Embedding Algorithms for Quantum Annealers with Chimera and Pegasus Connection Topologies
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Abstract
We propose two new algorithms – Spring-Based MinorMiner (SPMM) and Clique-Based MinorMiner (CLMM) – which take as input the connectivity graph of a Quadratic Unconstrained Binary Optimization (QUBO) problem and produce as output an embedding of the input graph on a host graph that models the topology of a quantum computing device. As host graphs, we take the Chimera graph and the Pegasus graph, which are the topology graphs of D-Wave’s 2000 qubit (first introduced in 2017) and 5000 qubit (expected 2020) quantum annealer devices, respectively. We evaluate our algorithms on a large set of random graph QUBO inputs (Erdős-Rényi \(G_{n,p}\), Barabási-Albert and d-regular graphs) on both host topologies against other embedding algorithms. For the Pegasus topology, we find that CLMM outperforms all other algorithms at edge densities larger than 0.08, while SPMM wins at edge densities smaller than 0.08 for Erdős-Rényi graphs, with very similar transition densities for the other graph classes. Surprisingly, the standard D-Wave MinorMiner embedding algorithm – while also getting slightly outperformed by SPMM for sparse and very dense graphs on Chimera – does not manage to extend its overall good performance on Chimera to Pegasus as it fails to embed even medium-density graphs on 175–180 nodes which are known to have clique embeddings on Pegasus.
1 Introduction
Study Parameters: we compare the performance of four embedding methods for three different QUBO graphs on the two main D-Wave host graph topologies.
On the Pegasus host graph, our Clique-based MinorMiner (CLMM) is a clear winner with our alternative Spring-Based MinorMiner (SPMM) algorithm edging out both CLMM and MM for very sparse graphs only. The relative ranking of the algorithms is the same across all three QUBO input classes with SPMM’s advantage at sparse graphs most pronounced for d-regular graphs. Somewhat surprisingly, a threshold edge density exists that is very similar for all three random graph classes (at about \(|E|/\left( {\begin{array}{c}|V|\\ 2\end{array}}\right) \approx 0.08\)) such that CLMM and SPMM win at edge densities larger and smaller than the threshold, respectively (E, V denote edges and nodes of the QUBO graph).
On the Chimera host graph, SPMM wins over MM and LAMM at sparse and dense graphs, whereas MM and LAMM perform slightly better at medium density graphs. Again, SPMM’s advantage at large sparse graphs is most pronounced for d-regular graphs.
On the Chimera host graph, all algorithms easily manage to embed the previously largest known embeddable clique (at 65 vertices), whereas on Pegasus only CLMM finds embeddings of cliques with more than 180 nodes. In fact, using SPMM for Chimera and CLMM for Pegasus we find largest embeddable cliques at sizes 65 and 185 respectively.
The paper is organized as follows: We introduce the concepts of QUBOs, embeddings, host graphs and other background material including related work in more detail in Sect. 2. We describe the embedding algorithms in Sect. 3, and give details about the experimental design in Sect. 4. We present our results for the Pegasus host graph in Sect. 5 and for the Chimera host graph in Sect. 6, before concluding in Sect. 7.
2 Background
2.1 Quadratic Unconstrained Binary Optimization (QUBO)
We remark that QUBOs are a class of NP-hard optimization problems; as we can use QUBOs to optimize the number of satisfied constraints in an instance of 0/1 Integer Programming – one of Karp’s original 21 NP-complete problems [21].
2.2 Solving QUBOs on Quantum Annealers
However, most users will have QUBO problems from their application domains with corresponding QUBO graphs that are far from being subgraphs of the host graph. In order to be able to solve QUBOs using a quantum annealer, the standard approach (see Fig. 1) is to find a minor embedding of the QUBO graph into the host graph [11] and to set the \(h_i, J_{ij}\) parameters accordingly [10]; i.e. one chains multiple qubits of the host graph with ferromagnetic couplings \(J_{ij} \ll 0\) to represent a single variable of a QUBO (indicated by shared colors in Fig. 1 (center)/(right)). The better the embedding algorithm, the more QUBO problems can be solved by an annealer. Designing and testing capable embedding algorithms that are able to embed a large set of QUBO graphs is thus crucial to expand the set of applications for a quantum device such as D-Wave. The same holds true for CMOS annealers, such as those of Hitachi [33, 36].
