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


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.

This is a preview of subscription content, access via your institution.


  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])


  1. 1.

    Di Molfetta, G., Arrighi, P.: A quantum walk with both a continuous-time limit and a continuous-spacetime limit. Quant. Inf. Process. 19(2), 47 (2020)

    ADS  MathSciNet  Article  Google Scholar 

  2. 2.

    Feynman, R.P.: Simulating physics with computers. Int. J. Theor. Phys. 21(6), 467 (1982)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Georgescu, I.M., Ashhab, S., Nori, F.: Quantum simulation. Rev. Mod. Phys. 86(1), 153 (2014)

    ADS  Article  Google Scholar 

  4. 4.

    Jordan, S.P., Lee, K.S., Preskill, J.: Quantum algorithms for quantum field theories. Science 336(6085), 1130 (2012)

    ADS  Article  Google Scholar 

  5. 5.

    Strauch, F.W.: Connecting the discrete-and continuous-time quantum walks. Phys. Rev. A 74(3), 030301 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  6. 6.

    Arnault, P., Debbasch, F.: Quantum walks and gravitational waves. Ann. Phys. 383, 645–661 (2017).

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. 7.

    Di Molfetta, G., Debbasch, F.: Discrete-time quantum walks: continuous limit and symmetries. J. Math. Phys. 53(12), 123302 (2012)

    ADS  MathSciNet  Article  Google Scholar 

  8. 8.

    Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum Walks in artificial electric and gravitational Fields. Phys. A Stat. Mech. Appl. 397, 157 (2014)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Arnault, P., Debbasch, F.: Quantum walks and discrete gauge theories. Phys. Rev. A (2016).

    Article  MATH  Google Scholar 

  10. 10.

    Di Molfetta, G., Pérez, A.: Quantum walks as simulators of neutrino oscillations in a vacuum and matter. New J. Phys. 18(10), 103038 (2016)

    Article  Google Scholar 

  11. 11.

    Arnault, P., Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum walks and non-Abelian discrete gauge theory. Phys. Rev. A 94(1), 012335 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  12. 12.

    Arnault, P., Debbasch, F.: Landau levels for discrete-time quantum walks in artificial magnetic fields. Phys. A Stat. Mech. Appl. 443, 179 (2016). URL

  13. 13.

    Di Molfetta, G., Brachet, M., Debbasch, F.: Quantum walks as massless Dirac fermions in curved space-time. Phys. Rev. A 88(4), 042301 (2013)

    ADS  Article  Google Scholar 

  14. 14.

    Arrighi, P., Facchini, F.: Quantum walking in curved spacetime: (3+1) dimensions, and beyond. Quant. Inf. Comput. 17(9-10), 0810 (2017). URL ArXiv:1609.00305

  15. 15.

    Succi, S., Fillion-Gourdeau, F., Palpacelli, S.: Quantum Lattice Boltzmann is a quantum walk. EPJ Quant. Technol. (2015).

    Article  Google Scholar 

  16. 16.

    Arrighi, P., Di Molfetta, G., Márquez-Martín, I., Pérez, A.: From curved spacetime to spacetime-dependent local unitaries over the honeycomb and triangular quantum walks. Sci. Rep. 9(1), 1 (2019)

    Article  Google Scholar 

  17. 17.

    Kogut, J., Susskind, L.: Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D 11(2), 395 (1975)

    ADS  Article  Google Scholar 

  18. 18.

    Zohar, E., Burrello, M.: Formulation of lattice gauge theories for quantum simulations. Phys. Rev. D 91(5), 054506 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  19. 19.

    Manighalam, M., Kon, M.: Continuum limits of the 1D discrete time quantum walk (2019)

Download references


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.

Author information



Corresponding author

Correspondence to Michael Manighalam.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


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 \(m=x,y\):

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

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

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

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

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

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 \(B=A_xB_y+B_xA_y\):

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

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):

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

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 \(A_m=R_z(\zeta '_{m})R_y(\theta _{0m})R_z(\phi _{m})\):

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

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

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

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 \(\nu '_{l_xl_yn_xn_y}\):

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

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 \(a_2=\phi _{y}+\zeta _{x}\)):

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

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:

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

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

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

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:

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

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

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

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 \(\pi \).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Manighalam, M., Di Molfetta, G. Continuous time limit of the DTQW in 2D+1 and plasticity. Quantum Inf Process 20, 76 (2021).

Download citation


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