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Redefining Lexicographical Ordering: Optimizing Pauli String Decompositions for Quantum Compiling

Qunsheng Huang, David Winderl, Arianne Meijer-van de Griend, Richie Yeung

TL;DR

This paper tackles efficient compilation of trotterized Hamiltonian evolutions under device connectivity constraints by introducing architecture-aware synthesis of Pauli gadgets. The core method propagates Clifford gates through the remainder of the Pauli polynomial and uses a Clifford tableau, enabling routing-free circuits and reduced CNOT counts. Empirical results show significant improvements in gate counts and circuit depth over Paulihedral and TKET across restricted and all-to-all architectures, with Trotter error remaining comparable on average. The approach offers practical gains for near-term quantum devices and lays groundwork for extensions to higher-order decompositions and optimized hardware mappings.

Abstract

In quantum computing, the efficient optimization of Pauli string decompositions is a crucial aspect for the compilation of quantum circuits for many applications, such as chemistry simulations and quantum machine learning. In this paper, we propose a novel algorithm for the synthesis of trotterized time-evolution operators that results in circuits with significantly fewer gates than previous solutions. Our synthesis procedure takes the qubit connectivity of a target quantum computer into account. As a result, the generated quantum circuit does not require routing, and no additional CNOT gates are needed to run the resulting circuit on a target device. We compare our algorithm against Paulihedral and TKET, and show a significant improvement for randomized circuits and different molecular ansatzes. We also investigate the Trotter error introduced by our ordering of the terms in the Hamiltonian versus default ordering and the ordering from the baseline methods and conclude that our method on average does not increase the Trotter error.

Redefining Lexicographical Ordering: Optimizing Pauli String Decompositions for Quantum Compiling

TL;DR

This paper tackles efficient compilation of trotterized Hamiltonian evolutions under device connectivity constraints by introducing architecture-aware synthesis of Pauli gadgets. The core method propagates Clifford gates through the remainder of the Pauli polynomial and uses a Clifford tableau, enabling routing-free circuits and reduced CNOT counts. Empirical results show significant improvements in gate counts and circuit depth over Paulihedral and TKET across restricted and all-to-all architectures, with Trotter error remaining comparable on average. The approach offers practical gains for near-term quantum devices and lays groundwork for extensions to higher-order decompositions and optimized hardware mappings.

Abstract

In quantum computing, the efficient optimization of Pauli string decompositions is a crucial aspect for the compilation of quantum circuits for many applications, such as chemistry simulations and quantum machine learning. In this paper, we propose a novel algorithm for the synthesis of trotterized time-evolution operators that results in circuits with significantly fewer gates than previous solutions. Our synthesis procedure takes the qubit connectivity of a target quantum computer into account. As a result, the generated quantum circuit does not require routing, and no additional CNOT gates are needed to run the resulting circuit on a target device. We compare our algorithm against Paulihedral and TKET, and show a significant improvement for randomized circuits and different molecular ansatzes. We also investigate the Trotter error introduced by our ordering of the terms in the Hamiltonian versus default ordering and the ordering from the baseline methods and conclude that our method on average does not increase the Trotter error.
Paper Structure (17 sections, 20 equations, 5 figures, 3 tables)

This paper contains 17 sections, 20 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries and Related Work
  3. Pauli Gadgets and Pauli Polynomials
  4. Trotter error
  5. Related algorithms
  6. Clifford Tableaus
  7. Methods
  8. Gate placement
  9. Heuristics
  10. Clifford tableau synthesis up-to-permutation
  11. Runtime Complexity Analysis
  12. CNOT Complexity Analysis
  13. Experiments
  14. Random Pauli polynomials
  15. UCCSD molecular ansatzes
  16. ...and 2 more sections

Figures (5)

  • Figure 1: Action of prepending Clifford gates $H$, $S$ and CNOT to an $n$ qubit Clifford tableau.
  • Figure 2: Step-by-step synthesis of time evolution operator $H=\alpha_0 IIZZ + \alpha_1 YXZX + \alpha_2 IIYZ + \alpha_3 YXZI$ from initial placement using methods in \ref{['subsec:gate_placement']}.$^{\ref{['note1']}}$
  • Figure 3: Unitary overlap between the exact time evolution operator and the synthesized circuit for 20 random Pauli polynomials (aggregated) and the molecules H2_P_631g and H4_P_sto3g.
  • Figure 4: Comparison of examined synthesis methods on CNOT count, circuit depth and 2-qubit gate depth for different IBM quantum computers.
  • Figure 5: The performance of the different synthesis methods on different metric for different size architectures with all-to-all connectivity (complete).