Quantum walks in polycyclic aromatic hydrocarbons Prateek Chawla1, 2, ∗ and C. M. Chandrashekar1, 2, y 1The Institute of Mathematical Sciences, C. I. T. Campus, Taramani, Chennai 600113, India 2Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India Aromaticity is a well-known phenomenon in both physics and chemistry, and is responsible for many unique chemical and physical properties of aromatic molecules. The primary feature con- tributing to the stability of polycyclic aromatic hydrocarbons is the delocalised π-electron clouds in the 2pz orbitals of each of the N carbon atoms. While it is known that electrons delocalize among the hybridized sp2 orbitals, this paper proposes quantum walk as the mechanism by which the delocalization occurs, and also obtains how the functional chemical structures of these molecules arise naturally out of such a construction. We present results of computations performed for some benzoid polycyclic aromatic hydrocarbons in this regard, and show that the quantum walk-based approach does correctly predict the reactive sites and stability order of the molecules considered. I. INTRODUCTION use a slightly simpler method that relies on the system's geometry. It requires a basis relative to which bond or- der can be calculated. We make the traditional choice The structure and properties of Benzene and other of C − C single bond in Ethane (H3C − CH3) to have arenes has been a subject of special interest in the field of bond order 1, and the C = C double bond of Ethene quantum chemistry for a long time. It is now known that (H C = CH ) to have a bond order of 2. This gives each of the six carbon atoms in Benzene have their or- 2 2 2 us a quantity known as Relational Bond Strength Order bitals hybridized into the sp state, and the 2pz orbitals (RBSO) [10], and provides a robust description of bond host a delocalized π-electron cloud of the molecule. This strength as long as the Born-Oppenheimer approxima- type of cyclic conjugated system has been a subject of tion is valid. This provides a useful tool to study the significant interest in the chemical sciences, and the de- bonds in an aromatic molecule, as a bond with bond or- scription of such systems is studied under the concept(s) der closer to 1 would have characteristics that are closer of (anti)aromaticity [1{5]. These molecules are charac- to a single bond, but in a bond with order closer to 2, terized by their electron-rich clouds, which stabilize the the double-bond characteristics will dominate. structure through cyclic resonant structures. While it is known that the delocalization does occur in the elec- With advancements in the field of quantum informa- tron cloud, the underlying physical process by which the tion and computation, a quantum walk has emerged as process takes place is an open question as of yet. one of the most efficient ways to model the controlled In order to understand the chemical characteristics of dynamics of quantum states and quantum particles [11{ the cyclic conjugated system, bond order of the bonds 15]. The quantum walk can be regarded as a quan- in the system is studied. Bond order is a widely used tum counterpart of random walks which have served as concept in chemical sciences, and works as a tool to help an efficient way to model dynamics defined by classical predict chemical behaviour. Certain reactions can take physics. Therefore, it is natural to explore the poten- place only for bonds of certain order (e.g. electrophilic tial of quantum walk to model quantum dynamics in and nucleophilic substitution reactions can only occur for the range of physical systems where quantum physics bond order ≥ 1). It can also help in explaining the re- plays a defining role. It is described in two main for- active tendencies of some molecules by consideration of malisms { the discrete-time quantum walk (DTQW) and the sum of bond order [6] of the atoms preferred by the the continuous-time quantum walk (CTQW). The dy- molecule. The strength of a chemical bond depends on namics of DTQW require a coin and a position Hilbert arXiv:2012.14463v1 [quant-ph] 28 Dec 2020 two things - the degree of overlap between the interact- space in order to be defined, however, CTQW dynam- ing orbitals, and the difference in energies of the atomic ics can be defined with only a position Hilbert space. A orbitals involved in bonding, which is reflected by bond quantum walk spreads quadratically faster than a clas- polarity. The bond polarity is used to qualify the ionic sical walk on its