# Phase Shift Analysis of Light Nucleon-Nucleus Elastic Scattering using
Reference Potential Approach

Lalit Kumar, Shikha Awasthi, Anil Khachi, O.S.K.S Sastri
Department of Physical and Astronomical Sciences, Central University of
Himachal Pradesh
Dharamshala, Himachal Pradesh-176215, Bharat(India)

###### Abstract

The neutron and proton scattering with either deuteron or stable alpha
particle can be modeled as a two particle system. In this paper, using Morse
function as reference potential, inverse potentials have been computationally
constructed directly from scattering phase shifts (SPS) data for light
nucleon-nucleus systems. The phase equation for various $\ell$-channels has
been numerically solved using 5th order Runge-Kutta (RK-5) method in each
iteration within the optimisation procedure to obtain best model parameters of
Morse function that minimise mean absolute percentage error (MAPE). The
inverse potentials for S-wave ($\ell=0$) of neutron-deuteron (nd) and proton-
deuteron (pd) systems have been obtained with MAPE of 1.95 and 2.05$\%$
respectively. Those corresponding to various P and D channels of n-alpha
($n\alpha$) and p-alpha ($p\alpha$) systems, have been determined to less than
1.53 and 2.17$\%$ respectively. The obtained(experimental) resonance energies
for p1/2 and p3/2, from their partial cross-sections plots, respectively are
4.1(4$\pm$1) and 0.93(0.89) for n-alpha and 5.21(5$\pm$2) and 1.96(1.96) for
p-alpha system. While total cross-section for $n\alpha$ has been found to be
matching with values available in literature, that of $p\alpha$ is seen to be
following the correct trend.

Keywords: $n\alpha$ scattering, $p\alpha$ scattering, pd scattering, nd
scattering Phase Shift, Morse potential, Cross section, Phase Function Method
(PFM).

##  1 Introduction

Scattering is a phenomenon that is essential in understanding many nuclear
properties. Numerous theoretical models have been developed to emphasise the
underlying physics behind these scattering results. The scattering of light
nuclei with nucleons and among themselves is crucial for understanding the
underlying interaction and providing structural information. Several groups
have conducted substantial research on both nucleon-deuteron [1, 2, 3, 4] and
nucleon-alpha systems [5, 6, 7, 8, 9, 10], with results of elastic scattering
phase shifts reported both experimentally and theoretically. In the cluster
model [11, 12, 13, 14, 15] description of nuclei, the study of
nucleon-$\alpha$ elastic scattering as a two-body problem at low energy is
important. Alpha-particles have a stable and symmetric structure and hence its
deformation due to a collision with a nucleon can usually be ignored in the
first approximation. Further, since it has no spin, there are no coupled
equations in the collision analysis, and polarisation phenomena are limited to
the nucleon. It contains no excited states below at least 20 MeV, therefore
there will be no inelastic scattering to confound the understanding of
collision processes at lower energies.

To study elastic scattering among light nuclei, generator coordinate method
(GCM) or the resonating group method (RGM) were usually employed. For elastic
scattering phase shifts (SPS), incorporation of phenomenological two-body
interactions in this model results in strong agreement with experimental
evidence [16, 17]. To examine nucleon-$\alpha$ elastic scattering below 18
MeV, Satchler et al. [18] employed an optical potential model and Dohet-Early
and Baye [19] used the formalism of unitary correlation operator method and
found good agreement with experimental results [20]. Using the phase function
method (PFM), Laha et. al., [21, 22, 23] suggested a simple phenomenological
potential model based on Manning-Rosen potential for nucleon-$\alpha$ elastic
scattering.

Buck et al., [24], have argued that local potential can be considered for
nucleon-nucleus and nucleus-nucleus systems without resorting to the
resonating group method (RGM) [25]. They also came to the conclusion that non-
local potential is primarily of the same type as local potential. Using a
local Gaussian potential as the model of interaction, they were able to
produce scattering phase shifts (SPS) for the $\alpha-\alpha$ and
$\alpha-{}^{3}He$ systems with reasonable success. The model interaction
obtained on solving the time independent Schrodinger equation (TISE) for
Gaussian potential using matrix method [26] in conjunction with variational
Monte-Carlo (VMC) [27] for $\alpha-\alpha$ system in SPS [28] that match well
with experimental data. We adopted a unique technique [29] to get SPS, in
which PFM is directly used in optimisation procedure to get model parameters
that minimise mean absolute percentage error (MAPE) between simulated and
experimental data. The efficacy of this method has been utilised to compile
data for alpha-alpha scattering [30] from various sources and performing MAPE
analysis that resulted with resonant peaks in close match with experimental
data. Recently, we have presented the phase function analysis of the alpha-
nucleon systems using a Gaussian potential [31]. Utilising the Morse function
as zeroth reference potential [32] to obtain the model of interaction directly
from the experimental SPS, inverse potential for nucleon-deuteron systems [33,
34] for energy range upto 10 MeV have been obtained. In this paper, we extend
the study to all available energies for nucleon-deuteron (nd) and proton-
deuteron (pd) and obtain inverse potentials for n-alpha ($n\alpha$) and
p-alpha ($p\alpha$) systems for low energies upto 18 MeV.

##  2 Methodology

The Morse function chosen as reference potential, is given by

$V(r)=V_{0}\left(e^{-2(r-r_{m})/a_{m}}-2e^{-(r-r_{m})/a_{m}}\right)$ (1)

where the model parameters $V_{0}$, $r_{m}$ and $a_{m}$ reflect strength of
interaction, equilibrium distance at which maximum attraction is felt and
shape of the potential respectively.
For $p\alpha$ & pd systems, Coulomb interaction that needs to be included is
chosen as

$V_{Coul(r)}=\frac{z_{1}z_{2}e^{2}}{r}erf(\beta r)$ (2)

for $p\alpha$, $\beta=0.609$fm-1 [35], pd $\beta=0.441$fm-1
The Schr$\ddot{o}$dinger wave equation for a particle undergoing scattering
having energy E and orbital angular momentum $\ell$ is given by

$\frac{\hbar^{2}}{2\mu}\frac{d^{2}u_{\ell}(k,r)}{dr^{2}}+\big{[}k^{2}-\ell(\ell+1)/r^{2}\big{]}u_{\ell}(k,r)=V(r)u_{\ell}(k,r)$
(3)

where $k=\sqrt{E/(\hbar^{2}/2\mu)}$, and $(\ell(\ell+1)\hbar^{2})/{2\mu
r^{2}}$ is centrifugal potential. Where,
$\mu_{n\alpha}=\frac{m_{n}*m_{\alpha}}{m_{n}+m_{\alpha}}$ is the reduced mass
for $n\alpha$ and $\mu_{p\alpha}=\frac{m_{p}*m_{\alpha}}{m_{p}+m_{\alpha}}$ is
the reduced mass for $p\alpha$ with $m_{n}=939.5654MeV/c^{2}$,
$m_{p}=938.2720MeV/c^{2}$ and $m_{\alpha}=3728.7705MeV/c^{2}$. And
$\mu_{nd}=\frac{m_{n}m_{d}}{m_{n}+m_{d}}$,
$\mu_{pd}=\frac{m_{p}m_{d}}{m_{p}+m_{d}}$ with $m_{d}$ = 469.4590 MeV/$c^{2}$
respectively.

This second order differential equation has been transformed to the following
first order non-homogeneous differential equation of Riccati type [36, 37],
known as phase equation:

$\delta_{\ell}^{\prime}(k,r)=-\frac{V(r)}{k}\bigg{[}cos(\delta_{\ell}(k,r))\hat{j}_{\ell}(kr)-sin(\delta_{\ell}(k,r))\hat{\eta}_{\ell}(kr)\bigg{]}^{2}$
(4)

with initial condition $\delta_{\ell}(k,0)=0$.

