# Quadratic Regularization of Data-Enabled Predictive Control: Theory and
Application to Power Converter Experiments

Linbin Huang  Jianzhe Zhen  John Lygeros  Florian Dörfler  Automatic Control
Laboratory, ETH, Zürich 8092, Switzerland (e-mail: {linhuang, jizhen,
jlygeros, dorfler}@ethz.ch).

###### Abstract

Data-driven control that circumvents the process of system identification by
providing optimal control inputs directly from system data has attracted
renewed attention in recent years. In this paper, we focus on understanding
the effects of the regularization on the data-enabled predictive control
(DeePC) algorithm. We provide theoretical motivation and interpretation for
including a quadratic regularization term. Our analysis shows that the
quadratic regularization term leads to robust and optimal solutions with
regards to disturbances affecting the data. Moreover, when the input/output
constraints are inactive, the quadratic regularization leads to a closed-form
solution of the DeePC algorithm and thus enables fast calculations. On this
basis, we propose a framework for data-driven synchronization and power
regulations of power converters, which is tested by high-fidelity simulations
and experiments.

###### keywords:

Data-driven control, predictive control, robust optimization, regularization,
power converters.

##  1 Introduction

Data-driven control is a promising technique for complex systems, as it does
not rely on an accurate model of the system, which is in many cases difficult
to obtain and maintain. For example, in power system applications, the system
model is ever-changing due to different operation modes, uncertainties, and
relaying. This poses great challenges for conventional model-based design and
tuning, but may potentially be handled using data-driven control due to the
great availability of data (Huang et al., 2019b).

The concept of data-driven control dates back to 1980s when the techniques of
system identification and adaptive control became popular (Ljung, 1999;
Goodwin and Sin, 2014), and received renewed attention with the developments
of iterative feedback tuning (Hjalmarsson et al., 1998), virtual reference
feedback tuning (Campi et al., 2002), etc. Data-driven control based on
reinforcement learning became a popular approach as well, which learns a
policy directly from data (Recht, 2019). It is also possible to provide
robustness guarantees for data-driven control, e.g., by means of system-level
synthesis (Boczar et al., 2018). In recent years, multiple data-driven control
methods (e.g., Coulson et al. (2019a); De Persis and Tesi (2019); Berberich et
al. (2020)) have been proposed based on a result originally formulated by
Willems et al. (2005) and recently extended by Markovsky and Dörfler (2020) in
the context of behavioral system theory. This result is also known as the
Fundamental Lemma and shows that the subspace of input/output trajectories of
a linear time-invariant (LTI) system can be obtained from the column span of a
data Hankel matrix, thereby avoiding a parametric system representation.

Here we concentrate on a data-enabled predictive control (DeePC) algorithm
proposed in (Coulson et al., 2019a). Rather than a parametric model, in the
spirit of the Fundamental Lemma, the DeePC algorithm relies only on
input/output data measured from the unknown system to predict future
trajectories and compute safe and optimal control inputs for the system. The
DeePC algorithm has been successfully applied in many scenarios, including
power systems (Huang et al., 2019a, b), motor drives (Carlet et al., 2020),
and quadcopters (Elokda et al., 2019). It has been frequently observed that
regularization terms are very important to ensure good performance when the
system is subjected to disturbances. Coulson et al. (2019b) showed that
regularization provides distributional robustness against stochastic
disturbances. Xue and Matni (2020) arrive at similar conclusions via a robust
control formulation. Though, the robust performance still needs to be
investigated thoroughly under bounded disturbances.

In this paper, we provide theoretical motivation and interpretation for adding
a quadratic regularization term in DeePC from a robust optimization
perspective. We show how including a quadratic regularization is equivalent to
a min-max optimization problem that minimizes the worst-case performance for
bounded disturbance sets imposed on the input/output data. This suggests that,
unlike system identification methods that filter out the disturbances to fit a
certain model, the regularized DeePC algorithm directly provides robust
control inputs to the system with regards to the disturbances in data.
Moreover, the quadratic regularization allows one to obtain a closed-form
solution of the DeePC algorithm enabling the fast calculation of optimal
inputs. We exploit these properties of DeePC with quadratic regularization to
robustly control a grid-connected power converter to achieve data-driven
synchronization and power regulation. Our results are illustrated with high-
fidelity simulations and experiments.

##  2 Data-Enabled Predictive Control

###  2.1 Notation and Preliminaries on the Fundamental Lemma

Consider an $n{\rm th}$-order discrete-time LTI system

$\left\\{\begin{array}[]{l}{x_{t+1}}=A{x_{t}}+B{u_{t}}\\\
{y_{t}}=C{x_{t}}+D{u_{t}},\end{array}\right.\,$ (1)

where $A\in\mathbb{R}^{n\times n}$, $B\in\mathbb{R}^{n\times m}$,
$C\in\mathbb{R}^{p\times n}$, $D\in\mathbb{R}^{p\times m}$,
$x_{t}\in\mathbb{R}^{n}$ is the state of the system at $t\in\mathbb{Z}_{\geq
0}$, $u_{t}\in\mathbb{R}^{m}$ is the input vector, and
$y_{t}\in\mathbb{R}^{p}$ is the output vector.

The lag of the system (1) is defined by the smallest integer
$\ell\in\mathbb{Z}_{\geq 0}$ such that the observability matrix

$\mathscr{O}_{\ell}(A,C):={\rm{col}}(C,CA,...,CA^{\ell-1})\vspace{-0.5mm}$

has rank $n$, i.e., the state can be reconstructed from $\ell$ measurements.
Here
${\rm{col}}(a_{0},a_{1},...,a_{i}):=[a_{0}^{\top}\;a_{1}^{\top}\;\cdots\;a_{i}^{\top}]^{\top}$.
In a data-driven setting, $\ell$ and $n$ are generally not known, but upper
bounds can usually be inferred.

Consider $L,T\in\mathbb{Z}_{\geq 0}$ with $T\geq L>\ell$ and length-$T$ input
and output trajectories of (1): $u={\rm{col}}(u_{0},u_{1},\dots
u_{T-1})\in\mathbb{R}^{mT}$ and $y={\rm{col}}(y_{0},y_{1},\dots
y_{T-1})\in\mathbb{R}^{pT}$. We denote the set of all such trajectories as the
restricted behavior. For the inputs $u$, define the Hankel matrix of depth $L$
as

$\mathscr{H}_{L}(u):=\left[{\begin{array}[]{*{20}{c}}{{u_{0}}}&{{u_{1}}}&\cdots&{{u_{T-L}}}\\\
{{u_{1}}}&{{u_{2}}}&\cdots&{{u_{T-L+1}}}\\\ \vdots&\vdots&\ddots&\vdots\\\
{{u_{L-1}}}&{{u_{L}}}&\cdots&{{u_{T-1}}}\end{array}}\right]\,.$ (2)

Accordingly, for the outputs define the Hankel matrix $\mathscr{H}_{L}(y)$.
Consider the stacked matrix
$\mathscr{H}_{L}(u,y)=\left[\begin{smallmatrix}\mathscr{H}_{L}(u)\\\
\mathscr{H}_{L}(y)\end{smallmatrix}\right]$.

