What is Physics

1. States
2. Dynamics
3. Measurements

What is Computation

1. Data
2. Functions
3. Output

They are the same things

Studying one of these is intrinsically related to studying the other.

Classical Mechanics - Real numbers

1. Single particle
2. Real numbers - \(x\)
3. Hamiltonian/Lagrangian/Newtonain Dynamics - \(x^\prime = F(x)\)
4. Observe with a meter - \(M(x)\)
5. Multiple objects, repeat numbers

Classical Mechanics - Probability Distributions

1. Ensembles
2. Probability vectors - \(p=\mathbb{R}^n, \sum_{i=1}^{n}p_i = 1\)
   1. Vector space over convex combinations over real numbers
   2. Inner product gives overlap
3. Column stochastic matrices - \(T = n \times n\) square matrix, \(\sum_{j=1}^{n}T_{i,j} = 1, p^\prime = Tp\)
4. Observe the system with a meter - \(\bar{M}(p)\)

Quantum Mechanics - Complex numbers

1. Single particle

2. Complex numbers as probability amplitudes - \(|\psi\rangle=\mathbb{C}^n, \sum_{j=1}^{n} |c_i|^2 = 1\)

   1. Vector space over convex combinations over complex numbers
   2. Inner product gives overlap

3. Unitary matrices - \(U = n \times n\) square matrix, \(U^\dagger U = I_n = UU^\dagger\)

   • \(|\psi^\dagger\rangle\) = \(U |\psi\rangle\)

4. Measurements

   • Let \(P_m\) = \(|m \rangle\langle m|\)
   • \(p(m) = \langle \psi | P_m | \psi \rangle\)

5. Multiple particles, tensor product

   • \(|\psi_1\rangle \otimes |\psi_2\rangle\)
   • \(U_1 \otimes U_2\)
   • \(p(m) = \langle \psi | P_m | \psi \rangle\)

2 Level systems

\[\cos(\theta) |0\rangle + \sin(\theta) e^{i\phi} |1\rangle\]

• Points on a sphere
• Unitaries are rotations
  • \(S_z\), \(S_x\), \(S_y\)
  • \(H\), \(T\)
• CNOT

Circuit Diagrams

• Serial Composition
• Parallel Composition
• Circuit Depth

Linear Search

Grover Search

1. Oracle
2. Diffusion