A product state combines independent single-qubit states into one composite quantum state. Like tossing two independently biased coins, its outcomes are all pairings of the constituent outcomes, and each composite amplitude is the product of two single-qubit amplitudes. section 4.2.1
Construct a two-qubit product state by:
- Represent each single-qubit state with two complex amplitudes, using probability magnitudes and phase directions. section 4.2.1
- Multiply every amplitude from the first state by every amplitude from the second, producing four amplitudes in outcome order. section 4.2.1
- Use polar-form arithmetic: multiply magnitudes and add phase directions. section 4.2.1
- Extend the pattern recursively across qubits when encoding geometric-sequence states with Hadamard and phase gates. section 8.2
Multiplying two single-qubit states
For p = 0.75, θ₀ = 0°, θ₁ = 60°, q = 0.5, φ₀ = 0°, and φ₁ = −120°, the resulting amplitudes are [(0.61237+0j), (−0.30619−0.53033j), (0.17678+0.30619j), (0.17678−0.30619j)]. The same list results from direct complex multiplication or from multiplying the probabilities and adding the phase directions. section 4.2.1
new_state = [first_state[0]*second_state[0], first_state[0]*second_state[1],
first_state[1]*second_state[0], first_state[1]*second_state[1]]
print([round(amp.real, 5)+1j*round(amp.imag, 5) for amp in new_state])
Inspect the compact Python construction that forms the four amplitudes by multiplying the two constituent amplitude lists. section 4.2.1
