Path-sum solution of the Weyl Quantum Walk in 3+1 dimensions

TL;DR

This paper presents an analytical path-sum solution for the Weyl quantum walk in 3+1 dimensions, providing explicit propagator expressions that reveal quantum interference effects on a Cayley graph.

Contribution

It introduces a novel analytical solution for the Weyl quantum walk in higher dimensions, leveraging path encoding and semigroup structures to derive the propagator.

Findings

Explicit analytical propagator for the 3+1D Weyl quantum walk

Demonstration of quantum interference effects in the walk

Framework for analyzing quantum walks on Cayley graphs

Abstract

We consider the Weyl quantum walk in 3+1 dimensions, that is a discrete-time walk describing a particle with two internal degrees of freedom moving on a Cayley graph of the group $\mathbb Z^3$, that in an appropriate regime evolves according to Weyl's equation. The Weyl quantum walk was recently derived as the unique unitary evolution on a Cayley graph of $\mathbb Z^3$ that is homogeneous and isotropic. The general solution of the quantum walk evolution is provided here in the position representation, by the analytical expression of the propagator, i.e. transition amplitude from a node of the graph to another node in a finite number of steps. The quantum nature of the walk manifests itself in the interference of the paths on the graph joining the given nodes. The solution is based on the binary encoding of the admissible paths on the graph and on the semigroup structure of the walk transition matrices.

Equations

$$|{\mathcal{N}}_x|\le k,$$

$${\mathcal{N}}_x:=\{y\in V\mid \exists \psi ,\phi \in {\mathbb{C}}^s,⟨y|⟨\psi |W|x⟩|\phi ⟩\ne 0\vee ⟨x|⟨\psi |W|y⟩|\phi ⟩\ne 0\},$$

$$\sum _{z\in V}\mathcal{E}(z,x)\mathcal{E}(z,y{)}^{†}=\sum _{z\in V}\mathcal{E}(x,z{)}^{†}\mathcal{E}(y,z)={\delta }_{xy}{I}_s,$$

$$\forall x\in V,|\{y\in V\}\mathcal{E}(x,y)\ne 0\vee \mathcal{E}(y,x)\ne 0|\le k.$$

$$|{\psi }_x(t+1)⟩=\sum _{y\in V}\mathcal{E}(y,x)|{\psi }_y(t)⟩,\forall x\in V,\forall t\in \mathbb{N},$$

$$W=\sum _{h\in S}{T}_h\otimes A_h.$$

$$|{\psi }_x(t)⟩$$

$$|{\psi }_x(t)⟩=\sum _{x^{\prime }\in {\mathcal{C}}_t(x)}\sum _{h_1,\dots ,h_t\in S}\delta (x^{-1}{x}^{\prime }{h}_1\cdots h_t)\mathcal{A}(h_1^{-1},\dots ,h_t^{-1})|{\psi }_{x^{\prime }}(0)⟩,$$

$${\mathrm{h}}_1=\left(\begin{array}{c}1\\ 1\\ 1\end{array}\right),{\mathrm{h}}_2=\left(\begin{array}{c}1\\ -1\\ -1\end{array}\right),{\mathrm{h}}_3=\left(\begin{array}{c}-1\\ 1\\ -1\end{array}\right),{\mathrm{h}}_4=\left(\begin{array}{c}-1\\ -1\\ 1\end{array}\right).$$

$$A_{{\mathrm{h}}_1}$$

$$A_{{\mathrm{h}}_2}$$

$$A_{{\mathrm{h}}_3}$$

$$A_{{\mathrm{h}}_4}$$

$${\mathrm{h}}_1:011,$$

$${\mathrm{h}}_{-1}:100,$$

$${\tilde{A}}_{b_1{b}_2{b}_3}=(\pm i{)}^{b_1\oplus b_2\oplus b_3}{B}_{b_1{b}_2},{\tilde{A}}_{b_1{b}_2{b}_3}:=({\zeta }^{\ast }{)}^{-1}{A}_{b_1{b}_2{b}_3},$$

$$B_{00}$$

$$B_{01}$$

$$B_{ab}{B}_{cd}=(-1{)}^{(c\oplus a)\cdot (d\oplus b)}{B}_{cb},$$

$$\tilde{\mathcal{A}}\left(w^{(1)},w^{(2)},w^{(3)}\right)$$

$$=(-1{)}^{\iota ((w^{(1)}\oplus S{w}^{(1)})\cdot w^{(2)})}(\pm i{)}^{\iota (w^{(1)}\oplus w^{(2)}\oplus w^{(3)})}{B}_{w_1^{(1)}{w}_t^{(2)}},$$

