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- 1. SAT P ROBLEM : A Q UANTUM A PPROACH Supervised by: M. R. Hooshmand Asl Advised by: S. A. Shahzade Fazeli Presented by: A. Shakiba University of Yazd ali.shakiba@stu.yazduni.ac.ir February 19, 2012 SAT P ROBLEM : A Q UANTUM A PPROACH 1 / 43
- 2. Quantum Complexity Theory EvolutionOver the past three decades, Quantum Computing has attractedextensive attention in the academic community.History 1973 Bennett proved any given Turing machine can be computed efﬁciently by a reversible one. 1980 Benioff described a microscopic Turing machine using Quantum Mechanics. 1982 Feynman suggested that computers behave quantum mechanically may be more powerful than classical ones. 1985 Deutsch formalized the ideas of Benioff and Feynman and proposed Quantum Turing machines. 1993 Yao demonstrated the equivalence between Quantum Turing machines and Quantum Circuits. SAT P ROBLEM : A Q UANTUM A PPROACH 2 / 43
- 3. A big bang!But just after pioneering work of Shor’s and Grover’s, QuantumComputing has intrigued more and more people.Some Big Bangs of Quantum Computing 1994 Shor discovered a polynomial-time algorithm for factoring problem; 1996 Grover found an algorithm for searching through a database in square root time. SAT P ROBLEM : A Q UANTUM A PPROACH 3 / 43
- 4. NP-completeness and Quantum ComputingFollowing the works of Shor and Grover, it is natural to ask whether allthe NP can be computed efﬁciently using a Quantum computer. By consulting Garey and Johnson’s book on intractability, let us choose Satisﬁability Problem from NP-complete to start attack, since it is a core of a large family of computationally intractable NP-complete problems. SAT P ROBLEM : A Q UANTUM A PPROACH 4 / 43
- 5. SAT ProblemA SAT problem tries to answer that is there any assignment of truthvalues to boolean variables, x1 , . . . , xn , such that f (x1 , . . . , xn ) = 1 for nf : {0, 1} → {0, 1}.SAT problem is consisted of Variables a set of n boolean variables, x1 , x2 , . . . , xn , Literals a set of literals, a variable or its negation, Clauses a set of m distinct clauses, C1 , C2 , . . . , Cm where each clause is disjunction of some literals.SAT problem tries to ﬁnd out whether there exists any assignment oftruth values to the variables which makes the following true C1 ∧ C2 ∧ . . . ∧ Cm SAT P ROBLEM : A Q UANTUM A PPROACH 5 / 43
- 6. Solving a SAT problemThroughout this presentation, the following instance of SAT problem isbeing solved.Assume Variables x1 , x2 , x3 , x4 , Clauses C1 = {x1 , x4 , x2 } , C2 = {x2 , x3 , x4 } , C3 = {x1 , x3 } , C4 = {x3 , x1 , x2 }.It seems, ﬁnding a quantum circuit is just enough to do the job. So Letus do it . . . SAT P ROBLEM : A Q UANTUM A PPROACH 6 / 43
- 7. How to Quantum Compute?In classical world,L OAD -RUN -R EADcycle is followed.But in Quantumworld, it is P REPARE -E VOLVE -M EASUREcycle. SAT P ROBLEM : A Q UANTUM A PPROACH 7 / 43
- 8. Quantum Computing 101: PreparePrepare Qubits are used instead of bits, Qubit may be a particle such as an electron, Spin up (blue) representing 1, Spin down (red) representing 0, Superposition (yellow) which involves spin up and spin down simultaneously. It is possible to prepare exponentially many inputs in the same amount of time. SAT P ROBLEM : A Q UANTUM A PPROACH 8 / 43