We note in passing that adiabatic quantum computing [15] – the theoretical inspiration for quantum annealer technology – is equivalent in power to standard gate-based quantum computing [2] that implements arbitrary unitary operations. However, the mapping challenge on gate-based quantum devices differs substantially from quantum annealers as logical variables are mapped only to single qubits and not to chains. To implement a gate between two non-neighboring qubits in a gate device, qubit states are swapped along paths of the host topology, giving a “time-dependent mapping”, sometimes called routing. Depending on the application, this can be done heuristically [12], with exact solvers [35], or using a swap network [26]. Comparing state-of-the-art approaches to equality constraints implementation on a quantum annealer [34] and on a gate-based quantum computer [8] shows, on a concrete application, how different the mapping problem is for the two platforms.
2.3 Minor Embeddings
- 1.
For each node v in \(V_P\), the set of nodes \(\varphi (v)\) induces a connected subgraph in H, called the chain of v.
- 2.
For every edge \(e =\{u,v\}\) in \(E_P\), there exist nodes \(\tilde{u}\in \varphi (u)\) and \(\tilde{v} \in \varphi (v)\) such that \(\{\tilde{u}, \tilde{v}\}\in E_H\).
- 3.
\(\varphi (v)\cap \varphi (u) = \emptyset \) for all \(u \ne v \in V_P\), i.e., each node \(\tilde{v}\) of the host graph H appears in the mapping of at most one node of the pattern graph P.
We call a mapping \(\varphi \) a chain mapping if it satisfies Condition . A chain mapping \(\varphi \) is called a semi-valid embedding if it satisfies Condition and is called a chain placement if it satisfies Condition . Only if all three conditions are satisfied do we have a minor embedding. Colloquially, we abbreviate minor embedding with just embedding.
Finding a minor-embedding is NP-complete [25] except for (small) fixed pattern graphs [30], and the best known algorithms [1] are exponential in \(|V_P|\) and the branch-width or tree-width of H (which is \(\varOmega (\sqrt{|V_H|})\) for current annealers). Research on minor-embedding for annealers has therefore focused on finding fast and hiqh-quality heuristics. Existing approaches can best be described along one of two trajectories: (i) iteratively modify a semi-valid embedding to reduce the number of multiply used nodes \(\tilde{v} \in V_H\) (the approach shared by the algorithms benchmarked in this paper), (ii) iteratively modify a chain placement to increase the number of represented edges \(e\in E_P\) (recently proposed [32] for King’s graphs, the topology of Hitachi CMOS annealers [33]).
Furthermore, good minor embeddings are known for highly structured pattern graphs such as cliques [6, 24], cartesian products thereof [37], bicliques [19], cubic grids [20] and cylindrical lattices (square-octogonal and triangular) [23].
2.4 Chimera and Pegasus Topologies
The host graphs of current and upcoming D-Wave annealers can be understood starting from an intersection graph of axis-parallel rectangles (the qubits):
In Pegasus [5], qubit rectangles are longer and connect to 12 orthogonal qubits. Furthermore, horizontal and vertical qubits are shifted asymmetrically, and have additional odd couplers that connect pairs of neighboring parallel qubits, such that qubits have degree 15. This results in cells that are connected by 4, 8, or 16 edges, see Fig. 2 (right). The Pegasus graph P16, such as in the upcoming D-Wave Advantage, has \(15\times 15\times 3\) cells, plus some partial cells on the border, for a total of 5640 qubits. We illustrate P4 in Fig. 3.
