In quantum computing, a qubit is the fundamental unit of quantum information. It is the quantum analogue of a classical bit, which can represent either a 0 or a 1. However, unlike classical bits, qubits can exist in a superposition of states, representing both 0 and 1 simultaneously.
To mathematically represent the states of a qubit, we use a vector-based formalism, typically in a two-dimensional complex vector space called the Hilbert space. The choice of a vector representation is due to several reasons:
Hilbert Space: Quantum mechanics describes the behavior of quantum systems using the language of linear algebra, which deals with vectors and vector spaces. The Hilbert space provides a mathematical framework to describe the possible states of a quantum system, including qubits. Each qubit can be represented as a vector in this Hilbert space.
Superposition: One of the defining features of qubits is the ability to exist in a superposition of states. In a superposition, a qubit can be in a combination of the 0 and 1 states, with certain probability amplitudes assigned to each state. Using vectors allows us to express these probability amplitudes and compute their combinations.
Quantum Gates: Quantum gates are the fundamental operations that manipulate qubits. They correspond to unitary transformations on the vectors representing the qubit states. By applying these operations, we can perform computations and transformations on qubits. The vector representation facilitates the application of these gates and allows us to track the state changes.
Measurement: When a qubit is measured, it collapses into one of the basis states (0 or 1) with a certain probability determined by the amplitudes in its superposition. The vector representation enables us to calculate these probabilities and determine the outcome of measurements.
By representing qubit states as vectors, we can leverage the mathematical tools of linear algebra to reason about and manipulate qubits, enabling the development of quantum algorithms and computations.