Hello, A new Rydberg atomic quantum computer is scheduled to be available in 2022, and Nasdaq and others are running ads celebrating the funding of atomic companies such as Atom Computing, which is attracting attention this year as the next new type of quantum computer.
Now, I have personally introduced the QuEra paper because it is of high quality and gives a good overview of the Rydberg atomic-type quantum computer. This time we will look at a paper on running an algorithm for a quantum computer called QAOA with two different parameters varied.
QAOA is an approximate algorithm performed using a quantum computer to solve an Ising model problem, which is a formulation of a discrete optimization problem based on a theory called quantum adiabatic calculation, in which the quantum state is changed step by step. In this article, we will look at a quantum simulator, which applies the mechanism of atoms to solve discrete optimization and graph problems using the motion of atoms.
A Rydberg atomic quantum computer is a machine that can use the low energy state called the ground state and the excited Rydberg state, in which the electrons are far away from the atomic nucleus. Rydberg atoms utilize a characteristic behavior of electrons within a certain radius, called the Rydberg radius, and have an interaction with other atoms called the Rydberg blockade, in which multiple electrons cannot coexist. This Rydberg state of energy can be described by an operator called the Hamiltonian, and quantum simulators can simulate atoms under the constraints of this Hamiltonian.
As of the beginning of 2022, a paper has been published in which a QAOA of up to 289 qubits has been performed.
Review: Quantum Optimization of Maximum Independent Set using Rydberg Atom Arrays
S. Ebadi1,∗ , A. Keesling1,2,∗ , M. Cain1,∗ , T. T. Wang1 , H. Levine1,‡ , D. Bluvstein1 , G. Semeghini1 , A. Omran1,2 , J.-G. Liu1,2 , R. Samajdar1 , X.-Z. Luo2,3,4 , B. Nash5 , X. Gao1 , B. Barak5 , E. Farhi6,7 , S. Sachdev1,8 , N. Gemelke2 , L. Zhou1,9 , S. Choi7 , H. Pichler10,11, S.-T. Wang2 , M. Greiner1,† , V. Vuletic´ 12,† , M. D. Lukin1,† 1Department of Physics, Harvard University, Cambridge, MA 02138, USA 2 QuEra Computing Inc., Boston, MA 02135, USA 3Department of Physics and Astronomy, University of Waterloo, Waterloo N2L 3G1, Canada 4Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada 5 School of Engineering and Applied Science, Harvard University, Cambridge, MA 02138, USA 6Google Quantum AI, Venice, CA 90291 7Center for Theoretical Physics, Massachusetts Institute of Technology,Cambridge, MA 02139 8 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA 9Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 10Institute for Theoretical Physics, University of Innsbruck, Innsbruck A-6020, Austria 11Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Innsbruck A-6020, Austria 12Department of Physics and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA ∗ These authors contributed equally to this work ‡ Current affiliation: AWS Center for Quantum Computing, Pasadena, CA 91125 † Corresponding authors
https://arxiv.org/pdf/2202.09372.pdf
The outline is the basic problem of solving a combinatorial optimization problem with a quantum computer dropped into a model called the Ising model.
A represents the Rydberg blockade of a single atom when the problem is dropped into the graph, and it affects within this radius.
B is the actual solution to the problem, which is red-white and corresponds to the quantum |g>,|r> states 01, respectively. This is how the solution to the maximum independent set problem is denoted when solved as a graph problem.
C is the encoding of the problem, where the atoms are stopped with optical tweezers and then rearranged with dynamic optical tweezers to actually place the atoms in the shape of the graph problem.
D is the time evolution calculation, where the optical tweezers are momentarily released and a laser is applied to control the state of the atoms. There are three parameters that can be changed depending on the time, while the distance parameter is fixed because it is set by the optical tweezers at the beginning.
E is the result of the calculation and represents the quantum state derived from the installed parameters.
Let us look at an example of QAOA with the above time evolution parameters.
QAOA performs this time evolution calculation updating the parameters as needed. The update policy is to calculate the parameters in such a way that the expected value of the overall Hamiltonian is reduced.
image from: https://arxiv.org/pdf/2202.09372.pdf
Check the Hamiltonian. The composition of the Hamiltonian here is important.
image from: https://arxiv.org/pdf/2202.09372.pdf
QAOA uses two Hamiltonians combined. Each time evolution is a superposition of solutions to the time-independent Schrodinger equation.
The above Hamiltonian uses a fixed time t. These two Hamiltonian configurations are common in discrete optimization, with the upper one called the driver and the lower one called the cost.
The driver determines how to search for a solution, and the cost is the problem setup for the problem to be solved.
We start from the initial eigenstate and settle on the final eigenstate.
Let's move on to the main topic of this paper, the parameter structure of time evolution. In this paper, two types of changes in time evolution parameters are used.
The first is the driver side, where time τ and phase φ are set.
The second is the bias term of the Hamiltonian on the cost side and the parameter control over the entire driver side.
image from: https://arxiv.org/pdf/2202.09372.pdf
In A, the discrete control is achieved by applying a time-evolving τ while changing φ when released; in B, the bias is controlled continuously, and the driver side eventually goes to 0, but it's almost a fixed value in the middle. It is named VQAA. It is a bit like analogue quantum annealing.
I think both experiments are interesting. As far as the Hamiltonian on the driver side is concerned, I think it will be tough to surpass the classical one, but I hope it will be fully programmable. That's all.