RNA-2QCFA: Evolving Two-way Quantum Finite Automata with Classical States for RNA Secondary Structures
aa r X i v : . [ c s . F L ] J u l RNA-2QCFA: Evolving Two-way Quantum Finite Automata with Classical States forRNA Secondary Structures
Amandeep Singh Bhatia , ∗ , Shenggen Zheng Chitkara University Institute of Engineering & Technology, Chitkara University, Punjab, India Center for Quantum Computing, Peng Cheng Laboratory, Shenzhen, China
E-mail: , ∗ [email protected] Recently, the use of mathematical methods and computer science applications have got significantresponse among biochemists and biologists to modeling the biological systems. The computationaland mathematical methods have an enormous potential for modeling the deoxyribonucleic acid(DNA) and ribonucleic acid (RNA) structures. The modeling of DNA and RNA secondarystructures using automata theory had a significant impact in the fields of computer science. It is anatural goal to model the RNA secondary biomolecular structures using quantum computationalmodels. Two-way quantum finite automata with classical states are more dominant than two-wayprobabilistic finite automata in language recognition. The main objective of this paper is onusing two-way quantum finite automata with classical states to simulate, model and analyze theribonucleic acid (RNA) sequences.
Keywords : ribonucleic acid, hairpin loop, formal languages, quantum finite automata, two-wayfinite automata with quantum and classical states, bio-molecular structures
I. INTRODUCTION & MOTIVATION
In recent years, the field of bioinformatics has gained much attention among research and academia communitiesto develop intelligent systems for simulating and analysis of molecular biology. Bioinformatics is a active, diverse andfast-growing research field. It is the application of information technology to store, process and analyse the biologicaldata, especially DNA, RNA, and protein sequences [1]. Presently, the focus is on developing probabilistic models toexamine the biological sequences at genome level. Till now, several methods based on automata theory, grammaticalformalism, learning theory and statistical theory have been introduced to modeling and analyzing the behaviour ofRNA, DNA and protein sequences [2]. The accurate modelling and prediction of the genomewide are the majorchallenges in bioinformatics [3].Nowadays, quantum computing is the new buzz word and become a hot research topic in industry, academic andR& D centres getting significant response and financial support from all directions. Quantum computing incorporateselements from physics, mathematics and computer science [4]. Quantum computers have the potential to tackle theproblems that would take classical computers millions of years. It promises to solve complex real-world problems suchas modeling financial risks, simulating chemistry and optimising supply chains. It would impact biologists to studythe 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 classicalmeans [5].The finite automata theory is one of the keystone of theoretical computer science [6]. The combination of quantummechanics and classical automata theory gives us quantum finite automata (QFA) [7, 8]. The concept of QFA modelswas introduced by Moore and Crutchfield [9] and Kondacs and Watrous [10] separately, soon after the discovery ofShors factoring algorithm [11]. QFA are abstract models of machines with finite memory for quantum computers,which play an important role in carrying out computation in real-time, i.e. the tape head takes exactly one step perinput symbol and moves towards the right direction only [12]. It is described as a quantum analogue of a classicalfinite automaton. It lays down the perception of quantum processors for executing quantum operations on readingthe inputs.Since then, there is a diversity of quantum automata models have been studied and investigated in all directionssuch as quantum finite automata, Latvian QFA (LQFA) [10], 1.5-way QFA [13], two-way QFA (2QFA) [10], quantumpushdown automata (QPDA) [14], quantum Turing machine (QTM) [15], quantum multihead finite automata (QMFA)[16, 17], multi-letter QFA [18], one-way quantum finite automata with classical states (1QCFA) [19], two-way quantumfinite automata with classical states (2QCFA) [20], quantum queue automata [21] and many more since last twodecades. These models are effective in examine the frontiers of computational properties and expressive power ofautomata. 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 [7].Bioinformatics introduces and utilizes biological computational algorithms for interpretation of biological processesbased on interaction between genomes [22]. The biological sequences are modeled using grammatical formalism toefficiently solve the bioinformatics computational problems such as prediction and classification of sequences, calculatemultiple alignments, sequences analysis and data mining [23]. DNA can be seen as recipe of an organism. It is doublestranded 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 ). Itis often single-stranded structure and folds around itself [24]. Some of the bases form mismatched nucleotides, whichresults in formation of loops of unpaired single strands at the center or end of a duplex.Kondacs and Watrous [10] proposed the notion of two-way quantum finite automata i.e. quantum version of two-waydeterministic finite automata. It has been proved that it is more dominant than classical counterparts for languagerecognition. 