The Unique Games Conjecture

1
The Unique Games Conjecture with Entangled
Provers is False
Julia Kempe Tel Aviv University Oded Regev Tel
Aviv University Ben Toner CWI, Amsterdam
2
Two-Prover One-round Games
Alice
Bob
t
s
a
b
accept
Verifier
reject

• The verifier sends one question to each prover
• Each prover responds with an answer from 1,,k
  (no communication allowed)
• The verifier decides whether to accept or reject
• The value of a game is the maximum success
  probability the provers can achieve, and is
  denoted by ?(G)
• The provers can have shared randomness but it
  cant help them

3
Unique Games

• We say that a game is unique if for each answer
  of the first prover there is exactly one good
  answer of the second prover and vice versa
  CaiCondonLipton90, FeigeLovász92
• In other words, the verifier accepts answers a,b
  iff b?(a) where ? is some permutation on k
  elements

1 2 3 4 5 k
1 2 3 4 5 k
1 2 3 4 5 k
1 2 3 4 5 k
4
Example the CHSH game

• The CHSH game ClauserHorneShimonyHolt69
• The verifier sends a random bit to each prover
• Each prover responds with a bit (so k2 here)
• The verifier accepts iff the XOR of the answers
  is equal to the AND of the questions
• Unique game
• The value of this game is 3/4

1
1


1
1
?
5
Unique Games Conjecture (UGC)

• In 2002, Khot conjectured that estimating the
  value of unique games is also hard
• Conjecture Khot02
• ??,?gt0 ?k such that it is NP-hard to determine
  whether a given unique game G with answer size k
  has ?(G)?1-? or ?(G)lt?
• Remarks
• It is crucial that ?gt0 since otherwise easy
• The PCP theorem parallel repetition shows
• this without the unique requirement

6
Implications of the UGC
7
Algorithmic Results on Unique Games

• Lots of algorithmic work on approximating ? for
  unique games Trevisan05, CharikarMakarychevM06,
  GuptaTalwar06, ChlamtacMakarychevM06
• Can be seen as attempts to disprove the UGC
• One of the best known results is
  CharikarMakarychevM06. Given any unique game G,
  their algorithm outputs a value ? s.t.
• 1-O((?logk)½) ? ?(G) ? 1-?
• This does not contradict the UGC, but instead
  tells us how big kk(?,?) needs to be in order
  for the conjecture to make sense

8
Games with Entangled Provers
Alice
Bob
t
s
a
b
accept
Verifier
reject

• These games are as before, except the provers are
  allowed to share an arbitrary entangled quantum
  state
• Originate in the works of EinsteinPodolskyRosen35
  , Bell64,

9
Games with Entangled Provers

• The entangled value of a game is the maximum
  success probability that entangled provers can
  achieve, and is denoted by ?(G)
• For instance, ?(CHSH)0.8536, which is strictly
  greater than ?(CHSH)0.75.
• This remarkable ability of entanglement to create
  correlations that are impossible to obtain
  classically (something Einstein referred to as
  spooky) is one of the most peculiar aspects of
  quantum mechanics

10
Games with Entangled Provers

• Why study this model?
• It was here first ?
• If we ever want to really use proof systems,
  then there is no physical way to guarantee that
  the provers dont share entanglement
• It might give us new insight on (non-entangled)
  games
• Despite considerable work, our understanding of
  this model is still quite limited

11
Games with Entangled Provers

• One of most important results is that of
  Tsirelson80 who showed that for the special
  case of unique games with k2, the entangled
  value is given exactly by an SDP and can
  therefore be computed efficiently (see also
  CleveHøyerTonerWatrous04)
• This is in contrast to the (non-entangled) value
  of unique games with k2 which is NP-hard to
  approximate (by Håstads hardness result for
  MaxCut)
• This SDP is used to determine that
  ?(CHSH)0.8536

12
Games with Entangled Provers

• The only other known result is by Masanes05 who
  shows how to compute ? for games with two
  possible questions to each prover and k2
• In all other cases, no method is known to compute
  or even approximate ?

