Consider the simplest partitioned QCA on a 1D-time 1D-space lattice of which time evolution rule is given by Figure 1 and Equation (1) (for TDSE-type QCA). This rule is governed by the 2 ´ 2 basic unitary matrix (which is called scattering unitary matrix [11] ) which operates on a vector consisting of functions at adjacent grid points.

Here is the x-component of Pauli matrices and.

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 and (the parameter of TDSE-type QCA) is given by

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

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

Here is the 2nd order central difference operator defined by

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 is given by

This is approximated according to individual time-discretization schemes of FDM. For example the simplest but less accurate scheme is Forward Euler method, where it is approximated as. In QCA however is approximated in a different way. Firstly is divided into two parts.

where

Then is approximated as a split-step form

Note that as have a 2 ´ 2 block diagonal form, their exponential can be explicitly calculated. In this way the time evolution for is

divided into two steps ,so we naturally define as the time step of QCA and is expressed as using this definition, which corres-

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 in general case Equation (3). This FDM-QCA correspondence is essentially the same as the first part of TEBD algorism we discuss later.

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 [23] [24] [25] . Although this approach lacks some exactness, it is sufficient to obtain right TDSE characteristics. More precise approach is to use a single scaling parameter on which other parameters (etc) are defined to depend properly such that limit exists [8] . Another approach is to use a “path integral” on a lattice to obtain a discrete Green function then take an appropriate limit [3] . Using these approaches one dimensional Dirac equation (for mass) can be derived in the

limit. Another recent study can be found in [26] . In this study we do

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 Figure 2. In the cases of (2) (3), the evolution rule at a boundary point(x = 0 or x = N − 1) in odd time is only to multiply the phase rotation factor as shown below. (We show only the case x = 0 as the case x = N − 1 is essentially the same).

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

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

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 [23] (2-step QCA of Section 6).

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 Figure 3. Concretely

(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 and they commute with each

other, approximately corresponds to the evolution

, namely it causes 2D free TDSE time evolution. We thus

generate multidimensional QCA for general dimension.

Applications of QCA to multidimensional cases are studied in [7] [8] [9] . In multidimensional QW, less straightforward (namely not direct product) models are mainly studied, where the number of internal states is not (D: the dimension of the space) but less than this (for example 2, 4) [24] [25] .

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 preserve this symmetry for the case of distinguishable particle systems and we can define for the subspace of indistinguishable particle systems. (Here U means global unitary matrix, not 2 ´ 2 local unitary matrix)Applications of QCA to multiparticle cases are studied in [3] [7] [8] [9] .

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 for its free particles.

(Here is the 1-particle Hamiltonian matrix elements, are the creation and annihilation operators for Boson).

If the one-particle time evolution rule is given not by a generator but by a finite time evolution matrix such as in QCA, the construction of its 2nd quantized formalism is done in a slightly different way, which though is consistent with the generator case. For example the evolution of 3-particle state is described as follows

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 is

18)

(local unitary matrices for other grid pairs have the same form).

We then apply the substitution rule to 1D free fermionic QCA system. Focusing on the grid pair, four states evolve as follows.

Here and are fermion creation operators fulfilling anti-commutation. Namely, the local unitary evolution matrix is

It should be noted that if periodic boundary condition is adopted for the grid pair, the off-diagonal element in Equation (20) must be replaced with considering that etc. (M is the total number of particles in all grid points.) Essentially same equation as Equation (20) is proposed in other literatures on multiparticle QCA [3] [8] [9] .

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 [4] . QCA with the nearest neighbor pair interaction was studied also by Meyer [3] and Boghosian [8] and Schumacher and Werner [9] in the form we present here.

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 [3] . Under this assumption, the 4 ´ 4 local unitary evolution matrix becomes

Here meansattraction, and means repulsion.

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 Figure 4 and Figure 5. A general wave function for the 2nd quantized form of QCA is approximated by MPS as

(for the 8 grid points case, m is the dimension of auxiliary spaces and the shape of tensor is (2,m) for i = 0, (m,2,m) for i = 1 to 6, (m,2) for i = 7), then upon the time evolution, each adjacent pair of are updated by applying the 4 ´ 4 unitary matrix U of Equation (21) as

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. for zero derivative or zero amplitude boundarycondition respectively.)

