Combining Topological Hardware and Topological Software: Color-Code Quantum Computing with Topological Superconductor Networks

We present a scalable architecture for fault-tolerant topological quantum computation using networks of voltage-controlled Majorana Cooper pair boxes and topological color codes for error correction. Color codes have a set of transversal gates which coincides with the set of topologically protected gates in Majorana-based systems, namely, the Clifford gates. In this way, we establish color codes as providing a natural setting in which advantages offered by topological hardware can be combined with those arising from topological error-correcting software for full-fledged fault-tolerant quantum computing. We provide a complete description of our architecture, including the underlying physical ingredients. We start by showing that in topological superconductor networks, hexagonal cells can be employed to serve as physical qubits for universal quantum computation, and we present protocols for realizing topologically protected Clifford gates. These hexagonal-cell qubits allow for a direct implementation of open-boundary color codes with ancilla-free syndrome read-out and logical $T$ gates via magic-state distillation. For concreteness, we describe how the necessary operations can be implemented using networks of Majorana Cooper pair boxes, and we give a feasibility estimate for error correction in this architecture. Our approach is motivated by nanowire-based networks of topological superconductors, but it could also be realized in alternative settings such as quantum-Hall–superconductor hybrids.

Popular Summary

One of the challenges when building a practical, large-scale quantum computer is that the information encoded in quantum bits (or qubits) is fragile and easily destroyed when connected to the noisy outside world. Topological quantum computing, which leans on properties of quantum particles that are impervious to outside change, promises to overcome this problem using both a hardware- and a software-based approach. Topological hardware is still expected to have some sources of error, however, so combining hardware and software will be necessary. This is not straightforward to implement; operations on physical qubits (hardware) are substituted by potentially different operations on logical qubits (software). We have designed the first architecture that combines the best of both worlds and offers topological protection via both hardware and software.

We establish a type of error-correcting code, known as a “topological color code,” as a seamless fit to hardware that is based on quasiparticles called Majorana fermions (fermions that are their own antiparticle). These codes have a set of transversal gates (logical gates that work in parallel) that coincide with topologically protected gates (known as Clifford gates), available from a particular manipulation of Majorana fermions called braiding. This enables us to carry over the topological protection of physical qubits to logical qubits. We give a complete description of our architecture, from physical ingredients to physical qubits, then to logical qubits, and finally to a large-scale quantum computer. We provide detailed protocols for the various gate operations, and we discuss their specific implementation in networks of topological superconductors on which experimental studies are currently focusing.

Our work lays out a scalable approach to fault-tolerant quantum computing and hopefully inspires future research that brings together condensed-matter physics and quantum information theory.

Figure 1

Overview of the design for a scalable fault-tolerant topological quantum computer. The basic building block is the Majorana Cooper pair box (a) consisting of a topological superconducting island with charging energy $E_C$ and Josephson energy $E_J$ hosting a pair of Majoranas ${\gamma }_1$ and ${\gamma }_2$. Parity measurements of the island are controlled by a gate voltage $V_g$. Multiple connected Majorana Cooper pair boxes form a topological superconductor network through which Majoranas can be moved and which allows for the measurement of $2n$-Majorana parity operators. A triangular lattice of hexagonal-cell qubits (b) allows for universal quantum computation with topologically protected Clifford gates. Fault tolerance is added by encoding hexagonal-cell qubits in diamond color codes (c) with transversal Clifford gates. These form a square lattice of logical qubits. Arranging qubits on a line (d) with a magic-state distillery and a cnot bypass completes the universal gate set with a logical $T$ gate and allows for cnot gates between any pair of data qubits with constant-time overhead.

Figure 2

Protocols for braiding operations in a double T junction, where red dots denote Majoranas and red lines connect the coupled superconducting islands (orange). Left diagrams: Braiding of ${\gamma }_1$ and ${\gamma }_2$ is achieved via a three-point turn in the left T junction. Right diagrams: To braid ${\gamma }_2$ and ${\gamma }_3$, first ${\gamma }_3$ is moved from the right island to the bottom right island. Then, ${\gamma }_2$ is moved to the right island by first connecting all three islands in the right T junction and then disconnecting the right island. Finally, ${\gamma }_3$ is moved to the center island.

Figure 3

Left diagram: Hexagonal cell hosting four Majoranas encoding one qubit. The single-qubit Clifford gates can be performed by braiding in the double T junction in the lower part of the cell. In a network of such cells, each cell has up to six neighbors. Right diagram: Such a hexagonal lattice can also be realized with only two different wire orientations using a brick-wall geometry of superconducting islands.

