Complexity Theory: Classical

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Complexity TheoryClassical Quantum

• A brief overview
• Debabrata Ghoshal

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Definition of Language

• Language is a set of strings over some
  alphabet
• L s e S s has a property P
• here s is a set of strings and S is the set
  of all strings over an alphabet S, including the
  empty string.
• Example S0,1
• P Max length of the string is 2 s has
  0,1,01 etc.
• L empty, 0, 1, 00,01,10,11
• NOTE L is the complement set of L

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Decision Problem and Language

• Set of languages form a class
• If all YES answers of a decision problem form a
  class D, then we say all NO answers form a
  class Co-D.
• Here Co-D is not D

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Language recognition generation

• Algorithm which is designed to recognize a
  string that belongs to a particular language is
  called Language recognition device.
• Algorithm which is designed to generate
  strings that belongs to a particular language is
  called Language generator.

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Models of Computation

• Deterministic Turing Machine
• Non-deterministic Turing Machine
• Oracle Turing Machine
• Probabilistic Turing Machine
• Quantum Turing Machine

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Turing Machine

• 
  

a
a
a



q0,q1 h
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Turing Machine

• T ( Q, S, ?, q0, F)
• H (L,N,R) Q (q0, q1) and S (
  a, )
• q a ?(q,
  a,h)
• q0 a
  (q1,,R)
• q0 (F
  ,N)
• q1 a
  (q0,a,R)
• q1
  (q0,,R)

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Transition function (Classical)

• Deterministic Turing Machine
• ? Q X S n X S n - 1 X Q X L,N,Rn n Tapes
• Non-deterministic Turing Machine
• ? Q X S n ? Power set (S n -1 X Q X
  L,N,Rn)
• Oracle Turing Machine
• Probabilistic Turing Machine (transition
  probability)
• ? Q X S n x S n -1 X Q X L,N,Rn ? 0,1
  ??1

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Church-Turing Thesis Universal Turing Machine

• Turing machines can simulate all reasonable
  computation model or device
• A language L is Turing decidable if for all x,
  the machine always halts and output Yes if x e
  L, and if x not e L the output is No.
• Universal Turing machine simulates all Turing
  machines.

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Halting Problem

• 
  NO
• YES
• 
  
• Halting problem is undecidable problem

Halt?
Halt
Start x
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Quantum Turing Machine

• Transition amplitude function
• ? Q X Sn X Sn -1 X Q X L,N,Rn ? C
• also ? ? 2 1
• and ? x yi , x and y are rational
• Transformation should be unitary and
• reversible.

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Scheme of Reversible Machine
I N P U T O U T P U T
C O M P
I N P U T 0 0 0
P M O C
I N P U T 0 0 0
C O P Y
From The Feynman Processor By Gerald J. Milburn
O U T P U T
0 0 0
O U T P U T
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Complexity Expression

• Complexity is expressed as the relation of the
  length of the input x and the amount of the
  resources required to answer if x e L
• Complexity is related to the model of Computation
• Deterministic Turing Machine (DTM) is one of the
  models.
• Quantum Turing Machine (QTM) is another.

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Rate of growth, Order of MagnitudeAsymptotic
Notation

• Big-O O f(n) and g(n) such that f(n) lt c.g(n)
  for cgt 0, Ngt 0
• and n gt N we say f(n) O(g(n)).
• Big-Omega O f(n) and g(n) such that f(n) gt
  c.g(n) for cgt 0, Ngt 0
• and n gt N we say f(n) O(g(n)). If fO(g)
  then g O(f)
• Little-o o When f O(g) but f ? O(g), f(n)
  grows strictly slower g(n) we say f(n) o(g(n)
• Little-Omega ? When f O(g) but f ? O(g), f(n)
  grows strictly faster g(n) we have f(n) ?g(n)
• Big Theta ? When fO(g) and f O(g) we say
  f(n) ?(n) i.e. upper bound is equal to lower
  bound

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Some examples

• Matrix Multiplication is O(n3)
• Using Strassens algorithm the complexity is
• T(n log27) T(n 2.807)
• Complexity of Eigenproblem
• O(n 3 (n log 2 n) log b) where Err. Bound 2-b
• http//citeseer.nj.nec.com/pan98complexity.html

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Time and Space Complexity

• The number of steps required to solve a problem
  is called the Time Complexity of the problem.
• The number of tapes required to solve a problem
  is called the Space Complexity of the problem.

