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

2. Algorithm : an efficient parallel algorithm approximately computing equilibrium values of a new kind of zero-sum games Complexity : special case: an efficient parallel algorithm for a new class of SDPs apply the algorithm to solve the open problem SQG=QRG(2)=PSPACE SQG=QRG(2)=PSPACE an extension of the QIP=PSPACE [JJUW10, Wu10] NO extra power with quantum in this model given RG(2)=PSPACE [FK97] Key Technique : enhanced Matrix Multiplicative Weight Update method

3. x accept, reject Parallel efficiency = Space efficiency [Bord77]

4. Payoff Matrix......….…… 0.5/ -0.5 Zero-Sum Zero-Sum games characterize the competition between players. Your gain is my Loss. equilibrium points The stable points at which people play their strategies, equilibrium points. Min-Max payoff = Max-Min payoff = equilibrium value There could be other forms! other forms! Normal form

5. Bob Alice Payoff Ref classical KM92, KMvS94] Time- efficient algorithms for classical ones (linear programming) [KM92, KMvS94] quantum [GW97] Time-efficient algorithms for quantum ones (semidefinite programming) [GW97] zero-sum games w/ interactions quantum version

6. Bob Alice Ref payoff classical [FK97]. (complicated, nasty) Efficient parallel algorithms for classical ones [FK97]. (complicated, nasty) Quantum Ones Quantum Ones: shown in this work.

7. Prover accept x, reject x Verifier x x

9. AM[poly] PSPACE. Both equal PSPACE. [LFKN92, S92, GS89] PSPACE. Both equal PSPACE. [LFKN92, S92, GS89] public randomnesspoly rounds

10. accept x, reject x no-prover verifier x x x yes-prover Two players

11. behavior at equilibrium points

12. IP=PSPACE RG(2)=PSPACE [FK97] RG=EXP [KM92, FK97] QIP=PSPACE [JJUW10, W10] QRG=EXP [GW07] QRG(2)=PSPACE ! This work: poly rounds quantum result: classical result:

13. Subsume and unify all the previous results. DQIP=SQG=QRG(2)=PSPACE First-principle proof of QIP=PSPACE. Double Quantum Interactive Proof (DQIP) (interacts with Alice, then Bob)

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

15. admissible quantum channels channels appropriately bounded Efficient parallel algorithm for all SDPs? No for general SDP unless NC=P [Ser91,Meg92]. Our result: Yes for this and more SDPs

16.  explicit steps NC  simple operations (NC) Finding the equilibrium point/value: beats … equilibrium point Potential Problem: Get into a cycle MMW MMW is a way to choose Alice’s strategy to break the cycle.Advantage Disadvantage density operators  Only good for density operators as strategies  Needs efficient implementation of response.  Nice responses so that not too many steps.

17. Finding good representations of the strategies

18. Find good representations Strategy inputs=>outputs strategy Min-Max payoff = Max-Min payoffCompute: density operator (net-effect of Alice) (net-effect of Alice) POVM measurement (net-effect of Bob) (net-effect of Bob) Come from a valid interaction! qubits Quantum operation

19. Find good representations snap-shot of density operators consistency condition

20. Finding good representations of the strategies Tailor the “transcript-like” representation into MMW Run many MMWs in parallel Rounding Penalization idea and the Rounding theorem Sol: Sol: Transcript Represetation Sol:

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

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

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

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

25. QIP = IP = PSPACE = RG(2) QIP(2) QMAAM MA NP RG(1) QRG(1) QRG = RG = EXP QRG(2) SQG RG(k) QRG(k)

26. QIP = IP = PSPACE = RG(2) QIP(2) QMAAM MA NP RG(1) QRG(1) QRG = RG = EXP QRG(2) SQG RG(k) QRG(k)

27. QIP = IP = PSPACE = SQG = QRG(2) = RG(2) QIP(2) QMAAM MA NP RG(1) QRG(1) QRG = RG = EXP RG(k) QRG(k)

28. QIP(2) QMAAM MA NP RG(1) QRG(1) QRG = RG = EXP RG(k) QRG(k) PSPACE

29. QIP(2) QMAAM MA NP RG(1) QRG(1) QRG = RG = EXP RG(k) QRG(k) PSPACE ?