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Quantum Complexity Classes

By: Larisse D. Voufo On: November 28 th , 2006 lvoufo@cs.indiana.edu. Quantum Complexity Classes. http://www.quantiki.org/wiki/images/4/46/PhotonIdentityCartoon.gif. Introduction. 1982 (Trend toward miniaturization and microcircuitry) , Paul Benioff & Richard Feynman :

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Quantum Complexity Classes

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  1. By: Larisse D. Voufo On: November 28th, 2006 lvoufo@cs.indiana.edu Quantum Complexity Classes http://www.quantiki.org/wiki/images/4/46/PhotonIdentityCartoon.gif

  2. Introduction • 1982 (Trend toward miniaturization and microcircuitry), Paul Benioff & Richard Feynman: Quantum Systems could perform computation. • 1985, David Deutch. Quantum Computer  Turing Machine  possibility of new Complexity of algorithms • Later On, Universality of Quantum Circuits  Machine independent notion of quantum complexity.

  3. Key quantum property for quantum complexity studies: Randomness of quantum measurement process  Algorithm performed by a quantum computer is probabilistic. (== multiple runs, different results)

  4. Probabilistic Computation vs. Quantum Computation. • Nondeterministic Computation (NC) = tree of configurations of NTM • Probabilistic Computation = NC where probabilities <--> edges and nodes.  Rules of Classical Probability. • Quantum Computation = NC where amplitudes <--> edges and nodes.  Rules of Quantum Probability.

  5. From Classical Complexity classes… • P – “easy”: languages decided by polynomial-time TMs • NP: languages decided by polynomial-time NTMs. • Guess an answer, verify in polynomial time. Is answer YES? • NP-hard: Every hard problem can be polynomially reduced to a problem in this class. • NP-complete:  NPC = NP-hard  NP NP-hard  P  {} => P = NP

  6. From Classical Complexity classes… • NPI: Problems in NP of intermediate difficulty • NPI = NP – P – NPC = NP – P – NP-hard • Co-NP: Like NP, but Answer is NO (counter-example based) NP  Co-NP No proof for: P  NP.

  7. From Classical Complexity classes… • AWPP: languages decided by Almost-Wide Probabilistic Polynomial-timeNTMs • PP: languages decided by polynomial-time NTMs where the majority of paths gives the correct answer. • P#P: functions that count the number of accepting paths through an NP machine. P  NP  AWPP  PP  P#P.

  8. From Classical Complexity classes… • IP: Problems solvable by an Interactive Proof System. • MA: languages decided by a bounded-error probabilistic Merlin-Arthur protocol. • BPP: Bounded-error Probabilistic Polynomial Time. “Problems that admit a probabilistic circuit family of polynomial size that always gives the right answer with prob > ½ + ”. • PSPACE: DPs that can be solved in polynomial-space, but may require exponential time.

  9. … toQuantum Complexity Classes: • BQP: Bounded-error Quantum Polynomial Time. “DPs that can be solved, with high probability, by polynomial-size quantum circuits”. • EQP (QP): Exact version of BQP

  10. … toQuantum Complexity Classes:  P  BPP  BQP  PSPACE • IP = PSPACE • NP  MA • BPP  MA  IP • BQP  P#P PSPACE • No firm proof for: BPP  BQP (in general) • If P = PSPACE, then P = AWPP “relative to oracle” • NP = AWPP “relative to oracle” • NP  PSPACE (checking if C(x(n), y(n)) = 1 for each y(m)) • NP  BQP ?

  11. … toQuantum Complexicity Classes: • BQNP ( = QMA) • QMA-complete • QIP • EQP  BQP  QMA  QIP

  12. Interactive Proof System: IP BPP ?, r, … Polynomial Number of Messages Proof (x  L)

  13. Merlin-Arthur Protocol: NP Deterministic Polynomial-time TM ?, r, … Constant Number of Messages

  14. Merlin-Arthur Protocol: MA BPP ?, r, … Constant Number of Messages

  15. Merlin-Arthur Protocol: QMA(C) • QMA-Completeness: ground state energy problem: (5-local hamiltonian). BQP ?, r, … Constant Number of Messages

