Ii : Meeting G. Classical warm. ups to

: 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 cases .

The won It of classical analogs address 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 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 . 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 , 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 . 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 - .

(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 . ! : 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 .