Figure 4

Quantum circuit for a cnot gate using parity measurements and an ancilla qubit initialized in the $|+⟩$ state. First, the parity operator ${\sigma }_z\otimes {\sigma }_z$ of the control and ancilla is measured, with outcome $m_1$. Next, the parity operator ${\sigma }_x\otimes {\sigma }_x$ of the ancilla and target is measured, with outcome $m_2$. Finally, the ancilla is measured in the computational basis ${\sigma }_z$ with outcome $m_3$. The three outcomes determine the final correctional operation ${\sigma }_z^{m_2}\otimes {\sigma }_x^{m_1+m_3}$ on the control and target, which can also be done by updating the Pauli frame.

Figure 5

Protocol for a cnot between two adjacent hexagonal-cell qubits using the quantum circuit in Fig. 4. In the cell occupied by the control qubit (red), an ancilla (blue) is initialized in the $|0⟩$ state and moved to the double T junction of the cell. (a) The ancilla and target (green) are rotated via a Hadamard gate. (b) The first two Majoranas of the control, $c_1$ and $c_2$ and ancilla $a_1$ and $a_2$ are moved onto a connected island, and the four-Majorana fermion parity $-a_1{a}_2{c}_1{c}_2$ is measured, corresponding to a two-qubit parity measurement ${\sigma }_z\otimes {\sigma }_z$ with outcome $m_1$. (c) The ancilla is moved back to the double T junction for another $H$ gate. (d) The ancilla and target parity $m_2$ is measured via a four-Majorana parity measurement in the right cell. (e) An $H$ gate is applied to the ancilla and target qubits in their respective double T junctions. (f) Finally, all qubits return to their initial positions, and the ancilla qubit is measured by measuring the two-Majorana fermion parity $i{a}_1{a}_2$ with outcome $m_3$.

Figure 6

Quantum circuit for a multitarget cnot gate using parity measurements and an ancilla qubit initialized in the $|+⟩$ state. First, the parity operator ${\sigma }_z\otimes {\sigma }_z$ of the control and ancilla is measured, with outcome $m_1$. Next, the parity operator ${\sigma }_x\otimes {\sigma }_x^{\otimes n}$ of the ancilla and $n$ targets is measured, with outcome $m_2$. Finally, the ancilla is measured in the computational basis ${\sigma }_z$, with outcome $m_3$. The three outcomes determine the final correctional operation ${\sigma }_z^{m_2}\otimes ({\sigma }_x^{m_1+m_3}{)}^{\otimes n}$ on the control and targets, which can also be done by updating the Pauli frame.

Figure 7

Six-qubit parity measurement in a triangular lattice of hexagonal-cell qubits. Six hexagonal-cell qubits, $Q_1–Q_6$, are arranged around an empty cell that is used for the measurement of the parity operator ${\sigma }_z^{\otimes 6}$. For clarity, the Majoranas of each qubit are colored red and blue in an alternating fashion. The first two Majoranas of each qubit are moved to this cell, such that 12 connected superconductors host 12 Majoranas. The total 12-Majorana parity of this island is precisely the six-qubit parity operator.

Figure 8

First three topological triangular color codes with code distances 3, 5, and 7 (where the smallest one is equivalent to the Steane code [63]). These 6.6.6 color codes are defined on a hexagonal lattice, where each vertex is a physical qubit and each face is an $X$-type and a $Z$-type stabilizer involving the surrounding qubits. Physical errors on a qubit affect the three different-colored stabilizers and edges surrounding the qubit. In the triangular lattice of hexagonal-cell qubits, the empty cell in the center of each face can be used for stabilizer measurement.

Figure 9

(a) Color codes feature transversal Clifford gates. While the logical Hadamard gate is simply $H_L=H^{\otimes n}$, the logical $S$ gate $S_L$ is a mixture of physical $S$ and $S^{†}$ gates. Using a bicoloration of the physical qubits, such that the sublattice involving the corner qubits is blue and the other one is orange, $S_L$ requires physical $S$ gates on blue qubits and an $S^{†}$ gate on orange qubits. The logical cnot gate corresponds to $n$ physical cnots between pairs of qubits from two triangles. (b) Since this requires the movement of ancilla qubits from one triangle to the other, this leaves some unused space in between, which can be used for the diamond color codes. These form a square lattice of logical qubits. (c) Universal fault-tolerant quantum computation can be achieved on a line of data qubits with a magic-state distillery and a cnot bypass.

