^{1}

^{2}

Linear fractional map type (LFMT) nonlinear QCA (NLQCA), one of the simplest reversible NLQCA is studied analytically as well as numerically. Linear advection equation or Time Dependent Schr ödinger Equation (TDSE) is obtained from the continuum limit of linear QCA. Similarly it is found that some nonlinear advection-diffusion equations including inviscid Burgers equation and porous-medium equation are obtained from LFMT NLQCA.

Quantum Cellular Automaton (QCA) [

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

The simplest is the QCA with constant U (independent from space and time). We then generalize it to the QCA with space dependent U as described by Equation (2).

Moreover QCA/QW with time dependent

In the NLQCA the basic 2 × 2 matrix depends on the amplitude of wave function. QCA’s fundamental and powerful properties are unitarity, locality, reversibility, and when we construct NLQCA, it is important to keep these properties. NLQCA was investigated by Meyer [

After this introduction, the rest of the article is organized as follows. In Section 2 we introduce LFMT phase rotation and define three typical types of it, type-0, type-1 and type-2. We also perform its fixed point analysis, which is a useful mean to investigate the characteristics of NLQCA. In Section 3 we introduce two kinds of reversible NLQCA using LFMT phase rotation, namely complex-LFMT NLQCA and real-LFMT NLQCA. In Section 4 we investigate the property of complex-LFMT NLQCA focusing on type-0. In Section 5 and 6 we investigate the property of real-LFMT NLQCA (type-0 in Section 5 and type-2 in Section 6).

Consider the following map (complex plane to the complex plane itself) which conserves its absolute value.

or

Here

Then

It is easily shown that LFMT phase rotations are closed with respect to inversion and composition. (see Equations (44) and (55) in Appendix A)

Generally A, B can be any function of r. However we restrict our discussion to the following 3 cases for simplicity. We discuss type-0 mainly in this study.

(1) type-0

(2) type-1

(3) type-2

Here,

[Definition]

For example, by using the above definitions,

as

The function which multiplies the constant k is written as [k]. Note that $ is not included in [

respectively. We will omit the function composition symbol

In LFMT phase rotation (type-0)

In small amplitude region this map becomes a linear map

can see that this map closes to a linear map at the limits of

Approaches from fixed point analysis are useful when we investigate the characteristics of complex-LFMT NLQCA. In general a map from a circle to the circle itself is called a circle map. LFMT phase rotation of Equation (4) is a circle map for any fixed r. Now we find fixed points of this circle map. The equation for the fixed points is

Apparently the Equation (10) is satisfied when

Therefore, Equation (11) has a real solution

This means the phase gain at the fixed point

The 2nd equation in the parentheses of Equation (13) is obtained by using Equation (11). If

because the product of the two denominators is

Therefore one fixed point is stable and the other fixed point is unstable

In this study, we investigate two kinds of NLQCA using LFMT phase rotation. The one is complex-LFMT NLQCA and the other is real-LFMT NLQCA. We refer the Time Dependent Schrödinger Equation (TDSE)-type QCA [

In complex-LFMT NLQCA, the 2 × 2 unitary matrix

In the above expression

and vary the phase of B. (As the change of the phase of A can be compensated by that of initial value of

In type-0 LFMT NLQCA, a singular amplitude point exists where

(Case 1) when imag(B) = 0 [000 in

Phase lock occurs in all space and the waveform reaches a flat pattern at the end.

(Case 2) when imag(B) is sufficiently small but not 0 [001,999 in

According to the magnitude relation between

(Case 3) when imag(B) is not small [050,075,100,900,925,950 in

A phase lock never occurs, and the phase is always rotating.

In

linear QCA part is

The 3-digit number DDD in the legend indicates that

When arg(B) = 0 (Case 0), the perfect phase lock occurs. At

(Case 2), the waveform behaves like an attractive Nonlinear Schrödinger Equation(NLS) (namely it does not diffuse as in free TDSE case) and condensates with a vibration. At

diffuses faster than in free TDSE case). As arg(B) goes away from 0 (Case 3), the waveform behaves like a free TDSE (see 050,075,100,900,925,950) probably because only the time averaged phase rotation speed is mainly effective to the qualitative behavior. Here we

mean NLS by the equation i

However even in the parameter region of Case 3, singular behaviors are observed at the neighbor of some special parameters which depend also on

And above this point (namely

the case of

Above these points, the noise level with reference to free TDSE becomes larger. In

the behavior is perfectly same as free TDSE(see 299) and when it exceed this point, the deviation from the free TDSE occurs (see 300,301,304) and the deviation becomes smaller as arg(B) goes away from the point (see 306,310). In the lower graph (t = 14500) of

of

because

and the mechanism of these singularities.

