# Does optimization via Hamiltonian evolution have an analogy like gradient descent?

I'm trying to find out, if there is a simplified concept to understand what is occuring during quantum annealing/ Falqon/ Hamiltonian evolution like algorithms.

During classical gradient descent algorithms, one can imagine a marble that rolls along a surface downhill that will eventually end up in a local minima.

Is there a similar analogy to what is happening during Hamiltonian evolution. We start in a global minima of an initial Hamiltonian $$H_0$$, perturb the energy landscape by some amount and then wait for quantum tunnelling to occur and repeat until we reach the desired minimum of a different hamiltonian $$H_1$$?

• @MarkSpinelli, Not in all algorithms but for quantum annealing we start in the all plus state and the initial Hamiltonian is -\sum_i X_i. Quantum Annealing is based off the adiabatic invariance theorem which I think means we must start in the ground state of a known system and try to stay in/close to the ground state of all the intermediate systems until we have reached the ground state of a desired system. Commented Aug 12 at 13:31
• Ah, sorry about that. I've edited for clarity. Commented Aug 12 at 14:49

Here is my analogy for adiabatic theorem explanation which I presented in my paper Finding the Optimal Currency Composition of Foreign Exchange Reserves with a Quantum Computer:

Assume we have a sheet of paper scattered with iron filings. This is our system described by the initial Hamiltonian $$\mathcal{H}_0$$. If no external force is applied, the system remains unchanged. Imagine that we slowly move a magnet under the sheet. The filings begin to move and follow the magnetic lines of force. In the end, we will see a typical pattern on the sheet - the iron filings oriented in the direction of the external magnetic field. This is the system described by the final Hamiltonian $$\mathcal{H}_1$$. The systems that exist when the magnet is only partially under the sheet are described by Hamiltonians $$\mathcal{H}(t)$$.

We now turn our attention to the requirement of carrying out the changes slowly. If we move the magnet quickly, some of the filings remain stuck because of friction between them and the sheet. Clearly, these filings resist the magnetic field and they have to have enough energy to do so. As additional energy is needed, the system is clearly not in its ground state. However, slow changes allow the filings to adapt to the increasing magnetic force without getting stuck, hence they need less energy. In other words, the system remains in the ground state all the time.

• "As additional energy is needed, the system is clearly not in its ground state". How the need of additional work implies not being in the ground state? Sorry my physics background is zero. +1 for an interesting analogy, though somewhat complicated one. Commented Aug 13 at 21:51
• @MonteNero: I meant that for an iron filling to be stuck to a paper and resist a magnetic force necessitates to have an additional energy in comparison with case the fillings just follow a magnetic field. The first configuration is less advantageous from energetic point of view, hence the system is not in ground state. Commented Aug 13 at 22:07

As you mentioned, the gradient descent method has a nice pictorial representation of a ball rolling down the hill of the optimization landscape.

In the quantum case, trying to find classical analogies or pictorial representations of various quantum phenomena is futile, as once you describe them in terms of classical phenomena, you lose all the unique features of "quantumness". For example, see a similar question where OP tried to find classical analogies of superposition and entanglement. For that question, forky40 gave a nice breakdown and explanation of why classical analogies can't fully convey quantum phenomena.

If you still want a simplistic picture of adiabatic evolution without invoking the adiabatic theorem or quantum phenomena like tunnelling, imagine that instead of a ball rolling, we have a cup of tea that we want to move from the initial environment to some other environment slowly enough so that the tea remains at rest at all times.

We have a cup of tea on a kitchen table, and we want to bring the cup to our office desk without spilling the tea. Initially, since nothing acts on the cup, its content is at rest (lowest energy state). Next, we pick up the cup and move it toward the office desk. If we move it slowly enough, the cup's content will remain at rest even though we transport it into a different environment (office desk). If we move the cup too quickly, the tea may get excited and spill, and it will never return to its original state because some tea was lost due to spilling outside the cup.

In quantum annealing there is two important analogies: simulated annealing and adiabatic theorem.

Simulated annealing start from very high temperature T, and thus thermal equilibrium is random state. Same way quantum annealing start from high quantum noise and thus equilibrium state is a quantum state with white noise only.

Adiabatic theorem states, that if you move from one equilibrium state at quantum system to another slowly enough then you stay at equilibrium. Equilibrium state is a ground state only without quantum noise. Same way as in simulated annealing only when temperature is zero is the equilibrium state of the system a ground state of a Hamiltonian.

• Do you have any intuition why the adiabatic theorem does what it does? Something like the quantum state being a marble and the evolution of the hamiltonian moving it about in some way? Commented Aug 13 at 0:05
• Adiabatic theorem wirks by tunneling. Act like a wave. Wave goes trough walls and barriers and do it fast in many cases. Commented Aug 13 at 5:53