Quantum ComputingPreethika Kumar

“Classical” Computing: MOSFET

CMOS Limitations (Wave-Particle Duality)

In the Quantum World…..

• Bits become qubits: “0”, “1” or “both”

• Unitary matrices become quantum gates:We have a universal set of gates

2 21 00 1 ; 1

0 1

Probability of measuring|1

Probability of measuring|0

IJunction

I

0 1

Schrödinger EquationNewton’s Law of the Little World

( )( )

ti t

t

H

2( ) e (0)i t

Gate

t H

U

Hamiltonian:2n 2n non-diagonal matrix

Typical Quantum Circuit

X

H

Z

H

S

0 1

1 0

0 1

1 0

U

1 11

1 12

0 10

2

0 11

2

U1 0 0 0

0 1 0 0

0 0 0 1

0 0 1 0

0 0

1 1

x x

x x

U

Single Qubit Hamiltonian

1H

bias tunneling

ti

tti

ti

ti

t

22

22

2222

22

22

22

22

22

22

sincossin

sinsincos

U

cos 2 sin 2

sin 2 cos 2

t i t

i t t

U

2

2

cos 2 sin 2 0 0

0 cos 2 sin 2 0

i t

i t

t i t e

t i t e

U

Single Qubit Hamiltonian

1H

bias tunneling

cos 2 sin 2 0 cos 2 0 sin 2 1;

sin 2 cos 2 1 sin 2 0 cos 2 1

t i t t i t

i t t i t t

U

22 2

2

0; 0 0 ; 1 1

0

i ti t i t

i t

ee e

e

U

2

0| ; 0 1 ; 1 0

0t

iNOT X i i

i

Two Qubit System

IBA2 HHHH A B

0

0

0

0

A B B A

B A B A

A A B B

A B A B

2H

|00 |01 |10 |11

00|

01|

10|

11|

2( ) e (0)i t

Gate

t H

U

Some Potential Challenges

• No Cloning Theorem - moving quantum data (fan-out)- quantum error correction (redundancy)

• Measurements collapse quantum states- closed quantum systems (coupling with environment)- quantum error correction (syndromes)

• Architectural layouts: limited interactions- gate operations - moving quantum data

IBA2 HHHH A B

0

0

0

0

A B B A

B A B A

A A B B

A B A B

2H

|00 |01 |10 |11

00|

01|

10|

11|

Goal: Find system parameters (mathematical solution)Constraints:

- Minimize control circuitry (closed system)- Fixed system parameters (design)

Research: Quantum Gates(Reducing the Hamiltonian)

Research: “Controlled” Gates(Reducing the Hamiltonian)

Fix A’s state – large A – can neglect effect of A.

BAB

BBA

BAB

BBA

H

00

00

00

00

|00 |01 |10 |11

00|

01|

10|

11|

BB

BBBH 1

BB

BBBH 2

Similar to

H

BABA

BBAA

ABAB

ABBA

H

0

0

0

0

|00 |01 |10 |11

00|

01|

10|

11|

Controlled- Hadamard Gate

11

11

2

1H

Barenco, et al., PRA 52, 3457 (1995)

Bias Pulse on Target

Time

T

emax

emin

Parameters : T = 7 ns = 25 MHz = 35.9 MHz min = 60.9 MHz max = 10.0 GHz

Research: Gates in Linear Nearest Neighbor Architectures (LNNA)

Want to do gate operations on qubit B

A B1 11 2 2

C

|0 B1 11 2 2

Method 1: Fix adjacent qubits (A and C) in the |0 state

|0

Method 2: Shut off the couplings (of qubit B with A and C)

A B11 2

C

A

B

C

A

B

C

=

Pulse 1 Pulse 2

A

B

C

U U U U

A

B

C

U U

A

B

C

U

2cos

2exp

2sin

2exp

2sin

2exp

2cos

2exp

ii

ii

U

Research: Gates in LNNA

1 2

1 2

1 2

1 2

1 2

1 2

1 2

1 2

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

B B

B B

B B

B B

B B

B B

B B

B B

Research: Gates in LNNAA B

1 2C

A = 0 A

= B = ?B =

C = 0 C

=

Approach will be used to implement controlled-unitary operations

000 001 010 011 100 101 110 111

Research: Mirror Inverse Operations

Research: Mirror Inverse Operations

Research: Mirror Inverse Operations

Research: Mirror Inverse Operations

Current Research

• Adiabatic Quantum Computing- optimization problems- hardware exists (DWave Systems)

• Quantum Neural Networks- designing QNNs (exploit quantum phenomena)- using QNNs for different applications to calculate parameters

• Fault-tolerant Quantum Computing- gate design without decoding