Discriminating Non-Isomorphic Graphs with an Experimental Quantum Annealer Zoe Gonzalez Izquierdo,1, 2, ∗ Ruilin Zhou,3 Klas Markstr¨om,4 and Itay Hen1, 2 1Department of Physics and Astronomy, and Center for Quantum Information Science & Technology, University of Southern California, Los Angeles, California 90089, USA 2Information Sciences Institute, University of Southern California, Marina del Rey, California 90292, USA 3Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois 60208, USA 4Department of Mathematics and Mathematical Statistics, Ume˚aUniversity, Ume˚a,Sweden We demonstrate experimentally the ability of a quantum annealer to distinguish between sets of non-isomorphic graphs that share the same classical Ising spectrum. Utilizing the pause-and-quench features recently introduced into D-Wave quantum annealing processors, which allow the user to probe the quantum Hamiltonian realized in the middle of an anneal, we show that obtaining thermal averages of diagonal observables of `classically indistinguishable' non-isomorphic graphs encoded into transverse-field Ising Hamiltonians enable their discrimination. We discuss the significance of our results in the context of the graph isomorphism problem. I. INTRODUCTION vides access to thermal properties of quantum, rather than classical, Hamiltonians. This new ability has al- Commercial quantum annealers [1{3] irrupted into the ready been tested with varying degrees of success [20{ quantum computing scene about a decade ago, bringing 22, 26]. along great promise|and just as much skepticism [4,5]. Another novel feature that has recently been intro- Whether these machines do or could in principle pro- duced into experimental quantum annealers is that of re- vide any advantages over their classical analogues|or verse annealing. Here, the initial Hamiltonian is set to be whether they could even be considered genuine quantum the Ising spin glass and the transverse field is ramped up processors [6{10]|is still a topic of much debate [11{14]. from zero to some intermediate value, and then brought While concrete evidence illustrating their usefulness back to zero again. This capability has been shown to as practical solvers of optimization problems is still help with the exploitation of regions of interest in the scarce [15, 16] the possibility that they can function as solution space of various optimization problems [27, 28]. efficient Boltzmann samplers [17{19] or as effective quan- In this work, we examine the possibility that mid- tum simulators [20{23] has been gaining traction. anneal probing through pause-and-quench may be uti- Alongside efforts to extend the range of potential ap- lized toward accomplishing an altogether different enter- plications for quantum annealers, improvements to these prise, namely, to solve instances of the Graph Isomor- devices are continuously being made. Notably, currently phism problem|the task of deciding whether two graphs available devices allow the user to exercise additional are isomorphic (i.e., identical up to vertex relabeling). control over various annealing parameters [24] and de- The possibility of using quantum annealers in this man- termine, to a certain extent, the schedule of the anneal, ner was first proposed almost a decade ago [29, 30] and which until recently could only be set to interpolate di- later experimentally attempted with somewhat limited rectly between a transverse-field at the beginning of the success on an older-generation experimental quantum de- anneal and a classical Ising spin glass at the end of it. vice that lacked the pause-and-quench features [31]. In particular, novel pause-and-quench capabilities enable Here, we study the power of quantum annealers some flexibility in determining the rates at which the clas- equipped with pause-and-quench capabilities to dis- sical Ising and transverse field Hamiltonians shift their re- tinguish between non-isomorphic graphs mapped onto spective strengths. This capability of pausing|the tem- transverse-field Ising Hamiltonians. We find that while arXiv:2005.01241v1 [quant-ph] 4 May 2020 porary halting of the anneal midway|deployed shortly measurements at the end of the anneal have only limited after the minimum gap, has already been demonstrated discrimination capabilities, mid-anneal measurements al- to increase the success rates of the quantum annealer low the extraction of sufficient information from the spec- when used as an optimizer [18, 25], by giving the system tra of strictly quantum Hamiltonians and are in turn time to thermalize after the minimum gap. powerful enough to distinguish all tested sets of non- Combining said pause with a sufficiently fast quench isomorphic graphs. to the end of the anneal, such that the evolution during The paper is organized as follows. In Sec.II, we provide the quench is highly non-adiabatic, allows the user to ef- some background on the Graph Isomorphism problem, fectively measure the system at the time of the pause, and the means for solving instances of the problem using rendering the annealer a quantum simulator, which pro- quantum annealers. Sec.III relays the technical details of our experiments, including information about the specific parameters used. Results are presented and discussed in Sec.IV, and conclusions and final remarks laid out in ∗ [email protected] Sec.V. 2 II. DISCRIMINATING BETWEEN GRAPHS Calculating the full quantum Ising spectrum of a given USING A QUANTUM ANNEALER graph requires in general the diagonalization of a 2n 2n matrix, which quickly becomes intractable as n increases.