Abstract
Quantum annealing is a proposed combinatorial optimization technique meant to exploit quantum mechanical effects such as tunneling and entanglement. Realworld quantum annealingbased solvers require a combination of annealing and classical pre and postprocessing; at this early stage, little is known about how to partition and optimize the processing. This article presents an experimental case study of quantum annealing and some of the factors involved in realworld solvers, using a 504qubit DWave Two machine and the graph isomorphism problem. To illustrate the role of classical preprocessing, a compact Hamiltonian is presented that enables a reduced Ising model for each problem instance. On random Nvertex graphs, the median number of variables is reduced from N^{2} to fewer than N log_{2} N and solvable graph sizes increase from N = 5 to N = 13. Additionally, error correction via classical postprocessing majority voting is evaluated. While the solution times are not competitive with classical approaches to graph isomorphism, the enhanced solver ultimately classified correctly every problem that was mapped to the processor and demonstrated clear advantages over the baseline approach. The results shed some light on the nature of realworld quantum annealing and the associated hybrid classicalquantum solvers.
Introduction
Quantum annealing (QA) is a proposed combinatorial optimization technique meant to exploit quantum mechanical effects such as tunneling and entanglement^{1}. Machines purportedly implementing a type of quantum annealing have recently become available^{2}. While the extent of “quantumness” in these implementations is not fully understood, some evidence for quantum mechanical effects playing a useful role in the processing has been appearing^{3,4,5}. Aside from the debate over quantumness, there are interesting questions regarding how to effectively solve a realworld problem using a quantum annealer. Quantum annealingbased solvers require a combination of annealing and classical pre and postprocessing; at this early stage, little is known about how to partition and optimize the processing. For instance, current quantum annealers have severe practical limitations on the size of problems that can be handled. Can the preprocessing algorithms be modified in order to improve scalability? A second question involves postprocessing. Quantum annealers provide solutions to an “embedded” version of a problem involving physical qubits. Postprocessing is generally required for translating these to solutions to the original problem involving logical qubits (aka variables). Occasionally, a chain of physical qubits representing a single variable resolves to an inconsistent state, a scenario known as a broken chain. Studies are needed regarding broken chains and the possibility of classical error correction during postprocessing.
This article presents an experimental case study of quantum annealing and some of the factors involved in realworld solvers, using a 504qubit DWave Two machine. An example of parsimonious preprocessing is considered, along with postprocessing majority voting. Through experiments on a 504qubit DWave Two machine, we quantify the QA success probabilities and the impact of the methods under study. We use the graph isomorphism (GI) problem as the problem of focus. The GI problem is to determine whether two input graphs G_{1,2} are in essence the same, such that the adjacency matrices can be made identical with a relabeling of vertices. This problem is an interesting candidate for several reasons. First, an accurate quantum annealingbased solver for GI has never been implemented. Second, quantum approaches can sometimes provide new insight into the structure of a problem, even if no speedup over classical approaches is achieved or even expected. Third, the GI problem is mathematically interesting; though many subclasses of the problem can be solved in polynomial time by specialized classical solvers, the run time of the best general solution is exponential and has remained at since 1983^{6,7}. The classical computational complexity of the problem is currently considered to be NPintermediate^{8} and the quantum computational complexity of the problem is unknown. Graph isomorphism is a nonabelian hidden subgroup problem and is not known to be easy in the quantum regime^{9,10}. Lastly, the GI problem is of practical interest. It appears in fields such as very largescale integrated circuit design, where a circuit’s layout graph must be verified to be equivalent to its schematic graph^{11} and in drug discovery and bioinformatics, where a graph representing a molecular compound must be compared to an entire database, often via a GI tool that performs canonical labeling^{6}.
