Quantum Computing based on Tensor Products Overview and Introduction

Transcription

1 Quantum Computing based on Tensor Products Overview and Introduction Guillermo Morales Luna Computer Science Section CINVESTAV-IPN 5-th International Workshop on Applied Category Theory Graph-Operad Logic Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 1 / 38

2 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 2 / 38

3 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 3 / 38

4 Abstract We present a short introduction to Quantum Computing (QC) from a procedural point of view. Rather, it is a course of parallel computing based on tensor products. We introduce primitive functions and the compositional schemes of QC. We use Tensor Product notation instead of the more conventional Dirac s ket notation. We introduce basic notions of Tensor Products and Hilbert Spaces and the qubits as points in the unit circle in the two-dimensional complex Hilbert space, then any word consisting of qubits lies in the corresponding unit sphere of the tensor product of these spaces. We illustrate the computing paradigm through the classical Deutsch-Josza algorithm. Then we show the quantum algorithm to compute the Discrete Fourier Transform in linear time and the famous polynomial-time Shor algorithm for integer factorization. We finish our exposition with a basic introduction to Quantum Cryptography and Quantum Communication Complexity. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 4 / 38

5 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 5 / 38

6 Overview of the whole course Contents 1 Introduction 2 Tensor Products 1 Vector and Space Products 2 Products of Linear Maps 3 Basic Notions on Quantum Computing 1 Measurement Principle 2 Qubits and Words of Quantum Information 3 Quantum Gates 4 Observables and the Heisenberg Principle of Uncertainty 5 Evaluation of Boolean Functions 6 Deutsch-Jozsa s Algorithm 4 Quantum Computation of the Discrete Fourier Transform Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 6 / 38

7 5 Shor Algorithm 1 A Short Refreshment of Number Theory 2 Quantum Algorithm to Calculate the Order of a Number 1 Elements whose Order is a Power of 2 2 Elements with Arbitrary Order 6 Quantum Cryptography: Key Agreement Protocols 1 Channels without Noise 2 Channels with Noise 7 Communication Complexity 1 Parities Addition 2 Congruent Functions with the Hamming Weight Map 3 Identity Checking 4 Inner Product 1 Using an Entangled Pair of Qubits 2 Algorithm without Entangled States 5 Deutsch-Josza Relation Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 7 / 38

8 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 8 / 38

9 General references Quantum Computing D. Bouwmeester, A. K. Ekert, and A. Zeilinger. The Physics of Quantum Information: Quantum Cryptography, Quantum Teleportation, Quantum Computation. Springer, Berlin, G. Brassard. Quantum communication complexity (a survey), G. Brassard, I. Chuang, S. Lloyds, and C. Monroe. Quantum computing. Proc. Natl. Acad. Sci. USA, 95: , Sept M. Nielsen and I. Chuang. Quantum Computation and Quantum Information. Cambridge University Press, Cambridge, Physics and Math R. Abraham, J. E. Marsden, and T. Ratiu. Manifolds, Tensor Analysis, and Applications. Springer-Verlag, Berlin, J. S. Bell. On the Einstein-Podolsky-Rosen paradox. Physics, 1:195, S. Lang. Algebra, 4-th Edition. Springer-Verlag, Berlin, Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 9 / 38

10 Special topics Classics D. Deutsch. Quantum Theory, the Church-Turing Principle and the Universal Quantum Computer. Proceedings of the Royal Society of London Ser. A, 400:96 117, D. Deutsch and R. Jozsa. Rapid solution of problems by quantum computation. Proceedings of the Royal Society of London Ser. A, 439: , A. S. Holevo. Some estimates of the information transmitted by a quantum communication channel. Probl. Peredachi Inf., 9:3, Also Probl. Inf. Transm. (USSR) 9, 177 (1973). P. W. Shor. Algorithms for quantum computation: Discrete logarithms and factoring. In IEEE Symposium on Foundations of Computer Science, pages IEEE, P. W. Shor. Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer. SIAM Journal on Computing, 26: , Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 10 / 38

11 Algorithmics and Crypto T. Hogg. Solving highly constrained search problems with quantum computers. Journal of Artificial Intelligence Research, 10:39 66, C. Lavor, L. Manssur, and R. Portugal. Shor s algorithm for factoring large integers. Mar S. J. Lomonaco. A Quick Glance at Quantum Cryptography. Cryptologia, 23:1 41, S. J. Lomonaco. A Talk on Quantum Cryptography or How Alice Outwits Eve. arxiv: quant-ph/ v1, Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 11 / 38

12 Quantum complexity E. Bernstein and U. Vasirani. Quantum complexity theory. In Proc. of the 25-th Annual ACM Symposium on Theory of Computing, AIP Conference Proceedings. Association of Computing Machinery, R. Cleve. An introduction to quantum complexity theory, Entanglement G. Brassard, R. Cleve, and A. Tapp. Cost of exactly simulating quantum entanglement with classical communication. Phys. Rev. Lett., 83: , H. Buhrman, R. Cleve, and W. van Dam. Quantum entanglement and communication complexity. SIAM J. Comp., 30(6): , March S. Massar, D. Bacon, N. Cerf, and R. Cleve. Classical simulation of quantum entanglement without local hidden variables. Phys. Rev. A, 63:52305, Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 12 / 38