$$\{\begin{array}{l}\iota (w^{(1)})=\frac{t-(x^3-x^{\prime 3})}{2},\\ \iota (w^{(2)})=\frac{t+(x^1-x^{\prime 1})}{2},\\ \iota (w^{(3)})=\frac{t+(x^2-x^{\prime 2})}{2}.\end{array}$$

$${\mathfrak{S}}_t^n(K_1,\dots ,K_n)$$

$$={\mathfrak{S}}_t(K_1)\times \cdots \times {\mathfrak{S}}_t(K_n),$$

$$|{\psi }_{\mathrm{x}}(t)⟩={{\zeta }^{\ast }}^t\sum _{{\mathrm{x}}^{\prime }\in {\mathcal{C}}_t(\mathrm{x})}\sum _{\mathrm{w}\in {\mathfrak{S}}_t^3(K_1,K_2,K_3)}\tilde{\mathcal{A}}(\mathrm{w})|{\psi }_{{\mathrm{x}}^{\prime }}(0)⟩,$$

$$|{\psi }_{\mathrm{x}}(t)⟩={{\zeta }^{\ast }}^t\sum _{{\mathrm{x}}^{\prime }\in {\mathcal{C}}_t(\mathrm{x})}\sum _{a,b=0,1}{c}_{ab}{B}_{ab}|{\psi }_{{\mathrm{x}}^{\prime }}(0)⟩,$$

$$c_{ab}(K_1,K_2,K_3)=\sum _{\mathrm{w}\in {\mathfrak{S}}_{t,ab}^3(K_1,K_2,K_3)}(-1{)}^{\iota ((w^{(1)}\oplus S{w}^{(1)})\cdot w^{(2)})}(\pm i{)}^{\iota (w^{(1)}\oplus w^{(2)}\oplus w^{(3)})}.$$

$$w_i\ge w_{i+1}\forall i=1,2,\dots ,|w|-1.$$

$$v\oplus {\mathfrak{S}}_t(H)=\underset{n\in I}{⨆}\mathfrak{W}(K,H,n),$$

$$\mathfrak{W}(K,H,n):=\{\begin{array}{ll}{\mathfrak{S}}_K(K-H+n){\mathfrak{S}}_{t-K}(n),& \text{if K≥H},\\ {\mathfrak{S}}_K(n){\mathfrak{S}}_{t-K}(H-K+n),& \text{otherwise},\end{array}$$

$$|\mathfrak{W}(K,H,n)|=D(t,\bar{r}t+(-1{)}^{\bar{r}}H,rt+(-1{)}^{r}K,n),$$

$$D(t,p,m,n)=\{\begin{array}{ll}\left(\genfrac{}{}{0ex}{}{m}{n}\right)\left(\genfrac{}{}{0ex}{}{t-m}{p-n}\right),& \text{if 0≤n≤m≤t and n≤p,}\\ 0,& \text{otherwise},\end{array}$$

$$v\oplus {\mathfrak{S}}_t(H)=\underset{n\in I}{⨆}{\pi }_v^{-1}(\mathfrak{W}(K,H,n))$$

$$\begin{array}{rl}|{\pi }_v^{-1}(\mathfrak{W}(K,H,n))|& =|\mathfrak{W}(K,H,n)|,\\ \iota ({\pi }_v^{-1}(\mathfrak{W}(K,H,n)))& =\iota (\mathfrak{W}(K,H,n)).\end{array}$$

$$\sum _{w\in {\mathfrak{S}}_t(H)}(\pm i{)}^{\iota (v\oplus w)}=f(K,H),$$

$$f(K,H):=(\pm i{)}^{|K-H|}\sum _{n\in I}(-1{)}^{n}D(t,\bar{r}t+(-1{)}^{\bar{r}}H,rt+(-1{)}^{r}K,n),$$

Full text

††journal: rsta

1 Introduction

Quantum walks (QWs) have been originally introduced as the quantum analog of classical random walks or Markov chains [1]. A discrete-time QW [2, 3, 4] describes a quantum particle jumping on a graph in discrete time steps, the “direction” of the jumping being conditioned on the internal state of the particle. The latter is represented in a finite dimensional Hilbert space—the coin space—while the Hilbert space associated with the positions on the lattice is generally infinite dimensional. The total Hilbert space of the walk is the tensor product of the above ones.

Since their first appearance, QWs received an increasing attention in the literature, where their main application is the design of quantum algorithms [5, 6, 7, 8]. As an example, efficient search algorithms [6, 9, 10] were devised exploiting the fact that the spread of a localized initial state after $t$ steps is proportional to $t$, whereas for the classical random walk the spread is proportional to $\sqrt{t}$. Remarkably, any quantum circuit can be implemented as a QW on a graph, proving that QWs can be used for universal quantum computation [11].

