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To the best of my knowledge, the gate notation forms the quantum programming. For instance, I use qiskit, pennylane, etc. products to see how the algorithms do their job. At the same time the "quantum" world relates to the Schrödinger equation. But how to construct solution based on this equation and then split it into gates? Pick the HHL paper - why does circuit looks like that? How to derive the same result from the Schrödinger equation?

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Recall that a time-dependent unitary operator (technically a one-parameter group) $U$ can be generated by a Hermitian operator $H$(the generator)

$$ U(t) = \exp(-itH) $$

This in turn gives us the solution

$$ | \psi(t) \rangle = U(t) | \psi(0) \rangle $$

to the corresponding Schrödinger equation

$$ i \hbar \partial_t | \psi(t) \rangle = H | \psi(t) \rangle $$

Well, a circuit is precisely the discretization of such a unitary $U$, defined as the product of unitary factors:

$$ | \psi(T) \rangle = \prod_{\tau=1}^{T} U(T - \tau) | \psi(0) \rangle $$

where each $U(T - \tau)$ is a gate.

In other words, we're not always necessarily concerned with the Schrödinger equation a circuit solves; we already know that it solves an equation by definition. Instead, we start with gates and try to figure out what we can do with them to solve our problem.

Of course, there are plenty of situations where we are concerned with the Hamiltonian, most obviously with simulation. In this case, we go the "traditional" direction of turning the (corresponding unitary of the) Hamiltonian into a circuit.

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