# quantum / Summary

A progressive list of facts about how quantum computing works.

• Qubits are vectors in a 2D vector space

  • the î equivalent is |0> (zero computational basis state)
  • the ĵ equivalent is |1> (one/excited computational basis state)

• All Qubit vectors have length 1 (normalised)

• Logic gates are unitary matrices

  • Unitary matrices preserve length
  • X/NOT gate
  • Hadamard gate

• Measurement: To Explain

• 2-qubit systems have 4 computational basis states: |00>, |01>, |10>, |11>

  • these can form linear combinations just like a 1-qubit system
  • e.g. α|00>+β|01>+γ|10>+δ|11>, where each coefficient is a complex number
  • The square sum of the amplitudes is still 1

• The controlled-not (CNOT) gate

  • If the first qubit is 1, it flips the second e.g. |10> -> |11>, |11> -> |10>
  • Otherwise it leaves the gate as is e.g. |00>, |01>
  • The second qubit becomes the result of XOR-ing the original two values

Open Questions

• Unitary Matrices
  • An intuitive proof for unitary matrices being the only length-preserving transformations
  • How does my straightforward map back to the traditional formulae? –
• What does it mean to have vectors & matrices containing complex numbers, in the Quantum computing context?