NP has log-space verifiers with fixed-size public quantum registers
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NP has log-space verifiers with fixed-size public quantum registers
ABUZER YAKARYILMAZFaculty of Computing
University of Latvia
A. C. CEM SAYDepartment of Computer Engineering
Boǧaziçi University
October 07, 2011TÕRVE
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𝑥∈𝑳?
VERIFIER
PROVER
An interactive proof system for a language
probabilistic machine
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𝑥∈𝑳?
VERIFIER
PROVER
unlimited computational power
Prover can cheat!
resource-bounded
An interactive proof system for a language
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Two criteria:Language has a proof system if
COMPLETENESSFor every , the verifier always accepts with high probability after interacting the prover
SOUNDNESSFor every and every , the verifier rejects with high probability after interacting
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Arthur-Merlin system (space-bounded)
₵ 0 1 # … 0 1 # $
… # 0 1 # …
Work tape (restricted)
1
Communicationcell
Random numbergenerator
Input tape (read-only)
… # 1 1 # …
Work tape (unlimited)
outcomesARTHUR
MERLIN
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Complexity classes is the class of languages recognized by a deterministic Turing machine in polynomial time. is the class of language recognized by a nondeterministic Turing machine in polynomial time.--- is the class of languages having an AM proof system with no error such that • the random number generator is removed and • the runtime of Arthur is restricted with polynomial time.Class is obtained, if the communication cell is removed as a further restriction. ---
- [Con89]A well-known open problem: Is equal to , or not?
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A new system: qAM
₵ 0 1 # … 0 1 # $
… # 0 1 # …
Work tape (restricted)
1
Communicationcell
A finite quantumregister
Input tape (read-only)
… # 1 1 # …
Work tape (unlimited)
outcomesÂRTHUR
MERLIN
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The finite quantum register• A quantum register is an -dimensional Hilbert space, , with
basis• , where
• A quantum state is a linear combination of basis states, i.e.• , where
• each is called the amplitude of being state and the probability of being in state is given by .
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The operations on the register• Initializing the register (a predefined quantum state)• Applying a superoperator () satisfying
,where• is an operation element • is the measurement outcome• [Optional] Each entry of is a rational number
𝜀(¿𝜓 ⟩)
=
𝑝1=~⟨𝜓 1∨
~𝜓 1 ⟩
𝑝2=~⟨𝜓 2∨
~𝜓 2 ⟩ =
𝑝𝑘=~⟨𝜓𝑘∨
~𝜓𝑘 ⟩
=
……
|~𝜓 1 ⟩√𝑝1
|~𝜓 2 ⟩√𝑝2
|~𝜓𝑘 ⟩√𝑝𝑘
……
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- -
-, a well-known -complete problem, is the collection of all strings of the form
such that , and ’s are numbers in binary , and there exists a set satisfying .---Ârthur can encode binary numbers into amplitudes of the states of the register and can also make addition and subtraction on them.
The strategy of Ârthur:Ârthur requests the set from Merlin and then tests
.
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Some details of the algorithm₵ 0 1 … 1 # 1 0 … 1 # … … 1 1 … 0 # $
… …𝑆 𝑎1
𝑎𝑛
(1000)
auxiliary value
to store
to store ’s
to store
|~𝜓|𝑤|⟩=( 13 )
|𝑤|(1𝑆0𝑇
)Initial state
Before reading $
( 13 )
|𝑤|+1
2(𝑆−𝑇 ) ⏟
( 13 )
|𝑤|+1
⏟
reject
Accept ()
• Member are accepted exactly.• Nonmembers are rejected with a probability at least .The error gap can be reduced to any desired value by usingconventional probability amplification techniques.
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-Any language in is log-space reducible to - [Pap94]:• Let be language in , then there exists a logarithmic space deterministic
algorithm that outputs for any given input string such that-.
---For any given input string , Ârthur can run the algorithm for - on .
-.
-
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Concluding remarks• A poly-time Ârthur can be simulated by a poly-time Arthur:
--
• In constant space [DS92,CHPW94,AW02]:- -----
(if arbitrary transition amplitudes are allowed)
• Is --? [Con93]• What is the relationship between
and --?
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References