The Riccati-Bessel & Riccati-Neumann functions for different $\ell$ are
obtained using

$\hat{j}_{\ell}(kr)=(-kr)^{\ell}\bigg{[}\frac{1}{(kr)}\frac{d}{d(kr)}\bigg{]}^{\ell}\frac{sin(kr)}{(kr)}$
(5)

and

$\hat{\eta_{l}}(kr)=-(-kr)^{\ell}\bigg{[}\frac{1}{(kr)}\frac{d}{d(kr)}\bigg{]}^{\ell}\frac{cos(kr)}{(kr)}$
(6)

respectively. The phase equation for

  1. (i)

$\ell$ = 0, S-wave :

$\delta^{\prime}_{0}(k,r)=-\frac{V(r)}{k}sin^{2}[kr+\delta_{0}(k,r)]$ (7)

  2. (ii)

$\ell$ = 1, P-wave :

$\displaystyle\delta_{1}^{\prime}(k,r)=-\frac{V(r)}{k}\bigg{[}cos(\delta_{\ell}(k,r))\bigg{(}\frac{sin(kr)}{kr}-cos(kr)\bigg{)}+$
(8) $\displaystyle
sin(\delta_{\ell}(k,r))\bigg{(}\frac{cos(kr)}{kr}+sin(kr)\bigg{)}\bigg{]}^{2}$

  3. and (iii)

$\ell$ = 2, D-wave :

$\delta_{2}^{\prime}(k,r)=-\frac{V(r)}{k}\big{[}cos(\delta_{2}(k,r))f(kr)-sin(\delta_{2}(k,r))g(kr)\big{]}^{2}$
(9)

where f(kr) is

$f(kr)=\left(\frac{3}{(kr)^{2}}-1\right)sin(kr)-\frac{3}{kr}cos(kr)$ (10)

and g(kr) is

$g(kr)=\left(\frac{-3}{(kr)^{2}}+1\right)cos(kr)-\frac{3}{kr}sin(kr)$ (11)

It is important to note that solving these equations for obtaining SPS
involves the potential directly, without the need for wavefunction. We have
implemented the $5^{th}$ order Runge-Kutta (RK-5) method for solving these
first order phase equations and optimised the model parameters by minimising
the mean absolute percentage error (MAPE)-value defined as

$MAPE=\frac{1}{N}\sum_{i=1}^{N}\frac{\lvert\delta^{exp}_{i}-\delta^{sim}_{i}\rvert}{\lvert\delta^{exp}_{i}\rvert}\times
100$ (12)

where $\delta^{exp}_{i}$ and $\delta^{sim}_{i}$ are the experimental and
obtained phase-shifts. The implemented code is shown as a flow chart in Fig.
1.

Figure 1: Flowchart showing implementation procedure for obtaining the inverse
potential

Once, SPS $\delta_{\ell}(E)$ are obtained for each orbital angular momentum
$\ell$, one can calculate the partial cross section $\sigma_{\ell}(E)$ & total
cross section $\sigma_{T}$ by using following formulae [31] :

$\sigma_{l}(E)=\frac{4\pi(2l+1)}{k^{2}}\sin^{2}({\delta_{l}(E)})$ (13)
$\sigma_{T}=\frac{4\pi}{k^{2}}\sum_{l}\left\\{(l+1)\sin^{2}(\delta_{l}^{+})+l\sin^{2}({\delta_{l}^{-}})\right\\}$
(14)

##  3 Results & Discussion

The various partial waves for both $n\alpha$ (${}^{5}He$) and $p\alpha$
(${}^{5}Li$) systems are $s$, $p_{3/2}$, $p_{1/2}$, $d_{5/2}$ and $d_{3/2}$.
The $n\alpha$ and $p\alpha$ scattering studies show sharp resonance for
$p_{3/2}$ and a slightly broader resonance for first excited $p_{1/2}$ state.
The experimental SPS data for all these states have been taken from Satchler
et al. [18].

###  3.1 Study of nucleon-alpha system:

The inverse potential parameters along with $MAPE$ values for $s$, $p$ and
$d$-states of $n\alpha$, $p\alpha$ systems obtained using our approach are
tabulated in Table 1.

Table 1: List of model parameters of Morse potential for various systems and
states along with corresponding MAPE values.

System | State | $\textbf{V}_{0}(\mathrm{\textbf{MeV}})$ | $r_{m}\left(\mathrm{fm}\right)$ | $a_{m}(\mathrm{fm})$ | MAPE(%)
---|---|---|---|---|---
n$\alpha$ | $\mathbf{s}$ | $37.66$ | $1.28$ | $0.90$ | $0.04$
$\mathbf{p}_{1/2}$ | $45.37$ | $2.13$ | $0.43$ | $0.25$
$\mathbf{p}_{3/2}$ | $58.30$ | $1.63$ | $0.50$ | $0.14$
$\mathbf{d}_{3/2}$ | $10.58$ | $2.54$ | $0.42$ | $1.53$
$\mathbf{d}_{5/2}$ | $20.22$ | $1.82$ | $0.50$ | $1.26$
p$\alpha$ | $\mathbf{s}$ | $33.24$ | $1.69$ | $1.22$ | $1.18$
$\mathbf{p}_{1/2}$ | $22.83$ | $1.61$ | $0.84$ | $1.95$
$\mathbf{p}_{3/2}$ | $65.22$ | $1.02$ | $0.66$ | $1.23$
$\mathbf{d}_{3/2}$ | $3.95$ | $0.44$ | $2.04$ | $2.17$
$\mathbf{d}_{5/2}$ | $28.25$ | $0.21$ | $1.07$ | $1.45$
nd | ${}^{2}S_{1/2}$ | $173.29$ | $2.53$ | $0.89$ | $1.95$
pd | ${}^{2}S_{1/2}$ | $33.46$ | $2.09$ | $1.33$ | $2.05$

The obtained SPS for $s$, $p_{1/2}$ and $p_{3/2}$-states along with
experimental data [18] are shown in Fig. 2 and a good agreement can be clearly
seen. It should be noted that for sake of convienience, SPS plot for $s$-state
is shifted by $180^{o}$ to avoid overlapping with other SPS plots.

Figure 2: Simulated and experimental SPS for $s$ and $p$-states of
$n\alpha$(left), $p\alpha$(right). The experimental data is taken from
Satchler et al. [18]

The SPS plots for $d_{3/2}$ and $d_{5/2}$ are plotted separately in Fig. 3 to
bring out their match with experimental data clearly, because the SPS for
these are small as compared to other states. Once again, the match between
obtained & experimental SPS is very good.

Figure 3: Simulated and experimental SPS for $d$-state of $n\alpha$(left) and
$p\alpha$(right). The experimental data is taken from Satchler et al. [18]

Figure 4: Interaction Potentials for $s$, $p$ and $d$-states of
$n\alpha$(left) and $p\alpha$(right) along with centrifugal term.

In Fig. 4, the total interaction potential including that of centrifugal term
are plotted for all the five states for both systems. It is clearly seen that,
the potentials of $n\alpha$ are slightly deeper in comparison to corresponding
states in $p\alpha$ system owing to the Coulomb term in the later. Using
obtained SPS, both partial cross section $\sigma_{l}(E)$ and total cross
section $\sigma_{T}$ for $n\alpha$, $p\alpha$ elastic scattering are
calculated for $E_{lab}$ values up to 18 MeV using Eqs.13-14. The partial
cross section for $p$ states are shown in Fig.5. For $n\alpha$ the resonance
peaks in center of mass energy for $p_{1/2}$ and $p_{3/2}$ are
observed(experimental) at $4.1(4\pm 1)$ MeV and $0.93(0.89)$ MeV and their
respective decay widths are $8.19(4\pm 1)$ MeV and $0.88(0.60)$ MeV. And for
p-$\alpha$ the resonance peaks in center of mass energy for $p_{1/2}$ and
$p_{3/2}$ are observed(experimental) at $5.21(5-10)$ MeV and $1.96(1.96)$ MeV
and their respective decay widths are $9.54(5\pm 2)$ MeV and $1.90(1.5)$ MeV
[38, 39, 40].