From this section we investigate real-LFMT NLQCA. Firstly we explain the parameters A, B we use in the case of type-0 described by Equation (17).

As

We consider the following CPTA symmetry (Equations (18)-(21)). We find that the NLQCAs with a parameter which belongs to the same parameter groups behave qualitatively similarly.

・ C-inversion (see Equations (45) and (46) in Appendix A)

・ P-inversion (see Equations (45) and (46) in Appendix A)

・ T-inversion (see Equation (44) in Appendix A)

・ A-inversion (see Equation (45) in Appendix A)

These C, P, T, A have the following properties.

(2) A commutes with P, T, C

(3) P commutes with T (PT = TP)

(4) C “anti”-commute with T, P (CP = PCA, CT = TCA)

We define the 6 groups C, D, E, F, G, H so that ID1 and ID2 belongs to the same group if ID1 and ID2 can be transformed each other by C, P, T, A. IDs that belongs to the same group have qualitatively similar behaviors.

[Remark]

These symmetry formulas are same for the type-2, because A, B can be regarded as function of |$| and we can replace A with

The CPTA mapping for ID = (n, m) of Equation (17) is shown in

ID | type | P | T | A | PT | PA | TA | PTA | C | PC | TC | AC | PTC | PAC | TAC | PAC |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

50 | G | 54 | 54 | 54 | 50 | 50 | 50 | 54 | 52 | 56 | 56 | 56 | 52 | 52 | 52 | 56 |

51 | H | 53 | 53 | 55 | 51 | 57 | 57 | 55 | 51 | 57 | 57 | 55 | 51 | 53 | 53 | 55 |

52 | G | 52 | 52 | 56 | 52 | 56 | 56 | 56 | 50 | 50 | 50 | 54 | 50 | 54 | 54 | 54 |

53 | H | 51 | 51 | 57 | 53 | 55 | 55 | 57 | 57 | 51 | 51 | 53 | 57 | 55 | 55 | 53 |

54 | G | 50 | 50 | 50 | 54 | 54 | 54 | 50 | 56 | 52 | 52 | 52 | 56 | 56 | 56 | 52 |

55 | H | 57 | 57 | 51 | 55 | 53 | 53 | 51 | 55 | 53 | 53 | 51 | 55 | 57 | 57 | 51 |

56 | G | 56 | 56 | 52 | 56 | 52 | 52 | 52 | 54 | 54 | 54 | 50 | 54 | 50 | 50 | 50 |

57 | H | 55 | 55 | 53 | 57 | 51 | 51 | 53 | 53 | 55 | 55 | 57 | 53 | 51 | 51 | 57 |

60 | C | 00 | 64 | 64 | 04 | 04 | 60 | 00 | 06 | 66 | 02 | 02 | 62 | 62 | 06 | 66 |

61 | D | 07 | 63 | 65 | 05 | 03 | 67 | 01 | 05 | 67 | 03 | 01 | 61 | 63 | 07 | 65 |

62 | C | 06 | 62 | 66 | 06 | 02 | 66 | 02 | 04 | 60 | 04 | 00 | 60 | 64 | 00 | 64 |

63 | D | 05 | 61 | 67 | 07 | 01 | 65 | 03 | 03 | 61 | 05 | 07 | 67 | 65 | 01 | 63 |

64 | C | 04 | 60 | 60 | 00 | 00 | 64 | 04 | 02 | 62 | 06 | 06 | 66 | 66 | 02 | 62 |

65 | D | 03 | 67 | 61 | 01 | 07 | 63 | 05 | 01 | 63 | 07 | 05 | 65 | 67 | 03 | 61 |

66 | C | 02 | 66 | 62 | 02 | 06 | 62 | 06 | 00 | 64 | 00 | 04 | 64 | 60 | 04 | 60 |

67 | D | 01 | 65 | 63 | 03 | 05 | 61 | 07 | 07 | 65 | 01 | 03 | 63 | 61 | 05 | 67 |

70 | E | 70 | 74 | 74 | 74 | 74 | 70 | 70 | 76 | 76 | 72 | 72 | 72 | 72 | 76 | 76 |

71 | F | 77 | 73 | 75 | 75 | 73 | 77 | 71 | 75 | 77 | 73 | 71 | 71 | 73 | 77 | 75 |

72 | E | 76 | 72 | 76 | 76 | 72 | 76 | 72 | 74 | 70 | 74 | 70 | 70 | 74 | 70 | 74 |

73 | F | 75 | 71 | 77 | 77 | 71 | 75 | 73 | 73 | 71 | 75 | 77 | 77 | 75 | 71 | 73 |

74 | E | 74 | 70 | 70 | 70 | 70 | 74 | 74 | 72 | 72 | 76 | 76 | 76 | 76 | 72 | 72 |

75 | F | 73 | 77 | 71 | 71 | 77 | 73 | 75 | 71 | 73 | 77 | 75 | 75 | 77 | 73 | 71 |

76 | E | 72 | 76 | 72 | 72 | 76 | 72 | 76 | 70 | 74 | 70 | 74 | 74 | 70 | 74 | 70 |

77 | F | 71 | 75 | 73 | 73 | 75 | 71 | 77 | 77 | 75 | 71 | 73 | 73 | 71 | 75 | 77 |

00 | C | 60 | 04 | 04 | 64 | 64 | 00 | 60 | 66 | 06 | 62 | 62 | 02 | 02 | 66 | 06 |

01 | D | 67 | 03 | 05 | 65 | 63 | 07 | 61 | 65 | 07 | 63 | 61 | 01 | 03 | 67 | 05 |

02 | C | 66 | 02 | 06 | 66 | 62 | 06 | 62 | 64 | 00 | 64 | 60 | 00 | 04 | 60 | 04 |

