I Introduction & Motivation
In recent years, the field of bioinformatics has gained much attention among research and academia communities to develop intelligent systems for simulating and analysis of molecular biology. Bioinformatics is a active, diverse and fastgrowing research field. It is the application of information technology to store, process and analyse the biological data, especially DNA, RNA, and protein sequences 2 . Presently, the focus is on developing probabilistic models to examine the biological sequences at genome level. Till now, several methods based on automata theory, grammatical formalism, learning theory and statistical theory have been introduced to modeling and analyzing the behaviour of RNA, DNA and protein sequences 3 . The accurate modelling and prediction of the genomewide are the major challenges in bioinformatics 4 .
Nowadays, quantum computing is the new buzz word and become a hot research topic in industry, academic and R& D centres getting significant response and financial support from all directions. Quantum computing incorporates elements from physics, mathematics and computer science 5 . Quantum computers have the potential to tackle the problems that would take classical computers millions of years. It promises to solve complex realworld problems such as modeling financial risks, simulating chemistry and optimising supply chains. It would impact biologists to study the possible ways to interact and fold proteins with one another and chemists to model interactions between drugs. We can examine some properties quickly using quantum superposition principle and entanglement than with classical means 6 .
The finite automata theory is one of the keystone of theoretical computer science 7 . The combination of quantum mechanics and classical automata theory gives us quantum finite automata (QFA) 8 ; 9 . The concept of QFA models was introduced by Moore and Crutchfield 50 and Kondacs and Watrous 12 separately, soon after the discovery of Shor’s factoring algorithm 10 . QFA are abstract models of machines with finite memory for quantum computers, which play an important role in carrying out computation in realtime, i.e. the tape head takes exactly one step per input symbol and moves towards the right direction only 11 . It is described as a quantum analogue of a classical finite automaton. It lays down the perception of quantum processors for executing quantum operations on reading the inputs.
Since then, there is a diversity of quantum automata models have been studied and investigated in all directions such as quantum finite automata, Latvian QFA (LQFA) 12 , 1.5way QFA 13 , twoway QFA (2QFA) 12 , quantum pushdown automata (QPDA) 55
, quantum Turing machine (QTM)
15 , quantum multihead finite automata (QMFA) 17 ; 40 , multiletter QFA 16 , oneway quantum finite automata with classical states (1QCFA) 14 , twoway quantum finite automata with classical states (2QCFA) 1 , quantum queue automata 100 and many more since last two decades. These models are effective in examine the frontiers of computational properties and expressive power of automata. Quantum computers are more powerful than probabilistic Turing machines and even Turing machines. Therefore, quantum computational models can be consider as generalizations of its physical models 8 .Bioinformatics introduces and utilizes biological computational algorithms for interpretation of biological processes based on interaction between genomes 18 . The biological sequences are modeled using grammatical formalism to efficiently solve the bioinformatics computational problems such as prediction and classification of sequences, calculate multiple alignments, sequences analysis and data mining 19 . DNA can be seen as recipe of an organism. It is double stranded and made from four different monomers called nucleotides (A, C, G ,T) representing adenine (A), cytosine (C), guanine (G) and thymine (T). RNA is like DNA except the base thymine (T) is replaced by base uracil (U). It is often singlestranded structure and folds around itself 34 . Some of the bases form mismatched nucleotides, which results in formation of loops of unpaired single strands at the center or end of a duplex.
