Approximate Hamiltonian Simulation Algorithm for Efficient Fluid Quantum Simulations

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.

• The paper introduces an approximate Hamiltonian simulation framework that truncates long-range gates via an AQFT, significantly reducing quantum circuit depth.
• It incorporates a momentum operator truncation scheme that prunes negligible entangling operations from O(n²) to O(n), mitigating hardware-induced decoherence.
• Experimental evaluations on a 10-qubit system demonstrate high macroscopic fidelity with correlation coefficients above 0.93 in key flow variables.

Approximate Hamiltonian Simulation Algorithm for Efficient Fluid Quantum Simulations

Context and Motivation

Hamiltonian simulation for quantum fluid dynamics aims to realize the exponential state-space efficiency of quantum computation for physically relevant, high-dimensional flow systems. However, in the NISQ era, current quantum hardware is fundamentally limited by a combination of circuit depth constraints, hardware resource availability, and error accumulation—primarily due to the quadratic scaling of two-qubit gates in key subroutines such as the quantum Fourier transform (QFT) and momentum operator implementation. This work introduces an approximate Hamiltonian simulation framework that strategically truncates and optimizes circuit structure, explicitly targeting efficient quantum simulation of fluid flows while mitigating both gate depth and cumulative decoherence errors (2604.17489).

Methodological Contributions

The authors propose a two-pronged strategy, combining an approximate quantum Fourier transform (AQFT) with a momentum operator truncation scheme. The AQFT selectively removes long-range controlled-phase gates $CR_k$ for indices $k > b$ (where $b$ is a tunable threshold parameter), a choice motivated both by the diminishing physical impact of exponentially small phase rotations and the prohibitive physical requirement for long-range interactions on current hardware. To partially recover phase coherence lost from the removal of these gates, single-qubit $R_z$ phase compensation is introduced, computed as an average over plausible control qubit states.

Momentum operator truncation leverages the commutative nature of Hamiltonian terms, decomposing evolution into a product of single- and two-qubit gates. By introducing a threshold $\epsilon_{th}$, only terms with $|\tilde{\theta}_{ij}| \geq \epsilon_{th}$ are retained, while those below are pruned, drastically reducing the number of entangling operations from $O(n^2)$ to $O(n)$.

The overall resource gains are substantial: AQFT reduces the circuit depth for the QFT component from $O(n^2)$ to $O(n\log n)$ or $k > b$0 (with careful selection of the truncation threshold), while momentum operator truncation removes the majority of two-qubit gates responsible for high-frequency contributions in the flow field.

Theoretical Analysis of Error

Key to this approach is a quantitative analysis of error scaling. The algorithmic error induced by the AQFT grows linearly with the number of qubits $k > b$1, stemming from the systematic removal of long-range phase gates. Truncating the momentum operator introduces a higher-order $k > b$2 error, reflecting the explosion in discarded high-frequency entangling gates as the system size increases. Empirically and theoretically, the total error is well-controlled for moderate system sizes and is substantially below the catastrophic error rates one would expect from untruncated, deep-circuit simulations on NISQ hardware.

Crucially, the error-resource trade-off is made explicit: more aggressive truncation (larger $k > b$3 or smaller $k > b$4) yields lower resource requirements but at increased numerical error, while conservative truncation preserves accuracy but leads to hardware-induced decoherence and loss of simulation fidelity as $k > b$5 increases (2604.17489).

Experimental Evaluation

Simulations were conducted on the Songshan supercomputer quantum simulator targeting a 10-qubit, two-dimensional unsteady divergent flow problem. Despite the deterministic errors from truncation, simulation fidelity remains high at the macroscopic level: correlation coefficients of $k > b$6 (density), $k > b$7 ($k > b$8-momentum), and $k > b$9 ($b$0-momentum) were attained compared to the ideal “untruncated” quantum simulation baseline. The optimized circuits exhibit marked resilience in preserving bulk hydrodynamic behavior, with errors mostly affecting fine-scale or high-frequency details.

Notably, the study reveals that for qubit counts $b$1, unoptimized circuits rapidly succumb to cumulative hardware errors, approaching complete decoherence well before algorithmic errors from truncation would dominate. Thus, circuit compression via truncation is not merely beneficial but essential for any hope of scalable, accurate quantum fluid simulation in the NISQ setting.

Implications and Future Directions

The developed methods establish a pragmatic framework for fluid dynamics simulation on restricted quantum resources. While the present approach is tailored to Hamiltonians with quadratic spectra (e.g., Laplacian operators as in Schrödinger-type equations for fluids), the core truncation and compensation principles extend naturally to other quantum simulation settings involving spatial or spectral transforms, including other classes of PDEs fundamental to physics and engineering.

The authors note several future directions:

• Extension to Nonlinear and Non-Hermitian Systems: The resource savings provided by these truncation strategies could enable simulation of more general fluid dynamics, including nonlinear and dissipative phenomena, by affording higher-order Trotterization or additional ancillary encodings before hardware errors become dominant.
• Adaptive Truncation and Variational Preparation: Treating the truncation threshold as a hyperparameter opens the door for adaptive strategies, where thresholds are tuned dynamically based on available hardware fidelity, error correction, or even feedback from the physical flow field.
• Compilation and Circuit-Level Optimization: Integrating truncation with advanced quantum compiling (e.g., topology-aware routing, error mitigation post-processing) could further reduce physical overhead.
• Engineering Application: The demonstrated balance between macroscopic fidelity and resource efficiency paves the way for quantum acceleration in turbulence, multiphase flows, or reacting flows, especially for lower-resolution or real-time engineering simulators.

Conclusion

This work provides a rigorous and actionable methodology for quantum fluid simulation on current and near-term quantum processors by strategically compressing quantum circuit resources via approximation and truncation. By making explicit the trade-off between algorithmic error and hardware-induced decoherence, the framework renders previously intractable fluid simulation problems feasible within hardware-limited regimes and offers a template for scalable quantum simulation in broader physical domains (2604.17489).