Simple questions about QSVT

Demos
1

I have read the QSVT tutorial QSVT in Practice | PennyLane Demos and I have a question.

I have now understood the basic ideas of qsvt to simulate an polynomial function, but I don’t understand how this can be expanded to perform a matrix function, for example, from the original sum_even_odd_circ() function to the latter linear_system_solver_circuit() . Is there any mathematical theory to consider?

Also, I would like to now how can I expand the original circuit that writes:

block_encoding = qml.BlockEncode(x, wires=[0])
projectors = [qml.PCPhase(angle, dim=1, wires=[0]) for angle in phi_qsvt]

poly_x = qml.matrix(qml.QSVT, wire_order=[0])(block_encoding, projectors)
qsvt_y_vals.append(np.real(poly_x[0, 0]))

so that I may do it in an easier way?

Thanks very much.

2

Hi @Chenjia , welcome back to the Forum!

I’ll try to write up an explanation that makes it clearer how QSVT connects to solving a linear system.

In the meantime, what do you mean by “expanding the original circuit”? Could you please clarify with an example or description?