2QCFA can recognize some context-free, context sensitive languages and all regular languages. But,atleast O ( log n ) 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 [20] introduced two-way quantum finiteautomata with classical states (2QCFA). Its computational power lies in between 1QFA and 2QFA, but still morepowerful than its classical variants. It has been proved that 2QCFA is more powerful than two-way probablisticfinite automata (2PFA). It can recognize the palindrome language L = { ww r | w ∈ { a, b } ∗ } , where w r is the reverseof w in exponential time with one-sided error, but 2PFA cannot be designed for L with bounded error. Zheng etal. [25] studied the state succinctness of 2QCFA. Zheng et al. [26] investigated that 2QCFA can be designed for L = { xcy | x, y ∈ { a, b } ∗ , | x | = | y | , Σ = { a, b, c }} with bounded error in polynomial time, but 2PFA takes anexponential time with bounded error. Qiu et al. [27] proved that the class of languages recognized by 2QCFA areclosed under union, intersection, complement and the reversal operation.Motivated from the above-mentioned facts, we have transcribed RNA secondary structures in the form of formallanguages and modeled them using two-way quantum finite automata with classical states. The main objective isto 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 beprocessed in linear time with one-sided bounded error. The organization of rest of this paper is as follows: Subsectionis devoted to prior work. In Sect. 2, some preliminaries are given. The notion of two-way quantum finite automatawith classical states is given in Sect. 3. In Sect. 4, the RNA secondary structures (hairpin loop, pseudoknot anddumbbell structures) are transcribed in formal languages and modeled using two-way quantum finite automata withclassical 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 andformal grammar have been found in literature. The structures of DNA and RNA are represented using the concept ofclassical automata theory. In 1984, Brendel and Busse [28] transcribed nucleic acid sequences as words over the inputalphabet of nucleotides and formulated that genomes can be described in formal language theory. Sung [29] modelledthe RNA pseudoknots using context sensitive grammar. In 1992, Searls [2] formulated RNA and DNA sequences suchas pseudoknot, inverted and tandem repeat using indexed grammar. Later, Searls [30] used string variable grammarto represent DNA sequences.Roy et al. [31] proposed the concept of micron automata processor to find the conserved sequences in protein ormultiple DNA sequences. Cai et al. [32] represented the pseudoknot biomolecular structures of RNA using parallelcommunicating grammar. Barjis et al. used finite automata as modeling tool for formulation and simulation ofproduction of proteins. Mizoguchi et al. [33] modeled different classes of pseudoknot structure with stochastic multiplecontext-free grammar. Kuppusamy and Mahendran [34] presented the notion of matrix insertion-deletion system andused to analyze and model the RNA secondary structures such as stem and loop, pseudoknot, attenuator, internalloop, bulge loop and kissing hairpin. Recently, Bhatia and Zheng [35] transcribed the chemical reactions in from offormal languages and modeled them using two-way quantum finite automata. Fernau et al. [36] introduced small sizeuniversal matrix insertion grammar to simulate the computation of DNA. Krasinski et al. [37] described the restrictedenzyme in DNA with circular mode pushdown automata.Khrennikov and Yurova [38] modeled the behavior of protein structures using classical automata theory and inves-tigated the resemblance between the quantum systems and modeling behavior of proteins. Quantum omega automatacan be used to model the behavior of chemical reactions in biological systems [39]. Soreni et al. [40] shown pro-grammable three symbol three state finite automata and carried out biomolecular computations parallelly on surface.Lin and Shah [41] represented the patterns in DNA sequences using statistical finite automata. Cavaliere et al. [42]and Rothemund used Turing machine and pushdown automata to analyze the action of a restricted enzyme in DNA.Bhatia and Kumar [24] shown the modeling of double helix, hairpin and internal loops using linear time 2QFA.Recently, Duenas-Diez and Perez-Mercader [43] designed molecular machines for chemical reactions. It has beendemonstrated that chemical reactions transcribed in formal languages, can be recognized by Turing machine withoutusing biochemistry.