13 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 13 / 38

14 Hilbert Spaces Basic notions Complex field. C Vector space. H: Non-empty set. 0 H Addition. + : H H H. (H, +) Abelian group Scalar multiplication. : C H H. Distributive w.r.t. addition Inner product. : H H C. Sesquilinear form. Positive definite. Norm. 2 : H R +, x 2 = x x. Completeness. Every Cauchy sequence is convergent. Autoduality. For each T H exists y H: T(x) = y x. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 14 / 38

15 Geometrical properties: m-dimensional Hilbert spaces Canonical basis. e j = (δ ij ) i<m Unit sphere. E m = {v H 1 = v H v =: v v } Unitary map. U : H H linear s.t. M H M = 1 mm. U Em : E m E m Tensor products Spaces. dim(u) = n & dim(v) = m dim(u V) = nm. U V U V. The difference consists of entangled states. Vectors. x U & y V x y U V. Maps. S : U 1 V 1 & T : U 2 V 2 S T : U 1 U 2 V 1 V 2. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 15 / 38

16 Measurement Principle In current state v = (v i1 ) i<m = m 1 i=0 E m, for each i < m, with probability v i1 2 the following is performed: The index i is output and the computing control is transferred to the state e i. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 16 / 38

17 Qubits and Quregisters Spaces and basis H 1 = C 2 ; H n = H n 1 H 1 dim(h n ) = 2 n e 0 = [1 0] T and e 1 = [0 1] T basis in H 1. (e εn 1 ε 1 ε 0 ) εn 1,...,ε 1,ε 0 {0,1} basis in H n. Qubits. z E 2 unit sphere in H 1. Quregister. z 1 z n 1 = z i, i N, qubits z 1 z n 1 E 2 n H n Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 17 / 38

18 Quantum speed-up Conventional Dirac s ket notation ε n 1 ε 1 ε 0 := e εn 1 ε 1 ε 0 = e εn 1 e ε1 e ε0 =: ε n 1 ε 1 ε 0 (1) Any state in H n, σ(z) = ε {0,1} n z εe ε is determined by 2 n coordinates. If U : H n H n is a quantum operator, the target state σ(uz) consists also of 2 n coordinates. A calculus involving an exponential number of terms is performed in just one step of the quantum computation. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 18 / 38

19 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 19 / 38

20 Function Evaluation V = {0, 1}: binary digits Each index in the set {0,..., 2 n 1} corresponds to a string ε = (ε n 1,..., ε 1, ε 0 ) V n which in turn corresponds to e ε H n. Let f : V n V m be a function. A quantum algorithm U f : H n+m H n+m computes f if U f : e ε 0 e ε ɛe f(ε) where ɛ = 1. A final measurement on last qubits provides the value f(ε), with probability 1. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 20 / 38

21 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 21 / 38

22 Deutsch-Jozsa s Algorithm Let V = {0, 1} be the set of classical truth values. Deutsch-Jozsa s problem Decide, for a given f : V V, whether it is constant or balanced in just one computing step. Given f, let U f be the matrix s.t. U f (e x e z ) = (e x e (z+f(x)) mod 2 ). We have H 2 U f H 2 : e 0 e 1 εe S e 1 where H 2 is Hadamard s operator, ε { 1, 1} is a sign and S is a signal indicating whether f is balanced or not. S coincides with f(0) f(1). Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 22 / 38

23 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 23 / 38

24 Quantum Computation of the Discrete Fourier Transform Given f = n 1 j=0 f(j)e j C n, its discrete Fourier transform is n 1 DFT(f) = ˆf 1 n 1 ( ) 2πijk = exp f(k) n n e j C n. j=0 k=0 DFT is linear transform and, w.r.t. ( ( the canonical basis, it is represented by the unitary matrix DFT = 1 2πijk )) n exp n jk Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 24 / 38

25 If n = 2 ν, H ν = C n, and by identifying each j [[0, 2 ν 1]] with ε j = ε j,ν 1 ε j,1 ε j,0 : DFT(e εj ) = ν (e 0 + exp ( ) ) πij e 1 2 k k=0 ( ) ) ( ) ) ( ) ) = (e 1 πij 0 +exp e 1 (e 1 πij 0 +exp e 1 (e 1 πij 0 +exp 2 2 ν 1 e 1 (2) The products appearing in this tensor product suggest the operators Q k : H 1 H 1 and their controlled versions: Q k = 1 0 ( ) πi 0 exp 2 k, Q c kj = exp ( πi ) j 2 k. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 25 / 38

