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Applications of Quantum Computers - Quantum Computing - Lecture Slides, Slides of Computer Science

Acharya Nagarjuna UniversityComputer Science

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: Applications of Quantum Computers, Classical Computers, Quantum Computers, Significantly Faster, Factorization Problems, Exponential, Classical Computers, Non Polynomial Problems, Unstructured Search, Circuit

Typology: Slides

2012/2013

Uploaded on 03/23/2013

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Applications of Quantum Computers
•Classical computers today are fast.
•However, in some cases, quantum computers
are significantly faster.
–For example, Shor’s algorithm can solve semiprime
factorization problems with exponential speedups
over classical computers.
–Grover’s algorithm can achieve polynomial
speedups in large, non-polynomial problems using
unstructured search.
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Download Applications of Quantum Computers - Quantum Computing - Lecture Slides and more Slides Computer Science in PDF only on Docsity!

Applications of Quantum Computers

  • Classical computers today are fast.
  • However, in some cases, quantum computers

are significantly faster.

  • For example, Shor’s algorithm can solve semiprime factorization problems with exponential speedups over classical computers.
  • Grover’s algorithm can achieve polynomial speedups in large, non-polynomial problems using unstructured search.

Superposition and Quantum Computers

  • A classical bit is represented by a

classical entity, like a current of

electrons

  • Thus, it is confined to two

discrete states, “0” and “1”

  • It is relatively easy to determine

its state.

Superposition and Search

  • We can use the Hadamard transform to create an even superposition between the |0> and |1> state in one qubit.
  • If I perform a collective Hadamard transform to a system of two qubits, the system as a whole can represent |00>, |01>, |10>, and |11>.
  • For this reason, quantum bits store more information than classical bits – I can represent 2ⁿ classical states with n qubits

Quantum Unstructured Search (“Grover”)

  • Grover’s Algorithm uses this property of quantum information to perform an unstructured search more quickly
  • The initial input qubits are superposed to represent all possible solutions
  • The Oracle operation tags the phase of the solution states in this superposition
  • Another circuit then changes the phase information (which is hidden) into amplitude information (which we can detect).
  • This process is iterated √N times (as opposed to N iterations in classical logic) to maximally amplify the states.

H

H

H

H

ORACLE

H

H H Grover Loop (* √N)

Applications of Grover Algorithm

  • Grover’s algorithm can provide quadratic speedup in NP Complete problems:
  • Examples of NP Complete problems are:
    • Graph Coloring
    • Maximum Clique
    • Satisfiability
    • Travelling Salesman
    • DNA Sequencing
    • Scheduling
    • Sudoku

THE ORACLE

The Oracle

  • Since the oracle operation is iterated √N times, a decrease in cost of one basic gate for the oracle would decrease the cost of the entire Grover loop by many more gates.
  • The number of input qubits is also important, because the Grover Circuit must be iterated 2⁽n/2⁾ times.
  • For example, the Grover Circuit for SEND MORE MONEY costs 17 thousand trillion trillion more basic gates with the less efficient method

Graph Coloring

  • Graph coloring is an NP complete problem
  • It involves finding a “good” coloration for a system of n nodes connected by e edges
  • No two nodes connected by an edge can have the same color

1

3

2

V V2 V V V5 V V V8 V V V4 V V V5 V V V

One color / node

Good Coloring

Hogg’s Method

  • In this method, each assignment of a color to a node is represented by a qubit, v1-v9.
  • No two elements of a row can coexist, because only one color can be assigned to a node. I use NAND gates to ensure this
  • If, for example, nodes one and two are connected, we must do: v1 NAND v4, v NAND v5, and v3 NAND v6.

Reversible Logic and Quantum Cost

  • An alternative classical logic implementation is called AND – EXOR logic.
  • It is reversible because you can determine inputs from the outputs.
  • This kind of logic is easier to simulate with most quantum technologies.

X

Y

X

X Y

Y Z

Y Y Z

X X

X

Z Z '

Quantum Cost

Gates Cost in Basic Gates

Quantum NOT 1 Hadamard Gate 1 CNOT Gate 1 3 input Toffoli Gate 5

N input Toffoli Gate

Best Case: 32n- Worst Case: (2n+1) -

N-Bit Toffoli Cost

  • I have used two estimates to calculate the cost of a

toffoli gate.

  • 32m – 96, plus one garbage bit, for m > 5
  • 2⁽m+1⁾ – 3 , where m is the number of controlling bits
  • The difference between these costs should be

underscored.

  • For example, one technique for building the SEND

MORE MONEY oracle costs about 100,000 basic

gates with the best case method and over a googol

with the worst case method.

COST DERIVATION FOR GRAPH

COLORING

Graph Coloring

  • Graph coloring is an NP complete problem
  • It involves finding a “good” coloration for a system of n nodes connected by e edges
  • No two nodes connected by an edge can have the same color

1

3

2