efficient-clifford-t-synthesis

name	efficient-clifford-t-synthesis
description	Efficient Clifford+T synthesis methodology for small-angle rotations with application to Trotterization - reducing T gate cost from O(log 1/δ) to Õ(θ²/δ) for small angles in fault-tolerant quantum compilation.
version	1.0.0
author	arxiv:2605.31544 (Bothe et al.)
created	2026-06-02T00:00:00.000Z
arxiv_id	2605.31544
category	quantum-compiling
activation_keywords	["Clifford+T synthesis","small-angle rotation","fault-tolerant quantum compilation","Trotterization","T gate optimization","magic state distillation","quantum circuit synthesis"]

Efficient Clifford+T Synthesis for Small-Angle Rotations

Overview

This methodology addresses a critical challenge in fault-tolerant quantum compilation: the high overhead of T gates in Clifford+T synthesis of rotation gates. The key breakthrough shows that T gate cost can be dramatically reduced for small rotation angles, which is particularly important for algorithms like Trotterization that are dominated by small-angle rotations.

Key Innovation

Previous belief: Clifford+T rotation synthesis had a high cost independent of rotation angle θ, requiring O(log 1/δ) T gates.

New finding: For small angles, T cost reduces to Õ(θ²/δ), returning to existing O(log 1/δ) results in worst case.

Technical Framework

Angle-Dependent Synthesis

1. Small-angle optimization:

   • T gate cost: Õ(θ²/δ) instead of O(log 1/δ)
   • Significant reduction when θ is small
   • Worst-case fallback to standard methods

2. Quasi-probability methods:

   • Further reduces total T cost by orders of magnitude
   • Small overhead in sample complexity
   • Quasi-probability mixtures of Clifford+T fallback channels

Trotterization Application

• Gate cost in small step limit: Becomes constant as Trotter step size approaches zero
• Order-of-magnitude reduction: Even for large step sizes
• Resource estimation: New θ-dependent formulas for fault-tolerant algorithms

Implementation Components

Core Synthesis Algorithm

Input: Rotation angle θ, target precision δ
Output: Clifford+T circuit with optimized T gate count

1. Assess angle magnitude:
   - If θ small: use small-angle synthesis (Õ(θ²/δ) T gates)
   - If θ large: use standard synthesis (O(log 1/δ) T gates)

2. Apply quasi-probability decomposition:
   - Generate fallback channels
   - Optimize sample complexity vs T cost tradeoff

3. Resource estimation:
   - Calculate θ-dependent T gate requirements
   - Estimate magic state distillation resources

Resource Estimation Formulas

New θ-dependent formulas for:

• T gate count estimation
• Magic state resource requirements
• Sample complexity bounds

Applications

Primary Applications

1. Trotterized Hamiltonian simulation