Figure 5: Plots of partial cross section w.r.t $E_{Lab}$ energies of resonant
states $p_{1/2}$ and $p_{3/2}$. for $n\alpha$(top) and $p\alpha$(bottom)
systems. Figure 6: Total cross-section ($\sigma_{T}$) for $n\alpha$(left) and
$p\alpha$(right) systems. Continuous line is from our work and empty black
circles are from [41].

###  3.2 Study of nucleon-deuteron systems:

An extensive analysis on elastic nucleon-deuteron scattering has been done by
Huber $etal.$ [1] and J. Arvieux [2] separately for laboratory energies
ranging from $1-19$ $MeV$ for nd and $1-46.3$ $MeV$ for pd and these are
considered as standard data for these systems. The model parameters for Morse
potential obtained by using PFM for ${}^{2}S_{1/2}$ ground state of nd and pd
scattering have been tabulated in Table 1. The mean absolute percentage error
(MAPE) has also been given along with.
In Fig. 7, the obtained SPS are compared with experimental data [1] on the
left side and the corresponding inverse potential for the doublet S-wave
ground state of nd is shown. The potential is attractive in nature even though
the experimental phase shifts are negative. This is because, for nd system an
extra phase shift of $180^{o}$ is added [3] while determining the inverse
potential. Similarly, the SPS and inverse potential for pd system are shown
Fig. 8 (left) and Fig. 8 (right) respectively. Once again, the depth of pd
system is observed to be less than that of nd owing to the extra contribution
due to Coulomb potential.

Figure 7: SPS(left) and interaction potential(right) for $nd$ system.

Figure 8: SPS(left) and interaction potential(right) for $pd$ system.

##  4 Conclusion

The inverse potentials for nd, pd, $n\alpha$, $p\alpha$ systems have been
obtained using Morse function as reference. The model parameters are optimised
to reduce the mean absolute percentage error between phase shift obtained,
from solving phase equation using RK-5 method, and experimental data. The
computed phase shifts for nd and pd systems have been found to be closely
matching with those of Huber & Arvieux respectively. In case of $n\alpha$ and
$p\alpha$ systems, not only are the scattering phase shifts matching well with
those of Satchler, but the resonance frequencies for the first two scattering
$p$-states for both systems are in excellent match with the experimental data
as compared in Table 2.

Table 2: Resonance parameters for ${}^{5}He$ and ${}^{5}Li$ determined from
partial cross section plots.

State |  ${}^{5}He$ [42] |  ${}^{5}He$ (Our) |  ${}^{5}Li$ [42] |  ${}^{5}Li$ (Our)
---|---|---|---|---
$E_{r}(MeV)$ | $\Gamma_{c.m.}(MeV)$ | $E_{r}(MeV)$ | $\Gamma_{c.m.}(MeV)$ | $E_{r}(MeV)$ | $\Gamma_{c.m.}(MeV)$ | $E_{r}(MeV)$ | $\Gamma_{c.m.}(MeV)$
$p_{1/2}$ | $4\pm 1$ | $4\pm 1$ | 4.1 | 8.19 | 5-10 | $5\pm 2$ | 5.21 | 9.54
$p_{3/2}$ | 0.89 | $0.60\pm 0.02$ | 0.93 | 0.88 | 1.96 | $\approx 1.5$ | 1.96 | 1.90

## References

  * [1] D. Hüber, et al., Few-Body Systems 19, 4,175-193 ,(1995).
https://doi.org/10.1007/s006010050025

  * [2] J. Arvieux, Nucl. Phys. A 221, 2, 253-268, (1974).
https://doi.org/10.1016/0375-9474(74)90317-0

  * [3] C. R. Chen, et al., Phys. Rev. C, 39, 4, 1261, (1989).
https://doi.org/10.1103/PhysRevC.39.1261

  * [4] S. Ishikawa, Few-Body Systems, 32, 4, 229-48, (2003).
https://doi.org/10.1007/s00601-003-0001-7

  * [5] H. Kanada, S. Naagata, S. Otsuki, Y. Sumi, Prog. Theor. Phys., 30, 475-493, (1963).
https://doi.org/10.1143/PTP.30.475

  * [6] P.E. Hodgson, Adv. Phys., 25, 1-39, (1958).
https://doi.org/10.1080/00018735800101157

  * [7] C.L. Critchfield, D.C. Dodder, Phys. Rev., 76, 602, (1949).
https://doi.org/10.1103/PhysRev.76.602

  * [8] J.L. Gammel, R.M. Thaler, Phys. Rev., 109, 2041, (1958).
https://doi.org/10.1103/PhysRev.109.2041

  * [9] L. Brown, W. Haeberli, W. Tr¨achslin, Nucl. Phys. A, 90, 339-352, (1967).
https://doi.org/10.1016/0375-9474(67)90238-2

  * [10] A. Houdayer, et al., Phys. Rev. C, 18, 1985, (1978).
https://doi.org/10.1103/PhysRevC.18.1985

  * [11] Y.C. Tang, in Topic in Nuclear Physics II, Lecture Notes in Physics, 145, 571-692, (1981).
https://doi.org/10.1007/BFb0017231

  * [12] R. Kamouni, D. Baye, Nucl. Phys. A, 791, 68-83, (2007).
https://doi.org/10.1016/j.nuclphysa.2007.04.009

  * [13] T.A. Tombrello, G.C. Phillips, Nucl. Phys., 20, 648-662, (1960).
https://doi.org/10.1016/0029-5582(60)90204-2

  * [14] A.N. Mitra, V.S. Bhasin, B.S. Bhakar, Nuclear Physics (Netherlands) Divided into Nucl. Phys. A and Nucl. Phys. B, 38, 316-321, (1962).
https://doi.org/10.1016/0029-5582(62)91037-4

  * [15] E. van der Spuy, Nucl. Phys., 8, 381-414 (1956).
https://doi.org/10.1016/S0029-5582(56)80056-4

  * [16] J. Berger, et al., Phys. Rev. Lett., (1976); Erratum, 37, 1651, (1976).
https://doi.org/10.1103/PhysRevLett.37.1195

  * [17] L.G. Votta, P.G. Roos, N.S. Chant, R. Woody, Phys. Rev. C, 10, 520–528, (1974).
https://doi.org/10.1103/PhysRevC.10.520

  * [18] G.R. Satchler, L.W. Owen, A.J. Elwyn, G.L. Morgan, R.L. Walter, Nucl. Phys. A, 112, 1, (1968).
https://doi.org/10.1016/0375-9474(68)90216-9

  * [19] J. Dohet-Eraly, D. Baye, Phys. Rev. C, 84, 014604 (2011)
https://doi.org/10.1103/PhysRevC.84.014604

  * [20] G.L. Morgan, R.L. Walter, Phys. Rev., 168, 1114, (1968).
https://doi.org/10.1103/PhysRev.168.1114

  * [21] U. Laha, J. Bhoi, Phys. Rev. C, 91, 034614, (2015).
https://doi.org/10.1103/PhysRevC.91.034614

  * [22] J. Bhoi, U. Laha, Brazilian Journal of Physics, 46, 129-132, (2016).
https://doi.org/10.1007/s13538-015-0388-x

  * [23] A.K. Behera, J. Bhoi, U. Laha, B. Khirali, Commun. Theor. Phys., 72, 075301, (2020).
https://doi.org/10.1088/1572-9494/ab8a1a

