STAR-Magic Mutation: Even More Efficient Analog Rotation Gates for Early Fault-Tolerant Quantum Computer

Abstract

We introduce STAR-magic mutation, an efficient protocol for implementing logical rotation gates on early fault-tolerant quantum computers. This protocol judiciously combines two of the latest state preparation protocols: transversal multi-rotation protocol and magic state cultivation. It achieves a logical rotation gate with a favorable error scaling of $\mathcal{O}(θ_L^{2(1-Θ(1/d))}p_{\text{ph}})$, while requiring only the ancillary space of a single surface code patch. Here, $θ_L$ is the logical rotation angle, $p_{\text{ph}}$ is the physical error rate, and $d$ is the code distance. This scaling marks a significant improvement over the previous state-of-the-art, $\mathcal{O}(θ_L p_{\text{ph}})$, making our protocol particularly powerful for implementing a sequence of small-angle rotation gates, like Trotter-based circuits. Notably, for $θ_L \lesssim 10^{-5}$, our protocol achieves a two-order-of-magnitude reduction in both the execution time and the error rate of analog rotation gates compared to the standard $T$-gate synthesis using cultivated magic states. Building upon this protocol, we also propose a novel quantum computing architecture designed for early fault-tolerant quantum computers, dubbed ``STAR ver.~3". It employs a refined circuit compilation strategy based on Clifford+$T$+$φ$ gate set, rather than the conventional Clifford+$T$ or Clifford+$φ$ gate sets. We establish a theoretical bound on the feasible circuit size on this architecture and illustrate its capabilities by analyzing the spacetime costs for simulating the dynamics of quantum many-body systems. Specifically, we demonstrate that our architecture can simulate biologically-relevant molecules or lattice models at scales beyond the reach of exact classical simulation, with only a few hundred thousand physical qubits, even assuming a realistic error rate of $p_{\text{ph}}=10^{-3}$.

Figures (13)

• Figure 1: Quantum circuit for implementing an analog $Z$-rotation gate $\hat{R}_{z,L}(\theta_L)$ on a single logical qubit. $M_Z$ denotes a destructive $Z_L$-measurement on an encoded qubit. The sign of the resulting rotation angle is randomly determined with equal probability, depending on the measurement outcome.
• Figure 2: Quantum circuit for implementing an analog multi-Pauli rotation gate $\hat{R}_{P,L}(\theta)$ on any encoded state $\ket{\psi}_L$. In this setup, we can choose any Pauli string operator $\hat{P}$. The sign of the resulting rotation angle is randomly determined with equal probability, depending on the outcome of the destructive $X$-measurement.
• Figure 3: Intuitive picture of the transversal multi-rotation (TMR) protocol. The transversal multi-Pauli rotation gate (green) produces superpositions of logical resource states with different logical angles and different stabilizer eigenvalues. By performing stabilizer measurement (yellow), we can post-select only the target resource state, discarding other erroneous resource states. However, under physical noises, erroneous resource states rarely appear in the output state (orange), leading to a finite error rate of the TMR protocol.
• Figure 4: Schematic of the TMR protocol on a rotated surface code with code distance $d=5$. (a) First, the ancilla state is initialized in $\ket{+}_L$, then multiple rotation gates are applied along the logical $Z_L$ operator. Each multi-$Z$ rotation gate can be performed by combining nearest-neighbor CNOT gates and single-qubit $Z$-rotation gates. (b) After applying the transversal multi-Pauli rotation gates, we perform a post-selection on the hatched stabilizers to remove critical errors that bring leading logical errors. The remaining errors detected by other stabilizers can be corrected in the subsequent quantum error correction procedure.
• Figure 5: STAR-magic mutation and the previous approaches Akahoshi2023Toshio2024 for directly executing a logical rotation gate with an arbitrary angle. In the original proposal of the STAR architecture (STAR ver. 1) Akahoshi2023, any analog rotation gate is executed using an injection protocol based on the [[4,1,1,2]] subsystem code (green circle). Whereas, STAR ver. 2 employs a more refined approach based on the TMR protocol Toshio2024. In this approach, the initial trials of RUS are executed using the TMR and PCEC protocols (pink circle). Once the RUS angle exceeds a specific threshold, the subsequent RUS trials are executed by using the injection protocol in STAR ver. 1. On the other hand, STAR-magic mutation (or STAR ver. 3) adopts another type of adaptive switching during the RUS process. In this protocol, as with STAR ver. 2, we perform the RUS trials using the TMR and PCEC protocols as long as the RUS angle is smaller than a predefined threshold $\theta_{\text{th}}$ (analog stage). However, in STAR ver. 3, PCEC is modified to suppress the higher-order errors in resource states as discussed in Sec. \ref{['sec:higher-order']}. Then, once the RUS angle $\theta_{\text{RUS}}$ gets greater than or equal to the threshold angle $\theta_{\text{th}}$ (digital stage), we perform the next rotation gate using a standard $T$-gate synthesis (purple and blue circles). In this stage, each $T$-gate is executed by preparing a magic state via MSC or MSD.

Theorems & Definitions (1)

• Theorem 1: Asymptotic behavior of STAR-magic mutation under the supply of noiseless magic states