Quantum Computing
Nick Bonesteel
Discovering Physics, Nov. 16, 2012
0
q
0
H
0 i 1
0 i 1
2
f
1
2
1
H
Uf
H
What is a quantum computer, and
what can we do with one?
A Classical Bit: Two Possible States
0
1
0
A Classical Bit: Two Possible States
0
1
1
Single Bit Operation: NOT
0
NOT
y
x
1
0
x y
0 1
1 0
Single Bit Operation: NOT
0
NOT
y
x
1
1
x y
0 1
1 0
A Quantum Bit or “Qubit”
0
1
0
A Quantum Bit or “Qubit”
0
1
1
A Quantum Bit or “Qubit”
0
1
0
A Quantum Bit or “Qubit”
0
0  1
2
1
1
2
0
1

2
1
A Quantum Bit or “Qubit”
0
Quantum superposition
of 0 and 1
0  1
2
1
1
2
0
1

2
1
A Quantum Bit or “Qubit”
0
0  1
2
1
1
A Quantum Bit or “Qubit”
0
0  1
0  1
2
2
1
1
2
0
1

2
1
A Quantum Bit or “Qubit”
0
0  1
0  1
2
2
1
0
A Quantum Bit: A Continuum of States
0
q
0  1
0  1
2
2
1
cosq
0

sinq
1
A Quantum Bit: A Continuum of States
0
q
0 i 1
0 i 1
2
2
f
1
cos
q
2
Actually, qubit
states live on the
surface of a
sphere.
0  sin
q
2
e
if
1
A Quantum Bit: A Continuum of States
0
But the circle is
enough for us
today.
q
0  1
0  1
2
2
1
cosq
0

sinq
1
A Quantum NOT Gate
0
X
0
0  1
0  1
2
2
1
1
A Quantum NOT Gate
0
0  1
0  1
2
2
1
0
X
1
1
X
0
A Quantum
0
X
NOT Gate
0  1
0  1
2
2
0 1
2
0
1
A Quantum
0
1
X
X
NOT Gate
0  1
0  1
2
2
0 1
2
0 1
2
0
1
Hadamard Gate
0
1
H
H
0
0  1
0  1
2
2
0 1
2
0 1
2
1
Hadamard Gate
0
1
H
H
0
0  1
0  1
2
2
0 1
2
0 1
2
1
Hadamard Gate
0
1
H
H
0
0  1
0  1
2
2
0 1
2
0 1
2
1
Hadamard Gate
0
1
H
H
0
0  1
0  1
2
2
0 1
2
0 1
2
1
Hadamard Gate
0 1
2
H
0
0  1
0  1
2
2
1
0
H is its own inverse
0 1
2
H
1
Hadamard Gate
0
H
0
0  1
0  1
2
2
0 1
1
2
H is its own inverse
1
H
0 1
2
Hadamard Gate
0 1
2
H
0
0  1
0  1
2
2
1
0
H is its own inverse
0 1
2
H
1
Hadamard Gate
0
H
0
0  1
0  1
2
2
0 1
1
2
H is its own inverse
1
H
0 1
2
Fair Coin
Trick Coin
Balanced Function
f ( 0)  0
f (1 )  1
or
f ( 0)  1
f (1 )  0
Unbalanced Function
f ( 0)  0
f (1 )  0
or
f (0)  1
f (1 )  1
A Two Qubit Subroutine to Evaluate f(x)
x
0
Uf
x
f (x)
A Two Qubit Subroutine to Evaluate f(x)
Input x can be either 0 or 1
x
0
Initialize to state “0”
Uf
x
f (x)
Output is f(x)
A Two Qubit Subroutine to Evaluate f(x)
Input x can be either 0 or 1
x
Uf
1
This qubit can also be in state “1”
x
f (x)
A Two Qubit Subroutine to Evaluate f(x)
Input x can be either 0 or 1
x
Uf
1
This qubit can also be in state “1”
x
Bar stands
for “NOT”
f (x)
0 = 1,
1=0
A Two Qubit Subroutine to Evaluate f(x)
x
0
Uf
x
x
f (x)
1
Balanced
f ( 0)  0
or
f (1 )  1
f (0)  f (1),
x
Uf
f (x)
Unbalanced
f ( 0)  1
f (1 )  0
f ( 0)  0
f (1 )  0
f (1)  f (0)
f (0)  f (1),
or
f (0)  1
f (1 )  1
f (0)  f (1)
A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
H
1
H
Uf
H
A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
H
1
H
Uf
0
1
H
A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
H
1
H
Uf
H
 0  1  0  1 
A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
H
1
H
0
Uf
H
 0 1  1  0 1 
A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
0
H
1
H

f (0)  f (0)
Uf
 1
H
f (1)  f (1)

A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
0
H
1
H

f (0)  f (0)
Uf
 1
H
f (1)  f (1)
Only ran Uf subroutine once, but f(0) and f(1) both
appear in the state of the computer!

