Table of contents
Introduction
Plan
Information about this web version
Topics
Some mathematical preliminaries
0.1
Euclidean vectors
0.2
Vector spaces
0.3
Bras and kets
0.4
Daggers
0.5
Geometry
0.6
Operators
0.7
Outer products
0.8
The trace
0.9
Some useful identities
I Foundations
1
Quantum interference
1.1
Two basic rules
1.2
Quantum interference: the failure of probability theory
1.2.1
The double slit experiment
1.3
Superpositions
1.4
Interferometers
1.5
Qubits, gates, and circuits
1.6
Quantum decoherence
1.7
Computation: deterministic, probabilistic, and quantum
1.8
Computational complexity
1.9
Outlook
1.10
Remarks and exercises
1.10.1
A historical remark
1.10.2
Modifying the Born rule
1.10.3
Complex numbers
1.10.4
Many computational paths
1.10.5
Distant photon emitters
1.10.6
Physics against logic?
1.10.7
Quantum bomb tester
1.10.8
More time, more memory
1.10.9
Quantum Turing machines
1.10.10
Polynomial = good; exponential = bad
1.10.11
Big O
1.10.12
Imperfect prime tester
1.10.13
Imperfect decision maker
2
Qubits
2.1
Composing quantum operations
2.2
Quantum bits, called “qubits”
2.3
Quantum gates and circuits
2.4
Single qubit interference
2.5
The square root of NOT
2.6
Phase gates galore
2.7
Pauli operators
2.8
From bit-flips to phase-flips, and back again
2.9
Any unitary operation on a single qubit
2.10
The Bloch sphere
2.10.1
Drawing points on the Bloch sphere
2.11
Composition of rotations
2.12
A finite set of universal gates
2.13
Remarks and exercises
2.13.1
Unknown phase
2.13.2
One of the many cross-product identities
3
Quantum gates
3.1
Physics against logic, via beamsplitters
3.2
Quantum interference, revisited (still about beam-splitters)
3.3
The Pauli matrices, algebraically
3.4
Unitaries as rotations
3.5
Universality, again
3.6
Some quantum dynamics
3.7
Remarks and exercises
3.7.1
Orthonormal Pauli basis
3.7.2
Pauli matrix expansion coefficients
3.7.3
Linear algebra of the Pauli vector
3.7.4
Matrix Euler formula
3.7.5
Special orthogonal matrix calculations
3.7.6
Phase as rotation
3.7.7
Geometry of the Hadamard
3.7.8
Swiss Granite Fountain
3.7.9
Dynamics in a magnetic field
4
Measurements
4.1
Hilbert spaces, briefly
4.2
Back to qubits; complete measurements
4.3
The projection rule; incomplete measurements
4.4
Example of an incomplete measurement
4.5
Observables
4.6
Compatible observables and the uncertainty relation
4.7
Quantum communication
4.8
Basic quantum coding and decoding
4.9
Distinguishability of non-orthogonal states
4.10
Wiesner’s quantum money
4.11
Quantum theory, formally
4.11.1
Quantum states
4.11.2
Quantum evolutions
4.11.3
Quantum circuits
4.11.4
Measurements
4.12
Remarks and exercises
4.12.1
Projector?
4.12.2
Knowing the unknown
4.12.3
Measurement and idempotents
4.12.4
Unitary transformations of measurements
4.12.5
Optimal measurement
4.12.6
Alice knows what Bob did
4.12.7
The Zeno effect
5
Quantum entanglement
5.1
A small history
5.2
One, two, many…
5.3
Quantum theory, formally (continued)
5.3.1
Tensor products
5.4
Back to qubits
5.5
Separable or entangled?
5.6
Controlled-NOT
5.6.1
The Bell states, and the Bell measurement
5.6.2
Quantum teleportation
5.6.3
Thou shalt not clone
5.7
Other controlled gates
5.7.1
Controlled-phase
5.7.2
Controlled-U
5.7.3
Phase kick-back
5.7.4
Universality, revisited
5.7.5
Density operators and the like
5.8
Why qubits, subsystems, and entanglement?
