Is it possible to implement any random Hamiltonian using quantum circuit

Question

Are there any restrictions on implementing the evolution of any random Hamiltonian? Suppose I want to implement Rabi oscillations using a quantum circuit, I initialize the state and the Hamiltonian evolution is implemented in the quantum circuit using trotterization. Suppose I make some modifications in the Hamiltonian, say I couple a spin with the Rabi Hamiltonian $$ H = \lambda (S_{z} \otimes H_{Rabi}), $$ and I want to understand the joint evolution of a bipartite initial state using this particular Hamiltonian. My first question is that (though Hermitian) the modified Hamiltonian may not be physically realizable and is a made-up one, but I see no restriction on implementing the evolution of this Hamiltonian in a circuit using trotterization. So I am wondering, is it possible to implement any random Hamiltonian or made-up Hamiltonians using quantum circuits?

Answer 1

I think you are asking whether a quantum computer can efficiently simulate the evolution of any Hamiltonian, as long as the Hamiltonian is represented by a hermitian matrix. There are some details about this, as evidenced in this question.

An important consideration is whether your arbitrary made-up Hamiltonian will be local. Here "locality" doesn't necessarily mean spatial locality; rather, it means that the hermitian matrix can be written as a sum of simpler hermitian matrices that act on only a subset of the $n$ qubits.

For example, if we have:

$$H=H_1+H_2+\ldots +H_m,$$

with each $H_i$ acting only on, say, $k$ qubits, and $m\le {n\choose k}$, then we can always write the time evolution of $H$ as:

$$e^{iHt}\approx \left(e^{-iH_1\delta t}e^{-iH_2\delta t}\ldots e^{-iH_m\delta t}\right)^{t/\delta t},$$

and accordingly we can trotterize this by doing a little bit of $H_1$, then a little bit of $H_2$, ... then a little bit of $H_m$, and then iterating $t/\delta t$ times.

However many interesting "made up" Hamiltonians may not be written locally. Nonetheless, there are some more complicated ways to address certain sparse Hamiltonians - these generalize the class of local Hamiltonians to those in which the vast majority of entries in the hermitian matrix $H$ are zero, even though $H$ itself could be highly non-local. The sparse Hamiltonian is transformed and reduced to a local Hamiltonian, which then can be simulated/trotterized.

So, the short answer to your question "is it possible to [efficiently] implement any random Hamiltonian or made-up Hamiltonians using quantum circuits" is, if the random Hamiltonian is local enough or sparse enough, then yes, we can efficiently simulate the Hamiltonian. If, however, the Hamiltonian is neither local nor sparse, then there might not be an efficient simulation.