III. TWO-WAY FINITE AUTOMATA WITH QUANTUM AND CLASSICAL STATES
Ambainis and Watrous [20] presented the notion of two-way 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 classicaland the internal state may be a (mixed) quantum state.
Definition 1. [20] A 2QCFA is defined as a nonuple (
S, Q, Σ , Θ , δ, q , s , S acc , S rej ), where • S is a finite set of classical states, • Q is a finite set of quantum states, • Σ is an input alphabet such that Γ = Σ ∪ { , $ } , where $ are left and right-end markers, respectively, • Θ defines the evolution of the quantum portion of the internal state, S \ ( S acc ∪ S rej ) × Γ → U ( H ( Q )) ∪ P ( H ( Q )) (1)where P ( H ( Q )) and U ( H ( Q )) representing the projective measurements and unitary operators over Hilbert space H ( Q ) with set Q . Therefore, Θ( s, ϑ ) is equivalent to either projective measurement or a unitary evolution. • δ defines the evolution of classical states. If Θ( s, ϑ ) ∈ P ( H ( Q )), then δ is defined as S \ ( S acc ∪ S rej ) × Γ × E → S × {← , ↑ , →} , (2)where E denotes possible set of eigenvalues E = { e , e , ..., e n } and projector set { P ( e j ) : j =1,2,..., n } and P ( e j )is the projector onto the eigenspace. {← , ↑ , →} shows the head movement towards left, stationary and right sideof the input tape, respectively. If Θ( s, ϑ ) ∈ U ( H ( Q )), then δ is defined as S \ ( S acc ∪ S rej ) × Γ → S × {← , ↑ , →} , (3) • q is an initial quantum state q ∈ Q , • s is an initial classical state s ∈ S , • S acc , S rej are the set of accepting and rejecting states respectively, ( S acc , S rej ⊆ S ).The computation procedure of 2QCFA to process a given input string w is as follows: Initially, the classical andquantum state are s and | q i and head is positioned on left-end marker $ . The state of 2QCFA is changed accordingto Θ( s , $). On reading the input symbol σ ∈ Σ, the classical state is changed according to δ ( s, σ ) and quantum stateis changed according to Θ( s, σ ). The tape head is moved according to direction d = {← , ↑ , →} . • If Θ( s , σ ) = U ( H ( Q )), then the classical state s is transformed to s according to δ ( s , σ ) = ( s , d ), quantumstate is transformed as | q i = U | q i and head movement is determined by d . • If Θ( s , σ ) = P ( H ( Q )), then the projective measurement is carried out on | q i . Suppose, P = { P , P , ..., P n } with possible eigenvalues { e j } nj =1 . After performing the measurement, we get a result e j ∈ E with probability p j = h q | P j | q i and the quantum state is transformed as P j | q i / √ p j . The classical state is changed as δ ( s , σ ) =( s , d ).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 S acc ( w ) when the computation is halted and automataenters the classical accepting state S acc , otherwise it is said to be rejected with probability S rej ( w ). Consider alanguage L ⊂ Σ ∗ , it is said to be accepted with one-sided error ǫ by 2QCFA M Q if the probability of acceptance P r [ M Q accepts w ] = 1, ∀ w ∈ L , otherwise it is said to be rejected with one-side error if P r [ M Q rejects w ] ≥ − ǫ if w / ∈ L . 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 two-way 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 [6], quantum information and computation [44, 45].
A. Hairpin Loop
Hairpin is the primary unit secondary structure in RNA molecules. It plays a crucial role in various biologicalprocesses such as DNA transposition, DNA recombination, gene expressions and RNA-protein recognition. A hairpinloop consists of a base-paired stem formed in single-stranded nucleic acids and ends to form unpaired nucleotide bases[46]. 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 L h = { x ∈ { a, u, g, c } ∗ | x = x r } to form base pairing, where x r is x inthe reverse order. The detailed proof for palindrome language L = { w ∈ { a, b } ∗ | w = w r } can be find in Ambainisand Watrous [20] paper. FIG. 1: Representation of Hairpin loop structure
Theorem 1.