- 9. Quantum Computing 101: EvolveEvolve A small number of qubits in superposition state can carry an enormous amount of information: A mere 1, 000 particles all in superposition state can represent every number from 0 to 21000 − 1 (about 10300 numbers). A quantum computer would manipulate all those data in parallel, For instance, by hitting the particles with laser pulses. SAT P ROBLEM : A Q UANTUM A PPROACH 9 / 43
- 10. Quantum Computing 101: MeasureMeasure When the particles states are measured at the end of the computation, all but one random version of the 10300 parallel states vanish. Clever manipulation of the particles could nonetheless solve certain problems very rapidly, such as factoring a large number. SAT P ROBLEM : A Q UANTUM A PPROACH 10 / 43
- 11. Logical gates AND and OR are not reversible!Reversibility is a MustAs the evolution of a quantum computer needs to be unitary, it must bea reversible, norm-preserving computation.By this constraint, we need to compile the classic circuit of SATproblem into an equivalent quantum one.Let’s use the results of Benioff’s(1980) and Bennett’s(1973) works. SAT P ROBLEM : A Q UANTUM A PPROACH 11 / 43
- 12. Reversible classical gatesDeﬁnitionNOT gate (UNOT ) |u X |u u u 0 1 1 0 SAT P ROBLEM : A Q UANTUM A PPROACH 12 / 43
- 13. Reversible classical gates (Cont’d)DeﬁnitionControlled-NOT gate (UCN ) u v u u⊕v |u • |u 0 0 0 0 |v |u ⊕ v 0 1 0 1 1 0 1 1 1 1 1 0 SAT P ROBLEM : A Q UANTUM A PPROACH 13 / 43
- 14. Reversible classical gates (Cont’d)DeﬁnitionToffoli gate (UToffoli ) u v w u v w ⊕ uv 0 0 0 0 0 0 0 0 1 0 0 1 |u • |u 0 1 0 0 1 0 |v • |v 0 1 1 0 1 1 |w |w ⊕ uv 1 0 0 1 0 0 1 0 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 0 SAT P ROBLEM : A Q UANTUM A PPROACH 14 / 43
- 15. A few words on notationSome Notation To represent a gate, e.g. AND gate acting on bits u and v and saving the result in bit w, we use UAND (u , v , w ). Circuits are represented, without loss of generality, as a product of gates acting on bits. The priority of gates to apply in a circuit is from right to left. In other words, the rightmost gate is applied ﬁrst and so on. SAT P ROBLEM : A Q UANTUM A PPROACH 15 / 43
- 16. Now constructing reversible AND and OR gatesUsing reversible gates; UNOT , UCNOT , andUToffoli ; it is possible toconstruct reversible AND and OR gates. UOR (u , v , w ) = UCNOT (u , w )UCNOT (v , w )UToffoli (u , v , w ) UAND (u , v , w ) = UToffoli (u , v , w ) |u |u |u |u |v OR |v |v AN D |v|w |w ⊕ uv |w |w ⊕ uv SAT P ROBLEM : A Q UANTUM A PPROACH 16 / 43
- 17. What else do we need? Dust bits To do reversible computation, we should not destroy even a single bit of information. So to save the result of a gate, there is a need for dust bit(s). Moreover, another gate to copy the value of an input bit to a dust bit for clauses with just one literal is needed. |u |uUCOPY (u , v ) = UCNOT (u , v ) COP Y |v |u SAT P ROBLEM : A Q UANTUM A PPROACH 17 / 43
- 18. Figuring out the size of workspaceTheoremFor a SAT problem with m clauses in n variables, the maximumnumber of dust qubits is no more than nmBut to construct the circuit, the total number of dust qubits should beknown:ProblemHow many dust bits do we need to construct a reversible circuit forsome SAT problem? SAT P ROBLEM : A Q UANTUM A PPROACH 18 / 43
- 19. What’s the shape of the workspace?The workspace is assumed as a register with R bits.Deﬁnition The truth assignment is represented by ﬁrst n bits (INPUT), For 1 ≤ i ≤ n, bit i represents boolean variable xi . Dust bits are added at the end of the input bits, µ bits. The last dust bit saves the truth value of SAT problem corresponding to the truth assignment, The rest of dust bits are used at the intermediate stages of computation. Bits are indexed from 1 to R.where R = n + µ SAT P ROBLEM : A Q UANTUM A PPROACH 19 / 43
