Continuous time limit of the DTQW in 2D+1 and plasticity

Abstract

A Plastic quantum walk admits both continuous time and continuous spacetime. The model has been recently proposed by one of the authors in Di Molfetta and Arrighi (Quant Inf Process 19(2): 47, 2020), leading to a general quantum simulation scheme for simulating fermions in the relativistic and non-relativistic regimes. The extension to two physical dimensions is still missing and here, as a novel result, we demonstrate necessary and sufficient conditions concerning which discrete time quantum walks can admit plasticity, showing the resulting Hamiltonians. We consider coin operators as general 4 parameter unitary matrices, with parameters which are functions of the lattice step size \(\varepsilon \). This dependence on \(\varepsilon \) encapsulates all functions of \(\varepsilon \) for which a Taylor series expansion in \(\varepsilon \) is well defined, making our results very general.

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Notes

  1. 1.

    This constraint reduces to the constraint obtained for \(\theta _{0}\) in Ref. [19] when the 1D limit is taken, i.e., \(\zeta _{0y},\theta _{0y},\phi _{0y},\delta =0\) (see Eq. (A7) of Ref. [19])

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Acknowledgements

The authors acknowledge inspiring conversations with Pablo Arrighi, Tamiro Villazon, and Pieter W. Claeys. This work has been funded by the Pépinière d’Excellence 2018, AMIDEX fondation, project DiTiQuS and the ID #60609 grant from the John Templeton Foundation, as part of the “The Quantum Information Structure of Spacetime (QISS)” Project.

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Appendices

Appendix A W expansion

We wish to expand \({\widehat{W}}^\tau \) to first order in \(\varDelta _t\). We begin by expanding \(\widehat{S_m}C_m\) up to \(O(\varDelta _t^2)\), where \(m=x,y\):

$$\begin{aligned} \begin{aligned} \widehat{S_m}C_m&=e^{i\delta _m} [R_z(\zeta '_{m})R_y(\theta _{0m})R_z(\phi _{m})-\frac{i \varDelta _t}{2}(\zeta _{1m} \sigma _z R_z(\zeta '_{m})R_y(\theta _{0m})R_z(\phi _{m})\\&\quad +\theta _{1m} \sigma _y R_z(-2\zeta '_{m})R_z(\zeta '_{m})R_y(\theta _{0m})R_z(\phi _{m})\\&\quad +\phi _{1m} R_z(\zeta '_{m})R_y(\theta _{0m})R_z(\phi _{m}) \sigma _z)+O(\varDelta _t^2)]\\&=e^{i\delta _m}(A_m-\frac{i \varDelta _t}{2} B_m+O(\varDelta _t^2)) \end{aligned} \end{aligned}$$
(78)

Now we can combine the product of \(\widehat{S_x}C_x\) and \(\widehat{S_y}C_y\) up to \(O(\varDelta _t)\):

$$\begin{aligned} \begin{aligned} {\widehat{W}}=\widehat{S_x}C_x\widehat{S_y}C_y&=e^{i\delta _x}\left( A_x-\frac{i \varDelta _t}{2} B_x+O(\varDelta _t^2)\right) e^{i\delta _y}\left( A_y-\frac{i \varDelta _t}{2} B_y+O(\varDelta _t^2)\right) \\&=e^{i(\delta _x+\delta _y)}\left( A_xA_y-\frac{i\varDelta _t}{2}(A_xB_y+B_xA_y)+O(\varDelta _t^2)\right) \\&=e^{i\delta }(A-\frac{i\varDelta _t}{2}B+O(\varDelta _t^2)) \end{aligned} \end{aligned}$$
(79)