A language L h = { x ∈ { a, u, g, c } ∗ | x = x r } , where x r is x in the reverse order, representing hairpin loopbiomolecule structure can be recognized by 2QCFA with one-sided error in exponential time, which cannot be recognizedby 2PFA.Proof. U a = U u = 15 − , U g = U c = 15 − (4)The idea of the proof is as follows. It consists of two phases. First we define a 2QCFA for L using quantum registerconsisting three orthogonal states. In second phase, we modify the 2QCFA such as natural mapping is performedfrom three-dimensional Euclidean space to unit sphere in two-dimensional Hilbert space H . We construct a 2QCFA M for the language L h = { x ∈ { a, u, g, c } ∗ | x = x r } with three quantum states { q , q , q } , where q is an initialstate. M has two unitary matrices U a = U u and U g = U c defined as follows. The automaton M proceeds as follows:Consider an input string w = w , w , ..., w n , the tape squares are indexed by 0 and n +1 consist both end-markers $ , respectively. The computation process of M starts with quantum state | q i is as follows. As while-loop 2 isimplemented, the input tape head traverse each input symbol and performs U a or U u on the quantum state (dependsupon whether the input symbol is a or u and performs U g or U c on the quantum state (depends upon whether theinput symbol is g or c , respectively. Suppose W i denote the matrix ( U a or U u ) and ( U g or U c ), as defined in (4),depending upon w i is ( a or u ) and ( g or c ). The automaton M changes its state after executing loop 2 as β | q i + β | q i + β | q i (5)for ( β , β , β ) T = 15 n W n ...W (1 , , T . As we repeat the subroutines 3, the input tape head is moved towards the leftdirection until the right-end marker is read, then shift the tape head one position to the right. Now, in loop 4, theinverses of U a and U g are performed, the quantum state is changed as γ | q i + γ | q i + γ | q i (6) TABLE I: Details of the 2QCFA for L Repeat the following endlessly:1. Set the initial quantum state q 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 U a or U u on the quantum state,respectively.(2.2).If the presently read symbol is g or c , execute U g or U c 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 right-end marker symbol $ , do the following:(4.1).If the presently read symbol is a or u , apply U − a or U − u on the quantum state, respectively.(4.2). If the presently scanned symbol is g or c , execute U − g or U − c 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 q , then it is rejected.Initialize the variable z =05. While the presently symbol scanned is not right-end marker 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. for ( γ , γ , γ ) T = W − n ...W − W n ...W (1 , , T . The quantum state is measured and M is said to be rejected withprobability P rej = γ + γ , else it collapses to initial state q . If w is a palindrome, then P rej = 0, else P rej > − n .Initialize the variable z equal to 0, which is stored in classical state. Finally, on executing the while-loop 5, the inputstring is said to be accepted if the loop is terminated with z = 0. It is known from standard result in probabilitytheory that probability of reaching the location n +1 is 1 n + 1 . On flipping the k coins, the probability of acceptance P acc = 1 / k ( n + 1). If the algorithm is repeated indefinitely, then the probability of rejectance is P r [ M rejects w ] = X i ≥ (1 − P acc ) i (1 − P rej ) i P rej = P rej P acc + P rej − P accP rej (7)and accepting probability is
P r [ M accepts w ] = X i ≥ (1 − P acc ) i (1 − P rej ) i +1 P acc = P acc − P acc P rej P acc + P rej − P accP rej (8)If the input string w ∈ L h , then the probability of M accepting w is 1. Suppose k ≥ max { log , − log ǫ } , it can bechecked that if w / ∈ L h , the M rejects the input string with probability atleast 1 − ǫ . B. Pseudoknot Structure
A pseudoknot structure is a double-hairpin structure that forms an extended quasi-continuous helix structure anddouble connecting loops [47]. It is formed when pairs are created between the bases outside and inside of a hairpin orinternal loop. Pseudoknot structure plays a crucial role in RNA functions such as regulation of splicing and translationand ribosome frameshifting [48]. It is considered as a key component of ribozymes or ribosomal RNAs. Fig 2 describesthe pseudoknot secondary structure. A closer look at pseudoknot structure shows a similarly with constructs of naturallanguage (i.e. dependencies are forced to cross) such as { a n g m u n c m | n, m ≥ } [49]. Thus, the number of a ’s is equalto the number of u ’s and correspondingly the number of g ’s is equal to the number of c ’s. Theorem 2.
