Time Evolution of Quantum Systems

The goal of simulating the dynamics of a quantum system is to understand how certain observable quantities evolve with time. Since the density matrix of our system of qubits can be written as

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

we see that tracking the time evolution of the density matrix is equivalent to tracking the time evolution of all of the expectation values $\{⟨A_n⟩\}$. If the equation of motion for the density matrix is linear, then the expectation values will also satisfy some linear system of equations

$$\frac{\mathrm{d}}{\mathrm{d}t}⟨A_n⟩=\sum _m{M}_{nm}⟨A_m⟩.$$

To determine this matrix $M$ explicitly, we can insert the expression for the density matrix into Liouville's equation of motion for the density matrix,

$$\frac{\mathrm{d}}{\mathrm{d}t}\rho =-i\left[H,\rho \right],$$

where $H$ is the Hamiltonian of the system, in order to find

$$\sum _m\frac{\mathrm{d}}{\mathrm{d}t}⟨A_m⟩A_m=-i\sum _m⟨A_m⟩\left[H,A_m\right].$$

Taking the trace of both sides of this equation against $A_n^{†}$, we find

$$\frac{\mathrm{d}}{\mathrm{d}t}⟨A_n⟩=-\frac{i}{\mathcal{D}}\sum _m\mathrm{Tr}\left[A_n^{†}\left[H,A_m\right]\right]⟨A_m⟩,$$

which therefore identifies

$$M_{nm}=-\frac{i}{\mathcal{D}}\mathrm{Tr}\left[A_n^{†}\left[H,A_m\right]\right]$$

If our system is subject to noise in addition to Hamiltonian dynamics, one must simply make the replacement

$$-i\left[H,A_m\right]\to \mathcal{L}\left(A_m\right),$$

where $\mathcal{L}$ is the full Lindblad time evolution operator,

$$\mathcal{L}=-i\left[H,\cdot \right]+\sum _i{\gamma }_i\left[L_i\cdot L_i^{†}-\frac{1}{2}\left\{L_i^{†}{L}_i,\cdot \right\}\right]$$

with jump operators $\left\{L_i\right\}$. Repeating the above steps with this replacement, we find

$$M_{nm}=\frac{1}{\mathcal{D}}\mathrm{Tr}\left[A_n^{†}\mathcal{L}\left(A_m\right)\right]=⟨⟨A_n||\mathcal{L}||A_m⟩⟩$$

which identifies $M$ as simply the matrix representation of the superoperator $\mathcal{L}$ in the basis of traceless operators.

Reducing the Complexity of the Equations of Motion

We see that the dynamical evolution of a quantum system can be understood by studying the time evolution of the expectation values of a given basis of traceless operators. The number of operators in such a basis will typically grow exponentially with the size of the physical system, making an exact simulation of quantum systems impractical beyond even relatively small system sizes. It is therefore necessary to make some sort of approximation when simulating most quantum systems of practical interest.

Raqet implements such a simplification by

1. Identifying a collection of operator expectation values which should be "unimportant" in some appropriate sense, and can therefore be omitted from the simulation

2. Determining how to modify the equations of motion in the absence of these unimportant expectation values.

How Raqet accomplishes these two tasks is explained in the following two sections.