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Lecture 10: Quantum Computing
 Basic Quantum Physics
 Motivation: Waves versus particles, interference
experiments
 Quantum Notation, Jargon, and Definitions
 Quantum computing
 Quantum logic gates
 Quantum software
 Quantum hardware
R. Rao: Lecture 10 – Quantum Computing
1
Classical double-slit experiment #1
 Gun shoots identical particles




Large particles like tennis balls
All identical
With random direction
At a slow firing rate
R. Rao: Lecture 10 – Quantum Computing
 Probability of detecting
particles for two slits is sum
of individual slit probabilities:
 P12 = P1 + P2
2
Classical double-slit experiment #2
 Source generates water waves
 At any intensity value
 No reflection from absorber
 Detector measures wave
intensity
  (wave height)2
 Waves interfere at absorber
R. Rao: Lecture 10 – Quantum Computing
 Wave intensity I12 = |h1 + h2|2
 = |h1|2 + |h2|2 + 2|h1||h2|cos
 h1 and h2 are complex numbers
3
Double-slit with electrons
 Gun shoots electrons





Individual particles
Indestructible
All identical
With random direction
At a slow firing rate
R. Rao: Lecture 10 – Quantum Computing
 Detector sees individual electrons




P12  P1 + P2
P12 = |1 +  2|2 (interference!)
1 and  2 are complex numbers
Electrons exhibit wave behavior?
4
Measurement: Watching the electrons
 Electrons scatter light
 Put a light at the back
side of the slit wall
 Watch where the
electron goes
R. Rao: Lecture 10 – Quantum Computing
 Light on: no interference!
 P12 = P1 + P2
 Light off: interference!
 P12 = |1 +  2|2
5
First principles of quantum mechanics
1. The probability of an event is given by the square of the absolute value
of the probability amplitude for that event:
P  probability
  probability amplitude (complex)
P  | |2
2. When an event can occur several ways, the probability amplitude for the
event is the sum of the individual probability amplitudes:
  1 + 2
(no measurement  interference)
P  |1 + 2 |2
(probability of event e.g. electron at backstop)
3. If you measure and determine which of the possible alternatives an
experiment takes, then the probability of the event is the sum of the
probabilities for each alternative:
P12 = P1 + P2 = |1|2 + |2|2 (measurement at slits  no interference)
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Other Examples of Quantum Phenomena
 Spin of an electron: spin up or spin down
 Can be set to a continuum of values but collapses to up or down
when measured with a magnetic field
 Polarization of a photon: horizontal or vertical
 Measured using a calcite crystal; can be set to a continuum of values
but collapses to horizontal or vertical upon measurement
 Energy levels of an ion: excited or ground state
 Can be put in a continuum of states in between these two but
collapses to excited or ground state when measured
 Just as the state of a transistor can represent a bit (0 or 1),
the state of a quantum system (e.g. spin) represents a qubit
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Quantum Notation
 Classical notation for a bit: x = 0 or x = 1 (only 2 values)
(0,1)
x=1
(1,0)
x=0
 Dirac notation for a quantum bit x (Qubit |x>) e.g. spin of an
electron: up = |1>, down = |0>, continuum of possible values |x>
|1>
1
0  
0 
0 
1  
1
x  c1 0  c2 1
Qubit |x>
|0>
Example: c1 and c2 are two real numbers
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Quantum Jargon
 A qubit in a quantum system can exist in a linear
superposition of basis states (“eigenstates”) |0> and |1>:
x  c1 0  c2 1
 c1 
  
 c2 
c1  c2
2
2
1
 c1and c2 are complex numbers: c1= a1 + b1i ; c2= a2 + b2i
 i is the square root of –1:
R. Rao: Lecture 10 – Quantum Computing
i  1 (i.e. i 2  1)
9
Cliff’s notes on complex numbers
 Consider a complex number c = a + bi
 Complex conjugate of c is c* = a - bi
 Amplitude of c = c
2
 a 2  b 2 = amplitude of c*
 Squared amplitude of c = a2 – (-b2) = a2 – i2b2 = c*c
 c can also be written as c = Aei = A (cos  + i sin ) where:
A  a 2  b2 is the amplitude,  is the phase of c,
cos  
a
a b
2
2
and sin  
R. Rao: Lecture 10 – Quantum Computing
b
a 2  b2
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The Effect of Measurement
 If you measure the quantum system, the qubit (superposition
of states) collapses to one of the basis states |0> and |1>
x  c1 0  c2 1  measure  either 0 or 1
 c1and c2 are called probability amplitudes because
probability of getting |0> or |1> upon measurement depends
on their squared amplitudes:
Prob 0   c1  a  b
2
2
1
2
1
Prob 1   c2
 Since Prob(|0>) + Prob(|1>) = 1,
R. Rao: Lecture 10 – Quantum Computing
c1  c2
2
2
2
 a22  b22
1
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Unitary matrices
 Matrix of complex numbers:
A  cij
c11 c12 
E.g., A  

