Theoretical Background

Here we briefly explain the theoretical ideas behind Raqet. While these ideas apply generally to any local quantum system, we focus here on the case of a system of interacting spin-1/2 particles (or qubits), since this is the type of system for which Raqet is currently implemented.

For the following discussion we consider the case of an \( N \)-qubit system, with Hilbert space \( \mathcal{H} \cong \mathbb{C}^{2^N} \) of dimension \( \mathcal{D} = 2^N \). The fundamental object of interest for any quantum system is the density matrix \( \rho \), with which we can compute the expectation value of any observable \( \mathcal{O} \) as

\[ \langle \mathcal{O} \rangle \equiv \mathrm{Tr}\left[\mathcal{O} \rho \right] \]

We denote by \( \{ A_n \} \) an orthonormal basis of the space of all traceless operators on the Hilbert space, normalized as

\[ \langle \langle A_n || A_m \rangle \rangle = \frac{1}{\mathcal{D}}\mathrm{Tr}\left[ A_n^{\dagger} A_m \right] = \delta_{nm}, \]

where the double bracket notation indicates an inner product on the space of operators. Any operator in our Hilbert space can be expressed as a linear combination of these operators and the identity,

\[ \mathcal{O} = \frac{1}{\mathcal{D}}\mathrm{Tr}\left[\mathcal{O} \right] I + \sum_{m=1}^{\mathcal{D}^2-1} a_{m} A_{m} \]

This includes the density matrix itself, which, taking into account the definition of the expectation value, can be written

\[ \rho = \frac{1}{\mathcal{D}} \left[ I + \sum_m \langle A_m \rangle A_m \right] \]

For a system of interacting qubits, it is customary to choose this basis to be the (Hermitian) elements of the Pauli group

\[ \{ A_n \} = \{ \sigma_1^{\alpha\left( n \right)} \sigma_2^{\beta\left( n \right)} \dotsb \sigma_N^{\omega\left( n \right)} \} \]

where \( \alpha, \beta, \ldots, \omega \in \{ I, x, y, z \} \). Performing the time evolution of the density matrix in an approximate manner is the main goal of Raqet.