a multiprover interactive proof system ... - quantum-lab.org vidick.pdfsutd and cqt, singapore....
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A multiprover interactive proof system forthe local Hamiltonian problem
Thomas VidickCaltech
Joint work with Joseph FitzsimonsSUTD 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
โข NP = { decision problems โdoes ๐ฅ have property ๐?โ
that have polynomial-time verifiable proofs }
โข Ex: Clique, chromatic number, Hamiltonian path
โข 3D Ising spin
โข Pancake sorting, Modal logic S5-Satisfiability, Super Mario, Lemmings
โข Cook-Levin theorem: 3-SAT is complete for NP
โข Consequence: all problems in NP have local verification procedures
โข Do we even need
the whole proof?
โข Proof required to guarantee
consistency of assignment
0 1 0 1 1 01 10 10 1
โ๐ฅ, ๐ ๐ฅ = ๐ถ1 ๐ฅ โง ๐ถ2 ๐ฅ โง โฏโง ๐ถ๐ ๐ฅ = 1?
๐ถ10 ๐ฅ = ๐ฅ3 โจ ๐ฅ5 โจ ๐ฅ8 ?๐ฅ3?
0 ๐ฅ5?
0 ๐ฅ8?
0
Graph ๐บ โ 3-SAT formula ๐๐บ 3-colorable โ๐ satisfiable
Is ๐บ 3-colorable?
Multiplayer games: the power of two Merlins
โข Arthur (โrefereeโ) asks questions
โข Two isolated Merlins (โplayersโ)
โข Arthur checks answers.
โข Value ๐ ๐บ = supMerlins Pr[Arthur accepts]
โข Ex: 3-SAT game ๐บ = ๐บ๐
check satisfaction + consistency
๐ SAT โ ๐ ๐บ๐ = 1
โข Consequence: All languages in NP have truly local verification procedure
โข PCP Theorem: poly-time ๐บ๐ โ ๐บ๐ such that ๐ ๐บ๐ = 1โน๐ ๐บ๐ = 1
๐ ๐บ๐ < 1โน๐ ๐บ๐ โค 0.9
0 1 0 10 10 1
โ๐ฅ, ๐ ๐ฅ = ๐ถ1 ๐ฅ โง ๐ถ2 ๐ฅ โง โฏโง ๐ถ๐ ๐ฅ = 1?
๐ถ10 ๐ฅ = ๐ฅ3 โจ ๐ฅ5 โจ ๐ฅ8 ?
๐ถ10? ๐ฅ8?
0,0,01
Local verification of quantum proofs
โข QMA = { decision problems โdoes ๐ฅ have property ๐โ
that have quantum polynomial-time verifiable quantum proofs }
โข Ex: quantum circuit-sat, unitary non-identity check
โข Consistency of local density matrices, N-representability
โข [Kitaevโ99,Kempe-Regevโ03] 3-local Hamiltonian is complete for QMA
โข Still need Merlin to
provide complete state
โข Today: is โtruly localโ
verification of QMA problems possible?
|๐โฉ
๐ป = ๐๐ป๐, each ๐ป๐ acts on 3 out of ๐ qubits. Decide:
โ|ฮโฉ, ฮ ๐ป ฮ โค ๐ = 2โ๐ ๐ , or
โ|ฮฆโฉ, ฮฆ ๐ป ฮฆ โฅ ๐ = 1/๐(๐)?
โ ฮ , ฮ ๐ป1 ฮ +โฏโจฮ|๐ป๐ ฮ โค ๐?
โจฮ|๐ป10|ฮโฉ?
Is ๐ โ ๐๐๐Id > ๐ฟ ?
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
โข Quantum Arthur exchanges quantum
messages with quantum Merlins
Quantum Merlins may use
shared entanglement
โข Value ๐โ ๐บ = supMerlins Pr[Arthur accepts]
โข Quantum messages โ more power to Arthur
[KobMatโ03] Quantum Arthur with non-entangled Merlins limited to NP
โข Entanglement โ more power to Merlinsโฆ and to Arthur?
โข Can Arthur use entangled Merlins to his advantage?
Quantum multiplayer games
Measure ฮ = {ฮ ๐๐๐ , ฮ ๐๐๐}
โข No entanglement:
๐ ๐บ๐ = 1 โ ๐ SAT
โข Magic Square game: โ 3-SAT ๐,
๐ UNSAT but ๐โ ๐บ๐ = 1!
โข Not a surprise: ๐โ ๐บ โซ ๐ ๐บ
is nothing else than Bell inequality violation
โข [KKMTVโ08,IKMโ09] More complicated ๐ โ ๐บ๐ s.t. ๐ SAT โ ๐โ ๐บ๐ = 1
โ Arthur can still use entangled Merlins to decide problems in NP
โข Can Arthur use entangled Merlins to decide QMA problems?
