An Introduction to Quantum Information and Applications

Transcription

1 An Introduction to Quantum Information and Applications Iordanis Kerenidis CNRS LIAFA-Univ Paris-Diderot

2 Quantum information and computation Quantum information and computation How is information encoded in nature? What is nature s computational power? Moore s law: quantum phenomena will appear by 2020 Rich Mathematical Theory Advances in Classical Computer Science Advances in Theoretical & Experimental Physics Advances in Information Theory

3 Quantum information and computation The power of Quantum Computing Quantum algorithm for Factoring and Discrete Logarithm [Shor 93] Unconditionally Secure Key Distribution [Bennett-Brassard 84] Quantum computers unlikely to solve NP-complete problems [Bernstein Bennett Brassard Vazirani 94]

4 Outline ) Introduction to the model Superdense Coding Teleportation 2) Basic algorithms Deutsch-Jozsa Ideas for Factoring 3) Cryptography Key Distribution 4) Communication Complexity Quantum fingerprints Exponential Separations

5 Quantum States Quantum bit is a unit vector in a 2-dim. Hilbert space " 0 = $ % # 0& ', = " $ 0% # ', & ( ) = 2 " $ % ', # & 2 0 ( ) = 2 " % $ ' # & A quantum state on logn qubits is a unit vector in Inner product:

6 Measurements on Quantum States A measurement of in an orthonormal basis is a projection onto the basis vectors and Pr[outcome is b i ] = Examples

7 Measurements on Quantum States A measurement of in an orthonormal basis is a projection onto the basis vectors and Pr[outcome is b i ] = Examples φ = a a, { ( ), 2 0 ( )} Prob[outcome ( )] = 2 a a Prob[outcome ( 2 0 )] = 2 a a 2 2 = 2 + a 0a = 2 a 0a

8 Measurements on Quantum States A measurement of in an orthonormal basis is a projection onto the basis vectors and Pr[outcome is b i ] = Examples Note that

9 Measurements on Quantum States A measurement of in an orthonormal basis is a projection onto the basis vectors and Pr[outcome is b i ] = IMPORTANT REMARK What is the final state after the measurement? The state changes to the basis state Hence, no more information in it about the a i s. If I repeat the measurement I always get the same basis vector.

10 Unitary Evolution Unitary matrix: inner product/length preserving, linear NOT gate X Phase Flip gate Z

11 Unitary Evolution cont. Hadamard Gate 0 H " ( ), H H " 2 0 ( ) Control NOT gate Example H 0 0 H on st ### ( ) 0 CNOT ## ( )

12 Superdense Coding Transmitting 2 bits with qubit Alice and Bob share the above state Alice wants to transmit the bits b b 2 to Bob ( ) Alice Z b X b2 Bob Let b b 2 =0 H M M ( ) CNOT "" Z if b= """ ( ) ( ) 2 0 ( ) 0 X if b 2= "" " H on st """ ( )

13 Teleportation Teleporting a qubit with 2 bits Alice and Bob share the state ( ) Alice wants to transmit an unknown qubit to Bob M Alice b 2 H M b Bob Z b X b2 ( a a ) ( CNOT ) ## ( 2 a a 0 + a a 0 ) H # 2 00 ( a a ) ( a 0 + a 0 ) ( a 0 0 a ) + 2 ( a 0 a 0 ) Z b X b 2 ### ( a a )

14 Outline ) Introduction to the model Superdense Coding Teleportation 2) Basic algorithms Deutsch-Jozsa Ideas for Factoring 3) Cryptography Key Distribution 4) Communication Complexity Quantum fingerprints Exponential Separations

15 Quantum Queries Let f: X -> Y Goal: Does f have a certain property? Classical Query: "What is the value of f(x)?" x O "" f f (x) Example: Is f linear or far from linear? 3 Queries u.a.r.: f(x),f(y),f(x+y). Check f(x)+f(y)=f(x+y) Quantum Query x b O "" f x b f (x) But, quantum operations are linear! H " x 2 n / 2 x {0,} n 0 O "" f x 2 n / 2 x {0,} n f (x)

16 Deutsch-Jozsa Algorithm Let f: {0,} n -> {0,} Goal: Is f identically zero or balanced? O Classical Query: x "" f f (x) deterministic: 2 n- + randomized: k queries, error probability 2 -k Quantum x b O "" f x b f (x) Z on 2nd """ H " H " x 0 2 n / 2 x {0,} n O "" f ( ) f (x) x f (x) 2 n / 2 x {0,} n ' if f = 0 ) ( a y y,with a 0 = 0, ) * y {0,} n x f (x) 2 n / 2 x {0,} n O "" f if f balanced ( ) f (x ) x 0 2 n / 2 x {0,} n

17 More Algorithms - Simon's problem Let f: {0,} n -> {0,} n Promise: f(x)=f(x+a) and f(x) f(y), y x+a (2-periodic) Goal: Find a Randomized: 2 n/2 Quantum: O(n), by finding each time a random y, st. y.a=0 - Period Finding [Shor94] Let f: Z N -> C Promise: f is periodic Goal: Find period Quantum: Easy algorithm, based on Fourier Transform Factoring = Period Finding! - Seach an unordered list: O( n) queries [Grover97]

18 2)Algorithms: Open Problems Find New Algorithms Graph Isomorphism? Lattice Problems? Hidden Subgroup Problems? other... Exponential speedup (possibly) Factoring, Discrete Log, Pell's Equality,... Quadratic speedup (provably) Grover's Search, Quantum walk-based algorithms,...

