Quantum Approximate Optimization Algorithms for Maximum Cut on Low-Girth Graphs

The quantum approximate optimization algorithm was introduced in [6], which is a variational quantum algorithm that requires $2p$ parameters: $(\gamma_{1},\gamma_{2},\ldots,\gamma_{p})$, $(\beta_{1},\beta_{2},\ldots,\beta_{p})$. The input is an $n$-qubit string $z$ and the goal is to find an approximate ground state of cost function operator $C$, where $C\ket{z}=C(z)\ket{z}$. It does so by preparing the state $\ket{\bm{\gamma},\bm{\beta}}$ on the ground state $\ket{s}$ of mixing operator $B$. QAOA prepares the state

where $U(B,\beta)=e^{-i\beta B}$ and $U(C,\beta)=e^{-i\gamma C}$.

The expectation of the cost function $C$ is $\bra{\bm{\gamma},\bm{\beta}}C\ket{\bm{\gamma},\bm{\beta}}$. When $p$ approaches infinity, Eq. (2) can be seen as Trotterization of the adiabatic theorem, and thus it can reach the minimum of the cost function operator $C$ [6]. In practice, for a fixed value of $p$, we can measure $\ket{\bm{\gamma},\bm{\beta}}$ in computational basis and optimize $\bm{\gamma}$ and $\bm{\beta}$.

Given a graph $G=(V,E)$, the cost function of MaxCut is to evaluate how many edges are cut due to the partition of vertices. If the qubits of vertices $u$ and $v$ in edge $(u,v)$ are different, then $Z_{u}Z_{v}=-1$ and count one edge to the cost function. If the qubits are the same, $\frac{1}{2}(-Z_{u}Z_{v}+1)=0$. Thus the operator can be written as:

where $Z_{u}$ is Pauli $Z$ operator on qubit $u$.

Since the constant $1/2$ in $C_{MC}$ only introduces a global phase that does not influence measurements, we can instead use the scaled cost function operator

And the mixing operator $B$ equals

Note that QAOA is a local algorithm where the expectation $\bra{\bm{\gamma},\bm{\beta}}Z_{u}Z_{v}\ket{\bm{\gamma},\bm{\beta}}$ on each edge only depends on the edges whose distance from edge $(u,v)$ are no more than $p$ and qubits on them. Therefore, edges in the same $p$-neighborhood subgraph have the same expectation values. That is, $\bra{\bm{\gamma},\bm{\beta}}Z_{u}Z_{v}\ket{\bm{\gamma},\bm{\beta}}=\bra{\bm{\gamma},\bm{\beta}}Z_{u^{\prime}}Z_{v^{\prime}}\ket{\bm{\gamma},\bm{\beta}}$ if the edges $(u,v)$ and $(u^{\prime},v^{\prime})$ have the same neighborhood subgraph $g$, in which case we denote the value by $\bra{\bm{\gamma},\bm{\beta}}ZZ(g)\ket{\bm{\gamma},\bm{\beta}}$. We can categorize edges in $E$ according to different subgraphs $g$, and the expectation becomes

where the summation is over all possible $p$-neighborhood subgraphs and $w_{g}$ is the number of edges $(u,v)$ whose $p$-neighborhood subgraph is $g$. The cut fraction is then

where $f_{g}$ is the proportion of edges with $p$-neighborhood subgraph $g$.

We will also provide a brief introduction to the $X$-Ramanujan graphs in [19], the class of expander graphs that is studied by QAOA. First, we introduce the notion of spectral radius:

Then, we introduce the definition of additive product graph. The additive product graph are generated by a set of graphs $A_{1},\ldots,A_{c}$, called plain atoms, that share a common vertex set. By choosing a fixed starting vertex $v_{1}$, the vertices and edges of additive product graph are defined as follows, with one example illustrated in Figure 3.

Note that different choices of $v_{1}$ generate isomorphic additive product graphs. In Figure 3, one example generated by Definition 2 is depicted, where the generators $A_{1},\ldots,A_{c}$ bring corresponding local structures within the additive product graph. Note that an additive product graph may not be regular. A simple example is when $G$ is a complete bipartite graph $K_{\Delta_{1},\Delta_{2}}$, and $A_{1},\ldots,A_{c}$ represent the associated single-edge graphs on $G$’s vertex set, where $c=\Delta_{1}\Delta_{2}$. It can be observed that $A_{1}\oplus A_{2}\cdots\oplus A_{c}$ construct an infinite ($\Delta_{1},\Delta_{2}$)-biregular tree, where any node in an odd level has degree $\Delta_{1}$ and any node in an even level has degree $\Delta_{2}$. See Figure 2 for an illustration of the case of $(\Delta_{1},\Delta_{2})=(3,4)$.

