Download JQI Fellows - University of Maryland, College Park

Quantum Computation
and the Bloch Sphere
Fred Wellstood
Joint Quantum Institute
and
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
Two key properties of quantum systems
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 a and b are complex numbers
  a | 0   b |1 
"macroscopic quantum superposition"
if this was
observable in
a macroscopic
object
+
live
dead
|1 
|0
But you don’t see such states in everyday objects:
"Schrodinger's cat paradox" (Schrodinger, 1935)
Two key properties of quantum systems
Quantum Entanglement (Schrodinger, 1935)
Multiple quantum systems can exist in entangled super-position
states in which the state of an individual system has no welldefined physical meaning.
Example: suppose
  | 0  a | 1 b
"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.
1
Example: suppose    | 0 a | 1 b  | 1 a | 0 b 
2
State cannot be written as  a  b (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
Quantum Entanglement
1
   | 0  a | 1 b  | 1  a | 0 b 
2
if this were the
state of two
macroscopic
objects
+
(dead, live)
and
(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!
Brief Introduction to Quantum Computing
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>|0>
this is like clearing the memory of a classical computer.
Next perform a mathematical or logical operation by applying a timedependent perturbation H1 (t ) that drives the system into a new state
according to: i   H   H (t )
t
o
1
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.
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
n
A 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.
n
A quantum computer can access superposition states
2
and entangled states. With n qubits, this gives of order 2 states.
Key Question: Can a useful quantum computer be built in practice?
Answer: Definitely maybe.
Main Experimental Challenge: Noise and interactions with other
quantum systems (the outside world) eventually destroys delicate
quantum superposition states. This is called decoherence.
Quantum Computing with one qubit
Consider one qubit with energy eigenstates |0> and |1>.
We will need to be able to put it into superposition states:
  a | 0   b |1 
- probability amplitudes a and b can be complex numbers
- state must be normalized to unity so a  b
2
2
1
- an overall phase factor has no effect, so we can choose a to be real
- then define a  cos( / 2)
b  ei sin(  / 2)
i
a  b  cos( / 2)  e sin(  / 2)
2
2
2
2
 cos 2 ( / 2)  sin 2 ( / 2)  1
can always write a superposition state in the form:
  a | 0   b | 1   cos( / 2) | 0   ei sin(  / 2) | 1 
Superposition States are Points on the Bloch Sphere
z
|0>

 
 
i
  cos  | 0  e sin   | 1 
2
2
y

x
|1>
sphere with
radius R=1
…..this is the
“Bloch Sphere”
Superposition States as Points on the Unit Sphere
Example:  = 0
z
|0>
 0
x
 
 
  cos  | 0   e i sin   | 1 
2
2
0
0
 cos  | 0  e i sin   | 1 
2
2
| 0 
y
Superposition States as Points on the Unit Sphere
Example:  = ,   0
z
|0>
 
 
 
  cos  | 0   e i sin   | 1 
2
2
 
 
 cos  | 0  e i 0 sin   | 1 
2
2
|1 
y
x
|1>
Superposition States as Points on the Unit Sphere
Example:  = /2,  = 0
z
|0>
| 0   |1 
2
  /2
x
|1>
 
 
i
  cos  | 0  e sin   | 1 
2
2
 
 
i0
 cos  | 0  e sin   | 1 
4
4
| 0   |1 

2
y
Superposition States as Points on the Unit Sphere
Example:  = /2,  = /2
z
|0>
  /2
| 0   |1 
2
 
 
  cos  | 0  ei sin   | 1 
2
2

i
 
 
2
 cos  | 0  e sin   | 1 
4
4
| 0  i | 1 

2
| 0  i | 1 
2
  /2
x
|1>
y
There are an infinite number of states on the Bloch sphere,
..... but we can choose a "digital" subset for computing
o  0
x 
1  1
0 1
2
 x 
0 1
2
0 1
0 i 1
2
2
y 
0 i 1
2
 y 
0 i 1
2
note: one classical bit has 21
possible states (0 and 1).
One of these qubits has of
1
2
order ~2 accessible states
Example: Number of states for n= 2 qubits
 o1  0 1
 oo  0 0
 1o  1 0
 x0
 0 1
 
