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 Graph 𝐻 → 3 -SAT formula 𝜒 𝐻 3-colorable ⇔ 𝜒 satisfiable • Consequence: all problems in NP have local verification procedures 0 1 0 1 0 1 0 1 1 1 1 0 • Do we even need 0 the whole proof? 𝑦 8 ? 0 0 𝑦 5 ? 𝐷 10 𝑦 = 𝑦 3 ∨ 𝑦 5 ∨ 𝑦 8 ? 𝑦 3 ? • Proof required to guarantee consistency of assignment ∃𝑦, 𝜒 𝑦 = 𝐷 1 𝑦 ∧ 𝐷 2 𝑦 ∧ ⋯ ∧ 𝐷 𝑛 𝑦 = 1? Is 𝐻 3-colorable?

Multiplayer games: the power of two Merlins 0 1 0 1 0 1 0 1 • Arthur (“referee”) asks questions • Two isolated Merlins (“players”) 0,0,0 1 • Arthur checks answers. 𝐷 10 ? 𝑦 8 ? • Value 𝜕 𝐻 = sup Merlins Pr[Arthur accepts] • Ex: 3-SAT game 𝐻 = 𝐻 𝜒 𝐷 10 𝑦 = 𝑦 3 ∨ 𝑦 5 ∨ 𝑦 8 ? check satisfaction + consistency ∃𝑦, 𝜒 𝑦 = 𝐷 1 𝑦 ∧ 𝐷 2 𝑦 ∧ ⋯ ∧ 𝐷 𝑛 𝑦 = 1? 𝜒 SAT ⇔ 𝜕 𝐻 𝜒 = 1 • Consequence: All languages in NP have truly local verification procedure • PCP Theorem: poly-time 𝐻 𝜒 → 𝐻 𝜒 such that 𝜕 𝐻 𝜒 = 1 ⟹ 𝜕 𝐻 𝜒 = 1 𝜕 𝐻 𝜒 < 1 ⟹ 𝜕 𝐻 𝜒 ≤ 0.9

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 𝐼 = 𝑗 𝐼 𝑗 , each 𝐼 𝑗 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) ? |𝜔〉 • Still need Merlin to provide complete state 〈Γ|𝐼 10 |Γ〉 ? • Today: is “truly local” verification of QMA problems possible? Is 𝑉 − 𝑓 𝑗𝜒 Id > 𝜀 ? ∃ Γ , Γ 𝐼 1 Γ + ⋯ 〈Γ|𝐼 𝑛 Γ ≤ 𝑏?

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 multiplayer games • Quantum Arthur exchanges quantum messages with quantum Merlins Quantum Merlins may use shared entanglement • Value 𝜕 ∗ 𝐻 = sup Merlins Pr[Arthur accepts] Measure Π = {Π 𝑏𝑑𝑑 , Π 𝑠𝑓𝑘 } • 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?

The power of entangled Merlins (1) The clause-vs-variable game 0,0,0 • No entanglement: 1 𝜕 𝐻 𝜒 = 1 ⇔ 𝜒 SAT 𝐷 10 ? 𝑦 8 ? • Magic Square game: ∃ 3-SAT 𝜒 , 𝜒 UNSAT but 𝜕 ∗ 𝐻 𝜒 = 1! 𝐷 10 𝑦 = 𝑦 3 ∨ 𝑦 5 ∨ 𝑦 8 ? ∃𝑦, 𝜒 𝑦 = 𝐷 1 𝑦 ∧ 𝐷 2 𝑦 ∧ ⋯ ∧ 𝐷 𝑛 𝑦 = 1? • Not a surprise: 𝜕 ∗ 𝐻 ≫ 𝜕 𝐻 is nothing else than Bell inequality violation 𝐻 𝜒 s.t. 𝜒 SAT ⇔ 𝜕 ∗ • [KKMTV’08,IKM’09] More complicated 𝜒 → 𝐻 𝜒 = 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 (2) A Hamiltonian-vs-qubit game? • Given 𝐼 , can we design 𝐻 = 𝐻 𝐼 s.t.: ⇒ 𝜕 ∗ 𝐻 ≈ 1 ∃|Γ〉 , Γ 𝐼 Γ ≤ 𝑏 𝐼 10 ? 𝑟 8 ? ∀|Φ〉 , Φ 𝐼 Φ ≥ 𝑐 ⇒ 𝜕 ∗ 𝐻 ≪ 1 • Some immediate difficulties: 〈Γ|𝐼 10 |Γ〉 ? • Cannot check for equality ∃ Γ , Γ 𝐼 1 Γ + ⋯ 〈Γ|𝐼 𝑛 Γ ≤ 𝑏? of 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