And now we expand \({\widehat{W}}^\tau \) in powers of \(\varDelta _t\):

$$\begin{aligned} \begin{aligned} {\widehat{W}}^\tau&=\left( \widehat{S_x}C_x\widehat{S_y}C_y\right) ^\tau \\&=e^{i\delta \tau }\left( A-\frac{i \varDelta _t}{2} B+O(\varDelta _t^2)\right) ^\tau \\&=e^{i\delta \tau }\left( A^\tau -\frac{i \varDelta _t}{2}(A^{\tau -1}B+A^{\tau -2}BA+\cdots +ABA^{\tau -2}+BA^{\tau -1})+O(\varDelta _t^2)\right) \\&=e^{i\delta \tau }\left( A^\tau -\frac{i \varDelta _t}{2}\sum _{j=0}^{\tau -1}A^{\tau -1-j}BA^j+O(\varDelta _t^2)\right) \\&=(e^{i\delta }A)^\tau \left( 1-\frac{i \varDelta _t}{2}A^{-1}\sum _{j=0}^{\tau -1}A^{-j}BA^j+O(\varDelta _t^2)\right) \end{aligned} \end{aligned}$$
(80)

Appendix B \(\{A,B\}\) expansion

We begin by expanding \(\{A,B\}\) in terms of \(A_x\), \(A_y\), \(B_x\), and \(B_y\), using \(A=A_xA_y\) and \(B=A_xB_y+B_xA_y\):

$$\begin{aligned} \begin{aligned} \{A,B\}&=A_xA_yA_xB_y+A_xB_yA_xA_y+A_xA_yB_xA_y+B_xA_yA_xA_y. \end{aligned} \end{aligned}$$
(81)

Now we expand further using \(B_m=\zeta _{1m}\sigma _z A_m+\theta _{1m} \sigma _y R_z(-2\zeta '_{m})A_m+\phi _{1m} A_m \sigma _z\) (where \(m=x\) or y):

$$\begin{aligned} \begin{aligned} \{A,B\}=&\zeta _{1y}A_xA_yA_x\sigma _zA_y+\theta _{1y}A_xA_yA_x\sigma _yR_z(-2\zeta _{0y}')A_y+\phi _{1y}A_xA_yA_xA_y\sigma _z\\&+\zeta _{1y}A_x\sigma _zA_yA_xA_y+\theta _{1y}A_x\sigma _yR_z(-2\zeta _{0y}')A_yA_xA_y+\phi _{1y}A_xA_y\sigma _zA_xA_y\\&+\zeta _{1x}A_xA_y\sigma _zA_xA_y+\theta _{1x}A_xA_y\sigma _yR_z(-2\zeta _{0x}')A_xA_y+\phi _{1x}A_xA_yA_x\sigma _zA_y\\&+\zeta _{1x}\sigma _zA_xA_yA_xA_y+\theta _{1x}\sigma _yR_z(-2\zeta _{0x}')A_xA_yA_xA_y+\phi _{1x}A_x\sigma _zA_yA_xA_y. \end{aligned} \end{aligned}$$
(82)

Using \(\sigma _zA_x=(-1)^\nu A_x\sigma _z\) and \(\sigma _zA_y=(-1)^{\nu +1} A_y\sigma _z\), it can be shown that the first and third columns of Eq. (82) cancel. Now we expand the remaining terms using \(A_m=R_z(\zeta '_{m})R_y(\theta _{0m})R_z(\phi _{m})\):

$$\begin{aligned} \begin{aligned} \theta _{1y}A_xA_yA_x\sigma _yR_z(-2\zeta '_{0y})A_y&=-\theta _{1y}R_z(-2\phi _{0y})\sigma _y\\ \theta _{1y}A_x\sigma _yR_z(-2\zeta '_{0y})A_yA_xA_y&=-\theta _{1y}R_z(2\zeta '_{0x}+2\phi _{0x}(-1)^\nu +2\zeta '_{0y}(-1)^\nu )\sigma _y\\ \theta _{1x}A_xA_y\sigma _yR_z(-2\zeta '_{0x})A_xA_y&=-\theta _{1x}R_z(2\zeta '_{0y}(-1)^\nu -2\phi _{0y}+2\phi _{0x}(-1)^\nu )\sigma _y\\ \theta _{1x}\sigma _yR_z(-2\zeta '_{0x})A_xA_yA_xA_y&=-\theta _{1x}R_z(2\zeta '_{0x})\sigma _y. \end{aligned} \end{aligned}$$
(83)

