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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.

Quantum Cellular Automaton (QCA) [

QCA can be regarded as a discrete mechanical system with a simple and elegant time evolution rule. Though it is simple, it is not just a toy method. It can simulate the real quantum system of matter. Moreover there are several ideas that QCA plays a key role in fundamental physics [

QCA can be also regarded as one of the approximation methods for solving the continuous Time Dependent Schrödinger Equation (TDSE) like Finite Difference Method (FDM). However QCA is unique in that it preserves the complete unitarity of quantum systems upon its time progression and it can be regarded as the discrete version of direct solution for quantum dynamics, not merely an approximation to the TDSE. TDSE emerges rather as an approximation in the zero wavenumber limit of the general QCA solution. By extending the one-particle QCA to many-particle QCA, we explore the possibility of the method in real quantum systems.

The Time Evolving Block Decimation (TEBD) [

Since the success of Density Matrix Renormalization Group (DMRG) [

The boson-fermion correspondence in the one dimensional quantum system is well known. However in studying QCA-TEBD formalism we notice that this can be derived in a rather simple and intriguing way. The main purpose of this study is to show this simple derivation and application to few body systems of boson. Numerical studies on applicable range of the “boson approximation” (approximation of the bosonic system by fermion) are also performed.

Consider the simplest partitioned QCA on a 1D-time 1D-space lattice of which time evolution rule is given by

Here

is the parameter of the TDSE-type QCA.

Though it is not straightforward to recognize intuitively, QCA gives a solution of the free particle TDSE in the zero wave number limit

The relation between mass

This relation can be obtained by several methods as we will mention later. We introduce a simple derivation using FDM for small

By replacing the spatial derivative with the spatial difference, Equation (2) becomes

Here

where S is the one-grid shift operator

(Here we use 6 ´ 6 matrices assuming that the system consists of 6-grid points with periodic boundary condition. Moreover unfilled matrix elements are assumed to be zero throughout this article.)

Therefore the time evolution for the time step

This

where

Then

Note that as

divided into two steps ,so we naturally define

ponds to Equation (3) for small

The QCA dynamics obeys such a simple rule above. However it requires more elaborate techniques to derive the exact relation between mass and QCA parameter

There are several approaches to obtain the continuous limit (namely PDE) of QCA. The most naive and straightforward one is to connect the discrete time to continuous time by the interpolation and then to take a wavenumber (k) expansion around k = 0 in the spatial direction [

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.

There are basically three easily implementable boundary conditions for QCA. These are illustrated in

For zero derivative boundary condition [(2)], phase rotation factor is 1 as

For zero amplitude boundary condition [(3)], phase rotation factor is

Note that the unitarity is always satisfied, because the probability increase and decrease are balanced between left and right boundary grid points in the case of (1) and they are zero in the case of (2) (3).

Boundary conditions and discontinuities (inhomogeneities) for QCA are firstly investigated by Meyer. He investigated more general 2-component QCA having two angle parameters. The scalar QCA we use is the simplest one having only one angle parameter, which corresponds to one of factors if his QCA is flattened (namely changed from 2-component to scalar by doubling the number of grid points) and is factorized [

It is straightforward to construct multi-dimensional QCA. We have only to use direct product of 2 ´ 2 local unitary 1D matrices to generate 2D matrices.

The rule is illustrated in

(where x, y are even if t is even, and x, y are odd if t is odd.)

As we know each U approximately corresponds to the evolution

or

other,

generate multidimensional QCA for general dimension.

Applications of QCA to multidimensional cases are studied in [

It is also straightforward to construct (non-interacting) multiparticle QCA. D- dimensional distinguishable M-particle system is equivalent to DM-dimensional 1-particle system. For indistinguishable particle systems, we have to restrict this space to symmetric or anti-symmetric subspace according to the statistics of the particles. Note that

If the one-particle time evolution rule is given by an infinitesimal time evolution matrix, namely, a generator or a Hamiltonian, it is straightforward to construct a 2nd quantized Hamiltonian

(Here

If the one-particle time evolution rule is given not by a generator but by a finite time evolution matrix

Namely we can apply the substitution rule

We now apply this substitution rule to 1D free bosonic QCA system where unitary transformation only between nearest neighbor grids occurs, and the one step evolution is given by

The explicit local unitary evolution matrix for the grid pair

(local unitary matrices for other grid pairs

We then apply the substitution rule to 1D free fermionic QCA system. Focusing on the grid pair

Here

It should be noted that if periodic boundary condition is adopted for the grid pair

Now we introduce an interaction between particles. For this purpose, it is reasonable to introduce an additional phase rotation factor by the potential caused by other particles just like the external potential case.

