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Abstract
In one-dimensional multiparticle Quantum Cellular Automaton (QCA), the approximation of the bosonic system by fermion (boson-fermion correspondence) can be derived in a rather simple and intriguing way, where the principle to impose zero-derivative boundary conditions of one-particle QCA is also analogously used in particle-exchange boundary conditions. As a clear cut demonstration of this approximation, we calculate the ground state of few-particle systems in a box using imaginary time evolution simulation in 2nd quantization form as well as in 1st quantization form. Moreover in this 2nd quantized form of QCA calculation, we use Time Evolving Block Decimation (TEBD) algorithm. We present this demonstration to emphasize that the TEBD is most natu-rally regarded as an approximation method to the 2nd quantized form of QCA.
Highlights
Quantum Cellular Automaton (QCA) [1] is a quantum version of cellular automaton (CA)
In one-dimensional multiparticle Quantum Cellular Automaton (QCA), the approximation of the bosonic system by fermion can be derived in a rather simple and intriguing way, where the principle to impose zero-derivative boundary conditions of one-particle QCA is analogously used in particle-exchange boundary conditions
We assume that only the amplitudes ψ ( x1, x2, ) at points where all {xi} are different comprise the full set of independent variables and amplitudes of other points ( x1 = x2 etc.) needed for evolution are evaluated by interpolation from other points We illustrate in Figure 6 the method of the boson approximation we propose for two-particle case, comparing with free fermionic QCA case
Summary
Quantum Cellular Automaton (QCA) [1] is a quantum version of (classical) cellular automaton (CA). When we use the usual hopping term in the Hamiltonian as the kinetic energy part, this first part of TEBD can be regarded as 2nd quantized form of QCA This QCA form seems to be a byproduct of the 2nd order Suzuki-Trotter decomposition or just an approximation method to the TDSE. ( ) ( ) Note that as δ 2 , δ 2 have a 2 × 2 block diagonal form, their expoeven odd nential can be explicitly calculated In this way the time evolution for ∆tFDM is divided into two steps ,so we naturally define. It requires more elaborate techniques to derive the exact relation between mass and QCA parameter θ in general case Equation (3) This FDM-QCA correspondence is essentially the same as the first part of TEBD algorism we discuss later. In this study we do 2 not get into detailed derivation leading to the Dirac equation, as we are interested here in nonrelativistic case, though we will discuss a relevant topic in the last supplementary section
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