Error correction of transversal CNOT gates for scalable surface code computation

TL;DR

This paper analyzes error correction for transversal CNOT gates between surface-code blocks, introducing three decoding strategies—single-update MWPM, hypergraph (HUF), and ordered decoding—and showing that ordered decoding preserves SCQM-like thresholds while maintaining graph-based decoding advantages. It extends the study to transversal teleportation, where decoding reduces to graph-based MWPM, and compares transversal implementations with lattice surgery under Pauli and erasure noise, highlighting potential overhead and performance trade-offs. The results indicate that transversal strategies can achieve competitive thresholds with favorable logical error rates, and erasure-noise scenarios further boost thresholds, providing hardware-relevant guidance for scalable fault-tolerant quantum computation using transversal gates.

Abstract

Recent experimental advances have made it possible to implement logical multi-qubit transversal gates on surface codes in a multitude of platforms. A transversal controlled-NOT (tCNOT) gate on two surface codes introduces correlated errors across the code blocks and thus requires modified decoding strategies compared to established methods of decoding surface code quantum memory (SCQM) or lattice surgery operations. In this work, we examine and benchmark the performance of three different decoding strategies for the tCNOT for scalable, fault-tolerant quantum computation. In particular, we present a low-complexity decoder based on minimum-weight perfect matching (MWPM) that achieves the same threshold as the SCQM MWPM decoder. We extend our analysis with a study of tailored decoding of a transversal teleportation circuit, along with a comparison between the performance of lattice surgery and transversal operations under Pauli and erasure noise models. Our investigation works towards systematic estimation of the cost of implementing large-scale quantum algorithms based on transversal gates in the surface code.

Figures (12)

• Figure 1: (a) A logical transversal CNOT operation between two rotated surface codes is performed by applying physical CNOT gates between each corresponding pair of data qubits of the SC states. (b) The transversal CNOT creates correlated errors between surface codes. Each SC is shown as a qubit set on which an error on a physical qubit can propagate to the other SC through the tCNOT.
• Figure 2: (a) A $d=5$ rotated surface code. (i) An $X$-logical operator, (ii) a single-qubit $Z$ error, with the corresponding anticommuting stabilizer measurements highlighted, and (iii) the X-decoding graph $G_X$ used to correct for $Z$ errors for one stabilizer measurement round. (b) A representation of $G_X$ generated by using $d$ rounds of stabilizer measurements on the underlying surface code. (i) Errors in the bulk create two defects to be matched together, (ii) An error at the boundary creates a single defect, and (iii) a string of data qubit and measurement errors (red) and its corresponding matching-obtained correction (grey); the correction restores the original logical state up to code stabilizers.
• Figure 3: (a) A 'binary-tree' tCNOT circuit of logical depth $M$. Each tCNOT is followed by $r$ rounds of stabilizer measurements (shaded yellow boxes). In this case a single error (for example an $X$ error on qubit $q$) can induce correlated errors that grow exponentially with circuit depth (marked by dashed red lines). (b) If $r << W$, each tCNOT cannot be decoded independently since weight $O(r)$ data errors can be misidentified as measurement errors, creating logical failures.
• Figure 4: Thresholds and logical failure rates under circuit-level depolarising Pauli noise for SCQMs and logical CNOTs. (left) HUF decoder performance for a 2SCQM and tCNOT. (centre) single-update and ordered MWPM decoder performance for a tCNOT. (right) MWPM decoder performance for a 2SCQM and lattice-surgery (LS) CNOT. Operations are plotted separately to prevent overcrowding. Thresholds are found by finite size scaling close to $p_t$ (translucent lines).
• Figure 5: Methods to perform a $T$ gate using fault-tolerant magic state state teleportation. (a) A logical state is teleported onto the original qubit using a CNOT, followed by a Clifford update. (b) In a lattice-surgery setting, the equivalent protocol in (a) can be optimized to reduce the number of joint parity measurements litinski_game_2019.