This article relates to previous works as follows. A preprint by King and McGeoch discusses tuning of quantum annealing algorithms, including the use of lowcost classical postprocessing majority voting similar to what is evaluated in this article^{12}. Our study goes further regarding preprocessing (designing a Hamiltonian to generate compact Ising models) and covers graph isomorphism rather than problems such as random notallequal 3SAT. A work by Rieffel et al. maps realworld problems such as graph coloring to a DWave quantum annealer^{13}. Regarding the graph isomorphism problem in particular, multiple attempts have been made using adiabatic quantum annealing. One of the first attempts assigned a Hamiltonian to each graph and conjectured that measurements taken during each adiabatic evolution could be used to distinguish nonisomorphic pairs^{14}. A subsequent experimental study using a DWave quantum annealer found that using quantum spectra in this manner was not sufficient to distinguish nonisomorphic pairs^{15}. A second approach converted a GI problem to a combinatorial optimization problem whose nonnegative cost function has a minimum of zero only for an isomorphic pair. The approach required problem variables and additional ancillary variables. It was numerically simulated up to N = 7 but not validated on a quantum annealing processor^{16}. An alternative GI Hamiltonian was proposed by Lucas^{17}.
Proposed Solver
Preliminaries
Before proceeding, we briefly define some key concepts. In quantum annealing, a system is first initialized to a superposition of all possible states as dictated by Hamiltonian H_{init}, then slowly evolved such that the initializing function becomes weaker and the energy function of interest, defined by the problem Hamiltonian H_{P}, becomes dominant. The timedependent combination of the two Hamiltonians is referred to as the system Hamiltonian H(t)^{18}:
where A(t) and B(t) are monotonic functions representing the annealing schedule up to time T. In this work, we focus on problem Hamiltonians H_{P}. The problem Hamiltonian represents an Ising model, where each spin variable s_{i} ∈ {−1, 1} is subject to a local field h_{i} and where variables interact pairwise with coupling strength J_{ij}^{19}:
Problems can alternatively be represented as quadratic unconstrained binary optimization (QUBO) problems, with binary variables x_{i} ∈ {0, 1} instead of spin variables and with the following cost function^{1}:
In some cases, a QUBO representation is a more convenient starting point. A QUBO problem can be converted to an Ising problem using the relation .
It is often the case that an Ising problem [h, J] involves variable interactions not supported by a quantum annealing processor graph. An example is a variable with degree higher than the maximum supported by the DWave Chimera architecture (six). One common strategy is to find a minor embedding of the problem graph in the processor graph^{19}, in which a set of physical qubits is used to represent each variable. Each qubit is strongly coupled to at least one other qubit in the set, in an effort to keep the entire set in a consistent state. These sets are commonly referred to as chains. We refer to a minorembedded version of a problem as an embedded Ising problem [h′, J′].
Baseline Hamiltonian
We first describe a baseline penalty Hamiltonian for the GI problem, building upon the Hamiltonian described in Lucas^{17}. The problem input is a pair of simple, undirected Nvertex graphs G_{1,2}. Penalties are applied such that the ground state energy is zero if the pair is isomorphic and greater than zero otherwise. The intent is for an energy minimization process (such as quantum annealing) to provide a solution to this decision problem.
The baseline Hamiltonian includes binary variables x_{u,i} ∈ {0, 1} for every possible mapping of a vertex u in G_{2} to a vertex i in G_{1}; in a solution, the variable is 1 if u is mapped to i and 0 otherwise. There are two types of penalties that can be applied to a vertex set mapping. One type penalizes mappings that are not bijective. For instance, a mapping in which vertices 1 and 2 in G_{2} are both mapped to vertex 1 in G_{1} is invalid; the term x_{1,1}x_{2,1} is assigned a positive penalty C_{1} such that if x_{1,1} and x_{2,1} both resolve to 1, the solution energy will necessarily be greater than zero. The second penalty type involves edge inconsistencies. An example of an edge inconsistency is when vertices u,v are mapped to i,j and an edge exists in one input graph and not the other (e.g. uv ∈ E_{2} while ij ∉ E_{1}). Edge inconsistencies are assigned a positive penalty C_{2}. Additional details regarding this style of Hamiltonian can be found in Lucas^{17}.