19 Outline ) Introduction to the model Superdense Coding Teleportation 2) Basic algorithms Deutsch-Jozsa Ideas for Unordered search and Factoring 3) Cryptography Key Distribution 4) Communication Complexity Quantum fingerprints Exponential Separations

20 3) Cryptography Current cryptography based on computational assumptions (e.g. hardness of factoring) Many such problems become insecure against a quantum adversary Can we use quantum interaction to achieve unconditionally secure cryptography?

21 Unconditional Key Distribution. Alice picks a secret key. She encodes each bit in one of two possible quantum ways and sends it to Bob. 2. Bob guesses the encoding and decodes each bit accordingly Remarks: - If Bob guesses correctly the encoding, then the decoding is perfect. If not, Bob gets a random bit. - Bob guesses correctly half the times.

22 Unconditional Key Distribution. Alice picks a secret key. She encodes each bit in one of two possible quantum ways and sends it to Bob. 2. Bob guesses the encoding and decodes each bit accordingly 3. Alice and Bob reveal publicly the encodings and keep only the bits on which they agree. (~ half) Remarks: - If there is no Eve, then they agree on the value of all these bits. - If Eve has got information about the key, then with high probability Alice and Bob will disagree on some bits.

23 Unconditional Key Distribution. Alice picks a secret key. She encodes each bit in one of two possible quantum ways and sends it to Bob. 2. Bob guesses the encoding and decodes each bit accordingly 3. Alice and Bob reveal publicly the encodings and keep only the bits on which they agree. (~ half) 4. Alice and Bob reveal publicly the values of half of the bits (/4 of the initial). - If they agree, they use the rest as the key (~ /4) - If they disagree in many bits, they throw it away

24 Unconditional Key Distribution Ψ 00 = 0, Ψ 0 =, Ψ 0 = ( ), Ψ = ( 2 0 ). Pick a,b {0,} n ( a: key b : encoding) Send each 2. Pick b {0,} n If b i =0 measure in If b i = measure in Denote outcome a i 3. Alice and Bob reveal publicly the encodings b,b. Keep the bits for which b i = b i (~ half) 4. Alice and Bob reveal publicly the values of a i = a i for half of the bits for which b i = b i - If they agree, they use the rest as the key (~ /4) - If they disagree in many bits, they throw it away

25 Unconditional Key Distribution Ψ 00 = 0, Ψ 0 =, Ψ 0 = ( ), Ψ = ( 2 0 ) Proof of Security (idea) Eve gets information, she disturbs the state (Heisenberg) Possible strategy: Eve picks encoding b E u.a.r and measures Alice's qubit. Let be the result. She sends it to Bob. Ψ a be If b A b B, Bob does not check, so Eve is not detected cheating If b A = b B and b E = b A, then Ψ a b E = Ψ a b A, so Eve is not detected If b A = b B and b E b A, then Alice Eve : measure in { 0, } Bob: measure in ( ) Ψ 0 = outcome 0, w.p./2, w.p./2 outcome { ( 2 0 ± )} +, w.p./2, w.p./2

26 Unconditional Key Distribution Ψ 00 = 0, Ψ 0 =, Ψ 0 = ( ), Ψ = ( 2 0 ) Proof of Security (idea) Eve gets information, she disturbs the state (Heisenberg) Possible strategy: Eve picks encoding b E u.a.r and measures Alice's qubit. Let be the result. She sends it to Bob. Ψ a be If b A b B, Bob does not check, so Eve is not detected cheating If b A = b B and b E = b A, then Ψ a b E = Ψ a b A, so Eve is not detected If b A = b B and b E b A and Alice and Bob check, then Alice Eve : measure in { 0, } Bob: measure in ( ) Ψ 0 = outcome 0, w.p./2, w.p./2 Overall, Pr[Eve is detected cheating]=/6 outcome { ( 2 0 ± )} +, w.p./2, w.p./2

27 Unconditional Key Distribution Ψ 00 = 0, Ψ 0 =, Ψ 0 = ( ), Ψ = ( 2 0 ) Proof of Security (continued) The optimal strategy of Eve is not much better than the one we described. (individual vs coherent attacks) The key is almost secure. We can distill a much stronger key by classical privacy amplification No assumptions on Eve s computational power!

