Adiabatic Quantum Computation

In this series we will be discussing about a fundamentally different paradigm to the universal quantum computer or gate model known as the Adiabatic Quantum Computation (AQC). In this article, I will be giving a naive or an easy to go understanding of AQC followed by a detailed explanation in the next article.

Most of us know about the Universal Gate Quantum Computers which follow the Gate model paradigm and it’s mathematically being shown that the adiabatic model is equivalent to the gate model, so we can find an equivalent algorithm which can be on one type of model for the other model and vice versa. Some of the outlining steps for the adiabatic model are:

  • Encoding the problem into a AQC compatible problem.
  • Preparing the initial state.
  • Annealing process
  • Measuring the answer.

The first thing we need to do is encoding the problem, that is translating the problem into one which a AQC can handle. One of such compatible notation is Boolean Satisfiability Problem. One of such mathematical notation which can be embedded on the Adiabatic Quantum Computer is the QUBO model. So the general idea to have the problem translated in such form and have a optimization problem which needs to be maximized or minimized. The optimization problem can be constrained or unconstrained.

Initial State:

For any type of Quantum Computation model, we need an initial state. For the AQC, the qubits have a certain connectivity. If the qubits are magnetic field direction (up or down depending on the flow of electric current being clockwise or anticlockwise, and the josephson junction creating a superpostion of the two using a superconducting quantum interference device. Below we see the connectivity of the qubits in D-Wave’s machine.



One important thing to note in the above column connectivity [Link] is that the D-wave’s system is not fully connected seeing the qubits 0,1,2,3 are not connected to each other. We get the idea from here that the above machine is a subset of a more powerful theoretical AQC but due to experimental limitations we are using the above architecture.

Annealing Process:

This is the main part where the computation actually happens. Let’s first understand the concept of annealing with an analogy. We take an ice in a cup and we start heating the ice. The goal of the above experiment is to melt the ice as slowly as possible that is when we have the water in the cup we should have zero disturbance or vibrations. If we do this process in an instantaneous manner we would see the ice being converted to water and the water being rushing to the walls of the cup because of the vibrations. What we are trying to do here is heat the water slowly and keep the tolerance to the max. The vibrations in the water state are the excitations and we need to reduce these so as to the least amount of error in the final state.

So we have this connected qubits (initial state) in the superconducting quantum interference device. We then slowly turn off the initial state and turn on the final state in a slow manner. So the process is starting from an initial state and transitioning into a mixed state of the initial and the final state energy finally stabilising into the final state. An alternative way is to start with both states on, but have the initial state be much, much stronger than the final part of the state, and then slowly turn off the very large, initial state. The below equation explains the above statement :



Here the Hamiltonian is made up of three terms. The Z and X being the spin matrices and h,J being the bias and coupler respectively. The above term can be grouped into two terms where the final state is being governed by the first two terms while the initial term is governed by the last term. We see the transition from the X basis state to the Z basis state. In the annealing process we start with the initial state being put together by the X spin state which is annealed to the Z spin state. So let say the h and J are very very small as compared to K initially and slowly we turn off the K and we are left with what we have in the first two states. What we are trying to achieve here is that we stay in the ground state of the corresponding Hamiltonian which is achieved cause of the slow procedure of the annealing.

Now in order to anneal properly so that we get the correct answer, we need to be sure you’re giving it enough time to settle without excitations. The above equation has time-dependent hamiltonian , or more specifically, the h, J, and K terms are time-dependent. We reach a 100% accuracy only when we run the adiabatic theorom for t=∞ . We can get close, by running it for a long time and able to reach an optimal solution with high accuracy.

The most stable state according to Quantum Mechanics are the ground states which have the lowest eigenvalue. Now as I stated before, when we are doing the annealing process we need to be in the ground state of the Hamiltonian, because if we move too quickly we are imparting energy to the system and causing excitations which would make the state jump in the excited spectra and would lead to errors.

Simulating the AQC process for the error correction :

There are measures for checking the correctness of the result when we have small Hamiltonians, and they have to do with tracking the actual energy of the Hamiltonian, finding its ground state eigenvector, and then comparing it with what your qubit state . The inner product of the two will show you your error. As we have access to the state vector all the time we can actually measure the error resorting to the mathematics but it mostly a theoretical protocol as in reality we would only be able to get the qubit state only after observing the state which would lead to collapsing and a change of state if there’s an error.



This is what the eigenspectrum of a qubit annealing simulation looks like [Link] .Here we see the energy on the y axis and the time on the x axis. The bottom most curve is the ground state and the curve above that is the first excited state. The part of the spectrum where we see the minimum gap between the energy of the ground state and the excited state is called the “minimum gap” and it most probably the state of the system where it can jump to the excited state instead of traversing in the ground state. Here is the part where we have to be cautious and traverse the annealing in a slow pace. The parts of the spectrum where the energy gaps are big enough, we can move at a faster pace without being going to the excited state.

So here we have looked into the introduction of the Adiabatic Quantum Computation in an easy understandable way and next we will be looking into the AQC in a detailed and advanced fashion in the next article of the series.

References:

Gaurav Singh
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Gaurav Singh
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