I Introduction & Motivation
Recently, the connection between complex reactions and their thermodynamics has got overwhelming response among research communities. Initially, in the 1970s, Conrad (1, ) processed the information of molecular systems and stated that complex biochemical systems cannot be analyzed on classical computers. Till now, artificial approaches use complex biomolecules or logic gates based reactiondiffusion systems to solve the problems 23 ; 24 ; 25 . Classical systems are not robust and capable to describe quantum systems. Some tasks that are impossible in classical systems can be realized in quantum systems. Quantum computing is concerned with computer technology based on the principles of quantum mechanics, which describes the behavior and nature of matter and energy on the quantum level (2, ). Quantum computation demonstrates the computation power and other properties of the computers based on principles of quantum mechanics.
Models of finite automata are abstract computing devices, which play a crucial role to solve computational problems in theoretical computer science. Classical automata theory is closely associated with formal language theory, where automata are ranked from simplest to most powerful based on their language recognition power 3 . Classical automata theory has been of significant importance due to its practical realtime applications in the development of several fields. Therefore, it is a natural goal to study quantum variants of classical automata models, which play an important role in quantum information processing.
The quantum automata theory has been developed using the principles of quantum mechanics and classical automata. Quantum computational models make it possible to examine the resources needed for computations. Soon after the brainstorm of Shor’s factorization quantum algorithm 4 , the first models of quantum finite automata (QFAs) have been introduced. Initially, Kondacs and Watrous 6 , and Moore and Crutchfield 5 proposed the concept of quantum automata separately. There is a diversity of quantum automata models have been studied since then and demonstrated in various directions such as quantum finite automata 5 , Latvian QFA, 1.5way QFA, twoway QFA (2QFA) 6 , quantum finite state machines of matrix product states 54 , quantum sequential machines 58
, quantum pushdown automata, quantum Turing machine, quantum queue automata
56 , quantum multicounter machines (QMCM), quantum multihead finite automata (QMFA), quantum finite automata with classical states (2QCFA) 36 ; 37 , quantum omega automata 55 , state succinctness of twoway probabilistic finite automata (2PFA), QFA, 2QFA and 2QCFA 51 ; 48 ; 49 , interactive proof systems with quantum finite automata 52 ; 53 and semiquantum twoway finite automata 46 ; 47 and many more since last two decades (50, ; 57, ; 8, ; 9, ; 10, ). These models are effective in determining the boundaries of various computational features and expressive power. Quantum computers are more powerful than Turing machines and even probabilistic Turing machines. Thus, mathematical models of quantum computation can be view as generalizations of its physical models.Computational biochemistry has been a rapidly evolving research area at the interface between biology, chemistry, computer science, and mathematics. It helps us to apply computational models to understand biochemical and chemical processes and their properties. A combination of chemistry and classical automata theory provides a constructive means of refining the number of objects that allow to understand the energetic cost of computation 11 . The research has been consistently grown in the field of chemical computing. There exist two ways to model complex chemical reactions: abstract devices and formal models based on multiset rewriting 13 . The complex chemical reaction networks carry out chemical processes that mimic the workings of classical automata models. Recently, DuenasDiez and PerezMercader 11 ; 12 have designed chemical finite automata for regular languages and chemical automata with multiple stacks for contextfree and contextsensitive languages. Further, the thermodynamic interpretation of the acceptance/rejection of chemical automata is given. It is useful to understand the energetic cost of chemical computation. They have used the onepot reactor (mixed container), where chemical reactions and molecular recognition takes place after several steps, without utilizing any auxiliary geometrical aid.
In classical automata theory, it is known that twoway deterministic finite automata (2DFA) can be designed for all regular languages. It has also been investigated that twoway probabilistic finite automata (2PFA) can be designed for a nonregular language in an exponential time 14 ; 15 . The research has consistently evolved in the field of quantum computation and information processing. In quantum automata theory, it has been proved that 2QFA can be designed for L with onesided bounded error and halted in linear time. Moreover, it has been demonstrated that 2QFA can be also designed for noncontext free language 6 . Hence, 2QFA is strictly more powerful than its classical counterparts based on language recognition capability.
