Nontrivial multi-product commutation relation toward reducing T-count in sequential Pauli-based computation

TL;DR

The paper tackles the challenge of reducing the $T$-count in Clifford$+T$ circuits for fault-tolerant quantum computing by introducing the nonlocal multi-product commutation relation (MCR) within a sequential Pauli-based computation. It formalizes MCR for four multi-Pauli rotation axes, shows how to construct sequential PBCs that satisfy MCR, and demonstrates how MCR enables reorderings beyond pairwise commutativity. A benchmarking framework based on quantum circuit unoptimization is used to reveal that current compilers do not exploit MCR, while an MCR-aware compiler can restore or surpass the original $T$-count, indicating substantial untapped potential for compiler design. The results suggest that integrating MCR-aware transformations into quantum compilers can meaningfully reduce $T$-count for both benchmark and practically relevant circuits, with implications for resource efficiency in FTQC. Overall, the work advances circuit optimization by expanding rewrite rules beyond local commutativity and points to scalable strategies and generalizations to broader circuit classes as fruitful directions.

Abstract

Quantum compilers that reduce the number of T gates are essential for minimizing the overhead of fault-tolerant quantum computation. Achieving further T-count reduction calls for identifying equivalent circuit transformation rules beyond those utilized in existing tools. In this paper, we rewrite any given Clifford+T circuit using a Clifford block followed by a sequential Pauli-based computation, and introduce a nontrivial, ancilla-free transformation rule, the multi-product commutation relation (MCR). MCR constructs gate sequences based on specific commutation properties among multi-Pauli operators, yielding seemingly non-commutative instances that can be commuted, thereby enabling gate orderings that cannot be derived from pairwise commutation alone. To evaluate whether existing compilers account for this commutation rule, we create a benchmark circuit dataset using quantum circuit unoptimization. This approach intentionally adds redundancy to the circuit while keeping its equivalence, allowing a quantitative evaluation of compiler performance by comparison with the original circuit. Our numerical experiments reveal that the transformation rule based on MCR is not yet incorporated into current compilers, despite their demonstrated effectiveness for T-count reduction. These findings suggest an untapped potential for further T-count reduction by integrating MCR-aware transformations, paving the way for improvements in quantum compilers.

Figures (17)

• Figure 1: An overview of the quantum compiler benchmarking procedure for Clifford$+T$ circuits using MCR-based unoptimization. Starting from an input Clifford$+T$ circuit, we first reorder all $T$ gates with adjacent Clifford gates to separate the Clifford part and yield a sequential Pauli-based computation. Then, we apply the multi-product commutation relation (MCR) to the non-Clifford block, intentionally introducing redundant gates and yielding an unoptimized circuit that is functionally equivalent but has a larger $T$-count. A target quantum compiler then attempts to compress this circuit and produces an optimized output, whose $T$-count is compared with that of the original circuit. If the compiler fails to recover to the original optimal $T$-count, analyzing the gap makes it possible to improve the compiler's performance.
• Figure 2: Representation of quantum gates used in this paper. The rotation axis is indicated inside each box, and the rotation angle is shown outside. Gates with rotation angles of $\pm\pi/4$ (non-Clifford) are colored green, while those with $k \pi /2 \ (k \in \mathbb{Z}$, Clifford) are colored orange.
• Figure 3: Decomposition of the multi-Pauli rotation gate $R_{ZZZZ}\qty(\theta)$. The gate applies a rotation about the tensor product of four Pauli-$Z$ operators, and is decomposed into a standard circuit using a sequence of CNOT gates and a single-qubit $Z$-rotation.
• Figure 4: Decomposition of the three-qubit rotation gate $R_{YXZ}\qty(\theta)$. The gate implements a rotation about the tensor product of $Y$, $X$, and $Z$, and is decomposed into a standard circuit using local Clifford gates and a single $Z$-rotation.
• Figure 5: Transformation of a circuit segment containing one $T$ gate surrounded by Clifford gates. In the transformed circuit, the Clifford gates remain unchanged, and a multi-Pauli rotation gate $R_{XYX}\qty(-\pi/4)$ is appended at the right end.

Theorems & Definitions (7)

• Definition 1: $T$ layer
• Definition 2: Multi-product commutation relation
• Theorem 1
• proof
• Theorem 2: Construction of rotation axes satisfying MCR
• proof
• Definition 3: Quantum circuit unoptimization unoptimization