Additive product graphs can be easily constructed and proved to exhibit certain desirable spectral properties, by which they offer a family of explicit constructions of expander graphs. We refer to [19] for more details on this.

In the original QAOA paper [6], the authors let each layer of quantum circuit share the same parameter. Later a multi-angle version of ma-QAOA was proposed [11] in which each node $v$ or edge $(u,v)$ has an individual parameter in each layer $j$, namely $U(\beta,\gamma)=\prod_{j=p}^{1}(e^{i\sum_{v\in V}\beta_{jv}X_{v}}e^{i\sum_{(u,v)\in E}\gamma_{j,(u,v)}Z_{u}Z_{v}})$. By introducing more parameters, the model becomes more expressive and can in principle improve the performance of the algorithm. However, more parameters also imply a much larger search space which may make training harder. Indeed, from a dynamical Lie algebra point of view, more parameter sharing shrinks the generated Lie algebra, while smaller dimension makes the circuit less likely to have a barren plateau and easier to estimate its gradient; see [21, 10] for details. In this paper, we propose to have a structural parameter sharing scheme and apply it to additive product graphs.

The multi-angle quantum approximate optimization algorithm (ma-QAOA) was firstly introduced in Ref. [11]. It introduces more classical parameters into the standard QAOA framework. Typically, $C=\sum_{a}C_{a}$ and $B=\sum_{v}B_{v}$ represent the sum of a collection of clauses. The ma-QAOA assigns each individual summand of $B$ and $C$ its own parameter. The parameters consist of $2p$ vectors: $(\bm{\gamma}_{1},\bm{\gamma}_{2},\ldots,\bm{\gamma}_{p})$ and $(\bm{\beta}_{1},\bm{\beta}_{2},\ldots,\bm{\beta}_{p})$.

Through having more classical parameters, the ma-QAOA can enhance the performance of standard QAOA. In Ref. [11], ma-QAOA was simulated on a collection of one-hundred triangle-free 3-regular graphs with fifty vertices and one hundred triangle-free 3-regular graphs with 100 vertices. The result shows that ma-QAOA achieved 21.26% and 17.98% increase in percentage on average to those of 1-QAOA respectively. Furthermore, the Erdős–Rényi graphs with 9-12 nodes and various edge probabilities were tested for ma-QAOA with $p>1$ [7], demonstrating the advantages of ma-QAOA over QAOA for multiple layers.

In our analysis of additive product graphs, we aim to leverage the advantages of the ma-QAOA while reducing the number of parameters to ensure convergence. To achieve this, we categorize the edges in the additive product graph based on their constituent atoms. If an additive product graph is constructed by $c$ atoms, then there are $c$ types of edges. Specifically, edge type $C$ comprises edges of the following form:

For each type of edge, we assign a separate $\gamma$ parameter to its corresponding summand. For the mixing operator $B$, we employ a single angle parameter, consistent with the original QAOA framework. That is, in our settings, if the edge set $E$ in graph $G$ has $c$ categories, the parameters are: $(\bm{\gamma}_{1},\bm{\gamma}_{2},\ldots,\bm{\gamma}_{p}),\bm{\gamma_{i}}\in[0,2\pi]^{c}$, $(\beta_{1},\beta_{2},\ldots,\beta_{p}),\beta\in[0,2\pi]$.

In order to better understand the performance of QAOA on MaxCut, we need to explore classical MaxCut algorithms to provide appropriate benchmarks. Since QAOA is a local algorithm, the comparison between local classical algorithms and local quantum algorithms will be reasonable. Similar to local quantum algorithm, if a classical algorithm is $k$-local, then the spin of a vertex $u$ in graph is only related to the vertices within distance $k$ of $u$. The statement implies that for the output of the algorithm, the spins of points $u$ and $v$, which are separated by a distance greater than $2k$, are independent. The best classical local algorithms known to the authors are the $k$-local algorithm proposed by Barak and Marwaha [1] and the threshold algorithm used in Refs. [12, 9, 18]. There are also other classical algorithms. For example, Ref. [20] addressed MaxCut on Erdős–Rényi graphs, while Ref. [5] extended local tree-structure graphs to local treelike structures, where a fraction of the vertices has tree-like neighborhoods. Despite the graphs under study in this paper are not high-girth graphs, we can still use existing local classical algorithms to establish good benchmarks.