2.5 QUBO Random Graph Classes
To extend the range of embeddable QUBOs on current and next-generation devices, we benchmark embedding algorithms based on their performance in finding embeddings. Other metrics such as average or maximum chain lengths [29] are not a focus of this paper; hence the actual values of non-zero QUBO terms do not matter. Similarly, we only consider connected graphs (as one can always solve connected components independently) and do not consider any divide-and-conquer strategies [27]. We use three classes of random graphs as benchmarks:
(i) Erdős-Rényi graphs \(G_{n,p}\) [18], where edges are included in the graph i.i.d with probability p, (ii) Barabási-Albert graphs \(BA_{n,m}\) [3, 4], in which, starting from m isolated nodes, we insert \(m-n\) nodes one by one, connecting each to m existing nodes with preferential attachment proportional to the current degree distribution, (iii) random d-regular graphs, in which each node has degree d. By varying p, d and m, respectively, we generate graphs of various densities.
We chose these three graph classes in order to test our algorithms on a diverse set of graphs: Erdős-Rényi graphs have a binomial (Poisson for small p) degree distribution, Barabási-Albert graphs have a power-law degree distribution (modeling networks), and d-regular graphs have a constant degree distribution.
In the following Section, we briefly present existing algorithms that we either compare to or use as a subroutine in our algorithms, which then follow next.
3 Minor Embedding Heuristics
3.1 Existing Embedding Algorithms
- 1.
Remove the chain \(\varphi (v) \subseteq E_H\) from the existing chain mapping.
- 2.
Compute a node-weighted shortest paths tree in H from each non-empty chain \(\varphi (u)\), where u is a neighbor of v in P (\(\{u,v\} \in E_P\)). The node weights in H come with a high penalty term for using nodes in multiple chains.
- 3.
Choose an optimal node \(\tilde{v} \in V_H\) that minimizes the sum of distances according to the computed shortest paths trees. Extend \(\tilde{v}\) to a chain \(\varphi (v)\) by backtracking along the shortest paths trees, and re-add \(\varphi (v)\) to the chain mapping.
This naturally splits MinorMiner into two phases: First, MM completes a single loop over the vertices \(V_P\), after which the chain mapping \(\varphi \) is in fact a semi-valid embedding (in which chains might still share qubits).
Secondly, MM enters a fixing phase, where consecutive loops over nodes in \(V_P\) have the goal of fixing this semi-valid embedding. The algorithm restarts when there has been no progression for too many steps in a row^{2}, with limiting parameters on the total number of steps and number of restarts allowed. Thus, when the algorithm terminates, it might either return a valid embedding when it found one, or an empty embedding if it did not.
MM has a few other controls, such as the \(\texttt {initial\_chain}\) parameter. This parameter can be used to feed the algorithm an initial chain mapping, which is then used in the first phase of finding a semi-valid embedding. However, the algorithm still iterates over all nodes. When it reaches a node which was assigned an initial non-empty chain, it still deletes and replaces that chain with the procedure outlined above.
Clique Embedding. D-Wave has a host-specific clique embedding algorithm [6], which can quickly embed any clique up to a certain size \(c_{\mathrm {host}}\) into the Pegasus or the Chimera graph (this also implies an embedding algorithm for any graph with up to \(c_{\mathrm {host}}\) nodes). For Pegasus P16, the maximal clique size embeddable this way is \(c_{\mathrm {host}} = 180\), for Chimera C16 it is \(c_{\mathrm {host}} = 64\). Chains gained from this embedding have a very special shape: they are all paths which are “L-shaped” if drawn into the 2D-layout of the respective host graph, see Fig. 3.
3.2 Our Contribution
We propose, implement and compare two new algorithms: Clique-based MinorMiner (CLMM) and Spring-based MinorMiner (SPMM).
For CLMM, we construct an initial chain mapping for a subset of QUBO nodes, able to implement a coupling between any two chains of this node subset. For SPMM, we give an initial chain mapping for all QUBO nodes \(V_P\), based on a force-directed graph drawing of P. In the second approach, there are no guarantees for existing couplings between chains. We then pass this initial chain mapping to MinorMiner with the \(\texttt {initial\_chain}\) parameter.
Clique-Based MinorMiner (CLMM). For CLMM, we construct an initial chain mapping as follows: We run D-Waves clique embedding algorithm for a clique of size \(k = \min (|V_P|, c_{\mathrm {host}})\). The k chains found this way are assigned to k nodes of the QUBO graph, with the assignment depending on the density of P: If \(|E_P| / \left( {\begin{array}{c}|V_P|\\ 2\end{array}}\right) \ge 0.55\), they are assigned to the k nodes of lowest degree, otherwise to k random nodes. The remaining QUBO nodes are mapped to empty chains.