The field of chemistry and chemical computing plays a significant role in the evolution of computational models to mimic the behavior of systems at the atomic level. It is greatly influenced by the computing power of quantum computers. Motivated from the abovementioned facts, we have modeled chemical reactions in the form of formal languages and represented them using twoway quantum finite automata. The main objective is to examine how chemical reactions perform chemical sequence identification equivalent to quantum automata models without involving biochemistry or any auxiliary device. 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 definition of twoway quantum finite automata is given in Sect. 3. In Sect. 4, the chemical reactions are transcribed in formal languages and modeled using twoway quantum finite automata approach. Summary of work is given in Section 5. Finally, Sect. 6 is the conclusion.
i.1 Prior work
The field of chemical computation has rich and interesting history. Various researchers have represented the chemical computation using the concept of logic gates based reactiondiffusion systems and artificial intelligence approaches. In early 1970s, Conrad
1 differentiated the information processing on molecules from digital computing. Nearly a decade after, Okamoto et al. 16 proposed the concept of a theoretical chemical diode for cyclic enzyme systems. It has been proved that it can be used to analyze the dynamic behavior of metabolic switching events in biocomputer. In 1991, Hjelmfelt et al. 17designed neural networks and finite state machines using chemical diodes. It has been investigated that the execution of a universal Turing machine is possible using connecting chemical diodes.
In 1995, Tóth and Showalter 18 implemented AND and OR logic gates using reactiondiffusion systems, where the signals are programmed by chemical waves. It was the first empirical realization of chemical logical gates. In 1997, Magnasco 19 showed that logic gates can be constructed and executed in the chemical kinetics of homogeneous solutions. It has been proved that such constructions have computational power equivalent to Turing machine. Adamatzky and Lacy Costello 20 experimentally realized the Chemical XOR gate by following the same approach of Toth and Showlter in 2002. Further, Górecki et al. 21 constructed the chemical counters for an information processing in the excitable reactiondiffusion systems.
It is one of the most promising new areas of research. Some of the difficulties can be caused by connecting several gates together for advanced computation. Thus, recently, researchers started focusing on native chemical computation, i.e. without reactiondiffusion systems. In 1994, Adleman 22 proposed the concept of DNA computing and solved the Hamiltonian path problem by changing DNA strands. In 2009, Benenson 23 reviewed biological measurement tools for new generation biocomputers. Prohaska et al. 24 studied protein domain using chromatin computation and introduced chromatin as a powerful machine for chemical computation and information processing. In 2012, Bryant 25 proved chromatin computer as computationally universal and used to solve an example of combinatorial problem.
The structures of DNA and RNA are represented using the concept of classical automata theory 26 27 . Krasinski et al. 28 represented the restricted enzyme in DNA with pushdown automata in circular mode. Khrennikov and Yurova 29 modeled the behavior of protein structures using classical automata theory and investigated the resemblance between the quantum systems and modeling behavior of proteins. Bhatia and Kumar 30 modeled ribonucleic acid (RNA) secondary structures using twoway quantum finite automata, which are halted in linear time. Recently, DuenasDiez and PerezMercader designed molecular machines for chemical reactions. The native chemical computation has been implemented beyond the scope of logic gates i.e. with chemical automata 12 . It has been demonstrated that chemical reactions transcribed in formal languages, can be recognized by Turing machine without using biochemistry 11 .
Ii Preliminaries
In this section, some preliminaries are given. 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 3 , quantum information and computation 2 7
. Linear algebra is inherited from quantum mechanics to describe the field of quantum computation. It is a crucial mathematical tool. It allows us to represent the quantum operations and quantum states by matrices and vectors, respectively, that obey the rules of linear algebra. The following are the notions of linear algebra used in quantum computational theory:

Vector space (V) 7 : A vector space (V) is defined over the field of complex numbers consisting of a nonempty set of vectors, satisfying the following operations:

Addition: If two vectors and belong to V, then .

Multiplication by a scalar: If belongs to V, then , where .


Dirac notation 2 : In quantum mechanics, the Dirac notation is one of the most peculiarities of linear algebra. The combination of vertical and angle bars () is used to unfold quantum states. It provides an inner product of any two vectors. The bra and ket represent the row vector and column vector, respectively.
(1) where indicates the complex conjugate of complex number .

Quantum bit 10
: A quantum bit (qubit) is a unit vector defined over complex vector space
. In general, it is represented as a superposition of two basis states labeled and .(2) The probability of state occurrence
is and is . It satisfies that . The two complex amplitudes ( and ) are represented by one qubit. Thus, complex amplitudes can be represented by qubits. 
Quantum state 2 : A quantum state is defined as a superposition of classical states,
(3) where are complex amplitudes and are classical states for . Therefore, a quantum state can be represented as ndimensional column vector.
(4) 
Unitary transformation: In quantum mechanics, the transformation between the quantum systems must be unitary. Consider a state of quantum system at time t: transformed into state at time t’: , where complex amplitudes are associated by , where U denotes a time reliant unitary operator satisfies that , and 2 .

Hilbert space: A physical system is described by a complex vector space called Hilbert space 7 . It allow us to describe the basis of the quantum system. The direct sum or inner product of two subspaces, satisfying the following properties for any vectors:

Linearity: ()

Symmetric property:

Positivity: and iff , where .
where and .


Quantum finite automata (QFA) 34 : It is defined as a quadruple (), where

Q is a set of states,

is an input alphabet,

Hilbert space and is an initial vector such that ,

and is an acceptance projection operator on ,

denotes a unitary transition matrix for each input symbol ().
The computation procedure of QFA consists of an input string . The automaton works by reading each input symbol and their respective unitary matrices are applied on the current state, starting with an initial state. The quantum language accepted by QFA is represented as a function , where . The tape head is allowed to move only in the right direction. Finally, the probability of QFA being in an acceptance state is observed; i.e. indicating whether the input string is accepted or rejected by QFA. It is also called as a realtime quantum finite automaton.

Based on the movement of tape head, QFA is classified as oneway QFA, 1.5way QFA, and 2QFA. In 1.5way QFA, the tape head is permitted to move only in the right direction or can be stationary, but it cannot move towards the left direction. It has been proved that it can be designed for noncontext free languages, if the input tape is circular
35 . In this paper, we focused on the 2QFA model due to its more computational power than classical counterparts.Iii Twoway quantum Finite automata
A quantum finite automaton (QFA) is a quantum variant of a classical finite automaton. In QFA, quantum transitions are applied on reading the input symbols from the tape 5 . Twoway quantum finite automaton (2QFA) is a quantum counterpart of a twoway deterministic finite automaton (2DFA). In 2QFA, the tape head is allowed to move either in the left or right direction or can be stationary. The illustration of 2DFA is shown in Fig 1. It consists of a readonly input tape consisting of a sequence of cells, each one storing a input symbol. The readonly input tape is scanned by an input head, which is allowed to move in both directions or can be stationary. At each step of a computation, a finite state control should be in state from a finite set Q.
Definition 1.
5 A twoway quantum finite automaton is represented as sextuple , where

is a finite set of states.

is an input alphabet,

Transition function is defined by , where is a complex number, and represent the left, stationary and right direction of tape head.

, where represent the set of nonhalting, accepting, rejecting states respectively. The transition function must satisfies the following conditions:

Local probability and orthogonality condition:

First separability condition:

