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Linear Algebra & Quantum Computing

two parallel tracks · discovery through proof
Track LA — Linear Algebra Track 0 — Quantum Computing
Linear Algebra feeds Quantum Computing. Both require rigor, both require proof. Months 1–6 run parallel.
Linear Algebra Quantum Computing
the full map

Two tracks · parallel

Linear Algebra (months 1–2) is the prerequisite for quantum mechanics. Run them in parallel. After Linear Algebra is complete, quantum computing deepens continuously through month 6.

TRACK LA — months 1–2
Linear Algebra Foundations
Vector spaces, eigenvalues, inner products, unitary operators. The language of quantum mechanics.
AxlerFoundations
TRACK 0 — months 1–6
Quantum Computing Foundations
Qubits, entanglement, Bell inequalities, quantum algorithms, quantum information. UvA QCS preparation.
de WolfNielsen & Chuang
Track LA
Linear Algebra Foundations
why this track

The language of quantum mechanics

→ A quantum state is a unit vector in a complex vector space. A quantum gate is a unitary linear transformation. Measurement is projection onto eigenvectors. Complete fluency before quantum mechanics makes sense.
PHASE 01
Axler — Linear Algebra Done Right
Chapters 1–8 · your complete foundation
primary ▼
Ch. 1–2
Vector spaces and linear maps
Definition of vector spaces over ℂ and ℝ. Linear maps, kernels, ranges. Fundamental theorem: a linear map is determined by where it sends basis vectors.
week 1
Ch. 5–6
Eigenvalues and inner products
Eigenvalues, eigenvectors, diagonalization. Inner products, orthonormal bases, projections. The geometry of vector spaces.
week 2–3
Ch. 7–8
Operators on inner product spaces
Self-adjoint operators, normal operators, spectral theorem. Unitary transformations — operators that preserve quantum states.
week 4–5
application
Qubits and quantum gates
Translate linear algebra into quantum. Qubits as vectors. Gates as unitary matrices. Where theory becomes computation.
week 6
  • Vector spaces over ℂ Basis, dimension, linear independence.
    Ch. 1
  • Linear maps Matrix representation, matrix multiplication, change of basis.
    Ch. 2–3
  • Eigenvalues Characteristic polynomial, eigenspaces, when diagonalizable.
    Ch. 5
  • Inner products Dot products in ℂⁿ. Orthogonal and orthonormal bases. Gram-Schmidt.
    Ch. 6
  • Spectral theorem Unitary and self-adjoint operators. Every self-adjoint diagonalizable by unitary.
    Ch. 7–8 ★
End of Track LA: Complete mastery of vector spaces, eigenvalues, inner products, and unitary transformations. You see quantum states as vectors and gates as unitary matrices. Ready for quantum mechanics.
Track 0
Quantum Computing Foundations UvA QCS PREP
why this track

