Establishing the equivalence between Szegedy’s and coined quantum walks using the staggered model

Abstract

Coined quantum walks (QWs) are being used in many contexts with the goal of understanding quantum systems and building quantum algorithms for quantum computers. Alternative models such as Szegedy’s and continuous-time QWs were proposed taking advantage of the fact that quantum theory seems to allow different quantized versions based on the same classical model, in this case the classical random walk. In this work, we show the conditions upon which coined QWs are equivalent to Szegedy’s QWs. Those QW models have in common a large class of instances, in the sense that the evolution operators are equal when we convert the graph on which the coined QW takes place into a bipartite graph on which Szegedy’s QW takes place, and vice versa. We also show that the abstract search algorithm using the coined QW model can be cast into Szegedy’s searching framework using bipartite graphs with sinks.

Appendix

1.1 Proof of Theorem 4.1

Suppose that we have a well-defined nonregular flip-flop coined QW on a graph $\varGamma (V,E)$ with coin $C'$ being an orthogonal reflection. As an intermediate step, we use a staggered QW on a graph $\varGamma (V',E')$ with two tessellations equivalent to the coined QW. If $C'=C_1\oplus \cdots \oplus C_{|V|}$, the eigenvectors of $C'$ are the direct sum of eigenvectors of $C_v$ and zero vectors, for $1\le v\le |V|$. Let $\big | \tilde{\alpha }_x^{(v)} \big \rangle$, $0\le x < m_v$ be an orthonormal basis for the $(+1)$-eigenspace of $C_v$, and let $\big | \alpha _{x_v}^{(v)} \big \rangle$ be the corresponding eigenvectors of $C'$ obtained from $\big | \tilde{\alpha }_{x_v}^{(v)} \big \rangle$ by performing the necessary direct sums with zero vectors. If $C'$ is an orthogonal reflection of graph $\varGamma '(V',E')$, it can be written as

$$\begin{aligned} C'= & {} 2\,\sum _{v=1}^{{|V|}} \sum _{x_v=0}^{m_v-1} \big | \alpha _{x_v}^{(v)} \big \rangle \big \langle \alpha _{x_v}^{(v)} \big | - I, \end{aligned}$$

(36)

where the set of $(+1)$-eigenvectors $\big | \alpha _{x_v}^{(v)} \big \rangle$ has the following properties: (1) If the ith entry of $\big | \alpha _{x_v}^{(v)} \big \rangle$ is nonzero, the ith entries of the other $(+1)$-eigenvectors must be zero, and (2) vector $\sum _{v=1}^{{|V|}}\sum _{x_v=0}^{m_v-1} \big | \alpha _{x_v}^{(v)} \big \rangle$ has no zero entries. Then each $C_v$ can be written as

$$\begin{aligned} C_v= & {} 2\sum _{x=0}^{m_v-1} \big | \tilde{\alpha }_x^{(v)} \big \rangle \big \langle \tilde{\alpha }_x^{(v)} \big | - I \end{aligned}$$

(37)

and the set of vectors $\big | \tilde{\alpha }_x^{(v)} \big \rangle$ inherit properties (1) and (2) in $\mathcal{H}^{d_v}$. Each $C_v$ is an orthogonal reflection in $\mathcal{H}^{d_v}$, where $d_v$ is the degree of vertex v in $\varGamma (V,E)$, and has an associated $d_v$-graph $\varGamma _{C_v}$ tessellated by the $(+1)$-eigenvectors of $C_v$. $\varGamma _{C_v}$ is a union of $m_v$ disjoint cliques. If $C_v$ has only one $(+1)$-eigenvector ($m_v=1$), $\varGamma _{C_v}$ is a clique.

Graph $\varGamma '(V',E')$ is obtained from $\varGamma (V,E)$ by replacing each vertex $v\in V$ by graph $\varGamma _{C_v}$ gluing the vertices of $\varGamma _{C_v}$, which run from 0 to $d_v-1$, in one-to-one mapping with the labels of the coin directions at vertex v. The vertices of $\varGamma _{C_v}$ after the gluing process receive the labels of the basis vectors in $\big | \alpha _{x_v}^{(v)} \big \rangle$ with nonzero coefficients. For example, the vertices of the 3-clique in graph $\varGamma '$ in Fig. 6 have labels 1, 2, and 3 because they correspond to the basis vectors of the $(+1)$-eigenvector $\big | \alpha _1 \big \rangle =(\big | 1 \big \rangle +\big | 2 \big \rangle +\big | 3 \big \rangle )/\sqrt{3}$.

