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Quantum Information and the PCP Theorem

Quantum Information and the PCP Theorem. Ran Raz Weizmann Institute. PCP Thm [BFL,FGLSS,AS,ALMSS]: x 2 SAT can be proved by a poly- size proof that can be verified by reading only O(1) of its bits. PCP Thm [BFL,FGLSS,AS,ALMSS]: x 2 SAT can be proved by poly(n)

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Quantum Information and the PCP Theorem

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  1. Quantum Information and the PCP Theorem Ran Raz Weizmann Institute

  2. PCP Thm [BFL,FGLSS,AS,ALMSS]: • x 2 SAT can be proved by a poly- • size proof that can be verified by • reading only O(1) of its bits

  3. PCP Thm [BFL,FGLSS,AS,ALMSS]: • x 2 SAT can be proved bypoly(n) • blocks of lengthO(1)that can be • verified by reading only2blocks

  4. Same with one block is impossible • (under hardness assumptions) • even if each block is of almost • linear size

  5. x 2 SAT can be proved by • 1) a log-size quantum state |i and • 2) a classical proof p of poly(n) blocks of length polylog each • s.t., after measuring |i the • verifier needs to read only one • block of p

  6. Part I: • The Information of a Quantum State

  7. Information of a Quantum State: • A quantum state |i of n qubits is • described by 2n complex numbers. • However, a measurement only gives • n bits of information about |i • (and the rest is lost) • How much of the information in |i • can be used ?

  8. Holevo’s Theorem (1973): • If Bob encodes a1,..,an by |i s.t. • Alice can retrieve a1,..,an from |i • then |i is a state of ¸ n qubits. • If Alice retrieves each bit ai with • prob 1- then |iis a state of ¸ • [1-H()]¢n qubits • We can’t communicate n bits by • sending less than n qubits

  9. ANTV-Nayak’s Theorem (1999): • If Bob encodes a1,..,an by |i s.t. • 8 i Alice can retrieve ai from |i • then |i is a state of ¸ n qubits. • If Alice can retrieve each bit ai • with prob 1- then |iis a state • of ¸ [1-H()]¢n qubits • Holevo’s: Alice retrieves a1,..,an • Nayak’s: Alice retrieves only one ai • (of her choice)

  10. Our Result: • Bob can encode N=2n bits a1,..,aN • by a state |i of O(n) qubits, s.t. • 8 i, ai can be retrieved from |i • by a (one round) Arthur-Merlin • interactive protocol of size poly(n) • (with a third party, Merlin) • (classical messages) • (polynomially small error)

  11. Retrieving ai from |i: • Alice measures |i (gets result e) • and sends a question q=q(i,e) • Merlin answers by r. • Alice computes V(i,e,r) 2{0,1,err} • Completeness:8i,q9 r, V(i,e,r) = ai • Soundness:8i,q,r, V(i,e,r)2{ai,err} • (with high probability) • (q,rarepoly(n)classicalbits)

  12. Retrieving ai from |i: • Alice measures |i (gets result e) • and sends a question q=q(i,e) • Merlin answers by r. • Alice computes V(i,e,r) 2{0,1,err} • Completeness:8i,q9 r, V(i,e,r) = ai • Soundness:8i,q,r, V(i,e,r)2{ai,err} • (with high probability) • (q,rarepoly(n)classicalbits) • Bob is trustworthy(|iis correct) • Merlin knowsa1,..,aN

  13. More Generally: • 1) Any constant number of elements from a1,..,aN can be retrieved in the same way, by a protocol of size poly(n) • 2) Any k elements can be retrieved by a protocol of size k¢poly(n) • 3) Each ai can be 2 {1,..,N}

  14. A Dequantumized Protocol: • |iis not needed: • Bob can senda (poly-size) random • secret classical string , • If Merlin doesn’t know  • The protocol works as before

  15. Part II: • The Retrieval Protocol

  16. Multilinear Extension: • Given a0,..,aN (N=2n-1) • F = field of size n2 • A: Fn ! F, s.t.: • 1)8i 2 {0,1}n, A(i) = ai • 2)A is multilinear (deg(A) · n)

  17. Quantum Multilinear Extension: • A= multilinear extension of a0,..,aN • 1)|i is a state of poly(n) qubits • 2) When Alice measures |i, she gets z,A(z) for a random z 2 Fn(Merlin doesn’t know z)

  18. Retrieving A(i): • Alice knows A(z) and wants A(i) • l = the line through i,z (in Fn) • Al: l ! F = restriction of A to l • (deg(Al) · n)

  19. The Protocol: • Alice sends l, Merlin is required to • give Al : l ! F.Merlin answers by • g : l ! F (deg(g) · n) • If g(z)  Al(z) Alice rejects • Otherwise, Alice assumes A(i)=g(i) • If g  Althen w.h.p. g(z)  Al(z) • (since both are low degree) • Otherwise,A(i)is correct

  20. A Dequantumized Protocol: • |iis not needed: • Bob can send z,A(z), for a random • z 2 Fn(s.t., Merlin doesn’t knowz) • The protocol works as before