Recently, in Ref. [12] the authors use QWs in a derivation of quantum field theory from principles, from which it follows that the graph of the QW is the Cayley graph of a finitely presented group $G$. The QWs reproducing free quantum field theory in Euclidean space correspond to group $G=\mathbb{Z}^{d}$ that is Abelian, which is the most common case in the literature. In particular the simplest QW descending from the principles is the Weyl QW [12], reproducing the Weyl quantum field theory. The QW approach to quantum field theory has been successfully used in Refs. [13, 14, 15, 12, 16, 17, 18, 19, 20], where non-interacting field theories are studied in a simplified picture in terms of Quantum Walks.

In Sec. 2 we review the definition of QWs on Cayley graphs. In Sec. 3 we introduce the Weyl QW and solve its path-sum on the body-centered cubic three dimensional lattice.

2 Path-sum approach for quantum walks on Cayley graphs

In this section we review the notion of a quantum walk, focusing on the underlying graph structure, that for QWs representing a homogeneous dynamics is the Cayley graph of some group.

2.1 Quantum walks on Cayley graphs

In an abstract way, a group can always be described in terms of a set of generators. If the generators are viewed as an alphabet, each word over the alphabet is associated to an element of the group. More precisely, following [21], an alphabet is any non-empty set $S$ and a string over an alphabet $S$ is a map $w\colon\{1,2,\dots,n\}\to S$, for some non-negative integer $n$. The length of a string $w\colon\{1,2,\dots,n\}\to S$ is $\lvert w\rvert\coloneqq l(w)\coloneqq n$, and for $n=0$ there is a unique string denoted $\varepsilon$, called the empty string for the alphabet $S$. Let $S^{\star}$ denote the monoid generated from $S$ by the associative operation of string concatenation, having $\varepsilon$ as its unit. Introducing the alphabet $S^{-1}$ of formal inverses of $S$, we say that a word over $S$ is a string over $S\cup S^{-1}$ and a word is called reduced if it does not contain substrings of the form $xx^{-1}$, with $x$ in $S\cup S^{-1}$. The set of all reduced words $F(S)$ can be given a group structure where the group operation is represented by word juxtaposition, the unit is $\varepsilon$, and the inverse of a word $w$ is the word consisting of the inverses of the symbols of $w$ juxtaposed in the reverse order. The group $F(S)$ is called the free group over the alphabet $S$.

Any group $G$ has a presentation $\langle S|R\rangle$—$S$ being a set of generators and $R$ any subset of $F(S)$, whose elements are called relators111The relators $r\in R$ in the presentation $\langle S|R\rangle$ of a group $G$ are words over $S$ that are equivalent to $\varepsilon$, namely to the identity. $\varepsilon$—so that $G$ is isomorphic to $F(S)/\langle R^{F(S)}\rangle$, where $\langle R^{F(S)}\rangle$ denotes the normal closure222The normal closure $\langle R^{G}\rangle$ of a set $R$ in a group $G$ is the subgroup generated by $gRg^{-1}$ with $g\in G$. of $R$ in $F(S)$. Knowing a presentation $\langle S|R\rangle$ of a group, we can give a graphical representation of it, called Cayley graph, capturing both its algebraic and geometric properties. We have the following definition.

Definition 2.1** (Cayley Graph).**

Let $G$ be a group and let $\langle S|R\rangle$ be a presenttion of $G$. Then the Cayley graph of $G$ corresponding to the presentation $\langle S|R\rangle$, denoted $\Gamma(G,S)$, is the coloured directed graph $(G,E,\{c_{h}\}_{h\in S})$ such that the vertex set is $G$, the edge set is $E=\set{(g,gh)\mid g\in G,\,h\in S}$ and each $h\in S$ has an associated colour $c_{h}$.

We are interested in a discrete unitary evolution over such a graph. Formally we have the following definition.

Definition 2.2** (Quantum walk).**

Let $V$ be a countable set, then a (discrete-time) quantum walk on $V$ is a local unitary operator $W$ on the Hilbert space $\mathcal{H}=\ell^{2}(V)\otimes\mathbb{C}^{s}$, namely a unitary operator $W$ such that there exists $k\in\mathbb{N}$ and

[TABLE]

where the neighborhood $\mathcal{N}_{x}$ of $x\in V$ is defined as the set

[TABLE]

and $\{\ket{x}\}_{x\in V}$ is the canonical orthonormal basis in $\ell^{2}(V)$.

