Physics of Quantum Annealing - Hamiltonian and Eigenspectrum

so this is my third video in the explainer videos about quantum annealing and in this video I'm going to take a look at the underlying quantum physics of our processes so this video I'm going to take a look at two things one is the Hamiltonian and another is eigen spectra now don't get off by these names they're actually not too difficult to understand and I'll explain them in this video so a Hamiltonian is basically a mathematical description of some physical system and it describes it in terms of energies so it's basically a formula where you input the state of your system and the Hamiltonian gives you out the energy of that system and you can have hamiltonians for both quantum systems and classical systems so a classical example would be a table and an apple so there's two states of this system the apple can be sitting on the table and the apple can be sitting on the floor so the hamiltonian tells you that the state where the apple sitting on the table is a higher energy state than the state when the apple sitting on the floor so for our systems what that means is we can take in the state of all of our qubits put it into the hamiltonian and the hamiltonian gives us the energy of that state and so what quantum annealing is trying to do is find the state of all the qubits that gives us the lowest energy when put into the Hamiltonian so here's the formula that describes the Hamiltonian in quantum annealing so on the left-hand side that H stands for energy and that's what you're calculating now on the right hand side is split into two parts the first part is called the initial Hamiltonian and the lowest energy state of this initial Hamiltonian is where you've got all of your qubits in a superposition state of zero and one as an easy state to set up your system in and then the second part of this is the final have all Tony Anor or your problem Hamiltonian and basically the the lowest energy state of this is the answer to the problem that you've posed so this is what you're trying to find out about quantum annealing and this part of the Hamiltonian has got all of those biases and couplings that I talked about in my last video so the biases and couplings are basically the energy program that the user programs into the system and it defines the problem that they want to get solved if you want more information there's more of that in the last video so I suggest watching that in quantum annealing you start off in just the initial Hamiltonian state and then as you anneal you introduce the problem Hamiltonian which contains the biases and couplers and you ramp down the initial Hamiltonian and at the end of the anneal you're in the state which is just the problem Hamiltonian and the initial hamiltonians gone to zero an ideally what's happened is you've stayed in the minimum energy state throughout this quantum annealing procedure so at the end you're actually in the minimum energy state of the problem Hamiltonian and you valence of the problem that you wanted to solve so it's worth mentioning that the states in the initial Hamiltonian are all quantum states whereas the states in the final Hamiltonian are all classical States and so at some point during the anneal each of the qubits goes from being a quantum object to collapsing into a classical object so an eigen spectrum is a really useful way of visualizing the different energy states that you can be in in the quantum annealing procedure so it's basically a graph where the lowest energy state during the anneal is at the bottom and then any higher excited states or above it so the system starts in the lowest energy state which is well separated from any other energy level then as the problem Hamiltonian is introduced there are other energy levels that get closer and closer to this lowest energy level and the closer they get the larger the probability of the system jumping out of the lowest energy state into one of these excited States one point during the anneal there's a point where the first excited state approaches the ground state closely and then diverges away again and this point is called the minimum gap it's also known as an anti crossing now there are several mechanisms that allow the system to jump into a higher energy State one is just from the thermal energy of thermal fluctuations which you get from any physical system and another one is simply running the annealing procedure to fast which gives it gives the system energy and allows the state to jump from the ground state energy to an excited state annealing slowly enough to stay in the ground state is known as an adiabatic process and this is where the named adiabatic quantum computing comes from in reality for large problems the probability of staying in the lowest energy state is small for these systems however low energy states that are returned are still very useful as long as they're lower than those found by other classical techniques so in conclusion there's two things for Hamiltonian which tells you what energy each state in your system is and then the eigen spectrum which plots all of those entities over time during the anneal and it's important to know that for every single different problem that a user specifies there's a different Hamiltonian and a different corresponding eigen space so they have different features so the very hardest problems are the ones which have the smallest gaps so these concepts of the Hamiltonian and eigen spectrum are going to be very useful in my next video are going to explain how we measure quantum physics in these systems

Channel: D-Wave Systems

Views: 35,627

Rating: 4.9399624 out of 5

Keywords: Quantum Computer, Quantum Mechanics, high performance computing, D-Wave Systems, quantum annealing, science explainer, D-Wave, dwave, d'wave, d wave, quantum, annealing

Id: tnikftltqE0

Length: 6min 24sec (384 seconds)

Published: Mon Mar 28 2016