the local Hamiltonian problem Thomas Vidick Caltech Joint work - - PowerPoint PPT Presentation

A multiprover interactive proof system for the local Hamiltonian problem Thomas Vidick

Caltech Joint work with Joseph Fitzsimons

SUTD and CQT, Singapore

Outline

• 1. Local verification of classical & quantum proofs
• 2. Quantum multiplayer games
• 3. Result: a game for the local Hamiltonian problem
• 4. Consequences:

a) The quantum PCP conjecture b) Quantum interactive proof systems

Local verification of classical proofs

• NP = { decision problems “does 𝑦 have property 𝑄?”

that have polynomial-time verifiable proofs }

• Ex: Clique, chromatic number, Hamiltonian path
• 3D Ising spin
• Pancake sorting, Modal logic S5-Satisfiability, Super Mario, Lemmings
• Cook-Levin theorem: 3-SAT is complete for NP
• Consequence: all problems in NP have local verification procedures
• Do we even need

the whole proof?

• Proof required to guarantee

consistency of assignment 0 1 0 1 1 0 1 1 0 1 0 1 ∃𝑦, 𝜒 𝑦 = 𝐷1 𝑦 ∧ 𝐷2 𝑦 ∧ ⋯ ∧ 𝐷𝑛 𝑦 = 1?

𝐷10 𝑦 = 𝑦3 ∨ 𝑦5 ∨ 𝑦8 ? 𝑦3? 𝑦5? 𝑦8?

Graph 𝐻 → 3-SAT formula 𝜒 𝐻 3-colorable ⇔ 𝜒 satisfiable Is 𝐻 3-colorable?

Multiplayer games: the power of two Merlins

• Arthur (“referee”) asks questions
• Two isolated Merlins (“players”)
• Arthur checks answers.
• Value 𝜕 𝐻 = supMerlins Pr[Arthur accepts]
• Ex: 3-SAT game 𝐻 = 𝐻𝜒

check satisfaction + consistency

𝜒 SAT ⇔ 𝜕 𝐻𝜒 = 1

• Consequence: All languages in NP have truly local verification procedure
• PCP Theorem: poly-time 𝐻𝜒 →

𝐻𝜒 such that 𝜕 𝐻𝜒 = 1 ⟹ 𝜕 𝐻𝜒 = 1 𝜕 𝐻𝜒 < 1 ⟹ 𝜕 𝐻𝜒 ≤ 0.9

0 1 0 1 0 1 0 1

∃𝑦, 𝜒 𝑦 = 𝐷1 𝑦 ∧ 𝐷2 𝑦 ∧ ⋯ ∧ 𝐷𝑛 𝑦 = 1? 𝐷10 𝑦 = 𝑦3 ∨ 𝑦5 ∨ 𝑦8 ?

𝐷10? 𝑦8? 0,0,0 1

Local verification of quantum proofs

• QMA = { decision problems “does 𝑦 have property 𝑄”

that have quantum polynomial-time verifiable quantum proofs }

• Ex: quantum circuit-sat, unitary non-identity check
• Consistency of local density matrices, N-representability
• [Kitaev’99,Kempe-Regev’03] 3-local Hamiltonian is complete for QMA
• Still need Merlin to

provide complete state

• Today: is “truly local”

verification of QMA problems possible?

|𝜔〉

𝐼 = 𝑗 𝐼𝑗, each 𝐼𝑗 acts on 3 out of 𝑜 qubits. Decide: ∃|Γ〉, Γ 𝐼 Γ ≤ 𝑏 = 2−𝑞 𝑜 , or ∀|Φ〉, Φ 𝐼 Φ ≥ 𝑐 = 1/𝑟(𝑜)? ∃ Γ , Γ 𝐼1 Γ + ⋯ 〈Γ|𝐼𝑛 Γ ≤ 𝑏?

〈Γ|𝐼10|Γ〉?

Is 𝑉 − 𝑓𝑗𝜒Id > 𝜀 ?