Polygons induced by $\big | \alpha _x^{(v)} \big \rangle ,\,\forall v,x$ tessellate $\varGamma '(V',E')$ because polygons induced by $\big | \alpha _{x}^{(v)} \big \rangle$, $0\le x<m_v$ exactly cover the graphs $\varGamma _{C_v}$ that replace vertices v. Tessellation $\alpha$ covers all vertices and all edges that were added via $\varGamma _{C_v}$, for all v. This tessellation does not cover the edges of $\varGamma '(V',E')$ that were inherited from the original graph $\varGamma (V,E)$.

Tessellation $\beta$ is made of size-2 polygons that cover the edges of $\varGamma '(V',E')$ that were inherited from the original graph $\varGamma (V,E)$. This tessellation has |E| polygons, and the set of those polygons has a one-to-one mapping with an independent set of $(+1)$-eigenvectors of S in the computational basis, which are given by vectors

$$\begin{aligned} \big | \beta _{v,j} \big \rangle \,=\, \frac{1}{\sqrt{2}}\left( \big | v,j \big \rangle +\big | v',j' \big \rangle \right) \end{aligned}$$

(38)

using the notation of Eq. (4), where $v\in V$ and $0\le j\le d_v-1$. The cardinality of the independent set of $(+1)$-eigenvectors of S is |E|. The shift operator of the nonregular flip-flop coined QW is

$$\begin{aligned} S\,=\, 2 \sum \big | \beta _{v,j} \big \rangle \big \langle \beta _{v,j} \big |-I \end{aligned}$$

(39)

where the sum runs over the set of independent $(+1)$-eigenvectors $\big | \beta _{v,j} \big \rangle$ (the sum has |E| terms). S is an orthogonal reflection because the set of independent $(+1)$-eigenvectors has nonoverlapping nonzero entries and the sum of those eigenvectors has no zero entries in the computation basis of $\varGamma '(V',E')$. Polygons induced by $\big | \beta _{v,j} \big \rangle$ form a perfect matching of $\varGamma '(V',E')$.

The union of tessellations $\alpha$ and $\beta$ covers all edges and is a well-defined staggered QW having one vertex in each polygon intersection. Using Proposition 4.3 of Ref. [21], this staggered QW can be cast into the extended Szegedy’s framework. $\square$

1.2 Proof of Theorem 4.2

Consider the staggered QW model on the line graph $L(\varGamma )$ equivalent to Szegedy’s QW on $\varGamma (X,Y,E)$. $L(\varGamma )$ has 2|Y| vertices. The polygons of the staggered model are induced by

$$\begin{aligned} \big | \alpha _x \big \rangle= & {} \sum _{y\in Y} \sqrt{p_{x y}} \, \big | f(x,y) \big \rangle , \end{aligned}$$

(40)

$$\begin{aligned} \big | \beta _y \big \rangle= & {} \sum _{x\in X} \sqrt{q_{y x}}\, \big | f(x,y) \big \rangle , \end{aligned}$$

(41)

where f is the bijection between E and the vertices of $L(\varGamma )$ as described in Ref. [21] and vectors $\big | \alpha _x \big \rangle ,\big | \beta _x \big \rangle$ belong to Hilbert space $\mathcal{H}^{2|Y|}$.

Tessellation $\alpha$ is induced by the orthogonal reflection

$$\begin{aligned} U_0= & {} 2\sum _{x\in X} \big | \alpha _x \big \rangle \big \langle \alpha _x \big | - I. \end{aligned}$$

(42)

By using a proper choice of f, matrix $\big | \alpha _x \big \rangle \big \langle \alpha _x \big |$ is a direct sum of zeros matrices and a $d_x\times d_x$ matrix $M_x$, which has no zero entries. Define

$$\begin{aligned} C_x\,=\, 2 M_x - I. \end{aligned}$$

(43)

Then

$$\begin{aligned} U_0 \,=\, \bigoplus _{x\in X} C_x. \end{aligned}$$

(44)

Operator $U_1$ is equal to the one described in the proof of Theorem 3.2 and is given by Eq. (27) because the assumptions about vertices $y\in Y$ are equal to the ones in Theorem 3.2. Then $U_1$ commutes basis vectors and $U_1^2=I$.

