The Standard Computational Bases

In the previous article, I covered the Bloch sphere, one of the most helpful visual tools for understanding qubits. The Bloch sphere allows us to represent a qubit’s state as a point in three-dimensional space. While this point represents the qubit’s state, it’s often even more useful to consider its position in relation to one of the three primary axes on the sphere – x, y, and z. These axes are not arbitrary; each one is linked to a unique set of orthonormal basis vectors, commonly referred to as computational bases.

In quantum computing, a computational basis provides a standardised framework to measure or affect a qubit’s state. If you’re unfamiliar with the term “orthonormal basis vectors,” consider reviewing this helpful Wikipedia page on orthonormality. These bases are “computational” because they lay the groundwork for every computation we perform in quantum computing.

While any set of orthonormal basis vectors could theoretically serve as computational bases in quantum computing, there are three primary ones that are used most frequently: the computational basis, the Hadamard basis, and the circular basis. Each of these serves a distinct purpose in interpreting and manipulating qubit states.

The Computational Basis

The computational basis, aligned with the z-axis of the Bloch sphere, is the foundation of quantum measurement. It’s the most common basis for reading the results of quantum computations because it directly corresponds to binary classical states, making it easier to translate quantum states into familiar classical outputs. The computational basis consists of two column vectors and , often written in Dirac notation as |0⟩ and |1⟩. These two states represent the “north” and “south” poles of the Bloch sphere, and they correspond to the conventional binary states, 0 and 1. When a qubit is measured in this basis, it “collapses” to one of these two poles, giving us a straightforward binary result.

The Hadamard Basis

Next, the Hadamard basis is associated with the x-axis of the Bloch sphere, which represents a state along the sphere’s equator rather than at its poles. This basis is one of the most powerful tools in quantum computing because it allows us to use any real number as a qubit’s state, not just the discrete values of 0 or 1. For a deeper dive into why this is significant, check out my article on quantum vs classical computing. The Hadamard basis enables superdense coding, a technique that leverages quantum superposition to represent classical bits in more complex ways. 

The Hadamard basis consists of two vectors and , denoted in Dirac notation as |+⟩ and |-⟩. These states lie along the equator of the Bloch sphere, representing the ability of qubits to exist in a superposition of 0 and 1 simultaneously. This property underlies many of quantum computing’s most promising applications, such as quantum parallelism, where the system explores multiple possibilities at once, and quantum encryption, where superpositions contribute to secure data encoding.

The Circular Basis

Finally, the circular basis aligns with the y-dimension of the Bloch sphere and introduces an essential concept called phase. In quantum mechanics, a qubit’s phase describes the relative angle or orientation of its wave function in its quantum state. This phase does not impact the probability of measuring the qubit in a particular state; rather, it influences how qubits interact with each other, particularly in superpositions and entangled states. 

This phase-based interference is foundational for many quantum algorithms and gate operations, such as the Hadamard gate and controlled-phase gates, which rely on phase manipulation to perform complex calculations. The circular basis uses two vectors and , usually written as |i⟩ and |-i⟩ in Dirac notation, to describe these states. This notation represents the relative phase of the qubit, allowing for fine-tuned control over quantum states and facilitating advanced computational processes like error correction and phase estimation.

Conclusion

Understanding the computational bases—computational, Hadamard, and circular—is essential for anyone delving into quantum computing. These bases provide different frameworks for interpreting and manipulating qubit states, whether through straightforward binary measurement, superposition, or phase manipulation.

Each basis has a unique role: the computational basis is the go-to for classical readouts, the Hadamard basis introduces superpositions that allow for more complex computations, and the circular basis enables control over phase, crucial for advanced algorithms and gate operations.

By mastering these foundations, we gain a toolkit to understand and perform quantum operations more effectively, unlocking the full potential of qubits for quantum computing tasks. Whether you’re analysing measurement outcomes or engineering interference patterns, these computational bases form the core of your journey through quantum mechanics and computation.