Quantum Computing for Computer Scientists

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					           Quantum Computing
                  for
           Computer Scientists


                Research by Forrest Briggs
Advisors: Ran Libeskind-Hadas, Chris Stone, John Townsend
     Aknowledgements

Profs. Ran Libeskind-Hadas, John
Townsend, Chris Stone, Peter Saeta

Quantum Computation and Quantum
Information by Nielsen and Chuang
 Why should you care about QC?
• It’s inevitable as transistors shrink
• There are quantum algorithms that are faster than any
  known classical algorithm
• It sheds light on fundamental issues in theoretical
  computer science
• Quantum information processing is here now
• There are abstract parts of quantum computing that will
  not depend on how they are physically realized in the
  future
• We can simulate quantum computers now
• It’s time to start designing algorithms and gaining insight
  from theoretical inquiry
                          The Qubit
                                                        c0 
      | ψ >= c0 | 0 > +c1 |1 >                   | ψ >=  
                                                         c1 


€   |Ψ> is the state of the system. It is a superposition of the basis
                                  €
           vectors |0> and |1> with complex coefficients.
            A Qubit in Nature




Before encountering the magnet, the atom is in the state
                     1        1
              | ψ >=    |↑> +    |↓>
                      2        2
                  Multiple Qubits
                          Two Qubits
    | ψ >= c00 | 00 > +c01 | 01 > +c10 |10 > +c11 |11 >



                           N Qubits
     | ψ >= c0 | 0 > +c1 |1 > +... + c2 N −1 | 2 N −1 >



€
            Unitary Operators

Every transformation you apply to the state of a
quantum computer is a unitary operator, from a
single gate to an entire algorithm.

An operator A is unitary iff A† = A-1.

Unitary operators are by definition, invertible.
                Unitary Operators


    A unitary operator on a single qubit is just a 2x2 matrix.



            a0                    w       xa0  b0 
    | ψ1 >=              | ψ 2 >=             =  
             a1                   y       z  a1   b1 



€            €
              Unitary Operators
A unitary operator on 3 qubits that acts as a universal logic
gate.
                     1   0 0 0 0 0 0 0
                                       
                     0   1 0 0 0 0 0 0
                     0   0 1 0 0 0 0 0
                                       
                     0   0 0 1 0 0 0 0
                     0   0 0 0 1 0 0 0
                                       
                      0   0 0 0 0 1 0 0
                                       
                     0   0 0 0 0 0 0 1
                                       
                     0   0 0 0 0 0 1 0



              €
      Measurement is Probabilistic
    Qubit Example:
               1       1
        | ψ >=    |0>+    |1 >
                2       2
    If the state of a system is
    |Ψ> = c0|0> + c1|1> + … + cn|n>,
€   then after measuring it, |Ψ> = |i> with
    probability |ci|2.

    Other kinds of measurements are possible.
Take a moment to reflect on the
         weirdness:
 Basically the same rules to we
 have seen apply to any physical
 system.

 Completely isolated systems in the
 universe are deterministic, but any
 measurement is probabilistic.
Quantum Circuits
                        1 1 1 
                     H=        
                         2 1 −1


                1   0 0 0
             €
                          
                 0   1 0 0
          UCN =           
                0   0 0 1
                0   0 1 0
                



    €
       Classical Universality
Unitary operators are invertible, so quantum
computation must be reversible (no NAND).

We can construct a quantum circuit for any classically
computable function.

A quantum computer can be simulated on a classical
computer (although not efficiently).

The set of computable functions is the same.
     Quantum Universality
We can approximate any unitary operator on a single
qubit as a product of a constant number of gates from
a universal set.

We can approximate unitary operators on many
qubits, but not always efficiently.

It is not possible to prepare arbitrary quantum states
efficiently.
             Complexity Theory
BPP - The class of decision problems that can be solved with
bounded probability of error in polynomial time on a TM.

BQP - The class of decision problems that can be solved with a
bounded probability of error with a polynomial sized quantum
circuit (which can be generated by a TM).

P  BPPBQP PSPACE EXP

Factoring is in NP, BQP, probably not in BPP.
  Quantum Fourier Transform
                                              T
Input:     | ψ >= [c0       c2    ... c N ]
                                               T
Output:    | ψ '>= [ f 0     f2   ...   fN ]
  €                  1 N 2 πijk/N
                fj =    ∑ e
                     N k=0
 €
Runtime: O(log n)          FFT: O(n log n)

Not as good as you might think: we can’t
     €
prepare arbitrary input states.
       QFT Circuit




         1     0 
    Rk =     2 πi/2 k 
         0 e          



€
                     Factoring

Shor’s algorithm is O(n2 log(n) log(log n))
The best known classical algorithm is O(exp[n1/3 log2/3 n])

Shor’s algorithm reduces to order finding (solve xr = 1 mod z).
Order finding reduces to phase estimation (find an eigenvalue).
Phase estimation reduces to the quantum fourier transform.
Quantum Search Algorithm

 Given:
 f:{0,…,N}{0,1}
 f(x) = 1 if x = a, 0 otherwise

 We can find a in O( N ) time.



         €
That’s just the tip of the iceberg…
• Quantum information (encryption,
  compression, teleportation)
• Physical realizations
• Error correcting codes
• Quantum programming languages
• Applications (better AI, simulation of
  quantum mechanical systems, etc).

				
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Description: Quantum computer is a kind of obey quantum mechanics for high-speed mathematical and logical operations, storage and processing of Quantum Information Physics device. When a device processing and calculation of quantum information, quantum algorithm is running, it is a quantum computer. Quantum computer concept stems from the reversible computer studies. The study of reversible computing aims to solve the problem of energy consumption in the computer.