The power of entangled Merlins (1)The clause-vs-variable game
๐ถ10 ๐ฅ = ๐ฅ3 โจ ๐ฅ5 โจ ๐ฅ8 ?
๐ถ10? ๐ฅ8?
0,0,01
โ๐ฅ, ๐ ๐ฅ = ๐ถ1 ๐ฅ โง ๐ถ2 ๐ฅ โง โฏโง ๐ถ๐ ๐ฅ = 1?
โข Given ๐ป , can we design ๐บ = ๐บ๐ป s.t.:
โ|ฮโฉ, ฮ ๐ป ฮ โค ๐ โ ๐โ ๐บ โ 1
โ|ฮฆโฉ, ฮฆ ๐ป ฮฆ โฅ ๐ โ ๐โ ๐บ โช 1
โข Some immediate difficulties:
โข Cannot check for equality
of reduced densities
โข Local consistency โ global consistency
(deciding whether this holds is itself a QMA-complete problem)
โข [KobMat03] Need to use entanglement to go beyond NP
โข Idea: split proof qubits between Merlins
๐ป10? ๐8?
โ ฮ , ฮ ๐ป1 ฮ +โฏโจฮ|๐ป๐ ฮ โค ๐?
โจฮ|๐ป10|ฮโฉ?
The power of entangled Merlins (2)A Hamiltonian-vs-qubit game?
โข [AGIKโ09] Assume ๐ป is 1D
โข Merlin1 takes even qubits,
Merlin2 takes odd qubits
โข ๐โ ๐บ๐ป = 1 โ โ|ฮโฉ, ฮ ๐ป ฮ โ 0?
โข Bad example: the EPR Hamiltonian ๐ป๐ = ๐ธ๐๐ โจ๐ธ๐๐ |๐,๐+1 for all ๐
โข Highly frustrated, but ๐โ ๐บ๐ป = 1!
๐4? ๐5?
โจฮ|๐ป4|ฮโฉ?
๐ป4
โจฮ|๐ป5|ฮโฉ?
๐ป5
The power of entangled Merlins (2)A Hamiltonian-vs-qubit game?
+ + +๐ป1 ๐ป3 ๐ป๐โ1+ ++๐ป2 ๐ป4
+ + ++ ++
โ ฮ , ฮ ๐ป1 ฮ +โฏโจฮ|๐ป๐ ฮ โค ๐?
๐3?
The difficulty
?
The difficulty
Can we check existence of global state
|ฮโฉ from โlocal snapshotsโ 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 ๐ป on ๐ qubits, design 5-player ๐บ = ๐บ๐ป such that:
โข โ|ฮโฉ, ฮ ๐ป ฮ โค ๐ โ ๐โ ๐บ โฅ 1 โ ๐/2
โข โ|ฮฆโฉ, ฮฆ ๐ป ฮฆ โฅ ๐ โ ๐โ ๐บ โค 1 โ ๐/๐๐
โข Consequence: the value ๐โ ๐บ for ๐บ with ๐ classical questions, 3 answer qubits,
5 players, is ๐๐๐ด-hard to compute to within ยฑ1/๐๐๐๐ฆ(๐)
โ Strictly harder than non-entangled value ๐(๐บ) (unless NP=QMA)
โข Consequence: ๐๐๐ผ๐ โ ๐๐๐ผ๐โ 1 โ 2โ๐, 1 โ 2 โ 2โ๐ (unless ๐๐ธ๐๐ = ๐๐๐ด๐ธ๐๐)
๐, ๐, ๐?๐โฒ, ๐โฒ, ๐โฒ?
The game ๐บ = ๐บ๐ป
โข ECC ๐ธ corrects โฅ 1 error
(ex: 5-qubit Steane code)
โข Arthur runs two tests (prob 1/2 each):
1. Select random ๐ปโ on ๐๐ , ๐๐ , ๐๐
a) Ask each Merlin for its share of ๐๐ , ๐๐ , ๐๐
b) Decode ๐ธ
c) Measure ๐ปโ
2. Select random ๐ปโ on ๐๐ , ๐๐ , ๐๐
a) Ask one (random) Merlin for its share of ๐๐ , ๐๐ , ๐๐.
Select ๐ โ ๐, ๐, ๐ at random; ask remaining Merlins for their share of ๐๐
b) Verify that all shares of ๐๐ lie in codespace
โข Completeness: โ|ฮโฉ, ฮ ๐ป ฮ โค ๐ โ ๐โ ๐บ โฅ 1 โ ๐/2
๐ธ๐๐
โ ฮ , ฮ ๐ป1 ฮ + โฏโจฮ|๐ป๐ ฮ โค ๐?
|ฮโฉ
๐3, ๐5, ๐8
๐5 โจฮ|๐ป10|ฮโฉ?
