Matrices

Quantum Computing • Unit 3

Matrices represent quantum operations and transformations. From basic matrix operations to unitary matrices, this unit provides the tools needed to describe how quantum gates manipulate qubits.

Core Concepts

Matrix fundamentals – entries, dimensions, and notation for real and complex matrices
Matrix arithmetic – addition, subtraction, and scalar multiplication
Matrix-vector multiplication – how matrices transform vectors (and quantum states)
Matrix multiplication – composing transformations and building quantum circuits
Transpose and conjugate transpose – the operations that define Hermitian and unitary matrices
Identity and unitary matrices – the reversible transformations of quantum mechanics

Key Equations

$(AB)_{ij} = \sum_k A_{ik} B_{kj}$

$A^\dagger = (A^*)^T$

$U^\dagger U = UU^\dagger = I$

$(AB)^\dagger = B^\dagger A^\dagger$

Real-World Applications

→Representing quantum gates as unitary matrices
→Computing the effect of quantum circuits
→Understanding reversibility in quantum computation
→Verifying that operations preserve probability

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