Quantum Computationand the Bloch Sphere

Fred Wellstood

Joint Quantum Instituteand

Center for Nanophysics and Advanced Materials Department of Physics

University of Maryland, College Park, MD

(April 10, 2009)

Outline

1. Two key properties of quantum systems:Superposition and Entanglement

2. Brief introduction to quantum computing

3. One qubit quantum computing - 1-qubit states and the Bloch sphere- counting 2-qubit states- 1-qubit operations- Rabi oscillations

4. Controlled Not (CNOT) - a 2-qubit logic operation

5. Conclusions

The Principle of Superposition

Suppose |0> and |1> are two allowed quantum states of a system, then the system can exist in any linear superposition of these states where and are complex numbers 1|0|

But you don’t see such states in everyday objects: "Schrodinger's cat paradox" (Schrodinger, 1935)

if this was observable in a macroscopic object

+

live dead

"macroscopic quantum superposition"

Two key properties of quantum systems

0|1|

Quantum Entanglement (Schrodinger, 1935)

Multiple quantum systems can exist in entangled super-position states in which the state of an individual system has no well-defined physical meaning.

ba 1|0|

baba 0|1|1|0|2

1

"a" is in |0> and "b" is in |1>. This is a "separable state" or "product state". Both systems are in well-defined state: not entangled.

Example: suppose

State cannot be written as (some state of system "a" times some state of system "b"). "b" is not in a well-defined state of its own, but depends on "a". If you measure this system and find "a" is 0, then "b" will be 1. If you find "a" is 1, then "b" will be 0. You will never find a and b are both 1. You can't say "sometimes it is in 01 and sometimes 10". Its in both 01 and 10. entangled

ba

Example: suppose

Two key properties of quantum systems

if this were the state of two macroscopic objects

+

and(dead, live) (live, dead)

Key fact: Superposition and entanglement are unobservable in everyday "macroscopic" objects due to their interaction with other degrees of freedom and the external world. Energy dissipation and "noise" causes transitions between states. "Dephasing" causes the relative phase between the terms in the state to change. The more objects are entangled or superposed, the faster this "decoherence" tends to happen. These strange states are what make a quantum computer powerful!

baba 0|1|1|0|2

1Quantum Entanglement

A quantum computer: a computer that uses coherent quantum mechanical properties of multi-particle systems to do calculations.

A quantum computer would be built from quantum bits or "qubits".

A "qubit" is just a quantum system with two energy levels ("two-level system"). You can call the levels | > and | >. Or |0> and |1>.

The first step in operating the computer would be to prepare all N~1000 or more qubits in a well defined state, such as:

|0>|0>|0>|0>.......... |0>|0>this is like clearing the memory of a classical computer.

Next perform a mathematical or logical operation by applying a time-dependent perturbation that drives the system into a new state according to:

Brief Introduction to Quantum Computing

)(1 tHHt

i o

)0(

)(1 tH

Note: the time-evolution of the state is completely deterministic! With the right perturbation you could flip the 3rd qubit from 0 to 1.

Apply perturbations H2(t), H3(t) ... Hm(t) until the m steps of the calculation are complete. Of course you will need to do some work beforehand to figure out what H' corresponds to what mathematical or logical operation.

The system is left in a well-defined state ... but it is typically a superposition of classical (0&1) states.

The state of each qubit is then measured, producing the result of the calculation: a string of 0's and 1's. This corresponds to selecting one outcome from the superposition... a non-deterministic step... which means you might not get the answer your looking for!

The entire calculation may then have to be repeated from the start if another possible outcome is needed.

Key Question: Can a useful quantum computer be built in practice?Answer: Definitely maybe.

So what's the big deal.... how can you get anything useful out of this? A quantum computer can access superposition states and entangled states... this is a huge set of accessible states that lets the computer do some things much faster than a classical computer:

- find the factors of large numbers quickly and break RSA encrypted messages (Shor's algorithm, exponentially faster) - simulate other quantum systems, search databases

Main Experimental Challenge: Noise and interactions with other quantum systems (the outside world) eventually destroys delicate quantum superposition states. This is called decoherence.

nA classical computer with an n-bit memory can access 2 states. Example: for n=2 bits the 22 = 4 states are 00, 01, 10 and 11.

A quantum computer can access superposition statesand entangled states. With n qubits, this gives of order states.2

2n

Consider one qubit with energy eigenstates |0> and |1>.