Note that QCA with external potential was firstly studied by Meyer [

Here we discuss the simplest case, namely the cases where the nearest neighbor interaction is included. We assume that interaction occurs as an additional phase rotation only when two particles exist in the neighboring grids. In the context of QLGA (two-component QCA), this additional phase rotation corresponds to the phase shift by the collision between the left-going and the right going particles [

Here

Note that in this simplest case, the structure of evolution scheme is kept same as the structure of free fermion case. After preparing this form, we can apply a MPS approximation and a usual TEBD algorism illustrated by

(for the 8 grid points case, m is the dimension of auxiliary spaces and the shape of tensor

Here SVD is applied then the subspace corresponding to small singular values is truncated in order to keep the dimension of auxiliary space to the given value (m in this case).

For zero derivative or zero amplitude boundary condition, at an odd time, the end points are updated by applying the 2 ´ 2 diagonal unitary matrix

Finally in this section, we compare our method with the ordinal way of reaching the TEBD algorithm. Basically so called hopping term representing kinetic energy part in evenly-spaced-grid-base (or site-base) quantum models such as Hubbard model or fermionized XXZ model is derived from the FDM-ap- proximation of kinetic energy term. The FDM-approximated Hamiltonian matrix of one-particle TDSE is given by

And, its 2nd quantized Hamiltonian for non-interacting fermions is

(The 1st term is so called hopping term.

By adding neighboring interaction term and dropping external potential term and constant term for simplicity, we have fermionized XXZ model [

If the anisotropy parameter

The XXZ spin model and its equivalent fermionized version are well studied [

According to the TEBD algorism, we decompose the Hamiltonian into two parts.

Time evolution during the small time

As terms

(This is QCA-like evolution). As each factor is finite matrix, we can obtain easily its matrix representation using the standard matrix representation of creation and annihilation operator as follows.

We see the exact correspondence of the hopping term parameter in the model Hamiltonian

As shown in Equation (18) and Equation (20), the grid pair evolution matrix for bosonic QCA is infinite size matrix, whereas that of fermionic QCA is reduced to 4 ´ 4. It is desirable if bosonic QCA is well approximated by a QCA with small degree of freedom as in the fermionic QCA.

We propose here boson approximation by fermionic QCA (or QCA with a hard core condition) when occupation number per grid is small. We mean by the hard core condition that at most one-particle can reside in one grid point. (We not necessarily mean

In order to make it easy to understand, we compare with the free fermion case, where no interpolation is needed, namely we set the amplitudes where

symmetric with respect to exchange of

For 2-particle Boson approximation case,

Under this assumption, we have the following evolution rule.

This implies, the 4 by 4 Unitary matrix in 2nd quantization formalism changed from that of Fermion case as follows

As mentioned before, this 4 ´ 4 unitary matrix is the same as that of the fermion system (fermionized XXZ) with nearest neighbor attractive interaction

We give another possible interpretation of Equation (34). The boson approximation Equation (34) can be obtained by applying coarse graining to Equation (18) using the following seemingly reasonable weight matrix for the adjacent grid pair subspace. Namely the 11 - 11 component of Equation (34) (=1) is obtained also by

Here, coarse graining means that three states of the adjacent grid pair, namely

Here we show examples of the QCA-TEBD application with the boson approximation. In our simulations, we adopt the minimal auxiliary space dimension for MPS which can describe any 1-slater wavefunction (namely

We can verify that

For example, in 2-particle case, using the standard representation of Fermion creation and annihilation operators

we have

We simulated 2-particles in a one-dimensional box using imaginary time evolution.

