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

a new quantum game  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] SDPs  Show a class of SDPs admits efficient parallel algorithm.  Enlarge the range to apply the Multiplicative Weight Update Method (MMW) Multiplicative Weight Update Method (MMW).

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

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 Payoff Matrix......….…… 0.5/ -0.5 There could be interactions! interactions!

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]

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

Prover accept x, reject x Verifier x x

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

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

IP=PSPACE RG(2)=PSPACE [FK97] RG=EXP [KM92, FK97] QIP=PSPACE [JJUW10, W10] QRG=EXP [GW07] Multiplicative Weight Update Method QRG(2)=PSPACE !

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

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

admissible quantum channels 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.

Finding the equilibrium point/value: beats … equilibrium point Get into a cycle MMW MMW is a way to choose Alice’s strategy.Advantage Disadvantage  explicit steps NC  simple operations (NC) 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 Min-Max payoff = Max-Min payoffCompute: density operator POVM measurement Come from a valid interaction!

Find good representations Quantum Coin Flipping Kitaev: Quantum Coin Flipping

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

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

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

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

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.

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 ?