We will need to be able to put it into superposition states:

Quantum Computing with one qubit

1|0|

- state must be normalized to unity so 122

- probability amplitudes and can be complex numbers

- then define )2/cos( )2/sin( ie

1)2/(sin)2/(cos

)2/sin()2/cos(

22

2222

ie

- an overall phase factor has no effect, so we can choose to be real

1|)2/sin(0|)2/cos(1|0| ie

can always write a superposition state in the form:

|0>

|1>

z

y

x

Superposition States are Points on the Bloch Sphere

sphere with radius R=1 …..this is the “Bloch Sphere”

1|

2sin0|

2cos

ie

|0>

z

y

x

0|

1|2

0sin0|

2

0cos

1|2

sin0|2

cos

i

i

e

e

0

Example: = 0

Superposition States as Points on the Unit Sphere

|0>

z

y

x

1|

1|2

sin0|2

cos

1|2

sin0|2

cos

0

i

i

e

e

Example: =

|1>

Superposition States as Points on the Unit Sphere

|0>

z

y

x

2/

Example: = /2, = 0

2

1|0|

1|4

sin0|4

cos

1|2

sin0|2

cos

0

i

i

e

e

2

1|0|

|1>

Superposition States as Points on the Unit Sphere

Superposition States as Points on the Unit Sphere

|0>

z

y

x

2/

Example: = /2, = /2

2

1|0|

1|4

sin0|4

cos

1|2

sin0|2

cos

2

i

e

e

i

i

2

1|0|

|1>

2/

2

1|0| i

2

1 0 x

11 0o

2

1 i 0 y

2

1 i 0 y

2

1 0 x

2

1 0

2

1 i 0

There are an infinite number of states on the Bloch sphere, ..... but we can choose a "digital" subset for computing

note: one classical bit has 21

possible states (0 and 1).

One of these qubits has of order ~22 accessible states1

Example: Number of states for n= 2 qubits

2

1 0

2

1 0 ixy

1111

00oo

2

1 00xo

011 o

101 o

2

1 00xo

2

1 i 00yo

2

1 00

iyo

2

1 011 x

2

1 011 x

2

1 0

2

1 0xx

2

1 011

iy

2

1 011

iy

2

1 0

2

1 0 iyx

02

1 00

x 1

2

1 01

x

2

1 0

2

1 0xx

+18 more states with -x, +y and -y in first index = 36 product states

2

11 001

e

But that's not all....... there are also entangled states (can’t be written as product) such as:

and many more such entangled combinations, for example

2

11 003

ie

2

11 002

e

2

11 i 004

e

There are way more 2-qubit states (more than 40)than 2 bit states (just 4)....

The total number of such quantum states rises super-exponentially with the number of qubits

1 12

10100

2

1

2

1

2

110 xx

Consider again just one qubit. There are just a few operations needed to go from one state to any other.

Example: Phase gate: |0> |0> |1> ei|1>

|0> +|1> |0> +ei1>

Example: NOT operation: |0> |1> |1> |0>

|0> +|1> |1> +|0>

Quantum Computing with one qubit

Example: operation:

NOT

1|0|2

1|0>

1|0|2

1|1>

All operations must work on superposition states!

|0>

z

y

x

0|

1|2

0sin0|

2

0cos

1|2

sin0|2

cos

i

i

e

e

Example: = 0

rotate about the x-axis by Such a rotation would also change |1> to |0>.NOT, or "x-pulse" or H'=x

Starting from |0>

|1>

Single qubit NOT operation as rotation on the Bloch sphere

|0>

z

y

x

2/

Example: = /2, = 0

2

1|0|

1|4

sin0|4

cos

1|2

sin0|2

cos

0

i

i

e

e

2

1|0|

|1>

starting from |0>rotate about the y-axis by A -pulse, or y, orTwo such rotations would produce a NOT

NOT

Single qubit operation as rotation on the Bloch sphereNOT

|0>

z

y

x

Example: = /2, = /2

2

1|0|

1|4

sin0|4

cos

1|2

sin0|2

cos

2

i

e

e

i

i

2

1|0|

|1>

2/

rotate about the z-axis by will increase phase of any state by /2. This is a“/2 phase gate” or z

2

1|0| i

Single qubit control operations as rotations on the Bloch sphere

- Basic Idea: Use a Rabi Oscillation- Consider a 2-level system with energy splitting E. - Apply power (a perturbation) continuously at frequency f = E/h.

0

1

E=hf

Start in 0 and Apply power for short time --> Small amplitude to be in 1

0

1

E=hf

Keep applying power --> eventually system pumped entirely into 1 (NOT gate or -pulse)

0

1

E=hf

Keep applying power --> system pumped back down to 0(stimulated emission)

System cycles back and forth between 0 and 1 deterministically at well-defined rate (Rabi frequency ) set by power and tuning. Stopping power at appropriate time can produce NOT or

But how can you get the state to rotate on the Bloch sphere?