Figure 10

(a) Fault-tolerant $ZZ$-parity ($XX$-parity) measurement between two diamond color-code qubits by lattice surgery [34], denoted by black lines crossing the neighboring boundaries. First, new three- and four-qubit green stabilizers are introduced, and new eight-qubit stabilizers are obtained by merging red plaquettes along the boundary. These stabilizers are measured along with all other stabilizers in order to obtain the $ZZ$ parity ($XX$ parity), where the new green boundary stabilizers are only measured in the $Z$ basis ($X$ basis) and the red eight-qubit stabilizers only in the $X$ basis ($Z$ basis). (All stabilizers are explicitly shown in Fig. 21 in Appendix pp3.) The product of the green boundary stabilizers is precisely the two-qubit parity. Finally, the stabilizers are returned to their initial configuration before the lattice surgery. (b) Protocol for a fault-tolerant cnot using lattice surgery. A logical ancilla is initialized in the $|+⟩$ state. The $ZZ$ parity between the control and ancilla is measured, and the ancilla is moved through the cnot bypass to the target. Finally, the $XX$ parity between the ancilla and target is measured, and the ancilla is read out. The length of this protocol scales linearly with the distance between the control and target. (c) Lattice-surgery-based cnot protocol with constant-time overhead. The $ZZ$ parities between the control and three ancilla qubits in the $|+⟩$ state are measured simultaneously using the three lattice surgeries indicated in the figure. Next, the $XX$ parity between ancilla 3 and the target is measured. Finally, ancilla 3 is read out in the $Z$ basis, while ancillas 1 and 2 are measured in the $X$ basis. In the presence of measurement errors during syndrome read-out, this protocol scales logarithmically with the distance between the control and target.

Figure 11

State injection algorithm: A cnot between a qubit $|\psi ⟩$ and a magic state $|m⟩=(|0⟩+e^{i\pi /4}|1⟩)/\sqrt{2}$, followed by a measurement of $|m⟩$ with outcome $m_z$ and a correctional $S^{m_z}$ gate, is equivalent to a $T$ gate on the qubit.

Figure 12

Code injection procedure which encodes an unknown physical state $|\psi ⟩$ (gray qubit) into a logical state $|{\psi }_L⟩$. First, the stabilizer state in the left panel is prepared by measuring all the stabilizers shown. Finally, we cease measuring the green stabilizers at the bottom boundary and start measuring the red stabilizers.

Figure 13

Inflation protocol for transversal multitarget cnot gates with four logical qubits. This protocol rearranges the physical qubits such that the qubits involved in transversal multitarget cnot gates are now close to each other, i.e., every first physical qubit of each of the four logical qubits, every second, etc. This protocol can be used for the multitarget cnots required for magic-state distillation, e.g., using inflation of 15 qubits arranged on a $4\times 4$ grid for 15-to-1 distillation.

Figure 14

A Majorana Cooper pair box as a basic building block of the topological hardware. Top diagram: A pair of Majorana zero modes ${\gamma }_1$ and ${\gamma }_2$ at the ends of a $p$-wave superconductor can be effectively obtained by depositing an $s$-wave superconductor with strong spin-orbit coupling on top of a material with a single spin-polarized conducting channel, such as a semiconducting nanowire in a magnetic field, a quantum anomalous Hall insulator, or a 2DEG in a strong magnetic field. Bottom diagram: A Majorana Cooper box requires the addition of charging energy $E_C$ and Josephson energy $E_J$ on the mesoscopic superconducting island. A top gate that is capacitively coupled to the superconducting island imposes a certain total charge on the island governed by the gate voltage $V_g$ and the (fixed) charging energy. Furthermore, a bulk superconductor is Josephson coupled to the mesoscopic island through a gate-tunable Josephson junction, which tunes the Josephson energy and imposes a certain phase on the island.

Figure 15

Parity-to-charge conversion in the Majorana Cooper pair box, as described in Ref. [35], and energy levels of the Majorana Cooper pair box $H_C+H_J$ as a function of gate voltage $V_g$ and for different ratios $E_J/E_C$. In the Majorana regime $E_J\gg E_C$, charging energy is negligible, and the spectrum is insensitive to $V_g$. The ground state is given by nearly degenerate states of opposite parity (blue and orange), where the maximum separation $\mathrm{\Delta }E$ vanishes exponentially in $E_J/E_C$, whereas the distance to the first excited states increases with $\sqrt{E_J{E}_C}$. As $E_J$ is decreased by decreasing the coupling to the bulk superconductor, the ground-state degeneracy is lifted in the intermediate regime. Here, varying the gate voltage distinguishes the parity states through their differential capacitance $C=\partial ⟨\widehat{N}⟩/\partial V_g$, which is larger for the orange parity than for the blue parity. Finally, in the Coulomb regime $E_J\ll E_C$, the spectrum is given by parabolas with well-defined charge number $N$. If $V_g$ is tuned to a minimum of a charge parabola, the two lowest-energy parity states are separated by $E_C$, which can be used to impose a certain parity on the island. The intermediate regime can also be understood from the emergence of avoided crossings between charge states of equal parity as $E_J$ is increased.

