Error and Resource Estimates of Variational Quantum Algorithms for Solving Differential Equations Based on Runge-Kutta Methods

TL;DR

The paper develops a rigorous framework to quantify error sources and resource costs for variational quantum algorithms solving differential equations using Runge-Kutta methods. It derives analytical bounds for truncation errors and shot-noise, and provides minimal-resource estimates for achieving a target accuracy, both with and without measurement noise. The authors validate the framework on a simple $1$D ODE and on the Black-Scholes PDE, showing problem-dependent optimal RKMs (order $p=4$ for the ODE, order $p=2$ for the PDE) and highlighting substantial practical resource challenges on current hardware. The work offers a systematic resource-optimization toolkit for applying Runge-Kutta-based variational methods to DEs and quantum simulations.

Abstract

A focus of recent research in quantum computing has been on developing quantum algorithms for differential equations solving using variational methods on near-term quantum devices. A promising approach involves variational algorithms, which combine classical Runge-Kutta methods with quantum computations. However, a rigorous error analysis, essential for assessing real-world feasibility, has so far been lacking. In this paper, we provide an extensive analysis of error sources and determine the resource requirements needed to achieve specific target errors. In particular, we derive analytical error and resource estimates for scenarios with and without shot noise, examining shot noise in quantum measurements and truncation errors in Runge-Kutta methods. Our analysis does not take into account representation errors and hardware noise, as these are specific to the instance and the used device. We evaluate the implications of our results by applying them to two scenarios: classically solving a $1$D ordinary differential equation and solving an option pricing linear partial differential equation with the variational algorithm, showing that the most resource-efficient methods are of order 4 and 2, respectively. This work provides a framework for optimizing quantum resources when applying Runge-Kutta methods, enhancing their efficiency and accuracy in both solving differential equations and simulating quantum systems.

Figures (9)

• Figure 1: The quantum circuit evaluating the elements of $A$ and $C$ as given in Eqs. \ref{['eq: A evaluation']} and \ref{['eq: C evaluation']}. The controlled unitary $U_{k,i}$ is one of the $\sigma_{k,i}$. Depending on if one is evaluating $A$ or $C$, the controlled unitary $U_{l,j}$ is another $\sigma_{l,j}$ or one of the Pauli strings $\sigma_m$ (in which case we take $l=N_V+1$) that constitute the Hamiltonian, respectively .
• Figure 2: A log plot comparing the condition number obtained from the toy model with $\tilde{\theta}=1/2$ and different upper bounds due to $N_V^2$, $N_V^3$ and $N_V^4$. We sampled 100 different matrices $A$ according to the toy model and calculated the resulting condition number for each sample. The orange line shows the median and the gray shaded area shows the range between the 0.16 and 0.84 quantiles of the condition number estimates. We used the Frobenius norm.
• Figure 3: A plot comparing the norms $\|A\|$, $\|C\|$ and $\|A^{-1} C\|$ obtained from the toy model with the functions $f_1(N_V)=N_V$ and $f_2(N_V)=\sqrt{N_V}$. In order to estimate $\|A^{-1} C\|$, we did 100 samples for $A$ and $C$ and calculated the resulting norm for each sample. The orange line shows the median and the gray shaded area shows the range between the 0.16 and 0.84 quantiles of the estimates for $\|A^{-1} C\|$. For $\|A\|$, we used the Frobenius norm and for $\|C\|$ and $\|A^{-1} C\|$ the $2$ norm.
• Figure 4: The function $\text{Lip}\left(\tilde{\theta}_1,\tilde{\theta}_2\right)$ defined in Eq. \ref{['eq:lipfunction']} plotted for two different drawings of random matrix and vector parameters according to the toy model from Eqs. \ref{['eq:toy model A']} and \ref{['eq:toy model C']}. We fixed $N_V=25$ and varied the parameters $\tilde{\theta}_1$ and $\tilde{\theta}_2$ between $0$ and $10$.
• Figure 5: A plot comparing the sensitivity of the cost function with respect to different parameters. The intersecting point in the middle is the value of the cost function where all parameters are chosen default as given in Table \ref{['tab:estimates classical']} as well as $p=2$ as a default. In each graph, we are changing one parameter, while keeping all the other parameters at default. We are changing the parameter by multiplying with the scaling factor $x$ given as the abscissa. The graphs for $M$ and $K$ overlap as well as the graphs for $L_{fy}$ and $b_{max}$. The continuous red line for $p$ is just for visualization purposes, as $p$ is integer.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Theorem 3
• Corollary 1
• Theorem 4
• Theorem 5
• Corollary 2
• Lemma 1
• Lemma 2
• Lemma 3