In Lucas^{17}, couplings can incur zero, one, or two penalties. Here, we require that couplings be penalized no more than once, in order to achieve a simple set of coupler values (e.g. {0, 1} instead of {0, 1, 2}) amenable to quantum annealing. Specifically, an edgerelated penalty is applied to a coupling only if there is not a vertex mapping penalty. As an example, if vertices u,v are mapped to i,j and i = j, then the bijection has been violated and coupling x_{u,i}x_{v,j} will incur a vertex mapping penalty (C_{1}); therefore, an additional edge inconsistency penalty (C_{2}) need not be applied. Edge inconsistency penalties are only applied if i ≠ j and u ≠ v. The set of required coupler values can be further simplified by setting C_{1} = C_{2} > 0. The complete Hamiltonian is
where the i ≠ j and u ≠ v conditions represent the main modifications to Lucas^{17}.
Compact Hamiltonian
The approach embodied in the baseline Hamiltonian H_{1} suffers from a severe lack of scalability. For Nvertex input graphs, it requires N^{2} logical variables. Moreover, due to the limited direct connections between qubits in the DWave Chimera architecture, problems are often given a minor embedding into the processor working graph. This typically involves replicating variables across multiple qubits. Thus, the qubit requirements can reach O(N^{4}). Problems mapped in this way to a ~500qubit processor tend to be limited to N = 5 or 6. We now investigate whether a more effective Hamiltonian can be designed. The idea is that many variables and interactions are unnecessary and information indicating so can be leveraged up front during the requisite preprocessing. Note that an isomorphic mapping requires the vertices in each matched pair to have the same degree. Thus, degree information can be used to decide whether two vertices are eligible to be matched. We propose a compact Hamiltonian H_{2} that avoids creating variables for vertices of different degree. A second, minor simplification deals with isolated vertices (degree = 0). If G_{1,2} each have k isolated vertices, an isomorphic mapping of such vertices is trivial and thus no variables or penalties for those vertices need be modeled. If G_{1,2} have a different number of isolated vertices, then they also have a different number of nonisolated vertices and existing variables and penalties for those will suffice. Thus we only create variables and penalties for vertices with degree greater than zero.
Given the two enhancements, the total number of variables required is where is the multiplicity of the set d_{i} containing vertices of degree i. In the worst case of regular graphs of degree r greater than zero, all nodes have the same nonzero degree and thus the simplifications provide no benefit—N^{2} variables are still required. Many realworld graphs are not regular and for these the benefits can be large. The entire compact Hamiltonian H_{2} is
Figure 1 shows an example of problem instances generated using the baseline and the compact Hamiltonians on the same input; it illustrates how the number of variables and the number of nonzero interactions—as seen in the offdiagonal entries in the associated QUBO matrix Q—can be much smaller when using the compact Hamiltonian. The variable scaling of each Hamiltonian is quantified and compared in Results.
H_{2} was validated using a software solver (DWave System’s isingheuristic version 1.5.2) that provides exact results for problems with low tree width. In exhaustive testing of all 2^{12} N = 4 pairs and 2^{20} N = 5 pairs, the ground state energy of H_{2} was confirmed to be zero for isomorphic cases and greater than zero for nonisomorphic.
As a part of a quantum annealingbased solver, an algorithm can be employed that accepts a graph pair as input and uses the Hamiltonian to generate an associated QUBO problem (later converted to an Ising problem). The proposed algorithm using the compact Hamiltonian H_{2} is presented (in pseudo code) in Algorithm 1.
Algorithm 1  Highlevel algorithm for generating a QUBO problem using the proposed compact Hamiltonian H_{2}.