03 | D | 65 | 01 | 07 | 67 | 61 | 05 | 63 | 63 | 01 | 65 | 67 | 07 | 05 | 61 | 03 |

04 | C | 64 | 00 | 00 | 60 | 60 | 04 | 64 | 62 | 02 | 66 | 66 | 06 | 06 | 62 | 02 |

05 | D | 63 | 07 | 01 | 61 | 67 | 03 | 65 | 61 | 03 | 67 | 65 | 05 | 07 | 63 | 01 |

06 | C | 62 | 06 | 02 | 62 | 66 | 02 | 66 | 60 | 04 | 60 | 64 | 04 | 00 | 64 | 00 |

07 | D | 61 | 05 | 03 | 63 | 65 | 01 | 67 | 67 | 05 | 61 | 63 | 03 | 01 | 65 | 07 |

As illustrated in

In this section we show simulated waveform of type-G/E/C. For these cases, evolution of the waveform can be explained almost by comparing to the corresponding continuous time counterpart equation we discuss later. In

0) waveform is set to the Gaussian form

simulated both in forward time (blue) and backward time (red) with a periodic boundary condition. Backward time evolution is performed using T-inversion formula (20).

Two-point-averaged

Type-G (

In type-E (

Type-C (

In this section we show that type-F and type-H can be understood from the view point of fixed point. In the case of real-LMFT NLQCA the meaning of a fixed point is slightly different from that in the case of complex-LMFT NLQCA we discussed before. In this case, a fixed point waveform does not keep still but propagates to ±45 deg. direction in the spacetime. Now we consider the (pseudo) fixed point equation for a fixed waveform moving to the left or right at the speed of one. Namely

or equivalently using

Here

As

(from Equations (45) (46) (48) in Appendix A), Equation (23) can be rewritten as

Obviously, in order for z to be a true fixed point or a pseudo fixed point,

where R, I denotes the set of real numbers and the set of pure imaginary numbers respectively. Sufficient (and presumably necessary) conditions for Equation (25) are

Namely, using Equation (24)

From Equation (17) and

In

and backward time (red) with a periodic boundary condition. Backward time evolution is performed using T-inversion formula (20). Two-point-averaged

waveforms around the start time

the NLQCA parameter

advection type linear QCA with advection speed

In type-H (

It is known that the continuum limit of the simplest QCA becomes linear advection equation or TDSE (see for example [

As

between the right side average and the left side average, p and (q/2) means the coeffi-

cients of ψ and

By inserting Equation (28) to Equation (29) we have

Therefore

As Equation (31) implies that

As NLQCA is unitary, its continuum limit must be unitary time evolution equation. The best candidate is

Note that the operators

Equation (33) can also be rewritten in the following forms.

or if we set

This implies

porosity

shown numerically that its continuum limit obeys a porous-medium equation with the degree of the porosity approximately 1.5.

In

Type-2 real-LFMT NLQCA is related to type-0 real-LFMT NLQCA with a space inversion (see Equation (48) in Appendix A) and the basic 2 × 2 matrix of type-2 real LFMT NLACA becomes the form of Equation (36) in small amplitude limit.

The linear QCA governed by Equation (36) is related to the TDSE-type linear QCA of which basic 2 × 2 matrix is given by Equation (37).

Both linear QCAs (Equations (36) and (37)) have essentially the same dispersion relation Equation (38) and basically behave as TDSE [

Note that according to the argument in [

And the eigenvalues of

eigenvectors of

However this time

It is not straightforward to represent type-2 real-LFMT NLQCA using continuum limit approach as in the case of type-0 NLQCA.

