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 .