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In this research paper (arXiv), the authors introduce a new algorithm to perform Hamiltonian simulation. The beginning of their abstract is

Given a Hermitian operator $\hat{H} = \langle G\vert \hat{U} \vert G\rangle$ that is the projection of an oracle $\hat{U}$ by state $\vert G\rangle$ created with oracle $\hat{G}$, the problem of Hamiltonian simulation is approximating the time evolution operator $e^{-i\hat{H}t}$ at time $t$ with error $\epsilon$.

In the article:

  • $\hat{G}$ and $\hat{U}$ are called "oracles".
  • $\hat{H}$ is an Hermitian operator in $\mathbb{C}^{2^n} \times \mathbb{C}^{2^n}$.
  • $\vert G \rangle \in \mathbb{C}^d$ (legend of Table 1).

My question is the following: what means $\hat{H} = \langle G\vert \hat{U} \vert G\rangle$? More precisely, I do not understand what $\langle G\vert \hat{U} \vert G\rangle$ represents when $\hat{U}$ is an oracle and $\vert G \rangle$ a quantum state.

Frederik vom Ende
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Adrien Suau
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1 Answers1

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You want to start by being careful with the sizes of the operators. $\hat U$ acts on $q$ qubits, and $\hat H$ acts on $n<q$ qubits. I believe that $|G\rangle$ is a state of $q-n$ qubits. So, what we really need to talk about is two distinct sets of qubits. Let me call them sets $A$ and $B$. $A$ contains $n$ qubits, and $B$ contains $q-n$ qubits. I'll use subscripts to denote which qubits the different operators and states act upon:

$$ \hat H_A=(\langle G|_B\otimes\mathbb{I}_A)\hat U_{AB}(|G\rangle_B\otimes\mathbb{I}_A) $$

DaftWullie
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