- 20. Exact counting of dust bitsAssume the series s1 , s2 , . . . , sm which represents index of the ﬁrstdust bit in workspace to compute t (Ci ), 1 ≤ i ≤ m. s1 = n + 1 s2 = s1 + card(C1 ) + δ1,card(C1 ) − 1 sk = sk −1 + card(Ck −1 ) + δ1,card(Ck −1 ) 3 ≤ k ≤ mwhere ik represents the number of literals in Ck and ik represents thenumber of negations in Ck . Then the index of the last dust bit would be sf = sm + card(Cm ) + δ1,card(Cm ) − 1And the number of total dust bits is µ = sf − n SAT P ROBLEM : A Q UANTUM A PPROACH 20 / 43
- 21. Size of Workspace For the instance of SAT problem we are solving, the workspace is consisted of 10 dust bits. s1 = n + 1 = 5, s2 = s1 + card(C1 ) + δ1,card(C1 ) − 1Problem =7Assume x1 , x2 , x3 , x4 as s3 = s2 + card(C2 ) + δ1,card(C2 )variables and clauses = 10 C1 = {x1 , x4 , x2 } , s4 = s3 + card(C3 ) + δcard(C3 ) C2 = {x2 , x3 , x4 } , = 12 C3 = {x1 , x3 } , sf = s4 + card(C4 ) + δcard(C4 ) − 1 C4 = {x3 , x1 , x2 } , = 14for SAT problem. SAT P ROBLEM : A Q UANTUM A PPROACH µ = 14 21 / 43
- 22. Reversible Circuit for SAT ProblemTo evaluate a SAT corresponding to a truth assignment, ﬁrst we needto evaluate each Ci ,Example UOR (1) = UNOT (2)UOR (2, 5, 6)UNOT (2)UOR (1, 4, 5), UOR (2) = UOR (4, 7, 8)UOR (2, 3, 7), UOR (3) = UNOT (3)UOR (1, 3, 10)UNOT (3), UOR (4) = UNOT (2)UOR (2, 12, 13)UNOT (2)UNOT (1) .UOR (3, 1, 12)UNOT (1).where R = 14. SAT P ROBLEM : A Q UANTUM A PPROACH 22 / 43
- 23. Reversible Circuit for SAT Problem (Cont’d)The evaluation is at last obtained byExample C1 ∧ C2 : UAND (1) = UAND (6, 8, 9), C1 ∧ C2 ∧ C3 : UAND (2) = UAND (9, 10, 11), C1 ∧ C2 ∧ C3 ∧ C4 : UAND (3) = UAND (11, 13, 14).where R = 14. SAT P ROBLEM : A Q UANTUM A PPROACH 23 / 43
- 24. Reversible Circuit for SAT Problem (Cont’d)Then the reversible circuit for evaluation becomes UC =UAND (11, 13, 14)UAND (9, 10, 11)UAND (6, 8, 9) .UNOT (2)UOR (2, 12, 13)UNOT (2)UNOT (1)UOR (3, 1, 12)UNOT (1) .UNOT (3)UOR (1, 3, 10)UNOT (3) .UOR (4, 7, 8)UOR (2, 3, 7) .UNOT (2)UOR (2, 5, 6)UNOT (2)UOR (1, 4, 5)where R = 14. SAT P ROBLEM : A Q UANTUM A PPROACH 24 / 43
- 25. Quantum Mechanics Enters!Instead of workspace of R bits, a Hilbert space, H ⊗R , of dimension Ris deﬁned.ExampleLet |vin be deﬁned as |04 , 09 , 0 .By applying the Hadamard transform to |vin , all the values 0 . . . 24 − 1are came to existence uniformly. (14) |v ≡ UH (4)|vin 24 −1 1 = √ 4 ∑ |ei , 09 , 0 ( 2) i =0 1 = √ 4 ∑ |ε1 , ε2 , ε3 , ε4 , 09 , 0 ( 2) ε1 ,ε2 ,ε3 ,ε4 ∈{0,1} SAT P ROBLEM : A Q UANTUM A PPROACH 25 / 43
- 26. Computing the ORs (14) (14) (14) (14)By applying UOR (1), UOR (2), UOR (3), and UOR (4) in order, (14) (14) (14) (14)UOR (1)UOR (2)UOR (3)UOR (4)|v 1 (14) (14) (14) (14)= √ UOR (1)UOR (2)UOR (3)UOR (4) ∑ |ε1 , ε2 , ε3 , ε4 , 09 , 0 ( 2)4 ε1 ,ε2 ,ε3 ,ε4 ∈{0,1} 1= √ ∑ |ε1 , ε2 , ε3 , ε4 , ε1 ∨ ε4 , ε1 ∨ ε4 ∨ ε2 , ε2 ∨ ε3 , ε2 ∨ ε3 ∨ ε4 , ( 2)4 ε1 ,ε2 ,ε3 ,ε4 ∈{0,1}0, ε1 ∨ ε3 , 0, ε3 ∨ ε1 , ε3 ∨ ε1 ∨ ε2 , 0 ≡ |vis obtained. SAT P ROBLEM : A Q UANTUM A PPROACH 26 / 43
- 27. Applying ANDsBy applying AND to state |v , we obtain (14) (14) (14)UAND (11, 13, 14)UAND (9, 10, 11)UAND (6, 8, 9)|v 1= √ ∑ |ε1 , ε2 , ε3 , ε4 , ε1 ∨ ε4 , ε1 ∨ ε4 ∨ ε2 , ε2 ∨ ε3 , ε2 ∨ ε3 ∨ ε4 , ( 2)4 ε1 ,ε2 ,ε3 ,ε4 ∈{0,1}t (C1 ) ∧ t (C2 ), ε1 ∨ ε3 , t (C1 ) ∧ t (C2 ) ∧ t (C3 ), ε3 ∨ ε1 , ε3 ∨ ε1 ∨ ε2 , tε (C )≡ |v SAT P ROBLEM : A Q UANTUM A PPROACH 27 / 43