Kondacs and Watrous 12 proposed the notion of twoway quantum finite automata i.e. quantum version of twoway deterministic finite automata. It has been proved that it is more dominant than classical counterparts for language recognition. 2QCFA can recognize some contextfree, context sensitive languages and all regular languages. But, atleast qubits are needed to store positions of the input tape head, where n is the length of an input string. In order to get over the disadvantage of 2QCFA, Ambainis and Watrous 1 introduced twoway quantum finite automata with classical states (2QCFA). Its computational power lies in between 1QFA and 2QFA, but still more powerful than its classical variants. It has been proved that 2QCFA is more powerful than twoway probablistic finite automata (2PFA). It can recognize the palindrome language , where is the reverse of w in exponential time with onesided error, but 2PFA cannot be designed for L with bounded error. Zheng et al. 78 studied the state succinctness of 2QCFA. Zheng et al. 20 investigated that 2QCFA can be designed for with bounded error in polynomial time, but 2PFA takes an exponential time with bounded error. Qiu et al. 21 proved that the class of languages recognized by 2QCFA are closed under union, intersection, complement and the reversal operation.
Motivated from the abovementioned facts, we have transcribed RNA secondary structures in the form of formal languages and modeled them using twoway quantum finite automata with classical states. The main objective is to examine how RNA secondary structures perform sequence identification equivalent to quantum automata models. The crucial advantage of this approach is that chemical reactions in the form of accept/reject signatures can be processed in linear time with onesided bounded error. The organization of rest of this paper is as follows: Subsection is devoted to prior work. In Sect. 2, some preliminaries are given. The notion of twoway quantum finite automata with classical states is given in Sect. 3. In Sect. 4, the RNA secondary structures (hairpin loop, pseudoknot and dumbbell structures) are transcribed in formal languages and modeled using twoway quantum finite automata with classical states. Finally, Sect. 5 is the conclusion.
Ii Prior Work
During the last three decades, several representations of RNA and DNA sequences using automata theory and formal grammar have been found in literature. The structures of DNA and RNA are represented using the concept of classical automata theory. In 1984, Brendel and Busse 22 transcribed nucleic acid sequences as words over the input alphabet of nucleotides and formulated that genomes can be described in formal language theory. Sung 88 modelled the RNA pseudoknots using context sensitive grammar. In 1992, Searls 3 formulated RNA and DNA sequences such as pseudoknot, inverted and tandem repeat using indexed grammar. Later, Searls 23 used string variable grammar to represent DNA sequences.
Roy et al. 24 proposed the concept of micron automata processor to find the conserved sequences in protein or multiple DNA sequences. Cai et al. 25 represented the pseudoknot biomolecular structures of RNA using parallel communicating grammar. Barjis et al. used finite automata as modeling tool for formulation and simulation of production of proteins. Mizoguchi et al. 26 modeled different classes of pseudoknot structure with stochastic multiple contextfree grammar. Kuppusamy and Mahendran 27 presented the notion of matrix insertiondeletion system and used to analyze and model the RNA secondary structures such as stem and loop, pseudoknot, attenuator, internal loop, bulge loop and kissing hairpin. Recently, Bhatia and Zheng 101 transcribed the chemical reactions in from of formal languages and modeled them using twoway quantum finite automata. Fernau et al. 28 introduced small size universal matrix insertion grammar to simulate the computation of DNA. Krasinski et al. 29 described the restricted enzyme in DNA with circular mode pushdown automata.
Khrennikov and Yurova 30 modeled the behavior of protein structures using classical automata theory and investigated the resemblance between the quantum systems and modeling behavior of proteins. Quantum omega automata can be used to model the behavior of chemical reactions in biological systems 104 . Soreni et al. 31 shown programmable three symbol three state finite automata and carried out biomolecular computations parallelly on surface. Lin and Shah 32 represented the patterns in DNA sequences using statistical finite automata. Cavaliere et al. 33 and Rothemund used Turing machine and pushdown automata to analyze the action of a restricted enzyme in DNA. Bhatia and Kumar 34 shown the modeling of double helix, hairpin and internal loops using linear time 2QFA. Recently, DuenasDiez and PerezMercader 35 designed molecular machines for chemical reactions. It has been demonstrated that chemical reactions transcribed in formal languages, can be recognized by Turing machine without using biochemistry.
Iii Twoway Finite Automata with Quantum and Classical States
Ambainis and Watrous 1 presented the notion of twoway quantum finite automata with classical states (2QCFA). The computational power of 2QCFA lies in between 2QFA and 1QFA. In 2QCFA, the tape head position is classical and the internal state may be a (mixed) quantum state.
Definition 1.
1 A 2QCFA is defined as a nonuple , where

