Experimental Demonstration of High-Fidelity Logical Magic States from Code Switching

TL;DR

This work demonstrates a practical route to universal fault-tolerant quantum computation by preparing a high-fidelity logical magic state through code switching between color-code families: from the 15-qubit quantum Reed-Muller code to the 7-qubit Steane code. The protocol initializes two code blocks, applies a transversal logical T gate on the qRM block to generate a magic state, and teleports the state to the Steane code via a logical CNOT, followed by a destructive X measurement with optional Z corrections. The authors certify the encoded state using a sample-efficient two-copy Bell-basis method, achieving a rigorous lower bound on fidelity of 1 − O(10^-4) and demonstrating a logical infidelity below the leading physical error rates, thus below the pseudothreshold. The experiment, conducted on a 28-qubit trapped-ion processor with all-to-all connectivity, establishes a new state-of-the-art for logic-encoded magic-state fidelity and points toward practical, scalable routes for fault-tolerant quantum computation, including potential reductions in magic-state distillation overhead and extensions to higher-distance codes and other transversal non-Clifford gates.

Abstract

Preparation of high-fidelity logical magic states has remained as a necessary but daunting step towards building a large-scale fault-tolerant quantum computer. One approach is to fault-tolerantly prepare a magic state in one code and then switch to another, a method known as code switching. We experimentally demonstrate this protocol on an ion-trap quantum processor, yielding a logical magic state encoded in an error-correcting code with state-of-the-art logical fidelity. Our experiment is based on the first demonstration of code switching between color codes, from the fifteen-qubit quantum Reed-Muller code to the seven-qubit Steane code. We prepare an encoded magic state in the Steane code with $82.58\%$ probability, with an infidelity of at most $5.1(2.7) \times 10^{-4}$. The reported infidelity is lower than the leading infidelity of the physical operations utilized in the protocol by a factor of at least $2.7$, indicating the quantum processor is below the pseudo-threshold. Furthermore, we create two copies of the magic state in the same quantum processor and perform a logical Bell basis measurement for a sample-efficient certification of the encoded magic state. The high-fidelity magic state can be combined with the already-demonstrated fault-tolerant Clifford gates, state preparation, and measurement of the 2D color code, completing a universal set of fault-tolerant computational primitives with logical error rates equal or better than the physical two-qubit error rate.

Figures (8)

• Figure 1: Schematic representation of the $[[7,1,3]]$ Steane code in a 2D lattice including labels for the physical qubits.
• Figure 2: Schematic representation of the $[[15,1,3]]$ qRM code in a 3D tetrahedral lattice including labels for the physical qubits.
• Figure 3: The magic state preparation protocol at the logical level Daguerre:2024gjd. A qRM (quantum Reed-Muller) and a Steane codeblock are prepared in $|\bar{+}\rangle$ and $|\bar{0}\rangle$, respectively. The transversal logical $\bar{T}$ gate is applied, creating a logical magic state $|\bar{T}\rangle=\bar{T}|\bar{+}\rangle$ in the qRM code. The magic state is then teleported to the Steane codeblock by applying a logical $\overline{\text{CNOT}}$ gate, measuring the logical $\bar{X}$ operator and applying a logical $\bar{Z}$ correction if $\bar{X}=-1$. The output of the protocol is a logical magic state $|\bar{T}\rangle$ in the Steane codeblock.
• Figure 4: Logical $\overline{\text{CNOT}}$ gate between the Steane and qRM codeblocks, consisting of physical CNOT gates acting on the base of the tetrahedron as controls and on the Steane codeblock as targets.
• Figure 5: Single-copy (a) and two-copy experiments (b), respectively. In both experiments, the magic states are prepared using pre-selection as in Fig. \ref{['fig:logical_circuit']}. In (a) the logical $\bar{P}$ operators with $\bar{P}\in \{\bar{X},\bar{Y},\bar{Z}\}$ are measured, whereas in (b) the logical $\bar{X}_1\otimes \bar{Z}_2$ operator is measured. The overlap with the singlet $\epsilon$ (\ref{['eq:epsilon_def']}) is given by the fraction of measurement outcomes such that $(m_{\bar{X}},m_{\bar{Z}})=(-1,-1)$. In both cases, the logical operators can be determined either with an ideal round of error-correction or error-detection/post-selection.