position space and its dynamics can nature of the bond, while the degree of overlap quantifies be engineered using a set of evolution parameters, and its covalent nature [7,8]. therefore it has been used as a basis for design and im- A comprehensive method to calculate bond order from plementation of quantum algorithms and quantum sim- first principles was only available in 2017 [9]. As we in- ulations [16{29], to study problems such as graph iso- vestigate polycyclic aromatic rings in this work, we can morphism [30], quantum percolation [31{33], and to de- velop schemes for implementation of universal quantum computation [34, 35], among others. Quantum walks are thus very versatile tools indeed, and their practical signif- ∗Electronic address: [email protected] icance has been demonstrated by way of implementation yElectronic address: [email protected] in many quantum systems, such as NMR [36], integrated 2 photonics [37{39], ion traps [40, 41], and cold atoms [42]. of the Schr¨odingerequation is given by, In this work, we will make use of both the discrete- time and continuous-time quantum walk formalisms in HΓ = γL = γ(D − A); 8 order to study the structure, stability, and the relative −γ i 6= j; (i; j) 2 E chemical reactivity of different sites when exposed to an <> : (4) =) H = 0 i 6= j; (i; j) 2= E electrophile for each of the molecules considered. We also Γij > qualitatively establish that the order of stability of the :diiγ i = j; molecules arises naturally from a quantum walk-based framework. We thus propose the hypothesis that the Here D is a diagonal matrix, and each diagonal element electrons in the 2pz orbitals delocalize in the π-electron djj corresponds to the degree of vertex j. γ is a con- cloud via a quantum walk. This in turn should lead to- stant, A is the adjacency matrix (defined as in Eq. (1)), wards exploring use of quantum walks to model the dy- L is known as the Laplacian of Γ, and γ is the rate of namics in wide range of chemical compounds and com- transition for the graph. plexes. It may be observed from the expression in Eq. (4), The paper is divided into six sections. In Sec. II, we that HΓ is independent of time, and thus the solution discuss the discrete- and continuous-time quantum walks, to Eq. (3) is given by the time evolution operator U, and and elucidate briefly the specific variants of the quantum may be expressed as, walks used in this work. A quick recap of the concepts of relative bond strength order is given in Sec. III, and U = e−iHΓt; (5) the methods we use to model the dynamics in aromatic j (t)i = U j (0)i ; hydrocarbons are detailed in Sec.IV. We present our results in Sec.V, and conclude in Sec.VI. where the expression of is in the units of ~. Since the adjacency matrix A is always real and symmetric, due to it representing an undirected graph, so is the Hamil- II. QUANTUM WALK tonian - which ensures that U is necessarily a unitary operation. A. Continuous-time quantum walk The coefficients γ in the Hamiltonian in Eq. (4) are rates of transition between different nodes. Tradition- A quantum walker performing CTQW has its position ally, in unweighted graphs, the rates are all normalized space defined by a graph Γ(V; E), where V is the set of to 1. In the case of arenes, however, we use the values its vertices and E the set of edges. Given the cardinality of RBSO as mentioned in the beginning of this paper in N(N−1) Sec.I, and formally defined in Sec. III as the values of of V = jV j = N, jEj ≤ 2 . Let A be the matrix defined as, the weights of the graph. This corresponds to how eas- ily the electrons are able to move across the bonds, and physically represents a measure of the bond length, as ( 1 edge (i; j) 2 E bonds of higher order are shorter. A := : (1) ij 0 otherwise A, defined as such on Γ, is known as the adjacency ma- B. Discrete-time quantum walk trix of the graph, and offers a representation of Γ itself. It is a real-valued matrix, and the vertices are labeled The quantum evolution of a walker executing a DTQW by the computational basis states fj1i ; j2i ; :::; jNig. The on a one-dimensional lattice may be described on a quantum state of the entire graph is defined at an ar- Hilbert space H = HC ⊗ HP , where HC and HP are bitrary point of continuous time t by the wavefunction coin and position Hilbert spaces, respectively. The coin j (t)i, which is defined as, space in this case is a 2-dimensional space, and thus its basis set can be assumed to consist of two vectors j"i ; j#i, such that they are mutually orthogonal.