c
c
 21 22 
 Conjugate transpose of a matrix:
 A *T  c*ji
*
*


c
c
T
11
21
E.g.,  A *   *
* 
c
c
 12 22 
T
 A matrix U is unitary if U* U = I
 I is the identity matrix
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Why are unitary matrices important?
 c1 
x   
 c2 
 Suppose |x> is a qubit:
 |x> has length 1
c
1
2
 c2
2

1
 Suppose |y> = U|x> (i.e. transform |x> using U)
 Length of |y> is:
 * c
length(U x )  U 

 c
*
1
*
2
T
   c1 
  U    c1*

   c2 

 c1 
c U U    1
 c2 
*
2

*T
 Unitary matrices preserve length!
 Unitary transformations conserve probability
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Quantum Operations and Gates
 Quantum systems are described by Schrödinger’s wave equation
 Integrating this differential equation, we can show that the state |x>
of a quantum system evolves as:
 |x>new = U|x>old




 i t

where U  exp    dt ' H 
  0

U is a unitary matrix derived from Hamiltonian H
H is a unitary matrix that represents total energy of the system
Every quantum state |x> is a vector of unit length
Quantum gates map unit vectors |x> to unit vectors |y>
 For quantum computing, we design H so that U acts like a logic gate
 Quantum computers are deterministic, linear, reversible, and
unitary until a measurement is made
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1-bit quantum gates
 Example: NOT gate
 Check: UNOT is unitary
U NOT
F0 1I
G J
H1 0K
U NOT
U NOT
1I F
0I
F
G
G
J
H0K H1J
K
0I F
1I
F

G
H1J
KG
H0J
K
 1-bit gates:
S  phase shift gate
R  Rotation gate
Walsh-Hadamard gate
R. Rao: Lecture 10 – Quantum Computing
1 0I
F
U G J
H0 1K
1 F1 1I

G
J
1 1K
2H
S
UR
U WH
1 1 1 



2 1  1
15
1-bit quantum gates (cont.)
 There are infinitely many 1-
bit gates, corresponding to
the rotations about a sphere:
e.g. R gate
Another example:
Square root of NOT gate
U
U
R. Rao: Lecture 10 – Quantum Computing
NOT
1 i
 
2 2
 1  i
2 2
U
NOT
NOT
1 i
 
2 2
1 i
 
2 2
 0 1
  U NOT
 
 1 0
16
More Notation
 A qubit is given by a column vector:
 In Dirac notation, define a row vector:
 c1 
x   
 c2 

x  c1* c2*

 x and x are called " bra" and " ket" of x respective ly
 The “bra”-“ket” of x is the inner product:

x x  c
*
1
 c1  *
*


c    c1 c1  c2c2  1
 c2 
*
2

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Orthonormal basis states
 The vectors |x and |y are orthonormal
 iff they are orthogonal:
 and normalized:
 Example:
x y 0
x x  1 and
 1
0    and
 0
y y 1
 0
1   
1
 1
0 0  1 0   1 and
 0
 0
1 1  0 1   1
1
 0
0 1  1 0    0 and
 1
 1
1 0  0 1   0
 0
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What is a Hilbert Space?
 If S is a set of basis states, then the Hilbert space is the
space of functions from S to complex numbers: each
function produces a vector of complex numbers in this space
 Example: |0> and |1> define an orthonormal Hilbert space
whose vectors are of the form:
 c1 
x   
 c2 
 The states of a quantum system are vectors in a particular
Hilbert space
 Measurement of the system produces one of the orthogonal
axes (a basis state or “eigenstate”) of the Hilbert space
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Vector Products in Dirac Notation
 Vector Products of basis states:
 1
 0 1

