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Faster and shorter synthesis of Hamiltonian simulation circuits

Timothée Goubault de Brugière, Simon Martiel

TL;DR

This work tackles the efficient synthesis of Hamiltonian-simulation circuits expressed as sequences of Pauli rotations by introducing two greedy, Clifford-based heuristics that minimize either entangling gate count or entangling depth, with optional order preservation. It leverages a vector-encoded Pauli framework and Clifford conjugations to progressively reduce rotation support toward size $1$, forming Pauli networks that implement the rotations with fewer entangling resources. The results demonstrate substantial depth reductions (up to a factor of $4$ over state-of-the-art heuristics) on standard benchmarks, and the methods remain competitive on larger instances while supporting resynthesis of literature circuits. The algorithms are implemented in Rust with Python bindings and show strong potential for broader quantum-circuit optimization, albeit with hardware-aware transpilation overhead such as SWAP insertion. Overall, the approach provides fast, scalable, and practical tools for near-term quantum devices and generic circuit optimization.

Abstract

We devise greedy heuristics tailored for synthesizing quantum circuits that implement a specified set of Pauli rotations. Our heuristics are designed to minimize either the count of entangling gates or the depth of entangling gates, and they can be adjusted to either maintain or loosen the ordering of rotations. We present benchmark results demonstrating a depth reduction of up to a factor of 4 compared to the current state-of-the-art heuristics for synthesizing Hamiltonian simulation circuits. We also show that these heuristics can be used to optimize generic quantum circuits by decomposing and resynthesizing them.

Faster and shorter synthesis of Hamiltonian simulation circuits

TL;DR

This work tackles the efficient synthesis of Hamiltonian-simulation circuits expressed as sequences of Pauli rotations by introducing two greedy, Clifford-based heuristics that minimize either entangling gate count or entangling depth, with optional order preservation. It leverages a vector-encoded Pauli framework and Clifford conjugations to progressively reduce rotation support toward size , forming Pauli networks that implement the rotations with fewer entangling resources. The results demonstrate substantial depth reductions (up to a factor of over state-of-the-art heuristics) on standard benchmarks, and the methods remain competitive on larger instances while supporting resynthesis of literature circuits. The algorithms are implemented in Rust with Python bindings and show strong potential for broader quantum-circuit optimization, albeit with hardware-aware transpilation overhead such as SWAP insertion. Overall, the approach provides fast, scalable, and practical tools for near-term quantum devices and generic circuit optimization.

Abstract

We devise greedy heuristics tailored for synthesizing quantum circuits that implement a specified set of Pauli rotations. Our heuristics are designed to minimize either the count of entangling gates or the depth of entangling gates, and they can be adjusted to either maintain or loosen the ordering of rotations. We present benchmark results demonstrating a depth reduction of up to a factor of 4 compared to the current state-of-the-art heuristics for synthesizing Hamiltonian simulation circuits. We also show that these heuristics can be used to optimize generic quantum circuits by decomposing and resynthesizing them.
Paper Structure (25 sections, 13 equations, 4 figures, 7 tables, 3 algorithms)

This paper contains 25 sections, 13 equations, 4 figures, 7 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Outline
  3. Problem and framework
  4. Preliminaries
  5. Vectorizing Pauli operators.
  6. Conjugation by a CNOT.
  7. Conjugation by a H.
  8. Conjugation by a $\sqrt{X}$.
  9. Conjugation by a $S$.
  10. Encoding a set of Pauli operators.
  11. Pauli network synthesis
  12. On the hardness of those problems.
  13. Low entangling count synthesis
  14. Score function.
  15. Clifford chunks.
  16. ...and 10 more sections

Figures (4)

  • Figure 1: Commutation rules of one-qubit Clifford gates with a CNOT gate.
  • Figure 2: Visual comparison of the CNOT count and depth of typical circuits produced by the various methods on small instances. Those instances correspond respectively to entries LiH cmplt JW sto3g, H2O cmplt P sto3g, C2H4 cmplt JW sto3g, H2 cmplt P ccpvdz, and NH frz P sto3g of Tables \ref{['tab:tab_chem_1']} to \ref{['tab:tab_chem_depth_3']}.
  • Figure 3: Average count and depth resulting from the synthesis of a sequence of rotations over 40 qubits. Each point is averaged over 10 instances.
  • Figure 4: Average running time of the synthesis of a sequence of rotations over 40 qubits. Each point is averaged over 10 instances. Left data excluding method ph for clarity. Right same data including ph.

Theorems & Definitions (2)

  • Definition 1: Pauli network
  • Definition 2: Ordered Pauli network