A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
0
H
1
H

f (0)  f (0)
Uf
 1
H
f (1)  f (1)
If f is balanced: f(0) = f(1) and f(0) = f(1)

A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
0
H
1
H

f (0)  f (0)
Uf
 1
H
f (0)  f (0)
If f is balanced: f(0) = f(1) and f(0) = f(1)

A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
H
1
H
Uf
 0  1 
H
f (0)  f (0)

If f is balanced: f(0) = f(1) and f(0) = f(1)
A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
0
H
1
H

f (0)  f (0)
Uf
 1
H
f (1)  f (1)
If f is unbalanced: f(0) = f(1) and f(0) = f(1)

A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
0
H
1
H

f (0)  f (0)
Uf
 1
H
f (0)  f (0)
If f is unbalanced: f(0) = f(1) and f(0) = f(1)

A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
H
1
H
Uf
 0  1 
H
f (0)  f (0)

If f is unbalanced: f(0) = f(1) and f(0) = f(1)
A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
H
1
H
Uf
H
Balanced:
 0  1 
f (0)  f (0)

Unbalanced:
 0  1 
f (0)  f (0)

A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
H
1
H
Uf
H
Balanced:
0

f (0)  f (0)

Unbalanced:
1

f (0)  f (0)

A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
1
H
Uf
H
H
Measure top
qubit
Balanced:
0

f (0)  f (0)

Unbalanced:
1

f (0)  f (0)

A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
1
H
Uf
H
H
Measure top
qubit
Balanced:
0

f (0)  f (0)

Unbalanced:
1

f (0)  f (0)

A Quantum Algorithm (Deutsch-Jozsa ‘92)
0
1
H
Uf
H
H
Measure top
qubit
Balanced:
0

f (0)  f (0)

Unbalanced:
1

f (0)  f (0)

One qubit
0
0 1
2
H
0 1
2
Two qubits
0
0 1
2
0 1
H
H
0

0 1
2
2
0 1
2

1
2

00  01  10  11

Two qubits
0
0 1
2
0 1
H
H
0

0 1
2
2
0 1
2

1
2

00  01  10  11
0
1
2
3

Counting in binary
Three qubits
0
0
0 1
2
0 1
H
H
H
0

0 1
2
2
0 1
2
0 1
2

0 1
2

1
 000  001  010  011  100  101  110  111
3/ 2
2

Three qubits
0
0
0 1
2
0 1
H
H
H
0

0 1
2
2
0 1
2
0 1
2

0 1
2

1
 000  001  010  011  100  101  110  111
3/ 2
2
0
1
2
3
4
5
6
7

N qubits
0 1
H
H
H
0
0
0
2
0 1
2
0 1

0 1
2
0 1
H
0

0 1
2
N
2

2
0 1

2

1
2
N /2

0 1
2
0 00  0 01  010  011    111
0
1
2
3
… 2N-1

N qubits
0
0
0
H
H
H

0
1
2N / 2
0
H
0 1
2
0 1
2
0 1
2
0 1
2
 1  2  3    2N 1

Quantum superposition of
all possible input states!
N qubits
0
0
0
H
H
H

0
1
2N / 2
0
H
0 1
2
0 1
2
0 1
2
0 1
2
 1  2  3    2N 1

Quantum superposition of
all possible input states!
For N=250 the number of states
is roughly the number of atoms in
the universe!
N qubits
0
0
0
H
H
H
One function call
Uf
U f x  f (x)


0
1
2N / 2
0
H
 1  2  3    2N 1

Quantum superposition of
all possible input states!
For N=250 the number of states
is roughly the number of atoms in
the universe!
N qubits
0
0
0
H
H
H
One function call
Uf
U f x  f (x)


0
1
2N / 2
0
H
 1  2  3    2N 1

x can be any integer
From 0 to 2N-1
Quantum superposition of
all possible input states!
For N=250 the number of states
is roughly the number of atoms in
the universe!
N qubits
H
H
H
0
0
0
One function call
Uf
U f x  f (x)