  * [24] B. Buck, H. Friedrich, C. Wheatley, Nucl. Phys. A, 275, 246-268, (1977).
https://doi.org/10.1016/0375-9474(77)90287-1

  * [25] S.Y. Igashov, A.M. Shirokov, Bull. Russ. Acad. Sci. Phys., 71, 769–775 (2007).
https://doi.org/10.3103/S1062873807060044

  * [26] A. Sharma, S. Gora, J. Bhagavathi, O.S.K.S. Sastri, Am. J. Phys., 88, 1-5, (2020).
https://doi.org/10.1119/10.0001041

  * [27] A. Sharma, O.S.K.S. Sastri,Int. J. Quantum Chem., 121, e26682, (2021).
https://doi.org/10.1002/qua.26682

  * [28] A. Khachi, L. Kumar, A. Sharma, O.S.K.S. Sastri, J. Nucl. Phys. Mat. Sci. Rad. A., 9, 1-5, (2021).
https://doi.org/10.15415/jnp.2021.91001

  * [29] A. Khachi, L. Kumar, O.S.K.S. Sastri, Brazilian J. Phys., 52, 1-6, (2022).
https://doi.org/10.1007/s13538-022-01063-1

  * [30] A. Khachi, L. Kumar, O.S.K.S. Sastri, Phys. Atom. Nuclei 85, 382–391 (2022).
https://doi.org/10.1134/S106377882204007X

  * [31] L. Kumar, A. Khachi, O.S.K.S. Sastri, J. Nucl. Phys. Mat. Sci. Rad. A., 9, 215-221, (2022).
https://doi.org/10.15415/jnp.2022.92032

  * [32] M. Selg, Molecular Physics, 104, 2671-2684, (2006).
https://doi.org/10.1080/00268970600761754

  * [33] S. Awasthi, A. Khachi, L. Kumar, O.S.K.S. Sastri, J. Nucl. Phy. Mat. Sci. Rad. A., 9, 81-85., (2021).
https://doi.org/10.15415/jnp.2021.91014

  * [34] S. Awasthi, A. Khachi, O.S.K.S. Sastri, J. Nucl. Phy. Mat. Sci. Rad. A., 9, 223-228, (2022).
https://doi.org/10.15415/jnp.2022.92033

  * [35] V.I Kukulin et al., Nucl. Phys. A 586 (1995) 151-189.
https://doi.org/10.1016/0375-9474(94)00494-8

  * [36] F. Calogero, American Journal of Physics, (1968).
https://doi.org/10.1119/1.1975005

  * [37] V.V. Babikov, Sov. Phys. Usp., 10, 271, (1967).
https://doi.org/10.1070/PU1967v010n03ABEH003246

  * [38] A. Csoto, G.M. Hale, Phys. Rev. C, 55, 536, (1997).
https://doi.org/10.1103/PhysRevC.55.536

  * [39] J.E. Bond, F.W.K. Firk, Nucl. Phys. A, 287, 317-343, (1977).
https://doi.org/10.1016/0375-9474(77)90499-7

  * [40] A.I. Mazur, et al., Phys. Rev. C, 79, 014610, (2008).
https://doi.org/10.1103/PhysRevC.79.014610

  * [41] K. Shibata, J. Nucl. Sci. Technol., 27, 81-88, (1990).
https://doi.org/10.1080/18811248.1990.9731154

  * [42] F. Ajzenberg-Selove, Nucl. Phys. A 413(1),(1984) pp.1-168.
https://doi.org/10.1016/0375-9474(84)90650-X


aainstitutetext: Institute for Particle Physics Phenomenology, Department of
Physics, Durham University, Durham DH1 3LE, U.K.bbinstitutetext: Department of
Physics and Astronomy, University College London, Gower Street, London WC1E
6BT, UKccinstitutetext: High Energy Physics Group, Blackett Laboratory,
Imperial College, Prince Consort Road, London, SW7 2AZ, United Kingdom

# A quantum walk approach to simulating parton showers

Khadeejah Bepari b Sarah Malik a Michael Spannowsky c and Simon Williams
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

This paper presents a novel quantum walk approach to simulating parton showers
on a quantum computer. We demonstrate that the quantum walk paradigm offers a
natural and more efficient approach to simulating parton showers on quantum
devices, with the emission probabilities implemented as the coin flip for the
walker, and the particle emissions to either gluons or quark pairs
corresponding to the movement of the walker in two dimensions. A quantum
algorithm is proposed for a simplified, toy model of a 31-step, collinear
parton shower, hence significantly increasing the number of steps of the
parton shower that can be simulated compared to previous quantum algorithms.
Furthermore, it scales efficiently: the number of possible shower steps
increases exponentially with the number of qubits, and the circuit depth grows
linearly with the number of steps. Reframing the parton shower in the context
of a quantum walk therefore brings dramatic improvements, and is a step
towards extending the current quantum algorithms to simulate more realistic
parton showers.

††preprint: IPPP/21/32

##  1 Introduction

The emergence of quantum computers has brought a new paradigm to the field of
computation. The unique features of these devices has garnered attention from
various disciplines, including High Energy Physics (HEP), where the
computational challenges associated with taking, processing and analysing vast
amounts of data in collider experiments like the Large Hadron Collider (LHC)
requires innovative solutions. Quantum algorithms have been proposed to tackle
some of these challenges, including the simulation of collision events
bauer2019quantum ; PhysRevD.103.076020 ; li2021partonic  reconstruction of
charged particle tracks in the detectors das2020track ; T_ys_z_2020 ;
magano2021quantum , and event classification and analysis Blance:2020nhl ;
Blance:2021gcs ; Terashi_2021 ; Wu_2021 ; Araz:2021zwu ; Belis_2021 ;
armenakas2020application ; pires2021digital ; MottQuantum .

Collision events at the LHC typically involve hundreds of particles and can be
very complicated. Simulation of such events requires extensive modelling of
proton-proton interactions and the subsequent detector response to fully
uncover the underlying physics processes. Theoretical descriptions of these
collisions can be separated into several stages. Constituent partons in the
colliding protons can interact via large momentum transfer in the so-called
hard interaction. Due to the large interaction energies, such collisions have
the potential to probe new physics. Colour-charged particles produced as a
result of this hard interaction are likely to emit further partons, resulting
in a parton shower. The parton shower process evolves the system down in
energy from the hard interaction to the hadronisation scale,
$\mathcal{O}(\Lambda_{\textrm{QCD}})$. It is a perturbative process and can
involve many partons, thus being one of the most time consuming parts of the
generation of a collision event. Consequently, the development of quantum
algorithms for the calculation of the hard process PhysRevD.103.076020  and
the resultant parton shower bauer2019quantum ; PhysRevD.103.076020  is an area
of interest.

This paper presents a novel approach to simulating a many-particle, collinear
parton shower on a quantum device using a quantum walk (QW) framework. It is
structured as follows: Section 2 gives a brief introduction to the QW
framework, Section 3 contains the description of the proposed parton shower
algorithm, and Section 4 gives a summary and conclusions.

##  2 Quantum walks

The quantum random walk PhysRevA.48.1687 ; Aharonov2 ; Kempe ;
Rohde2012IncreasingTD  is the quantum analogue of the classical random walk
and defines the movement of a particle, the walker, which can occupy certain
$position$ states on a graph. Here we will consider only discrete-time random
walks, where a coin flip determines the movement of the walker at distinct
time steps. The state of the walker can therefore be defined by the position
of the walker, $x$, and the coin, $c$, as $|x,c\rangle$. The movement of the
walker through the graph is determined by two operations: the coin operation,
$C$, which determines the direction the walker will move, and the shift
operation, $S$, which propagates the walker to the next position.

