Theoretical Approximation Ratios for Warm-Started QAOA on 3-Regular Max-Cut Instances at Depth $p=1$

TL;DR

It is suggested that at $\theta=60^\circ$, warm-started QAOA is able to (effectively) recover the cut used to generate the warm-start, thus suggesting that in practice, this value could be a promising starting angle to explore alternate solutions in a heuristic fashion.

Abstract

We generalize Farhi et al.'s 0.6924-approximation result technique of the Max-Cut Quantum Approximate Optimization Algorithm (QAOA) on 3-regular graphs to obtain provable lower bounds on the approximation ratio for warm-started QAOA. Given an initialization angle $θ$, we consider warm-starts where the initial state is a product state where each qubit position is angle $θ$ away from either the north or south pole of the Bloch sphere; of the two possible qubit positions the position of each qubit is decided by some classically obtained cut encoded as a bitstring $b$. We illustrate through plots how the properties of $b$ and the initialization angle $θ$ influence the bound on the approximation ratios of warm-started QAOA. We consider various classical algorithms (and the cuts they produce which we use to generate the warm-start). Our results strongly suggest that there does not exist any choice of initialization angle that yields a (worst-case) approximation ratio that simultaneously beats standard QAOA and the classical algorithm used to create the warm-start. Additionally, we show that at $θ=60^\circ$, warm-started QAOA is able to (effectively) recover the cut used to generate the warm-start, thus suggesting that in practice, this value could be a promising starting angle to explore alternate solutions in a heuristic fashion.

Figures (17)

• Figure 1: A geometric depiction of the states $\ket{0_\theta}$ and $\ket{1_\theta}$ on the Bloch sphere. The blue half-circle, $\textbf{Arc}$, in the $xz$-plane denotes all the possible positions for $\ket{0_\theta}$ and $\ket{1_\theta}$ as $\theta$ varies from $0$ to $\pi$.
• Figure 2: The graph $G$ has 3 different edge neighborhood types, which we refer to as $H_1, H_2, H_3$ in this example. For each edge neighborhood type, the central edge is depicted in red. The degeneracies for each edge neighborhood type is also shown along with the edges associated with the degeneracy; note that the sum of the degeneracies adds up to the number of edges in $G$. In $G$, for the edge $e = \{1,5\}$, the depth-1 edge neighborhood $G_e$ is shown with solid black edges with the central edge $e$ in red; note that $G_e$ is of the same type as $H_1$.
• Figure 3: The three depth-1 edge-neighborhood types, labeled $g_4,g_5,g_6$, that can be found in cubic graphs. For each edge neighborhood type, the red edge denotes the central edge and the solid black lines denote the actual edges in the edge neighborhood; the gray dashed edges represent edges outside the edge-neighborhood that reside in the remainder of the cubic graph. Note that the other endpoints of the gray edges (not shown) are not necessarily distinct.
• Figure 4: An example graph $G$ is depicted with a 2-coloring with corresponding bitstring $b=010001$ (the yellow and green vertex colors correspond to 0 and 1 respectively). It has 6 different colored edge neighborhood types, which we refer to as $H_1,\dots, H_6$ in this example. For each colored edge neighborhood type, the central edge is depicted in red. The degeneracies for each edge neighborhood type is also shown along with the edges associated with the degeneracy; note that the sum of the degeneracies adds up to the number of edges in $G$. In $G$, for the edge $e = \{1,2\}$, the depth-1 colored edge neighborhood $G_e$ is shown with solid black edges with the central edge $e$ in red; note that $G_e$ is of the same type as $H_3$.
• Figure 5: Consider a 3-regular graph $G$ with a coloring/cut determined by a bitstring $b$ and let $e = (u,v) \in E(G)$ and suppose the colored edge neighborhood $G_e(b)$ looks as depicted at the top of the figure. We show that in such a case, regardless of the colors of the remaining nodes in $G$, that it must be that the overall cut $b$ is not locally optimal with respect to single-bitflips (and hence any $g \sim G_e(b)$ is a forbidden type of colored edge neighborhood under the restriction that the colored edge neighborhood is generated from a locally optimal cut). In the colored edge neighborhood, the bold lines correspond to edges that would necessarily be cut in accordance with $b$. Given $b$, let $m'$ denote the number of edges that are cut outside the underlying graph of $G_e(b)$. Observe that we can recolor the vertices that are incident to the central edge without changing the value of $m'$. In particular, let $b'$ be $b$ but with the bit corresponding to vertex $v$ being flipped. Observe that causes an improvement not only in the "local cut value" but the cut value of the entire graph.

Theorems & Definitions (66)

• Theorem 1: Informal Statement of Theorem \ref{['thm:lowerBoundVaryingKappa']}
• Lemma 2
• proof
• Proposition 3
• proof
• Proposition 4
• proof
• Proposition 5
• proof
• Corollary 6