Outline

• 1. Local verification of classical & quantum proofs
• 2. Quantum multiplayer games
• 3. Result: a game for the local Hamiltonian problem
• 4. Consequences:

a) The quantum PCP conjecture b) Quantum interactive proof systems

• Quantum Arthur exchanges quantum

messages with quantum Merlins Quantum Merlins may use shared entanglement

• Value 𝜕∗ 𝐻 = supMerlins Pr[Arthur accepts]
• Quantum messages → more power to Arthur

[KobMat’03] Quantum Arthur with non-entangled Merlins limited to NP

• Entanglement

→ more power to Merlins… and to Arthur?

• Can Arthur use entangled Merlins to his advantage?

Quantum multiplayer games

Measure Π = {Π𝑏𝑑𝑑, Π𝑠𝑓𝑘}

• No entanglement:

𝜕 𝐻𝜒 = 1 ⇔ 𝜒 SAT

• Magic Square game: ∃ 3-SAT 𝜒,

𝜒 UNSAT but 𝜕∗ 𝐻𝜒 = 1!

• Not a surprise: 𝜕∗ 𝐻 ≫ 𝜕 𝐻

is nothing else than Bell inequality violation

• [KKMTV’08,IKM’09] More complicated 𝜒 →

𝐻𝜒 s.t. 𝜒 SAT ⇔ 𝜕∗ 𝐻𝜒 = 1 → Arthur can still use entangled Merlins to decide problems in NP

• Can Arthur use entangled Merlins to decide QMA problems?

The power of entangled Merlins (1)

The clause-vs-variable game

𝐷10 𝑦 = 𝑦3 ∨ 𝑦5 ∨ 𝑦8 ?

𝐷10? 𝑦8? 0,0,0 1

∃𝑦, 𝜒 𝑦 = 𝐷1 𝑦 ∧ 𝐷2 𝑦 ∧ ⋯ ∧ 𝐷𝑛 𝑦 = 1?

• Given 𝐼 , can we design 𝐻 = 𝐻𝐼 s.t.:

∃|Γ〉, Γ 𝐼 Γ ≤ 𝑏 ⇒ 𝜕∗ 𝐻 ≈ 1 ∀|Φ〉, Φ 𝐼 Φ ≥ 𝑐 ⇒ 𝜕∗ 𝐻 ≪ 1

• Some immediate difficulties:
• Cannot check for equality
• f reduced densities
• Local consistency ⇏ global consistency

(deciding whether this holds is itself a QMA-complete problem)

• [KobMat03] Need to use entanglement to go beyond NP
• Idea: split proof qubits between Merlins

𝐼10? 𝑟8? ∃ Γ , Γ 𝐼1 Γ + ⋯ 〈Γ|𝐼𝑛 Γ ≤ 𝑏?

〈Γ|𝐼10|Γ〉?

The power of entangled Merlins (2)

A Hamiltonian-vs-qubit game?

• [AGIK’09] Assume 𝐼 is 1D
• Merlin1 takes even qubits,

Merlin2 takes odd qubits

• 𝜕∗ 𝐻𝐼 = 1 ⇒ ∃|Γ〉, Γ 𝐼 Γ ≈ 0?
• Bad example: the EPR Hamiltonian 𝐼𝑗 = 𝐹𝑄𝑆 〈𝐹𝑄𝑆|𝑗,𝑗+1 for all 𝑗
• Highly frustrated, but 𝜕∗ 𝐻𝐼 = 1!

𝑟4? 𝑟5?

〈Γ|𝐼4|Γ〉?

𝐼4

〈Γ|𝐼5|Γ〉?

𝐼5

The power of entangled Merlins (2)

A Hamiltonian-vs-qubit game? + + + 𝐼1 𝐼3 𝐼𝑜−1 + + + 𝐼2 𝐼4 + + + + + +

∃ Γ , Γ 𝐼1 Γ + ⋯ 〈Γ|𝐼𝑛 Γ ≤ 𝑏? 𝑟3?

The difficulty

?

The difficulty

Can we check existence of global state |Γ〉 from “local snapshots” only?

?