The threshold algorithm was first introduced in [12]. Intuitively, the algorithm simply flips the spin of each vertex based on the spins of its neighbors. For a parameter $\tau\in[d/2,d]$, if a vertex has more than $\tau$ neighbors with the same spin as itself, then the spin of that vertex is flipped. The pseudocode for this algorithm is shown in Algorithm 1. This work studied the cut fraction of $d$-regular triangle-free graphs and derived an expected cut fraction of $1/2+0.28125/\sqrt{d}$ with the threshold equal to $\lceil\frac{d+\sqrt{d}}{2}\rceil$. But the threshold given in [12] was for notational convenience and may not be the optimal. The optimal threshold can be obtained by directly analyzing the expression for the expected cut fraction as in Theorem 2.

Note that to maximize the expected cut fraction in Theorem 2, the threshold $\tau$ needs to be optimized. Hastings [9] numerically optimized the cut fraction of 1-step threshold algorithm and found it outperformed $\text{QAOA}_{1}$ for $3<d<1000$ except of $d=3,4,6,11$. Marwaha [18] extended the 1-step threshold algorithm to an n-step version and directly calculated the optimal threshold $\tau_{1},\tau_{2}$ for all $41<d<500$ by assuming $\tau_{1}=\tau_{2}$. The result showed that the 2-step threshold algorithm outperformed $\text{QAOA}_{2}$ for all $41<d<500$ and the performance of 2-step threshold algorithm stabilized at $1/2+0.417/\sqrt{d}$ for large $d$. The results were also computed for $2\leq d\leq 50$ without assuming $\tau_{1}=\tau_{2}$, and found that 2-step threshold algorithm outperformed $\text{QAOA}_{2}$ when $d>5$.

The algorithm in [1] by Barak and Marwaha achieved better bound than the threshold algorithm for 1-step and 2-step cases on high-girth regular graphs. The algorithm is done by assigning an independent random normal variable to each vertex and define the spin of each vertex according to the summation of these random variables. The pseudocode of this algorithm is shown in Algorithm 2. This algorithm achieved an expected cut fraction of at least $1/2+2/(\pi\sqrt{D})$ for $D$-regular graphs with girth $g>2k+1$.

In this section, we give a theoretical guarantee for the expected cut fraction of QAOA for MaxCut on additive product graphs. Specifically, we propose an iterative formula to analyze each expectation term $\bra{\bm{\gamma},\bm{\beta}}Z_{u}Z_{v}\ket{\bm{\gamma},\bm{\beta}}$ in Eq. (7). The analysis of the iterative formula is conducted within the framework of ma-QAOA discussed Section 2.3. When letting ${\gamma}_{i,1}={\gamma}_{i,2}=\cdots={\gamma}_{i,c}$, we will have the iterative formula for the standard QAOA.

First, we need to analyze how many types of subgraphs exist for a given construction of additive product graph and what the proportion ($f_{g}$ in Eq. (7)) is for each type. The type of subgraphs for $\text{ma-QAOA}_{p}$ depends on the edge $(u,v)$ in $\bra{\bm{\gamma},\bm{\beta}}Z_{u}Z_{v}\ket{\bm{\gamma},\bm{\beta}}$ and the neighbors within distance $p$ from vertices $u$ and $v$. Based on the construction of additive product graph, every vertex $sCu$ in $V$ is connected to its neighbors following the same pattern: Firstly $sCu$ has neighbors $sCv$ of the same layer if $\{u,v\}$ is an edge in $\underline{A}_{C}$; then $sCu$ is connected to its successions $sCuC^{\prime}v$ if $\{u,v\}$ is an edge in $\underline{A}_{C^{\prime}}$ and its predecessors $s^{\prime}C^{\prime}v$ if $s=s^{\prime}C^{\prime}v$ and $\{u,v\}$ is an edge in $\underline{A}_{C}$. As a consequence, all vertices $sCu$ exhibit identical neighborhood configurations within the additive product graph regardless of choice of $s$. So the following property will hold:

Since the number of types of subgraph is the number of edges in $\underline{A}_{1},\ldots,\underline{A}_{c}$, proportion $f_{g}$ for each type is then $\frac{1}{|E({\underline{A}_{1}})|+\cdots+|E({\underline{A}_{c}})|}$. And this yields the expectation of cut fraction (7) to be

where $E(\underline{A})$ represent the number of edges in graph $\underline{A}$.