Consider a combinatorial problem defined on a space of binary strings of length N that has m clauses. Each clause is a constraint satisfied by some assignment of the bit string. The objective function can be written as C ( z ) = ∑ α = 1 m C α ( z ) , where z = z ₁ z ₂⋯z _N is the bit string and C _α (z) = 1 if z satisfies the clause α, and 0 otherwise. QAOA maps the combinatorial optimization problem onto a 2 ^N -dimensional Hilbert space with computational basis vectors |z⟩ and encodes C(z) as an operator C diagonal in the computational basis.

At each call to the quantum computer, a trial state is prepared by applying a sequence of alternating quantum operators | β ⃗ , γ ⃗ 〉 p ≔ U B β p U C γ p … U B β 1 U C γ 1 | s 〉 , (1) where U _C (γ) = e ^−iγC is the phase operator; U _B (β) = e ^−iβB is the mixing operator, with B defined as the operator of all single-bit σ ^x operators; B = ∑ j = 1 N σ j x ; and |s⟩ is some easy-to-prepare initial state, usually taken to be the uniform superposition product state. The parameterized quantum circuit (1) is called the QAOA ansatz. We refer to the number of alternating operator pairs p as the QAOA depth. The selected parameters β ⃗ , γ ⃗ are said to define a schedule, analogous to a similar choice in quantum annealing.

Preparation of the state (1) is followed by a measurement in the computational basis. The output of repeated state preparation and measurement may be used by a classical outer-loop algorithm to select the schedule β ⃗ , γ ⃗ . We consider optimizing the expectation value of the objective function ⟨ C ⟩ p = 〈 β ⃗ , γ ⃗ | p C | β ⃗ , γ ⃗ 〉 p , as originally proposed in (Farhi et al., 2014). The output of the overall procedure is the best bit string z found for the given combinatorial optimization problem. Figure 1 presents a schematic pipeline of the QAOA algorithm. We emphasize that the task of finding good QAOA parameters is challenging in general, for example, because of encountering barren plateaus (Wang et al., 2021; Anschuetz and Kiani, 2022). Acceleration of the optimal parameters search for a given QAOA depth p is the focus of many approaches aimed at demonstrating quantum advantage. Examples include warm- and multistart optimization (Shaydulin et al., 2019a; Egger et al., 2020), problem decomposition (Shaydulin et al., 2019b), instance structure analysis (Shaydulin et al., 2021), and parameter learning (Khairy et al., 2020).

For studying the transferability of optimized QAOA parameters, we consider the MaxCut combinatorial optimization problem. Given an unweighted undirected simple graph G = (V, E), the goal of the MaxCut problem is to find a partition of the graph’s vertices into two complementary sets such that the number of edges between the two sets is maximized. In order to encode the problem in the QAOA setting, the input is a graph with |V| = N vertices and |E| = m edges, and the goal is to find a bit string z that maximizes C = ∑ j k ∈ E C j k , (2) where C j k = 1 2 − σ j z σ k z + 1 .

It has been shown in (Farhi et al., 2014) that on a 3-regular graph, QAOA with p = 1 produces a solution with an approximation ratio of at least 0.6924.

Calculating the approximation ratio for a particular MaxCut problem instance requires the optimal solution of the combinatorial optimization problem. This problem is known to be NP-hard, and classical solvers require exponential time to converge. For our experiments, we use the Gurobi solver (Gurobi Optimization, 2021) with the default configuration parameters, running the solver until it converges to the optimal solution. For our QAOA simulations, we use QTensor (Lykov et al., 2021), a large-scale quantum circuit simulator with step-dependent parallelization. QTensor simulates circuits based on a tensor network approach, and as such, it can provide an efficient approximation to certain classes of quantum states (Biamonte and Bergholm, 2017; Kardashin et al., 2021).

It was shown in the seminal QAOA paper (Farhi et al., 2014) that the expectation value of the QAOA objective function, ⟨C⟩ _p , can be evaluated as a sum over contributions from subgraphs of the original graph, provided its degree is bounded and the diameter is larger than 2p (otherwise, the subgraphs cover the entire graph itself). The contributing subgraphs can be constructed by iterating over all edges of the original graph and selecting only the nodes that are p edges away from the edge. Through this process, any graph can be deconstructed into a set of subgraphs for a given p, and only those subgraphs contribute to ⟨C⟩ _p , as also discussed in Section 2.

