, and plot the average charge for both even parity (solid curve) and odd parity (dashed curve). We use the parameter values $N_{g,a}=0$, ${\varepsilon }_C=10E_C$, $h=E_C/2$, and ${\varepsilon }_M=E_C/2$.

Figure 6

Differential capacitance of the ground state $C_{\text{diff}}$ (in units of $C_{\mathrm{\Sigma },D}$) as a function of the dimensionless induced charge $n_{g,1}$ for the system shown in the right panel of Fig. 2. Both even parity (solid curve) and odd parity (dashed curve) are shown. We use the parameter values $N_{g,a}=0$, ${\varepsilon }_C=10E_C$, $h=E_C/2$, ${\varepsilon }_M=E_C/2$, and $C_g/C_{\mathrm{\Sigma },D}=0.1$.

Figure 7

Diagrammatic representation of the topological states (degenerate ground states) of a hexon. The center two MZMs ${\gamma }_3$ and ${\gamma }_4$ fuse to even fermion parity, forming the ancillary pair of MZMs. The left and right pairs of MZMs both fuse to $a=0$ or 1, which correspond to even or odd fermion parity, respectively. These outer pairs of MZMs form the computational qubit. The fusion channel $a$ labels the qubit basis state.

Figure 8

Diagrammatic representation of ${\mathrm{\Pi }}_0^{\left(34\right)}{\mathrm{\Pi }}_0^{\left(13\right)}{\mathrm{\Pi }}_0^{\left(23\right)}{\mathrm{\Pi }}_0^{\left(34\right)}$ applied to the topological state of a hexon qubit. Pairs of MZMs are projected to the vacuum (even fermion parity) fusion channel to perform anyonic teleportations on the topological state space of the MZMs. The series of projections has the same effect as exchanging the positions of MZMs 1 and 2, i.e., it generates the braiding operator $R^{\left(12\right)}$. This provides a diagrammatic proof of Eq. (31), as originally given in Ref. [37].

Figure 9

An example generalizing the measurements of MZMs from Fig. 2 (using the same legend). The upper region shows a four-MZM measurement of the joint parity operator $-{\gamma }_1{\gamma }_3{\gamma }_{2^{\prime }}{\gamma }_{6^{\prime }}$. The lower region shows a two-MZM measurement of the parity operator $i{\gamma }_{2^{\prime \prime }}{\gamma }_{4^{\prime \prime }}$. The quantum dots (gray ellipses) and their couplings (yellow lines) to MZMs are defined by appropriately tuning a set of underlying gates (see Fig. 1). Note: the illustration is not drawn to scale. In practice, the length (horizontal direction on the figure) is much larger than the width $L\gg w$, so as to simultaneously optimize topological protection due to the length of the 1DTSs and suppression of QPP error rates by large charging energies. As a practical constraint, in order for the quantum dots connecting MZMs to remain coherent, the vertical separation of the MZMs connected to the same quantum dot must be shorter than the effective coherence length of that quantum dot. The same principles apply to subsequent figures.

Figure 10

A two-sided hexon architecture. Note: the illustration is not drawn to scale for the same reason as Fig. 9. The magnification shows a single two-sided hexon. Additional topological superconducting links and semiconducting structures allow appropriate measurements to manipulate and entangle two-sided hexons.

Figure 11

A linear hexon architecture. Note: the figure is not drawn to scale for the same reason as in Fig. 9. The length ${\ell }_{\text{c}}$ of the nontopological segments is much larger than the corresponding coherence length ${\xi }_{\text{c}}$ of the nontopological regions and the length ${\ell }_{\text{t}}$ of the topological segments is much larger than the coherence length $\xi$ of the topological regions. The legend used is the same as in Fig. 10. The magnification shows a single linear hexon. Additional topological superconducting links and semiconducting structures allow appropriate measurements to manipulate and entangle linear hexons.

Figure 12

Two equivalent circuits implementing the cnot gate. The control, ancillary, and target qubits are labeled $C$, $A$, and $T$, respectively. Gates labeled $H$ are Hadamard gates, the other boxes correspond to one- and two-qubit measurements as indicated by the corresponding Pauli operators. The left circuit implements a cnot up to $\{X,Z\}$ on qubits $C$ and $T$. As explained in text, the Hadamard operators can be commuted through to yield the simplified circuit shown on the right, up to {$X,Z,H$}.

