# Problem with the mathematical formulation of “qubitization”

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

The beginning of their abstract is

Given a Hermitian operator $$\hat{H} = \langle G\vert \hat{U} \vert G\rangle$$ 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.

You want to start by being careful with the sizes of the operators. $$\hat U$$ acts on $$q$$ qubits, and $$\hat H$$ acts on $$n qubits. I believe that $$|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)$$
• I am still trying to understand your answer but the sizes of the operators were definitely one of the points I missed! About your answer, what does $\vert G \rangle_B \otimes \mathbb{I}_A$ represent? A tensor product between a quantum state (a vector) and an operator (a matrix)? – Nelimee Dec 11 '18 at 13:39