Introduction to Quantum Computing

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Quantum Computation

Dr. Richard B. Gomez rgomez_at_gmu.edu

Introduction to Quantum Computing
Lecture 12 Guest Lecture Debabrata Ghoshal
George Mason University School of Computational
Sciences
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Quantum Complexity
Debabrata Ghoshal CSI 789 Fall 2005
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• What is Quantum Complexity ?

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            Quantum Complexity Andrews Chi-Chih
YaoTsinghua University, China Special Time and
Place Wednesday, Nov. 2, 2005100 p.m., AML
Building (215) Room C103

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Abstract of Yaos Talk

• With rapid advances in technology, it appears
  that computing and communication devices based on
  quantum principles may become available in the
  not too distant future. A central question is
  how much can quantum devices speed up computation
  and communication over classical devices? In this
  talk, we give an overview of quantum
  complexity, which addresses this question, and
  discuss significant recent developments in the
  field, including examples taken from cryptography
  and communication complexity.
•  

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Quantum Complexity involves

• Quantum Principles
• Computational Complexity
• Communication Complexity

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Computational Complexity
Class Relations
Models of Computation

Computational Classes
Decision Problem
Computability Theory
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Set Theory Notation
S1 S2 means S1 contained in S2 S1?S2
S1 S2 means S1 contained in S2 S1
S2 Union S1 U S2 x x e S1 or
S2 Intersection S1 n S2 x x e S1 and
S2 Difference S1 S2 x x e S1 and x
(not e) S2 Cartesian Product S1 X S2 (x ,
y) x e S1 and y e S2 Power Set 2s2 S1
S1 S2 this includes empty set.

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Set Theory Notation

xP(x) is a set of x when predicate P
is true. When x is a natural number
(N) we write the set as S 1,2,3
where each element x is a member of S
we write x ? S and each x is a natural
number.
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Set, Symbol, Strings, Alphabet

Cardinality of a set S is S. It is the number
of elements in a set S. All finite sets are
enumerable as each element of a finite set has
finite number of predecessor. Symbol letters,
digits String Juxtaposed symbols. Length of
a string S is S. Empty string e has a length
zero. Alphabet Finite set of
symbols
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Languages

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 (L bar) 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

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Recursion

Computability theory is explained by the notion
of recursion. A set is defined to be Recursively
Enumerable if every element of the set can be
computed by an algorithm. A set is defined to be
Recursive if each element and each non-member of
the set can be detected by an algorithm.
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Relations and Functions

A binary relation is a set of pair (x , y) where
x is an element of a set called Domain and y is
an element of a set called Range. is called
Equvalence Relation as it is a reflexive ( aRa
for a set S), Symmetric ( aRb implies bRa) and
transitive ( aRb and bRc imply aRc). lt
is an Order Relation as it is reflexive,
anti-symmetric and transitive.
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Relations and Functions

A relation is called a function when each
element of the domain A has unique value in the
range B and we say the function f maps A to B.
f A ? B here f(a) is called
the image of a under f and the range of f is the
image of its domain.

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Relations and Functions

A mapping f A ? B is One-to-One or Injective if
(x1, x2) e A x1 x2 and f(x1) f(x2) A
mapping f A ? B is Onto or Surjective if every y
e B there exist at least one x e A. A mapping
f A ? B is Bijective if it is both One-to-One
(Injective) and Onto (Surjective).
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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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Deterministic Turing Machine

TAPE WITH CELLS

TAPE HEAD CONTROL


a
a
a



q0,q1 h
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Turing Machine Transition function

Turing Machine T ( Q, S, ?, q0, h) State
Q (q0, q1,h) Alphabet S
(a, ) Moving Head M (L,N,R) q0
Initial state q1 A state h Halting state ?
Transition Function

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Turing Table An Example

Q S ?(Q, S, M) q0
a (q1,,R) q0 (h
,N) q1 a (q0,a,N) q1
(q0,,R)
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Turing Machine




q0 a q1,,R
a
a

q1 a q0,a,N
a






q0 a q1,,R
a







q1 a q0,,N









q0 q0,,R
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Turing Machine

q0 q0,h,N

This Turing machine writes symbol to all
symbol a. If is the blank symbol then it
functions as eraser.






q0 h

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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 tape-cell required to solve a
  problem is called the Space Complexity of the
  problem.
• Complexity is defined in terms of size of the
  input.