By (Markovsky and Dörfler, 2020, Corollary 19), the length-$L$ restricted
behavior equals the image of $\mathscr{H}_{L}(u,y)$ if and only if
rank$\left(\mathscr{H}_{L}(u,y)\right)=mL+n$. In words, the Hankel matrix
$\mathscr{H}_{L}(u,y)$ composed of a single length-$T$ trajectory parametrizes
all length-$L$ trajectories provided that (and only that)
rank$\left(\mathscr{H}_{L}(u,y)\right)=mL+n$. This result extends and includes
the original Fundamental Lemma (Willems et al., 2005, Theorem 1) which
requires controllability and persistency of excitation of order $L+n$ (i.e.,
$\mathscr{H}_{L+n}(u)$ must have full row rank) as sufficient conditions. The
result by Markovsky and Dörfler (2020) also extends to mosaic Hankel, Page,
and trajectory matrix structures.

These behavioral results can be leveraged for data-driven prediction and
estimation as follows. Consider $T_{\rm ini},N,T\in\mathbb{Z}_{\geq 0}$ as
well as an input/output time series
${\rm{col}}(u^{\rm{d}},y^{\rm{d}})\in\mathbb{R}^{(m+p)T}$ so that
rank$\left(\mathscr{H}_{T_{\rm ini}+N}(u^{\rm{d}},y^{\rm{d}})\right)=m(T_{\rm
ini}+N)+n$. Here the superscript “d” denotes data collected offline in a
single shot, and the rank condition can be met by choosing $u^{\rm{d}}$ to be
persistently exciting of sufficiently high order (assuming that the system is
controllable). Further partition the Hankel matrix $\mathscr{H}_{T_{\rm
ini}+N}(u^{\rm{d}},y^{\rm{d}})$ as

$\left[{\begin{array}[]{*{20}{c}}{{U_{\rm P}}}\\\ {{U_{\rm
F}}}\end{array}}\right]:=\mathscr{H}_{T_{\rm
ini}+N}(u^{\rm{d}})\,,\;\;\left[{\begin{array}[]{*{20}{c}}{{Y_{\rm P}}}\\\
{{Y_{\rm F}}}\end{array}}\right]:=\mathscr{H}_{T_{\rm ini}+N}(y^{\rm{d}})\,,$
(3)

where $U_{\rm P}\in\mathbb{R}^{mT_{\rm ini}\times H}$, $U_{\rm
F}\in\mathbb{R}^{mN\times H}$, $Y_{\rm P}\in\mathbb{R}^{pT_{\rm ini}\times
H}$, $Y_{\rm F}\in\mathbb{R}^{pN\times H}$, and $H=T-T_{\rm ini}-N+1$. In the
sequel, the data in the partition with subscript P (for “past”) will be used
to estimate the initial condition of the system, whereas the data with
subscript F will be used to predict the “future” trajectories. In this case,
$T_{\rm ini}$ is the length of an initial trajectory measured in the immediate
past during on-line operation, and $N$ is the length of a predicted trajectory
starting from the initial trajectory. Recall that the image of
$\mathscr{H}_{T_{\rm ini}+N}(u^{\rm{d}},y^{\rm{d}})$ spans all length-$(T_{\rm
ini}+N)$ trajectories, that is, ${\rm{col}}(u_{\rm ini},u,y_{\rm
ini},y)\in\mathbb{R}^{(m+p)(T_{\rm ini}+N)}$ is a trajectory of (1) if and
only if there exists $g\in\mathbb{R}^{H}$ so that

$\left[{\begin{array}[]{*{20}{c}}{{U_{\rm P}}}\\\ {{Y_{\rm P}}}\\\ {{U_{\rm
F}}}\\\ {{Y_{\rm
F}}}\end{array}}\right]g=\left[{\begin{array}[]{*{20}{c}}{{u_{\rm ini}}}\\\
{{y_{\rm ini}}}\\\ u\\\ y\end{array}}\right]\,.$ (4)

The initial trajectory ${\rm{col}}(u_{\rm ini},y_{\rm
ini})\in\mathbb{R}^{(m+p)T_{\rm ini}}$ can be thought of as setting the
initial condition for the future (to be predicted) trajectory
${\rm{col}}(u,y)\in\mathbb{R}^{(m+p)N}$. In particular, if $T_{\rm
ini}\geq\ell$, for every given future input trajectory $u$, the future output
trajectory $y$ is uniquely determined through (4) (Markovsky and Rapisarda,
2008).

###  2.2 Review of the DeePC algorithm

The DeePC algorithm proposed in Coulson et al. (2019a) directly uses
input/output data collected from the unknown system to predict the future
behaviour, and perform optimal and safe control without identifying a
parametric system representation. More specifically, DeePC solves the
following optimization problem to obtain the optimal future control inputs

$\begin{array}[]{cl}\mathop{{\rm{min}}}\limits_{g,\sigma_{y}\atop
u\in\mathcal{U},y\in\mathcal{Y}}&\;\;{\left\|u\right\|_{R}^{2}}+{\left\|{y-r}\right\|_{Q}^{2}}+{\lambda_{y}}{\left\|\sigma_{y}\right\|_{2}^{2}}\\\
{\rm s.t.}&\;\;\left[{\begin{array}[]{*{20}{c}}{{U_{\rm P}}}\\\ {{Y_{\rm
P}}}\\\ {{U_{\rm F}}}\\\ {{Y_{\rm
F}}}\end{array}}\right]g=\left[{\begin{array}[]{*{20}{c}}{{u_{\rm ini}}}\\\
{{y_{\rm ini}}}\\\ u\\\
y\end{array}}\right]+\left[{\begin{array}[]{*{20}{c}}0\\\ \sigma_{y}\\\ 0\\\
0\end{array}}\right]\,,\end{array}$ (5)

where the sets $\mathcal{U}\subseteq\mathbb{R}^{mN}$ and
$\mathcal{Y}\subseteq\mathbb{R}^{pN}$ describe the feasible region of the
input $u$ and output $y$ of the system (assumed to be polytopes),
$R\in\mathbb{R}^{mN\times mN}$ is the control cost matrix (positive definite),
$Q\in\mathbb{R}^{pN\times pN}$ is the output cost matrix (positive
semidefinite), $\sigma_{y}\in\mathbb{R}^{pT_{\rm ini}}$ is an auxiliary slack
variable to ensure feasibility of the least-square initial condition
estimation, $\lambda_{y}\in\mathbb{R}_{>0}$ is regularization parameter,
$r\in\mathbb{R}^{pN}$ is the reference trajectory for the outputs, $N$ is the
prediction horizon, ${\rm{col}}(u_{\rm ini},y_{\rm ini})$ consists of the most
recent input/output trajectory of (1) of length $T_{\rm ini}$, and
${\left\|a\right\|_{X}^{2}}$ denotes the quadratic form $a^{\top}Xa$.

DeePC involves solving the convex optimization problem (5) in a receding
horizon manner, that is, after calculating the optimal control sequence
$u^{\star}$, we apply
$(u_{t},...,u_{t+k-1})=(u_{0}^{\star},...,u_{k-1}^{\star})$ to the system for
$k\leq N-1$ time steps, then, reinitialize the problem (5) by updating
${\rm{col}}(u_{\rm ini},y_{\rm ini})$ to the most recent input and output
measurements, and setting $t$ to $t+k$, to calculate the new optimal control
for the next $k\leq N-1$ time steps. As usual in MPC, the control horizon $k$
is a design parameter.

