While the circuit class samples the matrices making up a quantum circuit, the backend performs the actual matrix multiplications which appear in the statevector simulation.
The statevector simulation takes an input state psi0, and propagates it to the final state psi1 by repeated left matrix multiplications. Assuming that our circuit has k steps, then we compute psi1 = U psi0 = U_k … U_1 psi0 where U_i are matrices of dimension (2^n, 2^n) for n qubit psi0 of shape (2^n,).
Tensors are simply generalized versions of scalars, vectors and matrices. We can imagine it to be a box with incoming and outgoing legs. While a scalar has no legs, a vector has one leg, and a matrix has one incoming and one outgoing leg. We can interpret matrix-vector multiplication as connecting the right leg of the matrix with the left leg of the vector. This will create an object with one free left leg, which represents a vector as expected. Similarly, matrix-matrix multiplication will led to an object with one incoming and one outgoing leg, so a matrix.
As explained in this resource, the ordering of the tensor contractions (think matrix multiplications) can influence the computational cost dramatically. In the EfficientBackend, we leverage optimizations based on the property of the quantum circuit, as well as optimizations performed by the library opt_einsum. This library computes the contractions for us.
quantum_gates.backends
We apply the GHZ algorithm on the two-qubit zero state as an example.
from quantum_gates.backend import EfficientBackend
backend = EfficientBackend(nqubit=2)
H, CNOT, identity = ...
mp_list = [[H, np.eye(2)], [CNOT]]
psi0 = np.array([1, 0, 0, 0])
psi1 = backend.statevector(mp_list, psi0) # Gives [1, 0, 0, 1] / sqrt(2)Not applying gates to each qubit will lead to errors. In each matrix product, the dimension of the combined Kronecker product has to match the dimension of psi.
from quantum_gates.backend import EfficientBackend
backend = EfficientBackend(nqubit=2)
H, CNOT = ...
mp_list1 = [[CNOT, np.eye(2)]] # Gates for 2 + 1 = 3 qubits -> wrong.
mp_list2 = [[CNOT]] # Gates for 2 qubits -> fine.
mp_list3 = [[np.eye(2), np.eye(2)], [H, H]] # Gates for 1 + 1 = 2 qubits -> fine.The backend class has a simple, generic public interface. This makes implementing a custom backend easy. One can optimize the backend performance by exploiting properties of a specific quantum algorithm, use another simulation method like Matrix Product State (MPS) simulation, or use another library as backend for the computations.