As a simple example, we construct a random walk following the approach in
Kempe . Consider a walker moving along a one-dimensional line according to an
unbiased coin (i.e, the walker has an equal chance of moving left or right
after the coin flip), with the walker originally positioned at $x=0$, see
Figure 1. The position of the particle on the line forms a Hilbert space
$\mathcal{H}_{P}$ spanned by integer values on the line,
{$|i\rangle:i\in\mathbb{Z}$}. The position space is augmented by the coin-
space, $\mathcal{H}_{C}$, which spans two basis states,
{$|~{}\uparrow~{}\rangle,|~{}\downarrow~{}\rangle$}, the up and down spin-
states of a fermion***The choice of using the up and down spin-states of a
fermion is useful when implementing quantum walks on qubit-based quantum
devices, such as those available on the IBM Q network IBMQ .. Therefore, the
walker occupies a total space of

$\mathcal{H}=\mathcal{H}_{C}\otimes\mathcal{H}_{P}.$ (1)

In the classical case, the coin operation is carried out by evaluating a
classical coin. Based on the outcome of this coin, the shift operator moves
the walker in the correct direction. Here we will attribute the coin state
$|\uparrow\rangle$ to the walker moving in the positive $x$ direction and the
$|\downarrow\rangle$ state to the walker moving in the negative $x$ direction.
After the step process is complete, the walker is either in the $x=-1$ or
$x=1$ position. In contrast to the classical case, the quantum coin operation
is based on a quantum coin. In this example, we will consider the Hadamard
coin,

$H=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1&-1\end{pmatrix},$ (2)

which gives an equal chance for the coin to be measured in each of the coin
states. The quantum coin operation puts the system into a superposition of the
basis states of the $\mathcal{H}_{C}$ space. The shift operation is then
performed, moving the walker into a superposition of the position states,
$x=-1$ and $x=1$. A measurement after the step collapses the wavefunction to
recover the classical case of the walker being in either the $x=-1$ or $x=1$
position.

Figure 1: One-dimensional walker at position $x=0$ can move either left or
right depending on the outcome of the coin flip, $|\downarrow\rangle$ and
$|\uparrow\rangle$ respectively.

The Hadamard coin used here is a balanced unitary coin operation†††Strictly
speaking, the Hadamard coin introduces a bias to the quantum walk through the
phase on the coin qubit. This is discussed in detail in Kempe  and references
therein. Here we remove this bias by using a symmetric initial state. and
therefore the coin and shift operations can be defined as a single unitary
transformation to the initial qubit state,

$U=S\cdot(C\otimes I),$ (3)

which is applied iteratively to represent the number of steps. For a quantum
walk of $N$ steps, the propagation of the walker is described by the
transformation $U^{N}$ Kempe . An example of running a linear, one-
dimensional, $N=100$ step, random walk for both the classical case and the
quantum case is shown in Figure 2. The classical case has been achieved by
measuring the $coin$ qubit at each step, removing the superposition from the
system. As expected, the classical walk yields a Gaussian distribution of
positions centred about the initial position of the particle, with the
variance $\sigma^{2}=N$. In contrast, the quantum random walk, where the
quantum interference between the intermediate steps of the walk process,
results in a distribution that is dramatically different from the classical
case. It can be shown Kempe ; Ambainis  that the variance of the quantum
random walk process goes as $\sigma^{2}\sim N^{2}$. This is a remarkable
attribute of the quantum random walker, which propagates quadratically faster
than the classical walker. The average distance of the walker from the initial
position is $\sigma=\sqrt{N}$ and $\sigma\sim N$ for the classical and quantum
walks, respectively.

Figure 2: Simulation of a 100-step random walk using the IBM Q 32-qubit
simulator 32_sim  for 100,000 shots for a classical random walk obtained by
measuring the coin state after each step, and a quantum random walk using a
symmetric initial position and a Hadamard coin. Only non-zero probabilities
are shown, as odd-numbered positions will have zero probability for this walk.

##  3 Quantum walk as a parton shower simulation

The parton shower Buckley:2011ms  evolves the energy scale of a scattering
event from the hard interaction down to the hadronisation scale through the
radiation of additional partons. The emissions are determined by splitting
functions which correspond to the different emission probabilities in the
shower. The shower content is then updated depending on which splitting
probability is chosen. Due to this probabilistic interpretation of parton
showers, the quantum walk mechanism provides a natural framework for the
simulation of parton showers: the emission probabilities correspond to the
coin flip probabilities, and updating the shower content depending on the
emission corresponds to the shift operation in the quantum walk framework.

In this Section we detail this novel quantum walk approach to simulating a
parton shower on a quantum device. Within this framework, the algorithm can
simulate a many-particle parton shower, and shows a remarkable improvement on
the number of shower steps that can be simulated in comparison to previous
quantum algorithms bauer2019quantum ; PhysRevD.103.076020 . The Section is
ordered as follows: Section 3.1 gives the theoretical outline of the toy model
used in the parton shower, Section 3.2 shows the implementation of a simple
parton shower with one particle type, Section 3.3 outlines the full collinear
parton shower and Section 3.4 discusses possible extensions to the algorithm
with advancements in quantum computers to simulate a realistic parton shower.

###  3.1 Theoretical outline of shower algorithm

We present a discrete, collinear parton shower using the quantum walk
framework. As with the parton shower algorithms presented in
PhysRevD.103.076020 ; bauer2019quantum , this algorithm utilises the ability
of the quantum device to remain in a superposition state throughout the
calculation. Consequently, all shower histories are calculated simultaneously
and are encoded in the final wavefunction, with a measurement projecting out a
specific quantity of the final state, e.g. the number of partons. This offers
a unique advantage over classical parton shower algorithms, which need to
calculate each shower history explicitly and store the information on a
physical memory device. Only after summing over all possible shower histories
can a physically meaningful quantity be extracted. The goal of this algorithm
is to create the foundation for the development of a full, general parton
shower by studying a simplified toy model that meets the capabilities of
current quantum simulators.

An emission is collinear if a parton with momentum $P$ splits into two
massless particles, which have parallel 4-momenta, such that the momentum
distribution is,

$\displaystyle p_{i}=zP,$ $\displaystyle p_{j}=(1-z)P,$ (4)

thus, $(p_{i}+p_{j})^{2}=P^{2}=0$ Taylor_2017 . In this algorithm, we use a
similar theoretical set-up as PhysRevD.103.076020 . In each shower step,
emission is determined by first ascertaining whether an emission occurred in
the step using the Sudakov factors, and then applying the relevant splitting
functions. The Sudakov factors for a QCD process are given by,

$\displaystyle\Delta_{i,k}(z_{1},z_{2})=\exp\big{[}-\alpha^{2}_{s}\int^{z_{2}}_{z_{1}}P_{k}(z^{\prime})dz^{\prime}\big{]},$
(5)

and are used to calculate the probability of non-emission Sudakov:1954sw . The
probability that no particles split for an arbitrary step $N$ in the shower
process, where $N$ particles can be present, is given by the total Sudakov
factor,

$\Delta_{\textrm{tot}}(z_{1},z_{2})=\Delta_{g}^{n_{g}}(z_{1},z_{2})\Delta_{q}^{n_{q}}(z_{1},z_{2})\Delta_{\overline{q}}^{n_{\overline{q}}}(z_{1},z_{2})$
(6)

where $n_{g}$, $n_{q}$ and $n_{\overline{q}}$ are the number of gluons, quarks
and antiquarks present in the step‡‡‡As the algorithm allows for steps with no
emissions, for a step $N$: $(n_{g}+n_{q}+n_{\overline{q}})\leq N$. As in
PhysRevD.103.076020 , only collinear splittings will be considered. The
emission probabilities are therefore calculated using the collinear splitting
functions outlined in Dokshitzer:1977sg ; Gribov:1972ri ; ALTARELLI1977298 .
The emission of a gluon from a quark is defined at leading order (LO) by,

$\displaystyle P_{q\rightarrow qg}(z)=C_{F}\frac{1+(1-z)^{2}}{z},$ (7)

where $C_{F}=4/3$ is calculated using colour algebra, and the quark and gluon
have momentum fractions $1-z$ and $z$ respectively. The gluon can self-couple,
and therefore can split to both a pair of gluons and a quark-antiquark pair.
At LO, the splitting functions for these emissions are,

$\displaystyle P_{g\rightarrow
gg}(z)=C_{A}\Big{[}2\frac{1-z}{z}+z(1-z)\Big{]},$ $\displaystyle
P_{g\rightarrow q\overline{q}}(z)=n_{f}T_{R}(z^{2}+(1-z)^{2}),$ (8)

where $C_{A}=3$ and $T_{R}=1/2$ are calculated using colour algebra, and
$n_{f}$ is the number of massless quark flavours.