Second separability condition:
For each , a 2QFA is said to be simplified, if there exists a unitary linear operator on the inner product space such that . The transition function is represented as
(5) 
where is a coefficient of in .
Consider an input string , written on the input tape enclosed with both endmarkers such as . The computation of 2QFA is progress as follows. The tape head is above the input symbol and the automaton is in any state . Then, the state of 2QFA is changed to with an amplitude and moves the tape head one cell towards right, stationary and in left direction according to . It corresponds to a unitary evolution in the innerproduct space .
A computation of a 2QFA is a chain of superpositions where denotes an initial configuration. For any , when the automaton is observed in a superposition state with an amplitude , then it has the form , where represents the set of configurations. The probability associated with a configuration is calculated by absolute squares of amplitude. Superposition is said to be valid; if the sum of the squared moduli of their probability amplitudes is unitary. In quantum theory, the time evolution is specified by unitary transformations. Each transition function prompts a transformation operator over the Hilbert space in linear time.
Iv Modeling of Chemical Reactions
Before, we recognize the chemical reactions using twoway quantum finite automata model, it is important to show how computational chemistry works. Fig 2. shows the illustration of language recognition by the chemical computation model. It consists of three parts: (i) a mixed container where the computation process occurs, (ii) an input translator that translates the chemical aliquots into input symbols and gives them consecutively depending upon the processing time, (3) a system to monitor the response of an automaton as a chemical criterion. Finally, the chemical computation produces welldefined chemical accept/ reject signatures for the input. For instance, if the number of a’s and b’s are equal in the input, then the chemical computation produces heat i.e. an input is said to be accepted. Otherwise, if no heat is released at the end of computation, then the input is said to be rejected by the system. The following are the construction of twoway quantum finite state machines of chemical reactions.
Theorem 1.
Twoway quantum finite automata can recognize all regular languages.
Proof.
The proof has been shown in 5 . ∎
iv.1 Chemical reaction1 consisting regular language
For an illustrative and visual implementation, we can choose a precipitation reaction in an aqueous medium such as
(6) 
If, during computation, a white precipitate of silver iodate is observed, then the input string is said to be accepted; if the solution is clear from precipitate, the string has been rejected because there was no reaction. Therefore, we have chosen the recipes of alphabet symbols a for potassium iodate () and b for silver nitrate (), quantitatively. Fig. 3 shows the chemical representation of symbols a and b, the bimolecular precipitation reaction 11 . If the precipitate AgIO is not presented in the solution, then the computation is said to be rejected. For example, the input string is said to be accepted due to presence of precipitate or, equally, the heat has been determined during computation. But, the input is said to be rejected due to absence of precipitate, or, precisely, the heat has not been observed. The Kleene star () operator is a set of infinite strings of all lengths over input alphabet as well as empty string (). Fig. 4 shows the corresponding theoretical 2QFA state transition graph to recognize .
Theorem 2.
A language representing precipitation reaction in Eq. (6) can be recognized by 2QFA.
Proof.
The idea of this proof is as follows. The initial state reads a rightend marker and moves the head towards the right direction. If there is no occurrence of symbol b, then it shows no precipitate, and the input is said to be rejected by the 2QFA. Similarly, on reading the symbol b, the state is changed into . If there is no occurrence of symbol a, then the state is transformed into rejecting state . If the input string contains at least one a and one b, then silver iodate is present during computation and it is said to be recognized by 2QFA. A 2QFA for is defined as follows:
where

where and are used to move the head towards the $ on reading a’s and b’s respectively. The states and are used to confirm the last symbol read by head is a and b, respectively. .

is an initial state, and .

The specification of transition functions are given in Table 1.
It can be noted that in 2QFA where transition matrices consist 0 and 1, i.e. basically a twoway reversible finite automata (2RFA). Therefore, 2QFA can be designed for all the languages accepted by 2RFA. In transition matrix, each column and row have exactly only one entry 1. Hence, the dot product of any two rows is equal to zero. It is known that the language recognition power of 2RFA is an equivalent to 2DFA.
Head Functions:  
∎
iv.2 Chemical reaction2 consisting contextfree language
Next, we have considered the contextfree language from Chomsky hierarchy satisfying the balanced chemical reaction between NaOH and malonic acid is as follows:
(7) 
The language generated by the abovementioned chemical reaction is consisting Dyck language of all words with balanced parentheses. 2QFA is designed for is as follows:
Theorem 3.
A language consisting Dyck language of all words with balanced parentheses can be recognized by 2QFA with probability 1, otherwise rejected with probability at least , where N is any positive number.
Proof.
The idea of this proof is as follows. It consists of three phases. First, the initial state reads a first symbol and both heads start moving towards the rightend marker $. If the input string starts with closed parentheses, then it is said to be rejected. On reading the leftend marker , the computation is split into N paths, denoted by . Each path possesses equal amplitude . Along the N different paths, each path moves deterministically to the rightend marker . Each computational path keep track of the open parentheses with respect to the closed parentheses. If at the end of computation, the excess of open parentheses is observed, then it is said to be rejected. It means pH value is greater than midpoint pH and intermediate gray tone is observed. Secondly, if there is an excess of closed parentheses, then the darkest gray tone is observed i.e. pH value is less than midpoint pH. It is said to be rejected by 2QFA with probability . If there is a balanced occurrence of open and closed parentheses, the input string is said to be accepted with probability 1. Hence, pH value is equal to midpoint pH and the lightest gray tone is observed at the end of computation. A 2QFA for is defined as follows:
where

, where is used to check whether the first symbol is an open parentheses or not, and are used to traverse the input string.

is an initial state, and .

