^{1}

^{2}

^{†}

^{3}

^{2}

^{†}

^{4}

^{2}

^{5}

^{6}

^{2}

^{2}

^{5}

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

These authors have contributed equally to this work and share first authorship

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

The quantum approximate optimization algorithm (QAOA) is one of the most promising candidates for achieving quantum advantage through quantum-enhanced combinatorial optimization. A near-optimal solution to the combinatorial optimization problem is achieved by preparing a quantum state through the optimization of quantum circuit parameters. Optimal QAOA parameter concentration effects for special MaxCut problem instances have been observed, but a rigorous study of the subject is still lacking. In this work we show clustering of optimal QAOA parameters around specific values; consequently, successful transferability of parameters between different QAOA instances can be explained and predicted based on local properties of the graphs, including the type of subgraphs (lightcones) from which graphs are composed as well as the overall degree of nodes in the graph (parity). We apply this approach to several instances of random graphs with a varying number of nodes as well as parity and show that one can use optimal donor graph QAOA parameters as near-optimal parameters for larger acceptor graphs with comparable approximation ratios. This work presents a pathway to identifying classes of combinatorial optimization instances for which variational quantum algorithms such as QAOA can be substantially accelerated.

Quantum computing seeks to exploit the quantum mechanical concepts of entanglement and superposition to perform a computation that is significantly faster and more efficient than what can be achieved by using the most powerful supercomputers available today (

In this work we demonstrate two related key elements of optimal QAOA parameter transferability. First, by analyzing the distributions of subgraphs from two QAOA MaxCut instance graphs, one can predict how close the optimized QAOA parameters for one instance are to the optimal QAOA parameters for another. Second, by analyzing the overall parity of both donor-acceptor pairs, one can predict good transferability between those QAOA MaxCut instances. The measure of transferability of optimized parameters between MaxCut QAOA instances on two graphs can be expressed through the value of the approximation ratio, which is defined as the ratio of the energy of the corresponding QAOA circuit, evaluated with the optimized parameters

While the optimal solution is not known in general for relatively small instances (graphs with up to 256 nodes are considered in this paper), it can be found by using classical algorithms, such as the Gurobi solver (^{1}

Based on the analysis of the mutual transferability of optimized QAOA parameters between all relevant subgraphs for computing the MaxCut cost function of random graphs, we show good transferability _{max} = 6, and use it to demonstrate that in order to find optimized parameters for a MaxCut QAOA instance on a large 64-, 128-, or 256-node random graph, under specific conditions, one can reuse the optimized parameters from a random graph of a much smaller size,

This paper is structured as follows. In _{max} = 6. We then extend the consideration to parameter transferability using graph parity as a metric, and we demonstrate the power of the proposed approach by performing optimal transferability of QAOA parameters in many instances of donor-acceptor graph pairs of differing sizes and parity. We find that one can effectively transfer optimal parameters from smaller donor graphs to larger acceptor graphs, using similarities based on subgraph decomposition and parity as indicators of good transferability. In

The quantum approximate optimization algorithm is a hybrid quantum-classical algorithm that combines a parameterized quantum evolution with a classical outer-loop optimizer to approximately solve binary optimization problems (

Consider a combinatorial problem defined on a space of binary strings of length _{1}
_{2}⋯_{
N
} is the bit string and _{
α
}(^{
N
}-dimensional Hilbert space with computational basis vectors |

At each call to the quantum computer, a trial state is prepared by applying a sequence of alternating quantum operators_{
C
}(^{−iγC
} is the phase operator; _{
B
}(^{−iβB
} is the mixing operator, with ^{
x
} operators;

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

Schematic pipeline of a QAOA circuit. A parametrized ansatz is initialized, followed by series of applied unitaries that define the depth of the circuit. Finally, measurements are made in the computational basis, and the variational angles are classically optimized. This hybrid quantum-classical loop continues until convergence to an approximate solution is achieved.

For studying the transferability of optimized QAOA parameters, we consider the MaxCut combinatorial optimization problem. Given an unweighted undirected simple graph

It has been shown in (

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 (

Solving a QAOA instance calls for two types of executions of quantum circuits: iterative optimization of the QAOA parameters and the final sampling from the output state prepared with those parameters. While the latter is known to be impossible to simulate efficiently for large enough instances using classical hardware instead of a quantum processor (

Optimizing QAOA parameters for a relatively small graph, called the donor, and using them to prepare the QAOA state that maximizes the expectation value ⟨_{
p
} for the same problem on a larger graph, called the acceptor, is what we define as

Optimal QAOA parameter concentration effects have been reported for several special cases, mainly focusing on random 3-regular graphs (

The central question of this manuscript is under what conditions the optimized QAOA parameters for one graph also maximize the QAOA objective function for another graph. To answer that question, we study transferability between subgraphs of a graph, since the QAOA objective function is fully determined by the corresponding subgraphs of the instance graph, as well as transferability between graphs of similar parities, in order to determine structural effects of graphs on effective transferability.