H
0
1
2N / 2
1
2N / 2
0

x can be any integer
From 0 to 2N-1
 1  2  3    2N 1

Quantum superposition of
all possible input states!
f (0)  f (1)  f (2)  f (3)    f (2 N  1)

Evaluate f(x) for all
possible inputs!
Massive Quantum Parallelism
H
Run program Uf once, get result for
H
all possible inputs!
Uf
H
1



f (0)  f (1)  f (2)  f (3)    f (2  1) 
2
H
0
0
0
N
0
N /2
Only one problem: When I measure this state I only learn the
value of f(x) for one input x. (No free lunch!)
However, people have shown that a quantum computer can use
quantum parallelism to do things no classical computer can do.
Prime Factorization
• Given two prime numbers p and q,
pxq=C
Easy
C
p, q
Hard
• Best known classical factoring algorithm scales as
time = exp(Number of Digits)
• Mathematical Basis for Public Key Cryptography.
Quantum Factorization
• In 1994 Peter Shor showed that a Quantum
Computer could factor an integer exponentially
faster than a classical computer!
time = (Number of Digits)
3
• Shor’s algorithm exploits Massive Quantum
Parallelism.
OK, so how do we make a
quantum computer?
Boolean Logic Gates
Not
x
NOR
y
x y
0 1
1 0
x
z
y
x
0
0
1
1
y
0
1
0
1
z
1
0
0
0
Any classical
computation can be
carried out using these
two gates
Transistor Logic
A
A
A
B
A B
The Integrated Circuit
Core i7: 731,000,000 transistors
Single Qubit Gates

U
U 
Controlled-NOT Gate
0
0
X
1
0
X
0
0
0
1
1
1
1
1
0
X
1
1
X
0
Universal Set of Gates
U
X
Quantum Circuit
X
X
U
X
U
X
Any quantum computation can be carried out using these
two gates
Dave
Wineland
Serge
Haroche
2012 Nobel Prize in Physics
State of the Art: Superconducting Qubits
From : “Quantum Computers,” T. D. Ladd et al., Nature 464, 45-53 (2010)
Nature 460, 240-244 (2009)
2 superconducting
qubits coupled by a
microwave resonator
High fidelity (~95%)
2-qubit gates on a time
scale of 30 ns.
Nature 460, 240-244 (2009)
2 superconducting
qubits coupled by a
microwave resonator
High fidelity (~95%)
2-qubit gates on a time
scale of 30 ns.
First steps toward a scalable quantum computer
8 mm
From: http://ibmquantumcomputing.tumblr.com/
The Real Problem: Decoherence!
The Real Problem: Decoherence!
Qubit
( 0  1 )  rest of the universe
The Real Problem: Decoherence!
Qubit
Everything else
( 0  1 )  rest of the universe
The Real Problem: Decoherence!
Everything else
Qubit
( 0  1 )  rest of the universe
Over time….
0  rest of the universe 0  1  rest of the universe1
The Real Problem: Decoherence!
Everything else
Qubit
( 0  1 )  rest of the universe
Over time….
0  rest of the universe 0  1  rest of the universe1
Quantum coherence of qubit is inevitably lost!
The Real Problem: Decoherence!
Everything else
Qubit
( 0  1 )  rest of the universe
Over time….
0  rest of the universe 0  1  rest of the universe1
Quantum coherence of qubit is inevitably lost!
Amazingly enough, quantum computing is still possible using
what is known as “fault-tolerant quantum computation.”
Coherence Times for Superconducting
Qubits
Threshold for faulttolerant quantum
computation.
From: “Superconducting Qubits Are Getting Serious”, M. Steffen, Physics 4, 103 (2011)
Coherence Times for Superconducting
Qubits
Threshold for faulttolerant quantum
computation.
This is what most of my
own research on quantum
computing is focused on.
From: “Superconducting Qubits Are Getting Serious”, M. Steffen, Physics 4, 103 (2011)
Conclusions
A deep question: Do the laws of
nature allow us to manipulate quantum
systems with enough accuracy to build
“quantum machines” ?
Conclusions
A deep question: Do the laws of
nature allow us to manipulate quantum
systems with enough accuracy to build
“quantum machines” ?
If the answer is “yes” (as it seems to
be), then quantum computers are
coming!