In this context a QW on the set $V$ can be specified providing the map $\mathcal{E}\colon V\times V\to\mathcal{M}_{s}(\mathbb{C})$ associating to each pair of sites $(x,y)$ a matrix $\mathcal{E}(x,y)\coloneqq\bra{y}W\ket{x}$, called transition matrix, representing the transition amplitude from $x$ to $y$. The unitarity requirement in terms of transition matrices amounts to

[TABLE]

$I_{s}$ denoting the identity on $\mathbb{C}^{s}$. The locality constraint, on the other hand, becomes

[TABLE]

Furthermore, a sequence $\psi\colon\mathbb{N}\to\mathcal{H}$ is a solution of the quantum walk $W$ if it satisfies the following update rule for a given initial condition $\psi(0)\in\mathcal{H}$:

[TABLE]

where $\ket{\psi_{x}(t)}\in\mathbb{C}^{s}$ is defined in such a way that $\ket{\psi(t)}=\sum_{x\in V}\ket{x}\ket{\psi_{x}(t)}$.

A QW $W$ carries a natural graph structure defined by the non-null transition matrices, namely the directed graph $(V,E)$ where $V$ is the vertex set and $E$ is the set of edges $E=\set{(x,y)\in V\times V}{\mathcal{E}(x,y)\neq 0}$. It has been shown [12] that the assumption of homogeneity of the QW $W$—\iethe vertices cannot be distinguished by the walk dynamics—entails that the underlying graph is actually the Cayley graph $\Gamma(G,S)$ of a group $G$. The group $G$ acts on $\ell^{2}(G)$ by means of the right-regular representation $g\mapsto T_{g}$ as $T_{g}\ket{f}\coloneqq\ket{fg^{-1}}$. The homogeneity condition entails also that the transition matrices are independent of the location: $\mathcal{E}(gh,g)=\mathcal{E}(g^{\prime}h,g^{\prime})$, $\forall g,g^{\prime}\in G$ and $\forall h\in S$, allowing the choice $\mathcal{E}(gh,g)=A_{h}\in\mathcal{M}_{s}(\mathbb{C})$. Hence the walk operator $W$ can be written as

[TABLE]

The scope of this paper is the study of the QW evolution in terms of a path-sum, recalling the Feynman’s formulation of Quantum Mechanics. Such an approach has been effective to obtain the exact analytic solution in position space in a number of cases [22, 23, 24, 25], giving a background for possible generalizations of the method on general graphs. In the present work we will review the method in the setting of Cayley graphs and we will present a solution for the Weyl QW.

2.2 Quantum walk evolution as a path-sum

Let us consider now a QW whose transition matrices are given by the map $\mathcal{E}\colon V\times V\to\mathcal{M}_{s}(\mathbb{C})$ and whose associated graph is $(V,E)$. From Eq. 1 one can readily write the evolution of a given initial configuration $\ket{\psi(0)}\in\mathcal{H}$ as a path-sum, which can be viewed as a discrete version of Feynman’s path-integral [26, 27]. The iteration of the one-step update rule (1) leads to an expression of the solution at time $t$ in terms of the sum over all the paths $\sigma$ of length $t$ with fixed endpoints $x^{\prime}$ and $x$, that is $\sigma=(e_{1},e_{2},\dots,e_{t})$, where $e_{i}\coloneqq(y_{i-1},y_{i})$ with the the identifications $y_{0}\equiv x^{\prime}$ and $y_{t}\equiv x$. We denote the set of all such paths as $\Lambda_{t}(x^{\prime},x)$ and $\mathcal{C}_{t}(x)\coloneqq\set{x^{\prime}\in G}{\Lambda_{t}(x^{\prime},x)\neq\emptyset}$ is the slice at time $t$ of the past causal cone of $x$. In this way the equation for the evolved state $\ket{\psi_{x}(t)}$ takes the form

[TABLE]

The situation of main interest for us is when the underlying graph of the walk is actually a Cayley graph $\Gamma(G,S)$. In such a case, the transition matrices depend only on the generators associated to the edges and, as before, we let $\mathcal{E}(xh,x)=A_{h}$. Notice that a path $\sigma$ is in $\Lambda_{t}(x^{\prime},x)$ if and only if there exist $h_{1},\dots,h_{t}\in S$ such that $x^{\prime}h_{1}\dotsm h_{t}=x$; therefore we obtain the expression

[TABLE]

where $\delta(x)=1$ if $x=\varepsilon$ and [math] otherwise, $\varepsilon$ being the identity element of $G$, and $\mathcal{A}(h_{1},h_{2},\dots,h_{t})=A_{h_{t}}A_{h_{t-1}}\dotsm A_{h_{1}}$.

In the following section, we will present an application of the path-sum method in the specific case of the Weyl QW in $3+1$ dimensions deriving the solution in position space with the aid of a binary encoding of paths, an approach which has already proven its efficacy in the case of the Dirac QW in $1+1$ dimensions [24] and of the Weyl QW in $2+1$ dimensions [25].

