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/lp/association-for-computing-machinery/the-effect-of-penalty-factors-of-constrained-hamiltonians-on-the-RlARYRuF0j
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- Association for Computing Machinery
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- Copyright © 2023 Copyright held by the owner/author(s).
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- 2643-6809
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- 2643-6817
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- 10.1145/3577202
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Abstract
The Efect of Penalty Factors of Constrained Hamiltonians on the Eigenspectrum in uantum Annealing CHRISTOPH ROCH, Ludwig-Maximilians-Universität in Munich, Germany DANIEL RATKE, Aqarios GmbH, Germany JONAS NÜSSLEIN, Ludwig-Maximilians-Universität in Munich, Germany THOMAS GABOR, Ludwig-Maximilians-Universität in Munich, Germany SEBASTIAN FELD, Department of Quantum and Computer Engineering and QuTech, Delft University of Technology, The Netherlands Constrained optimization problems are usually translated to (naturally unconstrained) Ising formulations by introducing soft penalty terms for the previously hard constraints. In this work, we empirically demonstrate that assigning the appropriate weight to these penalty terms leads to an enlargement of the minimum spectral gap in the corresponding eigenspectrum, which also leads to a better solution quality on actual quantum annealing hardware. We apply machine learning methods to analyze the correlations of the penalty factors and the minimum spectral gap for six selected constrained optimization problems and show that regression using a neural network allows to predict the best penalty factors in our settings for various problem instances. Additionally, we observe that problem instances with a single global optimum are easier to optimize in contrast to ones with multiple global optima. CCS Concepts: · Computing methodologies → Neural networks; · Mathematics of computing → Combinatoric problems; Computations on matrices ; · Theory of computation → Quantum information theory. Additional Key Words and Phrases: Quantum annealing, penalty factor, constrained hamiltonian, minimal spectral gap, optimization, clustering, regression, artiicial neural network, d-wave systems 1 INTRODUCTION Quantum computing hardware has increasingly emerged in the last years and there is now the possibility to tackle highly complex problems in a completely diferent way to classical solution methods. In particular, the computing devices of D-Wave Systems, which implement a quantum annealing algorithm in hardware, have been extensively studied to solve diverse kinds of optimization problems [8, 13, 15, 24, 30, 31]. The solution quality when using quantum hardware for highly complex problems depends on both, the hardware’s maturity but also on the problem formulation, respectively the quantum algorithm to be executed. For the latter, much research has been put into the development of Ising spin glass model formulations for various problem classes4[, 16, 22], which is the input type for D-Wave Systems’ hardware. However, as is typical of noisy intermediate-scale quantum (NISQ) devices, many hardware but also software related hyperparameters can be optimized to achieve a potentially better solution18 quality , 23, 27, [32]. Since applied quantum computing is Authors’ addresses: Christoph Roch, christoph.roch@ii.lmu.de, Ludwig-Maximilians-Universität in Munich, Munich, Germany; Daniel Ratke, daniel.ratke@aqarios.com, Aqarios GmbH, Munich, Germany; Jonas Nüßlein, jonas.nuesslein@ii.lmu.de, Ludwig-Maximilians-Universität in Munich, Munich, Germany; Thomas Gabor, thomas.gabor@ii.lmu.de, Ludwig-Maximilians-Universität in Munich, Munich, Germany; Sebastian Feld, S.Feld@tudelft.nl, Department of Quantum and Computer Engineering and QuTech, Delft University of Technology, Delft, The Netherlands. Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for proit or commercial advantage and that copies bear this notice and the full citation on the irst page. Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s). © 2022 Copyright held by the owner/author(s). 2643-6817/2022/12-ART https://doi.org/10.1145/3577202 ACM Trans. Quantum Comput. 2 • Roch et al. still in its infancy, not much is known about the hyperparameters of the Ising model and how to adjust them in order to execute the problem formulations eiciently on hardware. One set of these parameters are the penalty factors, respectively weights of constraint Ising Hamiltonians [19, 22]. Those penalty factors ensure that the constraints of a corresponding optimization problem are satisied. When this is not the case, a penalty value (or factor) is added to the solution energy which is intended to be minimized within the quantum annealing process.22Lucas ] presents [ many Ising Hamiltonians of constrained optimization problems. However, so far there is no rule of thumb on how to set the penalty factors eiciently, i.e. without trial and error or guessing them. In previous work we showed that optimizing those parameters with cross entropy leads to a scaling of the minimum spectral gap such that the solution quality of three problem classes when using the D-Wave Systems’ quantum annealing algorithm incr26 ease ]. Ho d [wever, since we did not see a correlation between the optimized penalty factors for the problem instances, one would have to apply the time-consuming cross entropy method for each instance individually. This motivates our investigation on the efect of penalty factors on the minimum spectral gap of six selected constrained Hamiltonians using machine learning (ML) techniques. We want to remedy the need for intensive parameter testing by using a neural network based regression model for predicting useful penalty factors and investigate correlations between the minimum spectral gap and the penalty factors, in order to give guidelines on how to set them for the investigated problem classes. The results show, that predicting the best penalty factors in our setting is indeed possible and that there are correlations to the minimal spectral gap between and within the corresponding problem classes. The paper is structured as follows. Section 2 provides background information on quantum annealing, Ising- type Hamiltonians and their eigenspectra. In addition the investigated constrained Hamiltonians are stated. Section 3 talks about related work and previous investigations on constrained Hamiltonians. Section 4 explains the experimental setup, which includes the data generation and applied ML evaluation methods. In Section 5 the results are shown and discussed while inally in Section 6 a summary and future work is given. 