# Questions about the Hamiltonian of a decay

In paper Simulating quantum systems on a quantum computer the author mention in section 3, simulating a decay to obtain the ground state, and give the Hamiltonian for that: $$H=H_1+H_2 + H_{1I}\otimes H_{2I}$$ My questions are:

• Can you explain this Hamiltonian more explicitly, or refer me to a paper doing that?
• Are there papers that elaborate this Hamiltonian in a quantum computer?
• Are there other Hamiltonians expressing the decay process?

Any answer will be more than appreciated, and thank you :)

This is a very generic description that captures essentially all possibilities of describing the Hamiltonian with a decay process (I supposed one should allow for a general $$H_{1,2}$$ rather than a simple tensor product of terms)).

The basic idea is that if you've got a system ("1"), then if you just describe using a Hamiltonian on that system, the evolution is unitary and so you do not get any decay. In particular, the purity of the system is preserved.

So, the way to describe decay using unitary evolution (i.e. the result of Hamiltonian dynamics) is to go to the "church of the larger Hilbert space". You add in an ancilla system ("2", often referred to as the bath) and have some (unspecified) interaction between the two systems. The overall evolution is unitary, and therefore purity preserving, but the purity of the system 1 by itself is not necessarily preserved. Hence, this can be thought of as some sort of decay.

Section 8.2.3 of Nielsen & Chuang covers this conceptually, although in a slightly different formalism.

What those Hamiltonians actually are is up to you. It's not uncommon to pick the ancilla system to be one or more harmonic oscillators (see the wikipedia page for more references). One reason is the infinite Hilbert space that it provides. Finite Hilbert spaces (eventually) lead to revivals of the original state, which is not good if you're trying to describe loss!

Are there papers that elaborate this Hamiltonian in a quantum computer?

This depends what you mean. Are you wanting the description of the noise process that the quantum computer is undergoing? Or are you wanting to implement your own noise process on a quantum computer? For the former, the whole point of noise is that there's an associated, uncontrolled, environment/bath. You will not get a description of that. The best you can hope for is the net effect on just the system. For the latter, you have to pick your noise Hamiltonian, approximate it (because the computer only has a finite Hilbert space) and simulate it using standard Hamiltonian simulation techniques. But there are probably better ways of implementing noise...