× A graph G = (V; E), is a pair of sets, with V being the Quantum annealers, which implement the above quan- set of vertices and E the set of edges (unordered pairs of tum Ising Hamiltonian H(G; s), could alternatively be vertices), such that E [V ]2 [32]. We denote the number used to extract valuable information without resorting of vertices V by n.⊆ Two graphs G and G0 are said to to calculating its entire spectrum [29]. This can be done j j 0 if the annealing process is stopped midway at some in- be isomorphic, G ∼= G , when there exists a bijection between them that preserves vertex adjacency, that is, termediate s value, 0 < s < 1, and measurements are one can be converted to the other by a simple relabeling then taken in order to obtain an estimation of thermal of its vertices. averages of physical observables. Determining whether two graphs are isomorphic is Experimentally testing the conjecture that physi- known as the Graph Isomorphism (GI) problem [33]. cal quantum annealers can distinguish non-isomorphic The best general case algorithm for solving it is quasi- graphs poses however two main challenges. The first is polynomial in graph size [34]. One way of solving the that currently available processors only allow diagonal GI problem is by comparing graph invariants: maps that measurements to be performed, restricting to some ex- take a graph as their argument and assign equal values tent the amount of information accessible to the experi- to isomorphic graphs. A specific method to construct menter. Moreover, to ensure that the quantum annealer graph invariants results from encoding the structure of identifies isomorphic graphs as such, one must also choose the graph on an Ising Hamiltonian [29]. To do this, to to measure quantities that remain unchanged under ver- z z every edge (i; j) in the graph we assign a term σi σj in the tex relabeling. Hamiltonian, where σz represents an individual Pauli-z In this study, we measure the classical energy E i ≡ matrix on spin i. Optionally, we can add an external, Hp , the z-magnetization Mz , the spin-glass order pa- h i h i homogeneous longitudinal field to the spins. We call this rameter [29], the `problem Hamiltonian' of the graph: s n 1 X z z 2 X X Q2 = σi σj ; (4) H (G) = σzσz + σz: (1) N(N 1) h i p i j i − i6=j (i;j)2E(G) i=1 and the next-nearest neighbor interaction energy [31] The invariant of interest is the spectrum of Hp, and 0 two graphs G and G are said to be co-Ising if Hp(G) X 0 2 2 z z and Hp(G ) have the same spectra. The spectrum of Ω = [A (G)]i;jσi σj : (5) i;j Hp is however not a complete invariant, meaning that non-isomorphic, co-Ising graphs exist [35], and trying to The second challenge that current quantum anneal- use Hp as a graph discriminator to solve the GI problem would therefore fail in the general case. ing technology faces is that it only allows measurements Quantum annealers offer the exciting possibility of ex- to be taken at the end of the anneal, where only the tending the set of Hamiltonian graph invariants into the classical Hp component has non-zero strength. This set- quantum regime. This can most simply be done by ting is appropriate for the standard usage of a quantum adding a quantum term, such as a homogeneous trans- annealer as an optimizer, when the objective is finding verse field, to the classical Ising model onto which the the ground state of Hp, and the transverse field is just a graph is mapped. In this case, the Hamiltonian can be means to arriving at it. However, as was discussed above, written as for distinguishing between non-isomorphic graphs, sam- pling from a thermal state of a classical Hamiltonian is H(G; s) = A(s)Hd + B(s)Hp(G); (2) insufficient in general. To illustrate why this is so, con- sider Fig.1, which shows ∆ E = EG13 EG13p for one of where s is a rescaled, dimensionless time parameter, − s [0; 1]. A(s) and B(s) are functions of s such that the pairs of graphs we try to distinguish, denoted G13 2 and G13p, with n = 13 (small enough to be directly di- A(0) B(0) 0 and B(1) A(1) 0, and Hd is the standard transverse' field driver Hamiltonian' in quantum agonalized).