We begin by analyzing the case of MaxCut instances on 3-regular random graphs for QAOA circuit depth p = 1, which have three possible subgraphs (Farhi et al., 2014; Brandao et al., 2018). Figure 2 (top row) shows the landscapes of energy contributions from these subgraphs, evaluated for a range of γ and β parameters. We can see that all maxima are located in the approximate vicinity of each other. As a result, the parameters optimized for any of the three graphs will also be near-optimal for the other two. Because any random 3-regular graph can be decomposed into these three subgraphs, for QAOA with p = 1, this guarantees that optimized QAOA parameters can be successfully transferred between any two 3-regular random graphs, which is in full agreement with (Brandao et al., 2018).

The same effect is observed for subgraphs of 4-regular; see Figure 2 (middle row). The optimized parameters are mutually transferable between all four possible subgraphs of 4-regular graphs. Notice, however, that the locations of exactly half of all maxima for the subgraphs of 4-regular graphs do not match with those for 3-regular graphs. This means that one cannot expect good transferability of optimized parameters across MaxCut QAOA instances for 3- and 4-regular random graphs if these optimal parameters are to be transferred directly. It has been recently shown in (Basso et al., 2022) that gamma parameters can be rescaled in order to generalize between different random d-regular graphs.

Focusing now on all five possible subgraphs of 5-regular graphs, Figure 2 (bottom row), we notice that, again, good parameter transferability is expected between all instances of 5-regular random graphs. Moreover, the locations of the maxima match well with those for 3-regular graphs, indicating good transferability across 3- and 5-regular random graphs.

We discuss parameter concentration for instances of random graphs in a later section; similar discussions can be found in (Brandao et al., 2018) and (Wurtz and Lykov, 2021) in the context of 3-regular graphs.

To further investigate transferability among regular graphs, we evaluate the subgraph transferability map between all possible subgraphs of d-regular graphs, d ≤ 8; see Figure 3. The top panel shows the colormap of parameter transferability coefficients between all possible pairs of subgraphs of d-regular graphs (d ≤ 8, 35 subgraphs total). Each axis is split into groups of d subgraphs of d-regular graphs, and the color values in each cell represent the transferability coefficient T(D, A) computed for the corresponding directional pair of subgraphs D, A, defined as follows. For every subgraph G, we performed numerical optimization with 200 steps, repeated 20 times with random initial points. This process results in 20 sets of optimal parameters of the form ( γ G i , β G i ) pairs, the best of which we will denote as (γ _G*, β _G*). Doing so for the donor subgraph D and the acceptor subgraph A, the transferability coefficient T(D, A) averages over the QAOA energy contribution of each ( γ D i , β D i ) on the acceptor subgraph A as follows: T D , A = 1 20 ∑ i = 1 20 A γ D i , β D i A γ A * , β A * , (3) where A(γ, β) is the QAOA MaxCut energy of subgraph A as a function of parameters (γ, β).

Instead of averaging over the 20 optimal parameters of the donor subgraph, we could have considered only the contribution of the donor’s best optimal parameters (γ _D*, β _D*) in the above equation. For most donors, however, these best parameters were universal and hence yielded high transferability to most acceptors. However, in practice, because of a lack of iterations or multistarts, we may converge to non-universal optimal parameters, resulting in the donor’s poor transferability with some acceptors. The likelihood of converging to these non-universal optima for random graphs is discussed in Supplementary Section SA. Universal and nonuniversal optimal parameters are discussed in detail in Section 4.

This inconsistency was discussed for 3-regular and 4-regular subgraphs earlier in this section. For example, half of the local optima of 3 regular subgraphs have good transferability to 4-regular subgraphs while the half yield poor transferability, as shown in Figure 2. Thus, to reflect practical considerations and avoid such inconsistency, we average over the contributions of 20 optimal parameters of the donor subgraph in Eq. 3. It is worth noting here that this averaging over 20 optimal parameters can result in poor transferability, as seen for donor subgraph #0 to acceptor subgraphs #2, #9, #20, and #35. For these cases, there is a considerable subset of the donor’s optimal parameters that lead to poor transferability. All considered subgraphs are shown in the bottom panel of Figure 3. Note that parameter transferability is a directional property between (sub) graphs, and good transferability from (sub) graph D to (sub) graph A does not guarantee good transferability from A to D. This general fact can be easily understood by considering two graphs with commensurate energy landscapes, for which every energy maximum corresponding to graph D also falls onto the energy maximum for graph A, but some of the energy maxima for graph D do not coincide with those of graph A.