A regularization term on $g$ can be included in the cost function of (5) to
ensure robustness in case of noisy data samples. For example, when stochastic
disturbances affect the output measurements, a two-norm regularization on $g$
coincides with distributional two-norm robustness in the trajectory space
(Coulson et al., 2019b). Other forms of regularization (one-norm penalty,
etc.) on $g$ can also be used to provide different types of robustness. It was
also reported in (Xue and Matni, 2020) that a robust control formulation of
(5) leads to a two-norm regularization.

##  3 Robustness Induced by Regularization

In this section, we provide a theoretical motivation and interpretation for
adding a quadratic regularization on $g$ in DeePC from a robust optimization
perspective. We start by noting that the explicit equality constraints in (5)
can be eliminated, leading to

$\begin{array}[]{l}\mathop{{\rm{min}}}\limits_{g\in\mathcal{G}}\;\;{\left\|U_{\rm
F}g\right\|_{R}^{2}}+{\left\|{Y_{\rm
F}g-r}\right\|_{Q}^{2}}+\lambda_{y}\left\|Y_{\rm P}g-y_{\rm
ini}\right\|_{2}^{2}\,,\end{array}$ (6)

where $\mathcal{G}=\\{g\in\mathbb{R}^{H}\ |\ U_{\rm F}g\in\mathcal{U},\ Y_{\rm
F}g\in\mathcal{Y},\ U_{\rm P}g=u_{\rm ini}\\}$ is assumed to be nonempty. We
rewrite (6) as

$\begin{array}[]{l}\mathop{{\rm{min}}}\limits_{g\in\mathcal{G}}\;\;\left\|Ag-b\right\|_{2}^{2}\,,\end{array}$
(7)

where $\mathcal{G}=\\{g\ |\ Gg\leq q\\}$ for some $G$ and $q$,

$A=\left[{\begin{array}[]{*{20}{c}}\lambda_{y}^{\frac{1}{2}}Y_{\rm P}\\\
R^{\frac{1}{2}}U_{\rm F}\\\ Q^{\frac{1}{2}}Y_{\rm
F}\end{array}}\right]\quad\text{and}\quad
b=\left[{\begin{array}[]{*{20}{c}}\lambda_{y}^{\frac{1}{2}}y_{\rm ini}\\\ 0\\\
Q^{\frac{1}{2}}r\end{array}}\right]\,.$

Notice that the solution of (7) will in general not be robust to disturbances
on $A$, which in our case contains the historical input/output data. As a
remedy, one can instead consider a robust version of (7) by solving the
following min-max optimization problem

$\mathop{{\rm{min}}}\limits_{g\in\mathcal{G}}\;\;\mathop{{\rm{max}}}\limits_{\|\Delta\|_{F}\leq\beta}\;\left\|(A+\Delta)g-b\right\|_{2}\,$
(8)

where $\Delta$ is the disturbance matrix affecting $A$,
$\|\Delta\|_{F}=\sqrt{\sum_{i,j}\Delta_{ij}^{2}}$ denotes the Frobenius norm,
and $\beta>0$ is the bound on $\|\Delta\|_{F}$. Eq. (8) minimizes the worst-
case cost with respect to the disturbances on the matrix $A$. Hence, the
minimizer $g^{\star}$ (and thus the optimal control input sequence
$u^{\star}=U_{\rm F}g^{\star}$) is robust against disturbances affecting the
data matrices ($U_{\rm P}$, $Y_{\rm P}$, and $Y_{\rm F}$).

The following result explicitly shows how the minimizer of (8) can be obtained
by solving a regularized version of (7) with a quadratic regularization on
$g$.

###### Theorem 1

(Robustness from quadratic regularization). If $g^{\star}\in\mathbb{R}^{H}$ is
a minimizer of

$\mathop{{\rm{min}}}\limits_{g\in\mathcal{G}}\;\left\|Ag-b\right\|_{2}^{2}+\lambda_{g}\left\|g\right\|_{2}^{2},\,$
(9)

then $g^{\star}$ minimizes (8) with

$\beta=\begin{cases}\frac{\lambda_{g}\left\|g^{\star}\right\|_{2}}{\left\|Ag^{\star}-b\right\|_{2}}&\text{if
$Ag^{\star}\neq b$}\\\
\lambda_{g}\left\|g^{\star}\right\|_{2}&\text{otherwise.}\end{cases}$ (10)

Moreover, if $g^{\star}\neq 0$, $\beta$ in (10) is strictly monotonic
increasing with $\lambda_{g}$ chosen in (9).

Consider the Lagrangian of (9)

$\mathcal{L_{Q}}(g,\mu_{Q})=\left\|Ag-b\right\|_{2}^{2}+{\lambda_{g}}{\left\|g\right\|_{2}^{2}}+\mu_{Q}^{\top}(Gg-q)\,,$
(11)

where $\mu_{Q}$ is the vector of the dual variables. Since $\mathcal{G}$ is
nonempty, there exists a solution $(g^{\star},\mu_{Q}^{\star})$ to the
Karush–Kuhn–Tucker (KKT) conditions of (9)

$\displaystyle 2{A^{\top}(Ag-b)}+2{\lambda_{g}g}+G^{\top}\mu_{Q}=0\,,$ (12)
$\displaystyle\mu_{Q}^{\top}(Gg-q)=0\,,$ (13) $\displaystyle Gg\leq q\,,$ (14)
$\displaystyle\mu_{Q}\geq 0\,.$ (15)

Therefore, the vector $g^{\star}$ is a minimizer of (9).

Following (Bertsimas and Copenhaver, 2018, Theorem 2) and (El Ghaoui and
Lebret, 1997), the min-max problem (8) can be equivalently reformulated as

$\mathop{{\rm{min}}}\limits_{g\in\mathcal{G}}\;\;\left\|Ag-b\right\|_{2}+{\beta}{\left\|g\right\|_{2}}\,.$
(16)

Consider the Lagrangian of (16)

$\mathcal{L}(g,\mu)=\left\|Ag-b\right\|_{2}+{\beta}{\left\|g\right\|_{2}}+\mu^{\top}(Gg-q)\,,$
(17)

where $\mu$ is the vector of the dual variables. By choosing $\beta$ in (10),
it can be verified that $(g^{\star},\mu^{\star},y^{\star},z^{\star})$, where

$(\mu^{\star},y^{\star})=\begin{cases}\left(\frac{\mu_{Q}^{\star}}{2\|Ag^{\star}-b\|_{2}},\frac{A^{\top}(Ag^{\star}-b)}{\|Ag^{\star}-b\|_{2}}\right)&\text{if
$Ag^{\star}\neq b$}\\\
\left(\frac{\mu_{Q}^{\star}}{2},0\right)&\text{otherwise,}\end{cases}$

$z^{\star}=\beta g^{\star}/\|g^{\star}\|_{2}$ if $g^{\star}\neq 0$, and
$z^{\star}=0$ otherwise, and $g^{\star}$ as before, satisfy the KKT conditions
of (16):

$\displaystyle y+z+G^{\top}\mu=0\,,$ (18)
$\displaystyle\|A\bar{g}-b\|_{2}\geq\|Ag-b\|_{2}+y^{\top}(\bar{g}-g),$
$\displaystyle\forall\bar{g}\in\mathbb{R}^{H}$ (19)
$\displaystyle\beta\|\bar{g}\|_{2}\geq\beta\|g\|_{2}+z^{\top}(\bar{g}-g),$
$\displaystyle\forall\bar{g}\in\mathbb{R}^{H}$ (20)
$\displaystyle\mu^{\top}(Gg-q)=0\,,$ (21) $\displaystyle Gg\leq q\,,$ (22)
$\displaystyle\mu\geq 0\,,$ (23)

where (19) and (20) follow from the definition of subgradient for the convex
functions $\|Ag-b\|_{2}$ and $\|g\|_{2}$, respectively. Thus, the vector
$g^{\star}$ is also a minimizer of (16) (and (8)).