Combining the Sudakov factors with the splitting functions defines the full
probability of emission,

$\textrm{Prob}_{k\rightarrow ij}=(1-\Delta_{k})\times P_{k\rightarrow ij}(z).$
(9)

In the QW framework, this probability is applied as a unitary rotation to the
coin qubit, defining the shower algorithm’s coin operation.

The proposed algorithm does not include kinematics. This allows for the
calculation to be implemented on currently accessible simulators, such as the
32-qubit IBM Q Quantum Simulator 32_sim . As a result, the shower evolution
cannot be determined by the kinematics of the shower particles, but instead
the evolution variable $z$ is evolved to lower and lower momenta,
exponentially with the number of steps. Section 3.4 outlines how a more
realistic parton shower could be constructed on future devices.

###  3.2 Implementation of a simple shower as a one-dimensional quantum walk

The implementation of a parton shower as a quantum walk follows the framework
of a simple quantum random walk outlined in Section 2. Here we define the coin
operation as a unitary rotation on the coin qubit corresponding to the
probability of emission, calculated using the Sudakov factors and the
subsequent splitting functions defined in Section 3.1. This rotation takes the
form,

$U_{c}=\begin{pmatrix}\sqrt{1-P_{jk}}&\sqrt{P_{jk}}\\\
\sqrt{P_{jk}}&\sqrt{1-P_{jk}}\end{pmatrix},$ (10)

where $P_{jk}=(1-\Delta_{i})\times P_{i\rightarrow jk}$ is the probability of
particle $i$ splitting to $j$ and $k$. The coin space, $\mathcal{H}_{C}$,
therefore spans the space {$|0\rangle$, $|1\rangle$} defined by the possible
measured states of the coin qubit. Here we define the $|0\rangle$ state as the
“no emission” state, and the $|1\rangle$ state as the “$emission$” state. The
position space, $\mathcal{H}_{P}$, now defines the number of particles present
in the shower and has been altered to include only zero and positive integers,
{$|i\rangle:i\in\mathbb{N}_{0}$}, as the parton shower cannot have a negative
number of particles. The shift operation controls from the coin qubit and
moves the walker in the correct direction. In order to apply the correct
splitting probabilities to the coin qubit, the number of particles present
must be determined. An efficient scheme has been created using a series of
cnot gates to determine the position of the walker.

To illustrate this simple shower, Figure 3 shows a schematic of a one-
dimensional quantum walk able to simulate a particle which can split to
produce another particle of the same type. In this simple shower, the number
of particles present is encoded in the position of the walker. Figure 3 uses a
two qubit basis for the position of the walker, ultimately allowing the
algorithm to simulate a maximum of 4 shower particles in the final state. The
number of particles that the algorithm can simulate increases exponentially
with the number of position qubits, $x$, as $2^{x}$. For this example, only
one splitting is possible, $i\rightarrow ii$, and as a result only one coin
qubit is needed to encode the splitting probability. If after the coin
operation, the coin qubit is in the $|1\rangle$ state, then the splitting has
occurred and the position of the walker is increased by one, thus increasing
the number of particles present in the shower by one. However, if the coin
operation yields a $|0\rangle$ state, then the walker does not move for this
simple example§§§Note that in Figure 3 the shift operation also shows the
ability to decrease the walker’s position. This is not needed for the simple
example of $i\rightarrow ii$ splittings, but will be useful later.. This step
can then be repeated for the number of discrete shower steps in the parton
shower, resembling the quantum random walk outlined in Section 2. Throughout
the calculation, the device is in a superposition state of all possible
outcomes of the coin and shift operations. At the end of the shower process,
the final state of the system is measured and projected onto a classical
state.

Figure 3: Template for a single step of a quantum walk as a parton shower,
with the ability to simulate a particle which can split to more particles of
the same type. Here, the “position check” determines the number of particles
present by assessing the position of the walker. The “coin” operation applies
the correct splitting probabilities depending on the position of the walker.
The “shift” operation moves the walker depending on the outcome of the coin
operation.

###  3.3 Implementation of a collinear parton shower

It is possible to extend the simple shower outlined in Section 3.2 to include
multiple parton types by increasing the dimension of the position space
$\mathcal{H}_{P}$, with the aim of developing a multi-particle, discrete,
collinear parton shower using the theoretical outline discussed in Section
3.1. The algorithm presented here considers a toy model comprised of a gluon
and one flavour of quark, and can simulate the corresponding splittings.

As shown in Section 3.2, a quantum walker in a one-dimensional position space,
$\mathcal{H}_{P}$, has the ability to simulate a single particle type.
Augmented by the coin space, $\mathcal{H}_{C}$, with dimension equal to the
number of possible splittings associated with the particle, the quantum walk
can simulate a simple parton shower comprising one particle type. Increasing
the dimension of the position space increases the number of particles which
can be simulated in the algorithm. Applying this mechanism to our toy model of
the parton shower, the position space, $\mathcal{H}_{P}$, is increased to two
dimensions to accommodate the simulation of gluons and quarks, counting gluons
in one dimension and quarks in the other. Note that we do not need to include
dimensions for both quarks and antiquarks as they are produced in conjunction
through the $g\rightarrow q\overline{q}$ splitting, thus instead we count
quark-antiquark pairs. Figure 4 shows a visualisation of how the walker’s
position on a 2D plot corresponds to the number of particles in the shower,
with gluons and quarks measured on the $x$ and $y$-axes of the walker’s 2D
lattice respectively. The coin space, $\mathcal{H}_{C}$, is increased to a
three-dimensional Hilbert space, with three coin qubit rotations corresponding
to the splitting functions in Equations 7 and 8. Controlled from the coin
register, the shift operations propagate the walker to reflect the production
of new particles in the shower step. A schematic of the quantum circuit is
shown in Figure 5. It should be noted that it is likely that more than one of
the coin qubits can be in the $|1\rangle$ state in a step. In these
situations, it is not clear which splitting kernel should be applied and
therefore the algorithm does not apply a shift operation to the walker. This
is realised by controlling from coin states that only have one coin qubit in
the $|1\rangle$ state, as shown in Figure 5.

(a)

(b)

Figure 4: Visualisation of a quantum walk as a parton shower comprising gluons
and quarks. The quantum walker’s position on a 2D plot corresponds to the
number of particles in the parton shower: (a) shows a parton shower using the
collinear splitting functions for quarks and gluons, (b) shows a parton shower
with modified splitting functions to show how the walker moves in the 2D
lattice. Figure 5: Schematic of the quantum circuit for a single step of a
discrete QCD, collinear parton shower with the ability to simulate the
splittings of gluons and one flavour of quark. The shower algorithm is split
into 3 distinct sections: (1) The position check determines the position of
the walker so that the correct Sudakov form factors are applied in the
splitting kernels, (2) the coin flip applies unitary rotations to a coin
register corresponding to the possible splitting kernels, (3) the shift
operation propagates the walker into correct direction to describe the
particle splitting in the shower step. This step is then repeated iteratively
to simulate a full shower process.

