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Non-Clifford Fusion: T-Gate Optimization for Quantum Simulation

Yingheng Li, Xulong Tang, Paul Hovland, Ji Liu

TL;DR

This work targets efficient Hamiltonian simulation on fault-tolerant quantum computers by reducing T-gate count and depth. It introduces Non-Clifford Fusion (NCF), a two-stage framework that groups Pauli strings and uses Clifford circuit conjugation to merge RZ rotations into single-qubit or two-qubit unitaries, enabling parallelized and more compact synthesis with $U3$ or multi-qubit block synthesizers. Across 13 benchmarks, single-qubit NCF achieves average reductions of about 57% in T-gate count and around 49% in T-gate depth and Clifford count compared with state-of-the-art baselines; two-qubit NCF further improves T-gate counts, though depth gains are limited by unitary width. The method leverages tableau-based Clifford conjugation, anticommuting/grouping strategies, and a sliding window to balance optimization opportunities with computational cost, offering a practical pathway to more scalable fault-tolerant Hamiltonian simulation.

Abstract

Hamiltonian simulation is a key quantum algorithm for modeling complex systems. To implement a Hamiltonian simulation, it is typically decomposed into a list of Pauli strings, each corresponds to an RZ rotation gate with many Clifford gates. These RZ gates are generally synthesized into a sequence of Clifford and T gates in fault-tolerant quantum computers, where the T-gate count and T-gate depth are critical metrics for such systems. In this paper, we propose NCF, a compilation framework that reduces both the T-gate count and T-gate depth for Hamiltonian simulation. NCF partitions Pauli strings into groups, where each group can be conjugated (i.e., transformed) into a list of Pauli strings that apply quantum gates on a restricted subset of qubits, allowing for simultaneous synthesis of the whole group and reducing both T-gate count and depth. Experimental results demonstrate that NCF achieves an average reduction of 57.4%, 49.1%, and 49.0% in T-gate count, T-gate depth, and Clifford count, respectively, compared to the state-of-the-art method.

Non-Clifford Fusion: T-Gate Optimization for Quantum Simulation

TL;DR

This work targets efficient Hamiltonian simulation on fault-tolerant quantum computers by reducing T-gate count and depth. It introduces Non-Clifford Fusion (NCF), a two-stage framework that groups Pauli strings and uses Clifford circuit conjugation to merge RZ rotations into single-qubit or two-qubit unitaries, enabling parallelized and more compact synthesis with or multi-qubit block synthesizers. Across 13 benchmarks, single-qubit NCF achieves average reductions of about 57% in T-gate count and around 49% in T-gate depth and Clifford count compared with state-of-the-art baselines; two-qubit NCF further improves T-gate counts, though depth gains are limited by unitary width. The method leverages tableau-based Clifford conjugation, anticommuting/grouping strategies, and a sliding window to balance optimization opportunities with computational cost, offering a practical pathway to more scalable fault-tolerant Hamiltonian simulation.

Abstract

Hamiltonian simulation is a key quantum algorithm for modeling complex systems. To implement a Hamiltonian simulation, it is typically decomposed into a list of Pauli strings, each corresponds to an RZ rotation gate with many Clifford gates. These RZ gates are generally synthesized into a sequence of Clifford and T gates in fault-tolerant quantum computers, where the T-gate count and T-gate depth are critical metrics for such systems. In this paper, we propose NCF, a compilation framework that reduces both the T-gate count and T-gate depth for Hamiltonian simulation. NCF partitions Pauli strings into groups, where each group can be conjugated (i.e., transformed) into a list of Pauli strings that apply quantum gates on a restricted subset of qubits, allowing for simultaneous synthesis of the whole group and reducing both T-gate count and depth. Experimental results demonstrate that NCF achieves an average reduction of 57.4%, 49.1%, and 49.0% in T-gate count, T-gate depth, and Clifford count, respectively, compared to the state-of-the-art method.
Paper Structure (27 sections, 1 theorem, 3 equations, 8 figures, 5 tables)

This paper contains 27 sections, 1 theorem, 3 equations, 8 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Hamiltonian Simulation
  4. Pauli String
  5. Tableau Representation
  6. Commutation and Anticommutation
  7. Generator and Generated Pauli Strings
  8. Clifford Circuit Conjugation
  9. Motivation and Opportunity
  10. Our Method
  11. NCF Grouping
  12. Single-qubit Grouping
  13. Two-qubit Grouping
  14. Clifford Circuit Generation
  15. Sliding Window Strategy
  16. ...and 12 more sections

Key Result

Lemma 4.1

Let $C$ be a Clifford circuit and let $P_1,\dots,P_m$ be Pauli strings on $n$ qubits. If the conjugated Pauli strings $P'_j := C P_j C^\dagger$ act non-trivially only on a fixed subset of qubits $S \subseteq \{1,\dots,n\}$ for $j \in \{1, \cdots, m \}$, then for any product $Q=\prod_{j=1}^m P_j^{a_j

Figures (8)

  • Figure 1: Overviews of NCF and the baseline.
  • Figure 2: An example of Hamiltonian simulation and Pauli String circuit.
  • Figure 3: Examples of Tableau representation and Clifford conjugation.
  • Figure 4: Examples of NCF applied to six Pauli strings for both single and two-qubit unitaries.
  • Figure 5: An example of conjugating four Pauli strings.
  • ...and 3 more figures

Theorems & Definitions (2)

  • Lemma 4.1
  • proof