Breaking even with magic: demonstration of a high-fidelity logical non-Clifford gate

TL;DR

This work demonstrates a practical, fault-tolerant route to universal quantum computation by eight-qubit magic-state preparation within a [[6,2,2]] code, enabling a fault-tolerant non-Clifford CH gate with infidelity below the corresponding physical gate. The authors experimentally realize the protocol on a trapped-ion processor, achieving a magic-state infidelity around 7×10^-5 and a CH-gate infidelity around 2.3×10^-4, breaking even on the non-Clifford two-qubit operation. They further show through circuit-level stabilizer simulations that self-concatenation scales to four magic states in a [[36,4,4]] code with extremely low projected logical errors, outlining a path to high-distance fault-tolerant computation via code switching and level-raising teleportation. Overall, the work provides a low-overhead, scalable framework for producing high-fidelity magic states and fault-tolerant non-Clifford gates, highlighting practical routes toward larger, robust quantum processors.

Abstract

Encoding quantum information to protect it from errors is essential for performing large-scale quantum computations. Performing a universal set of quantum gates on encoded states demands a potentially large resource overhead and minimizing this overhead is key for the practical development of large-scale fault-tolerant quantum computers. We propose and experimentally implement a magic-state preparation protocol to fault-tolerantly prepare a pair of logical magic states in a [[6,2,2]] quantum error-detecting code using only eight physical qubits. Implementing this protocol on H1-1, a 20 qubit trapped-ion quantum processor, we prepare magic states with experimental infidelity $7^{+3}_{-1}\times 10^{-5}$ with a $14.8^{+1}_{-1}\%$ discard rate and use these to perform a fault-tolerant non-Clifford gate, the controlled-Hadamard (CH), with logical infidelity $\leq 2.3^{+9}_{-9}\times 10^{-4}$. Notably, this significantly outperforms the unencoded physical CH infidelity of $10^{-3}$. Through circuit-level stabilizer simulations, we show that this protocol can be self-concatenated to produce extremely high-fidelity magic states with low space-time overhead in a [[36,4,4]] quantum error correcting code, with logical error rates of $6\times 10^{-10}$ ($5\times 10^{-14}$) at two-qubit error rate of $10^{-3}$ ($10^{-4}$) respectively.

Figures (8)

• Figure 1: Preparing and benchmarking magic states: (a) Schematic of the the self-dual $H_6$ [[6,2,2]] code with physical qubits depicted as vertices of a hexagon, with generating stabilizers and logical operators indicated by shaded shapes. (b) Schematic of state preparation protocol where an arbitrary state encoder encodes physical states $\ket{\psi_{0,1}}$ into the logical states of a $H_6$ code (circuit shown in (d)), followed by a check stage using an ancilla (physical) Bell/GHZ pair. Although the encoder circuit works for arbitrary inputs, the check stage requires that $\ket{\psi_{0,1}}$ be stabilized by transversal gates. Specific checks with for preparing two or one magic states for $\ket{\psi_{0,1}} \in \{\ket{H^+},\ket{Y^-}\}$ are shown in (e,f) respectively. As explained in the text, this protocol may be run on physical qubits, in which case the output is two "level-1" magic states in the [[6,2,2]] code, or it may be run one concatenation level higher, in which case the two wires represent logical qubits in two separate [[6,2,2]] blocks and the CNOTs in the encoding circuit are transversal and the output is four "level-2" magic states. (c) A fault-tolerant circuit for benchmarking the combined infidelity of two logical magic states on different code blocks. Twirling stages, described in the text, are used to decohere errors to obtain rigorous fidelity bounds. In these experiments, data is post-selected on having a $+1$ measurement outcome for the $\bar{Y}_2$ measurements on the second qubit of each block, as well as each of the x-syndrome measurements ($QED_X$ block). For preparing resource states offline in a fault tolerant quantum computing architecture, this post-selection would be replaced by a repeat-until-success protocol with modest additional spatial overhead.
• Figure 2: On H1-1, a variable number of logical $R_y(-\pi/4)$ rotations were performed on one of the logical qubits in a [[6,2,2]] code using magic states generated by our experimental protocol. The blue line is the model under the maximum likelihood estimates (MLE) for the model parameters given the likelihood function that arises from the observed data. Each orange line is a random sample from a Markov chain Monte Carlo (MCMC) sampling of the model parameters under the same likelihood function. The spread in the MCMC samples lines gives a sense for the variance in the model predictions given the data. Using MLE to estimate the model parameters yields an infidelity of $(7^{+3}_{-2})\times10^{-5}$ for a logical $R_y(-\pi/4)_L$ rotation, where the confidence interval is derived from MCMC.
• Figure 3: Decomposition of the controlled-Hadamard gate into $R_y(\pm \pi/4)$ rotations which can be implemented using magic states and a $CZ$ gate which is transversal in the [[6,2,2]] code.
• Figure 4: The failure probabilities for the $\ket{+}$ distillation protocol and its concatenation in circuit-level stabilizer simulations. A log plot shows the expected scaling rates of $O(p^2)$ and $O(p^4)$ respectively. Using linear regression, we obtain the estimates $26^{+3}_{-3} \times p^{2.08^{+3}_{-3}}$ and $1000^{+500}_{-500} \times p^{4.1^{+1}_{-1}}$ for the scaling of the logical error of the level 1 and level 2 protocols, respectively.
• Figure 5: A fault-tolerant state prep circuit for $\ket{00}_L$ in the [[6,2,2]] code. We used StabGraph amaro2019 to find a bipartite graph state representation of this stabilizer state. Qubits 0 and 2 are control qubits, and qubits 1,3,4, and 5 are target qubits. We only need to flag qubits 0 and 2 because hook errors on the "target qubits" are equivalent to weight one errors up to stabilizers and $Z$ logicals.