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NP has log-space verifiers with fixed-size public quantum registers

NP has log-space verifiers with fixed-size public quantum registers. A. C. CEM SAY Department of Computer Engineering Boǧazi ç i University. ABUZER YAKARYILMAZ Faculty of Computing University of Latvia. October 07, 2011 TÕRVE. An interactive proof system for a language. PROVER.

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NP has log-space verifiers with fixed-size public quantum registers

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  1. NP has log-space verifiers with fixed-size public quantum registers A. C. CEM SAY Department of Computer Engineering Boǧaziçi University ABUZER YAKARYILMAZ Faculty of Computing University of Latvia October 07, 2011 TÕRVE

  2. An interactive proof system for a language PROVER VERIFIER probabilistic machine

  3. An interactive proof system for a language PROVER unlimited computational power resource-bounded Prover can cheat! VERIFIER

  4. Two criteria: Language has a proof system if COMPLETENESS For every , the verifier always accepts with high probability after interacting the prover SOUNDNESS For every and every , the verifier rejects with high probability after interacting

  5. Arthur-Merlin system (space-bounded) Input tape (read-only) 1 MERLIN Communication cell outcomes ARTHUR Work tape (restricted) Work tape (unlimited) Random number generator

  6. Complexity classes is the class of languages recognized by a deterministic Turing machine in polynomial time. is the class of language recognized by a nondeterministic Turing machine in polynomial time. --- is the class of languages having an AM proof system with no error such that • the random number generator is removed and • the runtime of Arthur is restricted with polynomial time. Class is obtained, if the communication cell is removed as a further restriction. --- - [Con89] Awell-known open problem: Is equal to , or not?

  7. A new system: qAM Input tape (read-only) 1 MERLIN Communication cell outcomes ÂRTHUR Work tape (restricted) Work tape (unlimited) A finite quantum register

  8. The finite quantum register • A quantum register is an -dimensional Hilbert space, , with basis • , where • A quantum state is a linear combination of basis states, i.e. • , where • each is called the amplitude of being state and the probability of being in state is given by .

  9. The operations on the register • Initializing the register (a predefined quantum state) • Applying a superoperator () satisfying , where • is an operation element • is the measurement outcome • [Optional] Each entry of is a rational number = = … … … … =

  10. -- -, a well-known -completeproblem, is the collection of all strings of the form such that , and ’s are numbers in binary , and there exists a set satisfying . --- Ârthurcanencode binary numbers into amplitudes of the states of the register and can also make additionand subtractionon them. The strategy of Ârthur: Ârthur requests the set from Merlin and then tests .

  11. Some details of the algorithm … … Before reading $ Initial state auxiliary value to store to store ’s to store Accept () reject • Member are accepted exactly. • Nonmembers are rejected with a probability at least . • The error gap can be reduced to any desired value by using • conventional probability amplification techniques.

  12. - Any language in is log-space reducible to - [Pap94]: • Let be language in , then there exists a logarithmic space deterministic algorithm that outputs for any given input string such that -. --- For any given input string , Ârthur can run the algorithm for - on . -. -

  13. Concluding remarks • A poly-time Ârthur can be simulated by a poly-time Arthur: -- • In constant space [DS92,CHPW94,AW02]: -- ---- (if arbitrary transition amplitudes are allowed) • Is--? [Con93] • What is the relationship between and --?

  14. References

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