  16. Merlin-Arthur Protocol: QIP BQP Q- ?, r, … Polynomial Number of Messages Q- Proof (x  L)

  17. A model for quantum circuits: Facts: • Quantum gate: unitary transformation  reversible gate. • Classical Reversible Computer = special case of Quantum Computer. • x(n) y(n) = f(x(n)) <==> U: |xi>  |yi> • |00…0>  Deterministic final measurement

  18. 3 Issues with this model: • Universality • Complete Model <==> There exists no transformation in U(2n) that we cannot reach. • Simulation of a Q-computer using another Q-computer • complexity classes do not depend on the details of the hardware. • Simulating a quantum computer on a classical computer: Better characterize the resources needed. • A Classical Computer can still simulate a Q-Computer, despite a polynomial limit on memory space available.

  19. 3 Issues with this model: 3. Accuracy == growth of error in measurement as the quantum circuit size increases. • NO Polynomial-size circuit family (hard problems) w/ gates of exponential accuracy. • An idealized T-gate q-circuit (acceptable accuracy): Error Prob / gate  1/T. • Quantum Algorithm w/ prob > ½ +  (in the ideal case)  Gates w/ accuracy T < O(). • BQP can really solve hard problems <==> linear improvement of the accuracy of the gates (computation size T).

  20. More on Relationships between Complexity classes • P  BPP  BQP  AWPP  PP  PSPACE. • Bernstein and Vazirani: BQP  PSPACE • Adelman, Demarrais and Huang: BQP  PP • Fortnow and Rogers: BQP  AWPP

  21. Other Complexity Classes Vary from one literature to another… • UP, QPSV, NPSV, UPSV, etc…  Elham Kashefi’s PhD thesis (Imperial College London) • NQP, C=P, coC=P, etc…  Tarsem S. Purewal Jr (University of Georgia)

  22. Analyzing Quantum Algorithm Performances Over Classical Ones: • Non-exponential speedup: Eg: Grover’s Quantum Speed-up of the Search of an unsorted database. • “Relativized” Exponential Speed-up  Oracles  BPP  BQP “relative to oracle”. Eg: Simon’s exponential quantum speedup for finding the period of 2 to 1 function. Deutch’s algorithm. • Exponential Speed-up for “apparently” hard problems Eg: Shor’s factoring algorithm.

  23. References: • Tarsem S. Purewal Jr. “Revisiting a Limit on Efficient Quantum Computation”. ACM Southeastern Conference 2006, Melbourne, FL. March 10, 2006. Slides at http://www.cs.uga.edu/~purewal/slides/BQPinPPTalk.pdf • John Preskill. “Lecture Notes on Physics 229: Quantum Information and Computation”. Sept. 1998. California Institute of Technology. • Tarsem S. Purewal Jr. “Nondeterministic Quantum Query Complexity”. Combinatorics, Algorithms, and Theory Seminar (CATS), University of Georgia, Athens, GA. May 1, 2006. • Tarsem S. Purewal Jr. “5-local Hamiltonian is QMA-Complete”. Quantum Computing Journal Club, University of Georgia, Athens, GA. June 6, 2005. • Prof. Tony Hey. “Quantum Computing: an introduction”. Quantum Technology Center. http://www.qtc.ecs.soton.ac.uk/flecture.html • Artur Ekert, Patrick Hayden, Hitoshi Inamori. “Basic concepts in quantum computation”. converted to wiki format by Burgarth 22:37, 8 Jun 2005 (BST). http://www.quantiki.org/wiki/index.php/Basic_concepts_in_quantum_computation • Qbit.com. “Introduction to Quantum Theory”. http://www.quantiki.org/wiki/index.php/Introduction_to_Quantum_Theory • Elham Kashefi. “Complexity Analysis and Semantics for Quantum Computation”. November 26, 2003. http://web.comlab.ox.ac.uk/oucl/work/elham.kashefi/papers/phdelham.pdf • Tarsem S. Purewal Jr. http://www.cs.uga.edu/~purewal/vita.html • Lance Fortnow. “One Complexity Theorist's View of Quantum Computing”. http://people.cs.uchicago.edu/~fortnow/papers/quantview.pdf

  24. -- Thank You!  http://www.quantiki.org/wiki/images/thumb/d/da/UncertaintyCartoon.jpg/180px-UncertaintyCartoon.jpg

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