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EECS 598: Background in Theory of Computation. Igor L. Markov and John P. Hayes http://vlsicad.eecs.umich.edu/Quantum/EECS598. Outline. On computational models … More on terminology and notation Representing states in a computer Tensor products (again) - PowerPoint PPT Presentation
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EECS 598: EECS 598: Background in Theory of Background in Theory of
Computation Computation
Igor L. Markov and John P. Hayeshttp://vlsicad.eecs.umich.edu/Quantum/EECS598
OutlineOutlineOn computational models …More on terminology and notationRepresenting states in a computerTensor products (again)Postulates of Q.M. specified for Q.C.Handling the probabilistic nature of quantum
computation Entanglement
Parameters of algorithms…Parameters of algorithms…
Execution Time– Wall-clock time– Asymptotic time
Memory (size)Ease of programmingWorst, best, average, amortized, typicalMultiple objectives -> often no single winner
– Cost trade-offs
Principle of InvariancePrinciple of Invariance
Challenges for analysis of [time] complexity– Different hardware executes different #instr./sec– Need a uniform measure of time complexity
Solution: measure constant-time steps– Will be off by at most a constant
Principle of invariance– Two implementations of an algorithm (when executed
on two actual computers) will not differ in time complexity by more than a constant
– This is not a theorem !(unlike, e.g., mathematical induction)
Comparing algorithmsComparing algorithms
POI: A similar situation with memory POI gives hope to compare algorithms,
but does not provide a good mechanism What we want:
– Equivalence relations for algorithms (“as fast”)– Order relations (“as fast or faster”)
Possible ideas:– Count elementary operations– Count elementary units of memory– Are all operations counted alike?
Elementary OperationsElementary Operations
What are elementary operations?– Operations whose execution time is bounded by a constant
that does not depend on input values– Actual “seconds per operation” may be disregarded– But the very selection of elem. ops is still hardware-dependent !!!
What about +, -, *, /, <, >, =, ==, etc. of integers (or doubles)?– Yes, if integers have a bounded number of bits, e.g., 32 (or 64)– No, if the number of bits is not bounded
What about function calls? What about memory accesses, e.g., a[i] ?
Computational ModelsComputational Models A computational model is determined by
– data representation and storage mechanisms– available elementary operations
Most popular examples– Sequential and combinational Boolean circuits – EECS 270, 478– [Non-] Deterministic Finite Automata (DFA/NFAs) – EECS
376(476),478 – Push-down automata – EECS 376(476)– Turing machines – EECS 376(476)– C programs, C++ programs – EECS 280, 281(380), 477
Computation models originate in technologies, physics, biology, etc– Parallel and distributed computing– Optical and DNA computing– Analog computing– Quantum computing
Measures of Algorithm Efficiency Measures of Algorithm Efficiency Based on Allowed InputsBased on Allowed Inputs
Each set of input data gives a new measure of efficiency !– Best cases– Worst cases– Representative / typical inputs (“benchmarks”)
application-specific and domain-specific Averaged measures (give “expected efficiency”)
– Consider different inputs in independent experiments– Average over a particular distribution of inputs
Some inputs may happen more frequently than others Application-specific
– Formal “average case” Averaged over the uniform distribution of all allowed inputs
– Empirical evaluation by sampling Generate random samples of the target distribution Much faster than enumeration. Theorem: results are very close.
Asymptotic ComplexityAsymptotic Complexity
Recall that resource consumption is typically described by positive monotonic functions
Asymptotic complexity measures (“on the order of”)– Main idea:
Given f(n) and g(n), does the difference grow with n or stay the same ? Difference: f(n)/g(n). Only pay attention to the “limit behavior” @ n∞
If f(n)/g(n) const as n∞ then f(n) is at least as good as g(n)if f(n)/g(n) ≤ const as n∞ then f(n) is at least as good as g(n)
If f(x) is at least as good as g(x) and g(x) is at least as good as f(x),then the two functions are equivalent
– Coarser than counts of elementary “things” (larger equiv. classes) e.g., “200 N steps” and “3 N + log(N) steps” are now equivalent
– Much greater hardware independence
Complexity of ProblemsComplexity of Problems
In a given computational model,for a given problem…consider best possible algorithms– In fact, whole equivalence classs of algos
in terms of asymptotic worst-case complexity
Call that the complexity of the problem Can often consider the same problem
in multiple computational models– Will they have the same complexity?
Problem reductionsProblem reductions
Reduction of problem X to problem Y– Every instance of problem X translated into an instance
of problem Y– Can apply any algorithm to solve Y– Every solution to a translated instance
translated back Complexity of problem X is no more than sum of
– Complexity of translating instances– Complexity of Y– Complexity of translating solutions back
Simulating Computational ModelsSimulating Computational Models
Write a C program that simulatesa Turing machine
Write a C interpreter as a program for a Turing machine The computational models are equivalent
– Need to be careful about bit-sizes of basic types Modern Church-Turing thesis
– Whatever you can do in terms of computation, can be done on a Turing machine “as efficiently”
– “as efficiently” allows poly-time reductions (next slide) How would you prove/disprove this?
NP-complete problemsNP-complete problems
P is the class of problems whose solutions can be found in poly-time NP is the class of problems whose solutions can be verified in
polynomial time Polynomial-time reductions
– Translations take poly-time NP-complete problems are those to which every other problem in NP
reduces in poly-time NP=P? is an open question Orphan problems:
– Those in NP\P but not NP-complete, assuming NP!=P– Number factoring– Graph auto-morphism
So, how could you disprove the So, how could you disprove the modern Church-Turing thesis ?modern Church-Turing thesis ?
Find a problem that has different worst-case asymptotic complexity – In terms of Turing machines– In terms of, say, quantum computers
Need to define a computational model first! Back to Quantum Mechanics
– Measurements are not deterministic– So, need to include randomized algorithms– BPP (answers can be correct with probability ½+e)
More complexity classes:– PSPACE– Decidable problems
Back to Quantum MechanicsBack to Quantum Mechanics
Details of computational model should follow from the postulates of quantum mechanics
Postulate 1: State Space (complex, Hilbert)– Will need only finite-dim vector spaces
Postulate 2: Evolutions = Unitaries– Think of matrices
Postulate 3: Quantum Measurement– Two forms: Hermitian observables and ortho-normal
decompositions Postulate 4: Composite systems
– Tensor products
Computing with functionsComputing with functions
Functions form a Hilbert space– It is infinite-dimensional, much harder to study– Inner product is defined via integrals– Examples of Hermitian ops: differential operators– Analogues of matrices: “kernels”
Not nearly as nice as finite-dim matrices
How do we get finite-dim spaces?– Restrict allowed states to a finite-dim subspace
Special case: restrict functions to finite sets
– All notions map to common finite-dim notions
Tensor productsTensor products
A state of a quantum system is f(x) A state of a composite system is f(x,y) We would like some construction that would give
us the space of f(x,y) from spaces of g(x) and h(y) Tensor product
– Take bases: g1(x), g2(x),… and h1(x), h2(x),…
– Pair-wise product will be a basis of 2-var funcs
Tensor product of operators