NP has log-space verifiers with fixed-size public quantum registers ABUZER YAKARYILMAZ Faculty of Computing University of Latvia A. C. CEM SAY Department of Computer Engineering Boǧaziçi University October 07, 2011 TÕRVE
Jan 02, 2016
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
𝑥∈𝑳?
VERIFIER
PROVER
An interactive proof system for a language
probabilistic machine
𝑥∈𝑳?
VERIFIER
PROVER
unlimited computational power
Prover can cheat!
resource-bounded
An interactive proof system for a language
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
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
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?
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
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 .
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
……
- -
-, 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
.
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.
-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 .
-.
-
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 --?
References