The regular pattern of alternating clusters of high- and low-transferability coefficients in Figure 3 illustrates that the parameter transferability effect extends from 3-regular graphs to the entire family of odd-regular graphs, as well as to even-regular graphs, with poor transferability between the two classes. For example, the established result for 3-regular graphs is reflected at the intersection of columns and rows with the label “(3)” for both donor and acceptor subgraphs. The fact that all cells in the 3 × 3 block in Figure 3, corresponding to parameter transfer between subgraphs of 3-regular graphs, have high values, representing high mutual transferability, gives a good indication of optimal QAOA parameter transferability between arbitrary 3-regular graphs (Brandao et al., 2018).

Having considered optimal MaxCut QAOA parameter transferability between random regular graphs, we now focus on general random graphs. Subgraphs of an arbitrary random graph differ from subgraphs of random regular graphs in that the two nodes connected by the central edge can have a different number of connected edges, making the set of subgraphs of general random graphs much more diverse. The upper panel of Figure 4 shows the transferability map between all possible subgraphs of random graphs with node degrees d ≤ 6, a total of 56 subgraphs, presented in the lower panel. The transferability map can serve as a lookup table for determining whether optimized QAOA parameters are transferable between any two graphs.

Figure 4 reveals another important fact about parameter transferability between subgraphs of general random graphs. Subgraphs labeled as (i, j), where i and j represent the degrees of the two central nodes of the subgraph, are in general transferable to any other subgraph (k, l), provided that all {i, j, k, l} are either odd or even. This result is a generalization of the transferability result for odd- and even-regular graphs described above. Figure 4, however, shows that a number of pairs of subgraphs with mixed degrees (not only even or odd) also transfer well to other mixed-degree subgraphs, for example, subgraph #20 (3, 4) → subgraph #34 (4, 5). The map of subgraph transferability provides a unique tool for identifying smaller donor subgraphs, the optimized QAOA parameters for which are also nearly optimal parameters for the original graph. The map can also be used to define the likelihood of parameter transferability between two graphs based on their subgraphs. As was the case for random regular graphs (see Figure 2), we see clustering of optimal parameters for subgraphs of random graphs in Figure 5.

We will now demonstrate that the parameter transferability map from Figure 4 can be used to find small-N donor graphs from which the optimized QAOA parameters can be successfully transferred to a MaxCut QAOA instance on a much larger acceptor graph. Initially, we consider three 256-node acceptor graphs to be solved and three 6-node donor graphs; see Figure 6. Table 1 contains the details of the donor and acceptor graphs, including the total number of edges, their optimized QAOA energies, the energy of the optimal classical solution, and the approximation ratio. Graphs 1 and 4 consist exclusively of odd-degree nodes, graphs 2 and 5 contain roughly the same amount of both odd- and even-degree nodes, and graphs 3 and 6 contain exclusively even-degree nodes. The optimized QAOA parameters for the donor and acceptor graphs were found by performing numerical optimization with 20 restarts, and 200 iterations each. Additionally, we use a greedy ordering algorithm and an RMSprop optimizer, with a learning rate of 0.002. Table 2 shows the results of the corresponding transfer of optimized parameters from the donor graphs ##1–3 to the acceptor graphs ##4–6, correspondingly. The approximation ratios obtained as a result of the parameter transfer in all three cases show only a 1%–2% decrease compared with those obtained by optimizing the QAOA parameters for the corresponding acceptor graphs directly. These examples demonstrate the power of the approach introduced in this paper.

To extend our analysis of parameter transferability between QAOA instances, we perform transferability of optimal parameters between large sets of small donor graphs to a fixed, larger acceptor graph. In particular, we transfer optimal parameters from donors ranging from 6 to 20 nodes to 64-, 128-, and 256-node acceptor graphs. Figure 7 shows the approximation ratio as we increase the number of donor graph nodes. These donor graphs were generated starting with graphs of exclusively odd-degree nodes and sequentially increasing the number of even-degree nodes until graphs of exclusively even-degree nodes were obtained. For each increasing number of node in a graph, 100 donor graphs were generated and each of their 20 sets of optimal parameters (20 multistarts) were transferred to the acceptor graph. We see that there are a few cases for which we achieve an approximation ratio that is comparable to the native approximation ratio for each of the acceptor graphs. Most notably, we can achieve good transferability of optimal parameters to larger (i.e., 256-node) acceptor graphs without having to increase the size of our donor graph. Each row of Figure 7 corresponds to an increasing acceptor graph size, while each column corresponds to the parity of the acceptor graph (a formal definition and study of parity follow in the next section), with a transition from odd to even parity in graphs going from left to right. For the fully odd and fully even acceptor graphs, we notice a bimodal distribution in approximation ratios. Remarkably, for even acceptor cases, the bimodal distribution has one mode centered around the mean (white dot) and one above the mean. This points to the fact that, regardless of donor graph parity, one can achieve better parameter transferability when transferring optimal parameters to acceptor graphs with even parity. We see this transition from odd to even acceptor graphs in the way the bimodal distribution shifts, there being a monomodal distribution for the cases where the acceptor graphs are neither even nor odd.

