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Parallel approximation of min-max problems

with applications to classical and quantum zero-sum games. Parallel approximation of min-max problems. University of Michigan. Xiaodi Wu. Joint work with Gus Gutoski at IQC, University of Waterloo. What is the talk about?.

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Parallel approximation of min-max problems

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  1. with applications to classical and quantum zero-sum games Parallel approximation of min-max problems University of Michigan Xiaodi Wu Joint work with Gus Gutoski at IQC, University of Waterloo

  2. What is the talk about? • A parallel (classical) algorithm for finding optimal strategies for a new quantum game. • DQIP=PSPACE, and thus, • SQG=QRG(2)=PSPACE • an extension of the QIP=PSPACE[JJUW10] • Show a class of SDPs admits efficient parallelalgorithm. • Enlarge the range to apply the • Multiplicative Weight Update Method (MMW).

  3. Parallel algorithm and our concern Parallel efficiency = Space efficiency [Bord77] x accept, reject

  4. Game Theory 101

  5. Game Theory 101 Payoff Matrix Zero-Sum games characterize the competition between players. Your gain is my Loss. The stable points at which people play their strategies, equilibrium points. Min-Max payoff = Max-Min payoff There could be interactions!

  6. Refereed games Time- efficient algorithms for classical ones (linear programming) [KM92, KMvS94] Alice Payoff Ref Bob Time-efficient algorithms for quantum ones (semidefinite programming) [GW97]

  7. Refereed games Efficient parallel algorithms for classical ones [FK97]. (complicated, nasty) Alice payoff Ref Quantum Ones: Open Until now! Bob

  8. Motivation: Complexity Theory Prover x accept x, reject x x Verifier

  9. Motivation: Complexity Theory

  10. Motivation: Complexity Theory Both equal PSPACE. [LFKN92, S92, GS89] AM[poly]

  11. Motivation: Complexity Theory yes-prover x verifier accept x, reject x x x no-prover

  12. Motivation: Complexity Theory

  13. Motivation: Complexity Theory QIP=PSPACE [JJUW10, W10] Multiplicative Weight Update Method IP=PSPACE QRG=EXP [GW07] QRG(2)=PSPACE ! RG=EXP [KM92, FK97] RG(2)=PSPACE [FK97]

  14. Our Results Subsume and unify all the previous results along this line. QIP inside SQG [GW05] DQIP=SQG=QRG(2)=PSPACE directly applicable to general protocol. first-principle proof of QIP=PSPACE.

  15. Our Results public-coin RG ≠ RG unless PSPACE=EXP In contrast to public-coin IP (AM[poly])=IP

  16. Our Results admissible quantum channels appropriated bounded Efficient parallel algorithm for above SDP. There cannot be an efficient parallel approximation scheme for all SDPs unless NC=P [Ser91,Meg92]. Our result adds considerably to the set of SDPs that admit parallel solutions.

  17. one-page tutorial forMultiplicative Weight Update Method Finding the equilibrium point/value: equilibrium point Hard to apply directly! Get into a cycle • explicit steps • simple operations (NC) MMW is a way to choose Alice’s strategy. … Advantage beats Disadvantage • Only good for density operators as strategies • Needs efficient implementation of response. • Nice responses so that not too many steps.

  18. Technical Difficulties Finding good representations of the strategies

  19. Find good representations Come from a valid interaction! strategy strategy Min-Max payoff = Max-Min payoff density operator POVM measurement Compute:

  20. Find good representations Transcript Representation Kitaev: Quantum Coin Flipping

  21. Technical Difficulties Finding good representations of the strategies Tailor the “transcript-like” representation into MMW Run many MMWs in parallel Penalization idea and the Rounding theorem

  22. Penalization idea and Rounding theorem Fits in the min-max form Penalty= + + trace distance trace distance trace distance relaxed transcript valid transcript

  23. Penalization idea and Rounding theorem Goal: if Alice cheats, then the penalty should be large! Bures metric Penalty Advantage Bures metric >= trace distance + Bures metric fidelity trick

  24. Technical Difficulties Finding good representations of the strategies Tailor the “transcript-like” representation into MMW Run many MMWs in parallel Penalization idea and the Rounding theorem Finding response efficiently in space Call itself as the oracle! Nested!

  25. Finding response efficiently in space Given Alice’s strategy, • purify it, and get rid of Alice Now deal with a special case, where Bob plays with “do-nothing” Charlie Call itself to compute Bob’s strategy, • and then the POVM. WE ARE DONE!

  26. The universe as we know it QRG = RG = EXP QRG(k) SQG RG(k) QRG(2) QIP = IP = PSPACE = RG(2) QRG(1) QIP(2) RG(1) QMA AM MA NP

  27. The universe as we know it QRG = RG = EXP QRG(k) RG(k) SQG QRG(2) QIP = IP = PSPACE = RG(2) QRG(1) QIP(2) RG(1) QMA AM MA NP

  28. The universe as we know it QRG = RG = EXP QRG(k) RG(k) QIP = IP = PSPACE = SQG = QRG(2) = RG(2) QRG(1) QIP(2) RG(1) QMA AM MA NP

  29. The universe as we know it QRG = RG = EXP QRG(k) RG(k) PSPACE QRG(1) QIP(2) RG(1) QMA AM MA NP

  30. The universe as we know it QRG = RG = EXP ? QRG(k) RG(k) PSPACE QRG(1) QIP(2) RG(1) QMA AM MA NP

  31. The End? PSPACE

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