Quantum algorithm for simulating the wave equation

We present a quantum algorithm for simulating the wave equation under Dirichlet and Neumann boundary conditions. The algorithm uses Hamiltonian simulation and quantum linear system algorithms as subroutines. It relies on factorizations of discretized Laplacian operators to allow for polynomially improved scaling in truncation errors and improved scaling for state preparation relative to general purpose quantum algorithms for solving linear differential equations. Relative to classical algorithms for simulating the $D$-dimensional wave equation, our quantum algorithm achieves exponential space savings and achieves a speedup which is polynomial for fixed $D$ and exponential in $D$. We also consider using Hamiltonian simulation for Klein-Gordon equations and Maxwell's equations.

Figure 1

For a given initial wave packet and a given scatterer, we would like to estimate the resulting spatial distribution of wave intensity resulting at some later time $t$. In particular, one may wish to know the total intensity captured by a detector occupying some region of space. This can be estimated using a quantum simulation in which the wave function directly mimics the dynamics of the solution to the wave equation. The final intensity in the detector region is equal to the probability associated with the corresponding part of the Hilbert space, which can be estimated from the statistics resulting from a projective measurement.

Figure 2

To implement Dirichlet boundary conditions in a discretized square region with a square hole, one adds self-loops as illustrated above. The thick red self-loops at the corners have weight two. All other edges (self-loops and otherwise) have weight one. This prescription was used in the numerical examples of Sec. 5. To implement Neumann boundary conditions, one omits all self-loops.

Figure 3

Shape preserving on line segment Dirichlet. Here, we consider the case of a rigidly translating wave packet as described by (35). We can see two different views of the same wave packet starting in the middle point in a box with size 20, where space is represented by the $x$ axis while in the $y$ axis we have the time and the units are meters and seconds, respectively. We can see the packet going back and forward between the extremes of the box. Although its wave amplitude is preserved in time, when the wave packet arrives at the end points, the amplitude reflects simultaneously with the propagation's direction. The red color gives us the positive amplitude against the blue one with negative value. In (b) the amplitude height is plotted in the $z$ axis and its units are meters. In this example, we choose lattice spacing $a=0.2469$ and Gaussian wave packet of width $\sigma =1.6$.

Figure 4

Spreading wave on line segment Dirichlet. In these figures we kept with the same parameters used for the previous plots, changing only the initial condition for ${\vec{\varphi }}_E$. Now, we can see the wave spreading equally for the both sides, reflecting in the boundary and then meeting themselves again in the center, but with the amplitude inverted. The units are the same used in the previous plots, meters and seconds.

Figure 5

Standing wave. Here, we consider a standing wave, which can be described analytically by $\varphi \left(x,t\right)=cos\left(\omega t\right)sin\left(\pi x\right)$. This can be simulated by Schrödinger's equation if we work with ${\vec{\varphi }}_0=sin\left(\pi x\right)$ and $d{\vec{\varphi }}_0/dt=0$ as long as we start with $t=0$. The units are the same ones used in the previous figures.

Figure 6

Wave packet in a cavity. Here, the initial state is a Gaussian wave packet, but now in a two-dimensional region with nontrivial boundary. Specifically, we simulate scattering of the wave packet off a square object with Dirichlet boundary conditions. This is implemented as a square hole in the underlying discrete lattice. These four views represent the same wave packet in different time instants, where $t_a>t_b>t_c>t_d$. As in the one-dimensional example, we worked with Dirichlet boundary conditions; however, the shape is not preserved. Here, the box has size 10 in both axes, and we choose $a=0.1563$ and $\sigma =0.4$.

Figure 7

This linear combination of values at neighboring lattice sites produces a discrete approximation to the Laplacian with errors of order $a^2$ satisfying the conditions of Theorem t2. Specifically, one obtains ${\nabla }^2\varphi (x,y)-\frac{a^2}{20}{\left({\nabla }^2\right)}^2+O\left(a^6\right)$. Thus, the operator $\mathcal{D}$ in Theorem t2 is in this case ${\nabla }^2$.