The power of entangled Merlins (2) A Hamiltonian-vs-qubit game? 𝐼 5 𝐼 4 • [ AGIK’09] Assume 𝐼 is 1D 𝑟 4 ? 𝑟 3 ? 𝑟 5 ? 𝐼 4 𝐼 2 𝐼 𝑜−1 𝐼 1 𝐼 3 + + + + + + • Merlin 1 takes even qubits, 〈Γ|𝐼 4 |Γ〉 ? 〈Γ|𝐼 5 |Γ〉 ? Merlin 2 takes odd qubits ∃ Γ , Γ 𝐼 1 Γ + ⋯ 〈Γ|𝐼 𝑛 Γ ≤ 𝑏? • 𝜕 ∗ 𝐻 𝐼 = 1 ⇒ ∃|Γ〉 , Γ 𝐼 Γ ≈ 0 ? • Bad example: the EPR Hamiltonian 𝐼 𝑗 = 𝐹𝑄𝑆 〈𝐹𝑄𝑆| 𝑗,𝑗+1 for all 𝑗 + + + + + + • Highly frustrated, but 𝜕 ∗ 𝐻 𝐼 = 1 !

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 = 𝐻 𝐼 such that: ⇒ 𝜕 ∗ 𝐻 ≥ 1 − 𝑏/2 • ∃|Γ〉 , Γ 𝐼 Γ ≤ 𝑏 • ∀|Φ〉 , Φ 𝐼 Φ ≥ 𝑐 ⇒ 𝜕 ∗ 𝐻 ≤ 1 − 𝑐/𝑜 𝑑 𝑗, 𝑘, 𝑙 ? 𝑗′, 𝑘′, 𝑙′ ? • 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 = 𝑅𝑁𝐵 𝐹𝑌𝑄 )

|Γ〉 The game 𝐻 = 𝐻 𝐼 𝐹𝑜𝑑 • ECC 𝐹 corrects ≥ 1 error (ex: 5 -qubit Steane code) • Arthur runs two tests (prob 1/2 each): 𝑟 5 𝑟 3 , 𝑟 5 , 𝑟 8 𝑟 5 Select random 𝐼 ℓ on 𝑟 𝑗 , 𝑟 𝑘 , 𝑟 𝑙 1. a) Ask each Merlin for its share of 𝑟 𝑗 , 𝑟 𝑘 , 𝑟 𝑙 b) Decode 𝐹 𝑟 5 〈Γ|𝐼 10 |Γ〉 ? c) Measure 𝐼 ℓ ∃ Γ , Γ 𝐼 1 Γ + ⋯ 〈Γ|𝐼 𝑛 Γ ≤ 𝑏? Select random 𝐼 ℓ on 𝑟 𝑗 , 𝑟 𝑘 , 𝑟 𝑙 2. a) Ask one (random) Merlin for its share of 𝑟 𝑗 , 𝑟 𝑘 , 𝑟 𝑙 . Select 𝑡 ∈ 𝑗, 𝑘, 𝑙 at random; ask remaining Merlins for their share of 𝑟 𝑡 b) Verify that all shares of 𝑟 𝑡 lie in codespace • Completeness: ∃|Γ〉 , Γ 𝐼 Γ ≤ 𝑏 ⇒ 𝜕 ∗ 𝐻 ≥ 1 − 𝑏/2

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

Soundness: cheating Merlins (2) • Goal: show ∀|Φ〉 , Φ 𝐼 Φ ≥ 𝑐 ⇒ 𝜕 ∗ 𝐻 ≤ 1 − 𝑐/𝑜 𝑑 • Contrapositive: 𝜕 ∗ 𝐻 > 1 − 𝑐/𝑜 𝑑 ⇒ ∃|Γ〉 , Γ 𝐼 Γ < 𝑐 → extract low-energy witness from successful Merlin’s strategies • Given: ? 1 • 5 -prover entangled state 𝜔 𝑉 𝑗 • For each 𝑗 , unitary 𝑉 𝑗 extracts 𝑟 𝑗 |𝜔〉 𝐸𝐹𝐷 Merlin’s answer qubit to 𝑟 𝑗 2 2 𝑉 𝑗 𝑉 ?? 𝑘 • For each term 𝐼 ℓ on 𝑟 𝑗 , 𝑟 𝑘 , 𝑟 𝑙 , unitary 𝑊 ℓ extracts {𝑟 𝑗 , 𝑟 𝑘 , 𝑟 𝑙 } • Unitaries local to each Merlin, but no a priori notion of qubit • Need to simultaneously extract 𝑟 1 , 𝑟 2 , 𝑟 3 , …

Soundness: cheating Merlins (3) We give circuit generating low-energy witness |Γ〉 from successful Merlin’s strategies 𝑟 1 𝑟 2

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