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					Quantum Computation

     Stephen Jordan
                Church-Turing Thesis
●   Weak Form: Anything we would regard as “computable” can
    be computed by a Turing machine.
●   Strong Form: Anything we would regard as efficiently
    computable can be computed in polynomial time by a Turing
            Models of Computation
●   Turing machines
    –   multiple tapes
    –   multiple read/write heads
●   Logic Circuits
●   Parallel Computation
●   All have been shown polynomially equivalent
    to Turing machines
                 Thesis Revised?

●   “Computers are physical objects and
    computations are physical processes. What
    computers can or cannot compute is determined
    by the laws of physics alone, and not by pure

                      -David Deutsch
        What Quantum Computers Are

●   A reasonable model of computation based on
    currently known physics
●   Apparently more powerful than the Turing
    –   can do prime factorization in polynomial time
●   The first challenge to the strong Church-Turing
    What Quantum Computers Aren't
●   Extant
●   A challenge to the weak Church-Turing thesis
●   Just like classical computers except smaller and
●   Analog
Relation To Other Models
    Quantum Church-Turing Thesis?
●   Many models of quantum computation:
    –   quantum turing machines
    –   quantum circuits
    –   adiabatic quantum computation
    –   measurement based quantum computation
    –   nonabelian anyons
●   All have equivalent power (BQP)
●   One exception: one clean qubit model
                        State of The Art
    –   Quantum Computers
         ●   many approaches
         ●   still in the laboratory

●   Quantum Cryptography
    –   fundamentally unbreakable
    –   commercialized
                 Earliest Inklings
 ●   At small scales the laws of classical mechanics
     break down and quantum mechanics takes over.
 ●   Can computers still work when their
     components reach this scale?

             ●   Yes: any computation can be made
                 reversible with minimal overhead.
             ●    Quantum computers can do
C. Bennett       reversible computation.
             ●   “The full description of quantum
                 mechanics for a large system with R
                 particles...has too many variables. It
                 cannot be simulated with a normal
                 computer with a number of elements
R. Feynman
                 proportional to R.” [1982]
             ●   An n-bit number can be factored in
                 time on a quantum computer. [1994]

 P. Shor
                 More Advantages
            ●   An unstructured database with N items
                can be searched in        time.

L. Grover

            ●   Quantum computers can efficiently
                simulate quantum systems.
            ●   Quantum computers cannot speed up all
            Quantum Mechanics
●   The state of a system is represented by a
    normalized complex vector.
●   Example: a bit
Dirac Notation
Inner Product
Two Bits
            Quantum Computing
●   Start with some state encoding your problem.
●   Example: factoring 9 = 1001
●   Apply some sequence of unitary time
●   Measure, and with high probability obtain a
    desired result, e.g. 3 = 0011
              Quantum Computing
●   2 questions about quantum computing
    1)How can we build a quantum computer?
      We'll ignore this.
    2)What can we do with them?
      We'll turn this into a precise question:

       For a problem of size n, how many
       computational steps do we need to solve it
       on a quantum computer?
             Computational Problems
●   Examples
    –   Find the prime factors of an n-digit number.
    –   Find the shortest route visiting n cities.
    –   Compute                   for given f.
●   Which problems can be solved with fewer steps
    on quantum computers than on classical
    computers for large n?
      Model of Computation: Quantum Circuits

●   Use only k-body
    interactions, “gates”
●   k=2 suffices
●   CNOT + one qubit
    gates suffice
●   only finite precision
           Family of Quantum Circuits
●   One quantum circuit
    for each input size
●   Trivial Example:
    bitwise XOR
               Circuit Complexity
●   Return to our original question:
    For a problem of size n, how many
    computational steps do we need to solve it
    on a quantum computer?

●   We can now make it precise:
    What is the minimum number of gates
    needed, as a function of n, in a family of
    quantum circuits which solves the problem?
    Problems with Circuit Complexity
●   Circuit complexity is notoriously difficult to evaluate
●   Explicit circuit families (algorithms) provide upper
●   Lower bounds are very difficult, even classically (e.g.
    P vs. NP)
                    Query Complexity
●   Many problems are
    naturally formulated in in
    terms of a blackbox f
    –   Find
    –   Find x s.t. f(x)=1
    –   Find x which minimizes f
●   Classical blackboxes can
    be made reversible,
    hence unitary
            An Easier Question
         For a given problem, how many black box
         queries do we need to solve it on a quantum
         computer, as a function of problem size?

●   Algorithms provide upper bounds.
●   Information arguments provide lower bounds.
●   Quantum speedups for several black box
    problems are known.
●   In many cases matching quantum lower bounds
    are known.
Bernstein-Vazirani Problem
Classical Algorithm
Phase Kickback
Bernstein-Vazirani Algorithm
           Classical Gradient Estimation
    ●   Classically, you need at least d+1 queries

    ●   Otherwise the system is underdetermined

●       Quantumly, one query suffices
●   Hadamard transform on n bits uses n Hadamard
●   Quantum Fourier Transform on n bits can be
    done using    gates
●   The transforms are on     amplitudes!
●   Inverse transforms are easy. Just take the
Minimizing a Quadratic Form
     Further Reading

●   Michael Nielsen and Isaac Chuang,
    Quantum Computation and Quantum
    Information (2000)
An Optical Analogy
An Optical Analogy
Lower Bounds
Lower Bounds by Polynomials
     Lower Bounds by Polynomials
●   After q queries, the amplitudes are polynomials
    of degree at most q, hence the p(1) is of degree
●   Recall that desired result     is some
    boolean function of the blackbox values
●   There is a minimal degree for a polynomial to
    match this function
Paturi's Theorem
Specific Lower Bounds
          Other Techniques
●   Quantum adversary methods
●   Reductions
                Further Reading
●   E. Bernstein and U. Vazirani, “Quantum
    complexity theory,” proceedings of STOC 1993
●   S. Jordan, “Fast quantum algorithm for
    numerical gradient estimation,” Phys. Rev.
    Lett. 95, 050501 (2005) [quant-ph/0405146]
●   R. Beals, H. Buhrman, R. Cleve, M. Mosca, and
    R. De Wolf. “Quantum lower bounds by
    polynomials,” Journal of the ACM, Vol. 48,
    No. 4 (2001) [quant-ph/9802049]

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