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

$$⟨\mathcal{O}⟩\equiv \mathrm{Tr}\left[\mathcal{O}\rho \right]$$

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

$$⟨⟨A_n||A_m⟩⟩=\frac{1}{\mathcal{D}}\mathrm{Tr}\left[A_n^{†}{A}_m\right]={\delta }_{nm},$$

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

$$\rho =\frac{1}{\mathcal{D}}\left[I+\sum _m⟨A_m⟩A_m\right]$$

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

$$\{A_n\}=\{{\sigma }_1^{\alpha \left(n\right)}{\sigma }_2^{\beta \left(n\right)}\cdots {\sigma }_N^{\omega \left(n\right)}\}$$

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