Generally, Quantum Computer are used to solve problems in the field of Cryptography and information science, although it&#39;s known that quantum computer can help us to simulate quantum systems as suggested by Richard Feynman, when he raised the idea of quantum computer, stating classical computers can never efficiently simulate an evolution of quantum system and proposed the first models for quantum computation. In this article, we will have insight on how quantum circuits can be used for better understanding of quantum system of particular interest. In example, we will walk through the time evolution of quantum state in the Schwinger Model , toy model used by physicist. The Schwinger model was proposed by theoretical physicist Julian Schwinger, and this toy model is generally used in the field of quantum chromodynamics, a sub-branch of particle physics mainly concerned with quarks, gluons and the strong nuclear force. Our toy model exhibits properties similar to those of QCD, such as charge confinement and chiral symmetry breaking, while being easy to work with, first because it only spans on 1+1 dimension, and it is exactly solvable in certain specific conditions. Below is the Lagrangian density function of Schwinger model, which finally determines the equations of motion of the system. The important point to take is, there are charged particles that can move on line, and there will be an electric field everywhere on the line. So, how are we going to simulate this &#34;Schwinger Model&#34; in our quantum computers? We will start with some pre-processings. The first step will be to limit our spatial dimensions, which is right now infinite, we will start by discretize our space into lattice and assume our &#34;particle&#34; can only be at one these lattice size, as we have decreased our Hilbert space, we will require less finite number of quits to simulate our system. The lattice will have certain characterising parameters describing the spacing between neighbouring lattice sites, and as this spacing goes to zero, we will get our ideal case solution or true physics of system. However, the lattice would still be infinite, so we also enforce periodic conditions, which will force lattice to wraps around on itself. Any finite lattice can now be handled, since it has a finite-dimensional Hilbert space, and in the limit of large lattices, we should recover the physics of our original model. Schwinger Model, requires one extra step, which I hope you all are familiar with known as the Jordan-Wigner transformation, which transforms our system of fermions and anti-fermions into a system with spin-(1/2) particles, where spin-up and spin-down corresponds to the anti-fermion and fermion being present or absent in particular way, to know more you can go through the literature. In the end, we will finally get something shown in the figure below. We will have a lattice with N “spatial site”. Each spatial site will have both a fermion and an anti-fermion site where they both can be present or absent.Then, each pair of adjacent particles will have a link between them that will carry an integer electric flux value. Furthermore, we stipulate that the electric flux at each site must be at most a particular cutoff in absolute value; this is yet another approximation for which one would approximate the real model by taking a limit of approximate cases. Physically, this is essentially cutting out states that have too much energy in the electric field. In the paper and here, we will let this cutoff be 1, so the electric flux values can only be +1, 0, or -1. In total, there will be 2N particle sites (N fermion, N antifermion, alternating) and 2N links. Shown below is a lattice with two spatial sites: Now we are familiar with Schwinger Model, in subsequent parts we will see how to simulate this model on quantum computer.

Leveraging Quantum Computers for Quantum Field Theory Calculations - part 1

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