It was shown in the seminal QAOA paper (_{
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 2_{
p
}, as also discussed in

We begin by analyzing the case of MaxCut instances on 3-regular random graphs for QAOA circuit depth

Landscapes of energy contributions for individual subgraphs of 3- (top row), 4- (middle row), and 5-regular (bottom row) random graphs, as a function of QAOA parameters

The same effect is observed for subgraphs of 4-regular; see

Focusing now on all five possible subgraphs of 5-regular graphs,

We discuss parameter concentration for instances of random graphs in a later section; similar discussions can be found in (

To further investigate transferability among regular graphs, we evaluate the subgraph transferability map between all possible subgraphs of _{
G*}, _{
G*}). Doing so for the donor subgraph

Transferability map between all subgraphs of random regular graphs with maximum node degree _{max} = 8, for QAOA depth

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 (_{
D*}, _{
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

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

The regular pattern of alternating clusters of high- and low-transferability coefficients in

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

Transferability map between all subgraphs of random graphs with maximum node degree _{max} = 6, for QAOA depth

Distribution of optimal parameters of subgraphs with node degrees of central nodes ranging from 1 to 6 (total 56). Each subgraph was optimized with 20 multistarts, each of which is plotted in the figure above.

We will now demonstrate that the parameter transferability map from

Demonstration of optimized parameter transferability between

Details of donor and acceptor graphs, including number of nodes, number of edges, and both QAOA, and classically optimized energies, along with their corresponding approximation ratios.

Graph | Nodes | Edges | QAOA energy | Energy (Opt) | Approx. Ratio |
---|---|---|---|---|---|

#1 | 6 | 7 | 4.6481 | 6.0 | 0.7746 |

#2 | 6 | 6 | 4.1272 | 5.0 | 0.8254 |

#3 | 6 | 9 | 5.7050 | 6.0 | 0.9508 |

#4 | 256 | 405 | 269.1192 | 363.0 | 0.7413 |

#5 | 256 | 461 | 301.7699 | 400.0 | 0.7544 |

#6 | 256 | 502 | 327.4132 | 430.0 | 0.7614 |

QAOA, energies from transferred optimal parameters from 6-node donor graphs to 256-node acceptor graphs, along with their corresponding approximation ratios. The values in parenthesis show the reduction in the approximation ratio.

Transfer | QAOA energy | Approx. Ratio |
---|---|---|

#1 → #4 | 226.2350 | 0.7334 (−1.0%) |

#2 → #5 | 293.8988 | 0.7347 (−2.6%) |

#3 → #6 | 323.8726 | 0.7753 (−1.0%) |

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.

Approximation ratios for QAOA parameter transferability between lists of 6- to 20-node donor graphs and

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 _{
even
} is the number of even nodes in graph

The computed transferability coefficients among each graph pair, sorted by their parity, are shown in _{
G
} = 0.8–1, and odd graphs, those with _{
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.

Transferability between 20-node random graphs as a function of the parity of degree of their vertices. The color of each block represents the average transferability of 100 graph pairs. As shown, graph pairs consisting of graphs of similar parity transfer well, while those of different parity transfer poorly.

This heatmap also suggests that the mutual transferability of a donor graph is not necessary for its good transferability with other random graphs, where _{
G
}(_{
G
} = 0.4–0.6. However, the results in

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

Energy landscapes of some 20-node graphs sorted by parity. Each subplot is the average energy landscape of 10 20-node random graphs with the specified parity.

Energy landscapes of most 20-node random graphs had either local minima or maxima at one of these 6 centers. Here we label those points for later reference in the text.

