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

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

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

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

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

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.

Prover accept x, reject x Verifier x x

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

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

behavior at equilibrium points

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:

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)

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

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

 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.

Finding good representations of the strategies

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

Find good representations snap-shot of density operators consistency condition

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:

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

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

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:

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

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

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

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

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

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