Leveraging Zero-Level Distillation to Generate High-Fidelity Magic States

TL;DR

This work tackles the significant spatiotemporal overhead of magic state distillation for universal fault-tolerant quantum computing by introducing (0+1)-level distillation, which combines zero-level distillation with the 15-to-1 protocol. The authors develop integration strategies, including zero-level integration in auto-correction blocks, teleportation-based transfer, and ancilla-height optimization, culminating in a seven-round two-level distillation circuit. Their analyses show that (0+1)-level distillation can reduce overhead by up to over 60-70% in various regimes and can lower total spatial costs by roughly 40-50% in Hamiltonian-simulation workloads, suggesting meaningful practical gains for FTQC architectures employing rotated surface codes. The results demonstrate the potential for substantial efficiency gains in large-scale quantum computations that demand many magic states, while also outlining considerations around failure rates of zero-level distillation and trade-offs between spatial and temporal resources.

Abstract

Magic state distillation plays an important role in universal fault-tolerant quantum computing, and its overhead is one of the major obstacles to realizing fault-tolerant quantum computers. Hence, many studies have been conducted to reduce this overhead. Among these, Litinski has provided a concrete assessment of resource-efficient distillation protocol implementations on the rotated surface code. On the other hand, recently, Itogawa et al. have proposed zero-level distillation, a distillation protocol offering very small spatial and temporal overhead to generate relatively low-fidelity magic states. While zero-level distillation offers preferable spatial and temporal overhead, it cannot directly generate high-fidelity magic states since it only reduces the logical error rate of the magic state quadratically. In this study, we evaluate the spatial and temporal overhead of two-level distillation implementations generating relatively high-fidelity magic states, including ones incorporating zero-level distillation. To this end, we introduce (0+1)-level distillation, a two-level distillation protocol which combines zero-level distillation and the 15-to-1 distillation protocol. We refine the second-level 15-to-1 implementation in it to capitalize on the small footprint of zero-level distillation. Under conditions of a physical error probability of $p_{\mathrm{phys}} = 10^{-4}$ ($10^{-3}$) and targeting an error rate for the magic state within $[5 \times 10^{-17}, 10^{-11}]$ ($[5 \times 10^{-11}, 10^{-8}]$), (0+1)-level distillation reduces the spatiotemporal overhead by more than 63% (61%) compared to the (15-to-1)$\times$(15-to-1) protocol and more than 43% (44%) compared to the (15-to-1)$\times$(20-to-4) protocol, offering a substantial efficiency gain over the traditional protocols.

Figures (20)

• Figure 1: Block diagrams depicting (i) $M_{ZZ}$, (ii) $M_{YZ}$, and (iii) $M_{ZZZ}$, respectively. Blue and green lines on the boundary of qubits represent logical operators involved in the lattice surgery operation, and the purple rectangle is used to connect qubits.
• Figure 2: Circuit of $\frac{\pi}{8}$ rotation along $P$ with a faulty $T^\dagger$ measurement (left) and its block diagram representation (right, $P = Z$).
• Figure 3: Single level 15-to-1 protocol implementation consisting of six rounds. Numbered cells represent data qubits. Blue cells represent the magic state which is accessible to the outside of the distillation factory. Black cells represent unused qubits.
• Figure 4: Size of data qubits. Logical $Z$ ($X$) operators on the boundaries are green (blue) colored.
• Figure 5: $\frac{\pi}{8}$ rotation along $Z_1Z_3Z_5$ (left) and logical operators on the boundary during the rotation (right).