Figure 1

Two circuits that can demonstrate non-Abelian statistics, by the initialization, braiding, and measurement of pairs of Majorana bound states (circles). Braiding is performed twice to flip the fermion parity of ${\gamma }_A$ and ${\gamma }_B$ (Ref. 13). Majoranas that can be coupled by Coulomb charging energy are connected by a thin line; the line is solid if the Majoranas are strongly coupled and dashed if they are uncoupled. A thick line indicates tunnel coupling of Majoranas. The $T$-shaped circuit of Ref. 11 (left column) requires control over tunnel couplings, while the $\pi$-shaped circuit considered here (right column) does not, because both readout and braiding involve a Majorana localized at a $T$ junction.Reuse & Permissions

Figure 2

(a) Minimal circuit for flux-controlled demonstration of non-Abelian Majorana statistics. Two large superconducting plates form a Cooper pair box in a transmission line resonator, i.e., a transmon qubit. Three smaller superconducting islands are embedded between the two transmon plates. Each superconducting island contains a nanowire supporting two Majorana bound states. At low energies, the three overlapping Majorana bound states at a $T$ junction form a single zero mode so that effectively the system hosts six Majorana bound states, labeled ${\gamma }_A$, ${\gamma }_B$, ${\gamma }_C$, ${\gamma }_D$, ${\gamma }_E$, and ${\gamma }_F$. The Coulomb couplings between the Majorana fermions can be controlled with magnetic fluxes ${\Phi }_k$. This hybrid device can measure the result of the braiding operation as a shift in the microwave resonance frequency when the fermion parity $i{\gamma }_A{\gamma }_B$ switches between even and odd. (b) Sequence of variation of fluxes during the initialization (steps 0–2), braiding (steps 3–8), and measurement (step 9). (c) Illustration of the steps required for initialization, braiding, and measurement. Fusion channels of pairs of Majorana fermions colored red, blue, and white are chosen to be the basis states in Eq. (4). To unambiguously demonstrate the non-Abelian nature of Majoranas, one needs to collect statistics of measurement outcomes when the adiabatic cycle describing the braiding operation (steps 3–8) is repeated $n$ times between initialization and measurement. The probabilities of observing changes in the cavity's resonance frequency $p_{\mathrm{flip}}$ for different values of $n$ should obey the predictions summarized in the table in (c). The sequence of probabilities shown in the table repeats itself periodically for larger values of $n$.Reuse & Permissions

Figure 3

(a) Minimal transmon circuit for fully flux-controlled topological qubit. The nanowires are placed in a triangular loop formed out of three $T$ junctions (Ref. 23). In this geometry, all single-qubit Clifford gates can be implemented. (b) Schematic overview of a Random Access Majorana Memory consisting of eight topological qubits. Compensating fluxes (dotted circles) are included between the topological qubits to ensure that the gauge-invariant phase differences in the different topological qubits are independent of each other (see Appendix ).Reuse & Permissions

Figure 4

Quantum circuits for universal quantum computation in the RAMM. In this figure, $p_1,p_2,p_3=\pm 1$ represent results of projective single- or multiqubit measurements, whose outcomes, carried by classical channels (double lines), determine postselected unitary operations. (a) cnot gate. Here $R_1=\exp \left[i\frac{\pi }{4}{\sigma }_x(1-p_1)\right]$, $R_2=\exp \left[i\frac{\pi }{4}{p}_2{p}_3{\sigma }_z\right]$, $R_3=\exp \left[i\frac{\pi }{4}{p}_2{p}_3{\sigma }_x\right]$, $R_4=\exp [-i\frac{\pi }{4}{p}_3{\sigma }_x]$ are all gates obtainable by braidings. (b) $\pi /8$ phase gate $T=\mathrm{diag}(1,\exp i\frac{\pi }{4})$, relying on distillation of the state $|A\rangle =\left(\right|0\rangle +\exp i\frac{\pi }{4}|1\rangle )/\sqrt{2}$. The required unitary operations are in this case $R_{\psi }=\exp [-i\frac{\pi }{8}{\sigma }_z(1-p_1)]$ and $R_A=R_1$. (c) Teleportation protocol. Here $R=\exp \left[i\frac{\pi }{4}{\sigma }_z(1-p_1{p}_2)\right]\exp \left[i\frac{\pi }{4}{\sigma }_x(1-p_3)\right]$. Apart from teleporting the unknown quantum state $|\psi \rangle$, the protocol leaves the remaining two qubits in an entangled Bell state $|\Psi \rangle$.Reuse & Permissions

