Introduction to Quantum Information Theoryand Applications to Classical Computer Science           Iordanis Kerenidis     ...
Quantum information and computation   Quantum information and computation        How is information encoded in nature?  ...
Quantum Information   Quantum bit (qubit): Carrier of quantum information       A quantum mechanical system, which can b...
Probability Theory – Quantum Information IBinary Random Var. X:   Quantum bit: unit vector in a 2-dim. Hilbert spaceRandom...
Probability Theory – Quantum Information IIEvolution by Stochastic Matrices Evolution by Unitary Matrices(preserve l1-norm...
Density matrix   Mixed state: Classical distribution over pure quantum states   Density matrix:                         ...
Entropy of Quantum states   Shannon Entropy: randomness in the measurement    Random variable X, distribution P   Von Ne...
Properties of von Neumann Entropy                     where   contains m qubits.   Conditional Entropy   Mutual infor...
Probability Theory - Quantum InformationRandom Variable X                   Quantum statePositive probabilities           ...
Amplitudes vs. Accessible Information   Exponential number of amplitudes    We can encode n bits                into log(...
Quantum vs. Classical Information    Communication complexity                               001110100011               Inp...
Quantum Information and Cryptography   Unconditionally Secure Key distribution [Bennett, Brassard 84]                    ...
Quantum Information & Classical Computer Science   Classical results via quantum arguments       Lower bound for Locally...
Locally Decodable Codes   Error Correcting Codes have numerous theoretical applications       Hash functions, expanders,...
Locally Decodable Codes (LDC)                   x      encode                         A code C : {0,1}n     {0,1}m is a   ...
Exponential Lower Bound for 2-query LDCsTheorem: A 2-query LDC has length m 2 ( n) [Kerenidis, deWolf 2003]      eq. A bin...
Exponential Lower Bound for 2-query LDCsProof:                n          m Let C : {0,1}    {0,1} be a 2-query LDC s.t. o...
Proof of the ClaimClaim:     From the quantum encoding                                       one can     recover any xi wi...
Quantum Decoding algorithmLet                                                be the matching for xi.Classical Decoding Alg...
Proof of Correctness   The decoder uses the state   Measurement in the basis   If                         then   If ...
Conclusions                                  Why is                           Quantum Computation                         ...
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Introduction to quantum information theory and applications to classical computer science by iordanis kerenidis

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Introduction to Quantum Information Theory and Applications to Classical Computer Science by Iordanis Kerenidis

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Introduction to quantum information theory and applications to classical computer science by iordanis kerenidis

