Quantum Algorithms for Biomolecular Solutions of the Satisfiability Problem on a Quantum Machine

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IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 7, NO. 3, SEPTEMBER 2008 215

Quantum Algorithms for Biomolecular Solutions of

the Satisfiability Problem on a Quantum Machine

Weng-Long Chang, Ting-Ting Ren, Jun Luo, Mang Feng, Minyi Guo, Senior Member, IEEE,*

and Kawuu Weicheng Lin

Abstract—In this paper, we demonstrate that the logic compu-tation performed by the DNA-based algorithm for solving gen-eral cases of the satisfiability problem can be implemented more efficiently by our proposed quantum algorithm on the quantum machine proposed by Deutsch. To test our theory, we carry out a three-quantum bit nuclear magnetic resonance experiment for solving the simplest satisfiability problem.

Index Terms—Molecular algorithms, quantum algorithms, the NP-complete problems, the satisfiability problem.

I. INTRODUCTION

S

INCE the publication of Deutsch’s [1] and Adleman’s [2] seminal articles, various quantum algorithms and DNA-based algorithms have been, respectively, proposed for many computational problems. So far, the most frequently cited quan-tum algorithms are Shor’s algorithms for solving factoring inte-gers and discrete logarithm [3] and Grover’s search algorithm [4] for unsorted databases. On the other hand, famous DNA-based algorithms are used to solve factoring integers [5] and the set-partition problem [6].

II. QUANTUMALGORITHMS FORBIOMOLECULARSOLUTIONS OF THESATISFIABILITYPROBLEM

A. All of the Possible Solutions to the Satisfiability Problem A clause is a formula of the form un∨ un−1· · · ∨ u2∨ u1,

where each uk for 1≤ k ≤ n is a Boolean variable or its

nega-tion. In general, a satisfiability problem includes a Boolean formula of the form C1∧ C2· · · ∧ Cm, where each Cj for

1≤ j ≤ m is a clause. Then, the question is to find values of the variables so that the whole formula has the value 1.

Assume that U is a set of 2n _{possible choices and equal}

to {un un−1· · · u2u1|∀uk ∈ {0, 1} for 1 ≤ k ≤ n} and each

element represents one of 2n _{combinational states for n}

Manuscript received April 23, 2007; revised May 13, 2008. Current version published August 29, 2008. This work was supported in part by the National Science Foundation of R.O.C. under Grant 2221-E-151-008 and Grant 96-2218-E-151-004, and in part by the National Natural Science Foundation of China under Grant 10774163 and Grant 2006CB921203. Asterisk indicates

corresponding author.

*W.-L. Chang is with the Department of Computer Science and Information Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung City 80778, Taiwan, R.O.C. (e-mail: changwl@cc.kuas.edu.tw).

K. W. Lin is with the Department of Computer Science and Information Engineering, National Kaohsiung University of Applied Sciences, Kaohsiung City 80778, Taiwan, R.O.C. (e-mail: linwc@cc.kuas.edu.tw).

T.-T. Ren, J. Luo, and M. Feng are with the State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China (e-mail: ttren@wipm.ac.cn; jluo@wipm.ac.cn; mangfeng@wipm.ac.cn).

M. Guo is with the School of Computer Science and Engineering, University of Aizu, Fukushima 965-8580, Japan (e-mail: minyi@u-aizu.ac.jp).

Digital Object Identifier 10.1109/TNB.2008.2002286

Boolean variables. For the sake of presentation, we sup-pose that u0

k is used to denote the value of uk to be zero

and u1

k means value of uk to be one. The jth element

in U can be represented as a unique computational state vector |u1 = u1,1 u1,2 · · · u1,2nT

1×2n, where u_1,j =

1∀ u1,h= 0 for 1≤ h = j ≤ 2n. The corresponding

compu-tational state vector for the first element, u0

n u0n−1· · · u02 u01,

in U is [ 1 0 · · · 0 ]T

1×2n, and the corresponding

compu-tational state vector for the last element, u1

n u1n−1· · · u12 u11,

in U is [ 0 0 · · · 1 ]T

1×2n. For the sake of presentation,

we assume that B is the set of the corresponding com-putational state vectors to the elements in U and B = {[ 1 0 · · · 0 ]T

1×2n · · · [ 0 0 · · · 1 ]T₁_×2n}. Because each

component in B is a coordinated vector, we span B = C2n

[1, 3, 4], where C2n

is a Hilbert space. This implies that the set B is an orthonormal basis in a Hilbert space.

