Quantum Complexity Theory

1
Quantum Complexity Theory

• CSI 789-004 Fall 2004
• Lecture Notes 6

2

Outline

• Complexity Theory ( Sets, Languages )
• Common Classical Complexity Classes
• Relationships of Common Classes
• Additional Classes and Relationships
• NP- Complete classes and reducibility
• Interactive Proof

3
Outline

• Quantum Complexity Classes
• NP- Complete problems and Quantum computation
• Open problems

4
Complexity Theory (Sets, Languages)

• The complement of recursive language is recursive
• If L and L (complement of L) are recursively
  enumerable then they are also recursive
• Union of two recursive languages is recursive
  

5

Complexity Theory (Sets, Languages)

• Let L be the set of all languages and Lre be the
  set of all recursively enumerable languages. It
  can be shown that
• L gt Lre i.e. the set of recursively
  enumerable languages cannot cover the set of all
  languages.
• In a sense this tells the limits of
• computability There is decision problem that
  does not have a Turing machine that accepts it.
  

6
Common Classical Complexity Classes
P Polynomial Time NP Non-deterministic
Polynomial Time PSPACE Polynomial Space NSPACE
Non-deterministic Poly. Space RP Randomized
Polynomial
BPP Bounded Probabilistic
Polynomial NP-Complete Non-determ. Polyno.
Complete EXP Exponential Time EXPSPACE
Exponential Space

7
Common Classical Classes

• P Class of problems solvable on a
  deterministic Turing machine in polynomial time
  is known as class P
• NP Class of problems solvable on a
  nondeterministic Turing machine in polynomial
  time is known as class NP
• These problems have efficiently verifiable
  solution in polynomial time.

8
Common Classical Classes

• PSPACE Class of problems solvable on a
  deterministic Turing machine in polynomial
• space is known as the class PSPACE
• NSPACE Class of problems solvable on a
  nondeterministic Turing machine in polynomial
  space is known as the class
• NSPACE

9
Common Classical Classes

• RP Classes of problems solvable on a
  probabilistic Turing machine in polynomial time
  with error bound between 0 and greater than equal
  to ½ is known as class RP (Randomized Polynomial)
• BPP Class of problems solvable on a
  probabilistic Turing machine in polynomial time
  with reasonable error bound is known as class BPP

10
Common Classical Classes

• In RP, if x is a member of class of languages L,
  then x is accepted by a probabilistic Turing
  machine with probability gt ½
• If x is not a member of class of languages L,
  then x is accepted by a probabilistic Turing
  machine with probability 0

11

Common Classical Classes

• In BPP, if x is a member of class of languages L,
  then x is accepted by a probabilistic Turing
  machine with probability gt 2/3
• If x is not a member of class of languages L,
  then x is accepted by a probabilistic Turing
  machine with probability lt 1/3

12


Common Classical Complexity Classes

• EXP (or EXPTIME) Class of problems solvable on a
  deterministic Turing machine in exponential time
  (2 (polynomial n) is known as class EXP
• EXPSPACE Class of problems solvable on a
  deterministic Turing machine in exponential space
  (2 (polynomial n) is known as class EXPSPAC

13

Relationships of classical classes

• P RP BPP PSPACE
• RP NP PSPACE EXP

14
Relationships of classical complexity classes

• 
  EXP
• 
  PSPACE
• 
  
• 
  NP
• BPP
• 
  RP
• 
  P
  
  

15

Some Other Classical Classes
PP Probabilistic Polynomial Time Co-NP
Complement of NP Co-RP Compliment of RP
ZPP Zero error Bounded
Probabilistic Polynomial NEXP Nondeterministic
Exponential Time (Polynomial
n) ( 2 2
)

16

Classical Complexity Classes

ZPP TRACTABLE RP
I N T R A C T A B L E

Co-RP




-? IS P NP ?
PP BPP NP Co-NP
P/N SPACE
P
EXP
NEXP
17

Million Dollar Question ?
The PNP? problem asks whether types P and NP
are (despite all appearances to the contrary) the
same. The expected answer is 'no'. However, if
any NP-complete problem turns out to be of type
P--- to have a polynomial time solution--- than
NP must equal P. We therefore expect all
NP-complete problems to be non-P, but no one can
yet prove this. http//www.claymath.org/Popu
lar_Lectures/Minesweeper/

18

Classical Circuit Complexity (P)

X1 X2
0/1 Xn
PRIMES e P
runs in O(log n) C
O(poly(n)) C computed in poly(n) on some Turing
Machine.
C
19

Classical Circuit Complexity (BPP)
random r bits
0/1 X if N e PRIMES,
C(N) 1 prob. 2/3 if N e PRIMES, C(N) 0
prob. 1/3

C
20
Class NP-Complete

• The hardest problems in NP
• Every problem in NP is reducible to it
• Traveling Salesman Problem is NP-complete.
  Hamiltonian Cycle is another NP-complete problem.
  They are reducible to one another.
  

21
Class NP-Complete

• Satisfiability (SAT) is NP-complete
• SAT(x) is True for some values of variable x.
• x is a propositional formula and it contains
• Boolean variables and operations of NOT,
• AND and OR

22
Reducibility

• There are two types of Reductions
• Karp or Many-one reduction Instance of a problem
  X and an algorithm in P is given. The output must
  be the instance of another problem Y.
• 2. Turing reduction (Cook-Levin) Algorithm of
  Problem Y can make arbitrarily many calls to an
  oracle for problem X.