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Multi-Tape Turing Machine

Control
Tape 1
Tape 2
Tape Head


Tape 3
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Transition function (Classical)

• Deterministic Turing Machine
• ? Q X S n ? 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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Quantum Turing Machine

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

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Churchs Thesis

Any physical computing device can be simulated
by a Turing machine in a number of steps
polynomial in the resources used by the
computing device
- Alonso Church This Thesis is still
valid for Classical computing machine. For
Quantum computing machine the Turing machine
should be replaced by Quantum Turing machine.
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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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Turing Acceptable and Turing Decidable Problems

• Turing Acceptable languages or problems
• are those which can be output from a Turing
  machine.
• Turing Decidable languages or problems are those
  which along with their complements are Turing
  Acceptable.

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Halting Problem

• Given a Turing Machine T and input string s,
• whether T eventually halts on input s?
• Infinite loop detection in a program
• Proof by diagonalization that the problem is
  undecidable

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Quantum Circuits

• Quantum gates are basically reversible
• gates in a very general sense.
• Quantum circuits are a collection of finite set
• of quantum gates.
• YAO has shown in 1993 that polynomial
• time quantum circuits and polynomial time
• QTM has same computational power.

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Rate of growthAsymptotic 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)). UPPER
  BOUND
• 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) LOWER BOUND
• 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 of Asymptotic Notation

Matrix Multiplication is O(n3) Using Strassens
algorithm the complexity is T(n log27) T(n
2.807) Complexity of Eigenproblem could
be O(n 3 (n log 2 n) log b) where Err. Bound
2-b Result as found in 1998

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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 called P.
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Classical Time Complexity class NP

NP L L L(NT),some Nondeterministic
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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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/Popul
ar_Lectures/Minesweeper/
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Time Complexity Classes
P Polynomial Llogarithmic NL
non-deterministic L, L2 (Square log) 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 Random. Poly. Co-RP Comp. BQP
Bounded Quantum Polynomial

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Space Complexity Classes

• 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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Randomized Algorithm

A randomized algorithm is an algorithm where
flipping a coin is allowed to generate random
numbers (pseudo-random number). These random
bits are part of input (auxiliary input)
introduced to have a better performance in the
average case. Here the worst case is so
unlikely that it is ignored.

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Randomized Algorithm

• Checking array elements for a particular
• value by a deterministic algorithm for all
• possible inputs takes long time.
• Checking randomly the array elements for
• desired value with high probability for
• whatever input is much quicker.

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Randomized Algorithm

• Las Vegas algorithm
• Output correct answer with some probability
  that the execution time may take longer.
• Monte Carlo algorithm
• Output quickly with small probability of error
  in output.

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Probabilistic Algorithm

Randomized algorithms (both Las Vegas and Monte
Carlo) are modeled by Probabilistic Turing
machines and form several computational
complexity classes RP No with certainty and
Yes with pr. ½ Co-RP Complement of RP BPP
Both Yes and No with some prob. ZPP Poly.
Average case with correct ans.
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Primality testing

There is a simple approach of Trial
Division. Divide N by integer 2, 3vN and see if
none of the trial divisors divides N then N is
prime. Running time is exponential in the length
of N.
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Probabilistic Algorithm
Primality Testing ( BPP,ZPP)

Miller-Rabin Test 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. is ½ at least.
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AKS algorithm

Primality Testing in P Agrawal, Kayal and
Saxena (India) found a new algorithm in 2002
known as AKS algorithm http//plus.maths.org/iss
ue22/news/prime/
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AKS Algorithm (2002)

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Primality testing algorithm class

This algorithm belongs to EXP class first.
Then we found Miller-Rubin test and the
algorithm is in BPP class. Now it is in class P
following the AKS algorithm.

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Complexity of Shors and Grovers algorithms

• Overall complexity is O(n3) but the success
  probability is guaranteed to be at least
  O(1/logn). n Input size
• Running Shors algorithm O(logn) times we get
  the required factor of n in time O((n3)logn)
  with high probability.
• 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

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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
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Classical Circuit Complexity (BPP)

random r bits
0/1 X if N e PRIMES,
C(N) 1 prob. ½ ? least if N e PRIMES,
C(N) 0 prob. ½ ? most
C
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Quantum Circuit Complexity (BQP)

Answer 0gt Xgt
C
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P BQP

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

In BPP, random bits 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 NP BQNP, BQNP-Complete

• 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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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
• 6.http//www.cs.jhu.edu/scheideler/courses/600.47
  1 S03/lecture_10.pdf
• 7.http//www.cs.berkeley.edu/vazirani/f02quantum.
  html
• 8.http//www.cs.berkeley.edu/aaronson/glossary.ht
  ml

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Classical Communication Complexity