Figure 16

Two Majorana Cooper pair boxes connected to the same bulk superconductor with Josephson energies $E_{J,1}$ and $E_{J,2}$, and top gate voltages $V_{g,1}$ and $V_{g,2}$, respectively. The islands are connected through the spin-polarized conducting channel, in which the interisland transmission probability $\tau$ can be tuned by a pincher gate.

Figure 17

Two-step protocol for moving Majoranas ${\gamma }_1$ and ${\gamma }_2$ from the left island, initially tuned to the Majorana regime, to the right island, initially tuned to the Coulomb regime with even parity. First, the two islands are coupled by increasing the transmission to $\tau =1$ and tuning the right island to the Majorana regime, shuttling ${\gamma }_2$ to the right island. The two islands now form a single connected superconducting island with Majoranas ${\gamma }_1$ and ${\gamma }_2$. In order to move ${\gamma }_1$ to the right island, the transmission is reduced back to $\tau =0$, and the left island is tuned to the Coulomb regime.

Figure 18

T-junction geometry consisting of three mesoscopic superconducting islands coupled through a three-terminal junction involving three pincher gates. Left diagram: If all three pincher gates are closed, the islands are decoupled and host a pair of Majoranas each. In dispersive read-out, tuning $V_{g,1}$ and $V_{g,2}$ is used to measure the parities $i{\gamma }_1{\gamma }_2$ and $i{\gamma }_3{\gamma }_4$, respectively. Right diagram: Opening the right and bottom pincher gate while tuning the bottom island into the Majorana regime moves ${\gamma }_3$ to the bottom island. Subsequently, opening the left pincher gate connects all three islands, where ${\gamma }_2$ is shared by all islands in the three-terminal junction. Connecting any of the top gates to a resonant circuit allows for dispersive read-out of the total parity $-{\gamma }_1{\gamma }_2{\gamma }_3{\gamma }_4$.

Figure 19

Logical error rate as a function of physical error rate obtained from Monte Carlo simulation with a lookup table decoder for triangular (solid line) and diamond (dashed line) color codes with code distances $d=3$ and $d=5$, and for the triangular color code with $d=7$. The sample size is between $10^7$ and $10^{10}$ trials for data point corresponding to high and low logical error rates, respectively. The upper line show the logical error rate without quantum error correction. The inset zooms into the crossover region around $p_{\mathrm{phys}}\sim 11\%$.

Figure 20

Protocol for three simultaneous cnots between three control qubits $Q_{c_1-c_3}$ and three target qubits $Q_{t_1-t_3}$ using the quantum circuit in Fig. 4. In the cell occupied by the control qubits (red), ancillas (blue) are initialized in the $|0⟩$ state (a) and moved to the double T junction of the cell for a Hadamard gate (b). The first two Majoranas of the control and ancilla qubits are moved onto a connected island, and the four-Majorana parity is measured (c), corresponding to a two-qubit parity measurement with outcome $m_1$. The ancillas are moved back to the double T junction for another $H$ gate (d). The third and fourth Majoranas $a_3$ and $a_4$ of each ancilla qubit are moved into the lower leg of their hexagonal cell, such that the remaining ancilla Majoranas can move to the target qubit cells for a four-Majorana parity measurement (e). The ancilla Majoranas are then moved back to the control cells for an $H$ gate (f). Finally, all qubits return to their initial positions (g), and the ancilla qubits are measured by measuring the two-Majorana parity $m_3$.

Figure 21

Stabilizers that are measured to obtain the $ZZ$ parity between two logical qubits using lattice surgery. In contrast to usual color-code stabilizer measurements, lattice surgery requires measurements where the support of $X$ and $Z$ stabilizers does not coincide. The left panel shows the required $Z$ stabilizers, and the right panel the $X$ stabilizers, which differ along the shared boundary. To obtain the $XX$ parity between two qubits, one simply has to swap $Z\leftrightarrow X$ in the protocol.

Figure 22

Quantum circuit corresponding to the constant-time overhead cnot gate in Fig. 10.

Figure 23

Survival time of triangular (dashed lines) and diamond (solid lines) color-code qubits until the logical error probability $p_{\mathrm{err}}$ reaches ${10}^{-6}$ for increasing code distances obtained from extrapolation of the numerical data (red). The labels on the right-hand side of the time axis show the survival time for a code cycle duration of $1\mu s$.

Figure 24

Logical error rate of the first three 4.8.8 and 6.6.6 triangular color codes. Even though the codes have the same code distance and error threshold, 4.8.8 color codes have a higher logical error rate but require fewer physical qubits per logical qubit.