Partially Fault-Tolerant Quantum Computation for Megaquop Applications

Abstract

Partially fault-tolerant quantum computing (FTQC) has recently emerged as a promising approach for the execution of megaquop-scale circuits with millions of logical operations. In this work, we demonstrate the strengths and the limitations of this approach by conducting quantum resource estimation (QRE) of the space--time-efficient analog rotation (STAR) architecture using realistic hardware specifications for superconducting processors, and compare it against the QRE of the full FTQC architecture. We show how the performance of the STAR architecture's protocols is affected by hardware improvements. We also reduce the space requirements for partial FTQC by developing a procedure leveraging code growth to decrease the size of a factory producing analog rotation states. Our results reveal a non-trivial dependence of the optimal pre-growth code distance on the rotation angle with respect to post-growth infidelity. Further, we analyze space--time trade-offs between the factory size and the error-mitigation overhead, and observe that in an application-agnostic setting, there is a Goldilocks zone for circuits in the regime of roughly $10^5$--$10^6$ small-angle rotation gates. We show that quantum simulation of 2D Fermi--Hubbard model systems is a particularly well-suited application for the STAR architecture, requiring only hundreds of thousands of physical qubits and runtimes on the order of minutes for modest system sizes. Due to its favourable algorithmic scaling to larger system sizes, utility-scale simulation of the 2D Fermi--Hubbard model could potentially be attained using partial FTQC.

Figures (28)

• Figure 1: Gate teleportation circuit for implementing the Pauli-product rotation $R_P(\theta)=\exp(-i\theta P/2)$, where $P$ is any multi-qubit tensor product of Pauli operators, by consuming the ancillary rotation resource state $|m_{\theta}\rangle$toshio2024practical. Here, $M_{P\otimes Z}$ denotes a multi-Pauli measurement of the operator $P\otimes Z$. If the ancilla measurement yields $-1$, then a correction by $P$ is required, which should be done in software. The state $\ket{\psi}$ undergoes a rotation of either $\theta$ or $-\theta$ depending on the outcome of the multi-Pauli measurement.
• Figure 3: Success probability of the multi-Pauli rotation protocol in the noise-free limit, for a target rotation angle $\theta_* = 10^{-3}$, and for various weights $m$ of the Pauli-rotation generator (as shown in the legend).
• Figure 4: Success rate of the transversal rotation state preparation protocol, and logical infidelity of the resource states prepared by this protocol, as a function of the physical error rate. The results of our simulation (shown as solid lines) are compared with those presented in Ref. Toshio2025PFTQC (shown as circles and dashed line) for $\theta_{*}=10^{-3}$ and $w=2$. (a) Success rates for $d = 11, 17$ (solid lines) obtained from our simulations. For comparison, success rates are shown for $d = 12, 18$ (circles) from Fig. 9 of Ref. Toshio2025PFTQC). (b) Logical infidelity from our simulation for $d = 7$ in comparison to the result for $d = 6$ presented in Fig. 7 of Ref. Toshio2025PFTQC). The discrepancies between the two results are primarily due to differences in code type and distance. This work uses a rotated surface code of odd distances, whereas the cited paper uses an unrotated surface code of even distances. Additionally, our use of odd distances requires an additional single qubit physical rotation along with the two-qubit rotations.
• Figure 5: Success rate of the transversal multi-rotation resource state preparation protocol as a function of the code distance, for the optimal post-selection method with $\theta_{*} = 10^{-4}$ and $w=2,3$ (indicated in the legend), and for target and desired hardware parameter specifications. Circles represent the simulation results, while dashed lines indicate fits. We find that the success rate is better for $w=3$ than for $w=2$.
• Figure 6: Infidelity of resource states prepared using the transversal multi-rotation protocol as a function of code distance, for the optimal post-selection method with $\theta_{*} = 10^{-4}$ and $w=2,3$ (indicated in the legend), and for target and desired hardware parameter specifications. The irregularities in the $w=3$ plots are due to the different weight of the final physical rotation.