There are several fundamental principles or axioms that dictate how quantum mechanical systems evolve over time. These principles or axioms are supported by various experimental or theoretical observations. Here, we will discuss these observations.

1. Unitary Transitions

In quantum systems, states are represented by unit vectors. Therefore, the transition of a quantum state from one to another must always be unitary (a special type of mathematical transformation). Unitary transformations preserve the properties of the system, meaning that the magnitude or norm of the system remains unchanged during these transformations.

This ensures that even as the quantum system evolves over time, the total energy or amount of information remains constant. In other words, unitary transformations keep the system’s evolution orderly and consistent, preventing information loss and ensuring energy conservation.

2. Schrödinger Equation

The Schrödinger equation is a fundamental equation that determines how quantum systems change over time. This equation mathematically describes the time evolution of a quantum system’s current state.

–iħ ∂|ψ(t)⟩/∂t = H|ψ(t)⟩

Let’s explain the variables here:

• |ψ(t)⟩ represents the state of the system at a given time.
• H is the operator representing the energy of the system, which dictates the behavior of the quantum system.
• i is the imaginary unit (known as the square root of -1 in mathematics).
• t is the time variable.

The solution to the Schrödinger equation provides a formula that shows how a system’s initial state evolves over time:

–|ψ(t)⟩ = U(t) |ψ(0)⟩

Where:

• U(t) is the operator that transitions the system from its initial state to a new state over time. This operator is unitary, meaning that it preserves the total energy and information as it evolves the system.

Finally, this unitary operator represents the computation steps in quantum computers. While classical computers rely on Boolean functions to perform basic logical operations, quantum computers use these unitary operators to manage the transition from input to output. Understanding how these operators work is crucial for grasping the fundamental logic and operation of quantum computing.

In summary, the unitary nature of quantum dynamics ensures the reliability of information processing and computation, while the Schrödinger equation provides the mathematical framework for these processes. Consequently, quantum computation, unlike classical computation, allows for the effective management and resolution of complex systems.