Phases

 Been struggling to understand the notion of phase with respect to qubits.

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...the electron propagates through space like a wave, it interacts at a point like a particle. This is known as wave-particle duality.

https://theconversation.com/explainer-what-is-wave-particle-duality-7414#:~:text=The%20energy%20of%20the%20electron,known%20as%20wave%2Dparticle%20duality.

I think of a phase as a quantum of time. In principle, time is continuous and hence 

endlessly divisible; but in dealing with fundamental particles, one holds time constant

with a wave function...

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GEMINI:

                                                             

                                                                           

In quantum computing, the phase of a qubit refers to a property related to the complex number that describes its state. It's a bit more nuanced than the classical concept of "on" or "off" for a bit.

Here's a breakdown:

* **Qubits and Superposition:** Unlike classical bits which are either 0 or 1, qubits can exist in a state called superposition. This means they can be a combination of both 0 and 1 simultaneously, represented by a complex number (α |0⟩ + β |1⟩).

* **Complex Numbers and Phase:** This complex number has two parts: amplitude (magnitude) and phase. The amplitude (|α|² and |β|²) determines the probability of measuring 0 or 1 when the qubit is measured. The phase (often denoted by φ) is encoded in the argument of the complex number.

* **Global vs. Relative Phase:** It's important to understand that the overall phase (φ) of a single qubit state doesn't physically affect the state itself. This is because adding a constant phase shift to all the complex amplitudes in a superposition doesn't change the probabilities of measuring 0 or 1. This global phase indifference is a quirk of quantum mechanics.

* **Relative Phase and Interference:** However, the relative phase between different qubits becomes crucial in multi-qubit systems. This relative phase can lead to constructive or destructive interference, affecting the overall outcome of quantum computations. 

**Thinking of Phase:**

Imagine a spinning coin. The state of the coin (heads or tails) is analogous to the classical bit (0 or 1). Now, imagine the coin can also be spinning at different speeds or directions. This additional information about the spin could be loosely compared to the phase of a qubit. While the spin itself doesn't determine heads or tails (like global phase), it can influence how the coin interacts with other spinning coins (like relative phase in multi-qubit systems). 

**Key Points:**

* Phase of a qubit is encoded in the complex number representing its state. 

* Global phase shift has no physical effect on a single qubit.

* Relative phase between qubits is crucial for understanding quantum algorithms.  *    

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It's a physical system, and one interacts with it through gates. The particles

- at near zero Kelvin - are zapped by lasers or some such!!    

Bloch sphere visualizer (kherb.io)