Approximation ratios of the 110 20-node graphs at the 6 points in parameter space identified in

The distribution of optimal parameters also explains poor transferability across random graphs of different parity. In

Furthermore, above-average transferability for all graph pairs can be attributed to universal parameters. As shown in

Comparison of subgraph similarity metric ^{2} graph pairs consisting of 20-node graphs. The color indicates the density of points. For most graph pairs,

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 _{
D
}, _{
D
}) and an acceptor graph _{
A
}, _{
A
}) as follows:_{
G
}(_{
D
}|⋅|_{
A
}| is the total number of subgraph pairs across graphs

In

Differences between subgraph similarity metric

Another approach to predicting transferability or similarity between two graphs is using their parity. In

Sorting of parity similarity metric

Thus, this metric penalizes graph pairs consisting of different parity graph pairs. Note that the lowest value of this metric is

Comparison of parity similarity metric ^{2} graph pairs consisting of 20-node graphs. The discrete columns occur because we cannot generate 20-node graphs with an arbitrary number of even-degree nodes.

To test our similarity metric for the data set shown in

For an increasing number of donor graph nodes, we see that parity can determine good transferability. For the case with subgraph transferability, we see that this does not depend on the number of nodes of the donor graph.

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

Finding optimal QAOA parameters is a critical step in solving combinatorial optimization problems by using the QAOA approach. Several existing techniques to accelerate the parameter search are based on advanced optimization and machine learning strategies. In most works, however, various types of global optimizers are employed. Such a straightforward approach is highly inefficient for exploration because of the complex energy landscapes for hard optimization instances.

An alternative effective technique presented in this paper is based on two intuitive observations: 1) The energy landscapes of small subgraphs exhibit “well-defined” areas of extrema that are not anticipated to be an obstacle for optimization solvers (see

With this in mind, the overarching idea of our approach is solving the QAOA parameterization problem for large graphs by optimizing parameterization for much smaller graphs and reusing it. We started with studying the transferability of parameters between all subgraphs of random graphs with a maximum degree of 8. Good transferability of parameters was observed among even-regular and odd-regular subgraphs. At the same time, poor transferability was detected between even- and odd-regular pairs of graphs in both directions, as shown in

A remarkable demonstration of random graphs that generalizes the proposed approach is the transferability of the parameters from 6-node random graphs (at the subgraph level) to 256-node random graphs, as shown in

Following the subgraph decomposition approach, we showed that one can determine good transferability between donor-acceptor graph pairs by exploiting their similarity based on parity. We see a good correlation between subgraph similarity and parity similarity. In the future, we wish to address the exploitation of graph structure to determine good donor candidates, since subgraph similarities involve overhead calculations of QAOA energies for each pair of donor-acceptor subgraphs.

One may notice that we studied parameter transferability only for

Another future direction is to determine whether the effects of parity of a graph hold for

This work was enabled by the very fast and efficient tensor network simulator QTensor developed at Argonne National Laboratory (

As a result of this work, finding optimized parameters for some QAOA instances will become quick and efficient, removing this major bottleneck in the QAOA approach and potentially removing the optimization step altogether in some cases, eliminating the variational nature of QAOA. Moreover, our approach will allow finding parameters quickly and efficiently for very large graphs for which it will not be possible to use simulators or other techniques. Our method has important implications for implementing QAOA on relatively slow quantum devices, such as neutral atoms and trapped-ion hardware, for which finding optimal parameters may take a prohibitively long time. Thus, quantum devices will be used only to sample from the output QAOA state to get the final solution to the combinatorial optimization problem. Our work will ultimately bring QAOA one step closer to the realization of quantum advantage.

The original contributions presented in the study are included in the article/

All authors listed have made a substantial, direct, and intellectual contribution to the work and approved it for publication.

This research was developed with funding from the Defense Advanced Research Projects Agency (DARPA). The views, opinions and/or findings expressed are those of the author and should not be interpreted as representing the official views or policies of the Department of Defense or the U.S. Government. AG, DL, XL, IS, JF, and YA are supported in part by funding from the Defense Advanced Research Projects Agency. EG was supported in part by the U.S. Department of Energy, Office of Science, Office of Workforce Development for Teachers and Scientists (WDTS) under the Science Undergraduate Laboratory Internships Program (SULI). This work used in part the resources of the Argonne Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC02-06CH11357.

Author XL was employed by Fujitsu Research of America, Inc.

The author IS declared that they were an editorial board member of Frontiers, at the time of submission. This had no impact on the peer review process and the final decision.

The remaining authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

The Supplementary Material for this article can be found online at:

The Gurobi solver provides classically optimal MaxCut solutions in a competitive speed with known optimization gap. For the purpose of this work, there is no particular reason to choose Gurobi over IPOPT or other similarly performing solvers.