To simulate a parton shower, the steps shown in Figure 5 are performed many
times, with only one splitting allowed to occur per step. Steps where no
emission occurred are dictated by the Sudakov form factors from Equation 6.
The system is kept in a superposition state throughout the algorithm, with a
measurement taking place only at the end of the calculation. Therefore, after
all the steps have been evaluated, the system is in a superposition of all
possible shower histories. This differs dramatically from classical parton
shower algorithms where each shower history must be individually calculated. A
physically meaningful quantity can only be extracted from a classical shower
algorithm once all possible shower histories have been summed over.
Consequently, the quantum algorithm approach to parton showers provides a
unique advantage over the classical approach.

The quantum parton shower algorithm with 31 shower steps has been run for
500,000 shots on the IBM Q 32-qubit Quantum Simulator 32_sim . The output from
the quantum simulator has been compared to a classical parton shower
algorithm, which follows the same theoretical framework as that outlined in
Section 3.1, simulating a toy model with one quark flavour and a gluon. Figure
6 shows the comparison between the quantum and classical parton shower
algorithms of the probability distributions of the number of gluons measured
at the end of the shower. This is shown for the scenario where there are zero
quark-antiquark pairs in the final state and the much less probable scenario
where there is one quark-antiquark pair in the final state. Due to the low
statistics for the $1q\overline{q}$ results, a further validation of the
parton shower algorithm has been carried out using modified splitting
functions to enhance the $g\rightarrow q\overline{q}$ and $q\rightarrow qg$
splittings. The results of this test are shown in Figure 7 and display good
agreement between the quantum and classical parton shower algorithms. The
probability of producing two quark-antiquark pairs is less than $10^{-5}$. For
both comparison runs, the classical algorithm has been executed for 31 shower
steps with $10^{6}$ shots of the algorithm.

The algorithm is implemented using 16 qubits to simulate a parton shower of 31
steps. This is a dramatic increase in the number of steps that can be
simulated on a quantum device in comparison to previous algorithms
bauer2019quantum ; PhysRevD.103.076020 , with almost a factor of two reduction
in the number of required qubits PhysRevD.103.076020 .

Figure 6: Probability distribution of the number of gluons measured at the end
of the 31-step parton shower for the classical and quantum algorithms, for the
scenario where there are zero quark anti-quark pairs (left) and exactly one
quark anti-quark pair (right) in the final state. The quantum algorithm has
been run on the IBMQ 32-qubit quantum simulator 32_sim  for 500,000 shots, and
the classical algorithm has been run for $10^{6}$ shots. Figure 7: Probability
distribution of the number of gluons measured at the end of the 31-step parton
shower for the classical and quantum algorithms with modified splitting
kernels, for the scenario where there are zero quark anti-quark pairs (left)
and exactly one quark anti-quark pair (right) in the final state. The quantum
algorithm has been run on the IBMQ 32-qubit quantum simulator 32_sim  for
100,000 shots, and the classical algorithm has been run for $10^{6}$ shots.

###  3.4 Towards a realistic parton shower

The parton shower algorithm described in Section 3.3 is a simplified, toy
model and has thus limited capability compared to state-of-the-art, classical
parton shower algorithms. However, the quantum algorithm leverages the unique
ability of the quantum computer to remain in a superposition state throughout
the calculation, enabling all shower histories to be calculated simultaneously
and providing a remarkable advantage over the classical algorithms. It is
interesting to consider how the parton shower algorithm will develop with
advancements in quantum technologies. Near-future devices with larger quantum
volume gambetta_2020 ; Jurcevic2020DemonstrationOQ  make it feasible to
imagine a practical parton shower algorithm on a quantum device.

An obvious extension to the algorithm proposed would be to include more
particle types and flavours. As described in Section 3.3, this is easily done
by increasing the dimension of the $\mathcal{H}_{P}$ and $\mathcal{H}_{C}$
spaces to include another particle and its corresponding splittings. It may
then be possible to extend the shower to include all quark flavours,
increasing the dimension of the walker’s lattice to seven: six quark
dimensions and one gluon dimension. To implement this circuit would require a
large number of qubits, with the number required for each particle type being

$n_{\textrm{qubits}}=\log_{2}N,$ (11)

where $N$ is the number of desired steps in the shower process. It is possible
to reduce the overall number of qubits in the system by removing redundant
areas in the quantum walker’s lattice. For example, in Figures 6 and 7, there
is only one quark-antiquark pair in the results. Therefore, all lattice sites
containing two or more quark-antiquark pairs could be removed to streamline
the circuit. However, this does reduce the generality of the circuit, and such
areas would have to be known a priori to running the device.

It is feasible to consider an algorithm that can simulate a parton shower for
calculations where a basis transformation is performed. For example, in
bauer2019quantum  a parton shower algorithm was proposed with two fermions
$f_{1}$ and $f_{2}$ and a scalar $\phi$. The algorithm considers a rotation
from the flavour basis, $f_{1/2}$, to a mass basis, $f_{a/b}$. The parton
shower calculation is then carried out in the mass basis, rotating back to the
flavour basis before measurement. It claims an advantage over classical
algorithms by simulating the quantum interference between the two fermions.
Due to the qubit requirement of the circuit, the full algorithm was restricted
to 2 steps in the shower process. It is possible to replicate the parton
shower algorithm from bauer2019quantum  in a quantum walk framework by
increasing the dimension of the position space $\mathcal{H}_{P}$ and the coin
space $\mathcal{H}_{C}$ to include the two fermions and one scalar, and their
corresponding splitting functions. The basis transformation can be reproduced
as a rotation across the fermion dimensions of the position space
$\mathcal{H}_{C}$. The shower would then follow the quantum walk process
outlined in Equation 3, with a final rotation across the fermion position
space to transform back to the flavour basis before measurement. This
algorithm would be able to run for many steps and would be a good test of the
quantum advantage claimed in bauer2019quantum .

Keeping track of particle kinematics in the parton shower algorithm outlined
in Section 3.3 is an important step towards emulating a realistic parton
shower. The current publicly accessible devices and simulators do not have
adequate quantum volume to include shower kinematics, but future devices may
have the capability to implement this. Within the quantum walk framework, it
is possible to consider extending the Hilbert space of the system to include a
kinematic space $\mathcal{H}_{K}$ such that the total space now has the form,

$\mathcal{H}=\mathcal{H}_{C}\otimes\mathcal{H}_{P}\otimes\mathcal{H}_{K}.$
(12)

The kinematic space $\mathcal{H}_{K}$ would comprise a discretised momentum
space that each shower particle could move in. Similarly to the position check
in Figure 5, conditional coin operations would then be used to apply the
correct splitting kernels to the coin qubits depending on the position of the
walker in the kinematic space $\mathcal{H}_{K}$. A schematic of a one particle
type parton shower, with kinematics included, is shown in Figure 8. It should
be noted that, in order to keep track of each particle’s momentum in the
shower, the kinematic space $\mathcal{H}_{K}$ will have to be extended at each
splitting. One can initialise the system to have the whole kinematic space at
the beginning of the algorithm, populating the space only in the event of a
splitting. However, this approach will lead to a large redundancy in the
circuit, an area which may have to be optimised in practice.

Figure 8: A schematic circuit diagram for a one particle type parton shower
with a discretised kinematic space. Here, $\mathcal{H}_{P}$, $\mathcal{H}_{K}$
and $\mathcal{H}_{C}$ are the position, kinematic and coin spaces
respectively, and $w$ is an ancillary register.

