Can amplitudes be added (or multiplied) or incremented or decremented? Does rotation about the Y axis do this?

[quantumadmin]

In quantum computing, amplitudes are complex numbers that represent the probability coefficients of quantum states. These amplitudes are manipulated through quantum gates to perform computations. Let's address your questions step by step:

Adding or Multiplying Amplitudes:
Quantum amplitudes cannot be simply added or multiplied in the same way as classical numbers. Quantum operations, represented by quantum gates, affect the amplitudes in a way that takes into account their complex nature. When two quantum states are combined, their amplitudes interfere in a quantum superposition, leading to constructive or destructive interference.

Incrementing or Decrementing Amplitudes:
Quantum gates like the Pauli-X gate (NOT gate) can flip the sign of amplitudes, effectively incrementing or decrementing their phase. However, the overall probability distribution, which is determined by the squared magnitudes of the amplitudes, remains unchanged.

Rotation About the Y-Axis:
Rotation about the Y-axis in the Bloch sphere representation does affect the amplitudes, but it's not a straightforward increment or decrement operation. The Y-axis rotation is typically performed using the Y gate or the more general Y-rotation gate. This operation changes the relative magnitudes and phases of the amplitudes, effectively causing a rotation in the quantum state's superposition.

For a single qubit, a Y-rotation gate applied to a quantum state |0⟩ would transform it into a superposition state (α|0⟩ + β|1⟩), where the amplitudes α and β are modified by the Y-rotation. The specific effect depends on the angle of rotation.

For example, a Y-rotation of π/2 radians (90 degrees) would transform |0⟩ into (1/√2)|0⟩ + (i/√2)|1⟩, effectively introducing a phase difference and changing the probability distribution.

Combining Amplitudes from Two Qubits:
Combining amplitudes from two qubits involves entanglement, which is a more complex phenomenon. Entanglement allows for correlations between qubits that cannot be explained by classical means. Quantum gates acting on entangled qubits can result in complex changes to their combined state, often involving non-local interactions that can't be easily explained in classical terms.

While there are quantum gates that can change the relative phases and magnitudes of amplitudes, the operations are inherently quantum and not directly analogous to classical arithmetic. Quantum gates are designed to manipulate amplitudes in ways that exploit the principles of superposition and entanglement, which are unique to the quantum realm.