NOT

Two-level System Dynamics

State of a system described by wavefunction that satisfies time-dependent Schrodinger’s Equation

H

ti

For a two-level system with Hamiltonian Ho that is being driven at frequency with a perturbing energy H’, we can write H in matrix form as:

11 )cos(

)cos(

0)cos(*

)cos(0

0

0'

EtV

tVE

tV

tV

E

EHHH oo

o

where: Eo = energy of ground state, E1 = energy of excited stateV cos(t) = <0|H’|1>

and where:

0

10|

1

01|

|

Two-level System Dynamics

Plug into Schrodinger’s Equation:

H

ti

1)cos(

)cos(

EtV

tVE

ti

ti

o

Write as two coupled equations:

1)cos(

)cos(

EtVt

i

tVEt

i o

Fairly nasty…guess solution of form: (this will always work!)

tE

i

tE

i

etB

etAo

1

)(

)(

Plug into Schrodinger’s Equation

notice that this says that the amplitude to be found in |1> will change based on amplitude to be in |0>

tE

itE

itE

i

tE

itE

i

o

tE

i

etBEetAtVetBt

i

etBtVetAEetAt

i

o

oo

11

1

)()()cos()(

)()cos()()(

1

tE

itE

i

o

tE

i

o

tE

i

etBtVetAEetAEt

tAei

ooo

1

)()cos()()()(

For the first equation, we find:

Clean things up:t

EEi o

etBtVt

tAi

1

)()cos()(

For simplicity, let’s assume we are on resonance ( )

tiettVBt

tAi

)cos()()(

oEE 1expand this term

titititi

ti

etBV

eee

tVB

ettVBt

tAi

21)(22

)(

)cos()()(

This term is changing very rapidly and is far from resonance at …so it can be dropped…. “rotating wave approximation”

)(2

)(

)(2

)(

tAi

V

t

tB

tBi

V

t

tA

)(2

)(2

2

2

tAV

t

tA

Assuming A(0) = 1, solution is:

2cos)( ttA

V

is the Rabi frequency

take another time derivative of the 1st equation and use 2nd to eliminate dB/dt

2sin)( titB

1|2sin0|2cos

2sin

2cos

)(

)(|

1

11

tE

itE

i

tE

i

tE

i

tE

i

tE

i

etiet

eti

et

etB

etA

o

oo

Take out an overall phase factor of tiEoexp

1|2sin0|2cos|

01 tEE

i

etit

prob

abil

ty

0

1

t

P0=||2

P1=||2

/2 /40

1|

2sin0|

2cos|

ie

Also notice this is now in the familiar “polar coordinate” form:

where andt2

01

tEE

1|2sin0|2cos

2sin

2cos

)(

)(|

1

11

tE

itE

i

tE

i

tE

i

tE

i

tE

i

etiet

eti

et

etB

etA

o

oo

Take out an overall phase factor of tiEoexp

1|2sin0|2cos|

01 tEE

i

etit

Rabi Oscillation on the Bloch Sphere

|0>

|1>

z

y

x 21|0|

21|0| i

d/dt 01

t

To make NOT gate, just drive system at resonance for total time of t =

All that stuff was 1 qubit operations. To be useful for computation, you need many qubits.

In particular, to do logic, you need to be able to control the state of one qubit based on the state of another. Look at what happens with 2.

input state output

state

|0,0> |0,0>

|0,1> |0,1>

|1,0> |1,1>

|1,1> |1,0>

Example, CNOT operation on |1,1> + |0,1> + |1,0>

yields: |1,0> + |0,1> + |1,1>

Two Qubit Operations

Controlled NOT or CNOT:Reversible two-qubit operation that flips the second qubit state if and only if the first qubit state is 1.

input state output

state

|0,0> |0,0>

|0,1> |0,1>

|1,0> |1,1>

|1,1> |1,0>

Notice that it just exchanges 10 and 11

.... Like a -pulse between 10 and 11

Implementing a CNOT in a 2-qubit system

|0,0>

|0,1> |1,0>

|1,1>Somehow arrange Hamiltonian so that 10 and 01 energy levels have splitting hfo..... different than splittings to 11 and 00.

Drive a Rabi oscillation (-pulse) at 01-10 resonant frequency fo.

01 and 10 flip, while 11 and 00 are unchanged

E=hfo

Conclusions

Given you a brief introduction to quantum computing. Used 1 and 2 qubit systems for simplicity

There is much more to this than covered here... including examples of physical systems that are being used as qubits ... and the many experiments that have been done.

Although there still is not a viable quantum computer in existence this is a very active area in theory and experiment.