2

 0 1 

2 

 o x  0 
 11  1 1

0


 0 1
 x1  
2

 xy
 o y
 0 i 1 

 0 
2 

 1 x
 0 1
 1 
2

 1 y
 0 i 1 

 1 
2 


1


 x x
 0 1
 
2

 0 1
 
2





 0 1 


2


 o x  0 
 o y
 1 x
 0 1
 1 
2

 1 y
 0 i 1 

 1 
2 

 0  1


2

 0  i 1 



2 

 0 i 1 

 0 
2 









 0  1  0  1

2 
2

 x x  




 0  1  0  i 1 




2 
2 

 x  y  
+18 more states with -x, +y and -y in first index = 36 product states
But that's not all....... there are also entangled states
(can’t be written as product) such as:
 e1 
0 0 11
2
 e2 
0 0 1 1
2
 e3 
0 0 i 1 1
2
 e4 
0 0 i 1 1
2
and many more such entangled combinations, for example
1
1
1
1
 x 0   1 x  0 0  1 0  1 1
2
2
2
2
There are way more 2-qubit states (more than 40)
than 2 bit states (just 4)....
The total number of such quantum states rises superexponentially with the number of qubits
Quantum Computing with one qubit
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>
a|0> +b|1>
a|0> +eib|1>
Example: NOT operation:
|0>
|1>
|1>
|0>
a|0> +b|1>
a|1> +b|0>
Example:
|0>
|1>
NOT operation:
1
 | 0   |1  
2
1
  | 0   |1  
2
All operations must work
on superposition states!
Single qubit NOT operation as rotation on the Bloch sphere
Example:  = 0
z
|0>
 
 
 
  cos  | 0   e i sin   | 1 
2
2
0
0
 cos  | 0  e i sin   | 1 
2
2
| 0 
y
Starting from |0>
x
|1>
rotate about the x-axis by .
Such a rotation would also
change |1> to |0>.
NOT, or "x-pulse" or H'=esx
Single qubit NOT operation as rotation on the Bloch sphere
Example:  = /2,  = 0
z
|0>
| 0   |1 
2
  /2
x
|1>
 
 
i
  cos  | 0  e sin   | 1 
2
2
 
 
i0
 cos  | 0  e sin   | 1 
4
4
| 0   |1 

2
y
starting from |0>
rotate about the y-axis by /2.
A /2-pulse, or sy, or NOT
Two such rotations would
produce a NOT
Single qubit control operations as rotations on the Bloch sphere
Example:  = /2,  = /2
z
|0>
| 0   |1 
2
 
 
  cos  | 0  ei sin   | 1 
2
2

i
 
 
2
 cos  | 0  e sin   | 1 
4
4
| 0  i | 1 

2
| 0  i | 1 
2
  /2
y
rotate about the z-axis by /2
will increase phase of any state
by /2.
This is a “/2 phase gate” or sz
x
|1>
But how can you get the state to rotate on the Bloch sphere?
- Basic Idea: Use a Rabi Oscillation
- Consider a 2-level system with energy splitting DE.
- Apply power (a perturbation) continuously at frequency f = DE/h.
1
1
1
DE=hf
DE=hf
0
0
Start in 0 and Apply
power for short time
--> Small amplitude to
be in 1
Keep applying power
--> eventually
system pumped
entirely into 1
(NOT gate or -pulse)
DE=hf
0
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 W) set by power and tuning. Stopping
power at appropriate time can produce NOT or NOT
Two-level System Dynamics
State of a system described by wavefunction  that satisfies
time-dependent Schrodinger’s Equation

 H
t
For a two-level system with Hamiltonian Ho that is being driven at
frequency w with a perturbing energy H’, we can write H in
matrix form as:
i
 Eo
H  H o  H '  
0
0 
Eo
V cos(w t ) 
0
V cos(w t )  
  


  
E1  V * cos(w t )
E1
0
 V cos(w t )

where: Eo = energy of ground state, E1 = energy of excited state
V cos(wt) = <0|H’|1>
and where:
1
0
a 
| 0   
| 1   
|    
 0
1
b 
Two-level System Dynamics
Plug into Schrodinger’s Equation:
 a 
 i
 
