A gentle introduction to
Quantum information/quantum computing
Physics 531
M. Saffman
2015.04.22
Why didn’t the qubit cross the road ?
Binary Data
Computers work with binary (base 2) data: 0 or 1
decimal binary
0
1
2
3
4
5
6
7
8
9
10
0
1
10
11
100
101
110
111
1000
1001
1010
0
1
Classical and quantum bits
Classical bit
0 or
1
Quantum bit = qubit
0 and 1
Bloch
sphere
ψ = a 0 +b1
| a |2 + | b |2 = 1
Quantum superposition
Superposition and entanglement
Two qubits:
Product State:
ψ
1
= a0 0 + a1 1 ,
ψ
2
= b0 0 + b1 1
ψ = (a0 0 + a1 1 ) ⊗ (b0 0 + b1 1 )
= a0b0 00 + a0b1 01 + a1b0 10 + a1b1 11
Classically we can only store one of four states at a time in a 2 bit
memory.
ψ
encodes four different states at one time.
With N qubits we can encode 2N states at one time.
Superposition and entanglement
It is also possible to create states that are not product states:
ψ = 00 + 11 ≠ ψ 1 ψ
2
Verschränkung
”entanglement”
Such a state is entangled, and cannot be described in terms of classical
bits – there is no local and realistic description of entangled states,
Einstein, Podolsky, Rosen 1935 (EPR paradox).
Quantum computers provide a speedup over classical machines.
It is not clear exactly where the speedup comes from.
The power of quantum computers appears to be intimately related to the
presence of entanglement. If there was no entanglement, we could use a
classical description of the machine.
Superposition and entanglement
It is also possible to create states that are not product states:
ψ = 00 + 11 ≠ ψ 1 ψ
2
Verschränkung
”entanglement”
Maximally entangled 2-qubit
state “Bell” state.
Classical data processing
Input data
Output data
0010111001111010
1110101001001100
0010011001111010
0101010101001100
0011100001111010
CPU
0001011011001100
0010110001111001
1110101101001100
0110111001000010
1000101001001100
Sequential data processing
Quantum data processing
Input data
ψ = a 00...00 + b 00...01 + c 00...10 + d 00...11 + ... 11...11
Quantum
CPU
Output data
ψ → ψ ' = f (ψ )
ψ ' = a ' 00...00 + b' 00...01 + c' 00...10 + d ' 00...11 + ... 11...11
The results for all possible input
data are computed in parallel
But the result is only
determined probabilistically
Running the computer
input
0
0
0
0
0
0
0
0
1
0
0
0
1
1
0
0
1
0
0
0
1
0
0
0
0
0
0
0
1
1
1
1
0
1
1
0
0
1
1
0
1
0
computation
Output state
(deterministic)
measurement
(probabilistic)
Quantum algorithms extract useful information from uncertain data.
What is it good for ?
Quantum computers will not replace general
purpose classical computers.
They are good for special types of problems:
Factoring
(RSA cryptosystem, and internet
data security)
Database searching
Simulating properties of materials, quantum chemistry
Foundations of quantum mechanics
RSA Public key cryptography
• Rivest, Shamir, Adleman (RSA) invented a public key
cryptosystem in 1977.
• Independently invented by C. Cocks in England in 1973 but
kept secret.
• There is a public key known to everyone and a private key.
• Messages are encrypted with the public key and broadcast.
• Only recipients who know the private key can decrypt the
message.
• This is widely used to protect personal data on the internet,
e.g. online shopping.
• The security of RSA relies on the difficulty of factoring large
numbers.
Factoring RSA Numbers
number
decimal
digits
prize
RSA-100
RSA-110
RSA-120
RSA-129
RSA-130
RSA-140
RSA-150
RSA-155
100
110
120
129
130
140
150
155
RSA-160
RSA-200
RSA-576
160
200
174
RSA-640
RSA-704
RSA-768
193
212 withdrawn
232 withdrawn
RSA-896
RSA-1024
RSA-1536
RSA-2048
270 withdrawn
309 withdrawn
463 withdrawn
617 withdrawn
factored (references)
Apr. 1991
Apr. 1992
Jun. 1993
Apr. 1994 (Leutwyler 1994, Cipra 1995)
Apr. 10, 1996
Feb. 2, 1999 (te Riele 1999a)
Apr. 6, 2004 (Aoki 2004)
Aug. 22, 1999 (te Riele 1999b, Peterson
1999)
Apr. 1, 2003 (Bahr et al. 2003)
May 9, 2005 (see Weisstein 2005a)
Dec. 3, 2003 (Franke 2003; see Weisstein
2003)
Nov. 4, 2005 (see Weisstein 2005b)
Jul. 1, 2012 (Bai et al. 2012, Bai 2012)
Dec. 12, 2009 (Kleinjung 2010, Kleinjung
et al. 2010)
Largest number known to have been factored:
RSA-768
=123018668453011775513049495838496272077285356959533479219732245215172640050
7263657518745202199786469389956474942774063845925192557326303453731548268507
9170261221429134616704292143116022212404792747377940806653514195974598569021
43413
=
3347807169895689878604416984821269081770479498371376856891243388982883793878
002287614711652531743087737814467999489
x
367460436667995904282 4463379962795263227915816434308764267603228381573
9666511279233373417143396810270092798736308917
Classical number field
sieve algorithm
1.9 (ln n )1 / 3 (ln ln n )2 / 3
time ~ e
RSA 768 took 1500
AMD64 years to
factor.
RSA 1536 would take
200 billion AMD64
years
Quantum Factoring
Exponentially faster factoring is possible
with a quantum computer.
This breaks RSA.
Peter Shor
1994
3
time ~ (ln n )
To date only demonstrated with numbers up to 21.
Factoring RSA-768 requires 1154 qubits (plus error correction).
No proof that a fast classical algorithm does not exist.
This is a Physics talk
– what do I care about internet shopping ?!@?
Quantum simulation
Feynman 1982: use a controllable quantum
system to simulate a quantum system of interest
(Int. J. Theor. Phys. 21, 467)
Quantum simulation
Feynman 1982: use a controllable quantum
system to simulate a quantum system of interest
(Int. J. Theor. Phys. 21, 467)
System Hamiltonian:
H physical
Model Hamiltonian:
H physical = H model + H extra
If Hmodel is a good description then simulations of Hmodel
may provide useful information.
However, the present state of the art for computer
simulations of ab-initio quantum dynamics is < 50 particles.
Important for semiconductor devices, quantum chemistry,
functional materials, designer medicines, foundations of
quantum mechanics, quantum-classical boundary, …
Quantum simulation
Feynman 1982: use a controllable quantum
system to simulate a quantum system of interest
(Int. J. Theor. Phys. 21, 467)
System Hamiltonian:
H physical
Model Hamiltonian:
H physical = H model + H extra
If Hmodel is a good description then simulations of Hmodel
may provide useful information.
Greiner lab, Harvard
However, the present state of the art for computer
simulations of ab-initio quantum dynamics is < 50 particles.
Important for semiconductor devices, quantum chemistry,
functional materials, designer medicines, foundations of
quantum mechanics, quantum-classical boundary, …
Idea: build a physical system that is accurately described
by Hmodel and watch it evolve.
49 atomic qubits, Saffman lab Madison
Quantum data processing
Input data
ψ = a 00...00 + b 00...01 + c 00...10 + d 00...11 + ... 11...11
Quantum
CPU
Output data
ψ → ψ ' = f (ψ )
ψ ' = a ' 00...00 + b' 00...01 + c' 00...10 + d ' 00...11 + ... 11...11
Quantum circuits
Frequently used gates
Qubit state
One-qubit gates
e
Frequently used gates
Two-qubit gates
CNOT gate
input
output
ct
ct
00
01
00
01
10
11
11
10
CNOT gate
c+t mod(2)
1