2 BACKGROUND 2.1 uantum Annealing Quantum annealing is a meta-heuristic most commonly known for solving optimization and decision problems [19, 21, 28]. While this meta-heuristic can also be simulated classically, it has been implemented in quantum hardware by companies such as D-Wave Systems. Those quantum annealers are designed to minimize a spin glass system, described by an Ising Hamiltonian in the following form: ︁ ︁ (�) (�) ( �) H = ℎ � + � � � (1) � �,� � � � � �>� (�) where � is the Pauli �-matrix operating on qubit �, ℎ is the independent energy or bias of qubit �and � are � � � � the interaction energies or couplings of �qubits and �. Within the fundamental process of quantum annealing, an initial Hamiltonian H with an easy-to-prepare minimal energy coniguration (or ground state) is physically interpolated to a problem Hamiltonian H whose minimal energy coniguration is sought, see Equation (2). The minimal energy coniguration of the problem Hamiltonian corresponds to the best solution of the deined problem. The physical principle on which the D-Wave computation process is based on can be described by a time-dependent Hamiltonian as follows ACM Trans. Quantum Comput. The Efect of Penalty Factors of Constrained Hamiltonians on the Eigenspectrum • 3 min Time Fig. 1. Simplified representation of an eigenspectrum, where the ground state is the blue line at the botom and the excited states are the ones above. The red circle marks the minimum spectral�gap. Figure adapted from [11]. min ! ! ︁ ︁ ︁ A(�) B(�) (�) (�) (�) ( �) H (�) = − � + ℎ � + � � � . (2) � � � �,� � � 2 2 � � �>� | {z } | {z } � H A(�) and B(�) are the anneal functions of D-Wave machines, with A(�) stating the tunneling energy and B(�) being the energy of the problem Hamiltonian at�time in units of Joules. The anneal functions must satisfy B(� = 0) = 0 and A(� = �) = 0, with� being the total evolution time. As the state evolution changes � =fr0om to � = �, the annealing process, describedH by(�) leads to the inal form of the Hamiltonian corresponding to the objective Ising problem that needs to be minimized. Therefore, the ground state of the initial Hamiltonian H (0) = H evolves to the ground state of the problem Hamiltonian H (�) = H . The measurements performed � � at time� deliver low energy states of the Ising Hamiltonian as stated in Equation (1). According to the adiabatic theorem 2], [if this process is executed suiciently slow and smooth � is (i.e., large) and the coherence is preserved long enough, the probability to acquire the ground state of the problem Hamiltonian is close1to ]. 1Ho [ wever, since no real-world computation can run in perfect isolation, the annealing process can sufer from non-adiabatic efects, i.e., thermal luctuations, which can lead the system to jump from the ground state to an excited state. The minimum distance between the ground state and the irst excited state ś the one with the lowest energy apart from the ground state ś throughout any point in the anneal process is called the minimum spectral�gap ofH (�) and is deined as min � = min [� (�) − � (�)] (3) min � 0 0≤�≤�; �≠0 where � (�) is the energy of any excited state and � (�) the energy of the ground state at time �[17]. By computing � 0 all those energy states and their corresponding eigenenergies one can analyze the eigenspectrum of a (relatively small) Hamiltonian and assess its minimum spectral gap, see Fig. 1 for an example eigenspectrum. However, every problem that one can specify has got a diferent Hamiltonian and therefore a diferent corresponding eigenspectrum. According to D-Wave Systems, the most diicult problems in terms of quantum annealing are generally those with the smallest spectral gaps 11].[For completeness, it should be noted that there is an alternative formulation to the Ising spin glass system that is used frequently. The so called quadratic unconstrained binary optimization (QUBO) formulation is mathematically equivalent to the Ising model and replaces each pauli (�) (�) �-operator � with a boolean variable � ; the conversion is as simple as setting � → 2� − 1 [3, 29]. The � � � � ACM Trans. Quantum Comput. Eigenvalues (Energy) 4 • Roch et al. D-Wave Systems annealer is also able to minimize the functional form of the QUBO formulation � �� , with � � ×� � ∈ {0, 1} being a vector of size � of binary variables and � ∈ R being a symmetric � × � real-valued matrix describing the interactions between the variables. Given� matrix , the annealing process tries to ind binary variable assignments � that minimize the objective function. 2.2 Constrained Hamiltonians In this section the six investigated problem Hamiltonians with constraints are stated. The constraints are weighted with the penalty factors in order to ensure valid solutions for the corresponding problem. 2.2.1 Minimum Exact Cover Problem.Within the Minimum Exact Cover Problem (MECP)�a = set{1, ..., �} and subsets � ⊆ � with�= 1, ..., � are given such that� = � . The task is to ind the smallest subset of the set of � � sets � , called�, for which the elements �ofare disjoint sets and the union of the elements � is of � . The QUBO Hamiltonian is stated in [22]. ︁ ︁ ︁ min� � + � 1 − � (4) � � � � �:� ∈� Here, � denotes the elements of� , while �denotes the subsets� . The second term, which represents the constraint, equals 0 if every element�ofis included exactly once, which implies that the�subsets of� are disjoint but their union includes every element � . of With the irst term, representing the objective function, the smallest number of subsets is sought. The eigenvalue of the ground state of this Hamiltonian � ·will �, wher bee � is the smallest number of subsets required for the complete union. The ratio of the penalty�factors and � can be determined by considering the worst case scenario, i.e., a small number of subsets with a single common element and whose union is � . To ensure this does not happen, one can set � · � < �. (5) The number of variables, respectively logical qubits needed for the Hamiltonian, scales linearly with the number of available subsets |� |. 