Parallel approximation of min-max problems

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 parallel (classical) algorithm for finding optimal strategies for a new quantum game.

What is the talk about?

DQIP=PSPACE, and thus,

SQG=QRG(2)=PSPACE an extension of the QIP=PSPACE [JJUW10] Show a class of SDPs admits

efficient parallel algorithm. Enlarge the range to apply theMultiplicative Weight Update

Method (MMW).

Parallel algorithm and our concern

x

accept,reject

Parallel efficiency = Space efficiency [Bord77]

Game Theory 101

Game Theory 101

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

Payoff Matrix

.

.

.

.

.. …. … … 0.5/ -0.5

There could be interactions!

Refereed games

Bob

Alice

PayoffRef

Time- efficient algorithms for classical ones (linear programming) [KM92, KMvS94]

Time-efficient algorithms for quantum ones (semidefinite programming) [GW97]

Refereed games

Bob

Alice

Ref payoff

Efficient parallel algorithms for classical ones [FK97]. (complicated, nasty)

Quantum Ones: Open Until now!

Motivation: Complexity Theory

Prover

accept x,reject x

Verifier

x

x

Motivation: Complexity Theory

Motivation: Complexity Theory

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

Motivation: Complexity Theory

accept x,reject x

no-prover

verifier

x

x

x

yes-prover

Motivation: Complexity Theory

Motivation: Complexity Theory

IP=PSPACE

RG(2)=PSPACE [FK97]

RG=EXP[KM92, FK97]

QIP=PSPACE [JJUW10, W10]

QRG=EXP [GW07]

Multiplicative Weight Update Method

QRG(2)=PSPACE !

Our Results

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]

Our Results

public-coin RG ≠ RG unless PSPACE=EXP

In contrast to

public-coin IP (AM[poly])=IP

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 admitparallel solutions.

one-page tutorial for Multiplicative Weight Update Method

Finding the equilibrium point/value:

beats

…

equilibrium point

Get into a cycle

MMW is a way to choose Alice’s strategy.

Advantage

Disadvantage

explicit steps simple operations (NC)

Only good for density operators as strategies Needs efficient implementation of response. Nice responses so that not too many steps.

Technical Difficulties

Finding good representations of the strategies

Find good representations

strategystrategy

Min-Max payoff = Max-Min payoffCompute:

density operator POVM measurement

Come from a valid

interaction!

Find good representations

Transcript Representation

Kitaev: Quantum Coin Flipping

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

relaxed transcript

Penalization idea and Rounding theorem

valid transcript

trace distance trace distance trace distance

Penalty=

+ +

Fits in the min-max form

Penalization idea and Rounding theoremGoal: if Alice cheats, then the penalty should be large!

trace distance fidelity trickBures metric Bures metricBures metric>=+Penalty

Ad

van

tag

e

Technical Difficulties

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, and get rid of Alice

and then the POVM.

The universe as we know it

QIP = IP = PSPACE = RG(2)

QIP(2)

QMA AM

MA

NP

RG(1)

QRG(1)

QRG = RG = EXP

QRG(2)

SQG RG(k)

QRG(k)

The universe as we know it

QIP = IP = PSPACE = RG(2)

QIP(2)

QMA AM

MA

NP

RG(1)

QRG(1)

QRG = RG = EXP

QRG(2)SQG

RG(k)

QRG(k)

The universe as we know it

QIP = IP = PSPACE = SQG = QRG(2) = RG(2)

QIP(2)

QMA AM

MA

NP

RG(1)

QRG(1)

QRG = RG = EXP

RG(k)

QRG(k)

The universe as we know it

QIP(2)

QMA AM

MA

NP

RG(1)

QRG(1)

QRG = RG = EXP

RG(k)

QRG(k)

PSPACE

The universe as we know it

QIP(2)

QMA AM

MA

NP

RG(1)

QRG(1)

QRG = RG = EXP

RG(k)

QRG(k)

PSPACE ?

The End?

PSPACE