Table of contents
Introduction
Plan and intended audience
Using the e-book
Acknowledgements
Some mathematical preliminaries
0.1
Complex numbers
0.2
Euclidean vectors and vector spaces
0.3
Bras and kets
0.4
Daggers
0.5
Geometry
0.6
Operators
0.7
Eigenvalues and eigenvectors
0.8
Outer products
0.9
The trace
0.10
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
Many computational paths
1.10.4
Distant photon emitters
1.10.5
Quantum Turing machines
1.10.6
More time, more memory
1.10.7
Big-O
1.10.8
Polynomial = good; exponential = bad
1.10.9
Imperfect prime tester
1.10.10
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
2.7.1
From bit-flips to phase-flips, and back again
2.8
Any unitary operation on a single qubit
2.9
The Bloch sphere
2.9.1
Drawing points on the Bloch sphere
2.10
Composition of rotations
2.11
A finite set of universal gates
2.12
Remarks and exercises
2.12.1
Unknown phase
2.12.2
One of the many cross-product identities
3
Quantum gates
3.1
Physics against logic, via beam-splitters
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
Quantum bomb tester
3.7.2
Orthonormal Pauli basis
3.7.3
Pauli matrix expansion coefficients
3.7.4
Linear algebra of the Pauli vector
3.7.5
Matrix Euler formula
3.7.6
Special orthogonal matrix calculations
3.7.7
Phase as rotation
3.7.8
Geometry of the Hadamard
3.7.9
Swiss Granite Fountain
3.7.10
Dynamics in a magnetic field
4
Measurements
4.1
Hilbert spaces, briefly
4.2
Complete measurements
4.3
The projection rule, and 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
Distinguishing 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 other things to come
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
5.9.9
Tensor products in components
5.9.10
Hadamard transforms in components
5.9.11
The Schmidt decomposition
6
Density matrices
6.1
Definitions
6.2
Statistical mixtures
6.3
Some instructive examples, questions, and remarks
6.4
The Bloch ball
6.5
Subsystems of entangled systems
6.6
Mixtures and subsystems
6.7
Partial trace, revisited
6.8
Remarks and exercises
6.8.1
Some density operator calculations
6.8.2
Purification of mixed states
6.8.3
Pure partial trace
6.8.4
Maximally Bell
6.8.5
Spectral decompositions and common eigenbases
7
Quantum channels
7.1
Everything is (secretly) unitary
7.2
Random unitaries
7.3
Random isometries
7.3.1
Three-qubit codes
7.4
Evolution of open systems
7.5
Stinespring’s dilation and Kraus’s ambiguity
7.6
Single-qubit channels
7.7
Composition of quantum channels
7.8
Completely positive trace-preserving maps
7.9
Channel-state duality
7.10
The mathematics of “can” and “cannot”
7.11
Kraus operators, revisited
7.12
Correctable channels
7.13
Remarks and exercises
7.13.1
Purifications and isometries
7.13.2
The Markov approximation
7.13.3
What use are positive maps?
7.13.4
Partial inner product
7.13.5
The “control” part of controlled-NOT
7.13.6
Surprisingly identical channels
7.13.7
Independent ancilla
7.13.8
Order matters?
7.13.9
Unchanged reduced density operator
7.13.10
Cooling down
7.13.11
No pancakes
7.13.12
Pauli twirl
7.13.13
Depolarising channel
7.13.14
Toffoli gate
7.13.15
Trace preserving and partial trace
7.13.16
Complete positivity of a certain map
7.13.17
Duals
7.13.18
Trace, transpose, Choi
7.13.19
Tricks with a maximally mixed state
8
Stabilisers
8.1
Pauli groups
8.2
Pauli stabilisers
8.3
Single stabiliser states
8.4
Measuring Pauli stabilisers
8.5
Mathematical detour: normal subgroups
8.6
Pauli normalisers
8.7
Clifford walks on stabiliser states
8.8
Remarks and exercises
8.8.1
Measuring parity
8.8.2
The Pauli group of three qubits
8.8.3
Half commuting
8.8.4
One out of four stabilisers
8.8.5
Stabilisers and projectors
8.8.6
Abelian Pauli quotients
8.8.7
Equivalent projective measurements
II Applications
9
Quantum cryptography
10
Bell’s theorem
10.1
Quantum correlations
10.2
Hidden variables
10.3
CHSH inequality
10.4
Quantum correlations, revisited
10.5
Tsirelson’s inequality
10.6
Remarks and exercises
11
Quantum algorithms
11.1
Quantum Boolean function evaluation
11.1.1
A worked example
11.2
Hadamard and quantum Fourier transforms
11.3
More phase kick-back
11.4
Oracles, and our first quantum algorithm
11.4.1
Deutsch’s algorithm
11.5
The Bernstein–Vazirani algorithm
11.6
Grover’s search algorithm
11.7
Simon’s algorithm
11.8
Remarks and exercises
11.8.1
Shor’s algorithm
11.8.2
RSA
11.8.3
More complexity classes
11.8.4
Implementing reflections
11.8.5
Grover’s optimality
12
Approximation
12.1
Metrics
12.2
How far apart are two quantum states?
12.3
Fidelity
12.4
Approximating unitaries
12.5
Approximating generic unitaries is hard, but…
12.6
How far away are two probability distributions?
12.7
Dealing with density operators
12.8
Distinguishing non-orthogonal states, again
12.9
How accurate is accurate enough?
12.10
Remarks and exercises
12.10.1
Operator decompositions
12.10.2
More operator norms
12.10.3
Fidelity in a trace norm inequality
12.10.4
Hamming distance
12.10.5
Operator norm
12.10.6
Tolerance and precision
12.10.7
Statistical distance and a special event
12.10.8
Joint probability distributions
12.10.9
Distinguishability and the trace distance
13
Decoherence and basic quantum error correction
13.1
Decoherence simplified
13.2
Decoherence and interference
13.3
Quantum errors
13.4
Some errors can be corrected on some states
13.5
Repetition codes
13.6
Remarks and exercises
14
Quantum error correction and fault tolerance
15
Further topics and selected reading
Introduction to Quantum Information Science
11.2
Hadamard and quantum Fourier transforms