is a finite set of classical states,

is a finite set of quantum states,

is an input alphabet such that , where # and $ are left and rightend markers, respectively,

defines the evolution of the quantum portion of the internal state,
(1) where and representing the projective measurements and unitary operators over Hilbert space with set Q. Therefore, is equivalent to either projective measurement or a unitary evolution.

defines the evolution of classical states. If , then is defined as
(2) where E
denotes possible set of eigenvalues
and projector set =1,2,…,n} andis the projector onto the eigenspace.
shows the head movement towards left, stationary and right side of the input tape, respectively. If , then is defined as(3) 
is an initial quantum state ,

is an initial classical state ,

are the set of accepting and rejecting states respectively, .
The computation procedure of 2QCFA to process a given input string w is as follows: Initially, the classical and quantum state are and and head is positioned on leftend marker $. The state of 2QCFA is changed according to . On reading the input symbol , the classical state is changed according to and quantum state is changed according to . The tape head is moved according to direction .

If , then the classical state is transformed to according to , quantum state is transformed as and head movement is determined by d.

If , then the projective measurement is carried out on . Suppose, with possible eigenvalues . After performing the measurement, we get a result
with probability
and the quantum state is transformed as . The classical state is changed as .
Finally, each measurement outcomes are probabilistic and the classical state transitions may be also probabilistic. Thus, the 2QCFA is said to be accepted with probability when the computation is halted and automata enters the classical accepting state , otherwise it is said to be rejected with probability . Consider a language , it is said to be accepted with onesided error by 2QCFA if the probability of acceptance , , otherwise it is said to be rejected with oneside error if if .
Iv RNA Secondary Structures Modeling
In this section, we analyze, model and simulate the RNA secondary biomolecular structures such as hairpin loop, pseudoknot structure and dumbbell structure using twoway quantum finite automata with classical states (2QCFA). We assume that the reader is familiar with the classical automata theory and the concept of quantum computing; otherwise, reader can refer to the theory of automata 7 , quantum information and computation 62 ; 64 .
iv.1 Hairpin Loop
Hairpin is the primary unit secondary structure in RNA molecules. It plays a crucial role in various biological processes such as DNA transposition, DNA recombination, gene expressions and RNAprotein recognition. A hairpin loop consists of a basepaired stem formed in singlestranded nucleic acids and ends to form unpaired nucleotide bases 36 . It is named according to size and composition of loop. Fig 1 shows the representation of hairpin loop structure. Hairpin loop can be transcribed as language to form base pairing, where is x in the reverse order. The detailed proof for palindrome language can be find in Ambainis and Watrous 1 paper.
Theorem 1.
A language , where is x in the reverse order, representing hairpin loop biomolecule structure can be recognized by 2QCFA with onesided error in exponential time, which cannot be recognized by 2PFA.
Proof.
(4) 
The idea of the proof is as follows. It consists of two phases. First we define a 2QCFA for using quantum register consisting three orthogonal states. In second phase, we modify the 2QCFA such as natural mapping is performed from threedimensional Euclidean space to unit sphere in twodimensional Hilbert space . We construct a 2QCFA for the language with three quantum states , where is an initial state. has two unitary matrices and defined as follows. The automaton proceeds as follows:
Repeat the following endlessly:  
1.  Set the initial quantum state and shift the input tape head under the first input symbol. 
2.  While the presently symbol read is not $, do the following: 
(2.1).  If the presently examined symbol is a or u, apply or on the quantum state, respectively. 
(2.2).  If the presently read symbol is g or c, execute or on the quantum state, respectively. 
(2.3).  Shift the position of tape head one square towards the right. 
3.  Repeat the following subroutines: 
(3.1).  Move the input tape head towards the left direction until the rightend marker symbol # is reached. 
(3.2).  Shift the position of tape head one square towards the right. 
4.  While the presently symbol read is not leftend marker $, do the following: 
