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arXiv:2604.17489

Approximate Hamiltonian Simulation Algorithm for Efficient Fluid Quantum Simulations

Zhiyuan Zhang, Bolin Zhang, Yongguang Lv, Ruiqing He, Hengliang Guo, Jiandong Shang, Qiang Chen

quant-phphysics.comp-phphysics.flu-dynSubmitted Apr 19, 2026• Announced Apr 21, 2026• 0 citations
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Abstract

This work aims to address the bottleneck issues of hardware resource limitation and decoherence error in the Hamiltonian simulation of quantum fluids, which are caused by the standard quantum Fourier transform and the evolution of momentum operators, resulting in excessively deep circuits and excessive two-qubit gates. We propose an approximate operator optimization scheme aimed at reducing the circuit depth in Hamiltonian evolution. The proposed scheme successfully reduces the depth of analog circuits from $O(n^2)$ to $O(nlogn)$ or even $O(n)$ by eliminating $O(n^2)$ redundant two-qubit entangling gates. In this work, the numerical experiments are implemented on a supercomputing-oriented quantum simulator, simulating two-dimensional unsteady divergent flow. Experimental results demonstrate that although the truncation of high-frequency qubit coupling terms introduces deterministic theoretical errors, scaling at $O(n)$ for AQFT and $O(n^2)$ for momentum truncation, the optimized simulations successfully preserve the inherent macroscopic temporal evolution characteristics of the fluid in a 10-qubit simulation, achieving high correlation coefficients of $r$=0.933, $r$=0.941, and $r$=0.977 for density, X-momentum, and Y-momentum distributions respectively. Furthermore, we also analyzed the relationship between the algorithm truncation error and the hardware cumulative noise when the qubit number is extended to a higher level. This study proves that rationally adjusting truncation thresholds can establish an equilibrium point, preventing the hardware cumulative error from rapidly approaching 100% at the 20-30 qubit scale, providing a feasible engineering pathway for simulating complex fluid systems on real quantum devices in the future.

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This work aims to address the bottleneck issues of hardware resource limitation and decoherence error in the Hamiltonian simulation of quantum fluids, which are caused by the standard quantum Fourier transform and the evolution of momentum operators, resulting in excessively deep circuits and excessive two-qubit gates. We propose an approximate operator optimization scheme aimed at reducing the circuit depth in Hamiltonian evolution. The proposed scheme successfully reduces the depth of analog circuits from O(n2)O(n^2)O(n2) to O(nlogn)O(nlogn)O(nlogn) or even O(n)O(n)O(n) by eliminating O(n2)O(n^2)O(n2) redundant two-qubit entangling gates. In this work, the numerical experiments are implemented on a supercomputing-oriented quantum simulator, simulating two-dimensional unsteady divergent flow. Experimental results demonstrate that although the truncation of high-frequency qubit coupling terms introduces deterministic theoretical errors, scaling at O(n)O(n)O(n) for AQFT and O(n2)O(n^2)O(n2) for momentum truncation, the optimized simulations successfully preserve the inherent macroscopic temporal evolution characteristics of the fluid in a 10-qubit simulation, achieving high correlation coefficients of rrr=0.933, rrr=0.941, and rrr=0.977 for density, X-momentum, and Y-momentum distributions respectively. Furthermore, we also analyzed the relationship between the algorithm truncation error and the hardware cumulative noise when the qubit number is extended to a higher level. This study proves that rationally adjusting truncation thresholds can establish an equilibrium point, preventing the hardware cumulative error from rapidly approaching 100% at the 20-30 qubit scale, providing a feasible engineering pathway for simulating complex fluid systems on real quantum devices in the future.
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We present a full quantum algorithm for the lattice Boltzmann method for simulating fluid flows, the only such algorithm to implement both the streaming and collision substeps as unitary operators using an efficient number of qubits. We use Hamiltonian simulation as the main route underlying the algorithm, and show that it uses qubits that scale logarithmically in the flow Reynolds number but gates that scale only polynomially. We motivate the discussion by a brief overview of existing attempts at simulating classical fluids on quantum computers and present a pedagogical discussion on assigning quantum operators to classical variables in both the streaming and the collision substeps, after highlighting the incompatibility of the latter with the implementations of the streaming step that exists in the literature. We use the Bhatnagar–Gross–Krook ansatz for the collision term, representing the relaxation toward an equilibrium distribution. For nonlinear collisions, we use Kowalski's framework that links the nonlinear dynamics of a system to the evolution of bosonic modes, assigning a Carleman linearization order to the truncation in the Fock space of the bosons. We present the qubit and gate complexities, in terms of the chosen accuracy and the Reynolds number. In the Appendix, we work out the details of implementing the operators of the truncated bosonic Fock space in terms of single-qubit gates as well as the error scaling for a general polynomial driving function.
Quantum computers can efficiently simulate highly entangled quantum systems, offering a solution to challenges facing classical simulation of quantum field theories (QFTs). This paper presents an alternative to traditional methods for simulating the real-time evolution in QFTs by leveraging Hamiltonian truncation (HT). As a use case, we study the Schwinger model, systematically reducing the complexity of the Hamiltonian via HT while preserving essential physical properties. For the observables studied in this paper, the HT approach converges quickly with the number of qubits, allowing for the interesting physics processes to be captured without needing many qubits. Identifying the truncated free Hamiltonian’s eigenbasis with the quantum device’s computational basis avoids the need for complicated and costly state preparation routines, reducing the algorithm’s overall circuit depth and required coherence time. As a result, the HT approach to simulating QFTs on a quantum device is well suited to noisy-intermediate scale quantum devices, which have a limited number of qubits and short coherence times. We validate our approach by running simulations on a noisy-intermediate scale quantum device, showcasing strong agreement with theoretical predictions. We highlight the potential of HT for simulating QFTs on quantum hardware. Published by the American Physical Society 2024
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.