Next we prove the monotonic relationship between $\beta$ and $\lambda_{g}$.
Let $g_{1}$ be the minimizer of (9) with $\lambda_{g}=\lambda_{g1}>0$ (and the
minimizer of (16) with $\beta=\beta_{1}$), and $g_{2}$ the minimizer of (9)
with $\lambda_{g}=\lambda_{g2}>\lambda_{g1}$ (and the minimizer of (16) with
$\beta=\beta_{2}$). If $g_{1}\neq g_{2}$, according to the definitions of
$g_{1}$ and $g_{2}$, we have

$\left\|Ag_{1}-b\right\|_{2}^{2}+\lambda_{g1}{\left\|g_{1}\right\|_{2}^{2}}-\lambda_{g1}{\left\|g_{2}\right\|_{2}^{2}}<\left\|Ag_{2}-b\right\|_{2}^{2}\,,$
(24)
$\left\|Ag_{2}-b\right\|_{2}^{2}<\left\|Ag_{1}-b\right\|_{2}^{2}+\lambda_{g2}{\left\|g_{1}\right\|_{2}^{2}}-\lambda_{g2}{\left\|g_{2}\right\|_{2}^{2}}\,,$
(25)

leading to

$\lambda_{g1}({\left\|g_{1}\right\|_{2}^{2}}-{\left\|g_{2}\right\|_{2}^{2}})<\lambda_{g2}({\left\|g_{1}\right\|_{2}^{2}}-{\left\|g_{2}\right\|_{2}^{2}})\,,$
(26)

which indicates that
${\left\|g_{1}\right\|_{2}^{2}}>{\left\|g_{2}\right\|_{2}^{2}}$ because
$0<\lambda_{g1}<\lambda_{g2}$. Then, we have
${\left\|g_{1}\right\|_{2}}>{\left\|g_{2}\right\|_{2}}$. According to the
definitions of $\beta_{1}$ and $\beta_{2}$, we also have

$\left\|Ag_{1}-b\right\|_{2}+\beta_{1}{\left\|g_{1}\right\|_{2}}-\beta_{1}{\left\|g_{2}\right\|_{2}}<\left\|Ag_{2}-b\right\|_{2}\,,$
(27)
$\left\|Ag_{2}-b\right\|_{2}<\left\|Ag_{1}-b\right\|_{2}+\beta_{2}{\left\|g_{1}\right\|_{2}}-\beta_{2}{\left\|g_{2}\right\|_{2}}\,.$
(28)

It can then be deduced that $\beta_{1}<\beta_{2}$. If $g_{1}=g_{2}\neq 0$, we
have $\beta_{1}<\beta_{2}$ according to (10). Hence, $\beta$ is increasing
with the increase of $\lambda_{g}$ if $g^{\star}\neq 0$. This completes the
proof.

Theorem 1 shows that the robust solution of (7) can be obtained by simply
adding a quadratic regularization term, and thus the problem can be solved
efficiently. Moreover, the Frobenius norm of the disturbance matrix is bounded
by $\beta$, which is monotonically increasing in $\lambda_{g}$. The bound
$\beta$ is an a-posteriori guarantee obtained after solving (9). With a larger
$\lambda_{g}$, the disturbance set becomes larger including more possible
disturbances. Obviously, an overly large $\lambda_{g}$ gives a conservative
formulation.

In Theorem 1, we consider disturbances affecting $A$ (i.e., affecting the data
matrices $U_{\rm P}$, $Y_{\rm P}$, and $Y_{\rm F}$). The following result will
show that the quadratic regularization on $g$ also provides robustness against
disturbances on the vector $b$.

###### Corollary 2

If a vector $g^{\star}\in\mathbb{R}^{H}$ is a minimizer of (9), then
$g^{\star}$ minimizes

$\mathop{{\rm{min}}}\limits_{g\in\mathcal{G}}\;\;\mathop{{\rm{max}}}\limits_{\left\|[\Delta\;\xi]\right\|_{F}\leq\beta^{\prime}}\;\left\|(A+\Delta)g-(b+\xi)\right\|_{2}\,$
(29)

with

$\beta^{\prime}=\begin{cases}\frac{\lambda_{g}\sqrt{\left\|g^{\star}\right\|_{2}^{2}+1}}{\left\|Ag^{\star}-b\right\|_{2}}&\text{if
$Ag^{\star}\neq b$}\\\
\lambda_{g}\sqrt{\left\|g^{\star}\right\|_{2}^{2}+1}&\text{otherwise,}\end{cases}$
(30)

where $\Delta$ is the disturbance matrix affecting $A$, and $\xi$ is the
disturbance vector affecting $b$.

Consider an augmented decision variable $\bar{g}={\rm col}(g,-1)$. If
$g^{\star}\in\mathbb{R}^{H}$ is a minimizer of (9), then $\bar{g}^{\star}={\rm
col}(g^{\star},-1)$ minimizes

$\mathop{{\rm{min}}}\limits_{\bar{g}\in\mathcal{G}^{\prime}}\;\;\left\|A^{\prime}\bar{g}\right\|_{2}^{2}+{\lambda_{g}}{\left\|\bar{g}\right\|_{2}^{2}}\,,$
(31)

where $\mathcal{G}^{\prime}=\\{\bar{g}\ |\ \bar{g}={\rm col}(g,-1),\
g\in\mathcal{G}\\}$, and $A^{\prime}=[A\;b]$. Then, it follows from Theorem 1
that $\bar{g}^{\star}$ minimizes

$\mathop{{\rm{min}}}\limits_{\bar{g}\in\mathcal{G}^{\prime}}\;\;\mathop{{\rm{max}}}\limits_{\left\|\Delta^{\prime}\right\|_{F}\leq\beta^{\prime}}\;\left\|(A^{\prime}+\Delta^{\prime})\bar{g}\right\|_{2}\,,$
(32)

with $\beta^{\prime}$ given in (30). Then, (32) can be rewritten as (29) with
$\Delta^{\prime}=[\Delta\;\xi]$, that is, $g^{\star}$ minimizes (29).

Corollary 2 indicates that the quadratic regularization in (9) actually
provides robustness against disturbances affecting both $A$ and $b$. In other
words, the solution to (9) is robust to the disturbances affecting the data
matrices ($U_{\rm P}$, $Y_{\rm P}$, and $Y_{\rm F}$) and the initial output
trajectory $y_{\rm ini}$. However, the considered disturbance set (resulting
from the quadratic regularization) is conservative because some entries in $A$
(e.g., the entries of $U_{\rm P}$) and $b$ (e.g., the entries of $r$) are not
subjected to disturbances. Moreover, how the considered disturbances affect
input/output signals is also related to the scaling matrices, e.g.,
$Q^{\frac{1}{2}}$ in $A$. We will consider structured disturbance sets in
future work.