(4.1).  If the presently read symbol is a or u, apply or on the quantum state, respectively. 
(4.2).  If the presently scanned symbol is g or c, execute or on the quantum state, respectively. 
(4.3).  Shift the position of the input tape head towards the right. 
Perform the measurement on quantum state, if the outcome is not , then it is rejected.  
Initialize the variable z=0  
5.  While the presently symbol scanned is not rightend marker #, do the following: 
(5.1).  Replicate k coin flips. Initialize z=1, if all outcomes are not ”heads”. 
(5.2).  Shift the position of tape head one square towards the left. 
if z=0, it is said to be accepted. 
Consider an input string , the tape squares are indexed by 0 and n+1 consist both endmarkers # and $, respectively. The computation process of starts with quantum state is as follows. As whileloop 2 is implemented, the input tape head traverse each input symbol and performs or on the quantum state (depends upon whether the input symbol is a or u and performs or on the quantum state (depends upon whether the input symbol is g or c, respectively. Suppose denote the matrix ( or ) and ( or ), as defined in (4), depending upon is (a or u) and (g or c). The automaton changes its state after executing loop 2 as
(5) 
for =. As we repeat the subroutines 3, the input tape head is moved towards the left direction until the rightend marker is read, then shift the tape head one position to the right. Now, in loop 4, the inverses of and are performed, the quantum state is changed as
(6) 
for =. The quantum state is measured and is said to be rejected with probability , else it collapses to initial state . If w is a palindrome, then , else . Initialize the variable z equal to 0, which is stored in classical state. Finally, on executing the whileloop 5, the input string is said to be accepted if the loop is terminated with
. It is known from standard result in probability theory that probability of reaching the location
n+1 is . On flipping the k coins, the probability of acceptance . If the algorithm is repeated indefinitely, then the probability of rejectance is(7) 
and accepting probability is
(8) 
If the input string , then the probability of accepting w is 1. Suppose , it can be checked that if , the rejects the input string with probability atleast . ∎
iv.2 Pseudoknot Structure
A pseudoknot structure is a doublehairpin structure that forms an extended quasicontinuous helix structure and double connecting loops 37 . It is formed when pairs are created between the bases outside and inside of a hairpin or internal loop. Pseudoknot structure plays a crucial role in RNA functions such as regulation of splicing and translation and ribosome frameshifting 38 . It is considered as a key component of ribozymes or ribosomal RNAs. Fig 2 describes the pseudoknot secondary structure. A closer look at pseudoknot structure shows a similarly with constructs of natural language (i.e. dependencies are forced to cross) such as 61 . Thus, the number of a’s is equal to the number of u’s and correspondingly the number of g’s is equal to the number of c’s.
Theorem 2.
A language representing pseudoknot biomolecule structure can be recognized by 2QCFA in polynomial time with error probability .
Proof.
The idea of the proof is as follows. The construction of 2QCFA consists of three phases. Firstly, it checks whether the input is in form , if not, then the input string is said to be rejected. In second phase, we check that it is in form or not. If yes, then we check that it is in form . Since, 2QCFA can be designed for . Similarly, it can recognize and in polynomial time. Qiu 21 proved that the class of languages recognized by 2QCFA are closed under intersection, reversal, complement and union operations. For convenience, 2QCFA(polytime) notation is used to denote the class of languages recognized by 2QCFA in polynomial time with error probability . Therefore, if 2QCFA(polytime) and 2QCFA(polytime), then can be recognized by 2QCFA in polynomial time with .
Based on the above analysis, the prove of this theorem is described now more formally. Let
(9) 
be 2QCFA’s for recognizing in polynomial time with error probabilities for j=1, 2, where

,

We construct a 2QCFA for such that where

,

,

,

,

,

,


and are defined as

is defined as a transition function:
(10) 
if corresponds to unitary operator over , then extend the relating to , for and ,

if , which denotes projective measurement over and measurement is represented by projector set . Then, denotes an orthogonal measurement on represented by projection operators on , where represent projection operators obtain by extending with , for and represents mapping of projection operator to , i.e. an identity operator.


