Introduction to quantum computation

Collaboration:University of Illinois

Angbo Fang, Gefei Qian (Phys) theoretical modelingJohn Tucker (ECE) design of test structures

Milton Feng (ECE) semiconductor processing Utah State University

T.-C. Shen (Phys) STM donor patterning University of Utah

Rui Du (Phys) low-T measurements

YIA-CHUNG CHANG ( 張亞中 )Research Center for Applied Sciences (RCAS)Academia Sinica, Taiwan ( 中研院 應用科學中心 )

Quantum Algorithm vs. Classical Algorithm

Input: a single number a coherent superposition of many numbers

Register: bit: 0, 1 qubit: 0 =, 1= or (a 0+b1)Processing: sequential massive parallel

Time for solving QM: exponential linear

e.g., One qubit operation, H: 0 (0 + 1)/√2 (Hadamard Transform)

Do this on two qubits 0 0 (0 + 1)(0 + 1)/ 2 =(00 + 01+10 + 11)/ 2

The input now has 4 different binary numbers.

Similarly, perform the H-transform on N qubits can generate 2N different binary numbers

Applications: cryptography, data-base searching, teleportation,…etc.

Classical Quantum

Information Unit Bit: 0 or 1 Qubit: 0+ 1Single-Bit

NOT Gate

NOT: 0 1

1 0

22 Unitary Operation

0+ 1 0+ 1

Two-bit

XOR Gate

a, b a, ba

00 00

01 01

10 11

11 10

A, B A, BA

Measurement Result: 0 or 1

100% certainty!

0 with ||2 probability

1 with ||2 probability

0100

1000

0010

0001

CNOTU

Classical v. s. Quantum Computation

Any unitary operation on n qubits may be implemented exactly by composing single qubit and CNOT gates.

The DiVincenzo Criteria

1. A scalable physical system with well-characterized qubits.2. The ability to initialize the state of the qubits to a simple fiduci

al state, such as .3. Long relevant decoherence times, much longer than the gate

operation time.4. A “universal” set of quantum gates.5. A qubit-specific measurement capability.6. The ability to interconvert stationary and flying qubits.7. The ability to faithfully transmit flying qubits between specified

locations.

000

To build a workable large-scale quantum computer, we need

Principle of the SET transistor

Like a MOSFET, the single-electron tunnelling (SET) transistor consists of a gate electrode that electrostatically influences electrons travelling between the source and drain electrodes. However, the electrons in the SET transistor need to cross two tunnel junctions that form an isolated conducting electrode called the island. Electrons passing through the island charge and discharge it, and the relative energies of systems containing 0 or 1 extra electrons depends on the gate voltage.

An electron in a box

(a) When a capacitor is charged through a resistor, the charge on the capacitor is proportional to the applied voltage and shows no sign of quantization. (b) When a tunnel junction replaces the resistor, a conducting island is formed between the junction and the capacitor plate. In this case the average charge on the island increases in steps as the voltage is increased (c). The steps are sharper for more resistive barriers and at lower temperatures.

Advantages of Quantum Algorithms

Complexity Classical Quantum

Factoring an n-bit number

Searching M solutions out of N possibilities

])(lnexp[ 3/23/1 ncn

MN

(Shor))ln)(ln(ln2 nnn

(Grove)MN

Basic Procedure of Quantum Computation

1. Prepare an appropriate initial state in N-qubit space 2. Implement the desired quantum algorithm via a series of element

ary gate operations3. Measure the final state in an appropriate basis and extract the so

lution from measurement result by some simple classical computation