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Classical Time Complexity class P

• P L L L(T), some Turing machine T in
• 
  Polynomial time
• If x is the input and the size of input is
  described by
• x, then the class of problems solved by some
• algorithm within a number of steps bounded by
• F(x), where F is some fixed polynomial
  function, is
• in P.

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Classical Time Complexity class NP

• NP L L L(NT), some Non-deterministic
• Turing machine NT in Polynomial time
• 1. Guess a solution (certificate)
• 2. Verify the solution in Polynomial time

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Classical Time Complexity class NP-Hard

• At least as hard as any NP problem
• Solving a problem in polynomial time by an
  algorithm can translate to solve any other
  problem in NP.
• In other words, if every problem in NP can be
  polynomial time reducible to a language L, then L
  is in NP-hard

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Classical Time Complexity class NP-Complete

• Languages which are NP-hard and also NP are
  called NP-Complete.
• In NP-Complete problems, one problem can be
  restated to the problem of other.
• The solution of the other problem can be
  translated back to the solution of the first
  problem.

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Examples of NP-Complete problems 1

• Traveling Salesman Problem
• Given undirected weighted graph
• Find The minimum-cost path, starting from a
• vertex, visiting all other vertices
  once
• and ending at the starting vertex.
• Another form vertices are cities, path between
• Vertices are roads, weights are the distances.

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Examples of NP-Complete problems 2

• Hamiltonian Cycle Problem
• A simplified version of Traveling Salesman
  Problem. Here the undirected graph has no
  weights.
• This problem is about finding if the graph
  contains a hamiltonian cycle or not.

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Examples of NP-Complete problems 3

• Subset sum problem
• Given A set of integer and a target number.
• Find A subset of these integers adds up to
• target number

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Examples of NP-Complete problems 4

• 3-SAT boolean satisfiability problem
• Example (x1 or x2 or x4) and (x2 or x3 or x1)
  and ( x2 or x4 or x3) Three clauses with
  litererals ( variables or their negations form a
  3-CNF expression. Assigning True or False to each
  variables is it possible to test the expression
  is satisfied or not ( True or False )
• Cook-Levin Theorem

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Classical Space Complexity classes

• L (logarithmic), NL ( non-deterministic L), L2
  (Square log), PSPACE and NSPACE are different
  classes of Space Complexity
• Savitchs Theorem shows that
• PSPACE NSPACE
• Immerman-Szelepscenyi Theorem provides the
  corollary NSPACE (r) Co-NSPACE(r)

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Classical Complexity Classes

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

PP BPP NP Co-NP
P/N SPACE
P
EXP
NEXP
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Classical Complexity Classes

• P Polynomial PP Probabilistic Polynomial
• BPP Bounded Probabilistic Polynomial
• ZPP ZERO-ERROR Bounded PP
• NP Non-deterministic Polynomial
• Co-NP Complement of NP
• EXP Exponential NEXP Non-determ. Exp.
  
• RP Randomized Polynomial Co-RP Comp.
  

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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/Popular_Lectures/Mines
  weeper/

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Primality Testing ( BPP,ZPP)

• Miller-Rabin Test ( Ref. 4)
• 1. If n is even (n ? 2) then n is composite.
  Stop
• 2. Let n-12x m, m is odd and x gt 0
• 3. Choose random r e (1,n-1)
• 4. Compute rm, r2m, r4mr2xm rn-1 mod n
• 5. If rn-1 ? 1 then n is composite
• else If r2i m ? 1 and r(2 i1)m 1 for
  some i gt 0,
• then n is composite
• else n is prime
• end.
• IF n is prime, probability is 1, if composite
  the prob. ½ at least

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Primality Testing in P. (2002)

• Input integer n gt 1.
• 1. If (n ab for a e N and b gt 1), output
  COMPOSITE.
• 2. Find the smallest r such that or (n) gt 4 log2
  n.
• 3. If 1 lt (a, n) lt n for some a r, output
  COMPOSITE
• 4. If n r, output PRIME.
• 5. For a 1 to ? 2vf(r) log n ? do
• if ((X a)n ? Xn a (mod Xr - 1, n)),
  output
• 
  COMPOSITE
• 6. Output PRIME
• http//www.cse.iitk.ac.in/news/primality_v3.pdf

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Quantum AlgorithmComplexity of Shors algorithm

• Input Odd number n size of n is n
• Output A nontrivial factor of n
• 1. Choose arbitrary a e 1, 2, ,n-1
• 2. f gcd(a,n) by Euclid algorithm. If fgt1
  output f stop
• 3. Find the period r by quantum Algorithm
• if r is odd or ar -1(mod n) or ar 1(mod
  n) output failure and stop
• 4. Compute f gcd(n,ar/2 1) by Euclids
  algorithm. Output f and stop

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Quantum AlgorithmComplexity of Shors algorithm
(cont.)