B. Computational Space of Molecules for the Satisfiability Problem

The following biomolecular operations cited from [2] will be used to construct computational space of molecules for solv-ing the satisfiability problem with m clauses and n Boolean variables.

Definition 2.1: Given a set U ={un un−1· · · u2 u1|∀uk ∈

{0, 1} for 1 ≤ k ≤ n} and a Boolean variable uj, the

biomolecular operation, “Append-Head,” appends uj onto

the head of every element in the set U . The for-mal representation is written as Append-Head (U, uj) =

{uj un un−1· · · u2 u1|∀uk ∈ {0, 1} for 1 ≤ k ≤ n and uj ∈

{0, 1}}.

{0, 1} for 1 ≤ k ≤ n} and a Boolean variable uj, the

biomolecular operation, “Append-Tail,” appends ujonto the end

of every element in the set U . The formal representation is writ-ten as Append-Tail(U, uj) ={un un−1· · · u2 u1 uj|∀uk ∈

{0, 1} for 1 ≤ k ≤ n and uj ∈ {0, 1}}.

{0, 1} for 1 ≤ k ≤ n}, the biomolecular operation, “Discard (U )” sets U to be an empty set.

{0, 1} for 1 ≤ k ≤ n}, the biomolecular operation “Amplify (U,{Ui})” creates a number of identical copies, Ui, of the set

U , and then discard(U ).

{0, 1} for 1 ≤ k ≤ n} and a Boolean variable, uj, if

the value of uj is equal to one, then the biomolecular

extract operation creates two new sets, + (U, u1 j) =

{un un−1 · · · u1j · · · u2 u1|∀uk ∈ {0, 1} for 1 ≤ k = j ≤ n}

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216 IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 7, NO. 3, SEPTEMBER 2008

and−(U, u1

j) ={un un−1 · · · u0j · · · u2 u1|∀uk ∈ {0, 1} for

1≤ k = j ≤ n}. Otherwise, it produces another two new sets, + (U, u0_j) ={un un−1· · · u0j· · · u2 u1|∀uk ∈ {0, 1} for 1 ≤

k = j ≤ n} and −(U, u0

j) ={un un−1· · · u1j· · · u2 u1|∀uk ∈

{0, 1} for 1 ≤ k = j ≤ n}.

Definition 2.6: Given m sets U1· · · Um, the biomolecular

merge operation,∪(U1, . . . , Um) = U1∪ · · · ∪ Um.

{0, 1} for 1 ≤ k ≤ n}, the biomolecular operation “Detect (U )” returns a true if U = ∅. Otherwise, it returns a false.

{0, 1} for 1 ≤ k ≤ n}, the biomolecular operation “Read(U)” performs an arbitrary element in U . Even if U contains many different elements, the biomolecular operation can give an ex-plicit description of exactly one of them.

For solving the satisfiability problem with m clauses and n Boolean variables, the following biomolecular algorithm can be used to create all of the 2n _{possible choices. A set U is an}

empty set and is regarded as the input set of the DNA-based algorithm. The second parameter n in CombinationalStates(U , n) is to represent the number of Boolean variables.

Procedure CombinationalStates(U , n) (0a)Append-Tail(U1, u1n). (0b) Append-Tail(U2, u0n). (0c) U =∪(U1, U2). (1) For k = n− 1 downto 1 (1a) Amplify(U, U1, U2). (1b) Append-Tail(U1, u1k). (1c) Append-Tail(U2, u0k). (1d) U =∪(U1, U2). End For End Procedure

Lemma 2.1: For solving the satisfiability problem with m clauses and n Boolean variables, 2n _{possible choices created}

from the DNA-based algorithm, CombinationalStates(U , n), form an orthonormal basis of a Hilbert space (i.e., a complex vector space, C2n

).

