Quantum Processing Simulation

Dalibor Hrg

Vienna, June 18, 2004.

Moore’s law, classical computers

Technology and computation

NANOTECHNOLOGY

Development Impact on big mathematical questions (P=PSPACE,

P=NP), theoretical research! We still don’t know if quantum computers are

stronger than classical computers! von Neumann architecture? Quantum memory is

needed! (in progress) “Quantum cryptography” is demonstrated!

(problem with error corection codes and speed) “Quantum teleportation”, Quantum communication

methods (demonstrated, in progress)

Classical and Quantum computer

State of classical computer

of quantum computer

2x1x nx {0,1}j Bx bits qubits

in 2 bits: 2 1011 3

1 2, ,..., nx x x {0,1}j Bx

in 2 qubits: 1 1 1 1

2 2 2 200 01 10 11

2

1, 2,..., 1 2, ,..., , 1x x xn xnC Cx x x

Transformation of statesclassical computer quantum computer

Boolean circuits:

n nB B Unitary operators:

2 2n n

C C(Quantum circuits)

Classical and Quantum algorithms

Algorithms

?

classical

(C,C++,C#,… )

A problem

Asembler

Machine code

Boolean circuits

Pseudo code

quantum

Mathematicalmodel

Quantum circuits (?)

Grover, ShorDeutsch-Jozsa, Simon

EASY

HARD

Quantum algorithms Grover’s algorithm (1997.) - searching unsorted database of N elements in

steps - on classical computer, steps are needed - if sorted, there exist classical algorithm with steps Deutsch-Jozsa problem (1992.) - finding global property of some Boolean function with N variables (function is constant or balanced) - complexity of quantum algorithm - complexity of classical algorithm

( )O N( )O N

(log )O N

( )O N2( )NO

Grover’s algorithm

Deutsch-Jozsa problem

3 0n

3

( )( 1) 0 1

2 2n

f x z x

z x

z

0n

z

Function constant if:

Function balanced if:

3 0n

( )( 1)

2n

f x

x

For state amplitude is

QPS Application

Quantum register. State of a qubit is colored: (blue, state is 0), (red, state is 1), (green, superposition of 0 and 1).

All states of register are seen here!

Quantum gates (unitary operators). Act on selected qubits of quantum register.

Characteristics of the QPS

Windows application, C#, .NET Framework 1.1 Grover’s and Deutsch-Jozsa algorithm simulation

(up to 8 qubits). Implementation of the most useful operators (H,

Pauli X, Pauli Z, Oracle, WH, Grover). Easy to use interface (selecting qubits and

operators) For education and further research on quantum

algorithms (handy tool).

Simulation problems? Number of qubits: N Number of states in register: Needed memory for all states: Needed memory for Walsh-Hadamard (interference) operation:

Number of qubits Number of statesMemory for all

statesMemory for Walsh-

Hadamard operation

4 16 0.18 KB 2 KB

8 256 4 KB 0.5 MB

12 1024 80 KB 128 MB

16 65536 1.5 MB 32 GB

32 4294967296 160 GB 13.7 10 GB

64 1.84 10 1.23 10 GB 31.2 10 GB19 12

10

28

2N

2 ( ) 2 (8 ) BN Ndouble char N N 22 2 2 8 BN N Ndouble

Conclusion Quantum algorithms can be simulated, but

inefficiantly (memory used and time needed) on classical computers.

Impossibility to implement quantum parallelism is a main reason for inefficient simulation ( > 10 qubits on classical PC, 256-512 MB RAM).

QPS is useful in education and research (handy tool).