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arXiv:2503.11153

Hamiltonian Reordering for Shallower Trotterization Circuits

Cédric Ho Thanh

Published Mar 14, 2025• 0 citations
arXivHTMLPDF

Abstract

Quantum simulation is a popular application of quantum computing, but its practical realization is hindered by the technical limitations of current devices. In this work, we focus on preprocessing Hamiltonians before Trotterization to generate shallower evolution circuits, which are less prone to noise and decoherence. Specifically, we apply graph coloring techniques to reorder Pauli terms and increase "gate parallelism". We benchmark two coloring algorithms, and report the depth reduction and computational overhead. Then, we examine how these optimized circuits affect the performance of the Quantum Approximate Optimization Algorithm (QAOA). Our results show that shallower circuits lead to faster convergence and reach higher energy levels compared to their non-reordered counterparts.

DOI
10.48550/arxiv.2503.11153

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Quantum simulation is a popular application of quantum computing, but its practical realization is hindered by the technical limitations of current devices. In this work, we focus on preprocessing Hamiltonians before Trotterization to generate shallower evolution circuits, which are less prone to noise and decoherence. Specifically, we apply graph coloring techniques to reorder Pauli terms and increase "gate parallelism". We benchmark two coloring algorithms, and report the depth reduction and computational overhead. Then, we examine how these optimized circuits affect the performance of the Quantum Approximate Optimization Algorithm (QAOA). Our results show that shallower circuits lead to faster convergence and reach higher energy levels compared to their non-reordered counterparts.
Simulating the time evolution of a physical system at quantum mechanical levels of detail - known as Hamiltonian Simulation (HS) - is an important and interesting problem across physics and chemistry. For this task, algorithms that run on quantum computers are known to be exponentially faster than classical algorithms; in fact, this application motivated Feynman to propose the construction of quantum computers. Nonetheless, there are challenges in reaching this performance potential. Prior work has focused on compiling circuits (quantum programs) for HS with the goal of maximizing either accuracy or gate cancellation. Our work proposes a compilation strategy that simultaneously advances both goals. At a high level, we use classical optimizations such as graph coloring and travelling salesperson to order the execution of quantum programs. Specifically, we group together mutually commuting terms in the Hamiltonian (a matrix characterizing the quantum mechanical system) to improve the accuracy of the simulation. We then rearrange the terms within each group to maximize gate cancellation in the final quantum circuit. These optimizations work together to improve HS performance and result in an average 40% reduction in circuit depth. This work advances the frontier of HS which in turn can advance physical and chemical modeling in both basic and applied sciences.
The Quantum Approximate Optimization Algorithm (QAOA), which is a variational quantum algorithm, aims to give sub-optimal solutions of combinatorial optimization problems. It is widely believed that QAOA has the potential to demonstrate application-level quantum advantages in the noisy intermediate-scale quantum(NISQ) processors with shallow circuit depth. Since the core of QAOA is the computation of expectation values of the problem Hamiltonian, an important practical question is whether we can find an efficient classical algorithm to solve quantum mean value in the case of general shallow quantum circuits. Here, we present a novel graph decomposition based classical algorithm that scales linearly with the number of qubits for the shallow QAOA circuits in most optimization problems except for complete graph case. Numerical tests in Max-cut, graph coloring and Sherrington-Kirkpatrick model problems, compared to the state-of-the-art method, shows orders of magnitude performance improvement. Our results are not only important for the exploration of quantum advantages with QAOA, but also useful for the benchmarking of NISQ processors.
