Two-prover entangled games are NP hard Anand Natarajan (MIT) - - PowerPoint PPT Presentation

Two-prover entangled games are NP hard

Anand Natarajan (MIT) Thomas Vidick (Caltech) arXiv:1710.03062

Interactive proofs

MIP = PSPACE [Shamir ‘90] IP = NEXP [BFL ‘91]

Completeness: If YES, prover can convince verifier with Pr ≥ c (e.g. 2/3) Soundness: If NO, prover cannot convince verifier with Pr ≥ s (e.g. 1/3)

Interactive proofs

= PSPACE [Shamir ‘90] IP

Completeness: If YES, prover can convince verifier with Pr ≥ c (e.g. 2/3) Soundness: If NO, prover cannot convince verifier with Pr ≥ s (e.g. 1/3)

MIP (log n bit messages) = NP [ALMSS’98,AS’98]

|Ψ⟩

Quantum interactive proofs

MIP*

(c. verifier, c. messages, q. provers)

Still = PSPACE ! [JJUW’09] QIP:

(q. verifier, q. messages)

What about a quantum verifier, or quantum messages?

How powerful is it?

Entangled provers

• Entanglement makes provers more

powerful

• [Bell64, CHSH68]: T

wo-prover, one round protocol

– Best classical psuccess = 0.75 – With entanglement (1 EPR pair) psuccess ≈ 0.85

• This can hurt soundness!
• [CHTW04]: ⊕MIP*(2) ⊆ PSPACE

– ⊕MIP(2) = MIP = NEXP [Håstad’01] – Based on SDP characterization of quantum strategies due to Tsirelson

|Ψ⟩

Quantum interactive proofs

MIP*

(c. verifier, c. messages, q. provers)

Still = PSPACE ! [JJUW’09] QIP:

(q. verifier, q. messages)

What about a quantum verifier, or quantum messages?

Contains NEXP [IV’12] ⊕MIP*(3) contains NEXP [Vidick’13]

Two vs three provers

• Classically: reduce any MIP protocol to 2

provers using oracularization

• Quantumly: no general prover reduction

known

– Monogamy of entanglement: if Alice is maximally entangled with Bob, must share zero entanglement with Charlie! – [T

• ner’09] shows 3-prover versions of CHSH,
• dd-cycle game have lower entangled psuccess

|Ψ⟩

Results

MIP*

(c. verifier, c. messages, q. provers)

Still = PSPACE ! [JJUW’09] QIP:

(q. verifier, q. messages)

What about a quantum verifier, or quantum messages?

This work:

• Contains NEXP
• For log(n)-bit

messages, contains NP

Low-degree test

• Encode an NP-proof as a low-degree

polynomial g: !q

m → !q

– qm =Θ(n)

• Let x ∈ !q

m be random point,

s ⊆ !q

m random plane containing x

– Ask Alice for g(x) – Ask Bob for g|s as polynomial – Check that g|s (x) = g(x)

• Thm [RS’97]:

If pass with prob ≥ 1- ε, agree with some low-degree g on ≥ 1 –O(εc) fraction of points

s x

Soundness against entangled provers

• Entangled strategy:

– Shared state |Ψ⟩ – Points measurements: Ax

a ≥ 0, Σa Ax a = Id

– Planes measurements: Bs

h ≥ 0, Σh Bs h = Id

• Thm: If pass with prob ≥ 1- ε,

exists measurement {Mg} s.t. ⟨Ψ| Mg ⊗ Ax

g(x)|Ψ⟩ ≥ 1- O(εc)

s x

Induction

• Hypothesis: If pass test on !q

m-1 with

prob ≥ 1- ε, exists measurement {Mg} with g: !q

m-1 → !q s.t. ⟨Ψ| Mg ⊗ Ax g(x)|Ψ⟩ ≥ 1- δ

• Goal: If pass test on !q

m with prob ≥ 1- ε,

exists measurement Mg with g: !q

m → !q s.t.

