Hardness of Approximating Entangled Games

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Hardness of Approximating Entangled Games
QIP08, Delhi
Thomas Vidick UC Berkeley

• Joint work with Julia Kempe, Hirotada Kobayashi,
  Keiji Matsumoto and Ben Toner

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2-Prover Games

• Verifier samples q,q in Q according to p
• He sends q to A, q to B
• He receives a,a as answers
• Provers win iff V(a,aq,q) 1

Alice
Bob
Verifier
Game G (Q,A,p,V) of size G poly(Q)
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How hard is it to approximate ??
Alice
Bob
Input game G (Q,A,p,V) of size G poly(Q)
Computing ? is NP-hard (Cook-Levin) Approximating
? is NP-hard (PCP Thm.)
Approximating ? is NP-hard Hastad01
XOR-games
What is the situation in the quantum world?
Understand the power gained by entangled
provers. How does it affect the value of a game?
General classical games
Unique games
Approximating ? is NP-hard Khot02
CONJECTURE
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Entangled Games
?gt
The provers may share arbitrary entanglement
q
Value of the game ? Max. Winning Prob.
(over all the provers strategies and shared
?gt)

• Provers can now produce nonlocal correlations
• - Bell inequalities 1962
• Hardness of approximating ? is a major open
  question

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How hard is it to approximate ??
(entangled provers)
Computing ? is in P CHTW04 (there is a
semi- definite program)
XOR-games
What about general entangledgames?
Unique games
Can we also compute their value in P (is there an
SDP?)
Approximating ? is in P KRT07 (there is an
SDP)
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Our Results

• Theorem
• There is an e1/poly(Q) such that
  distinguishing between ?1 and ?1-e is NP-hard
  for games with
• Classical communication with three provers.
• Quantum communication with two provers.

• Corollary
• Unless PNP there are no SDPs of size poly(Q)
  which optimum approximates the value of entangled
  games (in contrast to XOR-games, unique games,
  single-prover quantum games).
• The hierarchy of PironioNavascuesAcin07 cannot
  yield polynomial-sized SDPs (see also
  DohertyLiangTonerWehner08)

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Outline of the proof

• Do a reduction from the classical setting

• Start with a classical 2-player game
• ? NP-hard to distinguish between ?1 and ? lt 0.01

• Transform into a quantum game such that
• ?1 ?1
• ?lt0.01 ?lt1-e

(by construction) (we show ?lt1-e
?lt0.01)

• Add tests that limit the provers use of
  entanglement
• ? They can do little better than using it as
  shared
• randomness

? It is NP-hard to distinguish between ?1 and
? lt 1- e
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The modified game
Alice
Bob
Alice
Bob
a
a
agt
agt
q
qgt
q
qgt
Verifier
Verifier
accept
accept
reject
reject

• With probability ½ do
• Classical Test play classical game send qgt,
  qgt and check answers
• Quantum Test send 0gtqgtqgt 1gtqgtqgt
• After answer, do C-SWAP on last two registers,
  measure first in gt basis, accept if gt

Classical provers q ? a, q ? a
Quantum provers qgt ? agt, qgt ? agt
0gtqgtqgt 1gtqgtqgt ? 0gtagtagt
1gtagtagt ? (0gt1gt)agtagt
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After you do the math

• The provers strategies can be described by
  orthogonal projectors for every
  q such that

• Classical test Prob. that the verifier
  measures a,a as
• answers to q,q

• Swap-test The Ws almost commute

? This is the key relation that we will use to
round to a classical strategy
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Special Case e0

• Ws are diagonal in the same basis

Set of projectors ? partition of space
Norm of product ? dimension of the intersection
We can easily simulate this classically by
sampling using shared randomness ? Classical
strategy has the same success prob. as the
quantum strategy
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General case e 1/poly

• Are the Ws almost diagonal in a common basis?
• If we can find W such that W W and W can be
  simultaneously diagonalized, then we are done
• Do almost commuting matrices nearly commute?
• Long-standing open question Halmos76
• Finally proven true for two hermitian matrices
  Lin97
• False for three or more hermitian matrices and
  the operator norm!

• But
• - Our matrices have a special form (projectors)
• - Our norm is not the operator norm
• ? It is an open question

• We get around this by using a different
  rounding technique
• ? we lose a factor O(Q4 e) in statistical
  distance

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Hardness of computing the value of entangled games
Computing ? is in P
2-prover quantum games
XOR-games
3-prover classical games
Unique games
Approximating ? to within 1/poly is NP-hard
Approximating ? is in P
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Open questions

• Increase the gap!
• A better gap could yield NEXP QMIP
• Prove that almost commuting projectors nearly
  commute
• Find a better rounding procedure
• Show some upper bounds
• All known bounds depend on the amount of
  entanglement used

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Thank you!