Outline

• 1. Checking proofs locally
• 2. Entanglement in quantum multiplayer games
• 3. Result: a quantum multiplayer game for the local

Hamiltonian problem

• 4. Consequences:

1. The quantum PCP conjecture 2. Quantum interactive proof systems

Result: a five-player game for LH

Given 3-local 𝐼 on 𝑜 qubits, design 5-player 𝐻 = 𝐻𝐼 such that:

• ∃|Γ〉, Γ 𝐼 Γ ≤ 𝑏

⇒ 𝜕∗ 𝐻 ≥ 1 − 𝑏/2

• ∀|Φ〉, Φ 𝐼 Φ ≥ 𝑐 ⇒ 𝜕∗ 𝐻 ≤ 1 − 𝑐/𝑜𝑑
• Consequence: the value 𝜕∗ 𝐻 for 𝐻 with 𝑜 classical questions, 3 answer qubits,

5 players, is 𝑅𝑁𝐵-hard to compute to within ±1/𝑞𝑝𝑚𝑧(𝑜) → Strictly harder than non-entangled value 𝜕(𝐻) (unless NP=QMA)

• Consequence: 𝑅𝑁𝐽𝑄 ⊊ 𝑅𝑁𝐽𝑄∗ 1 − 2−𝑞, 1 − 2 ⋅ 2−𝑞

(unless 𝑂𝐹𝑌𝑄 = 𝑅𝑁𝐵𝐹𝑌𝑄) 𝑗, 𝑘, 𝑙? 𝑗′, 𝑘′, 𝑙′?

The game 𝐻 = 𝐻𝐼

• ECC 𝐹 corrects ≥ 1 error

(ex: 5-qubit Steane code)

• Arthur runs two tests (prob 1/2 each):

1. Select random 𝐼ℓ on 𝑟𝑗, 𝑟𝑘, 𝑟𝑙

a) Ask each Merlin for its share of 𝑟𝑗, 𝑟𝑘, 𝑟𝑙 b) Decode 𝐹 c) Measure 𝐼ℓ

2. Select random 𝐼ℓ on 𝑟𝑗, 𝑟𝑘, 𝑟𝑙

a) Ask one (random) Merlin for its share of 𝑟𝑗, 𝑟𝑘, 𝑟𝑙. Select 𝑡 ∈ 𝑗, 𝑘, 𝑙 at random; ask remaining Merlins for their share of 𝑟𝑡 b) Verify that all shares of 𝑟𝑡 lie in codespace

• Completeness: ∃|Γ〉, Γ 𝐼 Γ ≤ 𝑏 ⇒ 𝜕∗ 𝐻 ≥ 1 − 𝑏/2

𝐹𝑜𝑑

∃ Γ , Γ 𝐼1 Γ + ⋯ 〈Γ|𝐼𝑛 Γ ≤ 𝑏?

|Γ〉

𝑟3, 𝑟5, 𝑟8 𝑟5

〈Γ|𝐼10|Γ〉?

𝑟5 𝑟5

• Example: EPR Hamiltonian
• Cheating Merlins share single EPR pair
• On question 𝐼ℓ = {𝑟ℓ, 𝑟ℓ+1}, all Merlins sends back both shares of EPR
• On question 𝑟𝑗, all Merlins send back their share of first half of EPR
• All Merlins asked 𝐼ℓ → Arthur decodes correctly and verifies low energy
• One Merlin asked 𝐼𝑗 = {𝑟𝑗, 𝑟𝑗+1} or 𝐼𝑗−1 = {𝑟𝑗−1, 𝑟𝑗}, others asked 𝑟𝑗
• If 𝐼𝑗, Arthur checks his first half with other Merlin’s → accept
• If 𝐼𝑗+1, Arthur checks his second half with otherMerlin’s → reject
• Answers from 4 Merlins + code property commit remaining Merlin’s qubit

Soundness: cheating Merlins (1)

𝐹𝑜𝑑 𝐹𝑜𝑑

• Goal: show ∀|Φ〉, Φ 𝐼 Φ ≥ 𝑐 ⇒ 𝜕∗ 𝐻 ≤ 1 − 𝑐/𝑜𝑑
• Contrapositive: 𝜕∗ 𝐻 > 1 − 𝑐/𝑜𝑑 ⇒ ∃|Γ〉, Γ 𝐼 Γ < 𝑐

→ extract low-energy witness from successful Merlin’s strategies

• Given:
• 5-prover entangled state 𝜔
• For each 𝑗, unitary 𝑉𝑗 extracts

Merlin’s answer qubit to 𝑟𝑗

• For each term 𝐼ℓ on 𝑟𝑗, 𝑟𝑘, 𝑟𝑙,

unitary 𝑊

ℓ extracts {𝑟𝑗, 𝑟𝑘, 𝑟𝑙}

• Unitaries local to each Merlin, but no a priori notion of qubit
• Need to simultaneously extract 𝑟1, 𝑟2, 𝑟3, …

Soundness: cheating Merlins (2)

𝑉𝑗

2

𝑉𝑗

1

𝐸𝐹𝐷 𝑟𝑗 |𝜔〉

? ??