0 1   0 1  
 0
 0 0
 0
 0 0

1 0   1 0  
 1
 1 0
 0 1
 This is the matrix for the NOT gate!
0 1  1 0  
 1 0
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Quantum Logic Gates: 1-bit gates
 NOT gate:
U NOT
0 1I
F
G J
H1 0K
U NOT
U NOT
1I F
0I
F
G
G
J
H0K H1J
K
0I F
1I
F

G
H1J
KG
H0J
K
U NOT 0  1
U NOT 1  0
 NOT gate in Dirac Notation:
U NOT  0 1  1 0
U NOT 0  0 1 0  1 0 0  1
U NOT 1  0 1 1  1 0 1  0
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2-bit Quantum Logic Gates
 Orthonormal Basis States:
00 , 01 , 10 and 11
 Controlled NOT gate:
A
0
0
1
1
B
0
1
0
1
A’
0
0
1
1
 1
 
 0
00   
0
 
 0
 
 0
 
 1
01   
0
 
 0
 
 0
 
 0
10   
1
 
 0
 
 0
 
 0
11   
0
 
 1
 
B’
0
1
1
0
 CN gate in Dirac Notation:
UCN  00 00  01 01  10 11  11 10
E.g. UCN 10  00  0  01  0  10  0  11 1  11
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Two-bit Quantum Operations
 U  00 00  01 01  10 11  11 10
CN
 Operation of CN gate on arbitrary qubits x:
If x  c0 00  c1 01  c2 10  c3 11 ,
U CN x  c0U CN 00  c1U CN 01  c2U CN 10  c3U CN 11
 c0 00  c1 01  c2 11  c3 10
 c0   c0 
   
 c1   c1 
 U CN   
c 
c2
   3
c  c 
 3  2
UCN switches the amplitudes of |10> and |11>
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Entanglement
 Two bits in a quantum system are entangled if measurement
of one is always correlated with measurement of the other
E.g. x  c0 00  c3 11
(c1 and c2 are zero)
 If you measure the second bit of x, you know the first bit
also: E.g. if second bit was 1, we know first bit must be 1

E.g. y  c0 01  c3 11
(c1 and c2 are zero)
 The qubits in y are not entangled: cannot determine value of first bit
based on value of second bit
 Entangled qubits cannot be factored into their components:

c0 00  c3 11  a1 0  b1 1 a2 0  b2 1 
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No Cloning Theorem
 There is no unitary transform that allows us to copy a qubit
 Proof: Suppose U is a copying matrix: U|c0> = |cc> for all states |c>
 Then,
if c 
1
1
( 0  1 ), U c 
( 00  11 )
2
2
but U c  cc 
1
1
( 00  01  10  11 ) 
( 00  11 )
2
2
 Illustration: Copy a bit using CNOT….yields an entangled state!
a 0 b1
0
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Three-bit Quantum Gates
 Controlled Controlled NOT (CCN) gate:
A CCN gate
A
A'
B
B'
C
C'
A
0
0
0
0
1
1
1
1
B
0
0
1
1
0
0
1
1
C
0
1
0
1
0
1
0
1
A’
0
0
0
0
1
1
1
1
B’ C’
0 0
0 1
1 0
1 1
0 0
0 1
1 1
1 0
 UCCN is an 8 x 8 matrix: 000 000   110 111  111 110
 UCCN is universal: we can form any Boolean function using
only CCN gates: e.g. AND if C = 0
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Premise of quantum computing
 Simulating an N-bit quantum system on a classical computer
requires an amount of computation exponential in N
 Need to track 2N complex amplitudes simultaneously
 Nature updates real systems in constant time
 Updates all the amplitudes simultaneously
 Uses quantum superposition, or quantum parallelism
 Quantum computation: Use nature to compute
 Use N qubits to represent 2N complex amplitudes
 Perform unitary operations on qubits
 Measure to get the output
 Harness quantum superposition to get exponential speedup
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Comparson of Classical versus Quantum Computing
Classical Computer
 N particles

2N
unique states
 Computations are sequential
 Select 1 state
 Operate on it
 Put it back into memory
 Example: N=3
 3 bits
 8 states
 Work on one number at a
time
 x = 101
Quantum Computer
 N particles
 2N unique states
 Computations occur in parallel
 States interact
 They are entangled
 Operate on all states at once
 Example: N=3
 3 qbits
 8 complex amplitudes
 Operator manipulates 8 at once
x
1
a 000  b 001  c 010 ...h 111