28 3) Cryptography: Open Problems Other Cryptographic Primitives Oblivious Transfer Coin Flipping Bit Commitment Practical Quantum Cryptography Commercial systems for QKD Classical cryptography secure against quantum

29 Outline ) Introduction to the model Superdense Coding Teleportation 2) Basic algorithms Deutsch-Jozsa 3) Cryptography Key Distribution 4) Communication Complexity Quantum fingerprints Exponential Separations

30 4) Communication Complexity Input x Goal: Output P(x,y) (minimum communication) Input y Examples: Is x=y?, Find an i such that x i y i Applications of Communication Complexity VLSI design, Boolean circuits, Data structures, Automata, Formula size, Data streams, Secure Computation

31 Quantum Communication Complexity Classical vs. Quantum Input x Goal: Output P(x,y) (minimum communication) Input y Examples: Is x=y?, Find an i such that x i y i Applications of Communication Complexity VLSI design, Boolean circuits, Data structures, Automata, Formula size, Data streams, Secure Computation

32 Encoding Information with Quantum states We can encode a string with logn qubits. Holevo s bound We cannot compress information by using qubits. We need n qubits to transmit n classical bits. Quantum communication can still be useful since in many communication problems the information that needs to be transmitted is small. (e.g. Equality)

33 Equality in Simultaneous Messages Input x Referee Is x=y? Input y Randomized algorithm (Complexity ) Alice and Bob use an error correcting code C with constant distance and size O(n). They each send bits of their strings C(x), C(y) Referee outputs Yes if C(x) i = C(y) i

34 Equality in Simultaneous Messages Quantum algorithm : (Complexity O(log n)) [BCWdW0] Alice and Bob use an error correcting code C with constant distance. They send the states Referee measures the state in the symmetric and alternating subspace of If x=y, then the states are equal. If x y, then the states are almost orthogonal.

35 Exponential Separations Two-way communication [BCW98]: exponential separation for zero error. [Raz99]: exponential separation for bounded error. [Gav07, RK]: between q. One-way and rand. Two-way One-way communication [BJK04]: exponential separation for a relation [GKKRdW07]: exponential separation for a partial function Simultaneous Messages [BCWdW0]: equality via fingerprints [BJK04]: exponential separation for a relation

36 The Hidden Matching Problem Input: x {0,} 2n Output: Input: a matching M on [2n] eg. {(,5),(2,6),(3,7),(4,8)} Theorem There exists a one-way quantum protocol with compl. O(logn) Any randomized one-way protocol has complexity Ω( n) %

37 Quantum algorithm for HM 4 Let M = {(,3),(2,4)} be Bob s matching. Alice sends the state 2 (( )x + ( )x ( )x ( ) x 4 4 ) Bob measures in the basis B = { + 3, 3, 2 + 4, 2 4 } and outputs ((,3),0) if he measures ((,3),) " ((2,4),0) " ((2,4),) "

38 Quantum algorithm for HM 4 Alice sends the state 2 4 i= ( ) x i i = 2 (( )x + ( )x 3 3 ) + 2 (( )x ( )x 4 4 ) Bob measures in the basis B = { + 3, 3, 2 + 4, 2 4 } Pr[outcome + 3 ] = 8 (( )x + ( )x 3 )2 Pr[outcome 3 ] = 8 (( )x ( )x 3 )2 Bob can compute the XOR of a pair of the matching with probability.

39 4) Communication Complexity Open Problems Quantum communication complexity of total functions Power of entanglement in communication complexity Communication Complexity with super-quantum resources.

40 Conclusions Quantum Information can be very powerful Algorithms Factoring, Unordered Search Quantum Walks, etc Communication Complexity Many exponential separations Total Functions Cryptography Unconditional Key Distribution Impossibility of Bit Commitment, OT Interactions with Complexity Theory & Physics Ronald's talk

41 Why is Quantum Computation important? Further Conclusions Quantum Information and Computation Computational power of nature Quantum Mechanics as an theory of information Advances in classical Computer Science Practical Quantum Cryptography Advances in Experimental Physics

42 Simon's Algorithm Let f: {0,} n -> {0,} n Promise: f(x)=f(x+a) and f(x) f(y), y x+a (2-periodic) Goal: Find a Randomized: 2 n/2 Quantum: O(n), by finding each time a random y, st. y.a= measure f (x) """" amplitude of H onst """ x n/2 x {0,} n 2 x + 2 x + a H " O " f x f (x) 2 n/2 x {0,} n ( ) x y y + 2 n+/2 y {0,} n + ( ) (x+a) y y 2 n+/2 Hence, we only measure y, s.t. a.y=0 Repeat O(n) times to get n linear independent y's. y {0,} n y = 2 n+/2 ( )x y + 2 n+/2 ( )(x+a) y = 2 n+/2 ( )x y [+ ( ) a y ]

43 Period Finding Algorithm Let f:z N -> C Let f: Z N -> C Promise: f is periodic Goal: Find period Tool: Quantum Fourier Transform: x QFT " " N N ω x y y, ω = e y {0,} n 2π i /N measure f (x) """" QFT N onst """ N /r N /r j=0 N x Z N j r + l measure """ k N /r, k [ 0,r ] x QFT " " N O " f r r i=0 N x Z N i N /r x f (x) If gcd(k,r)=, then gcd(kn/r,n)=n/r REMARK: Factoring reduces classically to period finding!!!