The physics of quantum computing

→ Quantum computing is genuinely different from classical. This track covers what makes it different: superposition, entanglement, interference, measurement, and quantum algorithms that leverage these properties. Preparation for UvA QCS and quantum research.
PHASE 01 · months 1–2
Quantum mechanics fundamentals
de Wolf inaugural + Nielsen & Chuang Ch. 2
foundations ▼
core
Qubits and superposition
A qubit is a unit vector in ℂ². Superposition: |ψ⟩ = α|0⟩ + β|1⟩. Amplitudes encode quantum information. No classical analog.
week 1
measurement
Born rule and collapse
Measuring gives |0⟩ with probability |α|², |1⟩ with probability |β|². Measurement collapses the superposition.
week 2
multi-qubit
Entanglement and tensor products
Two qubits live in ℂ⁴. Entangled states like (|00⟩ + |11⟩)/√2 cannot be written as products. Entanglement violates Bell inequalities.
week 3–4
operators
Quantum gates and unitarity
Quantum gates are unitary matrices. Pauli gates, Hadamard, CNOT. Unitarity preserves norms — evolution is reversible.
week 4–5
  • Qubits Unit vectors in ℂ². Bloch sphere representation.
    Nielsen 2.1
  • Superposition Linear combinations of basis states. No classical analog.
    Nielsen 2.1
  • Born rule Measurement probabilities from amplitudes.
    Nielsen 2.2
  • Entanglement Tensor products. |ψ⟩ ≠ |ψ₁⟩ ⊗ |ψ₂⟩ for entangled.
    Nielsen 2.3
  • Quantum gates Unitary matrices. Pauli, Hadamard, CNOT, phase.
    Nielsen 2.4
End of Phase 1: You understand qubits, superposition, entanglement, and gates. You see why quantum is different from classical.
PHASE 02 · months 2–3
Bell inequalities and quantum foundations
de Wolf + Preskill lecture notes
philosophy ▼
local realism
Bell's theorem
If the world is local and realistic, correlations obey Bell's inequality. Entangled states violate it. Proven experimentally. The world is non-local or non-realistic.
week 1–2
interpretation
Many-worlds and Copenhagen
Copenhagen: collapse is real. Many-worlds: branching universes, no collapse. Experiments don't decide. Many-worlds reveals: QC leverages interference between branches.
week 2–3
decoherence
Why we don't see superposition
Classical behavior emerges when systems interact with environment. QC fights decoherence, harnesses it, then protects against unwanted decoherence.
week 3–4
limits
No-cloning theorem
You cannot copy an unknown quantum state. Limits what QC can do but shows where it differs from classical.
week 4–5
  • Bell inequalities CHSH ≤ 2 for local realism. Quantum: CHSH ≤ 2√2.
    de Wolf
  • Experiments Aspect 1981, Bell tests, closing loopholes.
    experiment
  • Interpretations Copenhagen, many-worlds, pilot wave.
    philosophy
  • Decoherence Environment interaction causes apparent collapse. Not fundamental.
    Zurek
  • No-cloning Cannot copy unknown quantum state. Full proof.
    Nielsen 12.1
End of Phase 2: You understand Bell inequalities, why decoherence connects quantum to classical, and how QC leverages superposition while managing decoherence.
PHASE 03 · months 3–4
Quantum algorithms
Nielsen & Chuang Ch. 5–6 + de Wolf
algorithms ▼
Shor
Factoring in polynomial time
Quantum Fourier transform factors integers in polynomial time. Classical: exponential. Breaks RSA encryption. Why quantum computing matters.
week 1–2
Grover
Search with quadratic speedup
Search unsorted database in O(√N) instead of O(N). Works via amplitude amplification. Answer becomes more likely after iteration.
week 2–3
simulation
Simulating quantum systems
QC can efficiently simulate other quantum systems. Exponential speedup over classical. Applications: drug discovery, materials science.
week 3–4
complexity
BQP and quantum advantage
BQP: problems solvable in polynomial time on quantum. What problems are provably faster? Still open for many applications.
week 4–5
  • QFT Quantum Fourier transform. Core of Shor's algorithm.
    Nielsen 5.1
  • Shor's algorithm Full proof. Order finding, factoring. Why it works.
    Nielsen 5.2 ★
  • Grover's algorithm Amplitude amplification. Quadratic speedup. Optimality.
    Nielsen 6.1
  • Phase estimation Find eigenvalues of quantum operators. Basis for many algorithms.
    Nielsen 5.2
  • Quantum simulation Simulating Hamiltonian dynamics. Trotter formula.
    Nielsen 4.7
End of Phase 3: You understand Shor (why quantum breaks RSA) and Grover (quadratic speedup). You see how QC leverages superposition, interference, entanglement. You know what "quantum advantage" means.
PHASE 04 · months 4–6
Quantum information theory
Nielsen & Chuang Ch. 11–12 + de Wolf notes
★ foundation ▼
density matrices
Mixed states
Not all quantum states are pure |ψ⟩. Mixed states from ignorance or entanglement with environment. Described by density matrices ρ.
week 1–2
channels
Quantum channels and evolution
Channels are completely positive trace-preserving maps. How quantum information transforms, including decoherence. Kraus representation.
week 2–3
entropy
Von Neumann entropy
S(ρ) = -Tr(ρ log ρ). Quantifies quantum information. Maximum when maximally mixed, zero for pure states.
week 3–4
resource
Quantum information as resource
Entanglement, quantum correlations. Teleportation, dense coding, superdense coding. How to use quantum information for advantage.
week 5–6
  • Pure and mixed states Density matrix formalism. Trace, eigendecomposition.
    Nielsen 2.4
  • CPTP maps Completely positive, trace-preserving. Quantum channels.
    Nielsen 8
  • Kraus representation Every channel has Kraus decomposition. Operator-sum representation.
    Nielsen 8.2
  • Von Neumann entropy S(ρ) = -Σᵢ λᵢ log λᵢ. Measurement of quantum uncertainty.
    Nielsen 11.1 ★
  • Teleportation Send quantum state using classical bits and entanglement. Protocol and proof.
    Nielsen 1.3
  • Superdense coding Send 2 classical bits with 1 qubit using entanglement.
    Nielsen 1.4
End of Track 0: You understand quantum information theory. Density matrices, channels, entropy. Entanglement is a resource. Ready for UvA QCS coursework and quantum research.
pink, but lethal. 💗 — linear algebra and quantum computing · one foundation, two directions · UvA QCS preparation