26 Algorithm for the Fourier transform Input. n = 2 ν, f C n = H ν. Output. ˆf = DFT(f) H ν. Procedure DFT(n, f) 1 Let x 0 := H(e 0 ). 2 For each j [[0, 2 ν 1]], or equivalently, for each (ε j,ν 1 ε j,1 ε j,0 ) {0, 1} ν, do (in parallel): 1 For each k [[0, ν 1]] do (in parallel): 1 Let δ := R k ( εj k ) be the reverse of the chain consisting of the (k + 1) less significant bits. 2 Let y jk := x 0. 3 For l = 0 to k do { y jk := Q c2 (y jk, e δj,l ) } 2 Let y j := y j0 y j,ν 1. 3 Output as result ˆf = 2 ν 1 j=0 f jy j. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 26 / 38

27 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 27 / 38

28 Let n be an integer to be factored 1 Select an integer m such that 1 < m < n. 2 If gcd(n, m) = d > 1, then d is a non-trivial factor of n. 3 Otherwise, m is in the multiplicative group of remainders of n. 1 If m has an even order r, then k = m r 2 will be such that k 2 = 1 mod n, and (k 1)(k + 1) = 0 mod n. 2 By calculating gcd(n, k 1) and gcd(n, k + 1), one gets non-trivial factors of n. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 28 / 38

29 Biggest problem Calculate the order of a current element m in Φ(n) Let ν = log 2 n, ν is the size of n. O(n) = O(2 ν ), thus an exhaustive procedure has exponential complexity with respect to the input size. Shor s algorithm is based over a polynomial-time procedure in ν to calculate the order of an element. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 29 / 38

30 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 30 / 38

31 Quantum Cryptography: Key Agreement Protocols Two entities, Alice and Bob, should agree in private a common key. They may use two transmission channels Quantum channel Transmits just one-way, say from Alice to Bob. Classical channel Transmits bidirectionally. We will present the BB84 Protocol, without and with noise. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 31 / 38

32 Quantum computing elements E 0 = {e 0 0 = (1, 0), e0 1 = (0, 1)}: canonical basis of H 1 H(E 0 ) = E 1 = {e 1 0, e1 1 }: basis of H 1 obtained by applying Hadamard s operator to E 0. E 0 corresponds to a spin with vertical horizontal polarization, E 0 = {, }, while E 1 corresponds to a spin with oblique or NW NE polarization, E 1 = {, }. The same sequence of qubits can be measured either w.r.t. to E 0 or E 1. An eavesdropper can be detected quite directly! This is characteristic of Quantum Cryptography. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 32 / 38

33 Agenda 1 Abstract 2 Overview of the whole course 3 References 4 Hilbert Spaces 5 Function Evaluation 6 Deutsch-Jozsa s Algorithm 7 Quantum Computation of the Discrete Fourier Transform 8 Shor Algorithm 9 Quantum Cryptography 10 Communication Complexity Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 33 / 38

34 Communication Complexity The complexity of a communication process is determined by the minimum information quantity that should be transmitted, in order that the total information can be recovered by the receiving part, within a given context. Optimal Transmission Let us assume that three sets X, Y, Z are given and a function f : X Y Z. At some moment, Alice, who is a communicating part, possesses a point x X, Bob, who is a second part, possesses a point y Y and both parts should calculate z = f(x, y), by interchanging the minimum information quantity. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 34 / 38

35 Identity Checking Exact and obvious method If f : (x, y) χ = (x, y) is the characteristic function of the identity relation: f(x, y) = 1 if and only if x = y, then Alice and Bob should interchange n bits to calculate f(x, y). n is exponential with respect to its size! Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 35 / 38

36 An interesting question is whether an exact algorithm can be obtained with logarithmic complexity. The following theorem excludes the possibility to communicate more than k (classical) bits of information by transmitting k qubits. Holevo s Theorem The information quantity recovered from a register of qubits is upperly bounded by the value of von Neumann s entropy, which is bounded by Shannon s entropy. Both entropies coincide whenever the qubits are pairwise orthogonal. However, in Quantum Computing the use of the notion of entangled states improves the communication complexities of several procedures. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 36 / 38

37 Deutsch-Josza Relation Pseudotelepathy Game Given four sets X, Y, A and B, a relation R X and A B, and the fact that Alice and Bob are separated, far from each other, at a given moment Alice receives a point x X, Bob a y Y and they, trying to interchange the minimum information, should produce, respectively, a A and b B such that (x, y, a, b) R. Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 37 / 38

38 In particular, for n = 2 k a power of 2, X = {0, 1} n = Y, A = {0, 1} k = B R is Deutsch-Josza relation (x, y, a, b) R [ (H n (x, y) = 0 a = b) { H n (x, y) 0, n }] 2 (H n (x, y) = n 2 a b ) (3) If the points x and y of Alice and Bob coincide, then the sequences that they produce should coincide If x and y differ exactly in half of the bits, then the produced sequences should differ In any other case no restrictions on produced sequences Morales-Luna (CINVESTAV) QC based on Tensor Products 5-th Int. WS App. Cat. Th. 38 / 38