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Quantum Computing
0
1
q
f2
10 i2
10 i Uf
0
1
H H
H
Nick BonesteelDiscovering Physics, Nov. 16, 2012
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
0
1 0
x y
NOTx y0 11 0
Single Bit Operation: NOT
0
1 1
x y
NOTx y0 11 0
Single Bit Operation: NOT
A Quantum Bit or “Qubit”
0
1 0
0
1 1
A Quantum Bit or “Qubit”
0
1 0
A Quantum Bit or “Qubit”
0
1 0 12
1
2
1
2
10
A Quantum Bit or “Qubit”
0
1 0 12
1
2
1
2
10
Quantum superpositionof 0 and 1
A Quantum Bit or “Qubit”
0
1 1
2
10
A Quantum Bit or “Qubit”
0
1 0 12
1
2
1
2
10 2
10
A Quantum Bit or “Qubit”
0
1 0
2
10 2
10
A Quantum Bit or “Qubit”
A Quantum Bit: A Continuum of States
0
1sincos qq +0 1
q
2
10 2
10
A Quantum Bit: A Continuum of States
0
1
q
f2
10 i2
10 i
12
sin02
cosfqq ie
-+
Actually, qubit states live on the
surface of a sphere.
A Quantum Bit: A Continuum of States
0
1sincos qq +0 1
q
2
10 2
10
But the circle is enough for us
today.
A Quantum NOT Gate
X0 1
0
1
2
10 2
10
A Quantum NOT Gate
X0 1
0
1
2
10 2
10
X1 0
A Quantum NOT Gate 0
1
2
10 2
10
X02
10
A Quantum NOT Gate
X02
10
0
1
2
10 2
10
X12
10
Hadamard Gate
H02
10
0
1
2
10 2
10
H12
10
Hadamard Gate
H02
10
0
1
2
10 2
10
H12
10
Hadamard Gate
H02
10
0
1
2
10 2
10
H12
10
Hadamard Gate
H02
10
0
1
2
10 2
10
H12
10
Hadamard Gate
H 02
10
0
1
2
10 2
10
H 12
10
H is its own inverse
Hadamard Gate 0
1
2
10 2
10
H is its own inverse
H02
10
H12
10
Hadamard Gate
H 02
10
0
1
2
10 2
10
H 12
10
H is its own inverse
Hadamard Gate 0
1
2
10 2
10
H is its own inverse
H02
10
H12
10
Fair Coin
Trick Coin
Balanced Function
1)1(
0)0(
f
f
0)1(
1)0(
f
f
Unbalanced Function
0)1(
0)0(
f
f
1)1(
1)0(
f
f
or
or
Uf
x x
0 )(xf
A Two Qubit Subroutine to Evaluate f(x)
Uf
x x
0 )(xf
A Two Qubit Subroutine to Evaluate f(x)
Input x can be either 0 or 1
Output is f(x)Initialize to state “0”
)(xf
A Two Qubit Subroutine to Evaluate f(x)
Uf
x x
1
Input x can be either 0 or 1
This qubit can also be in state “1”
)(xf
Uf
x x
1
Bar standsfor “NOT”
0 = 1, 1 = 0
A Two Qubit Subroutine to Evaluate f(x)
Input x can be either 0 or 1
This qubit can also be in state “1”
Uf
x x
0 )(xfUf
x x
1 )(xf
1)1(
0)0(
f
f
0)1(
1)0(
f
f
Balanced Unbalanced
0)1(
0)0(
f
f
1)1(
1)0(
f
for or
),1()0( ff ),1()0( ff
A Two Qubit Subroutine to Evaluate f(x)
)0()1( ff )1()0( ff
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
Uf
0
1
H H
H
10
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
1010
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
101100
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
)1()1(1)0()0(0 ffff
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
)1()1(1)0()0(0 ffff
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)
Uf
0
1
H H
H
)1()1(1)0()0(0 ffff
If f is balanced: f(0) = f(1) and f(0) = f(1)
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
)0()0(1)0()0(0 ffff
If f is balanced: f(0) = f(1) and f(0) = f(1)
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
)0()0(10 ff
If f is balanced: f(0) = f(1) and f(0) = f(1)
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
)1()1(1)0()0(0 ffff
If f is unbalanced: f(0) = f(1) and f(0) = f(1)
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
)0()0(1)0()0(0 ffff
If f is unbalanced: f(0) = f(1) and f(0) = f(1)
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
)0()0(10 ff
If f is unbalanced: f(0) = f(1) and f(0) = f(1)
A Quantum Algorithm (Deutsch-Jozsa ‘92)
Uf
0
1
H H
H
)0()0(10 ff Unbalanced:
)0()0(10 ff Balanced:
A Quantum Algorithm (Deutsch-Jozsa ‘92)
)0()0(0 ff
)0()0(1 ff
Uf
0
1
H H
H
Unbalanced:
Balanced:
A Quantum Algorithm (Deutsch-Jozsa ‘92)
)0()0(0 ff
)0()0(1 ff
Uf
0
1
H H
H
Unbalanced:
Balanced:
Measure topqubit
A Quantum Algorithm (Deutsch-Jozsa ‘92)
)0()0(0 ff
)0()0(1 ff
Uf
0
1
H H
H
Unbalanced:
Balanced:
Measure topqubit
A Quantum Algorithm (Deutsch-Jozsa ‘92)
)0()0(0 ff
)0()0(1 ff
Uf
0
1
H H
H
Unbalanced:
Balanced:
Measure topqubit
A Quantum Algorithm (Deutsch-Jozsa ‘92)
One qubit
H02
10
2
10
H02
10
H02
10
111001002
1
2
10
2
10
Two qubits
H02
10
H02
10
111001002
1
2
10
2
10
Counting in binary0 1 2 3
Two qubits
H02
10
H02
10
H02
10
1111101011000110100010002
12/3
2
10
2
10
2
10
Three qubits
H02
10
H02
10
H02
10
1111101011000110100010002
12/3
2
10
2
10
2
10
0 1 2 3 4 5 6 7
Three qubits
1111101000100002
12/
N
H02
10
H02
10
H02
10
H02
10
N
2
10
2
10
2
10
2
10
0 1 2 3 2N-1 …
N qubits
1232102
12/
NN
H0
H0
H0
H0
Quantum superposition ofall possible input states!