3 Weyl quantum walk in $3+1$ dimensions

In the following we will use the vector notation to denotes points ${\bm{\mathrm{x}}}\in G=\mathbb{Z}^{3}$. The Weyl QW in $3+1$ dimensions, derived by D’Ariano and Perinotti [12], is a QW on the BCC lattice $\Gamma(G,S)$, where the vertex set can be chosen to be $G=2\mathbb{Z}^{3}\cup(2\mathbb{Z}^{3}+(1,1,1))$ and the generators can be represented by the following vectors (and their inverses):

[TABLE]

The walk unitary operator is written as $W=\sum_{{\bm{\mathrm{h}}}\in S}T_{{\bm{\mathrm{h}}}}\otimes A_{{\bm{\mathrm{h}}}}$, acting on $\ell^{2}(G)\otimes\mathbb{C}^{2}$, and the transition matrices are

[TABLE]

with $\zeta=\frac{1\pm i}{4}$ and ${\bm{\mathrm{h}}}_{-l}=-{\bm{\mathrm{h}}}_{l}$. The two possible choices for the coefficient $\zeta$ correspond to the two inequivalent QW solutions existing on the BCC lattice [12].

The intent of this paper is to provide an explicit expression for the propagator given in terms of sums over paths. To this end, it turns out to be very effective to adopt a binary description of paths, which amounts to find a three-bits binary encoding $b_{1}b_{2}b_{3}$ for the generators. A suitable choice that simplifies the evaluation of the contribution to the transition amplitude of each path is given by

[TABLE]

This choice is tantamount to the requirement

[TABLE]

where $\oplus$ denotes the sum modulo $2$ and $B_{b_{1}b_{2}}$ are the matrices

[TABLE]

satisfying the product rule given by

[TABLE]

where the notation $a\cdot b$ denotes the binary product of the two bits $a$ and $b$; throughout this paper, all the bit operations are also extended element-wise to binary strings. One can notice from Eq. 4 that the matrices $\tilde{A}_{b_{1}b_{2}b_{3}}$ generate, up to phases, a finite semigroup. The general product of $t$ transition matrices is then given by

[TABLE]

where $w^{(j)}\in\mathfrak{B}_{t}$ is the string made of the $j$-th bits of the encoding Eq. 3, $\mathfrak{B}_{t}$ being the set of all binary strings of length $t$. Here $w_{k}$ denotes the $k$-th bit of the string $w$. Finally we introduced the function $\iota(w)\coloneqq\sum_{k=1}^{\lvert w\rvert}w_{k}$ counting the number of $1$-bits present in the string $w$ and the left circular shift $S$ defined by $(Sw)_{i}=w_{(i\bmod t)+1}$, for all $i=1,\dots,t$.

At this point, we need a characterization of the lattice paths in terms of their binary description. As shown in Appendix A, the number of $1$-bits in each one of the encoding strings $w^{(j)}\in\mathfrak{B}_{t}$ is fixed by the starting point ${\bm{\mathrm{x}}}^{\prime}$ and by the ending point ${\bm{\mathrm{x}}}$ via the equations

[TABLE]

It is then convenient to define a special notation for the set of tuples of strings with a fixed number of $1$-bits, which are indeed in bijective correspondence with paths connecting ${\bm{\mathrm{x}}}^{\prime}$ to ${\bm{\mathrm{x}}}$ in $t$ steps:

[TABLE]

assuming for short $\mathfrak{S}_{t}(K)\equiv\mathfrak{S}_{t}^{1}(K)$. Thus, the path-sum of Eq. 2 can be rewritten as

[TABLE]

where $K_{1}=\frac{t+(x^{3}-x^{\prime 3})}{2},K_{2}=\frac{t-(x^{1}-x^{\prime 1})}{2},K_{3}=\frac{t-(x^{2}-x^{\prime 2})}{2}$ (notice that the signs here are consistent with Eq. 2 where the matrix associated to the step $h$ is exactly $A_{h^{-1}}$). From Eq. 5, we see that the resulting matrix depends only on the first bit of $w^{(1)}$ and the last bit of $w^{(2)}$. Therefore, defining $\mathfrak{S}_{t,ab}^{3}(K_{1},K_{2},K_{3})\subset\mathfrak{S}_{t}^{3}(K_{1},K_{2},K_{3})$ as the subset of triads $(w^{(1)},w^{(2)},w^{(3)})$ with $w^{(1)}_{1}=a$ and $w^{(2)}_{t}=b$, we can rewrite Eq. 6 as:

[TABLE]

We seek now an explicit expression for the coefficients $c_{ab}$. As a first step let us consider the sum $\sum_{w}(\pm i)^{\iota(v\oplus w)}$ alone with $\iota(w)=H$ for a given $H$ and for some fixed $v\in\mathfrak{S}_{t}(K)$. In order to compute this sum we have to classify the strings lying in the set $v\oplus\mathfrak{S}_{t}(H)$, according to the number of ones. It is convenient to define a canonical form for strings as follows.

