Quantum Homotopy Algorithm for Solving Nonlinear PDEs and Flow Problems

TL;DR

The potential of the hybrid quantum algorithm for simulating practical and nonlinear phenomena on near-term and fault-tolerant quantum devices is shown and the complexity estimates improve existing approaches in terms of scaling of matrix operator norms, condition number, simulation time, and accuracy.

Abstract

Quantum algorithms to integrate nonlinear PDEs governing flow problems are challenging to discover but critical to enhancing the practical usefulness of quantum computing. We present here a near-optimal, robust, and end-to-end quantum algorithm to solve time-dependent, dissipative, and nonlinear PDEs. We embed the PDEs in a truncated, high dimensional linear space on the basis of quantum homotopy analysis. The linearized system is discretized and integrated using finite-difference methods that use a compact quantum algorithm. The present approach can adapt its input to the nature of nonlinearity and underlying physics. The complexity estimates improve existing approaches in terms of scaling of matrix operator norms, condition number, simulation time, and accuracy. We provide a general embedding strategy, bounds on stability criteria, accuracy, gate counts and query complexity. A physically motivated measure of nonlinearity is connected to a parameter that is similar to the flow Reynolds number $Re_{\textrm{H}}$, whose inverse marks the allowed integration window, for given accuracy and complexity. We illustrate the embedding scheme with numerical simulations of a one-dimensional Burgers problem. This work shows the potential of the hybrid quantum algorithm for simulating practical and nonlinear phenomena on near-term and fault-tolerant quantum devices.

Figures (3)

• Figure 1: (a) Shows the continuous homotopic deformations of $\xi$ for increasing values of $q$, starting from $f(\phi)$ (solid black line for $q=0$) to $g(\phi)$ (solid magenta line for $q=1$). (b) The left section of this panel, shows the schematic of the time-evolving velocity field of the governing PDE, as computed by a fully classical Direct Numerical Simulations (DNS). The right panel shows the solution as approximated by a Quantum Homotopy Algorithm. The homotopy method decomposes the full velocity solution into (i) $\bar{u}_{0}$ which is the initial guess and solution of the chosen linear PDE known for all $(x,t)$ and (ii) $\sum_{p=1}^{M}\bar{u}_{p}$ which are the higher-order deformation terms that contribute to the nonlinear corrections to the guess solution. The horizontal, red dotted-line indicates a nonlinear time $t_{NS}$, which represents the upper bound on the time horizon up to which the solutions from the quantum homotopy algorithm is maintained within the required accuracy threshold $\varepsilon$ (exponential convergence), while preserving algorithm's complexity.
• Figure 2: The figure shows an example of a 4 qubit LCU circuit to implement an iterative explicit-time simulation for $\tau=2$ time steps. Step 1 initializes all the qubits using an efficient quantum state preparation circuit. The first register $q(a_{0})$ is the counter qubit bharadwaj2024compact. These qubits are initially set to one (representing the maximum number of time steps, in a binary format) and via successive bit-flip operations, this is counted down after each application of the LCU. The ancilla qubits $q(a_{1})$ and $q(a_{2})$ are used to implement the linear combination of the four unitaries $U_{0}$ to $U_{4}$. The last register $q(u)$ stores the velocity field at every time step. Step 2 is the iterative application of LCU performed to march forward in time. Step 3 performs the summation of each term of the homotopy series as per eq. \ref{['eq:deformations']} and finally the result is measured or post-processed in situ using a Quantum Post Processing circuit bharadwaj2023hybrid.
• Figure 3: (a) Shows the homotopy solutions computed for different orders of truncation (solid symbols) $M=10,20,40,60,80$, for $\hat{h}=-0.5$, $\nu=0.001$, $\Delta t = 5\times10^{-3}$ and $N_{g}=32$. The fully classical DNS solution (black solid line) is also plotted for reference. (b) Shows the homotopy solutions (solid symbols) of the time evolving flow field with respect to the classical DNS results (solid lines). (c) We plot here, the mean-squared-error of the homotopy solutions as a function of truncation order $M$, which is seen to decay exponentially. Further, the inset shows the exponential decay of error, also as a function of time-steps $\tau$, plotted for increasing $M$. (d) Shows the contour plot of the MSE as a function of both $\hat{h}\in [-1,-0.1]$ and $M\in\{10,20,40,60,80\}$, revealing that $\hat{h}=0.5$ tends to yield an overall convergence behavior. However, it is also clear that for higher orders of truncation $M$, smaller $\hat{h}$ yields a better accuracy of the solution.