2
10
2
10
2
10
2
10
N qubits
1232102
12/
NN
H0
H0
H0
H0
Quantum superposition ofall possible input states!
For N=250 the number of states is roughly the number of atoms in the universe!
2
10
2
10
2
10
2
10
N qubits
1232102
12/
NN
H0
H0
H0
H0
Quantum superposition ofall possible input states!
For N=250 the number of states is roughly the number of atoms in the universe!
Uf
)(xfxf U
One function call
N qubits
1232102
12/
NN
H0
H0
H0
H0
Quantum superposition ofall possible input states!
For N=250 the number of states is roughly the number of atoms in the universe!
Uf
)(xfxf U
One function call
x can be any integerFrom 0 to 2N-1
N qubits
1232102
12/
NN
H0
H0
H0
H0
Quantum superposition ofall possible input states!
Uf
)(xfxf U
One function call
)12()3()2()1()0(2
12/
NN
fffff Evaluate f(x) for all possible inputs!
x can be any integerFrom 0 to 2N-1
N qubits
Massive Quantum ParallelismH0
H0
H0
H0
Uf
)12()3()2()1()0(2
12/
NN
fffff
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.
Run program Uf once, get result for all possible inputs!
• Given two prime numbers p and q,
p x q = C Easy
C p, q Hard
• Best known classical factoring algorithm scales as
time = exp(Number of Digits)
• Mathematical Basis for Public Key Cryptography.
Prime Factorization
• In 1994 Peter Shor showed that a Quantum Computer could factor an integer exponentially faster than a classical computer!
time = (Number of Digits)
• Shor’s algorithm exploits Massive Quantum Parallelism.
3
Quantum Factorization
OK, so how do we make a quantum computer?
Boolean Logic Gates
x
yzx y
Not NOR
Any classical computation can be carried out using these two gates
x y0 11 0
x y z0 0 10 1 01 0 01 1 0
Transistor Logic
A A
A
BA B
The Integrated Circuit
Core i7: 731,000,000 transistors
Single Qubit Gates
U U
X
X
0 0
0 1
0 0
1 1
Controlled-NOT Gate
X
X
1 1
1 0
0 0
1 1
X
X
X
XU
U
Any quantum computation can be carried out using these two gates
Universal Set of Gates
UX
Quantum Circuit
2012 Nobel Prize in Physics
DaveWineland
SergeHaroche
State of the Art: Superconducting Qubits
From : “Quantum Computers,” T. D. Ladd et al., Nature 464, 45-53 (2010)
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
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.
8 mm
First steps toward a scalable quantum computer
From: http://ibmquantumcomputing.tumblr.com/
The Real Problem: Decoherence!
The Real Problem: Decoherence!
universetheofrest )10(
Qubit
The Real Problem: Decoherence!
universetheofrest )10(
Qubit Everything else
The Real Problem: Decoherence!
universetheofrest )10(
Qubit Everything else
Over time….
10 universetheofrestuniversetheofrest 10
The Real Problem: Decoherence!
universetheofrest )10(
Qubit Everything else
Over time….
10 universetheofrestuniversetheofrest 10
Quantum coherence of qubit is inevitably lost!
The Real Problem: Decoherence!
Qubit Everything else
Over time….
10 universetheofrestuniversetheofrest 10
Quantum coherence of qubit is inevitably lost!
Amazingly enough, quantum computing is still possible using what is known as “fault-tolerant quantum computation.”
universetheofrest )10(
Coherence Times for Superconducting Qubits
From: “Superconducting Qubits Are Getting Serious”, M. Steffen, Physics 4, 103 (2011)
Threshold for fault-tolerant quantum computation.
From: “Superconducting Qubits Are Getting Serious”, M. Steffen, Physics 4, 103 (2011)
Threshold for fault-tolerant quantum computation.
This is what most of my own research on quantum computing is focused on.
Coherence Times for Superconducting Qubits
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!
Conclusions