• Step 2 has complexity O(n3).
• Step 3 has several steps with the highest
  complexity O(n3).
• Step 4 has complexity O(n3).
• Overall complexity is O(n3) but the success
  probability is
• guaranteed to be at least O(1/logn).
• Running Shors algorithm O(logn) times we get
  the required
• factor of n in time O((n3)logn) with high
  probability.
• See Reference 1

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Quantum AlgorithmGrovers Search Algorithm

• Input A blackbox function f
• Output Any q e f such that f(q) 1 if it
  exists
• Step 1. Select random r with uniform
  distribution.
• If f(r) 1 exists then stop
• Step 2. Let m v2n 1 and let integer i e
  o,m-1
• Step 3. Using Hadamard Transform and Grovers
  
• Operator G, i times, prepare the
  initial
• Superposition (1/ v2n ) S rgt
• Step 4. Observe to get some q e f
  

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Quantum Algorithm (Cont.)Complexity of Grovers
algorithm

• The task of finding an element q e f by Grovers
  method with non-vanishing probability by using
  O(v2n) queries.
• Grovers quantum searching algorithm is optimal
• By Christof Zalka
• http//arxiv.org/PS_cache/quantph/pdf/9711
• /9711070.pdf
• See Reference 1

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Classical Circuit Complexity (P) Ref. see link
3 Lecture notes.

• 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
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Classical Circuit Complexity (BPP) Ref. see link
3 Lecture notes.

• random r
• bits
  0/1
• X
• if N e PRIMES, C(N) 1 prob. ½ ?
• if N e PRIMES, C(N) 0 prob. ½ ?

C
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Quantum Circuit Complexity (BQP) Ref. see link
3 Lecture notes.

• 
  Answer
• 0gt
• Xgt

C
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P BQP see ref. 4 link 3 Lecture
notes.

• If it is found that something is computed in
• Polynomial time, then it can be also computed
• by quantum polynomial time.
• This can be demonstrated by using Fredkin
• gate with three classical input bit. We can
• simulate NOT and AND gates (Universal
• family).

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BPP BQP see ref. 4 link 3 Lecture
notes.

• In BPP, random qubits are used in the circuit.
• So we have to get random qubits out of 0gt of
• the BQP circuits. If we apply Hadamard gate
• we will gate 0gt? (½)(0gt 1gt). But we have
• to measure it for every 0gt. If we attach a
• control-NOT to every output qubit we can
• avoid that and this is called Principle of
• Deferred Measurement.

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Quantum Circuit Lower Bound See ref. 1 and link
2.

• From Grovers algorithm it is already found that
  the lower bound for queries from the black-box
  function is O(2n) for quantum algorithm.
• There are other techniques to find the quantum
  lower bound. The adversary technique gives the
  lower bound O(2n/2)

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Quantum NP BQNP, BQNP-Complete Ref. see link 2.

• BQNP is the probabilistic analogue of NP.
• If a e L, then there is a string b, b poly
  a such that C(a,b) 1. If a e L then with
  similar b as above, C(a,b) 0. Here C is the
  checker.
• Kitaev recently proved QSAT (quantum analogue of
  3-SAT) is Complete for BQNP.
• To find other examples is open question.

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Complexity Theorist

• A mathematician is a device for converting
• coffee into theorems
• - Paul Erdos
• Is Complexity Theorist a Turing Device ?

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References

• Quantum Computing Mika Hirvensalo
• Quantum Computation and Quantum Information
  Michael A. Nielsen and Isaac L.Chuang
• Elements of the Theory of Computation Harry
  R.Lewis and Christos H. Papadimitriou
• Classical and Quantum Computation A. Yu.
  Kitaev, A. H. Shen, M.N. VYalyi

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Links

• 1.http//www.cs.jhu.edu/scheideler/courses/600.47
  1 S03/lecture_10.pdf
• 2.http//www.csee.umbc.edu/lomonaco/ams/lecture
  notes/Vazirani.pdf
• 3.http//www.cs.berkeley.edu/vazirani/f02quantum.
  html
• 4.http//www.cs.berkeley.edu/aaronson/glossary.ht
  ml