Complete Solver Flow
The highlevel flow of the proposed GI solver is shown in Fig. 2. The problem input is a graph pair G_{1,2}. In this article, the graph types considered are random, simple, undirected graphs. Graphs are generated using the ErdősRényi model^{20} G(N, p) where we set the probability of an edge being present p = 0.5. An advantage of G(N, 0.5) graphs for an initial study is that all graphs are equally likely. This type has been used in classical graph isomorphism work as well^{6}. In step 1, the input graphs and the Hamiltonian formulation of interest (e.g. H_{1} or H_{2}) are used to generate a QUBO problem which is then converted to an Ising problem [h, J]. An example of an algorithm for generating the QUBO problem is shown in Algorithm 1. The Ising problem is then compiled to a specific quantum annealing processor in step 2. A main task is to find sets of physical qubits to represent the problem variables (aka logical qubits); this is achieved by providing the J matrix and the processor working graph to the DWave findEmbedding() heuristic^{21}. Subsequently, the parameters of the embedded Ising problem [h′, J′] are set following certain strategies such as the use of a random spin gauge (see Methods). The embedded Ising problem, sometimes referred to as a machine instruction, is submitted to the quantum annealing machine along with several job parameters. The quantum annealing job is executed in step 3 and solutions are returned in the form of strings of twovalued variables. These solutions and energies are associated with the embedded problem, not the original Ising problem. Therefore a postprocessing step is necessary (step 4), in which the state of each qubit chain is plugged into the cost function of the Ising problem. A difficulty arises when the states of the qubits in a chain are inconsistent, a case referred to as a broken chain. In the proposed solver, broken chains can be handled by either discarding the associated solution, or by performing majority voting over each chain. The two strategies are compared empirically in Results. Given a solution to the original Ising problem, the solution energy can be calculated. If the lowest energy is zero, then the input pair can be declared isomorphic and no further jobs are necessary. Otherwise, a decision must be made whether to repeat the process from step 2 or to stop and declare that isomorphism could not be established.
Results
Ising Model Scaling
To compare the resource requirements of the two proposed Hamiltonians, 100 pairs of graphs are used as inputs to Step 1 of the solver flow (Fig. 2), where 50 pairs are isomorphic and 50 are nonisomorphic for each size up to N = 100. Since H_{1} models a variable for each possible vertex pair, N^{2} variables are required by definition. Ising problems generated using H_{2} are found to use fewer variables than H_{1}; scaling of the median problems fits to 0.748 N^{1.45}. Incidentally, this indicates that most problems have fewer variables than with the Gaitan et al. approach, which entails plus ancillary variables^{16}. The variable scaling is illustrated in Fig. 3. In addition to the number of variables, a second resource metric is the number of nonzero interactions between variables; dense interactions make the minor embedding problem more difficult. We find that the scaling of variable interactions has been improved from O(N^{4}) for H_{1} to O(N^{2.9}) for H_{2} (where R^{2} = 0.9991).
Embeddability
Next, we compare the embeddability of the two approaches, in other words the extent to which Ising problems can be minorembedded in a given processor graph. The processor of choice is the DWave Two Vesuvius6 processor housed at USC ISI. At the time of this writing, the working graph contains 504 qubits and 1427 couplers. Embedding is attempted using the DWave findEmbedding() heuristic^{21} with default parameter values such as 10 “tries” per function call. As shown in Fig. 4a, embeddings are found for the majority of problems only for sizes N ≤ 6 when using H_{1}, but sizes N ≤ 14 with H_{2} (Fig. 4a). The median number of qubits across all problems scales as O(N^{4.22}) for H_{1} and has been reduced to O(N^{3.29}) for H_{2} (Fig. 4b).
Experimental Quantum Annealing for Graph Isomorphism
The accuracy of the solver described in the previous section was measured via trials conducted on a DWave Two Vesuvius quantum annealing processor. Several alternative strategies were compared—the use of Hamiltonians H_{1} vs. H_{2}, running a single job per problem vs. multiple jobs and the use of chain majority voting during postprocessing. Note that by construction of the Ising models using a penalty Hamiltonian, problems with nonisomorphic input graphs cannot achieve a zero energy state, regardless of annealing results. The main challenge for the solver is to find the zero energy state for isomorphic pairs. Thus, we first focus on the isomorphic case. One hundred isomorphic pairs were input into the solver for each size N from 3 to 20.
For one strategy in particular the zero energy state was always eventually achieved—the use of Hamiltonian H_{2} combined with multiple jobs and chain majority voting. Thus, with this strategy there were no false negatives and classification accuracy reached 100% of the embeddable problems, as shown in Table 1. For the most difficult problem, the zero energy state was achieved on the 9^{th} job. All other strategies incurred false negatives. For the successful strategy, the expected total annealing time was calculated (as described in Methods). Results are shown in Fig. 5.