##  4 Applications to Power converters

Power converters have been extensively used in modern power systems to
accommodate large-scale renewable generation and high-voltage DC systems.
Conventional control schemes for power converters are model-based, however, an
accurate system model can rarely be obtained especially when the converter is
connected to a power grid. This may result in inferior performance or even
instabilities. In this section, we apply the DeePC algorithm with quadratic
regularization to perform model-free optimal control in grid-connected power
converters.

The DeePC algorithm has been successfully applied in grid-connected power
converters for data-driven oscillation damping in Huang et al. (2019a, b).
However, in these works, the converters still rely on conventional schemes to
achieve grid synchronization, power regulation, etc., and the DeePC algorithm
only provides the control signal to stabilize the system. Here, we present a
data-driven control framework for power converters to achieve grid
synchronization and power regulation simultaneously.

###  4.1 Data-Driven Synchronization and Power Regulation

Figure 1: A data-driven control framework for power converters based on DeePC
algorithm.

Fig. 1 shows a three-phase power converter which is connected to an AC power
grid via an LCL filter. A current control loop is used to enable fast current
tracking and current limitation. The control objectives of the converter
include grid synchronization and active/reactive power regulation. As
illustrated in Fig. 1, we choose the q-axis voltage component $V_{q}$, active
power $P_{E}$, and reactive power $Q_{E}$ to be the output signals of the
converter system. The DeePC algorithm provides optimal control inputs that
enter the converter system as internal frequency deviation $\Delta\omega$ and
current references $I_{d}^{\rm ref}$, $I_{q}^{\rm ref}$. Note that $u_{i,t}$
is the $i{\rm th}$ element of $u_{t}$ and $y_{i,t}$ the $i{\rm th}$ element of
$y_{t}$ in Fig. 1.

Before the DeePC algorithm is activated, input/output data measured from the
converter system needs to be collected to construct the Hankel matrices. Note
that a phase-locked loop (PLL) is used to stabilize the system when the DeePC
is not activated, but is then bypassed once the DeePC is activated. After
constructing the Hankel matrices, the optimal control inputs are obtained by
solving (9) in a receding-horizon manner. The control framework in Fig. 1
achieves grid synchronization by minimizing the trajectory of $V_{q}$ (i.e.,
making the control coordinate aligned with the voltage vector). The
active/reactive power regulations are achieved by giving references for the
output signals $P_{E}$ and $Q_{E}$. We set the reference vector to
$r=I_{N}\otimes{\rm col}(0,P_{0},Q_{0})$ to achieve the above control
objectives, where $\otimes$ denotes the Kronecker product,
$I_{N}\in\mathbb{R}^{N\times N}$ is the identity matrix, $P_{0}$ is the active
power reference value, and $Q_{0}$ is the reactive power reference value.

###  4.2 Closed-Form Solutions

The quadratic regularization in (9) not only provides robustness for the DeePC
algorithm, but also allows one to obtain the solution by quadratic
programming. Moreover, if there are no input or output constraints (or if they
are inactive at the optimal solution (33) below), we can compactly obtain the
closed-form solution of (9) as

$g^{\star}=M\left[{\begin{array}[]{*{20}{c}}2A^{\top}b\\\ u_{\rm
ini}\end{array}}\right]$ (33)

where $M\in\mathbb{R}^{H\times(H+mT_{\rm ini})}$ is defined by

$\left[{\begin{array}[]{*{20}{c}}M\\\
\bar{M}\end{array}}\right]=\left[{\begin{array}[]{*{20}{c}}2(A^{\top}A+\lambda_{g}I_{H})&U_{\rm
P}^{\top}\\\ U_{\rm P}&\bf{0}\end{array}}\right]^{-1}\,.$

The closed-form solution in (33) is simply a linear mapping of the initial
trajectory ${\rm col}(u_{\rm ini},y_{\rm ini})$ and the reference vector $r$
(Carlet et al., 2020; Alexandru et al., 2020). If it happens to satisfy any
input and output constraints, it provides a fast way to obtain the minimizer
of (9) and thus the optimal input sequence $u^{\star}=U_{\rm F}g^{\star}$.

###  4.3 Simulation Results

In our simulations, we use the base values $f_{\rm b}=50{\rm Hz}$, $S_{\rm
b}=1.50{\rm kW}$, and $U_{\rm b}=208{\rm V}$ for per-unit calculations. The
LCL parameters are: $L_{F}=0.04{\rm p.u.}$ (with resistance $R_{F}=0.03{\rm
p.u.}$), $L_{g}=0.04{\rm p.u.}$ (with resistance $R_{g}=0.03{\rm p.u.}$), and
capacitor $C_{F}=0.02{\rm p.u.}$. The local load is $R_{\rm Load}=2.23{\rm
p.u.}$. The PI parameters of the current control are $\\{0.15,10\\}$. The
parameters for DeePC are: $T_{\rm ini}=6$, $N=12$, $T=500$,
$\lambda_{y}=10^{4}$, $R=I_{3N}$, and $Q=400I_{3N}$. The input/output
constraints are ignored. The control horizon is $k=1$, that is, after applying
$u^{\star}_{0}$ to the system, we update the initial trajectory and then
calculate the optimal inputs for the next time step.

Fig. 2 shows the time-domain active power responses of the converter. Before
$0.5{\rm s}$, the system operates with $I_{d}^{\rm ref}=I_{q}^{\rm ref}=0$,
and a PLL is used to generate $\Delta\omega$. From $0.5{\rm s}$ to $1{\rm s}$,
the input/output data of the converter is collected to construct the Hankel
matrices. During this period, the output of the PLL and a white-noise signal
are added to the input signal $\Delta\omega$, i.e., $u_{1,t}$; a constant
setting value $0.2$ and a white-noise signal are added to $I_{d}^{\rm ref}$,
i.e., $u_{2,t}$; a constant setting value $0$ and a white-noise signal are
added to $I_{q}^{\rm ref}$, i.e., $u_{3,t}$. The white-noise signals (with
noise power $6\times 10^{-7}{\rm p.u.}$) ensure that the input trajectory is
persistently exciting. After collecting the data, the noise is switched off,
and $I_{d}^{\rm ref}$ is set back to 0. The PLL is then switched off and the
DeePC controller is switched on at $1.5{\rm s}$ with $Q_{0}=0$ and $P_{0}=0$.
Finally, active power reference $P_{0}$ steps from $0$ to $0.3{\rm p.u.}$ at
$2{\rm s}$.

Figure 2: Active power responses of the converter with different values of
$\lambda_{g}$. (a) Without measurement noise. (b) With measurement noise power
being $10^{-7}$ (p.u.).