Figure 5

Preparation of a nine-qubit 2D cluster state with a RAMM. The nine qubits (represented by circles) are arranged in a $3\times 3$ square logical lattice, and numbered from left to right and top to bottom. (a) The nine stabilizer operators $K_1,\cdots ,K_9$ necessary to prepare the 2D cluster state. They are products of Pauli matrices, involving all qubits connected by lines, with black and gray dots representing ${\sigma }_x$ and ${\sigma }_z$ operators, respectively. (b) The quantum circuit creating the 2D cluster state in a nine-qubit RAMM register, consisting in a sequence of projective multiqubit measurements of the nine stabilizers.Reuse & Permissions

Figure 6

The $\pi$-shaped transmon circuit discussed in the main text, reproduced here with labels of the ten Majorana bound states.Reuse & Permissions

Figure 7

(a) Part of the RAMM circuit showing two fully controllable topological qubits. Compensating fluxes are included between the topological qubits in order that the gauge-invariant phase differences in the different topological qubits are independent of each other. (b) Topological qubit formed by the six Majorana fermions. The five couplings ${\Delta }_1,\cdots ,{\Delta }_5$ [see Eq. (C5)] can all be individually controlled by the fluxes ${\Phi }_1,\cdots ,{\Phi }_5$. The parity of the two Majoranas coupled by ${\Delta }_1$ can be measured, as explained in Appendix 8b.Reuse & Permissions

Figure 8

Flux-controlled sequences of operations that realize single-qubit Clifford gates and projective measurement on the Pauli basis.Reuse & Permissions

Figure 9

Measurement of the six generators of the Steane code. This circuit can be realized directly in a RAMM architecture.Reuse & Permissions

Figure 10

Quantum circuit to measure the generators of the Steane code in a traditional architecture that allows only for single- and two-qubit gates, and single-qubit measurements. Each of the six generator measurements is realized using four cnot gates with an ancilla, which is in turn encoded using four physical qubits to avoid error propagation. This is the circuit we used to compare the error threshold with and without multiqubit measurements.Reuse & Permissions

Figure 11

Ratio of the Steane code error thresholds with and without multiqubit measurements as a function of the ratio between gate and storage errors, ${\epsilon }_{\mathrm{g}}/{\epsilon }_{\mathrm{st}}$. The solid, dashed, dotted, and dash-dotted curves correspond to ratios ${\epsilon }_{\mathrm{om}}/{\epsilon }_{\mathrm{st}}={\epsilon }_{\mathrm{dm}}/{\epsilon }_{\mathrm{st}}=1,2,5$, and 10, respectively.Reuse & Permissions

Figure 12

Ratio of the computational error thresholds with and without multiqubit measurements as a function of $M$. Here $\epsilon ={\epsilon }_{\mathrm{om}}={\epsilon }_{\mathrm{dm}}={\epsilon }_{\mathrm{g}}$, with $\epsilon /{\epsilon }_{\mathrm{st}}=1$ (solid), 5 (dashed), and 10 (dotted). The range of $M$ starts from 11, because of the condition $M-N_0-1\ge 0$.Reuse & Permissions

Figure 13

The dependence of the Coulomb coupling $U_k$ on the ratio $E_{\mathrm{J},k}/E_{\text{C},k}$. The solid black line shows the exact solution obtained using Mathieu functions (Ref. 17), and the dashed red line shows the approximation [Eq. (1)], which is valid in the asymptotic limit $E_{\mathrm{J},k}/E_{\text{C},k}\gg 1$.Reuse & Permissions