We also tested a wide variety of other density- and degree-based assignments, as well as a splitting or a multi-assignment of chains in exploratory runs. In contrast to these approaches, the presented (albeit simpler) settings performed significantly better and were thus used in the final experiments.
Spring-Based MinorMiner (SPMM). For SPMM, we construct an initial chain mapping as follows: (i) We use standard D-Wave layout functions to get a drawing of the Pegasus/Chimera host graph in the plane (cf. the host graphs in Fig. 2), and a tuned Fruchterman-Reingold algorithm (see below) to get a QUBO graph layout as well. (ii) We rescale both plane layouts to fit into a \([-1,1]\times [-1,1]\) square. (iii) We map each of the QUBO nodes v to the closest qubit node in Euclidean metric.
Fruchterman-Reingold [16] is a force-directed graph drawing algorithm that computes a plane layout based on two principles: nodes pairwise repel each other, but nodes connected by an edge at the same time attract each other. The strength with which the latter takes place can be set for each edge individually; smaller weights implying a smaller attraction. For an edge \(e=\{u,v\}\) we set \(\mathrm {weight}(e) = \left( 2|E_P|/|V_P|\right) ^2\cdot \left( \mathrm {deg}(u)\, \mathrm {deg}(v)\right) ^{-1}\), where the weighting by node degrees ensures that neighboring nodes with high degrees are not too close to each other (as, intuitively, their chains need more space in the host graph) and where the first term is a normalization factor (normalizing weights in regular graphs to 1).
While SPMM and LAMM have some similarities, we find a significant performance difference on Pegasus graphs due to SPMM’s improved use of edge weights for node attraction, substituting LAMM’s consecutive diffusion phase.
4 Experimental Design
We present the results of a large factorial-design experiment to compare our two algorithms Clique-based MinorMiner (CLMM) and Spring-Based MinorMiner (SPMM) with the established MinorMiner (MM) and the recently proposed Layout-Aware MinorMiner (LAMM). We test the algorithms on the random QUBO graph classes \(G_{n,p}\), Barabási-Albert, and d-regular. As host graphs, we use the D-Wave Pegasus host graph (used in the 5000 qubit model first out in 2020) as well as the previous Chimera topolgy (used until the 2000 qubit model).
For the Erdős-Rényi \(G_{n,p}\) graph model, we generate five random graphs for each combination of values \(n = \{1, \ldots , 425\}\) and \(p = \{.01, .02, \ldots , 1.00 \}\). While this would result in a total of \(5\cdot 450\cdot 100 = 212,500\) graphs, we actually reduced this number to around 26, 000 graphs by carefully pruning the set of graphs for a specific algorithm once it has become clear – based on results for smaller/larger values of n or p – that the algorithm will always/never find an embedding. \(G_{n,p}\) graphs have a sharp threshold of \(n\cdot p > \ln n\) of being connected [14].
For the d-regular graph model (on Pegasus), we generate five graphs each for all combinations of \(n = \{1, \ldots , 1200\}\) and \(d = \{3, \ldots , 183\}\), employing again a pruning mechanism. On Chimera, we also use five graphs and cut off at 380 vertices and maximum \(d = 64\) to account for the smaller host graph. Random d-regular graphs can be sampled quickly for \(d\le n/2\) [31] and uniformly at random for \(d\in \mathcal {O}(n^{1/3-\varepsilon })\) [22]; we sample \((n-d)\)-regular graphs as complements of d-regular graphs. d-regular graphs only exist for \(2|E|=n\cdot d\) even and \(d<n\).
For the Barabási-Albert graph model, we generate five graphs each for \(n = \{1, \ldots , 1200\}\) and \(m = \{1, \ldots , 110\}\) and employ pruning. The number of edges in \(BA_{n,m}\) is \((n-m)\cdot m \le (\tfrac{n-m+m}{2})^2= n^2/4\) by AM-GM, with equality for \(m = n/2\). Hence we get increasing graph density for m up to n/2, and we restrict ourselves to this regime. All graphs are constructed with Python’s networkx.
