Fault Tolerant Quantum Simulation via Symplectic Transvections

TL;DR

This work reframes fault-tolerant quantum simulation by synthesizing entire logical Trotter circuit blocks within a single stabilizer code block, rather than executing gates gate-by-gate. It uncovers a structural link between Trotter circuits and symplectic transvections, enabling both Clifford and non-Clifford Trotter circuits to be realized in a way that preserves stabilizer centralization and the Trotter pattern across logical and physical layers. The authors prove that Clifford Trotter circuits correspond to logical-to-physical transvections and extend this to arbitrary θ for non-Clifford Trotter circuits via a linear propagation decomposition, enabling fault-tolerant Hamiltonian simulation on any stabilizer code. They illustrate the approach with the [[8,3,3]] code and demonstrate feasibility on larger QLDPC codes, reporting a pseudo-threshold around 2.5e-3 in simulations, which suggests practical potential for algorithm-tailored, single-code-block fault tolerance in quantum simulation.

Abstract

Conventional approaches to fault-tolerant quantum computing realize logical circuits gate-by-gate, synthesizing each gate independently on one or more code blocks. This incurs excess overhead and doesn't leverage common structures in quantum algorithms. In contrast, we propose a framework that enables the execution of entire logical circuit blocks at once, preserving their global structure. This whole-block approach allows for the direct implementation of logical Trotter circuits - of arbitrary rotation angles - on any stabilizer code, providing a powerful new method for fault tolerant Hamiltonian simulation within a single code block. At the heart of our approach lies a deep structural correspondence between symplectic transvections and Trotter circuits. This connection enables both logical and physical circuits to share the Trotter structure while preserving stabilizer centralization and circuit symmetry even in the presence of non-Clifford rotations. We discuss potential approaches to fault tolerance via biased noise and code concatenation. While we illustrate the key principles using a $[[8,3,3]]$ code, our simulations show that the framework applies to Hamiltonian simulation on even good quantum LDPC codes. These results open the door to new algorithm-tailored, block-level strategies for fault tolerant circuit design, especially in quantum simulation.

Figures (7)

• Figure 1: Logical Clifford Trotter circuit on $k=3$ qubits. Here, $P$ refers to the Clifford Phase gate $R_Z(\frac{\pi}{2})$.
• Figure 2: Physical realization of Fig. \ref{['fig:qsk_clifford_3qubit']} on $[\![ 8,3,3 ]\!]$ code. Here, $P$ refers to the Clifford Phase gate $R_Z(\frac{\pi}{2})$.
• Figure 3: The updated physical circuit after considering the effect of a stabilizer to reduce the depth. By multiplying the original symplectic transvection with the stabilizer $S_4$, the support size of the symplectic transvection is reduced, leading to fewer CNOT gates and an overall reduction in the circuit depth.
• Figure 4: Non-Clifford logical Trotter circuit with 3 qubits, where the central Phase gate in Fig. \ref{['fig:qsk_clifford_3qubit']} is replaced by $R_z(\theta)$.
• Figure 5: Physical realization of non-Clifford logical Trotter circuit in Fig. \ref{['fig:3_qubit_non_Clifford_logical']} on $[\![ 8,3,3]\!]$ code. The red contributions in the decomposition illustrate the Clifford-transformed part ($\sin(\theta)$-weighted term), while the black contributions ($\cos(\theta)$-weighted term) correspond to the unchanged input operator. This decomposition demonstrates the propagation of Pauli operators through the central $R_z(\theta)$ gate, maintaining the consistency of logical operator mappings.

Theorems & Definitions (11)

• Definition 1
• Definition 2
• Lemma 1
• Theorem 2
• Corollary 3
• Remark 4
• Theorem 5
• Remark 6
• Remark 7
• Theorem 8