0
0

0
0
1
0
0
0
0
0
1
0

0
1

0
From CZ to CNOT
The CNOT gate can be realized with a controlled
phase plus Hadamards
0
1
|c>
0
|t>
1
1 0

0 −1
0 0

0 0 π
0
0

0
0
−1 0 

0 − 1
“Mach-Zehnder in Hilbert space”
Entanglement on demand
We can create entanglement with a simple quantum circuit.
0
0
(0
00
H
+ i 1 ) 0 = 00 + i 10
00 + i 11
CNOT
entanglement
Universal gate set
Any quantum circuit on N bits can be built up from a
universal set of one and two-bit gates.
The standard set:
Hadamard
Phase
π/8
CNOT
Circuit synthesis
Circuit decomposition with the universal gate set is always
possible but may not be efficient.
Toffoli (C2NOT)
Circuit implementation: sixteen 1 and 2 bit gates
from Chuang & Nielsen
Quantum half adder
Classical half-adder
Quantum half-adder
Algorithms
Q.C. does parallel processing, but the output is in a complicated
superposition state. The result is only defined probabilistically.
Nevertheless there are algorithms that can speed up some
particular problems.
These are often problems that are asymmetric:
it is difficult to find the correct answer
but easy to check if a purported solution is correct
Prominent examples: Shor’s factoring algorithm, quantum search
Quantum algorithm zoo
http://math.nist.gov/quantum/zoo/
Contents
Introduction
Algebraic and Number Theoretic Problems
Oracular Problems
Approximation and BQP-complete problems
Acknowledgments
References
Gate fidelity
An important question is how accurate do gate operations have to be for
reliable quantum computation?
von Neumann discovered in the 1950s that reliable classical computers can
be built from unreliable components. Redundancy, e.g. extra bits for parity
checking makes this possible.
Fault tolerant quantum computation is also possible.
The reliability of individual gates necessary for scalable, fault tolerant
computation is architecture dependent.
connectivity, qubit resources, encoding scheme
With a minimum of overhead and resources need gates good to ~0.0001.
Surface codes get by with errors > 0.01, but may require >104 extra qubits.
DiVincenzo checklist for quantum computing
The physical implementation of quantum computation,
Fortschr. Der Physik, 48, 771 (2000)
• well defined qubit
• qubit initialization
• universal gate set
• long coherence time
• measurement
Qubit platforms
trapped ions
superconductors
NV centers in
diamond
photons
neutral
atoms
quantum
dots
Scaling
entangled qubits
ions
D-Wave 512 “qubits”
no proof of quantum speedup
ongoing scientific debate…
superconductors
neutral atoms
photons
quantum dots
year
liquid
state NMR*
Why didn’t the qubit cross the road ?
It was already on the other side.