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