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. is the total particle number and here we assume it constant).

By adding neighboring interaction term and dropping external potential term and constant term for simplicity, we have fermionized XXZ model [27] [28] , where only nearest neighbor grid point of occupation number have a interaction through.

If the anisotropy parameter, this model corresponds to the XXX model for fermion where the interaction is two-body coulomb interaction.

The XXZ spin model and its equivalent fermionized version are well studied [27] . The phase diagram of the XXZ spin model in extended systems with the external magnetic field consists of 3 phases, ferromagnetic, paramagnetic and antiferromagnetic phases. When the magnetic field is zero, correspond to ferromagnetic (gapped), paramagnetic (gapless) and antiferromagnetic phases respectively and in the paramagnetic phase, quasi particles (magnon) behave as boson-like Tomonga-Luttinger liquid [29] [30] . The magnetic field in the XXZ spin model becomes the chemical potential when the model is fermionized. The method has been used for grand canonical systems. We however focus our application in finite system where the number of particles fixed.

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

Time evolution during the small time interval is done as follows (Suzuki-Trotter)

As terms in or commute with each other, we have

and

(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 to the QCA parameter, and the strength of correlation introduced by can be interpreted as the phase factor caused by the local potential at the grid point from the other electron in QCA. When, namely, it corresponds to the free Boson approximation case we will address in the next section.

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). We assume that only the amplitudes at points where all are different comprise the full set of independent variables and amplitudes of other points (etc.) needed for evolution are evaluated by interpolation from other points (set to the value of nearby point.) We illustrate in Figure 6 the method of the boson approximation we propose for two-particle case, comparing with free fermionic QCA case.

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 to zero. For example, in 2-particle free fermion case, must be anti-

symmetric with respect to exchange of. And we can obtain the evolution rule at a quadrilateral on the diagonal line as follows.

For 2-particle Boson approximation case, must be symmetric with respect to exchange of, but the amplitude cannot be given without some assumptions. We take an approximation to assume that

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 (namely fermionized XXX). This means that fermion-boson correspondence for this 1D quantum system is easily derived from the QCA-TEBD formulation.

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, are joined into one state (1,1) so that the occupation number per grid is kept less than 2 upon time evolution. Note that the original bosonic QCA Equation (18) does not conserve hard core condition due to the transition from to (0,2) or (2,0).

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 for M particle system). 2nd-quantized MPS-form wave function describing 1st-quantized 1- slater wave function for M particle N grid point system can be given as follows using representation matrices of creation and annihilation operators and orthonormal orbitals (i = 0 to: occupied orbital number).

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 Figure 7 we show converged density distributions for N = 64,256 and cases. For (free fermion), it converges to the state where the two particles occupy the ground and the 1st-exited (1-particle) states.

For (free boson approximation), it converges to the state where the two-particle reside in the same (1-particle) ground state.

Similarly we computed MPS wave function for the three particle system and the results are shown in Figure 8.

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 the ground state waveform does not depend on N, and when it depends on N. The case of is exceptional in that the waveform does not depend on N as if the particles were not interacting despite the fact that interaction is taken into account by non-zero parameter. This reflects the validity of the boson approximation.

In Figure 9 we show the ratio of sum of the truncated norms of singular values to that of all singular values for N = 64,256 and cases. Ratios

are sufficiently small and MPS approximation must be good for the case. Theoretically the ratio should be zero for the free fermion case, but small numerical error is observed.

In Figure 10, we show the converged density distribution of two-particle system.For the case, the discontinuity of the wave funcion can be seen at. In general the boson approximation wavefuncion and the real fermionic wavefunction are thought to be related by Equation (42) [31] .

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 the unitary matrix Equation (30) to the imaginary value. We performed a canonicalization of MPS state proposed by Vidal [21] at each simulation step, and in addition to this we performed an appropriate gauge transformation of MPS state corresponding to an additional evolution by the spatially constant chemical potential. In a MPS simulation of systems of fixed particle numbers, the chemical potential is theoretically irrelevant to the result, but it affects the robustness of the simulation and a small numerical error causes violation of particle number conservation leading to the grand canonical ground state.