The evolution operator is

$$\begin{aligned} U \,=\, U_1 \,U_0, \end{aligned}$$

(45)

where $U_0$ is the coin operator given by Eq. (44) and $U_1$ is the shift operator given by Eq. (27). U is an evolution operator of a nonregular flip-flop coined QW on the (multi)graph obtained in the following way: The polygons of tessellation $\beta$ have two vertices and form a perfect matching of $L(\varGamma )$. The remaining cliques belong to tessellation $\alpha$. Each clique of tessellation $\alpha$ must be converted into a single vertex. If two cliques of tessellation $\alpha$ are connected by an edge, the vertices that replace those cliques are adjacent. If two cliques are connected by more than one edge, the vertices that replace those cliques must be connected by more than one edge generating a nonregular |X|-multigraph. $\square$

1.3 Proof of Theorem 5.1

Proof

The method employed in the proof of Theorem 4.1 when the coin is an orthogonal reflection can be straightforwardly extended when the coin is a partial orthogonal reflection. In this case, we can convert a nonregular flip-flop coined QW on a graph $\varGamma (V,E)$ with coin $(-I)$ on the marked vertices and the Grover coin on the nonmarked vertices into an equivalent generalized staggered QW on $\varGamma '(V',E')$, which is obtained from $\varGamma (V,E)$ in the following way: A nonmarked vertex $v\in V$ is converted into $d_v$-cliques and a marked vertex v into disconnected $d_v$-graphs (empty $d_v$-graphs), where $d_v$ is the degree of vertex v. Tessellation $\alpha$ is partial, with polygons being the cliques associated with nonmarked vertices only. Tessellation $\beta$ is the same employed in the proof of Theorem 4.1.

The next step is to define a new graph $\varGamma ''(V',E'')$ by converting the empty $d_v$-graphs into complete graphs by adding new edges to $\varGamma '(V',E')$. Let $\tilde{\alpha }$ be an extension of partial tessellation $\alpha$ by adding new polygons corresponding to the new complete graphs. $\varGamma ''(V',E'')$ is the line graph of some bipartite graph $\varGamma (X,Y,\tilde{E})$ because the union of tessellations $\tilde{\alpha }$ and $\beta$ forms a two colorable Kraus partition of $\varGamma ''(V',E'')$.

We have defined a generalized staggered QW on $\varGamma '(V',E')$, which is equivalent to a generalized staggered QW on $\varGamma ''(V',E'')$ using partial tessellation $\alpha$ because the edges in $E''{\setminus } E'$ do not belong to any polygon. Following Ref. [21], we can obtain an equivalent Szegedy’s QW; the missing polygons in partial tessellation $\alpha$ create sinks in graph $\varGamma (X,Y,\tilde{E})$ by removing the directed edges coming out of the vertices in X associated with the missing polygons. The edges oriented to the sinks are kept. This process creates a new directed bipartite graph $\varGamma '(X,Y,\tilde{E}')$. Reference [21] showed that Szegedy’s QW on $\varGamma '(X,Y,\tilde{E}')$ with vectors $\big | \phi _x \big \rangle$ and $\big | \psi _y \big \rangle$ given by Eqs. (6) and (7) in uniform superposition is equivalent to the generalized staggered QW on $\varGamma '(V',E')$. Then the nonregular flip-flop coined QW on a graph $\varGamma (V,E)$ with coin $(-I)$ on the marked vertices and the Grover coin on the nonmarked vertices can be cast into Szegedy’s searching framework. $\square$

1.4 Proof of Theorem 5.2

Proof

This theorem is a corollary of Theorem 4.2, if we employ the method described in Ref. [21] of converting Szegedy’s QWs on bipartited graphs with sinks into generalized staggered QWs. To convert generalized staggered QWs into the coined QW model, a missing polygon is converted into coin $(-I)$. $\square$