- 28. Computational Complexity of Quantum SATTheoremFor a SAT problem with n boolean varibles and m clauses, theQuantum Computational Complexity is m T (UC ) = m − 1 + ∑ |Ck | + 2ik − 1 k =1 ≤ 4mn − 1where ik is the number of negations in clause Ck . SAT P ROBLEM : A Q UANTUM A PPROACH 28 / 43
- 29. MeasureBy measuring the ﬁnal qubit, the ﬁnal state is obtained as a densitymatrix, 7 9 ρ = |1 1| + |0 0| 16 16716 is less than a half. It means the probability that the state 1 is 7measured, is 16 . SAT P ROBLEM : A Q UANTUM A PPROACH 29 / 43
- 30. Amplitude AmpliﬁcationProblemHow to increase the probability of getting |1 , if there is one, to near 1?There is a technique, known as amplitude ampliﬁcation.It was ﬁrst used by Grover(1996) to do a search in haystack. Soon itwas generalized by Boyer et. al. to Quantum Ampliﬁcation technique.Theorem nLet f : {0, 1} → {0, 1} be a function with a unique x ∈ {0, 1} suchthat f (x ) = 1. By repeating Grover’s iterator for k √ times, x is found byprobability of 1 − O( 2n ) where k = 4θ − 1 ≈ π N. 1 π 2 4 SAT P ROBLEM : A Q UANTUM A PPROACH 30 / 43
- 31. Why not Grover’s?Using Grover’s ampliﬁcation technique, can lead into a quadratic √reduction in complexity, reducing from O(2n ) to O( 2n ).It’s obviously not polynomial.Today, there are two approaches known against NP-completeproblems in Quantum World: Exploiting the NP-complete problem’s structure to develop polynomial time algorithms. Developing new amplitude ampliﬁcation techniques.Cerf et. al. (2003) developed a Quantum algorithm based on exploitingthe structure of SAT and using nested Grover’s search algorithm which √achieved 2n x queries, where x < 1. SAT P ROBLEM : A Q UANTUM A PPROACH 31 / 43
- 32. Chaotic Amplitude AmpliﬁcationDeﬁnitionA logistic map is deﬁned as xn = fa (xn−1 ) = axn−1 (1 − xn−1 )where x ∈ [0, 1] and a ∈ [1, 4]. 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 0.5 1 1.5 2 2.5 3 3.5 4 SAT P ROBLEM : A Q UANTUM A PPROACH 32 / 43
- 33. Using the Right ChaosTheorem 1For logistic map with a = 3.71 and x0 = 2n , there exists k ∈ J suchthat xk > 1 where 2 J = {0, 1, . . . , n, . . . , 2n} 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 3.4 3.5 3.6 3.7 3.8 3.9 4 SAT P ROBLEM : A Q UANTUM A PPROACH 33 / 43
- 34. Exploiting Chaos with InterferenceFor logistic map fa (x ), m times composition of fa (x ) is denoted by mfa (x ).The ampliﬁer is deﬁned as m I − fa (ρ0 )σ3 ρm = 2where σ3 is the pauli-z matrix, 1 0 0 −1The probability of getting |1 , if one exists, is equal to Mm ≡ trP1 ρm SAT P ROBLEM : A Q UANTUM A PPROACH 34 / 43
- 35. Sum up!Using a quantum computer, a SAT problem is solvable in polynomialtime, but the probability of measuring the satisﬁable assignment is verylow, almost zero.To make the satisﬁable assignment measurable with a good probability,it should be ampliﬁed.Grover’s ampliﬁcation technique makes no more than a quadraticspeed-up for SAT problem,Chaotic ampliﬁcation technique proposed by Ohya et. al. makes thepossibility of polynomial speed-up. SAT P ROBLEM : A Q UANTUM A PPROACH 35 / 43
- 36. Sum up! (Cont’d) But all the nice apparatus mentioned here are on the paper! So what do they mean?Since implausible kinds of Physics seemsnecessary for constructing a computer able tosolve NP-complete problems quickly, it’s not toofar-fetched that some day one adopt n newprinciple “NP-complete problems are hard!” SAT P ROBLEM : A Q UANTUM A PPROACH 36 / 43