13
Our Results

• Theorem There exists an efficient algorithm
  that, given any unique game G outputs a value ?
  s.t.
• 1-6? ? ?(G) ? 1-?
• This gives for the first time a way to
  approximate ? for games with kgt2
• It shows that the analogue of the UGC for
  entangled provers is false
• Notice that our lower bound is independent of k,
  whereas in the non-entangled case, the lower
  bound is 1-f(?,k)

14
Techniques

• We prove our main theorem in two steps
• We formulate an SDP relaxation of ?
• Surprisingly, this is essentially identical to
  the Feige-Lovász SDP, often used as a relaxation
  of ?
• We then show how to take a solution to the SDP
  and transform it into a strategy for entangled
  provers
• We call this quantum rounding in analogy with
  the rounding technique used in SDPs

15
The Proof
16
Quantum Correlations

• If Alice and Bob share the n-dimensional
  maximally entangled state
  then they can perform a measurement as follows
• Each party chooses an orthonormal basis of Rn
• Each party obtains an outcome in 1,...,n
• If Alice uses the orthonormal basis (x1,,xn) and
  Bob uses the orthonormal basis (y1,,yn), the
  output has the joint distribution given by
• Notice that each partys marginal is uniform

17
Quantum Correlations

• For example, heres how to get (cos?/8)20.8536
  success probability in the CHSH game
• The provers share a 2-dimensional maximally
  entangled state
• They perform a measurement in a basis depending
  on their input

x0
x1
Input 0 Input 1
y0
y0
x0
y1
y1
x1
Bob
Alice
18
The Proof

• For simplicity, lets consider unique games for
  which there exists an optimal strategy in which
  each provers answer distribution is uniform over
  1,,k
• Theorem There exists an efficient algorithm
  that, given any uniform unique game G, outputs
  a value ? s.t.
• 1-4? ? ?(G) ? 1-?
• Proof Start by writing an SDP relaxation
• I will not show why this is a relaxation of the
  entangled value (the proof is not difficult)
• Let the value of this SDP be 1-? and output ?
• Our goal is to show that there exists a strategy
  that achieves success probability ? 1-4?

19
The SDP Relaxation

• A solution consists of k orthonormal vectors in
  Rn for each question (for some large n)
• In a good solution, the vectors should be
  aligned according to the permutation

20
Quantum Rounding The Idea

• Main Idea On input s, Alice performs the
  measurement given by . Similarly
  for Bob using the v vectors.
• However, is not a basis of Rn !
• Instead, the provers complete their k vectors to
  an orthonormal basis of Rn in an arbitrary way

21
Quantum Rounding The Idea

• New problem the probability that Alice obtains
  one of the first k outcomes is only k/n
  otherwise, with probability 1-k/n, her outcome is
  meaningless
• Luckily, if Alice gets a meaningless answer, then
  with high probability Bob also gets a meaningless
  answer
• This allows us to solve the problem by having
  both parties repeat the measurement process until
  they get a meaningful answer

22
Quantum Rounding

• Alices strategy On input s, performs the
  measurement given by the completion of
• to an orthonormal basis.
• If obtains an outcome in 1,,k, returns it.
• Otherwise, repeats the measurement again (with a
  fresh maximally entangled state)
• Bobs strategy is similar.

23
Quantum Rounding Analysis

• Fix some question pair (s,t)
• Assume for simplicity that the permutation ?st is
  the identity permutation.
• The contribution to the SDP is
• We will show that Alice and Bob have probability
  at least 1-4? to output the same value i

24
Quantum Rounding Analysis

• Alice and Bob have k1 possible outcomes in one
  measurement, with k1 signifying try again.
• The joint probability distribution is given by

1 2 j k k1
?
1 ? i ? k k1
k/n
?
k/n
25
Quantum Rounding Analysis

• The probability that in one measurement, both
  Alice and Bob output the same value is
• The probability that both of them try again is
  therefore at least
• The probability of success is therefore at least

26
Conclusions

• We showed that the entangled value of unique
  games can be well approximated (up to factor 6)
• We extend our results to d-to-d games
• Our result also implies a parallel repetition
  theorem for the entangled value of unique games

27
Open Questions

• Unique games
• Improve our factor 6, or even compute ? exactly
• So far we only know how to do it for k2 by
  Tsirelson80
• General games
• Can one compute ? exactly?
• Probably not. KempeKobayashiMatsumotoTonerVidick0
  7 show this for games with 3 provers, and for
  games with quantum communication.
• But what about approximating ? ?
• Mostly open (even with many provers, quantum
  communication, etc.)
• The most important open question in this area