A language L p = { a n g m u n c m | n, m ≥ } representing pseudoknot biomolecule structure can be recognizedby 2QCFA in polynomial time with error probability ǫ = ǫ + ǫ − ǫ ǫ .Proof. The idea of the proof is as follows. The construction of 2QCFA M consists of three phases. Firstly, it checkswhether the input is in form a + u + g + c + , if not, then the input string is said to be rejected. In second phase, we checkthat it is in form L = { a n g ∗ u n c ∗ | n ≥ } or not. If yes, then we check that it is in form L = { a ∗ g m u ∗ c m | m ≥ } . FIG. 2: Representation of psedoknot bio-molecular structure
Since, 2QCFA can be designed for L eq = { a n b n | n ≥ } . Similarly, it can recognize L and L in polynomial time. Qiu[27] proved that the class of languages recognized by 2QCFA are closed under intersection, reversal, complement andunion operations. For convenience, 2QCFA ǫ ( poly-time ) notation is used to denote the class of languages recognizedby 2QCFA in polynomial time with error probability ǫ ≥
0. Therefore, if L ∈ ǫ ( poly-time ) and L ∈ ǫ ( poly-time ), then L p = L ∩ L can be recognized by 2QCFA in polynomial time with ǫ = ǫ + ǫ − ǫ ǫ .Based on the above analysis, the prove of this theorem is described now more formally. Let M j = ( S j , Q j , Σ j , Θ j , δ j , q j, , s j, , S j,acc , S j,rej ) (9)be 2QCFA’s for recognizing L j in polynomial time with error probabilities ǫ j ≥ j =1, 2, where • S j = { s j, , s j, , ..., s j,m j } , • Q j = { q j, , q j, , ..., q j,n j } We construct a 2QCFA for L p such that M p = ( S, Q, Σ , Θ , δ, q , s , S acc , S rej ) where • S = S ∪ S ∪ { z ,i | i = 0 , , ..., n } , • Q = Q ∪ Q , • Σ = Σ ∪ Σ , • q = q , , • s = s , , • S acc = S ,acc ∪ S ,acc , • S rej = S ,rej ∪ S ,rej • Θ j and δ j are defined as1. Θ is defined as a transition function: S \ ( S ,acc ∪ S ,rej ) × Γ → U ( H ( Q )) ∪ P ( H ( Q )) (10)(a) if Θ ( s, σ ) ∈ U ( H ( Q )) corresponds to unitary operator over H ( Q ), then extend the Θ ( Q ) relating toΘ( s, σ ) | q ,i i = | q ,i i , for 0 ≤ i ≤ n and δ ( s, σ ) = δ ( s, σ ),(b) if Θ ( s, σ ) ∈ P ( H ( Q )), which denotes projective measurement over H ( Q ) and measurement is representedby projector set { P i } . Then, Θ( s, σ ) denotes an orthogonal measurement on H ( Q ) represented by { P ′ i }∪{ I } projection operators on P ( Q ) = P ( Q ∪ Q ), where P ′ i represent projection operators obtain by extending P i with P ′ i | q ,i i , for 0 ≤ i ≤ n and I represents mapping of projection operator to H ( Q ), i.e. an identityoperator.2. For any s ∈ S and σ ∈ Σ ∪ { , $ } (a) if Θ ( s, σ ) ∈ U ( H ( Q )) corresponds to unitary operator over H ( Q ), then extend the Θ ( Q ) relating toΘ( s, σ ) | q ,i i = | q ,i i , for 0 ≤ i ≤ n and δ ( s, σ ) = δ ( s, σ ),(b) if Θ ( s, σ ) ∈ P ( H ( Q )), which denotes projective measurement over H ( Q ) and measurement is representedby projector set { P k } , then Θ( s, σ ) is an orthogonal measurement on H ( Q ) represented by { P ′ k } ∪ { I } projection operators on P ( Q ) = P ( Q ∪ Q ), where P ′ k extend P k to H ( Q ) by defining P ′ k | q ,k i = 0, for0 ≤ k ≤ n .3. For any s ∈ S ,acc and σ ∈ Σ ∪ { , $ } (a) if σ = s, σ ) = I and δ ( s, σ ) = ( s, − I is an identity operator over H ( Q ),(b) else if σ = s, σ ) represents an orthogonal measurement by projectors {| q ,i i h q ,i | | q ,i ∈ Q } , δ ( s, σ )(1 , i ) = ( z ,i , δ ( z ,i , s , , z q,i , U ( q ,i , q , ), where U denotes a unitary operatorover H ( Q ) satisfy U | q ,i i = | q , i .Recall the languages L = { a n g ∗ u n c ∗ | n ≥ } and L = { a ∗ g m u ∗ c m | m ≥ } . In respect of 2QCFA M p designedabove, for any input string w ∈ { a, u, g, c } ∗ , we have considered the following cases: • if w ∈ L ∩ L , then