The specification of transition functions is given in Table 2.
∎
Head Functions: 
iv.3 Chemical reaction3 consisting contextsensitive language
To implement a chemical 2QFA for contextsensitive language, we have used BelousovZhabotinsky (BZ) reaction network for the nonlinear oscillatory chemistry 11 , which consists temporal oscillation in the sodium bromate and malonic acid system 33 as
(8) 
In 2019, DuenasDiez and PerezMercader 11 designed chemical Turing machine for BZ reaction network. The chemical reaction is fed sequentially to the reactor as , where n0. It is transcribed in formal language as . The symbol a is interpreted as a fraction of sodium bromate, b is used for malonic acid and symbol c is transcribed as a quantity of NaOH. It is known that is a contextsensitive language and cannot be recognized by finite automata or pushdown automata with a stack. Although, it can be recognized by twostack PDA. We have shown that can be recognized by 2QFA without using any external aid.
Theorem 4.
A language can be recognized by 2QFA in linear time. For a language and for arbitrary Ncomputational paths, there exists a 2QFA such that for , it accepts w with bounded error and rejects with probability at least .
Proof.
The design of proof for BZ reaction network is as follows. It consists of two phases. First, the 2QFA traverse the input to check the form . On reading the rightend marker $, the computation is split into N paths such that . Secondly, the first path is used to check the number of ’s and ’s are equal or not. The second path is used to check the initial part of an input string to identify it is in . On reading the rightend marker $, the both paths are split into N different paths with an equal amplitude . Finally, upon reading the rightend marker #, if the number of a’s and b’s and b’s and c’s are equal in respective computational paths. Then, all paths come into N
way Quantum Fourier transform (QFT) and either one acceptance state or rejectance states are observed. Suppose, if the input string is not in the corrected form, then all computation paths read the # at different times. Thus, their amplitudes do not cancel each other and the input string is said to be rejected with probability
. Otherwise, the input string is said to be recognized by 2QFA with probability 1. ∎V Summary
In summary, 2QFA model can be efficiently designed for balanced chemical reaction and BelousovZhabotinsky (BZ) reaction network with onesided error bound, which are halted in linear time. Table 3 shows the language recognition ability of different computational models. The classical 2DFA and 2PFA are known to be equal in computational power to oneway deterministic finite automata (1DFA) 45 ; 46 . It has been proved that 2PFA’s can be designed for nonregular languages in expected polynomial time. Additionally, it has been demonstrated that the chemical PDA can be designed for aforementioned chemical reactions with multiple stacks. The recognition of languages by native chemical automata cab be found in 12 ; 13 ; 14 . But, we have shown that 2QFA can recognize such chemical reactions, without any external aid. It has been proved that 2QFA is more powerful than classical variants, because it follows the quantum superposition principle to be in more than one state at a time on the input tape. For the execution, it needs atleast quantum states to store the position of tape head, where n denotes the length of an input string.
Languages  Class  2DFA/2PFA  Chemical FA/PDA  2QFA  

RL  ✓  ✓  ✓  

CFL  ✗ 

✓  

CSL  ✗ 

✓ 

RL, CFL and CSL stand for regular languages, contextfree languages and contextsensitive languages, respectively.
Vi Conclusion
The enhancement in many existing computational approaches provides momentum to molecular and quantum simulations at the atomic level. It helps to test new abstract approaches for considering molecules and matter. Previous attempts to model the aforementioned chemical reactions used finite automata and pushdown automata with multiple stacks. In this paper, we focused on wellknown languages of Chomsky hierarchy and modeled them using twoway quantum finite automata. The crucial advantage of the quantum approach is that these chemical reactions transcribed in formal languages can be parsed in linear time, without using any external aid. We have shown that twoway quantum automata are more superior than its classical variants by using quantum transitions. To the best of our knowledge, no such modeling of chemical reactions is performed using quantum automata theory so far. For the future purpose, we will try to represent complex chemical reactions in formal languages and model them using other quantum computational models.
Vii 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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