Putting it all together, we have the following for \(\{A,B\}\):

$$\begin{aligned} \begin{aligned} \{A,B\}=&-\theta _{1y}(R_z(-2\phi _{0y})+R_z(2\zeta '_{0x}+2\phi _{0x}(-1)^\nu +2\zeta '_{0y}(-1)^\nu ))\sigma _y\\&-\theta _{1x}(R_z(2\zeta '_{0x})+R_z(2\zeta '_{0y}(-1)^\nu -2\phi _{0y}+2\phi _{0x}(-1)^\nu ))\sigma _y \end{aligned} \end{aligned}$$
(84)

Appendix C Cross term constraint

In this section, we will deduce for which constraints from Sect. 4 do the resulting continuum limit PDEs include a cross derivative term. We will find that the only constraints which will have cross terms will be the pair \(\cos (\frac{\phi _{x}+\phi _{y}+\zeta _{x}+\zeta _{y}}{2})=\cos (\frac{a_1+a_2}{2})=0\rightarrow a_1+a_2=2\pi m +\pi \) and \(\cos (\frac{a_1-a_2}{2})=0\rightarrow a_1-a_2=2\pi t +\pi \). We first reiterate the definitions of \({\hat{\varGamma }}_{l_xl_yn_xn_y}\) and \(\nu '_{l_xl_yn_xn_y}\):

$$\begin{aligned} \begin{aligned} {\hat{\varGamma }}_{l_xl_yn_xn_y}&=R_z(\zeta _{x})\sigma _z^{l_x}\sigma _y^{n_x}R_y(\theta _{0x})R_z(\phi _{x})R_z(\zeta _{y})\sigma _z^{l_y}\sigma _y^{n_y}R_y(\theta _{0y})R_z(\phi _{y})\\ \nu '_{l_xl_yn_xn_y}&=\frac{(\partial _x)^{l_x}(\partial _y)^{l_y}(-\frac{i\theta _{1x}}{2})^{n_x}(-\frac{i\theta _{1y}}{2})^{n_y}}{l_x!l_y!n_x!n_y!} \end{aligned} \end{aligned}$$
(85)

Since cross terms have all \(n_{vm}\)s equal to zero and two \(l_{vm}\)s equal to one, we write the proportionality expression for \(\nu '_{l_{1x}l_{1y}00}\nu '_{l_{2x}l_{2y}00}{\hat{\varGamma }}_{l_{1x}l_{1y}00}{\hat{\varGamma }}_{l_{2x}l_{2y}00}\) (where \(a_1=\phi _{x}+\zeta _{y}\) and \(a_2=\phi _{y}+\zeta _{x}\)):

$$\begin{aligned} \begin{aligned}&\nu '_{l_{1x}l_{1y}00}\nu '_{l_{2x}l_{2y}00}{\hat{\varGamma }}_{l_{1x}l_{1y}00}{\hat{\varGamma }}_{l_{2x}l_{2y}00}\\&\quad \propto \partial _{x}^{l_{1x}+l_{2x}}\partial _{y}^{l_{1y}+l_{2y}}\\&\qquad \times R_z(\zeta _{x})\sigma _z^{l_{1x}}R_y(\theta _{0x})R_z(\phi _{x})R_z(\zeta _{y})\sigma _z^{l_{2x}}R_y(\theta _{0y})R_z(\phi _{y})\\&\qquad \times R_z(\zeta _{x})\sigma _z^{l_{1y}}R_y(\theta _{0x})R_z(\phi _{x})R_z(\zeta _{y})\sigma _z^{l_{2y}}R_y(\theta _{0y})R_z(\phi _{y})\\&\quad =\partial _{1x}^{l_{1x}+l_{2x}}\partial _{1y}^{l_{1y}+l_{2y}}\\&\qquad \times R_z(\zeta _{x})\sigma _z^{l_{1x}}R_y(\theta _{0x})R_z(a_1)\sigma _z^{l_{2x}}R_y(\theta _{0y})R_z(a_2)\\&\qquad \times \sigma _z^{l_{1y}}R_y(\theta _{0x})R_z(a_1)\sigma _z^{l_{2y}}R_y(\theta _{0y})R_z(\phi _{y})\\&\quad =\partial _{1x}^{l_{1x}+l_{2x}}\partial _{1y}^{l_{1y}+l_{2y}}\sigma _z^{l_{1x}+l_{1y}+l_{2x}+l_{1y}} R_z(\zeta _{x})\\&\qquad \times R_y((-1)^{l_{1y}+l_{2x}+l_{1y}}\theta _{0x})R_z(a_1)R_y((-1)^{l_{2x}+l_{1y}}\theta _{0y})\\&\qquad \times R_z(a_2)R_y((-1)^{l_{2y}}\theta _{0x})R_z(a_1)R_y(\theta _{0y})R_z(\phi _{y}) \end{aligned} \end{aligned}$$
(86)

