Quantum simulations of hydrodynamics via the Madelung transformation

Developing numerical methods to simulate efficiently nonlinear fluid dynamics on universal quantum computers is a challenging problem. In this paper, a generalization of the Madelung transform is defined to solve quantum relativistic charged fluid equations interacting with external electromagnetic forces via the Dirac equation. The Dirac equation is discretized into discrete-time quantum walks which can be efficiently implemented on universal quantum computers. A variant of this algorithm is proposed to implement simulations using current noisy intermediate scale quantum (NISQ) devices in the case of homogeneous external forces. High resolution (up to $N=2^{17}$ grid points) numerical simulations of relativistic and nonrelativistic hydrodynamical shocks on current IBM NISQs are performed with this algorithm. This paper demonstrates that fluid dynamics can be simulated on NISQs, and opens the door to simulating other fluids, including plasmas, with more general quantum walks and quantum automata.

Figure 1

Quantum part of the NISQ algorithm for DTQW to compute every ${|k\rangle }_T$.

Figure 2

Profiles of density $n$ (left) and velocity $u^1/u^0$ (right) at different times as functions of the position $x$, for an electric field $E=16$, charge $q=-1$, mass $m=64$, and initial maximum velocity $u_{\text{max}}=0.55$. The mesh size is $N=4096, \epsilon =2\pi /N$, T is an arbitrary time unit and L is an arbitrary length unit.

Figure 3

Evolution of the shock's density $n$ (left) and velocity $u^1/u^0$ (right) as functions of the position $x$ and time for different values of the electric field $E=0,6,12$, charge $q=-1$, mass $m=64$, and initial maximum velocity $u_{\text{max}}=0.55$. The mesh size is $N=4096, \epsilon =2\pi /N$, T is an arbitrary time unit, and L is an arbitrary length unit.

Figure 4

Shock's profiles of fluid density $n$ (upper left) and fluid velocity $u^1/u^0$ (upper right) as functions of position $x$ computed on different IBM's quantum processors, ibm$\_$qasm$\_$simulator and a classical computer, at $t=1.96$ (arbitrary unit). The lower panels show relative errors (lower left) and absolute error (lower right) between the ideal results obtained on the classical computer and the results obtained on the quantum processors and simulator. The simulation parameters are electric field $E=0.6$, charge $q=-1$, mass $m=6$, and initial maximum velocity $u_{\text{max}}=0.92$. The mesh size is $N=32, \epsilon =2\pi /N$, and L is an arbitrary length unit.

Figure 5

High-resolution shock's profiles of fluid density $n$ (upper left) and fluid velocity $u^1/u^0$ (upper right) computed on ibmq$\_$manila quantum processor, ibm$\_$qasm$\_$simulator, and classical computer at $t=2.5$ (arbitrary unit), with $N=2^{17}$ grid points, an electric field $E=2$, a charge $q=-1$, a mass $m=6$, an initial maximum velocity $u_{\text{max}}=0.92, \epsilon =2\pi /N$, L an arbitrary length unit, relative errors (lower left), and absolute error (lower right) between the ideal results obtained on a classical computer and the results obtained on quantum processors and simulator as functions of the position $x$.