In Fig. 2 (a), the measurement noise for the output signals is not considered,
and the converter has very fast and smooth response to the change of $P_{0}$
(without any overshoot) when choosing $\lambda_{g}=10$. Moreover, the active
power perfectly tracks the reference value in steady state, which is achieved
by penalizing the tracking error in the cost function of (9). The response
becomes slower by increasing $\lambda_{g}$ to $10^{3}$, and it becomes even
slower by further increasing $\lambda_{g}$ to $10^{4}$ (there also exist very
small steady-state tracking errors in these two cases). This is expected: the
increase of $\lambda_{g}$ indicates that the optimal control inputs are robust
with respect to a larger disturbance set according to Theorem 1. This leads to
conservativeness at the expense of performance if the disturbance set is
unnecessarily large. Hence, the tuning of $\lambda_{g}$ can also be considered
as a convenient way to change the response speed of the system. Conversely,
when choosing $\lambda_{g}$ to be too small, e.g., $\lambda_{g}=10^{-5}$, the
active power cannot effectively track the reference value. This is because the
data in the Hankel matrices contains the true nonlinear behaviors of the
system, and leads to prediction errors and inferior performance without proper
robustification.

Fig. 2 (b) shows the active power responses of the converter when considering
measurement noise on the output signals. It can be seen that by choosing
$\lambda_{g}=10$, the converter has effectively perfect tracking performance
to the change of power reference. With $\lambda_{g}=10^{-5}$, the tracking
error is larger than (a) because the disturbance set is not large enough to
cover the measurement noise. Fig. 2 (a) and (b) also plot the A-phase voltage
and current of the system with $\lambda_{g}=10$. It shows that the sinusoidal
waveforms are well controlled by the DeePC algorithm in all the cases.

###  4.4 Experiment Results

In this section, we provide experiment results of the converter system given
in Fig.1 to show the performance of the DeePC algorithm in realistic settings.
The experiments are based on an Imperix PEB platform (which contains IGBT
bridges, passive elements, analog sensors, etc.), and the DeePC algorithm is
implemented in an ARM-based processor (B-BOX RCP) which provides control
signals to the IGBT bridges (Imperix, 2020). The parameters of the experiments
are the same as those in the simulation setup. We ignore the input/output
constraints and use the closed-form solution of DeePC given in (33) to enable
fast calculation and convenient implementation.

Figure 3: Experiment results. (a) Responses during the data-collection period
(with white-noise signals injected into the system). (b) Responses when the
DeePC algorithm is activated, and the active power reference steps from 0 to
$0.3{\rm p.u.}$ at $0.08{\rm s}$.

The experiment results are shown in Fig. 3. In Fig. 3 (a), white-noise signals
are injected into the converter such that the input trajectory is persistently
exciting. It can be seen that the fluctuations of the active power and current
signals are acceptable during this data-collection period. Fig. 3 (b) displays
the time-domain responses of the converter when the DeePC algorithm is
activated. The active power reference steps from 0 to $0.3{\rm p.u.}$ at
$0.08{\rm s}$. It can be seen that the active power perfectly tracks the
change of the reference signal, with the rise time being $0.02{\rm s}$. The
voltage and current signals are pure sinusoidal signals in steady state,
without any power quality issues.

##  5 Conclusion

This paper focused on understanding the effects of the quadratic
regularization in DeePC. We explicitly show that the regularized DeePC is
equivalent to a min-max formulation which minimizes the worst-case cost for a
bounded disturbance set affecting the data Hankel matrices and the initial
trajectory. Our analysis highlights the importance of the quadratic
regularization in DeePC for robustness. In the absence of input and output
constraints, we derived a closed-form solution of the DeePC algorithm with
quadratic regularization for fast calculation. Based on the above features, we
proposed a framework for grid-connected converters to achieve data-driven
synchronization and power regulation, and we tested its performance with high-
fidelity simulations and experiments. Future work will investigate the
robustness of other forms of regularization and consider structured
disturbance sets.

The authors would like to thank Catalin Arghir for his help in the
experiments. This research was supported by the ERC under project OCAL
(contract number 787845) and ETH Zurich Funds.

## References

  * Alexandru et al. (2020) Alexandru, A.B., Tsiamis, A., and Pappas, G.J. (2020).  Data-driven control on encrypted data.  _arXiv preprint arXiv:2008.12671_.
  * Berberich et al. (2020) Berberich, J., Köhler, J., Muller, M.A., and Allgower, F. (2020).  Data-driven model predictive control with stability and robustness guarantees.  _IEEE Trans. Autom. control (Early Access)_.
  * Bertsimas and Copenhaver (2018) Bertsimas, D. and Copenhaver, M.S. (2018).  Characterization of the equivalence of robustification and regularization in linear and matrix regression.  _European Journal of Operational Research_ , 270(3), 931–942.
  * Boczar et al. (2018) Boczar, R., Matni, N., and Recht, B. (2018).  Finite-data performance guarantees for the output-feedback control of an unknown system.  In _2018 IEEE Conference on Decision and Control (CDC)_ , 2994–2999. IEEE.
  * Campi et al. (2002) Campi, M.C., Lecchini, A., and Savaresi, S.M. (2002).  Virtual reference feedback tuning: a direct method for the design of feedback controllers.  _Automatica_ , 38(8), 1337–1346.
  * Carlet et al. (2020) Carlet, P.G., Favato, A., Bolognani, S., and Dörfler, F. (2020).  Data-driven predictive current control for synchronous motor drives.  In _2020 IEEE Energy Conversion Congress and Exposition_ , 5148–5154. IEEE.
  * Coulson et al. (2019a) Coulson, J., Lygeros, J., and Dörfler, F. (2019a).  Data-enabled predictive control: In the shallows of the DeePC.  In _2019 18th European Control Conference (ECC)_ , 307–312. IEEE.
  * Coulson et al. (2019b) Coulson, J., Lygeros, J., and Dörfler, F. (2019b).  Regularized and distributionally robust data-enabled predictive control.  In _2019 IEEE 58th Conference on Decision and Control (CDC)_. IEEE.
  * De Persis and Tesi (2019) De Persis, C. and Tesi, P. (2019).  Formulas for data-driven control: Stabilization, optimality, and robustness.  _IEEE Trans. Autom. control_ , 65(3), 909–924.
  * El Ghaoui and Lebret (1997) El Ghaoui, L. and Lebret, H. (1997).  Robust solutions to least-squares problems with uncertain data.  _SIAM Journal on matrix analysis and applications_ , 18(4).
  * Elokda et al. (2019) Elokda, E., Coulson, J., Beuchat, P., Lygeros, J., and Dörfler, F. (2019).  Data-enabled predictive control for quadcopters.
  * Goodwin and Sin (2014) Goodwin, G.C. and Sin, K.S. (2014).  _Adaptive filtering prediction and control_.  Courier Corporation.
  * Hjalmarsson et al. (1998) Hjalmarsson, H., Gevers, M., Gunnarsson, S., and Lequin, O. (1998).  Iterative feedback tuning: theory and applications.  _IEEE control systems magazine_ , 18(4), 26–41.
  * Huang et al. (2019a) Huang, L., Coulson, J., Lygeros, J., and Dörfler, F. (2019a).  Data-enabled predictive control for grid-connected power converters.  In _2019 IEEE 58th Conference on Decision and Control (CDC)_ , 8130–8135. IEEE.
  * Huang et al. (2019b) Huang, L., Coulson, J., Lygeros, J., and Dörfler, F. (2019b).  Decentralized data-enabled predictive control for power system oscillation damping.  _arXiv preprint arXiv:1911.12151_.
  * Imperix (2020) Imperix (2020).  _PEB8032_.  URL https://imperix.com/products/power/peb.
  * Ljung (1999) Ljung, L. (1999).  System identification.  _Wiley Encyclopedia of Electrical and Electronics Engineering_ , 1–19.
  * Markovsky and Dörfler (2020) Markovsky, I. and Dörfler, F. (2020).  Identifiability in the behavioral setting.  Available online.
  * Markovsky and Rapisarda (2008) Markovsky, I. and Rapisarda, P. (2008).  Data-driven simulation and control.  _International Journal of Control_ , 81(12), 1946–1959.
  * Recht (2019) Recht, B. (2019).  A tour of reinforcement learning: The view from continuous control.  _Annu. Rev. Control Robot. Auton. Syst._ , 2, 253–279.
  * Willems et al. (2005) Willems, J.C., Rapisarda, P., Markovsky, I., and De Moor, B.L. (2005).  A note on persistency of excitation.  _Systems & Control Letters_, 54(4), 325–329.
  * Xue and Matni (2020) Xue, A. and Matni, N. (2020).  Data-driven system level synthesis.  _arXiv preprint arXiv:2011.10674_.


