Surface-Acoustic-Wave Computing of the Grover Quantum Search Algorithm with Metasurfaces

Wave-based computing has attracted extensive attention recently due to the benefits of parallel processing. In particular, several acoustic wave computing devices have been demonstrated to carry out classical algorithms and mathematical operations. Here, we extend acoustic wave computing to simulate a quantum algorithm, by proposing an integrated acoustic gradient metasurface system supporting spoof surface acoustic waves to implement the Grover quantum search algorithm. We show that this integrated metadevice can achieve a designed subdiffraction and transmission phase, which can be used to simulate operations used in a quantum algorithm, such as the Hadamard transformation and the inverse about the average. Numerical simulations demonstrate promising searching abilities of this device, including a quadratic speedup over classical algorithms and subwavelength searching accuracy. We anticipate that our results will inspire alternative design schemes for on-chip integrated metadevices for more quantum-inspired acoustic analog computations.

Figure 1

Diagram of Grover search algorithm in quantum and classical wave systems. (a) Evolution of relative amplitude of quantum states. $\overline{\mathbf{O}}$, $\overline{\mathbf{H}}$, and $\overline{\mathbf{R}}$ represent an oracle operation, a Hadamard transform, and a phase-inversion transform, respectively. The input signal is an equal superposition of all states. The amplitudes of these states are $\alpha =1/N$. $\overline{\mathbf{O}}$ marks the target state $\left|t\right.⟩$ and inverts the target amplitude from ${\alpha }_t$ to $-{\alpha }_t$. The IAA performs an inversion of all state amplitudes about the mean, $M$. Hence, the amplitude of the target state is amplified to $2M+{\alpha }_t$. (b) Illustration of Grover quantum searching in a classical wave system. The colored areas represent boards that transmit the wave with a designed refractive index. The structures between hard boundaries (thick black lines) represent half an iteration of a full-step search operation. In order to demonstrate the propagation of the reflected wave, we extend the half-iteration system into an array with the hard boundaries as axes of symmetry. The solid arrows below the boards represent the direction of wave propagation, while the dotted arrows below the boards represent the direction of propagation of the wave reflected by the mirrors. $\overline{\mathbf{O}}$ is implemented by a double ${\overline{\mathbf{O}}}^{1/2}$ board. The IAA is implemented by the sequence $\overline{\mathbf{F}}{\overline{\mathbf{R}}}^{1/2}\overline{\mathbf{F}}\overline{\mathbf{F}}{\overline{\mathbf{R}}}^{1/2}\overline{\mathbf{F}}$. The inverse target is achieved by ${\overline{\mathbf{O}}}^{1/2}{\overline{\mathbf{O}}}^{1/2}$. The position of the maximum peak of the output wave is located at $-y_t$. The extra negative sign comes from the spatial-inverse functionality of ${\overline{\mathbf{F}}}^2$.

Figure 2

Simulations with ideal acoustic GRIN distribution. The sound speed in and mass density of the background material are $c_0=343$ m/s and ${\rho }_0=1.293\mathrm{kg}/{\mathrm{m}}^3$, respectively. The parameters of the structures are $D=240\mathrm{mm}\approx 2.80\lambda$, $r_0=20\mathrm{mm}\approx 0.23\lambda$, $W=310\mathrm{mm}\approx 3.61\lambda$, and $y_t=-50\mathrm{mm}\approx -0.58\lambda$. We change $d$ to obtain different database sizes $N$. Here, $n_t=2.57$ and $n_u=1.5$. (a) Sound pressure distribution at 1.5 iterations; $d=30\mathrm{mm}\approx 0.35\lambda$. Here, in order to show the details in the $y$ direction more clearly, we increase the $y$-direction ratio of the image. The ideal target position (blue circle) is marked by $y=50\mathrm{mm}\approx 0.58\lambda$. The right inset shows the refractive index of $\overline{\mathbf{F}}$. (b)–(d) Sound pressure distribution in the direction perpendicular to the propagation direction. The red numbers on the iteration-order axis highlight the maximum-probability iteration order. $r_f^{\prime }$ denotes the iteration order with the maximum target amplitude in the simulation. We give the theoretical $r_f$ calculated from Eq. (4) in the top left inset.

Figure 3

Illustration of acoustic metasurfaces for the Grover quantum algorithm and its operations. (a) Structure of acoustic metasurface system for emulating Grover search. The left inset is a longitudinal section of a single unit cell; $w$, $a$, and $h$ denote the width, center spacing, and depth, respectively, of the resonators. $W=31a\approx 3.62\lambda$ is the width of the entire structure. The length of the $\overline{\mathbf{F}}$ board is $L=19a\approx 2.22\lambda$, and the lengths of the ${\overline{\mathbf{O}}}^{1/2}$ and ${\overline{\mathbf{R}}}^{1/2}$ boards are both $r_0=2a\approx 0.23\lambda$, with $a=10\mathrm{mm}\approx 0.12\lambda$. (b) Illustration of the ${\overline{\mathbf{O}}}^{1/2}$ board. The top part shows a cross section of the ${\overline{\mathbf{O}}}^{1/2}$ board between $-100$ and 100 mm. Here, $h_u=14.4\mathrm{mm}\approx 0.17\lambda$, and $h_t=17.7\mathrm{mm}\approx 0.21\lambda$. The middle part shows the wave propagation in the ${\overline{\mathbf{O}}}^{1/2}$ board. The dotted black lines highlight the position of the ${\overline{\mathbf{O}}}^{1/2}$ board. When the acoustic wave propagates above the area where the depth of the target resonator is $h_t$, the wavelength is shorter than in other areas, so that the wave at the target state has a phase delay compared with those at other places. The bottom part shows the phase of the output wave. (c) Illustration of the $\overline{\mathbf{F}}$ board. The top shows a cross section of the $\overline{\mathbf{F}}$ board. The middle shows the wave pressure distribution in the $\overline{\mathbf{F}}$ board when the board acts on a incident plane wave. The bottom depicts the resonator’s depth $h$ and the associated refractive index in the $\overline{\mathbf{F}}$ board.

Figure 4

SAW simulation of Grover search with acoustic metasurfaces. (a) Distribution of wave amplitude (pressure) from position $y=-150\mathrm{mm}$ to $y=150\mathrm{mm}$ for an $N=7$ database. Here, $r_f=1.52$. The width of each data point is $30\mathrm{mm}$. The acoustic wave is fixed precisely at the ideal target position at an iteration order of 1.5. The colored squares at the left show the activated data positions in the ${\overline{\mathbf{O}}}^{1/2}$ board. The orange square highlights the position of the target data. The white squares denote the inactivated positions. The graph on the right shows the measured results at an iteration order of 1.5. The distribution of the normalized pressure squared (power) is represented by a line graph. The solid line shows the result with thermal loss, and the dotted line shows the result without thermal loss. The maximum acoustic wave amplitude for both loss and lossless conditions appears near the target position. The bar chart describes the probabilities of each data point as targets are searched for with the Grover algorithm. These probabilities are calculated from the distribution of the normalized acoustic pressure squared without thermal loss. (b) Simulation of multiple targets. The width of each data point is $20\mathrm{mm}$, while the limit of the GRIN plane is around $0.4{\lambda }_{\min }=17.2\mathrm{mm}$.