Fault-Tolerant Stabilizer Measurements in Surface Codes with Three-Qubit Gates

TL;DR

This work shows that stabilizer measurements for unrotated surface codes can be fault-tolerant using three-qubit gates with a single auxiliary qubit, reducing circuit depth and potentially speeding QEC cycles. Under an optimistic three-qubit depolarizing noise model, the logical threshold improves from about $0.76\%$ (CZ-based) to about $1.05\%$ (CZZ-based), and logical error rates can drop by an order of magnitude. The authors develop a formal framework of distinguishable fault sets to prove FT behavior, numerically verify FT for unrotated codes with NE/NW CZZ readout, and perform memory experiments with Stim to compare CZ and CZZ schemes. They demonstrate a cross-over in qubit resource requirements around $p \approx 0.3\%$, where unrotated codes with CZZ can reach target logical error rates with fewer physical qubits, highlighting the practical potential of multi-qubit gates for fault-tolerant QEC across platforms. This work thus motivates further exploration of multi-qubit gates and alternative QEC codes that can leverage FT stabilizer readout with reduced resource overheads.

Abstract

Quantum error correction (QEC) is considered a deciding component in enabling practical quantum computing. Stabilizer codes, and in particular topological surface codes, are promising candidates for implementing QEC by redundantly encoding quantum information. While it is widely believed that a strictly fault-tolerant protocol can only be implemented using single- and two-qubit gates, several quantum computing platforms, based on trapped ions, neutral atoms and also superconducting qubits support native multi-qubit operations, e.g. using multi-ion entangling gates, Rydberg blockade or parallelized tunable couplers, respectively. In this work, we show that stabilizer measurement circuits for unrotated surface codes can be fault-tolerant using single auxiliary qubits and three-qubit gates. These gates enable lower-depth circuits leading to fewer fault locations and potentially shorter QEC cycle times. Concretely, we find that in an optimistic parameter regime where fidelities of three-qubit gates are the same as those of two-qubit gates, the logical error rate can be up to one order of magnitude lower and the threshold can be significantly higher, increasing from $\approx 0.76 \%$ to $\approx 1.05 \%$. Our results, which are applicable to a wide range of platforms, thereby motivate further investigation into multi-qubit gates as components for fault-tolerant QEC, as they can lead to substantial advantages in terms of time and physical qubit resources required to reach a target logical error rate.

Figures (11)

• Figure 1: Objects used to determine distinguishability of a fault set. The box represents some (Clifford) circuit with measurements. It is encoded in an error-correcting code with stabilizer generators $\{S_i\}_{i=1}^{n-k}$. During the circuit, sets of deterministic measurements specify the detectors $\{D\}$. For an error-correction gadget, these are typically the parities of consecutive stabilizer measurements. The exemplarily shown fault path $\mathbf{F}$ of weight $w=3$ can be efficiently propagated using Clifford simulation. This results in a vector of flipped detectors $\mathbf{D}$ and a final error $E$. The ideal syndrome of $E$ is $s(E) = (\langle S_i , E \rangle)_{i=1}^{n-k}$.
• Figure 2: Detail of stabilizer measurement circuits for the distance-$3$ rotated surface code. a) A $Z$ fault on the ancilla of a $Z$-stabilizer measurement can propagate to two data qubits, indicated by the blue highlighting. The propagated fault is detected in the next round of $X$-stabilizer measurements. b) and c) show the effect for two different orderings of the gates. In the rotated surface codes, we draw the $X$-($Z$-)logical operators as thick red (blue) lines and draw flipped detectors with a light green dot. b) If the last two gates act on the north (N) and west (W) qubit, the final $Z$ error is parallel to the $Z$-logical operator. There is another first order fault with the same syndrome that is logically inequivalent implying a non-distinguishable fault set $\mathcal{F}^{(1)}$. c) For an ordering towards the north (N) and east (E) qubit, however, faults with the same flipped detectors are stabilizer-equivalent. The fault set $\mathcal{F}^{(1)}$ is thus distinguishable.
• Figure 3: Detail of stabilizer measurement circuits for the distance $3$ unrotated surface code, implemented with three-qubit $\mathrm{CZZ}$ gates as described in the main text. a) For a uniform depolarizing noise model, there also exist elementary two-qubit $X$-faults on the support of $Z$-stabilizer generators - contrary to the two-qubit gate situation. These faults are captured by a subsequent $Z$-stabilizer measurement. b) and c) show the effect for two different orderings of the gates. In the unrotated surface codes, we draw the $X$-($Z$-)logical operators as thick red (blue) lines and draw flipped detectors with a light green dot. b) If the last three-qubit gates act on the horizontal (WE) and vertical (NS) qubits, the final $X$ error is parallel to the $X$-logical. There is another first-order fault with the same syndrome, which is logically inequivalent implying a non-distinguishable fault set $\mathcal{F}^{(1)}$. c) For a diagonal ordering towards the south-east (SE) and north-west (NW) qubits, however, faults of the same order with the same flipped detectors are stabilizer-equivalent. The fault set $\mathcal{F}^{(1)}$ is distinguishable.
• Figure 4: Logical error rates for implementations using $\mathrm{CZ}$ (circle, dashed) and $\mathrm{CZZ}$ gates (square, solid) in the NI model. The color coding represents codes with increasing distance. a) In a rotated surface code, using three qubit $\mathrm{CZZ}$ gates breaks fault tolerance which can be seen by the different scaling for small physical error rates $p$. Additionally, the logical error rate is higher, e.g. about an order of magnitude for distance-$9$ codes at $p \approx 10^{-3}$. b) In unrotated surface codes, however, the logical error rate barely changes when replacing $\mathrm{CZ}$s with $\mathrm{CZZ}$s (using the same error strength, but $3$- instead of two $2$- qubit depolarizing channels). For low $p$, we also observe a flattening of the scaling. This, however, is due to the suboptimal decomposition of elementary faults for decoding with pymatching, as explained in the main text. Error bars are standard Monte Carlo errors and can be smaller than the symbol used.
• Figure 5: Logical error rates for implementations using $\mathrm{CZ}$ (circle, dashed) and $\mathrm{CZZ}$ gates (square, solid). We assume idling noise strength of $p/10$. The color coding represents codes with increasing distance. a) Distance-$3$ and $5$ codes correct for all faults up to order $t$, resulting in a scaling of the logical error rate $p_L \propto p^{t+1}$ as expected for a fault-tolerant protocol. Due to fewer idling locations using three-qubit gates, the logical error rate is up to 50% lower for physical error rates in the range of $10^{-2}$ to $10^{-3}$. b) The threshold is increased from $p^{(\mathrm{CZ})}_{\mathrm{th}} \approx 0.76 \%$ to $p^{(\mathrm{CZZ})}_{\mathrm{th}} \approx 1.05 \%$. Decoded using beliefmatching with $d$ iterations of BP before matching. Error bars are standard Monte Carlo errors and can be smaller than the symbol used. Thresholds are obtained using the finite-size scaling ansatz described in the Supplemental Material.

Theorems & Definitions (1)

• Definition 1