Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
These are the Lecture Slides of Quantum Computing which includes Classical Computers, Quantum Computers, Significantly Faster, Factorization Problems, Exponential, Classical Computers, Non Polynomial Problems, Unstructured Search, Circuit Level Representation etc. Key important points are: Quantum Computation, Circuits and Algorithms, Quantum Circuit Model, Alternate Model, Upon Measurement, Mean to Compute, Very Hard, Exploiting Quantum, Efficiently Simulatable, Violation of Strong
Typology: Slides
1 / 22
This page cannot be seen from the preview
Don't miss anything!
On special offer
Michael A. Nielsen
What does it
mean to compute?
The Church-Turing-Deutsch principle
Models of quantum computation
Quantum circuit model
Classical Quantum
Single-qubit quantum logic gates
P 0 = 0 ; P 1 =i 1
P i
2
Toffoli gate
c 1
t t ⊕ c 1 ⋅ c 2
Control qubit 1 c 1
Target qubit
Control qubit 2 c 2 c 2
Now we are using Dirac notation to be used to complex problems
x
y
x
y
How to compute classical functions
on quantum computers
(^0) x
x
x (^) f f x( )
x
0
⊗m g x
f x ( )
U f
Example of using
symbolic Dirac
Notation:
Deutsch’s problem
Example: Deutsch’s problem
Given a black box computing a function f : (^) { 0 ,1} →{ 0,1}
Our task is to determine wheth er f is constant or balanced?
x
z (^) z ⊕f x( )
x
f
x
z (^) z ⊕f x( )
x
U f
Quantum algorithm for Deutsch’s problem
0
0 1
2
−^ f
H H
( ) ( )
(0) (1) 1 0 1 1
f f → − + −
(0) (1) 1 0 1 + 1 0 1
f f → − + − −
( ) ( ) ( ) ( )
(0) (1) (0) (1) 1 1 0 + 1 1 1
f f f f = ^ − + − ^ ^ − − −
Summary of the quantum circuit model
Inp u :t An n-bit string, x, representing an instance of some problem.
Examp le: x is a number to be factored.
Init ial state : 0 , where is some computable function of.
m m n
⊗
A circuit of single-qubit and controlled-not gates is applied to the qubits. The sequence of gates applied is under the control of an external classical computer, and may depend
upon the
Circ
pr
uit:
oblem instance x.
Some fixed subset of the qubits is measured in the
computational basis at the end of the computation, and the output constitutes the solution to the p
Reado
rob
ut:
lem.
For a decision problem, just the first qubit would be
read out, to indicate "yes" o
Examp
r "
le:
no".
QP : The class of decision problems solvable by a quantum circuit
of polynomial size, with polynomial classical overhead.
Quantum complexity classes
What is known: BPP ⊆ BQP ⊆ PSPACE
