# Quantum operations as sparse matrices

What are examples of quantum operators or evolutions of quantum states that can be modeled as sparse matrices?

• Hey mondibrah! I think the question is somewhat straightforwards - is there any other context that you could contribute? (e.g. why are you looking, examples of sparse matrices you've seen yourself, etc.) This could be really useful to frame the question :) – C. Kang Jun 29 at 19:49
• See qutip.org. It uses sparse matrices for all operators – Paul Nation Jun 29 at 22:48

Pretty much everything!

Any Hamiltonian that you want to simulate from e.g. condensed matter physics, is highly likely to be sparse by virtue of the fact that interactions are local.

Equally, any tensor product of Paulis is sparse.

In terms of quantum computing, most gates that you choose to use tend to be sparse, the notable exception being something like a Hadamard transform. (Of course, there's a trivial statement that when viewed from the correct basis, any matrix can be considered 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.