Leakage-resilient approach to fault-tolerant quantum computing with superconducting elements

Superconducting qubits, while promising for scalability and long coherence times, contain more than two energy levels, and therefore are susceptible to errors generated by the leakage of population outside of the computational subspace. Such leakage errors constitute a prominent roadblock towards fault-tolerant quantum computing (FTQC) with superconducting qubits. FTQC using topological codes is based on sequential measurements of multiqubit stabilizer operators. Here, we first propose a leakage-resilient procedure to perform repetitive measurements of multiqubit stabilizer operators, and then use this scheme as an ingredient to develop a leakage-resilient approach for surface code quantum error correction with superconducting circuits. Our protocol is based on swap operations between data and ancilla qubits at the end of every cycle, requiring read-out and reset operations on every physical qubit in the system, and thereby preventing persistent leakage errors from occurring.

Figure 1

Repetitive measurements of stabilizer operators via (a) standard scheme, and (b) our swap-based scheme. The ancilla register (A) gets measured and initialized at each cycle in the standard scheme, while the data register (D) never gets reset. Both the registers are measured in our scheme at alternate cycles. $U_{\mathrm{cycle}}$ denotes the sequence of gate operations required for the stabilizer measurement in the standard scheme, and those for swap-based scheme are denoted by $U_{\mathrm{cycle}}^{\mathrm{odd}}$ and $U_{\mathrm{cycle}}^{\mathrm{even}}$ for odd and even cycles repsectively.

Figure 2

(a) A schematic diagram of the architecture for measuring 2-qubit stabilizer operators. The circles denote superconducting qubits (data qubits denoted by “D” and ancilla qubits denoted by “A”), and lines denote required nearest-neighbor couplings. The stabilizer operators, $ZZ$ and $XX$, are measured via qubit-3 and qubit-4 respectively. (b) Standard scheme for repetitive measurements of two-qubit stabilizer operators $XX$ and $ZZ$. $H$ denotes the Hadamard gate, the vertical lines connected by filled circles denote cz gates, and the numbers denote the indices for the qubits in the same order as shown in Fig. 2. The gates inside the dashed rectangle represent $U_{\mathrm{cycle}}$ in Fig. 1.

Figure 3

The leakage-resilient scheme for stabilizer measurement where swap gates are replaced by cz and Hadamard gates as shown in Eq. (2). The gates inside the left dashed rectangle are repeated for odd cycles, while the gates inside the right rectangle are repeated for even cycles.

Figure 4

Results for the standard approach to repetitive measurements of two-qubit stabilizer operators, $XX$ (dashed red) and $ZZ$ (solid blue). (a) The readouts of two ancilla qubits (qubits 3 and 4) are shown for various consecutive cycles. (b) The solid black curve shows the probability that either of the two data qubits is in the $|2\rangle$ state at the end of every measurement cycle. The green (solid gray) curve shows the overlap between the state encoded in the data register and the prediction of it from the corresponding ancilla outputs at the end of each cycle.

Figure 5

Results for our swap-based approach to repetitive measurements of two-qubit stabilizer operators, $XX$ (dashed red) and $ZZ$ (solid blue). (a) The readouts of two ancilla qubits (qubits 3 and 4) are shown for various consecutive cycles. (b) The solid black curve shows the probability that either of two data qubits is in the $|2\rangle$ state at the end of every measurement cycle. The green (solid gray) curve shows the overlap between the state encoded in the data register and the prediction of it from the corresponding ancilla outputs at the end of each cycle.

Figure 6

A schematic diagram of a leakage-resilient approach to surface code quantum error correction. The vertical arrows denote the directions for the transfer for quantum state from data qubits to syndrome qubits via swap gates after each cycle.