2.2.2 Set Packing Problem. Within the Set Packing Problem (SPP) a�set = {1, ..., �} and subsets � ⊆ � with �= 1, ..., � are given. The diiculty lies in inding the maximum number � ofwhich subsets are all disjoint. Lucas [22] gives the following QUBO Hamiltonian. ︁ ︁ min− � � + � � � (6) � � � � �,�:� ∩� ≠∅ � � The second term is minimized only when all subsets are disjoint, while the irst term simply counts the number of included sets. Choosing the penalty factors � < � (7) ensures that it is never favorable to violate the constraint, represented by the second term. Note, that there will always be a penalty of at least � per extra set included. Just as the MECP, the SPP requir�eslogical qubits. 2.2.3 Minimum Vertex Cover Problem.The Minimum Vertex Cover Problem (MVCP) is deined as inding the minimal set of vertices, which include at least one endpoint of every edge of a graph. Giv � en = (a� ,graph � ) with a set of vertices � = {� , ..., � } and their respective edges �, then � = 1 if� is in the desired minimal set of 0 � � � vertices and� = 0 otherwise. Following [16], the QUBO formulation is given as ︁ ︁ min� � + � 1 − � − � + � � . (8) � � � � � �=0 (�,�) ∈� ACM Trans. Quantum Comput. The Efect of Penalty Factors of Constrained Hamiltonians on the Eigenspectrum • 5 While the irst term represents the objective function, i.e., it counts the number of vertices in the solution, the second term ensures the constraint that every edge in the graph is at least connected to one vertex of the minimal set of vertices in the solution. The number of variables, respectively logical qubits needed, scales linearly with |� |. In order to ensure valid solutions the penalty factors must be chosen accordingly � < �. (9) 2.2.4 Maximum Clique Problem. A clique is a subset of the vertices of a graph that are all connected to each other. The Maximum Clique Problem (MCP) is deined as inding the clique of � =a graph (� , � ) that has the largest number of vertices of all cliques. Following [5], the QUBO formulation is given as ︁ ︁ min− � � + � � � . (10) � � � �=0 ¯ (�,�) ∈� � = 1 if vertex� is included in the clique � and = 0 otherwise�. is the order of the graph and � denotes the set � � � of edges of the graph. While the irst term represents the objective function, i.e., it counts the number of vertices in the solution, the second term ensures the constraint that an edge of the complement set of�edges is not in the solution. The number of variables, respectively logical qubits needed, scales linearly with the number of vertices |� | in the graph. In order to guarantee valid solutions the penalty factors must be chosen accordingly � < �. (11) 2.2.5 Knapsack Problem. In the Knapsack Problem (KP),� items are given, each having a certain weight � and a certain value � . The items must be picked in a way that the total weight of the items is less than or equal to the knapsack capacity� and the sum of the corresponding item values is maximized. The QUBO Hamiltonian is stated in [22]. ! ! 2 2 � � � � ︁ ︁ ︁ ︁ min− � � � + � 1 − � + � �� − � � (12) � � � � � � �=0 �=0 �=0 �=0 Here, � for 1 ≤ � ≤ � is a binary variable, which is set to 1 if the inal weight of the knapsack � and 0 is otherwise. In addition the binary variable � is 1 if item �is part of the solution and 0 otherwise. The second and third terms enforce that the weight can only take exactly one value and that the weight of the items in the knapsack equals the value we claimed it did. The irst term sums the values of the items in the knapsack. The penalty parameters�, � are chosen according to � · sum(� ) < � (13) in order to penalize violations of the weight constraint. Note, that the penalty parameter range for KP given here varies from [22], which was revised in [25]. The number of variables requir � + �ed. is 2.2.6 Binary Integer Linear Problem. Within the Binary Integer Linear Problem (BILP) a v�ector of binary variables � = (� ..., � ) is given. BILP tries to ind the largest value � · �of , for some integer vector �, given the 1 � constraint� · � = �, with� being an� × � matrix and� being a vector with � components. The corresponding QUBO Hamiltonian is given as in [22], � � � ︁ ︁ ︁ min− � � � + � � − � � . (14) � � � �� � �=1 �=1 �=1 While the second term enforces the constraint � · � = �, the irst term maximizes the scalar product of the vectors � and �. When the coeicients� and � are integers, the penalty factors must be chosen accordingly � � � � · � < �. (15) ACM Trans. Quantum Comput. 6 • Roch et al. The number of binary variables, respectively logical qubits needed, scales linearly with the size � of the vectors and �. For more details, see [22]. 3 RELATED WORK Cofey came up with an Adiabatic Quantum Computing (AQC) framework to study the Knapsack Problem 7], in[ which he transformed the optimization problem to an Ising Hamiltonian and used small problem instances to assess the approach. He concludes that numerical and theoretical analysis of the minimum spectral gap in the anneal path is of great importance to improve AQC [7]. Later, Choi theoretically showed that by adjusting the energy penalty value of the Maximum Weighted Independent Set (MWIS) Ising Hamiltonian, one may change the quantum evolution from one that has an anti-crossing to one that does not have one, or vice versa, and therefore signiicantly inluence the minimum spectral gap [6]. Following these insights, Roch et al. proposed a cross entropy (CE) optimization method for adjusting the penalty factors of three constrained Hamiltonians (KP, MECP, SPP) in order to scale their minimum spectral gaps [26]. By doing so, an improved solution quality of inding the global optimum on D-Wave quantum annealers could be achieved, i.e. the probability of measuring the global optimum increased. However, the authors did not observe any correlations between the optimized penalty factors of individual problem instances, meaning that one has to apply the time-consuming CE method for each instance separately. That is the main motivation for us to analyze the energy spectrum, respectively the minimum spectral gap, of diferent constrained problem Hamiltonian classes with machine learning techniques in order to ind patterns and guidelines on how to set the corresponding penalty factors such that an overall improvement in solution quality can be achieved. 