Quantum Operations are visualized as Unitary matrices. This puts a limit to their sparseness. A Quantum Operation acting on $n$ qubits can be represented by Unitary Matrix of size $2^n \times 2^n$. However since these are unitary operations, every column of the Unitary must have $L2$ norm as $1$ and hence have atleast 1 non-zero value. This implies that out of the $2^n \times 2^n = 2^{2n}$ elements of the Unitary atleast $2^n$ must be non-zero.

Hence for a Quantum Operation acting on a given $n$ qubits. We can define a most sparse matrix criteria. Unitaries which have the minimum i.e. $2^n$ non-zero elements can be said to satisfy this criteria.
The most common/canonical Quantum Operations which satisfy the most sparse matrix criteria are:

1. Single Qubit Pauli Gates $X,Y,Z,I$ .
2. Single Qubit Phase Gates $S,T,R_1(\theta),R_Z(\theta)$ .
3. Multi-Qubit $CNOT$, $SWAP$ gate.
4. The most sparse matrix criteria can be said to be closed under the Tensor product operator as $\otimes$. Most Sparse matrix $\otimes$ Most Sparse matrix is also a Most Sparse matrix.
5. Other common examples of these gates can be the $Increment, Decrement$ and $Shift$ Unitaries.

We can also define a least sparse criteria which can be said to include Unitaries which have the maximum i.e. $2^2n$ non-zero elements. The most common/canonical Quantum Operations which satisfy the most sparse matrix criteria are:

1. Hadamard Operator $H$ for Single Qubit Operations.
2. Single Qubit Rotation Gates $R_x(\theta),R_y(\theta)$ for non-trivial $\theta$
3. Quantum Fourier Transform $QFT$ for Multi Qubit Operations is another operations whose matrix representation can be said to be least sparse.

Quantum Operations are visualized as Unitary matrices. This puts a limit to their sparseness. A Quantum Operation acting on $n$ qubits can be represented by Unitary Matrix of size $2^n \times 2^n$. However since these are unitary operations, every column of the Unitary must have $L2$ norm as $1$ and hence have atleast 1 non-zero value. This implies that out of the $2^n \times 2^n = 2^{2n}$ elements of the Unitary atleast $2^n$ must be non-zero.

Hence for a Quantum Operation acting on a given $n$ qubits. We can define a most sparse matrix criteria. Unitaries which have the minimum i.e. $2^n$ non-zero elements can be said to satisfy this criteria.
The most common/canonical Quantum Operations which satisfy the most sparse matrix criteria are:

1. Single Qubit Pauli Gates $X,Y,Z,I$ .
2. Single Qubit Phase Gates $S,T,R_1(\theta),R_Z(\theta)$ .
3. Multi-Qubit $CNOT$, $SWAP$ gate.
4. The most sparse matrix criteria can be said to be closed under the Tensor product operator as $\otimes$. Most Sparse matrix $\otimes$ Most Sparse matrix is also a Most Sparse matrix.
5. Other common examples of these gates can be the $Increment, Decrement$ and $Shift$ Unitaries.

We can also define a least sparse criteria which can be said to include Unitaries which have the maximum i.e. $2^{2n}$ non-zero elements. The most common/canonical Quantum Operations which satisfy the most sparse matrix criteria are:

1. Hadamard Operator $H$ for Single Qubit Operations.
2. Single Qubit Rotation Gates $R_x(\theta),R_y(\theta)$ for non-trivial $\theta$
3. Quantum Fourier Transform $QFT$ for Multi Qubit Operations is another operations whose matrix representation can be said to be least sparse.