5.9
Remarks and exercises
5.9.1
Entangled or not?
5.9.2
SWAP circuit
5.9.3
Controlled-NOT circuit
5.9.4
Measuring with controlled-NOT
5.9.5
Arbitrary controlled-U on two qubits
5.9.6
Entangled qubits
5.9.7
Quantum dense coding
5.9.8
Playing with conditional unitaries
5.10
Appendices
5.10.1
Tensor products in components
5.10.2
The Schmidt decomposition
6
Density matrices
6.1
Definitions
6.2
Statistical mixtures
6.3
A few instructive examples, and some less instructive remarks
6.4
The Bloch ball
6.5
Subsystems of entangled systems
6.6
Partial trace, revisited
6.7
Mixtures and subsystems
6.8
Partial trace, yet again
6.9
Remarks and exercises
6.9.1
Some density operator calculations
6.9.2
Purification of mixed states
6.9.3
Pure partial trace
6.9.4
Maximally Bell
6.9.5
Trace norm
6.9.6
Distinguishability and the trace distance
6.9.7
Spectral decompositions and common eigenbases
7
Quantum channels
7.1
Random unitaries
7.2
Random isometries
7.3
Evolution of open systems
7.4
Stinespring’s dilation and Kraus’s ambiguity
7.5
Single qubit channels
7.6
Composition of quantum channels
7.7
Completely positive trace-preserving maps
7.8
State-channel duality
7.9
The mathematics of “can” and “cannot”
7.10
Kraus operators, revisited
7.11
Correctable channels
7.12
Appendices
7.12.1
Isometries
7.12.2
The Markov approximation
7.12.3
What use are positive maps?
7.12.4
The Choi–Jamiołkowski isomorphism
7.12.5
Block matrices and partial trace
7.13
Remarks and exercises
7.13.1
Partial inner product
7.13.2
The “control” part of controlled-NOT
7.13.3
Surprisingly identical channels
7.13.4
Independent ancilla
7.13.5
Cooling down
7.13.6
Unchanged reduced density operator
7.13.7
Order matters?
7.13.8
Pauli twirl
7.13.9
Depolarising channel
7.13.10
Depolarising channel and the Bloch sphere
7.13.11
Complete positivity of a certain map
7.13.12
Toffoli gate
7.13.13
Duals
7.13.14
Trace, transpose, Choi
7.13.15
Purifications and isometries
7.13.16
Tricks with a maximally entangled state
7.13.17
Trace preserving and partial trace
7.13.18
Rotating Kraus operators
7.13.19
No pancakes
II Applications
8
Quantum cryptography
9
Bell’s theorem
9.1
Quantum correlations
9.2
Hidden variables
9.3
CHSH inequality
9.4
Quantum correlations, revisited
9.5
Tsirelson’s inequality
9.6
Remarks and exercises
10
Quantum algorithms
10.1
Quantum Boolean function evaluation
10.1.1
Example
10.2
Hadamard and quantum Fourier transforms
10.3
More phase kick-back
10.4
Oracles and query complexity
10.4.1
Deutsch’s algorithm
10.5
Three more quantum algorithms
10.5.1
The Bernstein-Vazirani algorithm
10.5.2
Grover’s search algorithm
10.5.3
Simon’s problem
10.6
Remarks and exercises
10.6.1
Shor’s algorithm
10.6.2
RSA
10.6.3
More complexity classes
10.6.4
10.6.5
11
Decoherence and basic quantum error correction
11.1
Decoherence simplified
11.2
Decoherence and interference
11.3
Evolution of density operators under decoherence
11.4
Quantum errors
11.5
Same evolution, different errors
11.6
Some errors can be corrected on some states
11.7
Repetition codes
11.8
Quantum error correction
11.9
Turning bit-flips into phase-flips
11.10
Dealing with bit-flip and phase-flip errors
11.11
Remarks and exercises
12
Quantum error correction and fault tolerance
13
Further reading
Introduction to Quantum Information Science
Chapter 8
Quantum cryptography
About
!!!TO-DO!!!
.