4 EXPERIMENTAL SETUP 4.1 Data Preparation In this section we describe the procedure of generating the data and computing the necessary information, i.e., penalty factor ratio, the minimum spectral gap and its location in time w.r.t. the anneal path, as input for the ML analysis methods. As mentioned in Section 2.2, each constrained Hamiltonian has a certain valid half-open interval for its penalty factor � . While we ixed the penalty factor � = 1, the sampling interval of penalty factor � was computed according to the problem-speciic constraint, see Equations (5), (7), ..., (15). Using the MECP as an example, the sampling interval � was for set to [� · � + 0.1, � · � + 5.0], with� being � ·� the number of sets of the problem instance. The MECP penalty factor ratio was calculated.via Note, that we restricted the in general half-open interval � toof [� · � + 0.1, � · � + 5.0]. Although one could theoretically use a larger sampling range, one has to consider that D-Wave Systems’ auto-scaling feature scales every Ising model weight and bias to a given hardware solver dependent range ℎ and of � [12]. This means that the largest � � � value in the Ising model (probably afected by the constraint penalty�factor ) is scaled to the upper bound of the range and everything else proportionally smaller. In conclusion, in order to get good and meaningful solutions, one should set the Ising model coeicients of the constraints large enough to ensure valid solutions, but small enough to maintain the importance of the individual Ising terms. For each problem instance we draw 50 evenly spaced penalty factors�for from the interval to see the rate of change of the minimum spectral gap and its location in the anneal path. The minimum spectral gap of each constraint Hamiltonian with its unique sampled penalty factors was then calculated according to D-Wave Systems’ anneal functions A(�) and B(�) of their Advantage 4.1 System10[]. Note, that we also incorporated D-Wave Systems’ auto-scaling feature during the generation of the data, before computing the minimum spectral gap, so that the experiments match the behaviour of the D-Wave hardware [12]. ACM Trans. Quantum Comput. The Efect of Penalty Factors of Constrained Hamiltonians on the Eigenspectrum • 7 In Figure 2 the corresponding preliminary raw plots of 25 random instances for each problem class are visualized, in order to understand the following experiments in Section 5. However, for analyzing the trends of the minimum spectral gap, its location in the anneal path and the penalty factor ratio with machine learning methods, we generated 1000 unique problem instances per problem class. Note that we restricted the size of the instances to eight variables, respectively logical qubits, since the numerical computation of the eigenspectrum of each instance is quite time consuming and increases with the number of variables. Note, in order to generate such large datasets of Ising problem instances and their minimum spectral gaps for training ML methods, we computed the eigenspectra of the comparatively smaller logical Ising problem instances, which difer to some extent from the hardware embedded ones. We address this topic in Appendix A. Fig. 2. Visualization of the relation of the minimum spectral gap and its location in the anneal path to the penalty factor ratio of 25 random instances per problem class. For each problem instance (represented by one color) 50 evenly spaced penalty factor pairs (represented by the dots) were sampled. Note, that within the plots some trajectories might be overlapping. The x-axis shows the penalty factor ratio, which is calculated problem-specific as stated in Section 4.1. The anneal path and the location of the minimum spectral gap is discretized ranging from 0 to 100, with 0 being the start time of the annealing, as given on the y-axis. The size of the minimum spectral gap is stated on the z-axis. As explained in Section 2.1, the larger the minimum spectral gap the higher the probability to stay in the ground state during the annealing. 4.2 Evaluation Methods 4.2.1 Clustering.To ind patterns and to also determine similarities between and within problem classes in the generated minimum spectral gap trajectories, DBSCAN, a density-based clustering method that identiies arbitrarily shaped clusters of large size and noise in high dimensional databases, 14].was DBSCAN used [typically clusters dense regions of points in the data space that are separated by regions of low density. The two main hyperparameters of DBSCAN are� and MinPts . For a detailed description of the DBSCAN procedure20 se].e In [ ACM Trans. Quantum Comput. 8 • Roch et al. a preprocessing step, the data was normalized in the range of 0 and 1 and the pairwise distance matrix of the three dimensional trajectories (size of the minimum spectral gap, penalty factor ratio and location) was given as input to DBSCAN. We used the Pearson correlation coeicient (PCC) between the pairwise distance matrix and the DBSCAN labeling as the clustering metric. 4.2.2 Regression. For predicting the penalty factors associated with the largest minimum spectral gap for a given problem instance, neural networks (NN) with dense layers, were used. The architecture was kept simple, with no hidden layer and ReLU being the activation function between the input and output layers. For the input layer the set of problem instances was lattened and normalized to a range [−of 1; +1]. Some problem instances difer in size (e.g., diferent amount of sets of diferent sizes, which resulted in difering vector length when lattened). Thus, all problem instances were zero-padded to the largest problem instance that was generated. The number of output neurons of the regression network was set to two, representing the two penalty factors � and �. We selected a stochastic gradient descent method (Adam) for optimizing the weights of the NN. The best results were obtained with a learning rate.001 of 0for MECP, MCP, MVCP, BILP, KP and 0.0005 for SPP. For training, a batch size of 50 was used for all problem classes except SPP, for which the batch size was set to 32. The data set was split into 90% training data set and 10% testing data set. To evaluate the regression model, the Root Mean Square Error (���� ) and the R-squared (� ) were used as performance indicators to tell how well the model can predict the best penalty factors in absolute terms and how well in general it can predict the value of the response variable in percentage terms, respectively. 