at the end of the computation process M p enters a state S acc with probability atleast(1 − ǫ )(1 − ǫ ). Then, L p is said to be recognized by M p with probability atleast (1 − ǫ )(1 − ǫ ) = (1 − ( ǫ + ǫ − ǫ ǫ )). • if w ∈ L , but w / ∈ L , then at the end of the computation M P enters a state S ,acc × S ,rej with probability atleast(1 − ǫ )(1 − ǫ ). Then, L p is said to be rejected by M p with probability atleast (1 − ǫ )(1 − ǫ ) = (1 − ( ǫ + ǫ − ǫ ǫ )). • if w / ∈ L , then the state of M p is changed to S ,rej and it is said to be rejected with probability atleast 1 − ǫ .Hence, if the languages L and L are said to be recognized by 2QCFA’s M and M in polynomial time with errorprobabilities ǫ , ǫ ≥
0, respectively. Then, L p = L ∩ L is said to be recognized by M p in polynomial time with errorprobability ǫ = ǫ + ǫ − ǫ ǫ . C. Dumbbell Structure
Dumbbell shaped RNA structure is formed by analogy of DNA dumbbells comprised of two-helical stems closed bytwo hairpin loop structures. It plays an important role in analysis of local structures in DNA. The loops on its bothsides restrict its enzymatic cleavage and stabilize the duplex. It is successfully applied to transcriptional regulation[50]. Fig 3 shows the representation of dumbbell shaped RNA secondary structure.
FIG. 3: Representation of dumbbell bio-molecular structure
Theorem 3.
A language L d = { a n u n g m c m | n, m ≥ } representing dumbbell biomolecule structure can be recognizedby 2QCFA with one-sided error probability in polynomial time.Proof. Ambainis and Watrous proved that a language L eq = { a n b n | n ∈ N } can be recognized by 2QCFA inpolynomial time, which can be recognized by 2PFA in exponential time [20]. Similarly, it can be proved that alanguage L d = { a n u n g m c m | n, m ≥ } can be recognized by 2QCFA M d in polynomial time with one-sided errorprobability. It consists of three phases. Firstly, it examines whether the input string is in form a + u + g + c + , if not,then the input string is said to be rejected. Otherwise, in second phase, 2QCFA simulates the initial part of string todetermine whether a n u n is in L d , by using g in the right side of u as the right-end marker $ . If not, the computationis said to be rejected. Otherwise, in third phase, the 2QCFA finally checks g m c m is in L d and u is used as a left-endmarker g ’s and c ’s are equal, then it is said to be recognized with one-sided error probability inpolynomial time, otherwise rejected. V. CONCLUSION
The enhancement in many existing computational approaches provides momentum to biological systems and quan-tum simulations at the gene expression levels. It helps to test new abstract approaches for considering RNA, DNAand protein sequences. Previous attempts to model the aforementioned RNA secondary structures used formal gram-mar and finite automata theory. In this paper, we focused on well-known structures of RNA such as hairpin loop,pseudoknot and dumbbell biomolecular structures and modeled them using two-way quantum finite automata withclassical states. The crucial advantage of the quantum approach is that these secondary structures transcribed in for-mal languages takes exponential time for L h and polynomial time for L p and L d , respectively. It has been shown thattwo-way quantum finite automata with classical states are more superior than its classical variants by using quantumpart of finite size. For the future purpose, we will try to represent complex RNA structures in formal languages andmodel them using other quantum computational models. Additional Information
Conflict of interest
The authors declare that they have no conflict of interest.
Acknowledgement
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