Table of contents
Introduction
Plan and intended audience
Using the e-book
How to cite this 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
0.11
Probabilities
I Foundations
1
Quantum interference
1.1
Two basic rules
1.2
The failure of probability theory
1.3
The double-slit experiment
1.4
Superpositions
1.5
Interferometers
1.6
Qubits, gates, and circuits
1.7
Quantum decoherence
1.8
Types of computation
1.9
Computational complexity
1.10
Outlook
1.11
Remarks and exercises
1.11.1
A historical remark
1.11.2
Modifying the Born rule
1.11.3
Many computational paths
1.11.4
Distant photon emitters
1.11.5
Quantum Turing machines
1.11.6
More time, more memory
1.11.7
Asymptotic behaviour: big-O
1.11.8
Polynomial is good, and exponential is bad
1.11.9
Imperfect prime tester
1.11.10
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
2.8
From bit-flips to phase-flips, and back again
2.9
Any unitary operation on a single qubit
2.10
The Bloch sphere
2.11
Drawing points on the Bloch sphere
2.12
Composition of rotations
2.13
A finite set of universal gates
2.14
Remarks and exercises
2.14.1
One simple circuit
2.14.2
Change of basis
2.14.3
Operators as unitary matrices
2.14.4
Completing an orthonormal basis
2.14.5
Some sums of inner products
2.14.6
Some circuit identities
2.14.7
Unitaries preserve length
2.14.8
Unknown phase
2.14.9
One of the many cross-product identities
3
Quantum gates
3.1
Beam-splitters: physics against logic
3.2
Beam-splitters: quantum interference, revisited
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
Calculating a Pauli rotation
3.7.9
Geometry of the Hadamard
3.7.10
Swiss Granite Fountain
3.7.11
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
4.12
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
Entanglement
5.1
A very brief history
5.2
From one qubit to two
5.3
Quantum theory, formally (continued)
Tensor products
5.4
More qubits, and binary representations
5.5
Separable or entangled?
5.6
Controlled-NOT
5.7
Bell states
5.8
Quantum teleportation
5.9
No-cloning, and other no-go theorems
5.10
Controlled-phase and controlled-U
5.11
Universality, revisited
5.12
Phase kick-back
5.13
Density operators, and other things to come
5.14
Remarks and exercises
5.14.1
Why qubits, subsystems, and entanglement?
5.14.2
Entangled or not?
5.14.3
Instantaneous communication
5.14.4
SWAP circuit
5.14.5
Controlled-NOT circuit
5.14.6
Measuring with controlled-NOT
5.14.7
Arbitrary controlled-U on two qubits
5.14.8
Entangled qubits
5.14.9
Quantum dense coding
5.14.10
Playing with conditional unitaries
5.14.11
Tensor products in components
5.14.12
Hadamard transforms in components
5.14.13
The Schmidt decomposition
II Further foundations
6
Bell’s theorem
6.1
Hidden variables
6.2
Quantum correlations
6.3
The CHSH inequality
6.4
Bell’s theorem via CHSH
6.5
Tsirelson’s inequality
6.6
Quantum randomness
6.7
Loopholes in Bell tests
6.8
Remarks and exercises
6.8.1
XOR games
6.8.2
XOR games for quantum key distribution
6.8.3
XOR games for randomness expansion
6.8.4
Prescribed binary randomness
6.8.5
Unbiasing bias
6.8.6
Proving Tsirelson’s inequality
6.8.7
Detector loophole
6.8.8
Locality loophole
6.8.9
Free-will loophole
7
Stabilisers
7.1
Pauli groups
7.2
Pauli stabilisers
7.3
Single stabiliser states
7.4
Measuring Pauli stabilisers
7.5
Normal subgroups
7.6
Pauli normalisers
7.7
Clifford walks on stabiliser states
7.8
Remarks and exercises
7.8.1
Measuring parity
7.8.2
The Pauli group of three qubits
7.8.3
Half commuting
7.8.4
One out of four stabilisers
7.8.5
Stabilisers and projectors
7.8.6
Abelian Pauli quotients
7.8.7
Equivalent projective measurements
8
Density matrices
8.1
Definitions
8.2
Statistical mixtures
8.3
Instructive examples
8.4
The Bloch ball
8.5
Subsystems of entangled systems
8.6
Mixtures and subsystems
8.7
Partial trace, revisited
8.8
Remarks and exercises
8.8.1
Some density operator calculations
8.8.2
Purification of mixed states
8.8.3
Pure partial trace
8.8.4
Maximally Bell
8.8.5
Spectral decompositions and common eigenbases
9
Quantum channels
9.1
Everything is (secretly) unitary
9.2
Random unitaries
9.3
Random isometries
9.4
Evolution of open systems
9.5
Stinespring’s dilation and Kraus’s ambiguity
9.6
Single-qubit channels
9.7
Composition of quantum channels
9.8
Completely positive trace-preserving maps
9.9
Channel-state duality
9.10
The mathematics of “can” and “cannot”
9.11
Kraus operators, revisited
9.12
Remarks and exercises
9.12.1
Purifications and isometries
9.12.2
The Markov approximation
9.12.3
What use are positive maps?