4.2.3 uantum Annealing HardwareFor . assessing the solution quality of the optimized penalty factors on real hardware, D-Wave’s Advantage 4.1 System was used. The solution quality is associated with the approximation #BKS ratio, which is calculated as follo Appr ws: ox. ratio= with#BKS being the number of times the Best #Measurements Known Solution (BKS) was measured and #Measurements being the total number of measurements (default 100). We used a fully connected graph embedding of size 8 of the D-Wave hardware and mapped the logical Ising problems to it. Since all random generated instances are of the same size (8 variables) each one its into the fully connected hardware embedding. The only diference might be, that couplers of the hardware are not used, if the logical Ising problem is not fully connected. In that case the hardware coupler is set to 0.0. Since previous work26 [ ] showed that the embedding also inluences the solution quality, we reused this same hardware embedding (with the same qubits and couplers) for each problem class to make them comparable regarding the approximation ratio depending on the quality/noise of the hardware qubits. Furthermore, following the D-Wave Systems’ guidelines, the chain strengthparameter was set to max(Ising model coeicient) +in 1 order to avoid broken qubit chains [9]. 5 EVALUATION AND DISCUSSION This section provides insight into our empirical evidence showing that optimizing the penalty factors of Ising model formulations leads to a larger minimum spectral gap and furthermore causes a better approximation ratio on the D-Wave Advantage system. 5.1 Clustering Analysis of Penalty Factor Ratio and Minimum Spectral Gap Since, it is hard to visually ind trends in the 1000 generated problem instances of each problem class, DBSCAN was used to cluster the generated trajectories as described in Section 4. In Figure 3, the DBSCAN clustering results of the minimum spectral gap and its location over the set of penalty factor ratios is plotted. The axis labeling is exactly as in Figure 2. In each subplot the mean trajectory and the 95% conidence interval, as error bars, of each Note, that in omitted experiments we successfully double-checked with random uniform sampled factors in order to ensure that the same results / trajectory trends are obtained. ACM Trans. Quantum Comput. Penalty factor ratio Penalty factor ratio Penalty factor ratio Penalty factor ratio Penalty factor ratio Penalty factor ratio The Efect of Penalty Factors of Constrained Hamiltonians on the Eigenspectrum • 9 found cluster is plotted. In addition the cluster label is annotated in text form. Table 1 shows the associated best found DBSCAN hyperparameters and the achieved Pearson correlation coeicient for each problem class, which was used as metric for the clustering. -1 0.8 2 3.0 0.35 0.7 2.5 0.30 0.6 2.0 0.5 0.25 0.4 1.5 0.20 0.3 1.0 0.15 0.2 -1 0.5 0.10 0.1 0 0.0 0.0 0 0 0 20 2 20 20 -1 40 40 0 40 60 60 60 80 80 80 0.65 0.2 0.2 0.70 0.3 0.3 0.75 0.4 0.4 0.80 0.5 0.5 0.85 0.6 0.6 0.90 100 0.7 100 0.7 100 0.95 0.8 0.8 1.00 0.9 0.9 (a) MECP (b) MVCP (c) MCP 0.08 1.0 0.5 0.07 0.06 0.8 0.4 0.05 0.6 0.3 0.04 0.03 0.4 0.2 0.02 0.1 0.2 0 0 0.01 -1 6 0.0 0.0 0.00 0 0 3 0 20 20 20 -1 40 40 40 -1 60 60 60 0.65 80 0.2 80 80 0.70 0.3 0.65 0.70 0.75 0.4 0.75 0.80 0.5 0.80 0.85 0.6 0.85 0.90 0.7 0.90 0.95 100 0.8 100 0.95 100 1.00 0.9 1.00 (d) KP (e) SPP (f) BILP Fig. 3. DBSCAN clustering results of the minimum spectral gap and its location over a set of penalty factor ratios. The labeling of the clustering is used to color the original minimum spectral gap trajectories and is additionally annotated to each cluster in text form. In order to visualize the 1000 problem instances in each subplot the mean trajector 95% y and the confidence interval, as error bars, of each cluster is shown. In general, the clustering of MVCP, MCP and SPP are similar in that they all exhibit multiple low lying clusters w.r.t the size of the minimum spectral gap (trajectories at the very bottom of each plot) and a location of around 60−90 in the discretized anneal path ranging fr −om 100,0see y-axis. Additionally, they all have a cluster growing in terms of the size of the minimum spectral gap with increasing penalty factor ratio and a comparatively early gap location of − 1020 (cf. Figures 3b, 3c and 3e). This means, that in theory the problem instances of these clusters can be optimized with a certain penalty factor ratio w.r.t the minimal spectral gap size. Moreover, we assume that the early spectral gap of those instances might be favourable, since coherence time of NISQ computers, like D-Wave Systems, is still limited. With an early minimum spectral gap the probability, to jump from the ground to an exited state due to natural quantum decoherence, might be decreased. Note, that for MCP there is one ACM Trans. Quantum Comput. g location min gmin location gmin location g location min g location min g location min gmin gmin min min min min 10 • Roch et al. Table 1. Best found DBSCAN hyperparameters for each problem class with the Pearson correlation coeficient between the pairwise distance matrix of the trajectories and the cluster labels as the metric of the clustering. Problem class � MinPts PCC MCP 1.17 6 -0.962 MVCP 0.34 6 -0.929 KP 0.55 48 -0.994 SPP 0.37 24 -0.889 BILP 0.46 12 -0.919 MECP 0.22 24 0.911 cluster (label −1) of problem instances with a comparatively large minimum spectral gap trajectory. Investigation showed, that those MCP instances represent fully connected graphs/cliques, which is a trivial MCP and result in an omitted constraint respectively penalty factor � in the Ising formulation, see Equation (10). Therefore it can