  1. 1. Introduction to Quantum Information Theoryand Applications to Classical Computer Science Iordanis Kerenidis CNRS – Univ Paris
  2. 2. Quantum information and computation Quantum information and computation  How is information encoded in nature?  What is nature’s computational power? Quantum algorithm for Factoring [Shor 93] Information-theoretic secure key distribution [Bennett-Brassard 84] Quantum computers probably won’t solve NP-complete problems [BBBV94]
  3. 3. Quantum Information Quantum bit (qubit): Carrier of quantum information  A quantum mechanical system, which can be in a state |0 , |1 or any convex combination of them. Physical Examples: Spin of an electron, Spin of a nuclear, Photon polarization, Two-level atom, Non-abelian anyon, … Here, a qubit is an abstract mathematical object, i.e. a unit vector in a 2D Hilbert space.
  4. 4. Probability Theory – Quantum Information IBinary Random Var. X: Quantum bit: unit vector in a 2-dim. Hilbert spaceRandom Variable X: Quantum state: on logn qubitsCorrelations Entanglement (more general than correlations!!!)
  5. 5. Probability Theory – Quantum Information IIEvolution by Stochastic Matrices Evolution by Unitary Matrices(preserve l1-norm) (preserve l2-norm)Measurement Measurement (Projective) A measurement of in an orthonormal basis is a projection onto the basis vectors and Pr[outcome is bi ] =
  6. 6. Density matrix Mixed state: Classical distribution over pure quantum states Density matrix: (hermitian, trace 1, positive) 1. contains all information about a measurement in any basis B={b1,…bn}. Pr[outcome is bi ] 2. Different ensembles can have the same
  7. 7. Entropy of Quantum states Shannon Entropy: randomness in the measurement Random variable X, distribution P Von Neumann Entropy: randomness in the best possible measurement Mixed quantum state , Density matrix , where E are the eigenvectors and i the eigenvalues. Von Neumann Entropy shares many properties with the Shannon Entropy.
  8. 8. Properties of von Neumann Entropy where contains m qubits. Conditional Entropy Mutual information Strong subadditivity Fano’s Inequality
  9. 9. Probability Theory - Quantum InformationRandom Variable X Quantum statePositive probabilities Complex amplitudesCorrelations Entanglement (more general than correlations!!!)Stochastic Matrices & Measurement Unitary Matrices & Measurements (in different bases)Probability Distributions Mixed states (distribution over pure quantum states)Shannon Entropy Von Neumann Entropy (similar properties)
  10. 10. Amplitudes vs. Accessible Information Exponential number of amplitudes We can encode n bits into log(n) qubits e.g. Only indirect access to the information via measurements No measurement can give us all the bits of x. Holevo’s bound : n qubits encode at most n bits.  Quantum information does not compress classical information  S( ) n
  11. 11. Quantum vs. Classical Information Communication complexity 001110100011 Input x Input y Goal: Output P(x,y) ( minimum communication ) Quantum communication can be exponentially smaller than classical communication complexity  Two-way communication [Raz99]  One-way communication [Bar-Yossef, Jayram, K.04, GKKRdW07]  Simultaneous Messages [Bar-Yossef, Jayram, K. 04]
  12. 12. Quantum Information and Cryptography Unconditionally Secure Key distribution [Bennett, Brassard 84] 001110100011100011 Random number Generators [id Quantique] Private Information Retrieval [K., deWolf 03, 04] Message Authentication, Signatures [Barnum et al. 02, Gottesman, Chuang 01] Quantum One-way functions [Kashefi, K. 05] Quantum cryptography in practice
  13. 13. Quantum Information & Classical Computer Science Classical results via quantum arguments  Lower bound for Locally Decodable Codes and Private Information Retrieval [K., deWolf 03]  Lattice Problems in NP coNP [Aharonov, Regev 04]  Matrix Rigidity [deWolf 05]  Circuit Lower Bounds [K.05]  Lower bounds for Local Search [Aaronson 03]
  14. 14. Locally Decodable Codes Error Correcting Codes have numerous theoretical applications  Hash functions, expanders, list decoding, hardcore predicates One interesting property: Local Decodability  Introduced by Katz-Trevisan. Applications of Locally Decodable Codes  Probabilistically Checkable Proofs (PCP)  Worst-case to Average-case reductions  Private Information Retrieval  Extractors  Hardness Amplification  Polynomial Identity Testing
  15. 15. Locally Decodable Codes (LDC) x encode A code C : {0,1}n {0,1}m is a C(x) (q, , )-locally decodable code if one can recover any bit xi with probability fraction errors ½+ by q queries, even if fraction of a1 aq the codeword is corrupted. q probes Locally Decodable Codes ↕Decoding Algorithm outputs “Smooth Codes”xi with prob. >1/2+ ↕ Private Information Retrieval
  16. 16. Exponential Lower Bound for 2-query LDCsTheorem: A 2-query LDC has length m 2 ( n) [Kerenidis, deWolf 2003] eq. A binary 2-server PIR scheme has complexity (n)Known Bounds Servers PIR k=1 (n) k=2, binary (n) k =2 O(n1/3) , (log n) k>2 O(n1/loglogn) , (log n)
  17. 17. Exponential Lower Bound for 2-query LDCsProof: n m Let C : {0,1} {0,1} be a 2-query LDC s.t. one can recover any bit xiwith probability ½+ . Claim: From the quantum encoding one can recover any xi with prob. ½+ . ( n) Hence, m 2
  18. 18. Proof of the ClaimClaim: From the quantum encoding one can recover any xi with prob. ½+ .Proof:1. The Decoder w.l.o.g. is similar the Hadamard Decoder. Hadamard Decoding Algorithm: i) Pick a random r {0,1}n and query (r, r+ei)-th bits ii) Output the XOR: x • r + x • (r+ei) = x • ei = xi Prob[correct decoding] >1-2 Decoder can tolerate a fraction of errors The distribution of queries must be ”close” to uniform. The decoder queries a random edge of a matching and XORs
  19. 19. Quantum Decoding algorithmLet be the matching for xi.Classical Decoding Algorithm:i) Query a random edgeii) Output: Prob[correct decoding] > ½+Quantum Decoding Algorithm:i) Measure the state in the basisii) Output xi = 0 if the result is xi = 1 if the result is Prob[correct decoding] >½+
  20. 20. Proof of Correctness The decoder uses the state Measurement in the basis If then If then
  21. 21. Conclusions Why is Quantum Computation important? Quantum Information and Computation  Rich Mathematical Theory  Computational power of nature  Advances in Theoretical Physics  Advances in classical Computer Science  Practical Quantum Cryptography  Advances in Experimental Physics
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