Continuous-Time Quantum Markov Chains And Discretizations Of p-Adic Schrödinger Equations: Comparisons And Simulations

TL;DR

The paper advances a comprehensive framework connecting discretizations of $p$-adic Schrödinger equations to continuous-time quantum Markov chains (CTQMCs), in parallel with $p$-adic heat equations and classical continuous-time Markov chains (CTMCs). By solving initial-value problems, constructing orthonormal bases, and employing two CTQMC construction schemes, it demonstrates how quantum dynamics on ultrametric spaces can be encoded into Markov-type transition amplitudes. Numerical simulations, especially with Bessel-potentials, show that the long-time quantum distribution $\chi$ can exceed the classical stationary distribution $p^{\mathrm{sta}}$, indicating a potential computational advantage of CTQMCs in these $p$-adic settings. The work also develops discretization strategies and links to quantum networks, providing a path toward scalable quantum-network models based on ultrametric physics. Overall, the study supports the view that $p$-adic and ultrametric structures yield physically meaningful, nonlocal quantum models with rich Markovian and network interpretations, and it outlines open conjectures about scaling limits from CTMCs/CTQMCs to their continuous counterparts.

Abstract

The continuous-time quantum walks (CTQWs) are a fundamental tool in the development of quantum algorithms. Recently, it was shown that discretizations of p-adic Schrödinger equations give rise to continuous-time quantum Markov chains (CTQMCs); this type of Markov chain includes the CTQWs constructed using adjacency matrices of graphs as a particular case. In this paper, we study a large class of p-adic Schrödinger equations and the associated CTQMCs by comparing them with p-adic heat equations and the associated continuous-time Markov chains (CTMCs). The comparison is done by a mathematical study of the mentioned equations, which requires, for instance, solving the initial value problems attached to the mentioned equations, and through numerical simulations. We conducted multiple simulations, including numerical approximations of the limiting distribution. Our simulations show that the limiting distribution of quantum Markov chains is greater than the stationary probability of their classical counterparts, for a large class of CTQMCs.

Figures (10)

• Figure 1: Numerical Simulation 1. Matrix $\boldsymbol{J}^{\left( l\right) }\left( \alpha\right)$. The parameters are $p=2$, $l=5$, $\alpha=1.2$. The CTMC has $32$ states organized in a finite tree $G_{5}$. The figure illustrates the ultrametric nature of matrix $\boldsymbol{J}^{\left( l\right) }\left( \alpha\right)$.
• Figure 2: Numerical Simulation 1. $p_{I,12}(t)$ versus $\pi_{I,12}\left( t\right)$. The state $I$ runs through $5$ states. The parameters are $p=2$, $l=5$, $\alpha=1.2$. The figure (a) shows that $\lim_{t\rightarrow\infty}p_{I,12}(t)\approx2^{-5}$, while figure (b) illustrates that $\lim_{t\rightarrow\infty}\pi_{I,12}\left( t\right)$ does not exist.
• Figure 3: Numerical Simulation 1. $\pi_{I,J}\left( t\right)$ for six different times: $t=0$,$1$, $200$, $500$, $1000$, $4000$, $10000$. The states $I$, $J$ run through $32$ states. The parameters are $p=2$, $l=5$, $\alpha=1.2$.
• Figure 4: Numerical Simulation 1. $\pi_{I,J}\left( 200\right)$ for six different values of the parameter $\alpha$. The other parameters are $p=2$, $l=5$. The states $I$, $J$ run through $32$ states. The values $\alpha=0$, $1$ correspond to a pole, respectively a zero, of $\Gamma\left( \alpha\right)$. When the value of $\alpha$ is near to $0$, $1$, or $\alpha\geq5$, the transitions of the CTQMC occur around the diagonal; the walker only performs very short walks around each state. While for other values of $\alpha$, the walker performs long walks around any state.
• Figure 5: Numerical Simulation 1. The limiting distribution (or longtime average) is defined as $\chi_{I,J}=\lim_{T\rightarrow\infty}\frac{1}{T} {\int\limits_{0}^{T}} \pi_{I,J}\left( t\right) dt.$A numerical approximation of this limiting distribution, $p=2$, $l=5$, $\alpha=1.2$ is shown in the figure. The states $I$, $J$ run through $32$ states. In the literature, this limiting distribution is compared with $p_{I}^{\text{sta}}=\lim_{t\rightarrow\infty}p_{I,J}\left( t\right) =p^{-l},$cf. Theorem \ref{['Theorem_2']}, in Appendix C. The simulation shows that $\chi_{I,J}>p_{I}^{\text{sta}}=2^{-5}$. This fact is interpreted as the computational power of the CTQMC is greater than that of the corresponding CTMC. The computation of the average used the time interval $\left[ 0,10000\right]$.

Theorems & Definitions (44)

• Remark 2.1
• Proposition 3.1
• proof
• Theorem 3.1
• Proposition 3.2
• proof
• Theorem 3.2
• Remark 9.1
• Lemma 9.1
• proof