  21. Part III: • The Exceptional Power of QIP/qpoly

  22. The Class QIP/qpoly: • IP:[B][GMR] x 2 L can be proved • by a poly-size interactive proof • QIP: [Wat] x 2 L can be proved by a poly-size quantum interactive proof • QIP/qpoly: x 2 L can be proved by • a poly-size quantum interactive proof with poly-size quantum advice

  23. Quantum Advice: • (captures quantum non-uniformity) • A (poly-size) quantum state |L,ni • given to the verifier as an advice • Alternatively, the verifier is a • quantum circuit with working space • initiated with|L,ni • [NY],[Aar]: Limitations on BQP/qpoly

  24. QIP/qpoly: • QIP/qpoly:x 2 L can be proved by • a poly-size interactive proof where • the verifier is a poly-size quantum • circuit with working space initiated • with an arbitrary state |L,ni • Our Result: • QIP/qpoly containsalllanguages

  25. Proof: • Denote ai2 {0,1}, ai =1 iff i 2 L • |L,ni= the quantum multilinear • extension of a0,..,aN (N=2n-1) • aican be retrieved from |L,niby • Arthur-Merlin interactive protocol • of size poly(n) • (one round, classical communication)

  26. Randomized Advice: • A (poly-size) random string , • chosen from a distribution DL,n, and • given to the verifier as an advice • Alternatively, the verifier is a • distribution over poly-size classical • circuits

  27. Randomized Advice: • A (poly-size) random string , • chosen from a distribution DL,n, and • given to the verifier as an advice • Alternatively, the verifier is a • distribution over poly-size classical • circuits • IP/rpoly:x 2 L can be proved by • a poly-size interactive proof where • the verifier is a distribution over • poly-size classical circuits • IP/rpoly containsalllanguages

  28. Part IV: • Quantum Versions of the • PCP Theorem

  29. PCP Thm [BFL,FGLSS,AS,ALMSS]: • x 2 SAT can be proved by a poly- • size proof that can be verified by • reading only O(1) of its bits

  30. PCP Thm [BFL,FGLSS,AS,ALMSS]: • x 2 SAT can be proved bypoly(n) • blocks of lengthO(1)that can be • verified by reading only2blocks

  31. Same with one block is impossible • (under hardness assumptions) • even if each block is of almost • linear size

  32. We Show: • x 2 SAT can be proved by • 1) a log-size quantum state |iand • 2) a classical proof p of poly(n) blocks of length polylog each • s.t., after measuring |i the • verifier needs to read only one • block of p

  33. We Show: • x 2 SAT can be proved by • 1) a log-size quantum state |iand • 2) a classical proof p of poly(n) blocks of length polylog each • s.t., after measuring |i the • verifier needs to read only one • block of p

  34. Naive Attempt: • a1,..,aN = classical PCP (N=poly(n)) • |i=quantum multilinear extension • of a1,..,aN O(log N) qubits • p = Merlin’s answers in the retrieval protocol • The verifier retrieves a • constant number of bits • by reading one block

  35. Problem: • The verifier can’t trust that |iis • aquantummultilinear extension • In the settings of communication or • quantum advice, the verifier could • trust that |i is correct. In the • setting of PCP, |i can be anything • e.g. |i is concentrated on a point

  36. Quantum Low Degree Test: • The verifier checks that |iis a • quantum encoding of a low degree • polynomial. This is done with the • aid of the classical proof • (or equivalently, a classical prover)

  37. Problem: • We are only allowed one query • How can we do both: • quantum low degree testand • retrieval of bits • We combine the two tasks using • ideas from[DFKRS]

  38. Part V: • Scaling up to NEXP

  39. Our Result (for L 2 NEXP): • x 2 L can be proved by • 1) a poly-size quantum state |i • 2) a classical proof p of exp(n) blocks of length poly each • s.t., after measuring |i the • verifier needs to read only one • block of p

  40. Our Result (for L 2 NEXP): • x 2 L can be proved by • 1) a poly-size quantum state |i • 2) a classical proof p of exp(n) blocks of length poly each • s.t., after measuring |i the • verifier needs to read only one • block of p

  41. Alternatively (for L 2 NEXP): • x 2 L has a 3 messages (MAM) • interactive proof, where the prover • is quantum in round 1 and classical • in round 2: • 1) Prover sends |i • 2) Verifier sends q • 3) Prover answers p(q)

  42. Models of 3 Messages Proofs: • IP(3):prover is classical • QIP(3): prover is quantum • The hybrid model: • HIP(3): prover is quantum in first round and classical in second

  43. Models of 3 Messages Proofs: • IP(3):prover is classical • QIP(3): prover is quantum • The hybrid model: • HIP(3): prover is quantum in first round and classical in second • IP(3) µ IP µ PSPACE • QIP(3) µ QIP µ EXP [KW] • Our result: • HIP(3) = NEXP

  44. Why the prover in our protocol • can’t be quantum in both rounds ? • A quantum prover can answer in • round 2, based on a measurement • of a state entangled to the state • given in round 1 • (fancy version of the EPR paradox)

  45. The End

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