The reason for this increased likeliness of good transferability to even acceptor graphs will be explored in future work. For now, we turn our focus to parity in graphs as an alternative metric for determining good transferability between donor-acceptor graph pairs, one that does not involve subgraph decomposition (and parameter transferability between individual subgraphs).

As mentioned previously, the transferability maps of regular and random subgraphs suggest that the parity of graph pairs may affect their transferability. Here, we define parity of a graph G = (V, E) to be the proportion of nodes of G with an even degree: π G = n even | V | , (4) where n _even is the number of even nodes in graph G. For this and upcoming sections, we focus on transferability between 20-node random graphs. That is, we perform optimal parameter transferability between 20-node donor and 20-node acceptor graphs. For every possible number of even-degree nodes (0, 2, 4, … , 20), we generated 10 graphs with distinct degree sequences, resulting in a total of 110 20-node random graphs, with maximum node degree restricted to 6.

The computed transferability coefficients among each graph pair, sorted by their parity, are shown in Figure 8. Each block in the heatmap represents the average transferability coefficient of 100 graph pairs constructed from 10 distinct donor graphs and 10 distinct acceptor graphs. We can see that even graphs, those with π _G = 0.8–1, and odd graphs, those with π _G = 0–0.2, transfer well among themselves. However, the transferability between even donors and odd acceptors, as well as between odd donors and even acceptors, is poor.

This heatmap also suggests that the mutual transferability of a donor graph is not necessary for its good transferability with other random graphs, where mutual transferability of a graph G is a measure of how well the subgraphs of G transfer among themselves. Formally, it is defined as MT G = ∑ d ∈ G ∑ a ≠ d ∈ G n G d n G a T d , a T G , (5) where {G} is the set of distinct subgraphs of graph G, n _G (i) is the number of edges in G having subgraph i, and T G = ∑ d ∈ G ∑ a ≠ d ∈ G n G d ⋅ n G a is the total number of subgraph pairs consisting of distinct subgraphs within G. Graphs with low mutual transferability are those whose subgraphs transfer poorly among themselves. This is true for graphs with a nearly equal number of odd-parity and even-parity subgraphs since subgraph pairs of different parity report poor transferability coefficients, as shown in Figures 2, 3. In our case, such graphs are likely to be mixed-parity graphs, in other words, those with π _G = 0.4–0.6. However, the results in Figure 8 show that these graphs have good transferability to nearly all random graphs in the data set.

This trend can be explained by analyzing the energy landscapes of subgraphs. Most even- and odd-regular subgraphs have 4 maxima, two of which are universal for all regular subgraphs, as discussed in Section 3.1; the same trends were also observed in random subgraphs (see Figure 9). Since energy landscapes of random graphs are sums of the energy landscape of its subgraphs, most 20-node random graphs share the same points, centers 1,2 in Figure 10, as their local or global optima, as shown in Figure 11. On the other hand, the remaining two nonuniversal optimal parameters of regular subgraphs are shared only across regular subgraphs of similar parity. This property is also emergent in random graphs. In Figure 11, only odd random graphs share centers 3, 4 as their local optima, while only even graphs share centers 5, 6 as their local optima. However, mixed parity contained a nearly equal number of odd and even subgraphs. Since nonuniversal maxima of even subgraphs are minima for odd subgraphs and vice versa, these nonuniversal local optima blur on the energy landscapes of mixed-parity graphs. As a result, these graphs’ landscapes contain only universal maxima, as shown in the fourth energy landscape in Figure 9. With only universal parameters as their optimal parameters, mixed-parity graphs should indeed transfer well to all random graphs, as shown in the middle columns of Figure 8.