Since each cross term will be proportional to \(\partial _x\partial _y\), we define the following matrix to contain the relevant parts of the above equation to our analysis:

$$\begin{aligned} \begin{aligned} {\hat{J}}_{l_{1x}l_{1y}l_{2x}l_{2y}}&=R_y((-1)^{l_{1y}+l_{2x}+l_{2y}}\theta _{0x})R_z(a_1)R_y((-1)^{l_{2x}+l_{2y}}\theta _{0y})R_z(a_2)R_y((-1)^{l_{2y}}\theta _{0x}) \end{aligned} \end{aligned}$$
(87)

For the cross derivative terms in Eq. (53) to cancel, the following must be true:

$$\begin{aligned} \begin{aligned}&{\hat{J}}_{1100}+{\hat{J}}_{1001}+{\hat{J}}_{0110}+{\hat{J}}_{0011}=0\\&\quad \rightarrow R_y(-\theta _{0x})R_z(a_1)R_y(\theta _{0y})R_z(a_2)R_y(\theta _{0x})\\&\qquad +R_y(-\theta _{0x})R_z(a_1)R_y(-\theta _{0y})R_z(a_2)R_y(-\theta _{0x})\\&\qquad +R_y(\theta _{0x})R_z(a_1)R_y(-\theta _{0y})R_z(a_2)R_y(\theta _{0x})\\&\qquad +R_y(\theta _{0x})R_z(a_1)R_y(\theta _{0y})R_z(a_2)R_y(-\theta _{0x})=0 \end{aligned} \end{aligned}$$
(88)

We see that either the first two terms can cancel when the \(\partial _y\) non-divergent constraint in Eq. (62) is imposed, or the second and fourth term can cancel when the \(\partial _x\) non-divergent constraint in Eq. (62) is imposed. When the \(\partial _y\) constraint is imposed again, the above equation reduces to the following:

$$\begin{aligned} R_y(-2\theta _y)-R_y(2\theta _y)=0\rightarrow \theta _y=n\pi \text { for} n=1, 2, 3, \ldots . \end{aligned}$$
(89)

Similarly, when the \(\partial _x\) constraint is imposed again, we recover the following:

$$\begin{aligned} R_y(-2\theta _x)-R_y(2\theta _x)=0\rightarrow \theta _x=m\pi \text { for} m=1, 2, 3, \ldots . \end{aligned}$$
(90)

Thus, we see that these cross derivative terms will cancel with either \(\theta _x\) or \(\theta _y\) equal to an integer multiple of \(\pi \). Therefore, most of the constraints will contain no cross terms. The only set of constraints which will have cross terms will be the pair \(\cos (\frac{\phi _{x}+\phi _{y}+\zeta _{x}+\zeta _{y}}{2})=\cos (\frac{a_1+a_2}{2})=0\rightarrow a_1+a_2=2\pi m +\pi \) and \(\cos (\frac{a_1-a_2}{2})=0\rightarrow a_1-a_2=2\pi t +\pi \), as there is no constraints on \(\theta _x\) or \(\theta _y\) equalling an integer multiple of \(\pi \).

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Manighalam, M., Di Molfetta, G. Continuous time limit of the DTQW in 2D+1 and plasticity. Quantum Inf Process 20, 76 (2021). https://doi.org/10.1007/s11128-021-03011-5

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Keywords

  • Plastic quantum walk
  • Discrete time quantum walk
  • Continuous time quantum walk
  • Lattice fermions
  • Quantum simulation