# Quantum State-Channel Duality for the calculation of Standard Model
scattering amplitudes.

Clelia Altomonte  Department of Physics, Strand, King’s College London, WC2R
2LS  Alan J. Barr<EMAIL_ADDRESS>Department of Physics, Keble
Road, University of Oxford, OX1 3RH
Merton College, Merton Street, Oxford, OX1 4JD

(21 September 2023, First submitted: 6 October 2022 (arXiv: 4530106)
qd Accepted for publication in Physics Letters B: 30 October 2023)

###### Abstract

Abstract: Recent instances of successful application of quantum information
techniques to particle physics problems invite for an analysis of the
mathematical details behind such connection. In this paper, we identify the
Choi-Jamiolkowski isomorphism, or state-channel duality, as a theoretical
principle enabling the application of the theory of quantum information to the
scattering amplitudes associated with Standard Model processes.

##  I  QUANTUM INFORMATION THEORY APPLIED TO HIGH ENERGY SYSTEMS

While the first suggestion to make simulations of Nature quantum mechanical
famously dates back to Richard FeynmanFeynman (1982), recent attempts to apply
the theory of quantum information to the study of high energy physics systems
have proven particularly successful. As a paradigmatic example quantum state
tomography, a procedure that allows full reconstruction of the density matrix
of a system by performing a complementary series of measurements on an
ensemble of identical copies of the system under scrutiny Nielsen and Chuang
(2010), is ideally applicable to colliders, where large numbers of events are
generatedBarr, Caban, and Rembieliński (2022); Fabbrichesi, Floreanini, and
Panizzo (2021); Afik and de Nova (2021); Ashby-Pickering, Barr, and
Wierzchucka (2022), and has been applied to numerical simulation studies of
various high energy particle physics systemsFabbrichesi, Floreanini, and
Panizzo (2021); Afik and de Nova (2021); Barr (2022); Ashby-Pickering, Barr,
and Wierzchucka (2022). Quantum algorithms, including quantum machine learning
techniques, have been developed for the recognition of Standard Model and
beyond signatures in dataMöttönen  _et al._ (2004); Delgado and Hamilton
(2022); Wu  _et al._ (2021), and for a more computationally economic
simulation of collider eventsGustafson  _et al._ (2022).

While these results verify the expected agreement between the two fields of
particle physics and quantum information (being that the Standard Model is
based on quantum field theory, which is a quantum theory), the mathematical
details behind such connection could be further exploited, leading to novel
insights into both fields. In this paper, we identify the Choi-Jamiolkowski
isomorphism Jiang, Luo, and Fu (2013), or state-channel duality, as a
theoretical principle enabling a systematic application of the theory of
quantum information to the calculation of Standard Model scattering
amplitudes, and consider it worthy of the attention of the particle physics
community for the following reasons.

On the one hand, the utility of this approach lies in pointing to a
mathematical relation in the form of an isomorphism to establish a rigorous
dictionary connecting Standard Model scattering amplitudes and the outputs of
quantum channels. The existence of such dictionary allows one to better
exploit techniques from quantum information theory in particle physics
processes which could be experimentally observed at working and near-term
colliders, hence providing a mathematical framework that has the potential to
address the following problems:

(i) testing the foundations of quantum mechanics in particle physics
colliders, by identifying the collider itself as a natural quantum-information
processing machine provided that quantum mechanics holds. In this paper, we
address this problem by providing, as an example, a tree-level process which
shows how the presented framework allows us to test this assumption;

(ii) performing tests of basic quantum predictions such as entanglement and
quantum state and process tomography in high energy regimes (cf. refs.Barr,
Caban, and Rembieliński (2022); Barr (2022); Ashby-Pickering, Barr, and
Wierzchucka (2022));

(iii) expanding theoretical modelling into regions where the Standard Model
theory is difficult to calculate perturbatively, and where phenomenological
models are currently employed, by having an alternative method for performing
calculations.

On the other hand, the scope of our approach does not overlap with other
quantum computing protocols currently applied to nuclear and particle physics,
such as quantum Hamiltonian evolution simulationsKlco, Roggero, and Savage
(2022); Bauer  _et al._ (2023a, b); Bañuls  _et al._ (2020), where the full
dynamics of the particle physics system is simulated with the aim of observing
non-perturbative effects. Resorting only to the knowledge of initial and final
state density matrices, our approach allows us to establish whether a quantum
channel which is equivalent to a given S-matrix can be constructed. This makes
it possible reliably to use quantum information tools to learn about the
Hamiltonian underlying the channel, and can potentially reveal evidence of new
physics beyond the Standard Model.

The present paper is organised as follows: in Section II, we assess the
conditions for Standard Model scattering processes to be suitable systems for
the theoretical and experimental study of quantum information, satisfying the
general conditions for successful quantum computing implementationNielsen and
Chuang (2010). In Section III, by applying state-channel duality we show that
Standard Model scattering amplitudes can be calculated as quantum channels. In
Section IV we present an example model to illustrate how helicity matrix
elements associated with tree-level electroweak interaction Feynman diagrams
can be reproduced by means of simple quantum channels. In Section V, we
introduce a dictionary connecting some selected particle physics processes and
quantum channels. Section VI is dedicated to the extension of state-channel
duality to higher-order corrections. Finally, in Section VII we comment on
some possible applications of the present work.

##  II CONSTRUCTING A QUANTUM INFORMATION DESCRIPTION OF STANDARD MODEL
SCATTERING AMPLITUDES

A general formulationNielsen and Chuang (2010) of the conditions for a system
to successfully implement quantum computation requires that it must be
possible to:

a. Robustly represent quantum information;

b. Perform a universal family of unitary
transformations;

c. Prepare a fiducial initial state;

d. Measure the output result;

where ‘successful implementation’ means that the system can be manipulated in
such a way as to correspond to a given quantum circuit. It is also worth
clarifying that particle physics scattering processes described by quantum
field theory and observed at colliders can be considered to implement quantum
computation, albeit in a passive way. In fact, while candidate systems for the
construction of physical quantum computers (e.g. optical systemsNielsen and
Chuang (2010)) should in general allow for an easy and more extensive
manipulation of the system’s parameters, the experimenters’ possibility of
intervention at colliders is limited to the possible use of polarised beams
and in determining other limited experimental settings. However, constructing
a rigorous quantum information description of Standard Model scattering
amplitudes is possible because the four conditions outlined above are
fulfilled. We consider each condition in the subsections below.

