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Quantum Computation
Stephen Jordan
Church-Turing Thesis
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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
machine.
Models of Computation
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Turing machines
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multiple tapes
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multiple read/write heads
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Logic Circuits
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Parallel Computation
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All have been shown polynomially equivalent
to Turing machines
Thesis Revised?
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“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
mathematics.”
-David Deutsch
What Quantum Computers Are
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A reasonable model of computation based on
currently known physics
Apparently more powerful than the Turing
machine
–
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can do prime factorization in polynomial time
The first challenge to the strong Church-Turing
thesis.
What Quantum Computers Aren't
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Extant
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A challenge to the weak Church-Turing thesis
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Just like classical computers except smaller and
faster
Analog
Relation To Other Models
Quantum Church-Turing Thesis?
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Many models of quantum computation:
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quantum turing machines
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quantum circuits
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adiabatic quantum computation
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measurement based quantum computation
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nonabelian anyons
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All have equivalent power (BQP)
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One exception: one clean qubit model
State of The Art
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Quantum Computers
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many approaches
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still in the laboratory
Quantum Cryptography
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fundamentally unbreakable
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commercialized
Earliest Inklings
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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?
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C. Bennett
Yes: any computation can be made
reversible with minimal overhead.
[1973]
Quantum computers can do
reversible computation.
Advantages?
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R. Feynman
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P. Shor
“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
proportional to R.” [1982]
An n-bit number can be factored in
time on a quantum computer. [1994]
More Advantages
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An unstructured database with N items
can be searched in
time.
L. Grover
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Quantum computers can efficiently
simulate quantum systems.
Quantum computers cannot speed up all
problems.
Quantum Mechanics
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The state of a system is represented by a
normalized complex vector.
Example: a bit
Dirac Notation
Inner Product
Two Bits
Dynamics!
Example
Measurement
Quantum Computing
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Start with some state encoding your problem.
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Example: factoring 9 = 1001
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Apply some sequence of unitary time
evolutions.
Measure, and with high probability obtain a
desired result, e.g. 3 = 0011
Quantum Computing
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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
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Examples
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Find the prime factors of an n-digit number.
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Find the shortest route visiting n cities.
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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
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Use only k-body
interactions, “gates”
k=2 suffices
CNOT + one qubit
gates suffice
only finite precision
required
Family of Quantum Circuits
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One quantum circuit
for each input size
Trivial Example:
bitwise XOR
Circuit Complexity
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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?
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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
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Circuit complexity is notoriously difficult to evaluate
Explicit circuit families (algorithms) provide upper
bounds
Lower bounds are very difficult, even classically (e.g.
P vs. NP)
Query Complexity
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Many problems are
naturally formulated in in
terms of a blackbox f
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Find
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Find x s.t. f(x)=1
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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?
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Algorithms provide upper bounds.
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Information arguments provide lower bounds.
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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
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Classically, you need at least d+1 queries
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Otherwise the system is underdetermined
Quantumly, one query suffices
Transforms
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Hadamard transform on n bits uses n Hadamard
gates
Quantum Fourier Transform on n bits can be
done using
gates
The transforms are on
amplitudes!
Inverse transforms are easy. Just take the
adjoint.
Minimizing a Quadratic Form
Further Reading
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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
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After q queries, the amplitudes are polynomials
of degree at most q, hence the p(1) is of degree
2q
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
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Quantum adversary methods
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Reductions
Further Reading
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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]