Ii : Meeting G. Classical warm. ups to quantum computing : I . Probabilistic classical computing BPD and MA TL . Reversible classical computing Next time : circuits BQP and QMA quantum , Axioms of quantum mechanics lead to important problems if to we want to use quantum mechanical systems build a - : even in the ideal of a noise less computer , case system The I . classical information extracted via measurement a distribution is probability . How this? can we formalize complexity theory around If 2. want of we to use a Hilbert space some system " " as a the- quantum memory register , fqumtum) transformations must be hence in unitary , , particular, invertible ) reversible . Is reversible computation feasible? : these Goats answer questions in classical - warm up cases . The won It of classical analogs address all the issues " " . the . are in quantum case Erg quantum States not just classical distributions and unitary Uk) is probability , group Uncountable, infinite . : - Also important later non idealized quantum computing . Need error and a theory of quantum correction fault tolerance . I . Probabilistic classical computing : classical is Informally a probabilistic algorithm any algorithm that is allowed access to coin or flips , , equivalently, random lait strings . : Two to this more formal equivalent ways make the definition I. Extend of Turing machine so the transition function can in addition to the machine 's , using internal state and read of the toss a fair coin memory , . " " 2. Resolve a non - deterministic Turing machine by flipping to a coin to decide how branch . Remarks : I . It doesn't matter so much if the coin is fair but if , . heads) 't it should at least be reasonable number . p( 1/2 , a are . So our could L . Coin tosses always independent algorithm do all of them at the beginning . Equivalent to choosing a ) random bit and the bits (uniformly string , using one by one as needed . " " } . Access to coin tosses does not change computable . It " " thus might change efficiently computable , violating church - thesis extended Turing . 4. a counts as . flipping coin one time step A probabilistic algorithm / Turing machine for a counting induces for x a - distribution problem , any input , pr bodily on { on a } ? For a decision each x a problem , input yields probability = on Oil No . distribution { ) { Yes , } Informally : A decision problem should be considered efficiently a probabilistically solvable if there's poly . time Turing machine the that gets correct answer with high probability . a OLE L is in Fix constant LAK . A decision problem " " - time if there BPPE ( bounded error probabilistic polynomial ) exists a PTM T and a polynomial pflxl) such that when x T terminates in at most ) input , phxl steps, and : ' - - > T answers . I if - Yes , then l E G) Lex) Yes w/ prob L . ' - - > T answers . I - then E Iii if x No , No l ) ( ( ) w/ prob L . So E is of a , probability wrong answer. ' : for OLE LE cha Fact any , = , . BPP , BPPE " " Why? Amplification of probability . from BPPE E BP Per ⑧ buoy definition i , times BPPEZBPPE repeat (enough ) and use majority rule . : - Ta ke Define BPP . away =BPPqz : decision L Equivalent formulation BPP is all problems decidable by an NP TM such that at most 1/3 of the branches report the wrong answer. Variants : RP , PP : RP same as BPP if the answer is Yes , except the • , PTM always reports the correct answer . : = what we if set E 1/2 PP get we . - But beware of . BPPO P . ( ZPP) is in BPP Examples : Primality testing. vig Miller - Rabin test . Input : instant number N in binary) Question : Is N ? prime it was shown to be In fact , in P . " at moon nun.. ::: : : . :*: ÷ . - : Merlin Arthur probabilistic analog of NP. Has same definition as before we use a BPP , except machine when a is believable Tuning to decide witness . " " " Name Merlin - Arthur is supposed to invoke a game? Multi - round NP (but constant ) games generalize for MA) to polynomial hierarchy . Reversibkdassicalcomputi#II. interested in Boolean functions Classically , computing f : o {0,1}m→ { , 13h Of these are not all . Is there a course, bijection way ' " to encode f inside of a bijection ? " " " " Even better we do this and ? , can locally uniformly : First C5AT . Instance : Boolean circuit C Question : Is C satisfiable? Mgr Boolean circuit is something like this ? " "" ' " "" of wine : :÷÷÷÷w'i# ¥JrCj:÷, c : info 1¥. Dani Dor ft t int ' o l { 0,135 o l 4011 C is =/ satisfiable . ( ( 0,1 , ) , so If we have can rid a : crossings , get w/ swap * =¥¥. ( SAT is NP - complete . (can reduce from SATI. Size of a circuit is Off gates) . - Is there some NP complete analog of CSAT for " " reversible circuits? Fix a a set of , gates , which bijection " " : g { 0113 → { 0,13 the . where n may vary with gate We to can wire gates from G build planar reversible circuits . ! : R c- a. G. fifthn ! T width 5 Da IF a 4th tf : - k { 0,135 { 0,135. : Can we find - OI an NP complete problems for circuits with gate set G ? . Call it RSAT (G) - . : A- Depends on G . this Why is unclear ? " Note we can 't fix C- 0,1 and ask if , y { } C- there exists x such that - So, Ba Rft) y ? Why not? Because the answer is always Yes ! Let D= { ) and define RSA Sym ( 01133 , TCG) as follows : Input : Reversible circuit R of width 2k " : Does there exist x { oil such that Question , ye ) - - - Rex o - - ) O - O , , yo Cy , , , ) ? W - k k : This NP - Claim is complete ..