Real and Fourier space readout methods: Comparison of complexity and applications to CFD problems

TL;DR

The paper tackles the challenge of reconstructing real-valued PDE solutions encoded in quantum states for CFD-type problems, focusing on readout methods that can preserve quantum advantages. It introduces and contrasts real-space readout (RSR), Fourier-space readout (FSR), an approximate real-space readout (ARSR), and QAE-based readouts (RSQAE, FSQAE), highlighting how Fourier-based strategies can achieve end-to-end speedups for smooth periodic functions due to $N$-independent shot costs. The work applies these methods to CFD benchmarks, demonstrating improved visualization of planar jet and lid-driven cavity flows and proposing a time-stepwise readout (TSR) strategy that combines efficient readout with an approximate PITE algorithm to solve nonlinear evolution equations like the 2D Burgers' equation with reduced circuit depth. Overall, the results indicate that ARSR, FSR, and FSQAE are strong candidates for near-/mid-term quantum devices, offering substantial advantages over naive real-space readout and enabling practical quantum acceleration of PDE solvers in CFD contexts.

Abstract

Quantum computing is a promising technology that accelerates the partial differential equations solver for practical problems. The reconstruction of solutions (i.e., the readout of quantum states) remains a crucial problem, although numerous efficient quantum algorithms have been proposed. In this paper, we propose and compare several efficient readout methods in the real and the Fourier space. The Fourier space readout (FSR) and the proposed approximate real space readout (ARSR) methods are currently the most efficient and practical ones for the purpose of reconstructing continuous real-valued functions. In contrast, the quantum amplitude estimation (QAE) based methods (especially in the Fourier space) are favorable for mid-term/far-term quantum devices. Besides, we apply the methods for benchmark solutions in computational fluid dynamics (CFD) and demonstrate great improvements compared to the conventional sampling method for large grid numbers. Equipped with efficient readout methods, we further show that a 2D Burgers' equation can be solved efficiently without using the expensive strategy of linearization. It suggests the potential quantum advantages for some practical applications on mid-term quantum devices.

Figures (24)

• Figure 1: (a) Quantum circuit for the real space readout. (b) Quantum circuit for the Fourier space readout with an integer parameter $m\le n$. The first $2^m$ Fourier coefficients $c_0,\ldots,c_{2^m-1}$ can be obtained by post-selecting the most significant $n-m$ qubits to be in $\ket{0}_{n-m}$ state.
• Figure 2: Quantum circuit for the RSR method with approximation parameters $m_1,\ldots,m_d$. Z-basis measurements are executed for the dominant qubits in each spatial dimension.
• Figure 3: Error plots for different methods in the example of a 2D linear combination of Gaussian functions. (a) Approximation errors (i.e., $N_{\text{shot}}\to \infty$) regarding $M_0\in \{2^1,2^2,\ldots,2^7\}$. The dashed lines denote the orders $O\left(1/M_0^{t}\right)$ with $t=1/2,2$ for the FSR and the ARSR methods, respectively. (b) Total errors regarding the number of shots $N_{\text{shot}}\in 10000\times \{4^0,4^1,\ldots,4^6\}$. For the FSQAE method, the $x$-axis denotes the number of queries to the oracle, and the plot is done by taking $\epsilon_0\in \{0.05, 0.02, 0.01, 0.005, 0.0025, 0.001, 0.0005\}$ in the RQAE algorithm. The dashed lines denote the orders $O\left(1/N_{\text{shot}}^{t}\right)$ with $t=1/2,1/4,1/3,1/3$ for the RSR, the FSR, the ARSR, and the FSQAE methods, respectively.
• Figure 4: Error plots for different methods in the example of a 2D trigonometric function. (a) Approximation errors (i.e., $N_{\text{shot}}\to \infty$) regarding $M_0\in \{2^1,2^2,\ldots,2^7\}$. The dashed line denotes the order $O\left(1/M_0^{t}\right)$ with $t=2$ for the ARSR method. (b) Total errors regarding number of shots $N_{\text{shot}}\in 10000\times \{4^0,4^1,\ldots,4^6\}$. For the FSQAE method, the $x$-axis denotes the number of queries to the oracle, and the plot is done by taking $\epsilon_0\in \{0.05, 0.02, 0.01, 0.005, 0.0025, 0.001\}$ in the RQAE algorithm. The dashed lines denote the orders $O\left(1/N_{\text{shot}}^{t}\right)$ with $t=1/2,1/2,1/3,1/3$ for the RSR, the FSR, the ARSR, and the FSQAE methods, respectively.
• Figure 5: Visualization of solution for a planar jet flow and comparison of different methods with $N_{\text{shot}}=1.6\times 10^5$. The 2D curls are calculated using the (reconstructed) velocity fields.

Theorems & Definitions (6)

• Remark 1: Determinations of the approximation parameters
• Remark 2: Difference between the real space and the Fourier space readouts
• Remark 3: Quantum advantages in CAE simulations
• Remark 4: $K$-uniform success probability
• Remark 5
• Remark 6: Improved order for the FSR method