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Level 2 · Complex numbers & multiple qubits · Unit 1 of 2

Complex numbers are turns

about 8 min · Free · No sign-in needed

Until now amplitudes were positive or negative: arrows pointing forward or backward along a line. An amplitude can also point sideways. Draw a single amplitude as an arrow in a plane: multiplying it by −1 turns it by 180°, and multiplying it by i turns it by 90°. That is all i is here: a quarter turn. Doing it twice, i × i = −1, is a half turn.

  1. 1. Predict
  2. 2. Run
  3. 3. Rules
  4. 4. Redraw
  5. 5. Explain
  6. 6. Done

Prediction 1 of 2

Simulated

Predict first, then run it.

1i−1−i

Amplitude of 1: 0.707 → × i → ?

Probability of 1?
The qubit is (|0⟩ + |1⟩)/√2, so the probability of 1 is 50%. Multiply only the amplitude of 1 by i. What is the probability of 1 now?

Clears this unit's progress and drawing record on this device.