The distribution of optimal parameters also explains poor transferability across random graphs of different parity. In Figure 11, the nonuniversal optimal parameters that maximize the MaxCut energy of odd random graphs, centers 3, 4 also minimize that of even random graphs. Similarly, the nonuniversal optimal parameters that maximize the MaxCut energy of even random graphs, centers 3, 4 also minimize that of odd random graphs. Consequently, transferring nonuniversal optimal parameters of even random graphs to odd random graphs and vice versa would result in poor approximation ratios, as evident in Figure 8.

Furthermore, above-average transferability for all graph pairs can be attributed to universal parameters. As shown in Figure 12, all graph pairs have a true similarity or transferability coefficient greater than 0.60. Such a high lower bound can be attributed to universal parameters. Going back to Eq. 3, good transferability depends on whether the donor’s optimal parameters ( γ D i , β D i ) optimize the acceptor graph as well. If most of the donor’s optimal parameters are universal, in other words, are in the vicinity of centers 1, 2 in Figure 10, then the transferability coefficient will be high, regardless of the acceptor graph. In fact, Supplementary Figure S2 shows that, on average, all graphs reported at least half of their 20 optimal parameters as universal. As a result, they transfer well to other random graphs.

We have used the transferability coefficient to test whether an acceptor shares the same optimal parameters as its donor. In practice, this quantity is unknown because it requires knowledge of the acceptor’s maximum energy. In earlier examples, we used the parity of graphs to explain transferability among random graphs, but the parity of a graph is just one emergent property from its subgraphs. Using subgraphs directly, we devise a subgraph similarity metric SS to predict the transferability ratio between a donor graph D = (V _D , E _D ) and an acceptor graph A = (V _A , E _A ) as follows: SS D , A = ∑ d ∈ D ∑ a ∈ A n D d n A a T d , a | E D | ⋅ | E A | , (6) where {G} is the set of distinct p = 1 subgraphs of G, n _G (g) is the number of edges in G that share the subgraph g, and |E _D |⋅|E _A | is the total number of subgraph pairs across graphs D and A. Hence, this similarity metric states that the transferability coefficient of a donor and acceptor is the average transferability coefficient of donor subgraph-acceptor subgraph pairs.

In Figure 12 we compare this similarity metric with the true similarity or transferability coefficient. While this result does reveal a linear correlation between the two quantities, the metric clearly underapproximates the transferability coefficient by 0.05 units on average. In fact, Figure 13 shows that graph pairs with mixed-parity graphs as donors report the highest inconsistencies. This poor performance results from their constituent subgraphs. As discussed in Section 3.4, mixed-parity graphs consist of a nearly equal number of odd and even subgraphs. When optimized, these donor subgraphs may have nonuniversal optimal parameters. When transferred to an acceptor subgraph, the resulting transferability coefficient may be either poor or good, depending on the parity of that acceptor subgraph. While these nonuniversal optima do affect the subgraph similarity metric SS, they do not affect true similarity. As shown in Figure 9, mixed-parity graphs’ optimal parameters are universal. Thus, they transfer well to any random graph, regardless of its parity. Therefore, the subgraph similarity metric underestimates true similarity because it fails to capture that most optimal parameters of mixed-parity graphs are universal.

Another approach to predicting transferability or similarity between two graphs is using their parity. In Section 3.4 we observed that two graphs of similar parity have a high transferability ratio. If this correlation was ideal, then results shown in Figure 8 would resemble those in Figure 14. The parity similarity metric PS corresponding to the latter figure is easy to compute: PS D , A = 1 − 0.29 ⋅ | Parity D − Parity A | . (7)

Thus, this metric penalizes graph pairs consisting of different parity graph pairs. Note that the lowest value of this metric is ≈ 0.71 , which is consistent with results from Figure 8. Figure 15 illustrates the performance of this new similarity metric. The plot contains discrete columns because it is not possible to generate 20 node graphs with an arbitrary number of even degree nodes. To ensure that the sum of a degree sequence is even, the associated graphs only vary in even degree nodes in increments of two, resulting in parity of 0.0, 0.1, 0.2, … 1.0. This discretization is also reflected in the similarity metric.

To test our similarity metric for the data set shown in Figure 7, we compare our metric with the approximation ratio. Figure 16 shows that as parity between donor-acceptor pairs approaches 1 (i.e., the donor and acceptor graphs have the same parity), we achieve a higher approximation ratio. Noticeably, we see that we can have a good approximation ratio even if our parity similarity does not dictate so. This can be attributed to the fact that we are exploiting only one structural feature from our graphs.

These results indicate that one can use a parity approach to determine good transferability between donor-acceptor pairs. Furthermore, one can generate a parity metric that caters to specific graphs (please refer to Supplementary Material).