Figure 13

A linear tetron architecture. Note: the illustration is not drawn to scale for the same reason as in Fig. 9. The length ${\ell }_{\text{c}}$ of the nontopological segments is much larger than the corresponding coherence length ${\xi }_{\text{c}}$ of the nontopological regions and the length ${\ell }_{\text{t}}$ of the topological segments is much larger than the coherence length $\xi$ of the topological regions. The legend used is the same as in Fig. 10. The magnification shows a single linear tetron. Additional topological superconducting links (gray) and semiconducting structures (orange) allow appropriate measurements to manipulate and entangle linear tetrons.

Figure 14

A two-sided tetron architecture. Note: the illustration is not drawn to scale for the same reason as in Fig. 9. The legend used is the same as in Fig. 10. The magnification shows a single two-sided tetron. Additional topological superconducting links and semiconducting structures allow appropriate measurements to manipulate and entangle two-sided tetrons.

Figure 15

Examples of designs for experiments that demonstrate some of the basic operating principles in our scalable quantum computing architectures. (a) Experimental test of long-distance coherent transport through floating 1DTSs. Two long wires are coated with a superconductor and tuned into the topological regime. If single electron transport is coherent, the hybridization of left and right quantum dots should show Aharonov-Bohm oscillations when changing the enclosed flux $\varphi$. (b), (c) Experimental test of QPP rate, MZM hybridization, and measurement functionality of a topological qubit. These single tetron configurations contain the minimal structure for a topological qubit. The left and right leads can be used to first tune the system into the topological regime by checking for zero-bias peaks associated with ${\gamma }_1$ and ${\gamma }_4$. The central dot allows one to measure $i{\gamma }_2{\gamma }_3$, e.g., using charge sensing. The apparatuses shown in (b) and (c) differ in fabrication details. In (b) the quantum dot is defined on the same nanowire as the MZMs in a region where the superconducting shell was etched away. A superconducting bridge joins the two superconducting shells. In (c), no superconducting bridge is needed. A nontopological region between ${\gamma }_2$ and ${\gamma }_3$ is created by etching away the semiconducting part of the nanowire, or possibly by gating. The quantum dot is defined in a nearby nanowire connected to the 1DTS composed of four joined wires [102]. Note that the distance between ${\gamma }_2$ and ${\gamma }_3$ has to be much larger than the superconducting coherence length. (d) For enhanced measurement flexibility, additional gates allow one to replace single-dot configurations by double-dot configurations.

Figure 16

Agreement between perturbative (solid black) and exact (dashed red) parity-dependent energies ${\varepsilon }_1^{\text{tot}}$ and ${\varepsilon }_2^{\text{tot}}$ in units of $E_C$. We use the parameter values $N_{g,a}=0$, $n_{g,2}=n_g^{*}$, $p=p_{12}{p}_{34}=1$, ${\varepsilon }_{C,a}=10E_C$, $h=E_C/2$, ${\varepsilon }_M=E_C/2$, $t_1=0.1E_C$, and $t_{j\ne 1}=0.2E_C$.

Figure 17

Energy ${\varepsilon }_{\beta }^{\text{tot}}$ as a function of $n_{g,1}$ for $n_{g,2}=n_g^{*}-0.01$. We use the parameter values $N_{g,a}=0$, ${\varepsilon }_C=10E_C$, $h=E_C/2$, ${\varepsilon }_M=E_C/2$, $t_1=0.1E_C$, and $t_2=0.2E_C$. The parity-independent energies ${\varepsilon }_0^{\text{tot}}$ (shown in blue dotted-dashed) and ${\varepsilon }_3^{\text{tot}}$ (shown in purple dotted-dashed). Parity-dependent energies ${\varepsilon }_1^{\text{tot}}$ (black) and ${\varepsilon }_2^{\text{tot}}$ (red) are shown with solid curves for even parity and dashed curves for odd parity.