Alternatives to a nonhomogeneous partial differential equation quantum algorithm

Recently, J. M. Arrazola et al. [Phys. Rev. A 100, 032306 (2019)] proposed a quantum algorithm for solving nonhomogeneous linear partial differential equations of the form $A\psi \left(\mathbf{r}\right)=f\left(\mathbf{r}\right)$. Its nonhomogeneous solution is obtained by inverting the operator $\widehat{A}$ along with the preparation and measurement of special ancillary modes. In this work we suggest modifications in its structure to reduce the costs of preparing the initial ancillary states and improve the precision of the algorithm for semidefinite operators. These achievements enable easier experimental implementation of the quantum algorithm.

Figure 1

Adaptation of the schematic representation of the quantum algorithm proposed in Ref. [6]. The operators $\widehat{A}, \widehat{Y}$, and $\widehat{Z}$ act on the nonhomogeneous state $|f\rangle$, single Fock state $|1\rangle$, and step function state $|s\rangle$, respectively. The system follows homodyne momentum measurements on auxiliary modes ($p_z$ and $p_y$), which, after postselecting them on $p_{z,y}=0$, returns the solution of the algorithm $|\mathrm{\Phi }\rangle =A^{-1}|f\rangle$.

Figure 2

Dilog graphs of $F\left(a\right)$ as a function of the dimensionless quantity $a$. Top panel: for different values of $d$, with $\mathrm{\Delta }=0$ and $L=20$. Bottom panel: for different values of $\mathrm{\Delta }$, with fixed $d=140$ and $L=20$. Both results were obtained through Eq. (15), which was deduced from Ref. [6].

Figure 3

Dilog graph of $F\left(a\right)$ as a function of $a$ for the first modified algorithm (Top) with $M=0,1$ and fixed $\mathrm{\Delta }=0.01$, and (Bottom) for three distinct values of evolution time $t=\{1.0,10.0,100.0\}$ with $\mathrm{\Delta }=\{0.01,1.0\}$. In this case, the projective measurement on the Fock states with postselection on the vacuum state replaces the homodyne detection of the linear momentum with postselection on $p=0$.

Figure 4

Dilog graph of $F\left(a\right)$ as a function of $a$ for the second modified algorithm: (Top) with $M=0,1$ and fixed $\mathrm{\Delta }=0.01$, and (Bottom) for three distinct values of evolution time $t=\{1.0,10.0,100.0\}$ with $\mathrm{\Delta }=\{0.0,0.1,1.0\}$.

Figure 5

Semilog graph comparing the absolute relative error taking the ideal $1/a$ output as reference, derived numerically using the results of Eq. (15) (the AKWL proposal), Eq, (24) (Proposal 1), and Eq. (33) (Proposal 2) for each proposal with $M=1,\mathrm{\Delta }=0.1$, and $t=10$, and the original algorithm for $L=7, \mathrm{\Delta }=0.1$, and $d=20$.

Figure 6

Particular solution of Poisson's Eq. (36), as a function of the radial variable $r$, with nonhomogeneous function $f\left(\mathbf{r}\right)=g(4,\mathbf{r})$. The solutions obtained by the original algorithm considering distinct values of $\mathrm{\Delta }=\{0.01,1.0\}$ are compared to the ones provided by Proposals 1 and 2 for the evolution time (a) $t=1.0$, (b) $t=10.0$, and (c) $t=100$ with $\mathrm{\Delta }=1$. The exact output solution is plotted for the sake of comparison.

Figure 7

Particular solution of the nonhomogeneous quantum harmonic oscillator Eq. (48) as a function of the position $x$ in which the nonhomogeneous function $f\left(x\right)$ is the wave function of the coherent state with amplitude $\alpha =2.5$. The solutions obtained by the original algorithm considering distinct values of $\mathrm{\Delta }=\{0.01,1.0\}$ are compared to the ones provided by Proposals 1 and 2 for the evolution time $t=1.0$ (top panel) and $t=5.0$ (bottom panel). The exact solution is plotted for the sake of comparison. This result shows that the modified versions of the algorithm have the time evolution as a powerful way to improve its precision.

Figure 8

Probability distributions of successfully detecting the state $|{\mathrm{\Delta }}_{p_y}\rangle |{\mathrm{\Delta }}_{p_z}\rangle$ in the auxiliary states for the current example, with $\alpha =2.5$, where the solutions are displayed in Fig. 7. Proposals 1 and 2 are displayed with fixed time $t=5$. In this plot Proposal 1 is multiplied by a factor of ${10}^3$ and the original algorithm is multiplied by a factor of 10. When $\mathrm{\Delta }$ is small, the behavior of the distributions are $O\left({\mathrm{\Delta }}^2\right)$ for the three cases, as expected.