Qfyilyi

In this research paper, the authors introduce a new algorithm to perform Hamiltonian simulation.

The beginning of their abstract is

Given a Hermitian operator $hat{H} = langle Gvert hat{U} vert Grangle$ that is the projection of an oracle $hat{U}$ by state $vert Grangle$
created with oracle $hat{G}$, the problem of Hamiltonian simulation is approximating the time evolution operator $e^{-ihat{H}t}$ 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 $hat{H} = langle Gvert hat{U} vert Grangle$? More precisely, I do not understand what $langle Gvert hat{U} vert Grangle$ represents when $hat{U}$ is an oracle and $vert G rangle$ a quantum state.

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 $|Grangle$ 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:

$$
hat H_A=(langle G|_Botimesmathbb{I}_A)hat U_{AB}(|Grangle_Botimesmathbb{I}_A)
$$