⟨Ψ| Mg ⊗ Ax

g(x)|Ψ⟩ ≥ 1- δ

Induction

• Hypothesis: If pass test on !q

m-1 with prob ≥

1- ε, exists measurement {Mg} with g: !q

m-1 →

!q s.t. ⟨Ψ| Mg ⊗ Ax

g(x)|Ψ⟩ ≥ 1- δ

• Implies: for most dim-(m-1) subspaces s,

exists {Ms

g} that is δ’-consistent with {Ax a}

• Goal: If pass test on !q

m with prob ≥ 1- ε,

exists measurement Mg with g: !q

m → !q s.t.

⟨Ψ| Mg ⊗ Ax

g(x)|Ψ⟩ ≥ 1- δ’’

• Problem: δ’’ > δ’ > δ!

Induction with self- improvement

• Hypothesis: If pass test on !q

m-1 with

prob ≥ 1- ε, exists measurement {Mg} with g: !q

m-1 → !q s.t. ⟨Ψ| Mg ⊗ Ax g(x)|Ψ⟩ ≥ 1- δ

• Implies: for most dim-(m-1) subspaces

s ⊆ !q

m, exists {Ms g} that is δ-consistent with

{Ax

a}

• Goal: If pass test on !q

m with prob ≥ 1- ε,

exists measurement Mg with g: !q

m → !q s.t.

⟨Ψ| Mg ⊗ Ax

g(x)|Ψ⟩ ≥ 1- δ

Self-improvement

• Given:

– {Ax

a} ε-”globally consistent”

– {Qg} η-consistent with {Ax

a}

Ex Σg Σa≠g(x) ⟨Ψ| Qg ⊗ Ax

a|Ψ⟩ ≤ η

• Construct improved {Sg}

– ε1/2-consistent with {Ax

a}

Ex Σg Σa≠g(x) ⟨Ψ| Sg ⊗ Ax

a|Ψ⟩ ≤ ε1/2

– η’ := 1 – ε1/2 – η complete Σg ⟨Ψ| Sg ⊗ Id |Ψ⟩ ≥ η’

y x {Ax

a}

{Ay

a}

{Sg}

Self-improvement: SDP

• Ag := Ex[Ax

g(x)]

– Note: Σg Ag may not be ≤ Id

• SDP (“majority vote”)

max Σg Tr[Tg Ag] s.t.Tg ≥ 0, Σg Tg ≥ Id

• Qg is primal feasible, so SDP

value ≥ 1 - η

• Define

Sg := Ex Ax

g(x) Tg Ax g(x)

y x {Ax

a}

{Ay

a}

{Sg}

Self-improvement: consistency

• Sg := Ex Ax

g(x) Tg Ax g(x)

• Ex Σg Σa≠g(x) ⟨Ψ| Sg ⊗ Ax

a|Ψ⟩

= Ex Ey Σg Σa≠g(x) ⟨Ψ| Ay

g(y) Tg Ay g(y) ⊗ Ax a|Ψ⟩

≤ O(ε1/2)

Self-improvement: completeness

• Ag := Ex[Ax

g(x)]

Sg = Ex Ax

g(x) Tg Ax g(x)

• Dual SDP

min Tr[Z] s.t. Z ≥ Ag, Z ≥ 0

• Complementary

slackness: Tg Ag = Tg Z Σg ⟨Ψ| Sg ⊗ Id|Ψ⟩ ≈ Σg ⟨Ψ| Tg Ag ⊗ Id |Ψ⟩ = Σg ⟨Ψ| Tg Z ⊗ Id |Ψ⟩ = ⟨Ψ| Z ⊗ Id |Ψ⟩ ≥ 1- η’

Applications and future work

• Superclassical hardness results

– In [NV’18] improved to QMA-hardness – [Ji’17,FJVY’18] for small soundness

• Entanglement-sound tests for locally testable

codes?

• Non-signaling soundness?
• Prover reduction for MIP*?

THE END

arXiv:1710.03062