C. Computational Space of Quantum Mechanical Solution for the Satisfiability Problem

A quantum bit (qubit) has two “computational basis vectors” |0 and |1 of the 2-D Hilbert space corresponding to the classi-cal bit values 0 and 1 [1], [3], [4], and an arbitrary state|ϕ of a qubit is a linearly weighted combination of the computational basis vectors (2.1):|ϕ = l1· |0 + l2· |1, where the weighted

factors l1and l2 ∈ C2are the so-called probability amplitudes,

with|l1|2+|l2|2 = 1. A collection of n qubits is called a

quan-tum register (qregister) of size n. It may include any of the 2n-D computational basis vectors, n qubits of size, or arbitrary superposition of these vectors [1], [3], [4].

D. Lipton’s DNA-Based Algorithms for Solving the Satisfiabil-ity Problem

Lipton’s DNA-based algorithm [7] for solving the satisfiabil-ity problem is described next. The symbol|Cj| in the following

algorithm is applied to represent the number of Boolean vari-ables and their negations in the jth clause in a formula.

Algorithm 2.1: Lipton’s DNA-based algorithm for solving the satisfiability problem.

(1) CombinationalStates(U, n) (2) For j = 1 to m do begin (3) Fori = 1 to|Cj| do begin

(4) If the ith element in the jth clause is one of n Boolean

variables uk, Then

(5) Ui= +(U, u1k) and U =−(U, u1k)

(6) Else

(7) Ui= +(U, u0k) and U =−(U, u0k)

(8) End If (9) End For (10) Discard(U ) (11) For i = 1 to|Cj| do begin (12) U =∪(U, Ui) (13) End For (14) End For

(15) If (Detect(U ) = = true) Then (15a) Read(U )

End If

(16) End Algorithm

Lemma 2.2: Algorithm 2.1, Lipton’s DNA-based algorithm, can be applied to solve the satisfiability problem with m clauses and n Boolean variables.

E. Introduction of Quantum Gates for Solving the Satisfiability Problem

The time evolution of the states of quantum registers can be modeled by means of unitary operators that are often referred to as quantum gates [1], [3], [4]. Therefore, a quantum gate can be regarded as an elementary quantum-computing device that performs a fixed unitary operation on selected qubits during a fixed period of time. TheNOTgate is a one-qubit gate and sets the only (target) bit to its negation. TheCNOT(controlledNOT) gate is a two-qubit gate and flips the second qubit (the target qubit) if and only if the first qubit (the control qubit) is one. TheCCNOT (controlled-controlled-NOT) gate is a three-qubit gate and flips the third qubit (the target qubit) if and only if the first qubit and second qubit (the two control qubits) are both one.

F. Constructing Quantum Networks for Solving the Satisfiabil-ity Problem

The operationsOR and AND are implemented by quantum circuits in Figs. 1 and 2, respectively. For evaluating a clause with the form un∨ un−1· · · ∨ u2∨ u1, three quantum

regis-ters|un· · · u1 , |yn· · · y1 , and |rnrn−1· · · r1r0 are needed.

Therefore, its evaluating computationis equal to (2.2)

|rnrn−1· · · r1|r0|yn· · · y1|un· · · u1 → |(rn ⊕ ¯yn• ¯rn−1)

· · · (r1⊕ ¯y1• ¯r0)|r0|yn· · · y1|un· · · u1

where• denotes operationANDof their negations of two Boolean variables{¯yk, ¯rk−1} for 1 ≤ k ≤ n. The first bit, r0, in the third

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CHANG et al.: QUANTUM ALGORITHMS FOR BIOMOLECULAR SOLUTIONS OF THE SATISFIABILITY PROBLEM ON A QUANTUM MACHINE 221

Fig. 5. Experimental spectra (a)–(c) of the three-qubit solution for the satisfia-bility problem after the readout on the first, second, and third qubits, respectively.

The subscripts are the phases (i.e., along the x or y axis) of the pulse, and the superscripts are the nuclei to which the pulses are applied. Then, we could obtain the total pulse sequence by connecting and optimizing the aforesaid pulses according to the quantum circuit.

Step 3 is the measurement, where a readout pulse is applied to each qubit to obtain the spectra.