X(x0,x1,x2xn-1) bits
Alice
Bob
Y(y0,y1,y2yn-1)
bits Several rounds of Communication
between Alice and Bob. Goal of Alice Or Bob is
to Compute F(X,Y)

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Classical Deterministic Protocol Equality
Function

EQ(X,Y) 1 if F(X) F(Y)
EQ(X,Y) 0 if F(X) ? F(Y) If
BOB has to acquire the value of EQ(X,Y) with
certainty, then n bits of Communication are
necessary.
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Probabilistic Protocol of Equality Function

Alice and Bob can flip coin and that determines
the outcome of the bit. Error probability e gt 0
is permitted. O(log(n) log(1/e)) bit
communication are sufficient.

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Probabilistic Communication Complexity Public
and Private

In Public model Random string is shared by Alice
and Bob. In Private model Alice and Bob have
their own Private Random string. Private model
can simulate Public model with extra O(log(n))
bits of communication and increase in probability
of error.
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Communication Complexity (CC) Protocol and
Function

Define Probabilistic Protocol CC is the
maximal number of bits communicated by Alice
and Bob. Maximum is taken over all inputs and all
designation of random string. The CC of function
is the minimum CC of a Protocol to compute the
function with err e.

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Two Other similar problems

Intersection Problem IN(X,Y)(x0 ? y0) ? (x1 ?
y1)..?(xn-1 ? yn-1) Inner Product
Problem IP(X,Y)(x0 ? y0) (x1 ? y1).. (xn-1
? yn-1) Probabilistic Protocol of IN and IP
requires O(n) bits of communication with small
error. ? AND ? OR
XOR
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Quantum Protocol

There is a priori system of n-qubits Alice has
some of them and Bob has the rest.
Initialize the qubits as 0gt. Alice and Bob can
perform unitary transformations on the qubits and
can exchange qubits and thus change ownership of
some qubit. Output is some measurement of Bobs
qubit.
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Quantum Communication Complexity (QCC with qubits)

• If Alice sends an n-qubit message to Bob,
• She cannot convey more than n-bits of
• Information Holevo bound
• This is also true for n-bit message with prior
• entanglement but not true for n-qubit
• message with prior entanglement.
• Bennett Wiesner 92
• (Superdense coding)

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Deutsch and Ekert

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• Figure (a) Two bits for the price of one
  starting from the bottom, Bob sends one particle
  of an entangled (EPR) pair to Alice who performs
  one of four operations on it with the quantum
  gate M Alice then returns the particle to Bob who
  measures the state of the joint (and still
  entangled) system with the quantum gate M to
  receive one of four possible messages (two bits
  of information), although only one particle
  (which can exist in only one of two states, i.e.
  carry only one bit of information) has been sent.
  Ouantum communication channels are represented by
  thin lines, classical channels by thick lines.
  (b) Ouantum teleportation again Bob sends one
  particle of an entangled state to Alice who
  measures the joint state of this with the unknown
  state . She then transmits (classically) this
  result to Bob. This information can be used to
  put Bob's remaining particle (the other half of
  the entangled pair) in the state .
• Extract from Physics World, June 1993
  

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QCC More results with qubits

Grovers Search problem (1996) shown to be O(vn)
for quantum protocol where in classical it is
O(n) In 1997 Burhman et. al. computed
the function f(x,y) x . y . (-1) x.y where
x (x0,x1) and y (y0,y1)
x0,y0 ? (0,1) and x1,y1 ? (-1, 1) With highest
probability - Exchanging altogether 2n bits of
information.
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QCC More results with qubits

They showed in the case of Alice and Bob sharing
two qubits in a maximally entangled state, the
probability of success Pquantum
0.85 Where as for classical probabilistic case
without entanglement, it cannot exceed
Pclassical 0.75 3 bits are necessary for
classical and 2 bits are sufficient for quantum
(both P 0.85).
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QCC More results with qubits

Ran Raz in 1999 showed another example of
Communication Complexity where Quantum Protocol
was used and the performance is exponentially
better than the best Classical Probabilistic
Protocol.
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Quantum Communication Complexity with qutrits

Reviewing the research paper Quantum
Communication Complexity protocol with two
entangled qutrits Caslav Brukner, Marek
Zukowski and Anton Zellinger - 2004

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Quantum Communication Complexity with qutrits

Research of Caslav et. al. showed the Quantum
solution of Communication Complexity exploiting
entanglement of two qutrits. Quantum solution
can enhance the efficiency in quantum protocol
over classical if and only if state violates
Bells inequality for two qutrits.