At t = 0 we set 2-particles at adjacent two grid points near the center position.

In

For

Similarly we computed MPS wave function for the three particle system and the results are shown in

In general the parameter of the interaction must be scaled properly when the grid spacing is changed in order to obtain the same continuum limit waveform. It is reasonable that when

In

are sufficiently small and MPS approximation must be good for the case. Theoretically the ratio should be zero for the free fermion case

In

where

Finally we provide here more detailed information about simulation methods

we adopted, though this is not the main purpose of this study. To perform imaginary time simulation, we set simply

In this section we explain how to perform the equivalent simulation by the 1st

quantized form of QCA. Of course in contrast to the above QCA-TEBD simulation, this can be performed only when the number of grids or particles is small (notorious exponential wall for large number problems) . But as it is simple and free from the particle number conservation problem, the result can be used as a reference to the QCA-TEBD simulation. Moreover there are no fundamental difficulties, in simulating bosonic or higher dimensional systems by the 1st quantized form. We already explained the relation between the 1st quantized QCA and the 2nd quantized QCA in the free particles case, we here explain how to treat the additional phase rotation caused by interactions in the 1st quantized QCA. Firstly we explain the 1D-2 particle case. One step evolution is given by Equation (43).

At an even or odd time the evolution rule Equation (43) is applied to each even or odd quadrilateral (namely red or blue quadrilateral in

The algorithm of the 1st quantized form of QCA is basically independent of particle statistics. The only procedural difference between boson and fermion is in symmetrization or anti-symmetrization at each simulation step. Without this anti-symmetrization however a decay from a fermionic state to a bosonic state occurs occasionally.

In more general 1D M-particle case,

In higher dimensional case, the free evolution part

In a case of higher dimension or many particles, the requirement for the magnitude of

one simulation step to

space and

the memory only for the simplex region

In the following, we show two imaginary time 1st quantized QCA simulations, one is 1D 4 particle fermionic and corresponding bosonic system, the other is 2D 2 particle fermionic and bosonic systems.

The Hamiltonians related to the fermionic and bosonic QCAs we simulate are

where

We show the result of 1D 4 particle and 2D 2particle imaginary time simulations in

the 1D fermionic system with

By seeing

In

In this supplementary section, we briefly discuss the possibility of multi-step QCA. Firstly we discuss QW and Dirac Cellular Automaton (DCA) [

to the corresponding QCA. This equivalence is easily shown by using the factorization form of the two-grid translationally invariant banded unitary matrix (namely multi-step QCA form) [

Namely their two components (up and down) are assigned to two amplitudes of adjacent grid points in the lattice of which the number of grid points are doubled from the original lattice. The 2 ´ 2-unit Z-transformation representation [

respectively. Here

shift in the flattened lattice and

mation representation of the one-grid shift matrix defined by Equation (6). Note that

Now we rewrite

Considering

where

(51) are added in order to clarify the correspondence between the expressions and the graphs in

By taking the logarithm of

The corresponding Hamiltonian is

where

The case

In order to be able to connect this QCA with the Dirac equation, in the wave number

the DCA case

DCA more suitable in connecting to Dirac equation.

As we explained above (Equations (50) (51) and

In this study we show that in one-dimensional multiparticle QCA, the approximation of the bosonic system by fermion (boson-fermion correspondence) can be derived in rather a 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 boson approximation, we calculate the ground state of 2 or 3-particle systems in a box using imaginary time QCA-TEBD simulation. Obtained ground states are indeed boson-like. We also perform imaginary time simulations by the 1st quantized form of QCA not only for fermionic system but also for bosonic system and show the applicable range of boson approximation (boson-fermion correspondence

This work was supported by Education Center for Next-generation Simulation Engineering, Toyohashi University of Technology and University-Community Partnership Promotion Center, Toyohashi University of Technology. We would like to thank Prof. Hitoshi Goto for his support and Dr. Akira Saitoh for his helpful advice.

Hamada, S. and Sekino, H. (2017) The Approximation of Bosonic System by Fermion in Quantum Cellular Automaton. Journal of Quantum Information Science, 7, 6-34. https://doi.org/10.4236/jqis.2017.71002