 Exponential Speedup
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Shor’s Quantum Factoring Algorithm
 Suppose you want to factor a number N
 Shor’s algorithm:
1. Pick random x < N.
2. Compute f = gcd(x,N); if f l, return f // f is a factor
3. Find the least r > 0 such that xr  1 (mod N).
4. Compute f1 = gcd(xr/2 – 1,N) ); if f1  l, return f1 // f1 is a factor
5. Compute f2 = gcd(xr/2 + 1,N); if f2  l, return f2 // f2 is a factor
6. Go to 1 and repeat
 Number of repetitions for finding a factor with prob > 0.5
is polynomial in length of N
 Hard part: Step 3. Find the least r such that xr  1 (mod N).
 r is the period of repetition of x1, x2,… (mod N).
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Quantum parallelism for finding period

Finding the period r of repetition of x1, x2,… (mod N).
1. Prepare an equal superposition of all values of r < q = N2
q 1
 r,0
q
r 1
2. Chose random x and compute (xr mod N) for all r simultaneously:
q

r , x r mod N
q
r 1
3. Apply quantum Fourier Transform UQFT to superposition of states:
q
r
exp(
2

ikr
/
q
)
k
,
x
mod N

q
r , k 0
4. Measure contents of register containing k to compute period r

See Shor’s paper and tutorials on class website for more details
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Grover’s Database Search Algorithm
 Problem: Search a random list of N items for a target item
xT such that the function P(xT) is true e.g. searching for a
key in DES
 Grover’s algorithm: Amplify amplitude of target item
1.
2.
3.
4.
5.
Prepare an equal superposition of all x
Invert the amplitude of xj if P(xj) = 1
Subtract all amplitudes from average amplitude

Repeat (2) and (3)
N times
4
Measure the result
 Quadratic speedup over classical search (O(N) steps).
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Quantum Hardware
 Suggested Possibilities: Ion Traps (Quantum dots), Cavity QED
(quantum electrodynamics), NMR: nuclear magnetic resonance
QC Features Ion Trap
Cavity QED
NMR
Qubit
Energy levels in an
ion within an electric
field
Polarization of a
photon in a cavity
Spin states of a
nucleus in a molecule
Preparation
Ion cooling
Prepare linearly
and circularly
polarized photons
Set average state of
spins in sample
Evolution
Apply laser pulses at
specific frequencies
Photon-photon
interactions
Apply radio
frequency pulses
Conditional
logic
Coupled vibrations of State of Cesium ion Nuclear spin-spin
trapped ions
in cavity & photon interactions in a
polarization
molecule
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Quantum Computing: Summary
 Basic Mechanism: Parallel computation along all possible computational
paths, with selective manipulation of probability amplitudes
 Main Features: Problem instances encoded as states of a quantum system
(e.g. spins of n electrons, polarization values of n photons etc.)
1. The system is put into a superposition of all possible states, each
weighted by its probability amplitude (= a complex number ci)
E.g. Qubits for 2 electrons = c1 |00> + c2 |01> + c3 |10> + c4 |11>
2. The system evolves according to quantum principles:
1. Unitary matrix operation: describes how superposition of states
evolves over time when no measurement is made
2. Measurement operation: maps current superposition of states to
one state based on probability = square of amplitude ci
E.g. probability of seeing output bits (00) is | c1|2
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Quantum Computing: Problems and Future Directions
 Problems:
 Decoherence: Environmental noise may inadvertently “measure” the
system, thereby disturbing the computation (current decoherence
time ~ 1 ms)
 Software solution: Error correcting codes may help ([Shor et al.])
 Scaling: All physical implementations so far (NMR, Cavity QED,
etc.) have failed to scale beyond a few qubits.
 Future Directions:
 Hardware Implementations: New physical substrates are needed that
allow manipulations of large numbers of qubits (superpositions of
states) with little or no decoherence
 New Algorithms: New ways of exploiting quantum parallelism are
needed that allow solutions to NP-complete problems
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5-minute break…
(Please fill out course evaluations)
Next: Student presentations!
R. Rao: Lecture 10 – Quantum Computing
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