Our experiments were executed on LANL’s Darwin Cluster [17] using a trivially parallel approach. Running times for individual graphs ranged from milliseconds to more than 10 min per graph, largely proportional to graph vertex and edge counts. Overall, the study consumed around 100, 000 core hours. We assess the different algorithms on whether they succeed in finding an embedding with the default parameters of MinorMiner, and not by running times, but overall we observed that running times were very comparable for all the tested algorithms.
5 Embeddings on the Pegasus Host Graph
We get a more in-depth understanding of performance difference by looking at the difference plot in Fig. 5. Its structure is similar to the individual performance plots, except the color bar ranges from green (positive) to white (zero) to purple (negative). A point (n, p) above (below) the blue line at \(p=0.08\) is assigned a color based on the number of embeddings found by CLMM (SPMM, respectively) minus the maximum of the number of embeddings found by the clique embedding algorithm or by MM. This way we capture the improvement SPMM gains for sparse graphs and the improvement CLMM gains on dense graphs in one plot. The transition between areas where CLMM and where SPMM are the respective best performing algorithms is sharp, around an edge density value of \(|E_P|/\left( {\begin{array}{c}|V_P|\\ 2\end{array}}\right) \approxeq p=0.8\). In combination, our algorithms manage to outperform the already existing algorithms at every value of p, gaining the most around \(p = 0.20\), and for \(p = 0.02\) where the graphs get sparse enough such that SPMM’s advantage over MM starts to get significant.
Figure 6 (bottom) shows that on d-regular graphs, performance of CLMM, SPMM and MM mirrors their performance on Erdős-Rényi and Barabási-Albert graphs. Since d-regular graphs only exist for even \(n\cdot d\), we omit odd n columns from the plot (but not from the experiments, see the concluding data in Sect. 7). SPMM again gains the biggest advantage on the sparsest graphs, namely \(d=3\), while CLMM outperforms MM on dense graphs, with a transition threshold at \(d=18\), \(n=233\), corresponding to an edge density of \(d/(n-1) \approx 0.078\).
Secondly, we look at the link between CLMM and SPMM’s performance and the sparsity of the graph. In embeddings for dense graphs, chains often form a path through a large part of the host graph, with few or no nodes of induced degree larger than two. We believe that providing initial “L-shaped” chains such as in CLMM may promote newly built chains to take on such shapes as well. On the other hand, for sparse graphs a well-chosen initial single-qubit chain such as in SPMM can enable short connections to neighboring chains, reducing the qubit footprint of a semi-valid embedding created after the first phase of MinorMiner.
6 Embeddings on the Chimera Host Graph
On Chimera, we only compare the three algorithms MM, LAMM and SPMM. We did not test CLMM in great detail, as MM performs very similar, and since preliminary observations could not find any improvements of CLMM over MM.
SPMM performs better than MM on graphs with \(p\ge 0.8 \) and graphs with \(p\le 0.3\). However, for \(0.3<p<0.8\), both algorithms perform comparably well. The performance difference between SPMM and LAMM is similar to the one between SPMM and MM. However, while SPMM still beats LAMM for \(p\le 0.2\), for larger p LAMM outperforms SPMM slightly.
Embedding Barabási-Albert and d-regular Graphs on Chimera. While SPMM delivers the best overall performance in embedding both Barabási-Albert and d-regular graphs on Chimera, the second place depends on the graph class (MM for Barabási-Albert, LAMM for d-regular graphs). In Fig. 8 (top), we show the performance difference between SPMM and MM on Barabási-Albert graphs. While MM outperforms SPMM slightly on \(m\ge 20\), the advantage of SPMM on small m is much more apparent, especially at \(m = 2\).
Summary of all experiments: We rank Algorithms based on the total number of found embeddings. Pegasus experiments are split into a sparse and a dense QUBO graph regime, given by the observed transition parameters for p, m, d. For comparison, we also give the number of possible embeddings via host-specific cliques.