not be optimized with a certain penalty factor ratio and is visualized as a horizontal trajectory in Figure 3c. MECP, KP, and BILP all show diferent clustering. As already seen in the preliminary raw data, MECP has instances with diferent trends in the clustering (cf. Figures 2a and 3a). One cluster −1) with (labelno slope and three clusters with upward (label 0 and 1) and downward (label 2) trends with increasing penalty factor ratio. KP shows two clusters with both having a small upward trend with increasing penalty factor ratio (cf. Figure 3d). Within the BILP clustering, all found clusters remain with the same minimum spectral gap size despite varying penalty factor ratios. Three clusters (lab −1, el7 and 8) have a relatively small minimum spectral gap and a late location (80 − 100) in the anneal path, while the remaining clusters got larger minimum spectral gaps within an early stage (20− 40) in the anneal path (cf. Figure 3f). Investigation showed, that BILP instances don’t difer in the on- and of- diagonal values in the Ising formulation, respeℎctiv andely � values of the Ising model. Therefore � � � no change in the ratio of penalty factors is possible, see Equation (15). 5.2 Regression Analysis of Penalty Factor Ratio and Minimum Spectral Gap In the next step, we investigate whether a neural network is able to predict the penalty factor ratio associated with the largest minimum spectral gap, given the problem instance as input. Figure 4 shows the ��r�esulting � and � coeicients while training our model, including the 95% conidence interval over 10 runs. Trivially, for KP, MVCP, MCP and SPP, the � coeicient reaches around .0981 ± 0.002 to 1.0 ± 0.0 at the end of training, since for these problems the best penalty ratio to choose is the maximal one, i.e., wher �eis factor set to the lowest possible value in our setting. Also ���the � converges to 0.0, which tells us that the distance between the predicted penalty factors made by our regression model and the actual best penalty factors in our setting is very small or null. Interestingly, training a good neural network model for MECP and BILP is more diicult. The best achieved� coeicient was .0918 ± 0.048 and 0.959 ± 0.012, respectively. This also relects in a comparatively worse ���� of 0.091 ± 0.048 and 0.072 ± 0.001. A possible reason for that can be observed in the corresponding clustering, see Figure 3. While KP, MVCP, MCP, SPP all have no or upward trends with increasing penalty factor ratio, MECP correspondingly contains upward and downward trends, which leads to diferent penalty factors being best. We presume that this leads to a slightly worse performance in predicting the best penalty factors in our setting. Regarding BILP no trajectory cluster has a slope, which leads to all penalty factor ratios being equally good w.r.t the size of the minimum spectral gap. We assume that this results in a similar behavior to the MECP for the NN model. ACM Trans. Quantum Comput. The Efect of Penalty Factors of Constrained Hamiltonians on the Eigenspectrum • 11 1.00 KP 0.8 MECP 0.95 SPP 0.6 MCP 0.90 MVCP KP BILP 0.85 0.4 MECP SPP 0.80 MCP 0.2 MVCP 0.75 BILP 0.0 0.70 0 500 1000 1500 2000 0 500 1000 1500 2000 Epoch Epoch (a) (b) Fig. 4. � (a) and ���� (b) coeficients while training our model including95% theconfidence interval over 10 runs, for the six problem types investigated. A large � and a small ���� coeficient are preferred. 5.3 Application of Predicted against Random Penalty Factors on uantum Annealing Hardware Subsequently, we test whether choosing the best penalty factor ratio in our predeined speciic range inluences the approximation ratio on the D-Wave Advantage 4.1 system and whether using a NN for predicting the penalty factors is beneicial compared to choosing them arbitrarily from the valid penalty factor range. For each given problem class a set of 100 random problem instances is generated and a penalty ratio is inferred from the neural network model. Next, the Ising model of those 100 problem instances is formulated (using the predicted penalty ratio) and run on the D-Wave machine. In Figure 5, the resulting approximation ratio of the model is shown as the red line including the 95% conidence interval over the 100 problem instances. Next, an experiment is started: For 50 iterations, a random penalty ratio is sampled from the predeined range and the Ising model is formulated and sent to the D-Wave machine. The running best approximation ratio is kept (green line) with a 95% conidence interval. Note, that even though the approximation ratio was calculated as stated Section 4.2.3, we used the average approximation ratio of the model as baseline and therefore .set 0, while it to 1 the approximation ratio of the randomly sampled penalty factors was set relative to that of the model, in order to better visualize at what point the random sampling process reaches or surpasses the baseline. For MVCP, MCP, and SPP the NN model performs comparatively very well. In the case of MVCP, it takes 43 iterations for the random process to reach the model on average. In the case of SPP it takes about 50 iterations to nearly reach the model while in the case of MCP the random process never reaches the model in the 50 samples that we generated. Of course, these three problems that work well are also the ones for which inference is easy (it consists of predicting the highest penalty value ratio). Regarding the other problem classes (MECP, KP, BILP) it takes on average just 5 iterations for the random process to outperform the model. However, it should be noted that the conidence interval of the model performance in these cases completely overlaps with the approximation ratio of the random process. Possible reasons can be found in the clustering results. Even though the trends of the minimum spectral gap trajectories in the KP clustering are similar to MVCP, MCP and SPP, the performance of the predicted optimal penalty factors is comparatively bad on real hardware (cf. Figure 3 and 5). We believe this is due to the small minimum spectral gap interval of KP problems, which is in.0the andrange 0.05 of (cf.0 Figure 2d). We assume that those small diferences in the minimum spectral gap size of KP Hamiltonians have no ACM Trans. Quantum Comput. RMSE 12 • Roch