Note that in NMR measurements, the frequencies and phases of NMR signals could clearly indicate the state the system

evolved to after the readout pulses had been applied. In our experiment, the phases of the reference of 13_{C spectra from}

a thermal equilibrium were adjusted to be in absorption (i.e., to be positive), and then the same phase corrections were used to determine the absolute phases of the experimental spectra of 13_{C after the algorithm was accomplished. In our}

case, the final state was (|000123+|101123)/ √

2 = (|0013+ |1113)|02/

√

2 which means the first and the third qubits are entangled. As the readout by NMR is a weak measurement, we have no state collapse after the measurement. Besides, only sin-gle quantum coherence can be detected in NMR. As a result, we have to employ some additional operations for detecting the output state (|000 > +|101 >)/√2. We may detect the second qubit directly by applying a π/2 readout pulse along the x-axis, yielding Fig. 5 (b). But for the first and third qubits, we need to disentangle them before measurement. To this end, we apply a CNOTgate, respectively, on the first and second qubits followed by anotherCNOTgate, respectively, on the second and first qubits to get the state (|000 > +|011 >)/√2. Then the first qubit can be read out by a single π/2 pulse along the x-axis, as shown in Fig. 5 (a). Similar steps applied to the third qubit result in the spectrum in Fig. 5 (c). It is evident that the experimental results are in good agreement with our theoretical prediction.

Therefore, due to the fact that NMR quantum operations are not made on individual nuclear spins, but on spin ensemble, there is a difference in the operation between Fig. 4 and our experiment. Some remarks must be addressed. First of all, the three-qubit NMR experiment that we have carried out suffices to make a comprehensive test for our theory, because we have achieved the key aspects of our theory. Although the simple case with three qubits did not reflect the efficiency of quantum com-putation for the SAT problem, we argue that, with more variables and clauses involved, the quantum computing efficiency would be more and more evident, which has been reflected in our pre-vious discussion about the computational complexity. Second, DNA computation does not involve entanglement, whereas en-tanglement does appear in our quantum treatment. The necessity of additional operations to disentangle the output qubits is not the intrinsic characteristic of our quantum mechanical treatment, but due to the unique feature of the NMR technique. Anyway, those additional operations have not changed the essence of our implementation.

VI. CONCLUSION

By using the mature technique of NMR, we have carried out a solution for the simplest satisfiability problem. The experimen-tal results are in good agreement with the theoreticalprediction.

REFERENCES

[1] D. Deutsch, “Quantum theory, the Church–Turing principle and the uni-versal quantum computer,” in Proc. Roy. Soc. Lond. Ser. A, 1985, vol. 400, pp. 97–117.

[2] L. Adleman, “Molecular computation of solutions to combinatorial prob-lems,” Science, vol. 266, pp. 1021–1024, 1994.

[3] P. W. Shor, “Algorithm for quantum computation: Discrete logarithm and factoring algorithm,” in Proc. 35th Annu. IEEE Symp. Found. Comput.

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222 IEEE TRANSACTIONS ON NANOBIOSCIENCE, VOL. 7, NO. 3, SEPTEMBER 2008

[4] L. K. Grover, “A fast quantum mechanical algorithm for database search,” in Proc. Twenty-Eighth Annu. ACM Symp. Theory Comput., 1996, pp. 212– 219.

[5] W.-L. Chang, M. Ho, and M. Guo, “Fast parallel molecular algorithms for DNA-based computation: factoring integers,” IEEE Trans. Nanobiosci., vol. 4, no. 2, pp. 149–163, Jun. 2005.

[6] W.-L. Chang, “Fast parallel DNA-based algorithms for molecular compu-tation: The set-partition problem,” IEEE Trans. Nanobiosci., vol. 6, no. 1, pp. 346–353, Dec. 2007.

[7] R. Lipton, “DNA solution of hard computational problems,” Science, vol. 268, pp. 542–545, 1995.

[8] D. G. Cory, M. D. Price, and T. F. Havel, “Nuclear magnetic resonance spectroscopy: An experimentally accessible paradigm for quantum com-puting,” Phyisca D, vol. 120, no. 1–2, pp. 82–101, 1998.