𝑉

𝑘 2

Soundness: cheating Merlins (3)

We give circuit generating low-energy witness |Γ〉 from successful Merlin’s strategies

𝑟1 𝑟2

Outline

• 1. Checking proofs locally
• 2. Entanglement in quantum multiplayer games
• 3. Result: a quantum multiplayer game for the local

Hamiltonian problem

• 4. Consequences:

1. The quantum PCP conjecture 2. Quantum interactive proof systems

Perspective: the quantum PCP conjecture

[AALV’10] Quantum PCP conjecture: There exists constants 𝛽 < 𝛾 such

that given local 𝐼 = 𝐼1 + ⋯ + 𝐼𝑛 , it is QMA-hard to decide between:

• ∃|Γ〉,

Γ 𝐼 Γ ≤ 𝑏 = 𝛽𝑛, or

• ∀|Φ〉, Φ 𝐼 Φ ≥ 𝑐 = 𝛾𝑛

PCP theorem (1): constant-factor approximations to 𝜕 𝐻 are NP-hard PCP theorem (2): Given 3-SAT 𝜒, it is NP-hard to decide between 100%-SAT vs ≤ 99%-SAT Quantum PCP conjecture*: constant-factor approximations to 𝜕∗(𝐻) are QMA-hard Our results are a first step towards: Kitaev’s QMA-completeness result for LH is a first step towards:

No known implication!

?

Clause-vs- variable game

Consequences for interactive proof systems

𝑀 ∈ 𝑁𝐽𝑄(𝑑, 𝑡) if ∃𝑦 → 𝐻𝑦 such that

• 𝑦 ∈ 𝑀 ⇒ 𝜕 𝐻𝑦 ≥ 𝑑
• 𝑦 ∉ 𝑀 ⇒ 𝜕 𝐻𝑦 ≤ 𝑡

𝑀 ∈ 𝑅𝑁𝐽𝑄∗(𝑑, 𝑡) if ∃𝑦 → 𝐻𝑦 such that

• 𝑦 ∈ 𝑀 ⇒ 𝜕∗ 𝐻𝑦 ≥ 𝑑
• 𝑦 ∉ 𝑀 ⇒ 𝜕∗ 𝐻𝑦 ≤ 𝑡
• [KKMTV’08,IKM’09]

𝑂𝐹𝑌𝑄 ⊆ (𝑅)𝑁𝐽𝑄∗ 1,1 − 2−𝑞

• [IV’13]

𝑂𝐹𝑌𝑄 ⊆ (𝑅)𝑁𝐽𝑄∗ 1,1/2

• Our result: 𝑅𝑁𝐵𝐹𝑌𝑄 ⊆ 𝑅𝑁𝐽𝑄∗ 1 − 2−𝑞, 1 − 2 ⋅ 2−𝑞
• Consequence: 𝑅𝑁𝐽𝑄 ≠ 𝑅𝑁𝐽𝑄∗ 1 − 2−𝑞, 1 − 2 ⋅ 2−𝑞

(unless 𝑂𝐹𝑌𝑄 = 𝑅𝑁𝐵𝐹𝑌𝑄)

• Cook-Levin:

𝑂𝐹𝑌𝑄 = 𝑁𝐽𝑄 1,1 − 2−𝑞

• PCP:

𝑂𝐹𝑌𝑄 = 𝑁𝐽𝑄(1,1/2)

Summary

• Design “truly local” verification pocedure for LH
• Entangled Merlins strictly more powerful than unentangled
• Proof uses ECC to recover global witness from local snapshots
• Design a game with classical answers for LH?

[RUV’13] requires poly rounds

• Prove Quantum PCP Conjecture*
• What is the relationship between QPCP and QPCP*?
• Are there quantum games for languages beyond QMA?

Questions

Thank you!