Equilibrium Value Method For the proof of

QIP=PSPACEXiaodi Wu

EECS, University of Michigan, Ann ArborJanuary, 2010

•The work was conducted while the author was visiting the Institute for Quantum •Computing, University of Waterloo, Ontario, Canada.

Interactive Proof System : Intuitive Picture

Prover

Verifier

input x

input x

Goal : determine whether x is in a language L

Computationally unbounded power.

Try to convince the verifier that x is in language L.

Probabilistic Polynomial time

power. To determine

whether x is in L.

Polynomial Rounds interactions of classical messages

Quantum Interactive Proof System

Quantum Prover

Quantum

Verifier

input x, still classical

input x

Goal : determine whether x is in a language L

Computationally unbounded

power = anything allowed

by quantum physics

Probabilistic Polynomial

time on quantum

computer.

Polynomial Rounds interactions of quantum messages

Previous Result:

IP=PSPACE [Lund, Fortnow, Karloff, and Nisan 1992; Shamir 1992]

Easy Direction : IP PSPACE, since all the messages are of polynomial size.

Hard Direction: PSPACE IP

µ

µ

Solved by a method called Arithmetizationwhich constructs a polynomial round interaction protocols for PSPACE-complete problem. Polynomial rounds is necessary !

What about Quantum Case

IP QIP, and thus we know PSPACE QIP The easier direction becomes hard in

quantum caseIt is an open problem for almost a decade to

show QIP=PSPACE.However, we do know something non-trivial

about the QIP in the very beginning. QIP=QIP(3)[KW00], 3 rounds are sufficient

for quantum caseQIP(3) EXP[KW00], by formulated as a SDP.

µ µ

µ

3 rounds Quantum Interactive Proof System

1 2 3

Qubits

efficient quantum circuits

all- power , any quantum circuits

Notations: linear algebra : complex Euclidean spaces : space of operators acting on : set of positive semidefinite operators

(or matrices) acting on

X ;Y ;ZL(X ) XPos(X )

X

D(X ) := f½2 Pos(X ) Tr(½) = 1gQuantum State:

One part of the whole state: Given

½2 D(X ­ Y )The X part of the state is and Y part of the state is TrY (½) TrX (½)

This is called Partial Trace

More precisely, we have partial trace be the unique linearmapping such that

TrY : L (X ­ Y ) ! L (X )

TrY (A ­ B) = Tr(B)A 8;A 2 L(X );B 2 L(Y )

Notations: continuedQuantum Operations: a operation maps a quantum state to a quantum state.

More precisely:

©: L(Y ) ! L (Z)

auxiliary space

(Complete Positivity) (Trace Preserving)

½ 0) ¾= (I X ­ ª )½ 0Tr(½) = Tr(¾) = 1

Quantum Measurement : an irreversible quantum operations defined by a set of positive operators such that .M = fM i 2 Pos(X )g

Pi M i = I X

The outcome k occurs with probability hMk;½i

3 rounds Quantum Interactive Proof System(revisit)

1 2 3

P ­ M ­ VQuantum State on

efficient quantum operations

all- power , any quantum operation

SDP?

RoadmapMatch from both sides

[KW00]

General Model for interaction

[GW07]

SDP formulation but with exponential size

Polynomial algorithm for SDP (IPM,

Ellipsoid) QIP in EXP [KW00]

PSPACE ?

How to parallelize ?

Matrix Multiplicative Weights Update

method[AHK05]

Parallelizable for some SDP and Equilibrium

Value ?

PSPACE =NC(poly)[Bord77]

One Possible Way

Roadmap(continued)

Bad News: old formulation of QIP still open

Good News: reformulation becomes solvable

SDP reformulation

[JJUW09], August 09

QIP=QMAM[MW05]

1 2 3Only one classical

bit is sent in the second

step

Simpler solvable SDP

QMAM in NC(poly) too complicated and technical

Starts with definition, simpler SDP but not that simple.Using MMW to solve SDP involves more complicated

steps.

Roadmap(continued)A Neater Proof is available [Wu09]

August, 09

Starts with QIP-Complete problem

[RW05]

Resulted in a simple Equilibrium Value

Problem

Quantum Circuit Distinguishability:

Given two short quantum circuits, distinguish their

distance between two promises.