Ranking | Erdős-Rényi | Barabási-Albert | d-regular | |
---|---|---|---|---|
Pegasus (dense) | 1. CLMM | 86,159 | 38,256 | 54,937 |
\(0.08 < p\; \le 1.00\) | 2. MM | 78,230 | 31,206 | 44,229 |
\(12 < m \le n/2\) | 3. SPMM | 70,512 | 24,435 | 35,375 |
\(18 < d\; \le n-1\) | 4. LAMM | 68,349 | 23,248 | 32,998 |
Clique | 81,530 | 30,420 | 49,410 | |
Pegasus (sparse) | 1. SPMM | 6,150 | 23,334 | 37,210 |
\(0.01 \le p\; \le 0.08\) | 2. MM | 6,039 | 21,814 | 35,866 |
\(2 \le m \le 12\) | 3. CLMM | 5,964 | 21,803 | 35,490 |
\(3 \le d\; \le 18\) | 4. LAMM | 6,047 | 18,681 | 35,374 |
Clique | 2,700 | 10,985 | 12,470 | |
Chimera | 1. SPMM | 33,793 | 10,874 | 16,506 |
2. LAMM | 33,688 | 10,217 | 16,132 | |
3. MM | 33,530 | 10,367 | 15,972 | |
Clique | 27,860 | 5,445 | 8,060 |
7 Conclusion
We studied the performance of two new embedding algorithms, Spring-based MinorMiner (SPMM) and Clique-based MinorMiner (CLMM), and contrasted these to existing embedding heuristics for the two different D-Wave host graph topologies Pegasus and Chimera. To the best of our knowledge, this is the first such study on the upcoming Pegasus topology. While we observed that the existing MinorMiner heuristic does not extend its overall good performance on Chimera to Pegasus, we show how to remedy the situation with our Clique-based and Spring-based MinorMiner variants, see Table 2.
In conclusion, we studied different random graphs to represent a wide variety of possible QUBO graphs and gave a detailed analysis of the performance of CLMM, SPMM, MM and the recent LAMM. A relative ranking of the algorithms based on the total number of found embeddings is given in Table 2. While SPMM and CLMM are the clear winners in their respective density domains, the order of the competitors can change depending on the graph class studied.
Though SPMM and CLMM outperform the standard algorithm MM, their simplicity is somewhat remarkable and of course they build upon the work of both the original MinorMiner paper [9] and its implementation [13] as a subroutine. We suggest that the MinorMiner parameter initial_chain be extended with ‘clique’ and ‘spring’ parameters to serve as calls to the respective CLMM and SPMM algorithms presented in this work.
Future Work. Future research directions are three-fold: First, we intend to add case studies of real-world QUBO instance graphs to include them in a full version of this paper, together with plots and results of all our experiments.
Secondly, we would like to study other (CMOS) host graphs [36] and compare our algorithms to simulated annealing-based approaches which were recently proposed in the literature [32] but not yet published as software.
Finally, applying various embedding algorithms to the same QUBO problem will result in embeddings with different characteristics, such as the distribution of chain lengths. These characteristics, in turn, will influence the chance of success and hence the overall time-to-solution of solving QUBO problems with a quantum annealer. Once the Pegasus architecture becomes available, it will be useful to compare embedding algorithms with respect to these metrics, as was done for Chimera before [29].
Footnotes
- 1.
We have \(\sigma _z = \left( {\begin{matrix}1 &{} 0 \\ 0 &{} \text {-}1\end{matrix}}\right) \), \( Id =\left( {\begin{matrix}1 &{} 0 \\ 0 &{} 1\end{matrix}}\right) \), and tensor product \(\sigma _z^{(i)} = Id ^{\otimes i-1} \otimes \sigma _z \otimes Id ^{\otimes n-i}\).
- 2.
Even if the algorithm is already in a state with a valid embedding, progression is measured for example in having a smaller maximal chain size.
- 3.
Especially compared to Chimera, where MM manages to find an embedding for \(K_{65}\), the largest embeddable clique, given that \(\mathrm {treewidth}(K_{65}) = 64 = \mathrm {treewidth}(C16)\).
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