Error-strcture-tailored fault-tolerance

Fault-tolerant gate design based on specific error properties.

Fault tolerance is widely regarded as indispensable for achieving scalable and reliable quantum information processing. However, the spacetime overhead required for fault-tolerant quantum computation remains prohibitively large. A critical challenge arises in many quantum algorithms with Clifford + $\varphi$ compiling, where logical rotation gates $R_{Z_L}(\varphi)$ serve as essential components. The Eastin-Knill theorem fundamentally limits these operations, preventing their transversal implementation in quantum error correction codes and necessitating resource-intensive workarounds through $T$-gate compilation combined with magic state distillation and injection.

In our recent work entitled Error-structure-tailored early fault-tolerant quantum computing, we consider error-structure-tailored fault tolerance, where fault-tolerance conditions are analyzed by combining perturbative analysis of realistic dissipative noise processes with the structural properties of stabilizer codes. Based on this framework, we design a 1-fault-tolerant continuous-angle rotation gate $R_{Z_L}(\varphi)$ in stabilizer codes, implemented via dispersive-coupling Hamiltonians.

Illustration of the main idea. For the Clifford+$\varphi$ circuits, unlike the usual approach to compile the $R_Z(\varphi)$ gates to many $T$ gates and then perform magic state distillation or cultivation, we directly prepare the $\ket{r_\varphi}:=R_{Z}(\varphi)\ket{+}$ ancilla based on the error-structure-tailored FT analysis and perform direct injection.

Our approach could circumvent the need for $T$-gate compilation and distillation, offering a hardware-efficient solution that maintains simplicity, minimizes physical footprint, and requires only nearest-neighbor interactions. Integrating with recent small-angle-state preparation techniques, we can suppress the gate error to $9|\varphi|\cdot\,p^2$ for small rotation angles $|\varphi|<0.2$ rad (where $p$ denotes the physical error rate). For current achievable hardware parameters ($p=1\cdot\,10^{-3}$), this enables reliable execution of over $10^7$ small-angle rotations when $|\varphi|\approx\,10^{-3}$, meeting the requirements of many near-term quantum applications. We estimate that, compared to the 15-to-1 magic state distillation and magic state cultivation approaches, our method reduces spacetime resource costs by factors of $1337.5\times$ and $43.6\times$, respectively, for a Heisenberg Hamiltonian simulation task under realistic hardware assumptions.

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