Solved by Matrix Multiplicative Weight Update Method

Two Key Ingredients

Matrix Multiplicative Weight Update Method

w1

w2

.

.

.

wn

n agents weights

Update weights according toperformance:

wit+1 Ã wi

t (1 + ¢ performance of i)

1$ for correct prediction

0$ for incorrect

Long History(cited from Sanjeev Arora)

N “experts” on TVCan we perform as good as the best

expert ?

The answer is Yes by using multiplicative weight update

Matrix Multiplicative Weight Update Method

Density operator Observation

hM (t);½(t) i = Tr(M (t);½(t))cost of round t

updated in this way

my performance any agent’s performance

some small gap

Proof Hint: use potential function Tr(W (t)) and matrix

inequality.

Equilibrium Value

C1 £ C2Consider C1, C2 are convex compact sets, function f is a bilinear function on

mina2C1 maxb2C2 f (a;b) =maxb2C2 mina2C1 f (a;b)

Moreover, there exists an equilibrium point (a¤;b¤)maxb2C2 f (a¤;b) = f (a¤;b¤) =mina2C1 f (a;b¤)equilibrium value ¤

Question: How to compute the equilibrium value ?

Pick a random series of points from C1, and then get the maximum over C2

a1a2....at

b1b2....bt

¸¤ = mina2C1

maxb2C2

f (a;b)· min

tmaxb2C2

f (at;b) =mint

f (at;bt)

a upper bound is obtained easily

How about the lower bound ? choose a better series

Equilibrium ValueIn our settings, C1 is the set of density operators, C2 is the set

The bilinear function is h¦ ;¥(½)i ;½2 C1; ¦ 2 C2 linear mapping

a1a2....at

b1b2....bt

a1

a2....at

b1

b2....bt

MMW helps to generate the

series

Intuitively thinking : Why this works?

Problem with the upper bound is it can be any large. MMW helps to make the value less than the “best agent”

plus small gap.

equilibrium point

Equilibrium Value

max¦ h¦ ;¥(½(t))i

make own “observation”use MMW to get

Get for the round t½(t)

½(t+1)substitute

Equilibrium Value

use equilibrium point (½¤; ¦ ¤)Considermaxb2C2 f (a¤;b) = f (a¤;b¤) =mina2C1 f (a;b¤)

approximated value

Conclusion: with precision , need rounds! ±

Convert QIP-Complete to Equilibrium Value ProblemGiven any two quantum mixed state circuits Q1, Q2, wants to distinguish between

This norm measures the distance between two circuitsor channel. It is called diamond norm.

Induced from L1 norm for super-operators:k©k1 =maxk½k· 1k©(½)k1k©k¦ =maxk½k· 1k©­ I (½)k1

To:

better representation of the distinguishing powerby using entanglement with auxiliary space.

Proved to be QIP-Complete when

a+b=2, 0<b<1<a<2 [RW05]

A formulation of equilibrium value simply follows!

The Converted Problem

two promises with constant gap!

constant precision will do the job!

two promises!

1.9

0.1

converted from Q0,Q1

equilibrium value

kf©A ¡ f©B k1min

The Conversion : sketch

diamond norm to fidelity

fidelity to L1 norm

max

“ – “ sign, min implied

k½k1 =max¦ h½;2I ¡ ¦ ithen we have a min-max form

Simulation by NC(poly)constant polynomial

polynomial time matrix operations

matrix operations in NC

Finally, polynomial compositions of NC(poly), still NC(poly) !

thus in PSPACE

ConclusionsCorollary: QIP=PSPACE

SDP reformulation

[JJUW09], August 09

A Neater Proof is available [Wu09]

August, 09

Use QIP=QMAMUse definition to

formulateSolve SDP by MMW

Use QIP-Complete Problem

Formulated as equilibrium value

Solved by MMW

Open Questions

How to make more applications of MMW method?

For quantum, QRG(2), QRG, QMA(2) candidates

In other fields, like algorithmic game theory…

MMW method for Convex Programming, under KKT conditions, or …

Thank You!Q&A