## Abstract

Quantum annealing is a promising technique which leverages quantum mechanics to solve hard optimization problems. Considerable progress has been made in the development of a physical quantum annealer, motivating the study of methods to enhance the efficiency of such a solver. In this work, we present a quantum annealing approach to measure similarity among molecular structures. Implementing real-world problems on a quantum annealer is challenging due to hardware limitations such as sparse connectivity, intrinsic control error, and limited precision. In order to overcome the limited connectivity, a problem must be reformulated using minor-embedding techniques. Using a real data set, we investigate the performance of a quantum annealer in solving the molecular similarity problem. We provide experimental evidence that common practices for embedding can be replaced by new alternatives which mitigate some of the hardware limitations and enhance its performance. Common practices for embedding include minimizing either the number of qubits or the chain length and determining the strength of ferromagnetic couplers empirically. We show that current criteria for selecting an embedding do not improve the hardware’s performance for the molecular similarity problem. Furthermore, we use a theoretical approach to determine the strength of ferromagnetic couplers. Such an approach removes the computational burden of the current empirical approaches and also results in hardware solutions that can benefit from simple local classical improvement. Although our results are limited to the problems considered here, they can be generalized to guide future benchmarking studies.

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## Notes

- 1.
H. Katzgraber, personal communication, 2016.

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## Acknowledgements

This research was supported by 1QBit. The authors would like to thank Clemens Adolphs, Hamed Karimi, Anna Levit, Dominic Marchand, and Arman Zaribafiyan for useful discussions, Robyn Foerster for valuable support, and Marko Bucyk for editorial help. We thank Helmut Katzgraber for reviewing the manuscript. We also acknowledge the support of the Universities Space Research Association (USRA) Quantum Artificial Intelligence Laboratory Research Opportunity program.

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## Appendix

### Appendix

### Theoretical parameter setting

As mentioned in Sect. 3.2.2, we generalize the approach presented by Choi [13] to theoretically set the parameters of the embedded graph, in which the connected subgraphs representing the logical qubits are not necessarily subtrees of the hardware graph. The procedure used in this paper is explained in Algorithm 1. It is worth mentioning that we do not discuss the proof for the validity of the theoretical approach. The key idea as presented by Choi [13] is to ensure that the ground state of the input graph before and after embedding matches.

Let us consider a conflict graph \(G_{\hbox {c}} =(V_{\hbox {c}}, E_{\hbox {c}})\) with logical local fields and couplings denoted by *h* and *J*, respectively. Assume that logical qubit \(i \in V_{\hbox {c}}\) is represented by \(n_i\) physical qubits forming the physical subgraph \(S_i = (V_{S_i}, E_{S_i})\). Further assume that \(\hbox {lnb}(i)\) is the set of neighbouring vertices of the logical qubit *i* in the logical graph \(G_{\hbox {c}}\) and \(\hbox {pnb}(l)\) is the set of neighbouring vertices of the physical qubit \(l \in V(S_i)\), \(\forall i \in V_{\hbox {c}}\), excluding the vertices representing the same logical qubit.

The theoretical approach detailed below has an iterative pre-processing step in which several logical qubits might be removed from the logical graph, since their optimal values can be inferred in advance. The parameters are then set on the reduced logical graph.

### Time-to-solution estimation

Since the quantum annealer is a stochastic solver, we consider the successive annealing runs as a sequence of binary experiments that might succeed in returning the ground state with some probability. Let us formally define \(X_1, X_2, \ldots , X_n\) as a sequence of random independent outcomes of *n* annealing runs, where \({\mathbbm {P}}(X_i=1)=\theta \) denotes the probability of observing the ground state at the *i*-th anneal. Defining *Y* as the number of successes observed in *n* anneals (\(Y=\sum _{i=1}^{n} X_i\)), we have \({\mathbbm {P}}(Y = y|\theta ) = {n \atopwithdelims ()y} (1-\theta )^{n-y}\theta ^y\) (\(Y|\theta \sim \hbox {Bin}(n, \theta )\)). That is, \(Y|\theta \) has a binomial distribution with parameters *n* and \(\theta \). The \(\hbox {R}_{99}\) then equals *n* such that \({\mathbbm {P}}(Y \ge 1|\theta ) = 0.99\). It is easy to verify that \(\hbox {R}_{99} = \log (1-0.99) / \log (1-\theta )\). Since the probability of success \(\theta \) is unknown, the challenge is to estimate \(\theta \).

We follow the Bayesian inference technique to estimate the probability of success for each instance *i* [23]. In the Bayesian inference framework, we start with a guess on the distribution of \(\theta \) known as prior and update it based on the observations from the device in order to get the posterior. Since the observations from the device have a binomial distribution, the proper choice of prior is a beta distribution which is the conjugate prior of the binomial distribution. This choice guarantees that the posterior also has a beta distribution. The beta distribution with parameters \(\alpha =0.5\) and \(\beta =0.5\) (the Jeffery prior) is chosen as prior since it is invariant under reparameterization of the space and it learns the most from the data [15].

Updating the Jeffery prior based on the data from the device, the posterior distribution denoted by \(\pi _i(\theta )\) is then

where *C* is the number of calls to the quantum annealer, *N* is the number of anneals in each call, and \(y_{ci}\) is the number of times that the ground state of instance *i* is observed at the *c*-th call.

To estimate the TTS (or \(\hbox {R}_{99}\)) for the entire population of instances with similar parameters, let us assume that there are *I* instances with similar properties, for example, with the same number of variables. We are interested in using the data on these *I* instances to estimate the TTS for the entire population of instances with the same number of variables. After finding the posterior distribution \(\pi _i(\theta )\) for all instances in set \(\{I\}\), we use the bootstrap methodology to estimate the distribution of the *q*-th percentile of the TTS. The procedure is described in Algorithm 2.

### Percolation threshold of the molecular similarity problem instances

The QUBO graphs of molecular similarity problem instances can be considered random graphs. A random graph is a collection of vertices with edges connecting pairs of them randomly. Newman et al. [31] and Callaway et al. [12] developed an approach based on generating functions to determine the statistical properties of random graphs with arbitrary degree distribution. Here we review their approach to determine the percolation threshold.

Let us denote the probability that a randomly chosen vertex has degree *k* by \(p_k\). Let us further define

where \(G_0(x)\) and \(G_1(x)\) are the generating functions for the probability distribution of vertex degrees and outgoing edges, respectively, and *z* is the average vertex degree. Callaway et al. [12] have shown that the percolation threshold, or critical probability, can be calculated as follows:

The details of the derivation can be found in [12, 31]. The key idea is that the critical probability is the point at which the mean cluster size goes to infinity.

The context in which we apply this criterion is on random graphs whose degree distribution is known. It is known because the degree distribution can be measured directly [31]. Below, we provide an example with a known degree distribution for which we calculate the critical probability.

*Example * Consider a graph similarity problem instance with 18 vertices and \(d \in [65, 75]\). The number of vertices with degree 11, 12, and 13 are, in respective terms, 8, 8, and 2. The distribution of vertex degrees can be generated by

Applying the formula above, the critical probability is \(p_{\hbox {c}} = 0.0934\).

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Hernandez, M., Aramon, M. Enhancing quantum annealing performance for the molecular similarity problem.
*Quantum Inf Process* **16, **133 (2017). https://doi.org/10.1007/s11128-017-1586-y

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### Keywords

- Quantum annealing
- Quantum optimization
- Molecular similarity
- Minor embedding
- Parameter setting
- Quadratic unconstrained binary optimization