9.12.4
Partial inner product
9.12.5
The “control” part of controlled-NOT
9.12.6
Surprisingly identical channels
9.12.7
Independent ancilla
9.12.8
Order matters?
9.12.9
Unchanged reduced density operator
9.12.10
Cooling down
9.12.11
No pancakes
9.12.12
Pauli twirl
9.12.13
Depolarising channel
9.12.14
Toffoli gate
9.12.15
Expressing vectors using the maximally mixed state
9.12.16
Complete positivity of a certain map
9.12.17
Duals
9.12.18
Trace, transpose, Choi
9.12.19
Entanglement witness
9.12.20
Almost Kraus decomposition
9.12.21
Tricks with a maximally mixed state
III Applications and reality
10
Quantum algorithms
10.1
Quantum Boolean function evaluation
10.2
More phase kick-back
10.3
Oracles
10.4
Deutsch’s algorithm
10.5
The Bernstein–Vazirani algorithm
10.6
Grover’s search algorithm
10.7
Simon’s algorithm
10.8
Phase estimation
10.9
Quantum Fourier transform
10.10
Hidden-order determination
10.11
Shor’s algorithm
10.12
Remarks and exercises
10.12.1
RSA
10.12.2
More complexity classes
10.12.3
Implementing reflections
10.12.4
Grover’s optimality
10.12.5
Picking out a single state
10.12.6
Writing an integer as a power
11
Quantum cryptography
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 apart are two probability distributions?
12.7
Dealing with density operators
12.8
Distinguishing non-orthogonal states, again
12.9
Approximate phase estimation
12.10
How accurate is accurate enough?
12.11
Remarks and exercises
12.11.1
Operator decompositions
12.11.2
More operator norms
12.11.3
Fidelity in a trace norm inequality
12.11.4
Hamming distance
12.11.5
Operator norm
12.11.6
Tolerance and precision
12.11.7
Statistical distance and a special event
12.11.8
Joint probability distributions
12.11.9
Distinguishability and the trace distance
13
Decoherence and recoherence
13.1
The three-qubit code
13.2
Towards error correction
13.3
Discretisation of quantum errors
13.4
Digitising quantum errors
13.5
Recoherence
13.6
The classical repetition code
13.7
Correcting bit-flips
13.8
Correcting phase-flips
13.9
Composing correctable channels
13.10
Correcting any single error: Shor [[9,1,3]]
13.11
Error-correcting assumptions
13.12
Remarks and exercises
13.12.1
Decoherence-free subspaces
13.12.2
Repetition encoding and majority voting failure
13.12.3
Correcting Pauli rotations with three qubits
13.12.4
More on Shor [[9,1,3]]
13.12.5
Distillation for Bell pairs
13.12.6
Composing quantum codes
14
Quantum error correction
14.1
The Hamming code
14.2
Linear codes
14.3
Quantum codes from classical
14.4
Logical operators …
14.5
… and error families
14.6
Logical operators (a different approach)
14.7
Error-correcting conditions
14.8
Code distance and thresholds
14.9
Encoding circuits
14.10
Encoding arbitrary states
14.11
Remarks and exercises
14.11.1
Error correcting conditions for the three-qubit code
14.11.2
The smallest
d=3
code, full stop
14.11.3
Hamming code encodings and decodings
14.11.4
Generator and parity-check matrices
14.11.5
A big parity check matrix
14.11.6
Using Tanner graphs
14.11.7
Five-qubit repetition code
14.11.8
An error in the Steane
[[7,1,3]]
code
14.11.9
Non-degenerate codes
14.11.10
The smallest
d=3
CSS code
14.11.11
CSS codes from a single matrix
14.11.12
Error-correcting conditions, algebraically
14.11.13
Steane error correction: towards fault tolerance
15
Fault tolerance
Appendix:
Further topics and selected reading
Introduction to Quantum Information Science
4.10
Wiesner’s quantum money