For completeness, nonisomorphic pairs were run as well, using Hamiltonian H_{2} and chain majority voting. Since in the worst case nine jobs were required to correctly classify the isomorphic pairs above, nine jobs were submitted for each nonisomorphic problem. One hundred nonisomorphic G(N, 0.5) problems were tested at each size between N = 3 to 14; of the 1200 problems, embeddings were found for 1186. In addition, pairs of isospectral nonisomorphic graphs (PINGs) were tested. All N = 5 PINGs were tested (150 permutations), as well as 100 random N = 6 PINGs. As expected, none of the nonisomorphic problems achieved a zero energy state and thus none were classified as isomorphic. In other words, there were no false positives.
Discussion
Several observations can be made from this case study. First, the formulation of the cost function (Hamiltonian) can have a noticeable impact on quantum annealing results. For the graph isomorphism problem, the baseline approach (embodied in Hamiltonian H_{1} and in [Lucas]^{17}) blindly creates QUBO variables for every possible vertex pair, whereas the proposed Hamiltonian H_{2} is more parsimonious. Variable requirements decreased from N^{2} to fewer than N log_{2} N (Fig. 3) on the graph type under study, allowing larger problems to be solved (Fig. 4 and Table 1). Along with Rieffel^{13}, this is one of the first quantum annealing studies to experimentally quantify the effect of alternative Hamiltonian formulations. One of the impacts of this observation is increased appreciation for the fact that all quantum annealingbased solvers are actually classicalquantum hybrids and that focus must be placed on effectively partitioning the processing and optimizing the classical portion. A caveat is in order—if the classical side is made to do too much work then the quantum annealing aspect becomes trivial and of little value. Further work is needed in identifying the specific strengths of annealing processors and in leveraging the two sides appropriately.
A second observation is that using chain majority voting during postprocessing can in some cases provide a benefit. Previously, such majority voting was evaluated for a different set of problems (scheduling) and was not found to provide a significant benefit^{13}. In our context, there were many problems for which the zero energy ground state solution was only achieved when using this postprocessing; without this form of error correction (in other words, when all solutions containing a broken chain were discarded), false negatives occurred. For instance, at N = 12, 53 of 83 embedded problems were solved on the first job without using chain majority voting and an additional 12 problems were solved by applying chain majority voting (Table 1). Classical error correction strategies other than majority voting should be explored and assessed in future studies and their costs quantified.
To our knowledge, the evaluated solver is the first validated, experimental implementation of a QAbased graph isomorphism solver. While it ultimately classified every embeddable problem correctly and demonstrated clear advantages over the baseline approach, it has serious limitations as a graph isomorphism solver. The problem sizes are not competitive with those handled by classical solvers, which can handle G(N, 0.5) graphs with thousands of vertices^{6} and even for the hardest graph types can handle hundreds of vertices before running into difficulty^{22}. Similarly, the scaling of the total annealing times (Fig. 5) is not competitive with classical scaling^{6}. Ultimately, new approaches are likely needed if quantum annealing is to contribute to graph isomorphism theory or practice. Fortunately, the case study provides some new insight into experimental quantum annealing and contributes methods that have relevance beyond the GI problem. It is hoped that the experimental evaluation of alternative Hamiltonian formulations adds to the understanding of the factors affecting quantum annealing performance and that the demonstration of majority voting raises new questions about the role of postprocessing for a variety of problems.
Methods
Quantum annealing experiments were performed on the DWave Two machine housed at USC ISI and operated by the USCLockheed Martin Quantum Computing Center. Experiments were conducted in October and November, 2014. The working graph of the machine’s Vesuvius6 quantum annealing processor consisted of 504 qubits and 1427 couplers during this period. The pattern of working qubits is shown in Figure 6. The qubit temperature was estimated by the manufacturer to be 16 ± 1 mK. Additional processor specifications include a maximum antiferromagnetic mutual inductance of 1.33 and amplitude of 7.5 ± 1 .