Definition 3.1**.**

We say that a binary string $w$ is in canonical form if the following condition holds:

[TABLE]

The computation of the coefficients $c_{ab}$ in Eq. (8) is based on the next three results whose proofs are presented in Appendix B.

Lemma 3.1**.**

Given $t,K,H\in\mathbb{N}$, with $K,H\leq t$, let $v\in\mathfrak{S}_{t}(K)$ be in canonical form, then we have

[TABLE]

with $I=\{0,1,\dots,\min\{K,H,t-K,t-H\}\}$. Defining $r=\vartheta(K-H)$ and $\bar{r}=1-r$, the size of each subset is given by

[TABLE]

and for all $w\in\mathfrak{W}(K,H,n)$ the set-bit count is given by $\iota(w)=\lvert K-H\rvert+2n$.

Corollary 3.1**.**

Given $t,K,H\in\mathbb{N}$, with $K,H\leq t$, for all $v\in\mathfrak{S}_{t}(K)$ and for any $\pi_{v}$ bitwise permutation of $\mathfrak{B}_{t}$ such that $\pi_{v}(v)$ is in canonical form, the following decomposition holds:

[TABLE]

Proposition 3.1**.**

Given $t,K,H\in\mathbb{N}$, with $K,H\leq t$, for all $v\in\mathfrak{S}_{t}(K)$ we have that

[TABLE]

with $I=\{0,1,\dots,\min\{K,H,t-K,t-H\}\}$.

The expression of the coefficients $c_{ab}$ of Eq. 8 can now be written as

[TABLE]

where, at this point, the sum is performed over the set $\mathfrak{S}_{t,ab}^{2}(K_{1},K_{2})$. In order to ease the calculation we can study separately the values of $\iota((w^{(1)}\oplus Sw^{(1)})\cdot w^{(2)})$ and their interplay with $f(K,H)$. The combinatorics for $\iota((w^{(1)}\oplus Sw^{(1)})\cdot w^{(2)})$ is the same as that of Ref. [25], and is here reviewed for the convenience of the reader. The idea is to find a classification of the binary strings in terms of the values taken by $\iota((w^{(1)}\oplus Sw^{(1)})\cdot w^{(2)})$. We first focus on the classification induced by $w^{(1)}\oplus Sw^{(1)}$ and then, since the only contributions to the result come from the $1$-bits of both $w^{(1)}\oplus Sw^{(1)}$ and $w^{(2)}$, we use $w^{(2)}$ to select a given number of $1$-bits from $w^{(1)}\oplus Sw^{(1)}$.

The combinatorics induced by $w^{(1)}\oplus Sw^{(1)}$ is provided by the following lemma (proved in Appendix B).

Lemma 3.2**.**

Let $v\in\mathfrak{S}_{t}(K)$. Then $v\oplus Sv\in\pi_{v}^{-1}\left\lparen\mathfrak{S}_{K}(n)\mathfrak{S}_{t-K}(n)\right\rparen$, for some $n\in\{0,1,\dots,\min\{K,t-K\}\}$.

The above classification is not sufficient in our case, since the summation of Eq. 13 requires the strings $w^{(1)}$ ad $w^{(2)}$ to have, respectively, the first and the last bit fixed. This means that we have to refine the counting taking into account such constraint: we denote $\mathfrak{T}_{aa^{\prime}}(t,K,n)$ the set of strings $v\in\mathfrak{S}_{t}(K)$ such that $v\oplus Sv\in\pi_{v}^{-1}\left\lparen\mathfrak{S}_{K}(n)\mathfrak{S}_{t-K}(n)\right\rparen$ with $v_{1}=a$ and $v_{t}=a^{\prime}$. The following lemma (also proved in Appendix B) concludes the counting for strings with fixed endpoints.

Lemma 3.3**.**

The number $u_{aa^{\prime}}(n)$ of binary strings $v\in\mathfrak{T}_{aa^{\prime}}(t,K,n)$ is given by

[TABLE]

with $n_{\textup{min}}(K)\leq n\leq n_{\textup{max}}(K)$ and $n_{\textup{min}}(K)\coloneqq\min\{1,K,t-K\}$,

[TABLE]

Finally we can focus on the joint classification given by the values of $\iota((v\oplus Sv)\cdot w)$ and of $\iota(v\oplus w)$. The final expression of the coefficients $c_{ab}$ is summarized by the following proposition, whose proof is reported in Appendix B.