et al. 1.2 1.2 1.2 1.0 1.0 1.1 0.8 0.8 1.0 0.6 0.6 0.9 0.4 0.4 0.8 0.2 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 Sample steps Sample steps Sample steps (a) MECP (b) MVCP (c) MCP 1.2 2.0 1.2 1.1 1.5 1.0 1.0 1.0 0.8 0.9 0.5 0.6 0.8 0.0 0.4 0 10 20 30 40 50 0 10 20 30 40 50 0 10 20 30 40 50 Sample steps Sample steps Sample steps (d) KP (e) SPP (f) BILP Fig. 5. Iterative process of randomly sampling penalty factors (for 50 iterations) and keeping the rolling best (green) versus the model (red) prediction, including the 95% confidence interval over 100 problem instances. Note, that the average approximation ratio of the model was used as baseline and set1.to 0, while the approximation ratio of the randomly sampled penalty factors was set relative to that of the model. efect when executing it on D-Wave’s hardware. Regarding MECP, a similar problem can be observed. Although the minimum spectral gap interval is by a factor of 10 larger than the one of KP (cf. Figure 2a), the diference of the gap size between the worst and best penalty factor ratio of a problem instance is very small (low slope), which leads to the same assumption as for KP. Since in the BILP clustering no slope is present, no optimization can be achieved here in theory, but also practically on real hardware. Since in this experiment the running best approximation ratio is kept for the random process, we now compare the average performance of the NN model prediction against the average of all 50 random samples from the iterative process of the 100 problem instances per problem class, as presented in the previous paragraph. As metric the approximation ratio is used again. Table 2 shows the corresponding results. It is clear that the model on average outperforms at least 12 .9% ± 7.7% and up to 167.1% ± 29.8% against the random process. Thus, from this perspective, the model should always be used for the six problems we analyzed here. 5.4 Correlation of Trajectory Clusters to the Number of Global Optima and to the Sparseness of the Problem Instances Furthermore, an interesting observation in terms of the number of optimal solutions could be made. Using the DBSCAN clustering that we found previously and grouping the clusters with no slope around the gap.0size of 0 ACM Trans. Quantum Comput. Random to inference ratio Random to inference ratio Random to inference ratio Random to inference ratio Random to inference ratio Random to inference ratio The Efect of Penalty Factors of Constrained Hamiltonians on the Eigenspectrum • 13 Table 2. Average superiority of the model against the Table 3. Correlation results of the merged DBSCAN trajectory average of all 50 random samples from the iterative sam- clusters, having no slope and being located around the gap size pling process over the 100 problem instances per problem of0.0, of Figure 3 to the problem instances having either exactly class in terms of approximation ratio. one or multiple optimal solutions. The Pearson correlation coeficient is shown. The numbers in the braces of the last table column represent the corresponding annotated cluster Avg. superiority of the model labels of Figure 3. Problem class vs the random sampling process MCP 167.1% ± 29.8% Problem class PCC Merged clusters MVCP 97.4% ± 26.2% MCP 1.0 {1, -1}, {0} KP 82.8% ± 55.7% MVCP 0.990 {1}, {0, 1, 2} SPP 87.3% ± 32.9% KP 0.419 {0}, {-1} BILP 12.9% ± 7.7% SPP 0.972 {0}, {1, -1} MECP 18.4% ± 11.2% BILP -0.958 {0, 1, 2, 4, 5 ,6}, {-1, 3, 8, 7} MECP -0.799 {0, 1, 2}, {-1} (cf. Figure 3) and correlating it to each problem instance having either exactly one optimal solution or multiple, the Pearson correlation coeicient shows in general very high correlations. As seen in Table 3 MCP, MVCP, SPP, and BILP all have extremely strong (either positive or negative) correlations, with MECP still having a rather highly negative correlation. Only KP is an exception, with a comparatively lower correlation with the clustering. It is quite interesting that having exactly one optimal solution leads to minimum spectral gap trajectories that can be optimized well, while having two or more optimal solutions results in a lat minimum spectral gap trajectory with a gap size of around .0.0This is the case for all problems with high correlations. It should be noted that correlating the clustering with the exact number of optimal solutions (e.g., with exactly 3 optimal solutions) reduces the overall Pearson correlation coeicient for all problems. Which in turn shows that the exact number of optimal solutions is comparatively rather insigniicant for the correlation. In addition we investigated the sparseness of the problem instances, in order to check if the trajectory clusters with their diferent trends also correlate with the sparseness of the problems. The sparseness of an problem #�������� −#��������������� instance (Ising matrix) was calculate�d��as������= � . Even though sparse problems #�������� are easier to embed into the sparse connectivity of D-Wave hardware and should therefore be easier to solve, since smaller (embedded) problems are produced, the minimum spectral gap trajectories and their trends did not correlate with the sparseness of the problems. The Pearson correlation coeicient was around 0 for each problem class. In general each cluster (with upwards, downwards, and also the lat trends) of the problem classes contained problem instances with diferent sparseness. 5.5 Correlation of Scaled Problem Instances to the Minimum Spectral Gap Lastly, we investigate the minimum spectral gap trajectories when scaling the problem instances up and down in terms of variables. Since for all previous experiments the problem instances were restricted to 8 variables (due to the time consuming computation of the eigenspectra), we now look into instances with 6, 8, and 10 variables for only the comparatively well performing problem classes MVCP, MCP and SPP. Note, that for problem instances of size 6 and 8, we generated 1000 instances per problem class and for size 10 only 500 instances per problem class. Figure 6 shows the composed DBSCAN clustering of problem instances of diferent size. Clusters of same problem size are identically coloured. The achieved Pearson correlation coeicient (with optimized DBSCAN ACM Trans. Quantum Comput. Penalty factor ratio Penalty factor ratio Penalty factor ratio 14 • Roch et al. hyperparameters) was −0.900, −0.992 and −0.901 for MVCP, MCP and SPP, respectively. One can clearly see, that the same trajectory trends occur regardless of the problem size. We therefore assume that the observed trends are size independent and will also hold for other problem sizes. 