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# Mathematical foundations for Quantum Computing

by | Dec 16, 2022 | Computing, Quantum Computing

Preface – This post is part of the Quantum Computing series.

The mathematical foundations of quantum computing are based on the principles of quantum mechanics, which is a fundamental theory in physics that describes the behavior of matter and energy at the atomic and subatomic scale. Quantum mechanics is based on the principles of wave-particle duality and quantum uncertainty, and it provides a mathematical framework for understanding the properties of quantum systems.

Some of the key mathematical concepts that are used in quantum computing include:

1. Wave functions: In quantum mechanics, the wave function is a mathematical function that describes the probability of finding a particle or system at a given position and orientation. The wave function is represented by the Greek letter psi (Ψ), and it is a key component of the quantum mechanical description of a system.
2. Operators: Operators are mathematical objects that act on functions to produce new functions. In quantum mechanics, operators are used to describe the physical quantities and properties of quantum systems, such as position, momentum, and energy.
3. Matrix algebra: Matrix algebra is a branch of mathematics that deals with matrices, which are rectangular arrays of numbers. Matrix algebra is an important tool in quantum computing because it is used to represent quantum states and quantum operations.
4. Linear algebra: Linear algebra is a branch of mathematics that deals with linear equations and vector spaces. Linear algebra is an important tool in quantum computing because it is used to describe the linear relationships between quantum states and quantum operations.
5. Group theory: Group theory is a branch of mathematics that deals with symmetry and the properties of groups, which are sets of elements that can be combined using certain operations. Group theory is an important tool in quantum computing because it is used to classify and understand the symmet

Here are a few more mathematical concepts that are used in quantum computing:

1. Tensors: Tensors are mathematical objects that generalize the concept of a matrix to higher dimensions. Tensors are used in quantum computing to represent quantum states and quantum operations in a compact and efficient way.
2. Eigenvalues and eigenvectors: In linear algebra, eigenvalues and eigenvectors are special types of scalars and vectors that are associated with linear transformations. Eigenvalues and eigenvectors are important in quantum computing because they are used to describe the properties of quantum states and quantum operations.
3. Fock space: In quantum mechanics, Fock space is a mathematical structure that is used to describe the states of systems with a fixed number of particles, such as atoms and photons. Fock space is an important concept in quantum computing because it is used to represent the states of quantum systems with a fixed number of qubits.
4. Unitary matrices: In linear algebra, a unitary matrix is a matrix that preserves the inner product of vectors. Unitary matrices are important in quantum computing because they are used to represent quantum operations that preserve the inner product of quantum states.
5. Schrödinger equation: The Schrödinger equation is a fundamental equation in quantum mechanics that describes the time evolution of a quantum system. The Schrödinger equation is used to predict the behavior of quantum systems and to understand the properties of quantum states.

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