23
Reducibility

• Proof of Karp-reducibility
• If L1 is reducible to L2 then
• 1. L2 is a member of P means L1 is a
• member of P.
• 2.L1 is not a member of P means L2 is not
• a member of P.
• 3. L2 is a member of NP implies L1 is a
• member of NP

24

Reducibility

• Cook-Levin Theorem
• SAT is a member of NP and SAT is NP-complete.
• SAT is reducible to 3-SAT.

25
Interactive Proof

• 
  Two Way
• 
  communication
• 
  Channel
• Power Polynomial
  Power Unlimited
• some randomness
  

Verifier
Prover
26
Interactive Proof

• The Prover has to send x with a property
• The Verifier has to validate that x has that
  property
• Here the output is from the Verifier - If it
  validates the property of x. Otherwise the
  Verifier rejects it

27
Interactive Proof

• If x has a property
• There exists a Prover to convince the Verifier
  to accept x with high probability
• If x does not have a property
• No Prover can convince a Verifier to accept x (
  except with small probability)

28
Example Interactive Proof

• Suppose two graphs are given Graph1 and
• Graph2
• Verifier chooses one of the graphs randomly and
  permutes it randomly. Then he sends it to the
  Prover.
• 2. The Prover has to identify if the graph sent
  by the Verifier is isomorphic to the two given
  graphs.

29
Class of Interactive Proof

• The class of Interactive Proof is known as IP
  class and it is shown that
• IP PSPACE
• And for constant m messages,
• IP(m) IP(2)

30



Quantum ComplexityQuantum Turing Machine

Bernstein and Vazirani defined Quantum Turing
machine with Finite sets Q Set of states S
Set of alphabet M Set of Left /Right head
movement L,R Transition Function ? Q X S X S
X Q X M? C (Here C is the set of complex numbers
and ? ? 2 1 and ? x yi , x ,y are rational )
31
Quantum ComplexityQuantum Turing Machine

• ?(q0,r,w,qf,m) means it is the amplitude of
• the event that the machine is in state q0
• Reading r writing w enters the state qf and
• move the head in direction m where m is either
• left or right.

32
Quantum ComplexityQuantum Turing Machine

• Classical Turing Machine configurations a
• and b are considered.
• A QTM can be in an arbitrary superposition
• aaßb where a2ß2 1.
• Here a is observed with probability a 2
• and b is observed with probability ß 2.

33

Quantum ComplexityQuantum Turing Machine

• In QTM, all the possible computation paths
• are taken simultaneously.
• All paths that have same final result will
  interfere.

34
Quantum ComplexityQuantum Turing Machine

• Let the amplitude is a for path p1-gt p2-gt p3
• Case 1. amplitude is also a for p1-gt p4-gt p3
• Case 2. amplitude is -a for p1-gt p4-gt p3
• For case 1 p1-gtp3 the prob. is (a a)2 4 a2
• For case 2 p1-gtp3 the prob. is (a- a)2 0
• Here sum of probabilities p1-gtp3 2 a2

35
Quantum Complexity Classes

• EQP Exact Quantum Polynomial time.
• A language is in class EQP if there exists a
• QTM such that given input x, whether x is a
• member of L is decided with probability 1.
• This is done by observing a distinguished
• tape cell in polynomial time p(size of x).
  

36

Quantum Complexity Classes
BQP Bounded error Quantum Polynomial time. A
language is in class BQP if there exists a QTM
such that given input x, whether x is a member
of L is decided with probability at least
2/3. This is done by observing a distinguished
tape cell in polynomial time p(size of x).

37

Quantum Complexity Classes

ZQP Zero error Quantum Polynomial time. A
language is in class ZQP if there exists a QTM
such that given input x, whether x is a member
of L is decided with probability 1 when the
probability of halting state is 1/2. This is done
by observing a distinguished tape cell and a
halting cell in polynomial time p(size of x).

38


Relationships of Quantum and Classical
Complexity Classes

• P EQP ZQP BQP
• BPP BQP
• ZPP ZQP
• Open Problems
• BPP EQP? NP BQP?
• BPP BQP?

39
Quantum Circuit Complexity (BQP)

Answer 0gt Xgt
C
40
NP-Complete Problem and Quantum Computation

• We take satisfiability problem
• f(x1,x2,x3xn) where x1,x2..xn are
• Boolean variables.
• We represent
• x1,x2,x3.xn,xn1gt where xn1 is the
• value of the output f.

41
NP-Complete Problem and Quantum Computation

• Initialize 0,0,0,.,0gt and then in one single
  step 2n configurations can be applied for x and
  the superposition becomes
• 1/v2n ? ? x1,x2..,0gt
• 2. Compute f by simulating deterministic
  reversible Turing Machine and resulting
  superposition
• 1/v2n ? ? x1,x2..,f(x1,x2xn)gt

42
NP-Complete Problem and Quantum Computation

• 3. Measurement of xn1 qubit provide some real
  number ? ( using some experimental step)
• If the value of f(xn1) is always 0 for 2n
  configuration we get ? -1 and conclude that f
  is not satisfiable.
• If at least one superpostion gets value 1
• then ? gt -1 and f is satisfiable
  

43
Complexity Zoo

• www.complexityzoo.com