Efficient Magic State Distillation by Zero-Level Distillation

TL;DR

This work tackles the high resource cost of magic state distillation (MSD) by introducing zero-level distillation, which performs distillation at the physical level on a square-lattice architecture using the Steane code and Hadamard-test verification, followed by teleportation to surface codes. It demonstrates a robust $p_L \approx 100 p^2$ scaling for the logical magic state and keeps circuit depth modest ($\sim$25), enabling substantial reductions in spatiotemporal overhead for early FTQC and compatibility with traditional multilevel distillation. An alternative direct-conversion route from the Steane code to the rotated surface code offers even lower qubit counts at the cost of larger depth ($\sim$42). Numerical simulations with realistic depolarizing noise support the quadratic scaling and high success probabilities, while discussions place zero-level distillation within the broader landscape of fault-tolerant quantum computing strategies such as (0+1)-level distillation and magic-state cultivation. The approach promises practical gains in both early and full-scale FTQC by reducing resource requirements for implementing non-Clifford gates.

Abstract

Magic state distillation (MSD) is an essential element for universal fault-tolerant quantum computing, which distills a high-fidelity magic state from noisy magic states using ideal (error-corrected) Clifford operations. For ideal Clifford operations, it needs to be performed on the logical qubits and hence incurs a large spatiotemporal overhead, which is one of the major bottlenecks for the realization of fault-tolerant quantum computers (FTQCs). Here we propose zero-level distillation, which prepares a high-fidelity logical magic state at the physical level, namely zero level, using physical qubits and nearest-neighbor two-qubit gates on a square lattice. We develop a zero-level distillation circuit and show that distillation can be made even more efficient than the conventional sophisticated approaches with logical level distillations. The key idea involves the Knill et al.-type distillation using the Steane code and its careful mapping to the square-lattice architecture with error detection. The distilled magic state on the Steane-code state is then teleported or converted to surface codes. We numerically find that the error rate of the logical magic state scales as approximately $100 \times p^{2}$ in terms of the physical error rate $p$. For example, with a physical error rate of $p = 10^{-4}$ ($10^{-3}$), the logical error rate is reduced to $p_{L} = 10^{-6}$ ($10^{-4}$), resulting in an improvement of 2 (1) orders of magnitude. This contributes to reducing both space and time overhead for early FTQC as well as full-fledged FTQC combined with conventional multilevel distillation protocols.

Figures (17)

• Figure 1: The Steane code. A qubit is located at each vertex. The red, blue, and green faces represent stabilizers.
• Figure 2: MSD circuits based on transversality of the $H$ gate. The encoding circuit creates a logical magic state encoded with the Steane code. The Hadamard test distills the magic state by measuring $H^{\otimes 7}$. The decoding circuit is based on the one-bit teleportation.
• Figure 3: (a) The zero-level distillation circuit. (b) The detailed circuits for encoding of noisy magic state, preparation of cat state, and distillation. The circuit in the blue box encodes a magic state encoded with the Steane code non-fault-tolerantly. The circuit in the red box prepares a cat state. The circuit in the green box is the distillation circuit utilizing the Hadamard test.
• Figure 4: Qubit arrangement and stabilizer deformation during encoding a magic state. The left figure represents the $X$-stabilizer deformation, and the right figure illustrates the $Z$-stabilizer deformation. Each dot represents a qubit: the blue dots and numbers correspond to the locations of the qubits shown in Fig. \ref{['fig:zero_circuit']}, and the light blue arrows indicate the transfer of qubits using one-bit teleportations. The blue bold edges indicate the CNOT gates, where the $+$ symbol on one side represents the target qubit.
• Figure 5: Qubit arrangement during encoding a cat state. Each dot represents a qubit: the orange dots and numbers correspond to the locations of the qubits in Fig. \ref{['fig:zero_circuit']}. The yellow arrows indicate the transfer of qubits using one-bit teleportations. The red bold edges indicate the CNOT gates where the $+$ symbol on one side represents the target qubit.