##  4 Summary

Simulating parton showers on quantum computers has been shown bauer2019quantum
; PhysRevD.103.076020  to have distinct advantages that exploit the unique
features of the quantum device. In classical parton showers, all possible
shower histories are calculated individually, stored on a physical memory
device and then analysed in their entirety to provide information on a
physical quantity. In contrast, the quantum device remains in a quantum state
throughout the calculation, constructing a wavefunction which comprises a
superposition of all possible shower histories. Consequently, all shower
histories are calculated simultaneously in a single calculation, removing the
requirement to store and track each shower history on physical memory.
However, simply porting over the classical parton shower implementations onto
a quantum device is computationally inefficient, requiring a large number of
qubits and only allowing up to 2 steps of the parton shower to be simulated on
current simulators PhysRevD.103.076020 .

This paper proposes a novel quantum walk approach to simulating parton showers
on a quantum computer that represents a significant improvement in the depths
of the shower that can be simulated and with far fewer qubits. We present a
quantum algorithm for the simulation of a collinear, 31-step parton shower
implemented as a 2D quantum walk, where the coin flip represents the total
parton emission probability, and the movement of the walker in the 2D space
represents an emission corresponding to either gluons or a quark-antiquark
pair. Reframing the parton shower in this quantum walk paradigm enables a
31-step shower to be simulated, a dramatic improvement over previous quantum
algorithms bauer2019quantum ; PhysRevD.103.076020  and with almost a factor of
two reduction in the number of required qubits PhysRevD.103.076020 . The
quantum walk approach thus offers a natural and much more efficient approach
to simulating parton showers on quantum devices. Furthermore, the algorithm
scales efficiently: the number of possible shower steps increase exponentially
with the number of qubits in the position registers; and the circuit depth
grows linearly with the number of steps, in contrast to previous algorithms
which grow quadratically, at best.

Acknowledgements: Sarah Malik and Simon Williams are funded by grants from the
Royal Society. We would like to acknowledge the use of the IBM Q for this
work. We thank Frank Krauss and Stefan Prestel for valuable discussions.

## References

  * (1) C. W. Bauer, W. A. de Jong, B. Nachman, and D. ide Provasoli, A quantum algorithm for high energy physics simulations, 2019.
  * (2) K. Bepari, S. Malik, M. Spannowsky, and S. Williams, Towards a quantum computing algorithm for helicity amplitudes and parton showers, Phys. Rev. D 103 (Apr, 2021) 076020.
  * (3) T. Li, X. Guo, W. K. Lai, X. Liu, E. Wang, H. Xing, D.-B. Zhang, and S.-L. Zhu, Partonic structure by quantum computing, 2021.
  * (4) S. Das, A. J. Wildridge, S. B. Vaidya, and A. Jung, Track clustering with a quantum annealer for primary vertex reconstruction at hadron colliders, 2020\.
  * (5) C. Tüysüz, F. Carminati, B. Demirköz, D. Dobos, F. Fracas, K. Novotny, K. Potamianos, S. Vallecorsa, and J.-R. Vlimant, Particle track reconstruction with quantum algorithms, EPJ Web of Conferences 245 (2020) 09013.
  * (6) D. Magano, A. Kumar, M. Kālis, A. Locāns, A. Glos, S. Pratapsi, G. Quinta, M. Dimitrijevs, A. Rivošs, P. Bargassa, J. Seixas, A. Ambainis, and Y. Omar, Quantum speedup for track reconstruction in particle accelerators, 2021\.
  * (7) A. Blance and M. Spannowsky, Quantum Machine Learning for Particle Physics using a Variational Quantum Classifier, [arXiv:2010.07335].
  * (8) A. Blance and M. Spannowsky, Unsupervised Event Classification with Graphs on Classical and Photonic Quantum Computers, [arXiv:2103.03897].
  * (9) K. Terashi, M. Kaneda, T. Kishimoto, M. Saito, R. Sawada, and J. Tanaka, Event classification with quantum machine learning in high-energy physics, Computing and Software for Big Science 5 (Jan, 2021).
  * (10) S. L. Wu, J. Chan, W. Guan, S. Sun, A. Wang, C. Zhou, M. Livny, F. Carminati, A. Di Meglio, A. C. Y. Li, and et al., Application of quantum machine learning using the quantum variational classifier method to high energy physics analysis at the lhc on ibm quantum computer simulator and hardware with 10 qubits, Journal of Physics G: Nuclear and Particle Physics (Jul, 2021).
  * (11) J. Y. Araz and M. Spannowsky, Quantum-inspired event reconstruction with Tensor Networks: Matrix Product States, JHEP 08 (2021) 112, [arXiv:2106.08334].
  * (12) V. Belis, S. González-Castillo, C. Reissel, S. Vallecorsa, E. F. Combarro, G. Dissertori, and F. Reiter, Higgs analysis with quantum classifiers, EPJ Web of Conferences 251 (2021) 03070.
  * (13) A. E. Armenakas and O. K. Baker, Application of a quantum search algorithm to high- energy physics data at the large hadron collider, 2020.
  * (14) D. Pires, P. Bargassa, J. Seixas, and Y. Omar, A digital quantum algorithm for jet clustering in high-energy physics, 2021.
  * (15) A. Mott, J. Job, J.-R. Vlimant, . D. Lidar, and M. Spiropulu, Solving a higgs optimization problem with quantum annealing for machine learning, Nature 550 (10, 2017) 375–379.
  * (16) Y. Aharonov, L. Davidovich, and N. Zagury, Quantum random walks, Phys. Rev. A 48 (Aug, 1993) 1687–1690.
  * (17) D. Aharonov, A. Ambainis, J. Kempe, and U. Vazirani, Quantum walks on graphs, Conference Proceedings of the Annual ACM Symposium on Theory of Computing (01, 2001).
  * (18) J. Kempe, Quantum random walks: An introductory overview, Contemporary Physics 44 (2003), no. 4 307–327, [https://doi.org/10.1080/00107151031000110776].
  * (19) P. Rohde, A. Schreiber, M. Štefaňák, I. Jex, A. Gilchrist, and C. Silberhorn, Increasing the dimensionality of quantum walks using multiple walkers, Journal of Computational and Theoretical Nanoscience 10 (2012) 1644–1652.
  * (20) IBM Research, Qiskit, an open source computing framework, .
  * (21) A. Ambainis, E. Bach, A. Nayak, A. Vishwanath, and J. Watrous, One-dimensional quantum walks, (New York, NY, USA), Association for Computing Machinery, 2001.
  * (22) IBM Q Team, IBM Q 32 Simulator v0.1.547, .
  * (23) A. Buckley et. al., General-purpose event generators for LHC physics, Phys. Rept. 504 (2011) 145–233, [arXiv:1101.2599].
  * (24) T. R. Taylor, A course in amplitudes, Physics Reports 691 (May, 2017) 1–37.
  * (25) V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics, Sov. Phys. JETP 3 (1956) 65–71.
  * (26) Y. L. Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics., Sov. Phys. JETP 46 (1977) 641–653.
  * (27) V. Gribov and L. Lipatov, Deep inelastic e p scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 438–450.
  * (28) G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nuclear Physics B 126 (1977), no. 2 298 – 318.
  * (29) J. Gambetta, Ibm’s roadmap for scaling quantum technology, Sep, 2020.
  * (30) P. Jurcevic, A. Javadi-Abhari, L. Bishop, I. Lauer, D. F. Bogorin, M. Brink, L. Capelluto, O. Gunluk, T. Itoko, N. Kanazawa, A. Kandala, G. Keefe, K. D. Krsulich, W. Landers, E. P. Lewandowski, D. McClure, G. Nannicini, A. Narasgond, H. Nayfeh, E. Pritchett, M. B. Rothwell, S. Srinivasan, N. Sundaresan, C. Wang, K. X. Wei, C. Wood, J.-B. Yau, E. Zhang, O. Dial, J. Chow, and J. Gambetta, Demonstration of quantum volume 64 on a superconducting quantum computing system, 2020.