Element distinctness revisited

Abstract

The element distinctness problem is the problem of determining whether the elements of a list are distinct, that is, if $x=(x_1,\ldots ,x_N)$ is a list with N elements, we ask whether the elements of x are distinct or not. The solution in a classical computer requires N queries because it uses sorting to check whether there are equal elements. In the quantum case, it is possible to solve the problem in $O(N^{2/3})$ queries. There is an extension which asks whether there are k colliding elements, known as element k-distinctness problem. This work obtains optimal values of two critical parameters of Ambainis’ seminal quantum algorithm (SIAM J Comput 37(1):210–239, 2007). The first critical parameter is the number of repetitions of the algorithm’s main block, which inverts the phase of the marked elements and calls a subroutine. The second parameter is the number of quantum walk steps interlaced by oracle queries. We show that, when the optimal values of the parameters are used, the algorithm’s success probability is $1-O(N^{1/(k+1)})$, quickly approaching 1. The specification of the exact running time and success probability is important in practical applications of this algorithm.

Appendix

In this appendix, we define the graph theory terms used in this work [30,31,32] and the staggered quantum walk [22,23,24].

A simple and undirected graph (simply graph) $\varGamma (V,E)$ is defined by a set $V(\varGamma )$ of vertices or nodes and a set $E(\varGamma )$ of edges so that each edge links two vertices and two vertices are linked by at most one edge. Two vertices linked by an edge are called adjacent. Two edges that share a common vertex are also called adjacent. A subgraph $\varGamma '(V',E')$, where $V'(\varGamma ')\subset V(\varGamma )$ and $E'(\varGamma ')\subset E(\varGamma )$, is an induced subgraph of $\varGamma (V,E)$ if it has exactly the edges that appear in $\varGamma$ over the same vertex set. If two vertices are adjacent in $\varGamma$, they are also adjacent in the induced subgraph. A bipartite graph is a graph whose vertex set V is the union of two disjoint sets X and $X'$ so that no two vertices in X are adjacent and no two vertices in $X'$ are adjacent. A clique is a subset of vertices of a graph such that its induced subgraph is complete. A maximal clique is a clique that cannot be extended by including one more adjacent vertex, that is, it is not contained in a larger clique. A clique of size d is called a d-clique. A clique can have one vertex. A partition of the vertex set into cliques is a collection of vertex–disjoint cliques, whose union is the vertex set. Some references in graph theory use the term “clique” as a synonym of maximal clique. We avoid this notation here. A clique graph $K(\varGamma )$ of a graph $\varGamma$ is a graph such that every vertex represents a maximal clique of $\varGamma$ and two vertices of $K(\varGamma )$ are adjacent if and only if the underlying maximal cliques in $\varGamma$ share at least one vertex in common. A line graph (or derived graph or interchange graph) of a graph $\varGamma$ (called root graph) is another graph $L(\varGamma )$ so that each vertex of $L(\varGamma )$ represents an edge of $\varGamma$ and two vertices of $L(\varGamma )$ are adjacent if and only if their corresponding edges share a common vertex in $\varGamma$. A proper coloring or simply coloring of a loopless graph is a labeling of the vertices with colors such that no two vertices sharing the same edge have the same color. A n-colorable graph is the one whose vertices can be colored with at most n colors so that no two adjacent vertices share the same color. This concept can be used for edges and other graph structures.

A graph tessellation ${\mathcal {T}}$ is a partition of the vertex set into cliques, that is, there are disjoint cliques $c_1, \ldots , c_{\left| {\mathcal {T}}\right| }$ such $\cup _{\ell =1}^{\left| {\mathcal {T}}\right| } c_\ell =V(\varGamma )$, where $\left| {\mathcal {T}}\right|$ is the tessellation size. An element $c_\ell$ of the tessellation is called a polygon (or tile). An edge belongs to the tessellation ${\mathcal {T}}$ if and only if its endpoints belong to the same polygon in ${\mathcal {T}}$. The set of edges belonging to ${\mathcal {T}}$ is denoted by ${\mathcal {E}}({\mathcal {T}})$. A graph tessellation cover of size n is a set of n tessellations ${\mathcal {T}}_1,\ldots ,{\mathcal {T}}_n$, whose union is the edge set, that is, $\cup _{j=1}^n\,{\mathcal {E}}({\mathcal {T}}_j)=E(\varGamma )$. A graph is called n-tessellable if there is a tessellation cover of size at most n. The tessellation cover number is the size of a smallest tessellation cover of $\varGamma$. A graph $\varGamma$ is 2-tessellable if and only if $K(\varGamma )$ is 2-colorable [24]. The definition of graph tessellation cover was introduced by Portugal et al. [22].

In its simplest form, a staggered quantum walk on a graph $\varGamma (V,E)$ with a graph tessellation cover ${\mathcal {C}}=\{{\mathcal {T}}_1,\ldots ,{\mathcal {T}}_n\}$ is a quantum walk driven by the unitary operator $U_{\mathcal {C}}=U_n\cdots U_2\cdot U_1$, where $U_j$ is associated with tessellation ${\mathcal {T}}_j$ and is defined by

$$\begin{aligned} U_j\,=\,2\sum _{\ell =1}^{\left| {\mathcal {T}}_j\right| } \left| c_\ell ^{(j)} \right\rangle \left\langle c_\ell ^{(j)} \right| - I, \end{aligned}$$

where

$$\begin{aligned} \left| c_\ell ^{(j)} \right\rangle \,=\, \frac{1}{\sqrt{\left| c_\ell ^{(j)}\right| }}\sum _{v\in c_\ell ^{(j)}}\left| v \right\rangle . \end{aligned}$$

The dimension of Hilbert space is |V|, and the computational basis is indexed by the vertices of $\varGamma$.

Ambainis’ graph [8] is a bipartite graph with ${N\atopwithdelims ()r}+{N\atopwithdelims (){r+1}}$ vertices. The vertices of the first set are r-subsets of [N] and of the second set are $(r+1)$-subsets. A vertex $v_1$ in the first set is adjacent to a vertex $v_2$ in the second set if and only if $|v_1\cap v_2|=r$. The graph $\varGamma$ defined in Sect. 2, on which the 2-tessellable staggered quantum walk takes place, is the line graph of Ambainis’ graph, which on the other hand is the clique graph of $\varGamma$. $K(\varGamma )$ is 2-colorable because Ambainis’ graph is bipartite.