xiaodi wu with applications to classical and quantum zero-sum games eecs, university of michigan...
TRANSCRIPT
Parallel approximation of min-max problems
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
What is the talk about?
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 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
Parallel algorithm
x
accept,reject
Parallel efficiency = Space efficiency [Bord77]
Payoff Matrix
.
.
.
.
.. …. … … 0.5/ -0.5
Zero-sum Game
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
= equilibrium value
There could be other forms!
Normal form
Refereed games
Bob
Alice
PayoffRef
Time- efficient algorithms for classical ones (linear programming) [KM92, KMvS94]
Time-efficient algorithms for quantum ones (semidefinite programming) [GW97]
zero-sum games w/ interactions
quantum version
Refereed games
Bob
Alice
Ref payoff
Efficient parallel algorithms for classical ones [FK97]. (complicated, nasty)
Quantum Ones: shown in this work.
Motivation: Complexity Theory
Prover
accept x,reject x
Verifier
x
x
Motivation: Complexity Theory
AM[poly]Both equal PSPACE. [LFKN92, S92, GS89]
PROOF VERIFICATION SYSTEM
public randomness poly rounds
accept x,reject x
no-prover
verifier
x
x
x
yes-prover
PROOF VERIFICATION W/ ZERO-SUM GAMES
Two players
PROOF VERIFICATION W/ ZERO-SUM GAMES
behavior at equilibrium
points
Known Results
IP=PSPACE
RG(2)=PSPACE [FK97]
RG=EXP[KM92, FK97]
QIP=PSPACE [JJUW10, W10]
QRG=EXP [GW07]
QRG(2)=PSPACE !This work:
poly rounds
poly rounds
quantum result:
classical result:
Subsume and unify all the previous results.
DQIP=SQG=QRG(2)=PSPACE
First-principle proof of QIP=PSPACE.
QIP SQG [GW05]
Our Results
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
Our Results
admissible quantum 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
Our Results
explicit steps simple operations (NC)
Matrix Multiplicative Weight Update Method (well-known powerful method)
Finding the equilibrium point/value:
beats
…
equilibrium pointPotential Problem:Get into a cycleMMW is a way to
choose Alice’s strategy to break the cycle.
Advantage
Disadvantage Only good for density operators as strategies Needs efficient implementation of response. Nice responses so that not too many steps.
Technical Ingredients
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)
POVM measurement (net-effect of Bob)
Come from a valid
interaction!
DQIP CIRCUIT
qubitsQuantum operation
Find good representations
Transcript Representation [Kitaev03]
snap-shot of density operators
consistency conditionconsistency conditionconsistency condition
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
Sol: Transcript Represetation
Sol:
relaxed transcript
Penalization idea and Rounding theorem
valid transcript
trace distance trace distance trace distance
Penalty=
+ +
Fits in the min-max form
violateconsistency
violateconsistency
violateconsistency
Penalization idea and Rounding theoremGoal: if Alice cheats, then the penalty should be large!
trace distance
fidelity trick
Bures metric Bures metricBures metric>=+Penalty
Ad
van
tag
e
invalidtranscript
validtranscript consistent consistent consistent
trace distance
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
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, and get rid of Alice
and then the POVM.
purification
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