For any and

if corresponds to unitary operator over , then extend the relating to , for and ,

if , which denotes projective measurement over and measurement is represented by projector set , then is an orthogonal measurement on represented by projection operators on , where extend to by defining , for .


For any and

if , then and , where I is an identity operator over ,

else if , then represents an orthogonal measurement by projectors , ; ; , where U denotes a unitary operator over satisfy .

Recall the languages and . In respect of 2QCFA designed above, for any input string w , we have considered the following cases:

if , then at the end of the computation process enters a state with probability atleast . Then, is said to be recognized by with probability atleast .

if , but , then at the end of the computation enters a state with probability atleast . Then, is said to be rejected by with probability atleast .

if , then the state of is changed to and it is said to be rejected with probability atleast .
Hence, if the languages and are said to be recognized by 2QCFA’s and in polynomial time with error probabilities , respectively. Then, is said to be recognized by in polynomial time with error probability . ∎
iv.3 Dumbbell Structure
Dumbbell shaped RNA structure is formed by analogy of DNA dumbbells comprised of twohelical stems closed by two hairpin loop structures. It plays an important role in analysis of local structures in DNA. The loops on its both sides restrict its enzymatic cleavage and stabilize the duplex. It is successfully applied to transcriptional regulation 39 . Fig 3 shows the representation of dumbbell shaped RNA secondary structure.
Theorem 3.
A language representing dumbbell biomolecule structure can be recognized by 2QCFA with onesided error probability in polynomial time.
Proof.
Ambainis and Watrous proved that a language can be recognized by 2QCFA in polynomial time, which can be recognized by 2PFA in exponential time 1 . Similarly, it can be proved that a language can be recognized by 2QCFA in polynomial time with onesided error probability. It consists of three phases. Firstly, it examines whether the input string is in form , if not, then the input string is said to be rejected. Otherwise, in second phase, 2QCFA simulates the initial part of string to determine whether is in , by using g in the right side of u as the rightend marker $. If not, the computation is said to be rejected. Otherwise, in third phase, the 2QCFA finally checks is in and u is used as a leftend marker #. If the number of g’s and c’s are equal, then it is said to be recognized with onesided error probability in polynomial time, otherwise rejected. ∎
V Conclusion
The enhancement in many existing computational approaches provides momentum to biological systems and quantum simulations at the gene expression levels. It helps to test new abstract approaches for considering RNA, DNA and protein sequences. Previous attempts to model the aforementioned RNA secondary structures used formal grammar and finite automata theory. In this paper, we focused on wellknown structures of RNA such as hairpin loop, pseudoknot and dumbbell biomolecular structures and modeled them using twoway quantum finite automata with classical states. The crucial advantage of the quantum approach is that these secondary structures transcribed in formal languages takes exponential time for and polynomial time for and , respectively. It has been shown that twoway quantum finite automata with classical states are more superior than its classical variants by using quantum part of finite size. For the future purpose, we will try to represent complex RNA structures in formal languages and model them using other quantum computational models.
Additional Information
Conflict of interest The authors declare that they have no conflict of interest.
Acknowledgement
S.Z. acknowledges support in part from the National Natural Science Foundation of China (Nos. 61602532), the Natural Science Foundation of Guangdong Province of China (No. 2017A030313378), and the Science and Technology Program of Guangzhou City of China (No. 201707010194).
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