[Paper Review] Faster and shorter synthesis of Hamiltonian simulation circuits

Timothée Goubault de Brugière, Simon Martiel|arXiv (Cornell University)|Apr 4, 2024

Simulation Techniques and ApplicationsDecision Sciences3 citations

TL;DR

This paper introduces greedy heuristics for synthesizing quantum circuits that implement sequences of Pauli rotations, optimizing for either entangling gate count or depth. By greedily selecting Clifford transformations that maximize leading identity accumulation, the method reduces circuit depth by up to a factor of 4 compared to state-of-the-art approaches, while remaining efficient and extensible to order-preserving synthesis.

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.

Motivation & Objective

• To address the challenge of minimizing entangling gate count and depth in quantum circuits implementing Pauli rotations for Hamiltonian simulation.
• To overcome limitations of existing methods that rely on codiagonalization and phase polynomial synthesis, which introduce additional entangling gates.
• To develop a greedy, low-overhead approach that simultaneously optimizes basis changes and gate count by tracking leading identities in Pauli operator encodings.
• To extend the method to order-preserving synthesis for broader applicability in generic quantum circuit optimization.
• To achieve competitive performance on small instances while maintaining low runtime, complementing existing large-structure optimization techniques.

Proposed method

• The method uses a vectorized representation of Pauli operators in $\mathbb{F}_2^{2n}$ to efficiently track Pauli terms and their conjugation under Clifford gates.
• A score function is defined to measure the increase in leading identities (I terms) on qubits after applying a two-qubit Clifford chunk, favoring reductions in non-Clifford gate count.
• The algorithm explores a set of 18 canonical single-CNOT Clifford chunks, including $CNOT \cdot (U_1 \otimes U_2)$ with $U_1 \in \{I, H, X\}$, $U_2 \in \{I, H, S\}$, to identify optimal basis changes.
• The greedy selection prioritizes chunks that maximize the score function, effectively reducing the number of non-trivial Pauli terms and enabling more efficient single-qubit rotation implementations.
• The method supports both depth-optimized and count-optimized variants by adjusting the cost function, and can be adapted to preserve rotation order for generic circuit synthesis.
• Post-processing removes redundant Clifford gates that do not affect the score function, ensuring minimal circuit overhead.

Experimental results

Research questions

• RQ1 Can a greedy, low-overhead heuristic outperform state-of-the-art methods in minimizing entangling gate depth for Pauli rotation sequences in Hamiltonian simulation?
• RQ2 How effective is the use of leading identity accumulation as a score function for guiding Clifford basis changes in circuit optimization?
• RQ3 To what extent can the method maintain or relax rotation ordering while still achieving significant depth and count reductions?
• RQ4 Can the proposed heuristics be effectively extended to generic quantum circuit optimization beyond Hamiltonian simulation?
• RQ5 How does the method scale in performance and efficiency compared to codiagonalization and phase polynomial synthesis for larger instances?

Key findings

• The proposed heuristics achieve up to a 4-fold reduction in circuit depth compared to the current state of the art for Hamiltonian simulation circuits.
• The method reduces entangling gate count effectively, with performance competitive on larger instances due to low execution time and greedy optimization.
• The score function based on leading identity accumulation enables efficient identification of optimal Clifford transformations without explicit codiagonalization.
• The approach remains effective even when rotation order must be preserved, enabling application to generic quantum circuit synthesis.
• The method complements existing large-structure optimization techniques like codiagonalization and phase polynomial synthesis, particularly excelling on small to medium circuits.
• Benchmark results confirm that the greedy strategy captures significant optimizations that are missed by more exhaustive or structured methods.