Simple undirected Nvertex graphs were constructed according to the ErdősRényi G(n, p) model^{20} with n = N and with the probability p of including each edge equal to 0.5. Nonisomorphic pairs were generated by creating two graphs as above and checking for nonisomorphism using the MATLAB graphisomorphism() function. Isomorphic pairs were generated by generating a single graph then applying a random permutation to arrive at the second graph. For each pair of input graphs, an Ising model was created using equation (4) or (5). Programming was performed using MATLAB R2014a win64 and the DWave MATLAB pack 1.5.2beta2. The current version of the DWave sapiFindEmbedding() function cannot directly embed Ising models with more than one connected component (i.e. a set of variables that interact only with each other and not any of the remaining variables); therefore, models with this characteristic were not included in the input data. When attempting to generate 100 input pairs for each size, such disconnected models occurred no more than 4 times for each size N ≥ 14. Similarly, the heuristic cannot accept models with fewer than two variables, so in the rare case of a trivial Ising problem with fewer than two variables (e.g. a nonisomorphic pair with no matching degrees), dummy variables were added to the problem.
The h_{i} values of the Ising problem were split evenly across each qubit in the associated chain in the embedded Ising problem. The J_{ij} values of the Ising problem were assigned to a single coupler connecting two variable chains in the embedded problem. The magnitudes of the embedded h′ and J′ were scaled together such that the maximum magnitude reached 20% of the full range supported by the processor; the range of the embedded h_{i}′ values was [−0.4, 0.4] and the range of the embedded J_{ij}′ values coupling different variables was [−0.2, 0.2]. This 20% value was determined empirically to provide good performance on the median difficulty problem at the largest sizes. Subsequently, the J_{ij}′ values connecting physical qubits within a chain were set to the maximum ferromagnetic value (−1). A single random spin gauge transformation^{2} was then applied to each embedded problem, with a gauge factor a_{i} ∈ {−1, 1} associated with each qubit and transformation h_{i}′ → a_{i} h_{i}′; J_{ij}′ → a_{i} a_{j} J_{ij}′. One job was submitted to the quantum annealer per embedded problem; some Ising problems were associated with multiple embedded problems and jobs. After each programming cycle, the processor was allowed to thermalize for 10 ms (the maximum supported by the machine). The annealing time was set to the minimum value of 20 μs. The number of annealing and readout cycles per programming cycle was 40000, which allowed the total job time to be within the limits of the machine (1 s). The readout thermalisation time was set to the default value of 0. Regarding error correction through majority voting of chains of physical qubits, ties were broken by choosing the spin up state. The probability of achieving the zero energy state on job k is denoted
When multiple jobs are required, we calculate the geometric mean in the style of Boixo et al.^{2}:
The total annealing time required to reach 0.99 probability of success was calculated by multiplying the annealing time by the expected number of annealing cycles (repetitions R) using the formula^{2}:
Additional Information
How to cite this article: Zick, K. M. et al. Experimental quantum annealing: case study involving the graph isomorphism problem. Sci. Rep. 5, 11168; doi: 10.1038/srep11168 (2015).
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Acknowledgements
We would like to thank Itay Hen for helpful discussions and suggestions and Federico Spedalieri for feedback on an early version of the manuscript. O.S. would like to thank Professor Samuel J. Lomonaco, Jr. for insight into the graph isomorphism problem. This material is based in part upon work supported by the Defense Advanced Research Projects Agency (DARPA) under Contract No. HR00111C0041. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government. Distribution Statement “A” (Approved for Public Release, Distribution Unlimited).
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K.Z. and O.S. conceived the research; O.S. performed mathematical modeling and developed an algorithm for generating an Ising model; K.Z. designed and implemented the solvers, conducted experiments and wrote the manuscript text; M.F. and K.Z. supervised the research; M.F. reviewed the manuscript and contributed revisions.
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Zick, K., Shehab, O. & French, M. Experimental quantum annealing: case study involving the graph isomorphism problem. Sci Rep 5, 11168 (2015). https://doi.org/10.1038/srep11168
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