Holonomic surface codes for fault-tolerant quantum computation

Surface codes can protect quantum information stored in qubits from local errors as long as the per-operation error rate is below a certain threshold. Here we propose holonomic surface codes by harnessing the quantum holonomy of the system. In our scheme, the holonomic gates are built via auxiliary qubits rather than the auxiliary levels in multilevel systems used in conventional holonomic quantum computation. The key advantage of our approach is that the auxiliary qubits are in their ground state before and after each gate operation, so they are not involved in the operation cycles of surface codes. This provides an advantageous way to implement surface codes for fault-tolerant quantum computation.

Figure 1

(a) A 2D lattice implementation of dynamic surface codes. (b) The stabilizer units (left) and syndrome extraction circuits (right) for measurement qubits $Z$ and $X$ in (a). (c) A 2D lattice implementation of holonomic surface codes, where we introduce two types of auxiliary qubits, shown as orange solid and open circles, respectively. (d) The stabilizer units for measurement qubits $Z$ and $X$ in (c).

Figure 2

(a) Implementation of holonomic single-qubit operations. (b) Implementation of the holonomic quantum gate $U_{\mathrm{sz}}$ between qubits $M$ and $D$. (c) Implementation of the holonomic swap gate between qubits $M$ and $D$ via three holonomic two-qubit $U_{\mathrm{sz}}$ gates (first between qubits $M$ and $B$, then between qubits $D$ and $M$, and finally between qubits $B$ and $D$).

Figure 3

The lower bound of the fidelity for the holonomic cnot gate (blue curve) and that for the dynamic cnot gate (green curve) vs the area error $\delta$. For $\delta \lesssim 0.152$, the fidelity of the holonomic cnot is larger than the threshold $99.4\%$ (marked by the red dotted line), but it requires $\delta \lesssim 0.041$ for the dynamic one.

Figure 4

Quantum circuit for a holonomic cnot gate acting on qubits $M$ and $D$, where qubit $B$ is initially prepared in $|0\rangle$. Here we choose qubit $D$ as the control qubit, so there are two Hadamard gates applied before and after the four $U_{\mathrm{sz}}$ gates. On the contrary, if we choose qubit $M$ as the control qubit, the two Hadamard gates should be applied to qubit $M$.

Figure 5

(a) The stabilizer unit for a measurement qubit in holonomic surface codes, where the four data qubits are denoted by $a,b,c$, and $d$. Here qubit $M$ is the measurement qubit and each qubit $B$ is an auxiliary qubit. (b) The stabilizer unit for a measurement qubit in dynamic surface codes.

Figure 6

Quantum circuit for the state recovery of ${|\psi \rangle }_1$ to ${|\psi \rangle }_f=a_{00}|00\rangle +a_{10}|01\rangle +a_{01}|10\rangle -a_{11}|11\rangle$. (a) The recovery is divided into two steps, i.e., the population recovery and the phase recovery, both of which are also accomplished with holonomic gates. (b) Explicit circuit for the population recovery. (c) Explicit circuit for the phase recovery.