Abstract The authors show that a large class of 2‐adic Schrödinger equations is the scaling limit of certain continuous‐time quantum Markov chains (CTQMCs). Practically, a discretization of such an equation gives a CTQMC. As a practical result, new types of continuous‐time quantum walks (CTQWs) on graphs using two symmetric matrices are constructed. The transport between nodes in one direction is described by one matrix, while the transport between nodes in the opposite direction. This construction includes, as a particular case, the CTQWs constructed using adjacency matrices. The final goal of this work is to contribute to the understanding of the foundations of quantum mechanics (QM) and the role of the hypothesis of the discreteness of space. The connection between 2‐adic QM and CTQWs shows that 2‐adic QM has a physical meaning. 2‐Adic QM is a nonlocal theory because the Hamiltonians used are nonlocal operators, and consequently, spooky actions at a distance are allowed. However, this theory is not a mathematical toy. The violation of Bell's inequality is experimentally confirmed, which implies that realism is allowed by this theory. The authors pointed out several new research problems connected with the foundations of QM.
Abstract The authors show that a large class of 2‐adic Schrödinger equations is the scaling limit of certain continuous‐time quantum Markov chains (CTQMCs). Practically, a discretization of such an equation gives a CTQMC. As a practical result, new types of continuous‐time quantum walks (CTQWs) on graphs using two symmetric matrices are constructed. The transport between nodes in one direction is described by one matrix, while the transport between nodes in the opposite direction. This construction includes, as a particular case, the CTQWs constructed using adjacency matrices. The final goal of this work is to contribute to the understanding of the foundations of quantum mechanics (QM) and the role of the hypothesis of the discreteness of space. The connection between 2‐adic QM and CTQWs shows that 2‐adic QM has a physical meaning. 2‐Adic QM is a nonlocal theory because the Hamiltonians used are nonlocal operators, and consequently, spooky actions at a distance are allowed. However, this theory is not a mathematical toy. The violation of Bell's inequality is experimentally confirmed, which implies that realism is allowed by this theory. The authors pointed out several new research problems connected with the foundations of QM.
Abstract 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. Unlike classical continuous-time Markov chains (CTMCs), the CTQMCs studied here do not satisfy the classical Chapman–Kolmogorov equation; this is a genuine quantum effect, due to the fact that the underlying unitary evolution composes at the level of probability amplitudes rather than probabilities, and we make precise the sense in which the term ‘quantum Markov chain’ is used. 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 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.
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. Unlike classical continuous-time Markov chains, the CTQMCs studied here do not satisfy the classical Chapman-Kolmogorov equation; this is a genuine quantum effect, due to the fact that the underlying unitary evolution composes at the level of probability amplitudes rather than probabilities, and we make precise the sense in which the term ``quantum Markov chain'' is used. 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.
We introduce a new class of p-adic Dirac equations, formulated in the standard axiomatic framework of quantum mechanics, in which the ordinary momentum operators are replaced by bounded non-local operators built from arbitrary integrable radial kernels. We diagonalize the free Dirac Hamiltonian in momentum space, construct its plane-wave solutions, and determine its spectrum, which is pure point, with eigenvalues of infinite multiplicity; in the massless case we describe the zero-energy subspace explicitly. We identify the charge-conjugation symmetry of the model: it is given by an antiunitary operator whose square is minus the identity, it exchanges the positive- and negative-energy subspaces, and in an external electromagnetic field it reverses the sign of both the charge and the vector potential. We then discretize the free equation in two ways, both giving genuine continuous-time quantum walks rather than discrete-time, coined walks: one on a countable graph and one on a finite, tree-like graph. In both cases the discretization is exact: the walk is the restriction of the p-adic Dirac evolution to an invariant subspace, and its Hamiltonian is a Dirac-type combination of weighted graph Laplacians with ultrametric weights. We prove that the Born-rule transition probabilities, summed over the internal degrees of freedom and averaged over the initial internal state, form a doubly stochastic matrix at every instant, and that, as for any non-trivial continuous-time quantum walk, this family of matrices is not a Markov semigroup. We discuss how the construction can serve as the foundation of a quantum network with four-component spinor degrees of freedom, complementing earlier, non-relativistic p-adic quantum neural networks, and prove global well-posedness of the associated nonlinear network equation. We close with a discussion of open problems.
This paper discusses a p-adic version of the infinite potential well in quantum mechanics (QM). This model describes the confinement of a particle in a p-adic ball. We rigorously solve the Cauchy problem for the Schrödinger equation and determine the stationary solutions. The p-adic balls are fractal objects. By dividing a p-adic ball into a finite number of sub-balls and using the wavefunctions of the infinite potential well, we construct a continuous-time quantum walk (CTQW) on a fully connected graph, where each vertex corresponds to a sub-ball in the partition of the original ball. In this way, we establish a connection between p-adic QM and quantum computing.
Quantum walks can be defined in two quite distinct ways: discrete-time and continuous-time quantum walks (DTQWs and CTQWs). For classical random walks, there is a natural sense in which continuous-time walks are a limit of discrete-time walks. Quantum mechanically, in the discrete-time case, an additional ``coin space'' must be appended for the walk to have nontrivial time evolution. Continuous-time quantum walks, however, have no such constraints. This means that there is no completely straightforward way to treat a CTQW as a limit of a DTQW, as can be done in the classical case. Various approaches to this problem have been taken in the past. We give a construction for walks on d-regular, d-colorable graphs when the coin flip operator is Hermitian: from a standard DTQW we construct a family of discrete-time walks with a well-defined continuous-time limit on a related graph. One can think of this limit as a ``coined'' continuous-time walk. We show that these CTQWs share some properties with coined DTQWs. In particular, we look at a spatial search by a DTQW over the two-dimensional (2D) torus (a grid with periodic boundary conditions) of size $\sqrt{N}\ifmmode\times\else\texttimes\fi{}\sqrt{N}$, where it was shown that a coined DTQW can search in time O(NlogN), but a standard CTQW takes $\mathrm{\ensuremath{\Omega}}(N)$ time to search for a marked element. The continuous limit of the DTQW search over the 2D torus exhibits the O(NlogN) scaling, like the coined walk it is derived from. We also look at the effects of graph symmetry on the limiting walk, and show that the properties are similar to those of the DTQW as shown in Krovi and Brun, Phys. Rev. A 75, 062332 (2007).
In this thesis, we introduce a p-adic version of the infinite potential well in quantum mechanics (QM). This model describes the confinement of a particle in a p-adic ball. We rigorously solve the Cauchy problem for the Schrödinger equation and determine the stationary solutions. The p-adic balls are fractal objects. By dividing a p-adic ball into a finite number of sub-balls and using the wavefunctions of the infinite potential well, we construct a continuous-time quantum walk (CTQW) on a fully connected graph, where each vertex corresponds to a sub-ball in the partition of the original ball. In this way, we establish a connection between p-adic QM and quantum computing. This thesis is based on [71].