๐5
๐5
โข Example: EPR Hamiltonian
โข Cheating Merlins share single EPR pair
โข On question ๐ปโ = {๐โ, ๐โ+1}, all Merlins sends back both shares of EPR
โข On question ๐๐ , all Merlins send back their share of first half of EPR
โข All Merlins asked ๐ปโ โ Arthur decodes correctly and verifies low energy
โข One Merlin asked ๐ป๐ = {๐๐ , ๐๐+1} or ๐ป๐โ1 = {๐๐โ1, ๐๐}, others asked ๐๐
โข If ๐ป๐ , Arthur checks his first half with other Merlinโs โ accept
โข If ๐ป๐+1, Arthur checks his second half with otherMerlinโs โ reject
โข Answers from 4 Merlins + code property commit remaining Merlinโs qubit
Soundness: cheating Merlins (1)
๐ธ๐๐ ๐ธ๐๐
โข Goal: show โ|ฮฆโฉ, ฮฆ ๐ป ฮฆ โฅ ๐ โ ๐โ ๐บ โค 1 โ ๐/๐๐
โข Contrapositive: ๐โ ๐บ > 1 โ ๐/๐๐ โ โ|ฮโฉ, ฮ ๐ป ฮ < ๐
โ extract low-energy witness from successful Merlinโs strategies
โข Given:
โข 5-prover entangled state ๐
โข For each ๐, unitary ๐๐ extracts
Merlinโs answer qubit to ๐๐
โข For each term ๐ปโ on ๐๐ , ๐๐ , ๐๐,
unitary ๐โ extracts {๐๐ , ๐๐ , ๐๐}
โข Unitaries local to each Merlin, but no a priori notion of qubit
โข Need to simultaneously extract ๐1, ๐2, ๐3, โฆ
Soundness: cheating Merlins (2)
๐๐2
๐๐1
๐ท๐ธ๐ถ ๐๐|๐โฉ
?
??๐๐2
Soundness: cheating Merlins (3)
We give circuit generating low-energy witness |ฮโฉfrom successful Merlinโs strategies
๐1๐2
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
[AALVโ10] Quantum PCP conjecture: There exists constants ๐ผ < ๐ฝ such
that given local ๐ป = ๐ป1 +โฏ+๐ป๐ , it is QMA-hard to decide between:
โข โ|ฮโฉ, ฮ ๐ป ฮ โค ๐ = ๐ผ๐, or
โข โ|ฮฆโฉ, ฮฆ ๐ป ฮฆ โฅ ๐ = ๐ฝ๐
PCP theorem (1):
constant-factor approximations
to ๐ ๐บ are NP-hard
PCP theorem (2): Given 3-SAT ๐,
it is NP-hard to decide between
100%-SAT vs โค 99%-SAT
Quantum PCP conjecture*: constant-factor
approximations to ๐โ(๐บ) are QMA-hard
Our results are a
first step towards:
Kitaevโs QMA-completeness result for LH is a first step towards:
No known implication!?
Clause-vs-variable
game
Consequences for interactive proof systems
๐ฟ โ ๐๐ผ๐(๐, ๐ ) if โ๐ฅ โ ๐บ๐ฅ such that
โข ๐ฅ โ ๐ฟ โ ๐ ๐บ๐ฅ โฅ ๐
โข ๐ฅ โ ๐ฟ โ ๐ ๐บ๐ฅ โค ๐
๐ฟ โ ๐๐๐ผ๐โ(๐, ๐ ) if โ๐ฅ โ ๐บ๐ฅ such that
โข ๐ฅ โ ๐ฟ โ ๐โ ๐บ๐ฅ โฅ ๐
โข ๐ฅ โ ๐ฟ โ ๐โ ๐บ๐ฅ โค ๐
โข [KKMTVโ08,IKMโ09]
๐๐ธ๐๐ โ (๐)๐๐ผ๐โ 1,1 โ 2โ๐
โข [IVโ13]
๐๐ธ๐๐ โ (๐)๐๐ผ๐โ 1,1/2
โข Our result: ๐๐๐ด๐ธ๐๐ โ ๐๐๐ผ๐โ 1 โ 2โ๐, 1 โ 2 โ 2โ๐
โข Consequence: ๐๐๐ผ๐ โ ๐๐๐ผ๐โ 1 โ 2โ๐, 1 โ 2 โ 2โ๐
(unless ๐๐ธ๐๐ = ๐๐๐ด๐ธ๐๐)
โข Cook-Levin:
๐๐ธ๐๐ = ๐๐ผ๐ 1,1 โ 2โ๐
โข PCP:
๐๐ธ๐๐ = ๐๐ผ๐(1,1/2)
Summaryโข Design โtruly localโ verification pocedure for LH
โข Entangled Merlins strictly more powerful than unentangled
โข Proof uses ECC to recover global witness from local snapshots
โข Design a game with classical answers for LH?
[RUVโ13] requires poly rounds
โข Prove Quantum PCP Conjecture*
โข What is the relationship between QPCP and QPCP*?
โข Are there quantum games for languages beyond QMA?
Questions
Thank you!