Eo
V cos(w t )  a 
 t   
 

b
E1
 i
 V cos(w t )
 b 


 t 

i
 H
t
Write as two coupled equations:
a
 Eoa  V cos(w t ) b
t
b
i
 V cos(w t )a  E1b
t
i
notice that this says that the
amplitude b to be found in |1>
will change based on amplitude
a to be in |0>
Fairly nasty…guess solution of form: (this will always work!)
a  A(t )e
b  B (t )e
E 
i  o  t
  
E 
i  1  t
  
Plug into
Schrodinger’s
Equation
 Eo 
 Eo 
 E1 



i
t

i
t

i




 t

   
  
i
A(t )e
 Eo A(t )e
 V cos(w t ) B(t )e   

t 

E 
E 
E 
i  o  t
i  1  t 
i  1  t
 
i
B(t )e      V cos(w t ) A(t )e     E1 B(t )e   

t 

For the first equation, we find:
ie
E 
i  o  t
  
A(t )
 Eo A(t )e
t
E 
i  o  t
  
 Eo A(t )e
E 
i  o  t
  
 V cos(w t ) B(t )e
E 
i  1  t
  
Clean things up:
 E1  Eo 
t
 
i 
A(t )
i
 V cos(w t ) B(t )e 
t
For simplicity, let’s assume we are on resonance ( w  E1  Eo)
expand this
A(t )
 iw t
i
 VB(t ) cos(w t )e
term
t
A(t )
i
 VB(t ) cos(w t )e iw t
t
e iw t  e i w t  i w t
 VB(t )
e
2



V
 B(t ) 1  ei 2w t
2

This term is changing very rapidly and is far from resonance at w…
so it can be dropped…. “rotating wave approximation”
A(t ) V

B (t )
t
2i
B (t ) V

A(t )
t
2i
 A(t )
V 
   A(t )
2
t
 2 
2
2
take another time derivative of the 1st
equation and use 2nd to eliminate dB/dt
Assuming A(0) = 1, solution is:
A(t )  cosWt 2
B(t )  i sin Wt 2
V
is the Rabi frequency
W

E 
E 


i  o  t 
i  o  t 
 a   A(t )e      cosWt 2e    
|      

 E1  
 E1  
i   t
i   t
b  

   
   
B
(
t
)
e



i
sin
W
t
2
e

 

 cosWt 2e
E 
i  o  t
  
| 0   i sin Wt 2e
E 
i  1  t
  
Take out an overall phase factor of exp  iEot  
probabilty
|   cosWt 2 | 0   i sin Wt 2e
 E E 
i  1 0  t
  
|1 
1
P0=|a|2
0
P1=|b|2
t
0
2 / W
4 / W
|1 
E 
E 


i  o  t 
i  o  t 
 a   A(t )e      cosWt 2e    
|      

 E1  
 E1  
i   t
i   t
b  

   
   
B
(
t
)
e



i
sin
W
t
2
e

 

 cosWt 2e
E 
i  o  t
  
| 0   i sin Wt 2e
E 
i  1  t
  
|1 
Take out an overall phase factor of exp  iEot  
|   cosWt 2 | 0   i sin Wt 2e
 E E 
i  1 0  t
  
|1 
Also notice this is now in the familiar “polar coordinate” form:
 
 
i
|   cos  | 0   e sin   | 1 
2
2
where   Wt
 E  E0  
and    1
t 
   2
Rabi Oscillation on the Bloch Sphere
z
|0>
d/dt w01
  Wt
| 0  i | 1  
y
x
| 0   | 1  
2
|1>
2
To make NOT gate,
just drive system at
resonance for total time
of t = /W
Two Qubit Operations
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.
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>
Example, CNOT operation on
a|1,1> + b|0,1> + g |1,0>
yields:
a|1,0> + b|0,1> + g |1,1>
Implementing a CNOT in a 2-qubit system
input state
output
state
|0,0>
|0,0>
|0,1>
|0,1>
|1,0>
|1,1>
|1,1>
|1,0>
|1,1>
|1,0>
|0,1>
|0,0>
DE=hfo
Notice that it just
exchanges 10 and 11
.... Like a -pulse
between 10 and 11
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
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.