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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?. - PowerPoint PPT Presentation
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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
