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Quantum Information and the PCP Theorem
Ran RazWeizmann Institute
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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
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PCP Thm [BFL,FGLSS,AS,ALMSS]:
x 2 SAT can be proved by poly(n)
blocks of length O(1) that can be
verified by reading only 2 blocks
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Same with one block is impossible
(under hardness assumptions)even if each block is of almost linear size
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x 2 SAT can be proved by 1) a log-size quantum state |i
and2) a classical proof p of poly(n)
blocks of length polylog eachs.t., after measuring |i the verifier needs to read only oneblock of p
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Part I:
The Information of a Quantum State
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Information of a Quantum State:
A quantum state |i of n qubits isdescribed 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 ?
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Holevo’s Theorem (1973):
If Bob encodes a1,..,an by |i s.t. Alice can retrieve a1,..,an from |ithen |i is a state of ¸ n qubits.If Alice retrieves each bit ai withprob 1- then |i is a state of ¸ [1-H()]¢n qubits
We can’t communicate n bits by sending less than n qubits
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ANTV-Nayak’s Theorem (1999):If Bob encodes a1,..,an by |i s.t. 8 i Alice can retrieve ai from |ithen |i is a state of ¸ n qubits.If Alice can retrieve each bit ai with prob 1- then |i is a state of ¸ [1-H()]¢n qubitsHolevo’s: Alice retrieves a1,..,an Nayak’s: Alice retrieves only one
ai
(of her choice)
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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 |iby a (one round) Arthur-Merlin interactive protocol of size
poly(n)(with a third party, Merlin)
(classical messages) (polynomially small error)
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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,q 9 r, V(i,e,r) = ai
Soundness: 8i,q,r, V(i,e,r)2 {ai,err}
(with high probability)
(q,r are poly(n) classical bits)
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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,q 9 r, V(i,e,r) = ai
Soundness: 8i,q,r, V(i,e,r)2 {ai,err} (with high probability)
(q,r are poly(n) classical bits)
Bob is trustworthy (|i is correct)
Merlin knows a1,..,aN
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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}
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A Dequantumized Protocol:|i is not needed:Bob can send a (poly-size)
randomsecret classical string , If Merlin doesn’t know The protocol works as before
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Part II:
The Retrieval Protocol
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Multilinear Extension:Given a0,..,aN (N=2n-1)
F = field of size n2
A: Fn ! F, s.t.:1) 8 i 2 {0,1}n, A(i) = ai
2) A is multilinear (deg(A) · n)
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Quantum Multilinear Extension:A= multilinear extension of
a0,..,aN
1) |i is a state of poly(n) qubits2) When Alice measures |i, she
gets z,A(z) for a random z 2 Fn
(Merlin doesn’t know z)
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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)
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The Protocol:Alice sends l, Merlin is required
to give Al : l ! F. Merlin answers byg : l ! F (deg(g) · n)If g(z) Al(z) Alice rejectsOtherwise, Alice assumes
A(i)=g(i)
If g Al then w.h.p. g(z) Al(z) (since both are low degree)Otherwise, A(i) is correct
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A Dequantumized Protocol:|i is not needed:Bob can send z,A(z), for a
randomz 2 Fn
(s.t., Merlin doesn’t know z)
The protocol works as before
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Part III:
The Exceptional Power of QIP/qpoly
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The Class QIP/qpoly:
IP: [B][GMR] x 2 L can be proved by a poly-size interactive proofQIP: [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
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Quantum Advice:
(captures quantum non-uniformity)A (poly-size) quantum state |L,nigiven to the verifier as an adviceAlternatively, the verifier is aquantum circuit with working spaceinitiated with |L,ni
[NY],[Aar]: Limitations on BQP/qpoly
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QIP/qpoly:
QIP/qpoly: x 2 L can be proved bya poly-size interactive proof
wherethe verifier is a poly-size quantumcircuit with working space
initiatedwith an arbitrary state |L,ni
Our Result:QIP/qpoly contains all languages
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Proof:
Denote ai 2 {0,1}, ai =1 iff i 2 L |L,ni = the quantum multilinear extension of a0,..,aN (N=2n-1)ai can be retrieved from |L,ni byArthur-Merlin interactive
protocolof size poly(n)(one round, classical
communication)
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Randomized Advice:A (poly-size) random string ,chosen from a distribution DL,n,
andgiven to the verifier as an adviceAlternatively, the verifier is a distribution over poly-size
classicalcircuits
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Randomized Advice:A (poly-size) random string ,chosen from a distribution DL,n, andgiven to the verifier as an adviceAlternatively, the verifier is a distribution over poly-size classicalcircuits
IP/rpoly: x 2 L can be proved bya poly-size interactive proof
wherethe verifier is a distribution overpoly-size classical circuits
IP/rpoly contains all languages
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Part IV:
Quantum Versions of the PCP Theorem
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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
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PCP Thm [BFL,FGLSS,AS,ALMSS]:
x 2 SAT can be proved by poly(n)
blocks of length O(1) that can be
verified by reading only 2 blocks
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Same with one block is impossible
(under hardness assumptions)even if each block is of almost linear size
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We Show:
x 2 SAT can be proved by 1) a log-size quantum state |i
and2) a classical proof p of poly(n)
blocks of length polylog eachs.t., after measuring |i the verifier needs to read only oneblock of p
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We Show:
x 2 SAT can be proved by 1) a log-size quantum state |i and2) a classical proof p of poly(n) blocks of
length polylog eachs.t., after measuring |i the verifier needs to read only oneblock of p
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Naive Attempt:a1,..,aN = classical PCP
(N=poly(n))|i = quantum multilinear
extension of a1,..,aN O(log N) qubitsp = Merlin’s answers in the
retrieval protocol
The verifier retrieves a constant number of bits by reading one block
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Problem:The verifier can’t trust that |i is a quantum multilinear extension
In the settings of communication orquantum advice, the verifier couldtrust that |i is correct. In the setting of PCP, |i can be anythinge.g. |i is concentrated on a point
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Quantum Low Degree Test:The verifier checks that |i is a quantum encoding of a low
degreepolynomial. This is done with
the aid of the classical proof (or equivalently, a classical
prover)
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Problem:We are only allowed one query How can we do both: quantum low degree test andretrieval of bits
We combine the two tasks using
ideas from [DFKRS]
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Part V:
Scaling up to NEXP
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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 eachs.t., after measuring |i the verifier needs to read only oneblock of p
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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 eachs.t., after measuring |i the verifier needs to read only oneblock of p
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Alternatively (for L 2 NEXP):
x 2 L has a 3 messages (MAM)interactive proof, where the
proveris quantum in round 1 and
classical in round 2:
1) Prover sends |i 2) Verifier sends q3) Prover answers p(q)
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Models of 3 Messages Proofs:
IP(3): prover is classicalQIP(3): prover is quantum
The hybrid model:HIP(3): prover is quantum in
first round and classical in second
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Models of 3 Messages Proofs:
IP(3): prover is classicalQIP(3): prover is quantum
The hybrid model:HIP(3): prover is quantum in first round and
classical in second
IP(3) µ IP µ PSPACEQIP(3) µ QIP µ EXP [KW]
Our result: HIP(3) = NEXP
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Why the prover in our protocol can’t be quantum in both rounds
?
A quantum prover can answer in round 2, based on a
measurementof a state entangled to the stategiven in round 1(fancy version of the EPR
paradox)
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The End