Various Hamiltonian simulation algorithms have been proposed to efficiently study the dynamics of quantum systems on a quantum computer. The existing algorithms generally approximate the time evolution operators, which may need a deep quantum circuit that is beyond the capability of near-term noisy quantum devices. Here, focusing on the time evolution of a fixed input quantum state, we propose an adaptive approach to construct a low-depth time evolution circuit. By introducing a measurable quantifier that characterizes the simulation error, we use an adaptive strategy to learn the shallow quantum circuit that minimizes that error. We numerically test the adaptive method with electronic Hamiltonians of the H_{2}O and H_{4} molecules, and the transverse field Ising model with random coefficients. Compared to the first-order Suzuki-Trotter product formula, our method can significantly reduce the circuit depth (specifically the number of two-qubit gates) by around two orders while maintaining the simulation accuracy. We show applications of the method in simulating many-body dynamics and solving energy spectra with the quantum Krylov algorithm. Our work sheds light on practical Hamiltonian simulation with noisy-intermediate-scale-quantum devices.
The quantum approximate optimization algorithm (QAOA) generates an approximate solution to combinatorial optimization problems using a variational ansatz circuit defined by parameterized layers of quantum evolution. In theory, the approximation improves with increasing ansatz depth but gate noise and circuit complexity undermine performance in practice. Here, we investigate a multi-angle ansatz for QAOA that reduces circuit depth and improves the approximation ratio by increasing the number of classical parameters. Even though the number of parameters increases, our results indicate that good parameters can be found in polynomial time for a test dataset we consider. This new ansatz gives a 33% increase in the approximation ratio for an infinite family of MaxCut instances over QAOA. The optimal performance is lower bounded by the conventional ansatz, and we present empirical results for graphs on eight vertices that one layer of the multi-angle anstaz is comparable to three layers of the traditional ansatz on MaxCut problems. Similarly, multi-angle QAOA yields a higher approximation ratio than QAOA at the same depth on a collection of MaxCut instances on fifty and one-hundred vertex graphs. Many of the optimized parameters are found to be zero, so their associated gates can be removed from the circuit, further decreasing the circuit depth. These results indicate that multi-angle QAOA requires shallower circuits to solve problems than QAOA, making it more viable for near-term intermediate-scale quantum devices.
Abstract Simulating quantum systems is believed to be one of the most important applications of quantum computers. On noisy intermediate-scale quantum (NISQ) devices, the high-level circuit designed by quantum algorithms for Hamiltonian simulation needs to consider hardware limitations such as gate errors and circuit depth before it can be efficiently executed. In this work, we develop a hardware-agnostic circuit optimization algorithm to reduce the overall circuit cost for Hamiltonian simulation problems. Our method employ a novel sub-circuit synthesis in intermediate representation and propose a greedy ordering scheme for gate cancellation to minimize the gate count and circuit depth. To quantify the benefits of this approach, we benchmark proposed algorithm on different Hamiltonian models. Compared with state-of-the-art generic quantum compilers and specific quantum simulation compiler, the benchmarking results of our algorithm show an average reduction in circuit depth by 16.5× (up to 64.1×) and in gate count by 7.8× (up to 23.7×). This significant improvement helps enhance the performance of Hamiltonian simulation in the NISQ era.
Simulating quantum systems is one of the most important potential applications of quantum computers. The high-level circuit defining the simulation needs to be compiled into one that complies with hardware limitations such as qubit architecture (connectivity) and instruction (gate) set. General-purpose quantum compilers work at the gate level and have little knowledge of the mathematical properties of quantum applications, missing further optimization opportunities. Existing application-specific compilers only apply advanced optimizations in the scheduling procedure and are restricted to the CNOT or CZ gate set. In this work, we develop a compiler, named 2QAN, to optimize quantum circuits for 2-local qubit Hamiltonian simulation problems, a framework which includes the important quantum approximate optimization algorithm (QAOA). In particular, we exploit the flexibility of permuting different operators in the Hamiltonian (no matter whether they commute) and propose permutation-aware techniques for qubit routing, gate optimization and scheduling to minimize compilation overhead. 2QAN can target different architectures and different instruction sets. Compilation results on four applications (up to 50 qubits) and three quantum computers (namely, Google Sycamore, IBMQ Montreal and Rigetti Aspen) show that 2QAN outperforms state-of-the-art general-purpose compilers and application-specific compilers. Specifically, 2QAN can reduce the number of inserted SWAP gates by 11.5X, reduce overhead in hardware gate count by 68.5X, and reduce overhead in circuit depth by 21X. Experimental results on the Montreal device demonstrate that benchmarks compiled by 2QAN achieve the highest fidelity.