Proposition 3.2**.**

For $t\geq 2$ the coefficients of Eq. 13 take the form

[TABLE]

where, defining $r=\vartheta(K_{1}-K_{2})$, we have that $\tau_{J}=\lvert K_{1}-K_{2}\rvert+2J$, and

[TABLE]

As a concluding remark of this section we point out that thanks to the binary encoding of paths, the counting is largely simplified providing a powerful tool working in the general case.

4 Conclusions

In this paper we provided an explicit analytical solution in position space of the propagator for the Weyl QW in $3+1$ dimensions. The method used here to analytically compute the Weyl QW path sum is grounded on:

A combinatorial analysis for the characterization of the admissible paths connecting two fixed sites of the graph in a given number of steps; 2. 2.

The semigroup structure of the walk transition matrixes.

The semigroup structure allows us to classify the admissible paths (or binary strings according to the first point) into equivalence classes according to the global transition matrix of the path. To aid the calculation, we adopted here an extremely powerful approach consisting in the translation of the geometric properties of lattice paths into algebraic properties of binary strings. Such binary description of paths was succesful also in the case of the Dirac QW in $1+1$ dimensions [24] and of the Weyl QW in $2+1$ dimensions [25]. In all these cases the practical computation of the propagator relies on the semigroup property of the transition matrices which allows one to determine all the paths contributing with the same matrix.

As a future perspective we propose the possibility to investigate such semigroup property of the transition matrices in the general non Abelian case. It would also be interesting to derive the general hypotheses in the framework of QWs on Cayley graphs for which the semigroup structure is recovered.

\aucontribute

Giacomo Mauro D’Ariano and Paolo Perinotti conceived and designed the study. Nicola Mosco carried out the calculations. Nicola Mosco and Alessandro Tosini drafted the manuscript. All the authors read, edited and approved the manuscript.

\competing

The author(s) declare that they have no competing interests.

\funding

This publication was made possible through the support of a grant from the John Templeton Foundation, ID # 60609 “Quantum Causal Structures”. The opinions expressed in this publication are those of the authors and do not necessarily reflect the views of the John Templeton Foundation.

Appendix A Relation between lattice points and binary strings

The approach we follow here requires to find a binary description of paths: we provided such description by choosing a suitable binary encoding in Eq. 3 which also simplifies the evaluation of functions involving products of matrices giving the general formula of Eq. 5. Then, we also need to obtain a criterion telling us which paths have the same endpoints ${\bm{\mathrm{x}}}^{\prime}$ and ${\bm{\mathrm{x}}}$; in order to do so, we employ the fact that the BCC lattice is Abelian: each path is described by a quadruple $(n_{\pm 1},n_{\pm 2},n_{\pm 3},n_{\pm 4})$, where $n_{\pm l}$ counts the number of steps in direction ${\bm{\mathrm{h}}}_{\pm l}$ and thus each path must satisfy the system

[TABLE]

The binary encoding and the quadruple $(n_{\pm 1},n_{\pm 2},n_{\pm 3},n_{\pm 4})$ are then related by the following constraints:

[TABLE]

where $\iota(w)\coloneqq\sum_{k=1}^{\lvert w\rvert}w_{k}$ denotes the set-bit count of the string $w$. The relation we seek can be easily obtained by defining the number of steps in the positive- and negative-coordinate direction

[TABLE]

$\vartheta$ being the Heaviside step function; then, for each $i$ it holds that

[TABLE]

Therefore, the set-bit counts of Eq. 19 are fixed by the coordinates through the equations

[TABLE]

Appendix B Proofs of the results

Proof of Lemma 3.1.

Suppose for definiteness that $K\geq H$. Given a string $w\in\mathfrak{S}_{t}(H)$ we can define the two substrings $(L_{K}w)_{i}=w_{i}$, for $i=1,\dots,K$, and $(R_{K}w)_{i}=w_{K+i}$, for $i=1,\dots,t-K$: we have that $L_{K}w\in\mathfrak{S}_{K}(H-n)$, $R_{K}w\in\mathfrak{S}_{t-K}(n)$ and $w\in\mathfrak{S}_{K}(H-n)\mathfrak{S}_{t-K}(n)$. Here for $A$ and $B$ sets of strings, the set $AB$ is formed by concatenating the elements of $A$ and $B$. Moreover if $n\neq m$, then $\mathfrak{S}_{K}(n)\cap\mathfrak{S}_{K}(m)=\emptyset$, which entails that the products $\mathfrak{S}_{K}(H-n)\mathfrak{S}_{t-K}(n)$ form a partition of $\mathfrak{S}_{t}(H)$, with $n\in I=\{0,1,\dots,\min\{H,t-K\}\}$:

[TABLE]

Thus the set $v\oplus\mathfrak{S}_{t}(H)$ decomposes as

[TABLE]

where in the third equation we have made use of the fact that $v$ is in canonical form. The set-bit count of $w\in\mathfrak{W}(K,H,n)$ can be straightforwardly computed:

[TABLE]

Finally, one can easily obtain the size of each factor:

[TABLE]

The case $K<H$ follows from a similar reasoning and we have the sought result. ∎

Proof of Corollary 3.1.

