How do I write a tensor product of conditional gates in matrix form?

Question

I am writing a program where I need to find the eigenstates of an operator that is a Kronecker product of conditional quantum gates. I am wondering how I would compute this product in matrix form as the conditional gates act on different qubits.

For example: Say I want to find the Kronecker product of some Pauli-X gate X_k that acts on qubit k and conditional phase gates CZ_i,j that have a control qubit i and target j:

X_1 ⊗ CZ_2,3 ⊗ CZ_3,4 ⊗ CZ_2,4

How would I compute this product so the result is a matrix? (Or in some form where I can find the eigenstates) For context, I am using Python and NumPy.

Answer 1

You can always write your operation as one gate at a time, and take the product. So, in your case, you'd be calculating a product $$ U=U_1U_2U_3U_4 $$ where, for example, $U_1=X_1=X\otimes 1\otimes 1\otimes 1$, $$U_2=CZ_{2,3}=I\otimes I\otimes I\otimes I-I\otimes |1\rangle\langle 1|\otimes(Z-I)\otimes I,$$ and so on. In particular, you can start to use the formulation I've given you for $U_2$ to work out how to apply CZ on any pair of qubits. For instance, $$ U_4=CP_{2,4}=I\otimes I\otimes I\otimes I-I\otimes |1\rangle\langle 1|\otimes I\otimes(Z-I). $$

Answer 2

pauli_X=[[0,1],[1,0]] cz=[[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,-1]] np.kron(np.kron(np.kron(pauli_X,cz),cz),cz)

This would compute $X_1 \otimes CZ_{2,3} \otimes CZ_{4,5} \otimes CZ_{6,7}$.

Addressing the confusion about what tensor product means. $A$ operating on $d_1$ qubits $(\mathbb{C}^2)^{\otimes d_1}$ and $B$ operating on $d_2$ qubits $(\mathbb{C}^2)^{\otimes d_2}$ means that $A \otimes B$ will operate on $(\mathbb{C}^2)^{\otimes d_1} \otimes (\mathbb{C}^2)^{\otimes d_2}$ which is $d_1 + d_2$ qubits. No overlapping indices between the $d_1$ and the $d_2$.

Edit: to address comment

In this case there are 7 qubits so the total Hilbert space is $(\mathbb{C}^2)^{\otimes 7} \simeq \mathbb{C}^{2^7}$ with basis indexed by the $2^7$ bit strings of length 7. From 0000000, 0000001, 0000010, etc all the way to 1111111.