0.8 0.8 3.0 0.7 0.7 2.5 0.6 0.6 2.0 0.5 0.5 0.4 0.4 1.5 0.3 0.3 1.0 0.2 0.2 0.5 0.1 0.1 0.0 0.0 0.0 0 0 0 20 20 20 40 40 40 60 60 60 80 80 80 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.6 0.5 0.6 0.5 0.6 0.7 0.7 0.7 0.8 100 0.8 100 0.8 100 0.9 0.9 0.9 (a) MVCP (b) SPP (c) MCP Fig. 6. DBSCAN clustering of the minimum spectral gap and its location over a set of penalty factor ratios. The problem instances of each problem class difer in size of 6 (blue), 8 (red) and 10 (yellow) variables. In order to visualize the 2500 problem instances in each subplot the mean and the 95% confidence interval, as error bars, of each cluster is ploted. 6 CONCLUSION In this work we analyzed the efect of the penalty factors on the minimum spectral gap of diferent constrained Hamiltonians. We showed that speciic penalty factor ratios can enlarge the minimum spectral gap, which in turn is relected by an improved approximation ratio on real quantum hardware. We were able to train regression models to predict the most suitable penalty factors in our setting, which on average performed.at 9%least ±7.7% 12 and up to 167.1% ± 29.8% better than the ones of the random process. This leads us to the conclusion that such learned models should always be used for the investigated problem classes in order to reach a better solution quality for the corresponding Hamiltonians. An additional inding was the high Pearson correlation of the number of optimal solutions with the clusters that were found with DBSCAN. The results showed that problem instances with exactly one optimal solution can be optimized well using a itting penalty factor ratio, while the ones with multiple equivalent optimal solutions cannot. Besides that, the ones which can be optimized well tend to have their minimum spectral gap early in the anneal path, compared to the ones, which cannot be optimized. Since in this work we restricted ourselves to constrained Hamiltonians with only one constraint, optimization problems with multiple constraints (as they appear in the real world more commonly) are of interest for future work. Moreover, w.r.t the step of data generation, eicient methods for pre-selecting promising penalty factor ratios need to be investigated, since the possible penalty factor ratio combinations increase (in the worst case) exponentially with the number of factors. Another interesting aspect for future work would also be to investigate the qubit precision capabilities of D-Wave Systems quantum annealing hardware from a practical point of view. Since the experiments showed that for some problem classes (MVCP, MCP, SPP) the best penalty factor ratio was the largest one, i.e., where the factor � was set to the minimal possible value in our setting, it would be interesting to see the rate of change in solution quality before quantization error efects of the digital analog converts occur [10], and no improvement is possible anymore. ACM Trans. Quantum Comput. g location min g location min gmin location gmin min gmin Penalty factor ratio Penalty factor ratio Penalty factor ratio The Efect of Penalty Factors of Constrained Hamiltonians on the Eigenspectrum • 15 A COMPARISON OF MINIMUM SPECTRAL GAPS OF LOGICAL AND EMBEDDED INSTANCES Since we computed the minimum spectral gaps of the non-embedded Ising problems, due to computational limitations (embedded problems might require more variables/qubits), we analysed, if the eigenspectra difer from each other on a small subset of problem instances with 6 logical variables for the SPP, MCP and MVCP (the problem classes which worked well in our evaluation). As already mentioned in Section 4.1 the D-Wave scaling function, in order to scale the logical Ising coeicients to the precision range of the hardware qubits and the D-Wave Anneal functions of the corresponding hardware were used. Thus, the only diference between the logical Ising problems and the embedded ones are the physical qubit chains, which occur when a logical problem is not directly embeddable in hardware. In order to analyse the diference of the logical and embedded instances we read out the actual embedding of the D-Wave hardware graph. The 6 logical variables of the Ising problem instances lead to 8 physical qubits on the hardware. Afterwards the minimum spectral gaps (for diferent penalty factors) of the now 8 variable Ising problems were computed with the same method as for the logical Ising problems. In Figure 7 the corresponding minimum spectral gap trajectories are plotted for 25 instances of each of the three problem classes. The results show that there is a diference in the overall size of the MSG, however the trends of the trajectories seem to stay the same as with the original (not embedded) Ising problem instances of Figure 2. We assume that the qubit chains (at least for such small problem sizes) should not alter the distribution of eigenvalues of the corresponding eigenspectrum. 0.7 0.5 0.5 0.6 0.4 0.4 0.5 0.3 0.3 0.4 0.3 0.2 0.2 0.2 0.1 0.1 0.1 0.0 0.0 0.0 0 0 0 20 20 20 40 40 40 60 60 60 0.2 80 0.2 80 0.2 80 0.3 0.3 0.3 0.4 0.4 0.4 0.5 0.5 0.5 0.6 0.6 0.6 0.7 0.7 0.7 0.8 100 0.8 100 0.8 100 0.9 0.9 0.9 (a) MVCP (b) SPP (c) MCP Fig. 7. Visualization of the relation of the minimum spectral gaps and its location in the anneal path to the penalty factor ratio of 25 random instances per problem class. For each simulated embedded problem instance (represented by one color) 50 evenly spaced penalty factor pairs (represented by the dots) were sampled. Note, that within the plots some trajectories might be overlapping. ACKNOWLEDGMENTS This work was funded by the German BMWK Project PlanQK (01MK20005I). ACM Trans. Quantum Comput. g location min g location min gmin location min gmin min 16 • Roch et al. 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Journal
ACM Transactions on Quantum Computing – Association for Computing Machinery
Published: Mar 2, 2023
Keywords: Quantum annealing