EECS 598: Background in Theory of Computation

1. EECS 598: Background in Theory of Computation Igor L. Markov and John P. Hayes http://vlsicad.eecs.umich.edu/Quantum/EECS598

2. Outline • On computational models … • More on terminology and notation • Representing states in a computer • Tensor products (again) • Postulates of Q.M. specified for Q.C. • Handling the probabilistic nature of quantum computation • Entanglement

3. Parameters of algorithms… • Execution Time • Wall-clock time • Asymptotic time • Memory (size) • Ease of programming • Worst, best, average, amortized, typical • Multiple objectives -> often no single winner • Cost trade-offs

4. Principle 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)

5. Comparing 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?

6. Elementary Operations • What are elementary operations? • Operations whose execution time is bounded by a constantthat 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] ?

7. Computational 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

8. Measures of Algorithm Efficiency Based 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.

9. Asymptotic Complexity • Recall that resource consumption is typically described by positivemonotonic 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)iff(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

10. Complexity of Problems • In a given computational model,for a given problem…consider best possible algorithms • In fact, whole equivalence classs of algosin terms of asymptotic worst-case complexity • Call that the complexity of the problem • Can often consider the same problemin multiple computational models • Will they have the same complexity?

11. Problem 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

12. Simulating 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?

13. NP-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

14. So, how could you disprove the 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

15. Back 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

16. Computing 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

17. Tensor 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