quantum computation
DESCRIPTION
Stephen Jordan. Quantum Computation. 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 machine. Models of Computation. - PowerPoint PPT PresentationTRANSCRIPT
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Quantum Computation
Stephen Jordan
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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 machine.
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Models of Computation
● Turing machines– multiple tapes
– multiple read/write heads
● Logic Circuits● Parallel Computation● All have been shown polynomially equivalent
to Turing machines
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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 mathematics.” -David Deutsch
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What Quantum Computers Are
● A reasonable model of computation based on currently known physics
● Apparently more powerful than the Turing machine– can do prime factorization in polynomial time
● The first challenge to the strong Church-Turing thesis.
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What Quantum Computers Aren't
● Extant● A challenge to the weak Church-Turing thesis● Just like classical computers except smaller and
faster● Analog
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Relation To Other Models
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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
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State of The Art
● Quantum Cryptography– fundamentally unbreakable
– commercialized
– Quantum Computers● many approaches● still in the laboratory
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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. [1973]
● Quantum computers can do reversible computation.C. Bennett
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Advantages?● “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]
R. Feynman
P. Shor
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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 problems.
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Quantum Mechanics
● The state of a system is represented by a normalized complex vector.
● Example: a bit
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Dirac Notation
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Inner Product
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Two Bits
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Dynamics!
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Example
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Measurement
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Quantum Computing● Start with some state encoding your problem.● Example: factoring 9 = 1001● Apply some sequence of unitary time
evolutions.● Measure, and with high probability obtain a
desired result, e.g. 3 = 0011
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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?
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● 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?
Computational Problems
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Model of Computation: Quantum Circuits
● Use only k-body interactions, “gates”
● k=2 suffices● CNOT + one qubit
gates suffice● only finite precision
required
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Family of Quantum Circuits● One quantum circuit
for each input size● Trivial Example:
bitwise XOR
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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?
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Problems with Circuit Complexity● 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)
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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
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An Easier QuestionFor 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.
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Bernstein-Vazirani Problem
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Classical Algorithm
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Phase Kickback
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Bernstein-Vazirani Algorithm
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Classical Gradient Estimation● Classically, you need at least d+1 queries
● Otherwise the system is underdetermined
● Quantumly, one query suffices
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Transforms
● 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.
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Minimizing a Quadratic Form
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Further Reading
● Michael Nielsen and Isaac Chuang, Quantum Computation and Quantum Information (2000)
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An Optical Analogy
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An Optical Analogy
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Lower Bounds
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Lower Bounds by Polynomials
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Lower Bounds by Polynomials
● 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
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Paturi's Theorem
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Specific Lower Bounds
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Other Techniques
● Quantum adversary methods● Reductions
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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]