- 37. Future of Quantum ComputingSimulating Quantum Physics: a fundamentalproblem for Chemistry, Nano-technology, andother ﬁelds.As transistors in microchips approach the atomicscale, ideas from quantum computing are likelyto become relevant for classical computing aswell.The most exciting possible outcome of quantumcomputing research would be to discover afundamental reason why quantum computersare not possible. Such a failure would overturnour current picture of the physical world,whereas success would merely conﬁrm it. SAT P ROBLEM : A Q UANTUM A PPROACH 37 / 43
- 38. ReferencesGeneral references for Quantum Mechanics and Computing N IELSEN , M., AND C HUANG , I. Quantum computation and quantum information. Cambridge Series on Information and the Natural Sciences. Cambridge University Press, 2000. K AYE , P HILLIP ; L AFLAMME , P HILLIP AND M OSCA , M ICHELE An introduction to quantum computing. Oxford University Press, 2007. B ERNSTEIN , E. J. Quantum complexity theory. PhD thesis, University of California, Berkeley, 1997. Retrieved October 22, 2011, from Dissertations & Theses: A&I.(Publication No. AAT 9803127). SAT P ROBLEM : A Q UANTUM A PPROACH 38 / 43
- 39. References (Cont’d)Literature on Classic and Quantum Complexity D EUTSCH , D. Quantum theory, the church-turing principle and the universal quantum computer. Proceedings of the Royal Society of London A 400 (1985), 97–117. C LEVE , R. An Introduction to Quantum Complexity Theory. eprint arXiv:quant-ph/9906111 (June 1999). WATROUS , J. Quantum Computational Complexity. ArXiv e-prints (Apr. 2008). SAT P ROBLEM : A Q UANTUM A PPROACH 39 / 43
- 40. References (Cont’d)Quantum Algorithms S HOR , P. W. Algorithms for quantum computation: Discrete logarithms and factoring. In Proceeding SFCS ’94 Proceedings of the 35th Annual Symposium on Foundations of Computer Science (1994), , pp. 124–134. G ROVER , L. K. A fast quantum mechanical algorithm for database search. In ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING (1996), ACM, pp. 212–219. C ERF , N. J., G ROVER , L. K., AND W ILLIAMS , C. P. Nested quantum search and structured problems. 61, 3 (Mar. 2000), 032303. SAT P ROBLEM : A Q UANTUM A PPROACH 40 / 43
- 41. References (Cont’d)Chaos Ampliﬁcation O HYA , M., AND M ASUDA , N. NP problem in quantum algorithm. eprint arXiv:quant-ph/9809075 (Sept. 1998). O HYA , M., AND VOLOVICH , I. V. New quantum algorithm for studying np-complete problems. Reports on Mathematical Physics 52, 1 (2003), 25 – 33. SAT P ROBLEM : A Q UANTUM A PPROACH 41 / 43
- 42. References (Cont’d)Linear Algebraic References H ORN , R. A., AND J OHNSON , C. R. Matrix Analysis. Cambridge University Press, 1985. Z HANG , F. Matrix theory: basic results and techniques. Springer, 1999. SAT P ROBLEM : A Q UANTUM A PPROACH 42 / 43
- 43. Thanks for your attention“If quantum states exhibit small nonlinearities during time evolution,then quantum computers can be used to solve NP-Complete problemsin polynomial time . . . we would like to note that we believe thatquantum mechanics is in all likelihood exactly linear, and that the aboveconclusions might be viewed most proﬁtably as further evidence thatthis is indeed the case.” Nonlinear Quantum Mechanics Implies Polynomial-Time Solution for NP-Complete and P Problems, Phys. Rev. Lett., Volume 81 (1998) pp. 39923995 — Dan Abrams and Seth Lloyd SAT P ROBLEM : A Q UANTUM A PPROACH 43 / 43

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