Let $v\in\mathfrak{S}_{t}(K)$ and let $\pi_{v}$ be a bitwise permutation such that $\pi_{v}(v)$ is canonical, then we have that

[TABLE]

Therefore, using the decomposition of Lemma 3.1, we can write

[TABLE]

Proof of Proposition 3.1.

Suppose that $K\geq H$ and let $v\in\mathfrak{S}_{t}(K)$, $w\in\mathfrak{S}_{t}(H)$. From Corollary 3.1 we know that $v\oplus w\in\pi_{v}^{-1}(\mathfrak{W}(K,H,n))$, for some $n\in I=\{0,1,\dots,\min\{H,t-K\}\}$, and $\iota(v\oplus w)=K-H+2n$. Therefore we can compute the sum of Eq. 12, obtaining finally the result:

[TABLE]

Proof of Lemma 3.2.

Let $v\in\mathfrak{S}_{t}(K)$. From Lemmas 3.1 and 3.1 we know that $v\oplus Sv\in\mathfrak{S}_{t}(2n)$, for some $n\in\{0,1,\dots,\min\{K,t-K\}\}$. We can split the string $\pi_{v}(v\oplus Sv)\eqqcolon c\oplus\pi_{v}Sv$ as

[TABLE]

Given that

[TABLE]

Therefore, since $\iota(c\oplus\pi_{v}Sv)=2n$, we have precisely

[TABLE]

and the result follows. ∎

Proof of Lemma 3.3.

A string in $\mathfrak{T}_{aa^{\prime}}(t,K,n)$ can be constructed by first arranging the $K$ $1$-bits in $n+aa^{\prime}$ slots and then by arranging the $t-K$ [math]-bits in $n+\bar{a}\bar{a}^{\prime}$ slots: the problem is then the same as counting the number of compositions of an integer [28]. We denote here as $C_{K,n}$ the number of $n$-compositions of an integer $K$:

[TABLE]

Since the $n+aa^{\prime}$ slots can be filled independently from the $n+\bar{a}\bar{a}^{\prime}$ ones, the cardinality of $\mathfrak{T}_{aa^{\prime}}(t,K,n)$ is simply given by the product

[TABLE]

Proof of Proposition 3.2.

Let us consider the case where $K_{1}\geq K_{2}$ as the other one is a simple variation of the following construction. First of all, recall the expression for the coefficients (13), implementing the result of Proposition 3.1:

[TABLE]

where $av_{n}a^{\prime}\in\mathfrak{T}_{aa^{\prime}}(t,K_{1},n)$ and $w\in\mathfrak{S}_{t-1}(K_{2}-b)$. For ease of discussion, we consider from now on the case where $a=a^{\prime}=0$, as the others can be treated in a very similar way.

We already know that the sum modulo $2$ of a pair of strings $0v_{n}0$ and $wb$ can be parametrized as $\iota(0v_{n}0\oplus wb)=|K_{1}-K_{2}|+2J$ for some $J$; yet we need to find a finer classification taking into account the values of $\iota((0v_{n}0\oplus v_{n}00)\cdot wb)$, for each fixed $v_{n}$. To such end, we consider the partition

[TABLE]

this particular construction corresponds to choose as reference strings

[TABLE]

where the text under braces denotes the length of the substring.

Since $\mathfrak{S}_{t}(K_{2})$ can be constructed by concatenating strings picked from the sets $\mathfrak{Z}_{b}^{(1)}(J,k)$ and $\mathfrak{Z}_{b}^{(0)}(J,k^{\prime})$, we can study separately the binary product and binary sum on them. Multiplying $p_{L}$ and $p_{R}$ with $\mathfrak{Z}_{b}^{(1)}(J,k)$ and $\mathfrak{Z}_{b}^{(0)}(J,k^{\prime})$ respectively, we obtain

[TABLE]

for all $w\in\mathfrak{Z}_{b}^{(1)}(J,k)\mathfrak{Z}_{b}^{(0)}(J,k^{\prime})$; whereas, for the binary sum we get

[TABLE]

In conclusion, the total number of ones for the sum modulo $2$ turns out to be

[TABLE]

which corresponds to the parameter $\tau_{J}$ of the statement of the proposition. Furthermore, since $f(\iota(0v_{n}0\oplus wb),K_{3})$ does not depend on $k$ nor $k^{\prime}$, we can sum separately over the strings in Eqs. 24 and 23 obtaining the function $w_{00b}^{(s)}$ of Eq. 17:

[TABLE]