Weng-Long Chang received the Ph.D. degree in

computer science and information engineering from the National Cheng Kung University, Tainan, Taiwan, R.O.C., in 1999.

He is currently an Associated Professor at the National Kaohsiung University of Applied Sciences, Kaohsiung, Taiwan. His current research interests in-clude DNA-based algorithms, quantum algorithms, and languages and compilers for parallel computing.

Ting-Ting Ren is currently working toward the Ph.D.

degree at Wuhan Institute of Physics and Mathemat-ics, Chinese Academy of Sciences, Wuhan, China.

Her current research interests include quantum computation, quantum algorithm, and nuclear mag-netic resonance (NMR) techniques.

Jun Luo received the Ph.D. degree in physics at

Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan, China.

He is currently an Associate Professor of Physics at the State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences. His current research interests include exper-imental fields of nuclear magnetic resonance (NMR) quantum computing and the enhancement of the sen-sitivity of NMR signals with spin-exchange optical pumping methods.

Mang Feng received the Ph.D. degree in condensed

matter physics at the Centre for Fundamental Physics, University of Science and Technology of China, Hefei, China.

He is currently a Full Professor at the Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathemat-ics, Chinese Academy of Sciences, Wuhan, China. His current research interests include quantum com-putation and quantum communication with atoms and solid-state materials, entanglement and decoherence: quantum information processing with decoherence-free subspace, quantum al-gorithms and their realization, investigation of quantum optics problem, such as quantum treatment of interaction between matter and radiation field (based on Jaynes–Cummings model), and other aspects concerning quantum mechanics and mathematical physics.

Minyi Guo (M’02–SM’07) received the Ph.D. degree

in computer science from the University of Tsukuba, Tsukuba, Japan.

Before 2000, he had been a research scientist of NEC Corporation, Japan. He is currently a Professor in the School of Computer Science and Engineer-ing, University of Aizu, Fukushima, Japan. He was also a Visiting Professor of Georgia State University, USA, Hong Kong Polytechnic University, University of Hong Kong, National Sun Yet-Sen University in Taiwan, University of Waterloo, Canada, and Univer-sity of New South Wales, Australia, and an Adjunct Professor at Shanghai Jiao Tong University, China. He is the Editor-in-Chief of the Journal of Embedded

Systems. He is also on the Editorial Boards of the Journal of Pervasive ing and Communications, International Journal of High Performance Comput-ing and NetworkComput-ing, Journal of Embedded ComputComput-ing, Journal of Parallel and Distributed Scientific and Engineering Computing, and International Journal of Computer and Applications. His current research interests include parallel

and distributed processing, parallelizing compilers, pervasive computing, em-bedded software optimization, molecular computing, and software engineering. He is the author or coauthor of more than 150 research papers published in international journals and conferences.

Dr. Guo was the General Chair, and the Program Committee or Organizing Committee Chair for many international conferences. He is the founder of the International Conference on Parallel and Distributed Processing and Applica-tions (ISPA) and the International Conference on Embedded and Ubiquitous Computing (EUC). He is a member of the Association for Computing Machin-ery (ACM), the Information Processing Society of Japan (IPSJ), and the Institute of Electrical, Information and Communication Engineers (IEICE).

Kawuu Weicheng Lin received the B.S. degree from

the Department of Computer Science and Informa-tion Engineering, NaInforma-tional Taiwan University (NTU), Taipei, Taiwan, R.O.C., in 1999 and the Ph.D. degree from the Department of Computer Science and Infor-mation Engineering, National Cheng Kung Univer-sity (NCKU), Tainan, Taiwan, 2006.

Since August 2007, he has been an Assistant Pro-fessor in the Department of Computer Science and Information Engineering, National Kaohsiung Uni-versity of Applied Sciences (KUAS), Kaohsiung, Taiwan. His current research interests include data mining and its applications, sensor technologies, and parallel and distributed processing.

Dr. Lin is a member of the Phi Tau Phi honorary society, and has won the Phi Tau Phi Scholastic Honor in 2006.

Quantum Algorithms for Biomolecular Solutions of the Satisfiability Problem on a Quantum Machine