Quantum Computing and Its Applications

Jia Jie NgohAdvisor : Prof. William J. Song

22

Overview• Introduction• Background

• Quantum Bits and Gates• Qubit Measurement• Experiment Environment

• Applications: Quantum Fourier Transform• Conclusion

3

Introduction• It becomes more challenging to drive the performance growth of

computing systems in conventional approaches, i.e., Moore’s Law.

• Quantum computing draws growing attention as an alternative solution to overcome the physical limitations of traditional computing systems and methods.

“How do quantum algorithms compare to classical computational methods?”

4

Quantum Bits• A quantum bit (or qubit) represents a datum in quantum computing.

• As opposed to a classical bit that can only be 0 or 1, the qubit stores the superposition status of 0 and 1 as follows.

| ⟩# = %| ⟩0 + (| ⟩1

| ⟩# = *+, -2 | ⟩0 + /01,23 -2 | ⟩1Bloch Sphere representation of a qubit:

5

Quantum Gates

Types Unitary Transform Functions

X gate 0 11 0 Bit flipping

Hadamard gate121 11 −1 Superposition

Controlled-NOT (CNOT)1 00 1

0 00 0

0 00 0

0 11 0

Controlling target qubit

• Quantum gates are used to manipulate qubits.Examples of Commonly Used Quantum Gates

6

Qubit Measurement• The observation of a qubit results in logical bits, 0 and 1, despite

its quantum property, e.g., superposition.• The observed result must be either 0 or 1 with the probability of

! " or # ", respectively; ! " + # " = 1.• Reading qubits inevitably incurs measurement errors.

Types Description

Gate error Errors occur in quantum gates, due to noise interference, e.g., Pauli error, thermal relaxation.

Readout error Errors occur in the process of probing qubits to determine its logical state.

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Results of Measuring Qubit• There are two possible outcomes from measuring qubit, ideal

measurement (no errors) and real measurement (with errors).

0.0

0.2

0.4

0.6

0.8

1.0

1

Prob

abilitie

s

Output

0.0

0.2

0.4

0.6

0.8

1.0

0 1

Prob

abilitie

s

Output

(a) Ideal measurement (b) Real measurement

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Experiment Environment• IBM provides public-use of cloud-based quantum computing

through online platform called IBM Q Experience.• Quantum programs are coded using Python in Qiskit, an open-

source quantum computing framework and then executed in real quantum processors.

Quantum programming Program offloading

User QiskitQuantum ProcessorResults report

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Complexity of Quantum Fourier Transform• Quantum Fourier Transform (QFT) has the complexity of O(n2)

compared to O(n2n) of Fast Fourier Transform (FFT).• A function f(x) can be evaluated for many different values of x

simultaneously for QFT owing to quantum parallelism.

FFT

O(2#(∑%&'() *%. ,%)) = / 2# ∑%&'# 2= /(02#)

QFT

#(#1')2 operationsarerequired =O(n2)

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Derivation of QFT Quantum Circuit

| ⟩# → 12' (

)*+

,-./012,341)/,-| ⟩6 = | ⟩0 +2,34+.1-| ⟩1 | ⟩0 +2,34+.1-;<1-| ⟩1 … | ⟩0 +2,34+.1<1>…1-| ⟩1

2'

• QFT quantum circuit is built based on the equation below.

Quantum circuit representation of QFT algorithm

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Code of QFT

(a) QFT program (b) Error modeling in Qiskitdef input_state(circuit, q, n):

#n-qubit input state for QFT that produces output 1s string

for j in range(n):circuit.h(q[j])circuit.u1(math.pi/float(2**(j)), q[j])

def qft(circuit, q, n):#n-qubit Quantum Fourier Transform on

qubits in circuitfor j in range(n):

for k in range(j):circuit.cu1(math.pi/float(2**(j-

k)), q[j], q[k])circuit.h(q[j])

z = 0.995004165 + 1j * 0.099833417z = z / abs(z)u_error = np.array([[1, 0], [0, z]])simulator_config = {“noise_params”:

{‘gate_time’: 0.00545,‘p_depol’: 0.001,‘p_pauli’: [0, 0, 0.01],‘U_error’: u_error,‘readout_error’: 0.0620

}}

• The following codes create quantum program of QFT.

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Scalability of QFT – 9 qubits

• Below is the result of running QFT circuit for 9 qubits.

0.00.20.40.60.81.0

0010

11111

0011

11111

0110

11111

0111

10111

0111

11101

0111

11111

1001

11101

1010

11111

1011

11101

1011

11111

1101

01101

1101

10111

1101

11110

1110

01111

1110

10101

1110

11010

1110

11101

1111

00011

1111

00111

1111

01111

1111

10111

1111

11001

1111

11011

1111

11101

1111

11111

Prob

abilitie

s

Output

(a) Real measurement

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Quantum Circuit of 9 qubits QFT• Below is the generated quantum circuit of QFT for 9 qubits.• IBM Q Experience supports up to 32 qubits, but Qiskit GUI can

handle only a handful number of qubits.

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QFT Extension: Shor’s Algorithm

• Below is the generated quantum circuit of Shor’s Algorithm.

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Result of running Shor’s Algorithm

(a) Ideal measurement (b) Real measurement

• Below is the result of running Shor’s Algorithm for 5 qubits.

0.0

0.2

0.4

0.6

0.8

1.00000

0

0001

0

0010

0

0011

0

Prob

abilitie

s

Output

0.0

0.2

0.4

0.6

0.8

1.0

0000

0

0000

1

0001

0

0001

1

0010

0

0010

1

0011

0

0011

1

Prob

abilitie

s

Output

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Conclusion• Quantum computers alone are not sufficient to completely replace

traditional computing systems.

• However, they may significantly reduce computational complexity for particular algorithms (e.g., Quantum Fourier Transform) and hence can be used as accelerators, i.e., Quantum Accelerator.

• The current limitations of quantum processors lie in reliability, scalability, and programmability support.

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References1) "IBM Q Experience", Quantumexperience.ng.bluemix.net, 2018.

[Online]. Available: https://quantumexperience.ng.bluemix.net/qx/experience. [Accessed: 11- Oct- 2018].

2) M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th ed. Cambridge University Press, 2010.

3) Monz, T., Nigg, D., Martinez, E., Brandl, M., Schindler, P., Rines, R., Wang, S., Chuang, I. and Blatt, R. (2016). Realization of a scalable Shor algorithm. Science, 351(6277), pp.1068-1070.