In this section, we try to understand the large amplitude behavior of type-2 NLQCA by relating it to the small amplitude behavior of type 0. Using both-sides inversion and

conjugation formula

(Note that it is important to consider the pair not of

for the adjacent grid points pair which causes sign alternating behavior of

We numerically examine the validity of the approximation in the case of inviscid Burges equation. Inviscid Burgers case (type-G) is the most promising for the above

continuum limit argument to be applicable because the PDE for

evolution too. (This can be easily verified by the fact that the flux

In

well, although a certain adjusting parameter

Note that we do not plot the solution of the inversed Burgers equation itself but plot its inverted values. Initial waveform for Burgers equation is

and its inverse is used for the inversed Burgers equation.

Linear fractional map type (LFMT) nonlinear QCA (NLQCA), is studied analytically as well as numerically. Firstly we introduce LFMT phase rotation which maps the complex plane to itself conserving its absolute value. We employ this LFMT phase rotation in

two ways in order to construct reversible NLQCA, namely complex-LFMT NLQCA and real-LFMT NLQCA. In order to categorize the qualitative behavior of the LFMT NLQCA, stability analysis around fix points is introduced. Complex- and Real-LFMT NLQCA are studied numerically using a simple model. Results are summarized and analyzed according to the category by the symmetry classification for real-LFMT NLQCA. We further study the continuum limit of the real-LFMT NLQCA analytically and verify it numerically. Linear advection equation or Time Dependent Schrödinger Equation (TDSE) is obtained from the continuum limit of linear QCA. Similarly it is found that nonlinear advection-diffusion equations including inviscid Burgers equation and porous-medium equation are obtained from real-LFMT NLQCA. Although it is already reported in the article [

This research was supported by TUT Programs on Advanced Simulation Engineering, Toyohashi University and University-Community Partnership promotion center, Toyohashi University. We would like to thank Prof. Hitoshi Goto for his support.

Hamada, S. and Sekino, H. (2016) Solution of Nonlinear Advection-Diffusion Equations via Linear Fractional Map Type Nonlinear QCA. Journal of Quantum Information Science, 6, 263- 295. http://dx.doi.org/10.4236/jqis.2016.64017

As mentioned in 2.1,

Scale transformation

Inverse transformation

By clearing the fraction of the left equation and replacing zz* with z'z'* the right equation is obtained. Namely

is obtained.

Rotation

Here A, B are constants, or A, B can be functions of |$| if |k| = 1.

[Proof]

Equation (45a) is the special case of the general formula

(Note that the factor u(k) must be factored out to the left, or $ (=evaluated value of the right side) would be changed.)

Equations (45c) and (45d) can be obtained from Equations (45b) and (45a) respectively by applying C from the right and using Equation (48). Note that if |k| = 1 Equations (45c) and (45d) can be obtained also by replacing A with

Both sides conjugation

A, B can be a function of |$|.

[Proof]

Right side conjugation

A, B can be a function of |$|. This formula means that type-0 LFMT phase rotation is related to type-2 LFMT phase rotation via complex conjugation (C).

[Proof]

[Remark]

A, B can be a function of |$|. Therefore especially for type-1 and type-2

are satisfied.

Left side conjugation

Note that the left side conjugation is equivalent to the both sides inversion if A and B are swapped (see Equation (52)). A, B can be a function of |$|.

[Proof]

It is obvious by applying the right side conjugation formula then the both side inversion formula.

Both side inversion

[Proof]

Both side inversion and conjugation

It is easily derived from both side inversion and both side conjugation formulas. Different from conjugation formula, A, B cannot be regarded as a general function of |$| in (52)-(54).

Composition of mappings

As A, B, A', B' are functions of |$|, it is closed. Especially for type-1

Therefore it is closed even when A, B, A', B' are restricted to fixed complex number.

Here we discuss extension of the discrete time LFMT phase rotation to the continuous time LFMT phase rotation. This discussion may be the foundation for the more complicated problem such as the continuum limit of the complex-LFMT NLQCA. It is well known that infinitesimal LFM is governed by Riccati-type equation [

Consider the differential equation for

By integrating for unit time, we obtain LFMT phase rotation.

Here A, B can be a function of |z|.

[Proof]

Assume that the continuous time extension of (58) obeys the following differential equation Equation (59) in the polar coordinate

Let

We have the following Riccati equation.

Riccati equation can be linearized by setting

And the time evolution is expressed as follows.

By setting

and using the relation

is obtained. Therefore the time evolution of

By comparing with Equation (5), (written again here as Equation (67))

We have

Reversely (a, b, c) can be obtained from (A, B) by using

as

From this and setting

[Remark]

The point where

the point where

we use

= 0, the sign does not affect the integrated result.

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