###  II.1 Robustly representing quantum information: the choice of basis

Condition a. rests on being able to encode particle physics observables such
as spin states into qudits (i.e. quantum mechanical $d$-dimensional bits of
informationNielsen and Chuang (2010)). This is verified by considering that in
quantum information theory it is commonplace to represent the state of a
qudit, a $d$-dimensional system described by a $d\times d$ density matrix, in
vectorial form (Bloch vector). Since the space of matrices is a vector space,
it is possible to find bases matrices to decompose any matrixBertlmann and
Krammer (2008).

As an example, in the generalized Gell-Mann matrix basis, the matrices
correspond to the standard $SU(N)$ generators ($d=N$), and induces a Bloch
vector with real components, which can readily be interpreted as expectation
values of measurable quantities (even though the SU($d$) generators do not
necessarily correspond to any physical state, combining them with the Identity
operator can result in a linear combination which can represent a physical
state density operator). Considering a $2$-dimensional state as an example,
the Pauli operators offer a complete Hermitian operator basis to parametrise
the one qubit space, and correspond to the Pauli group used to describe
physical observables of spin-$\frac{1}{2}$ particles Dogra, Dorai, and Arvind
(2018); Weyl (2014); Gell-Mann (1962). Because particle physics systems’
observables such as spin can also be described according to some group-
theoretic structure, it follows that they can easily be connected to quantum
state’s Bloch vector ‘robustly’, making them very interesting qudit
implementations.

It is worth noting that while spin is a particle physics observable that can
in principle be regarded as a good candidate for encoding quantum information,
having a both a physical interpretation and unitary representation,
alternative basis choices may prove more useful when trying to connect the
processes observed at colliders with a quantum information description (as an
example, cf. Appendix A for a detailed explanation of the helicity basis used
in Section IV).

###  II.2 Performing a universal family of unitary transformations

Because the Standard Model of particle physics is a quantum field theory and
is constructed on the unitary product group
$\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$, and qubit quantum
computers generally implement unitary operations, it is expected that an
interpretation of the Standard model in terms of quantum information is
achievable (thus satisfying  Condition b.).

As we will show in the following section, state-channel duality can be used to
prove a related point: even at the lowest perturbation order such translation
is achievable and hence quantum information machinery and tests (e.g. Bell
inequalities violation tests) can be rigorously applied to particle physics
processes and give meaningful results.

###  II.3 Preparing a fiducial initial state

Condition c. is satisfied by the proposal Afik and de Nova (2021) that
accelerators, especially those that allow for spin polarised beams, can be
used to study quantum information processes. On the quantum information side,
as mentioned above, the choice of basis needs to be made carefully,
eigenstates of unitary operators corresponding to physical observables are the
preferred choice.

###  II.4 Measuring the output result

Finally, colliders fulfill  Condition d. by construction, being designed and
engineered for the purpose of precise and reliable energy and momentum
measurements. We can take the past history of successful measurements carried
out at colliders as proof of this. In addition, as explained in section IV
below, specific particle physics processes (e.g. $t\bar{t}$ quarks leptonic
decays) provide particularly good candidates for quantum information
applications, allowing for reliable access to spin states.

##  III State-Channel duality applied to Standard Model scattering amplitudes

In quantum information theory, the initial and final states of a qudit are
connected through operations called quantum channels. The question we address
in this section is whether we can find quantum channels connecting the density
matrices describing the initial and final states of particle physics
processes. Our proposal is that the Choi-Jamiolkowski isomorphism, also known
as state-channel duality, allows us to give an affirmative answer and to apply
systematically many more quantum information procedures to particle scattering
(cf. Sections V-VII below).

However, before we proceed to show how it is applied to particle physics, it
is worth noticing that the expression “Choi-Jamiolkowski isomorphism” is often
indistinctly used to refer to three distinct isomorphisms: between a general
map and a general operator; between a positive map and a positive operator;
and finally between a completely positive map and a positive operatorJiang,
Luo, and Fu (2013). The latter is the most used and explored isomorphism, and
establishes that any channel, by which we mean one or more quantum operations
(i.e. $\mathcal{B}(\mathcal{H})$ is the bounded operator $\mathcal{B}$ on
Hilbert space $\mathcal{H}$) corresponding to a linear, completely positive,
trace preserving map
$\mathcal{E}:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}\left(\mathcal{H}^{\prime}\right)$,
connecting an input state space $\mathcal{H}$ of dimension $d$ and an output
state space $\mathcal{H}^{\prime}$ of dimension $d^{\prime}$ corresponds to a
bipartite state of the tensor product of the two.

To show this, we follow the exposition in ref.Ekert (2022). First, we pick
$d^{2}$-many basis matrices $|i\rangle\langle j|$ of
$\mathcal{B}(\mathcal{H})$, where $i,j\in\\{1,2\ldots,d\\}$ and use them to
characterise the density operator $\hat{\rho}$ on which the channel
$\mathcal{E}$ acts:

$\mathcal{E}(\hat{\rho})=\mathcal{E}\left(\sum_{ij}\hat{\rho}_{ij}|i\rangle\langle
j|\right)=\sum_{i}\hat{\rho}_{ij}\mathcal{E}(|i\rangle\langle j|).$ (1)

We then tabulate all the possible $\left(d^{\prime}\times d^{\prime}\right)$
matrices $\mathcal{E}(|i\rangle\langle j|)$ in $\mathcal{H}^{\prime}$ by
constructing the $\left(dd^{\prime}\times dd^{\prime}\right)$ block matrix
$\mathcal{H}\otimes\mathcal{H}^{\prime}$, which is called the Choi matrix
$\tilde{\mathcal{E}}$:

$\widetilde{\mathcal{E}}=\frac{1}{d}\left[\begin{array}[]{cccc}\mathcal{E}(|0\rangle\langle
0|)&\mathcal{E}(|0\rangle\langle 1|)&\mathcal{E}(|0\rangle\langle
2|)&\cdots\\\ \mathcal{E}(|1\rangle\langle 0|)&\mathcal{E}(|1\rangle\langle
1|)&\mathcal{E}(|1\rangle\langle 0|)&\cdots\\\ \mathcal{E}(|2\rangle\langle
0|)&\mathcal{E}(|2\rangle\langle 1|)&\mathcal{E}(|2\rangle\langle
2|)&\cdots\\\ \vdots&\vdots&\vdots&\ddots\end{array}\right]$ (2)

The above matrix $\tilde{\mathcal{E}}$ is just an equivalent representation of
the quantum channel
$\mathcal{E}:\mathcal{B}(\mathcal{H})\rightarrow\mathcal{B}\left(\mathcal{H}^{\prime}\right)$.

In quantum computing, state channel duality is mainly used to assess whether a
given channel is physically realisable by considering whether the system’s
Choi matrix is a density matrix (i.e. positive: $\mathcal{E}(\rho)\geqslant 0$
whenever $\rho\geqslant 0$ and trace preserving:
$\operatorname{tr}\mathcal{E}(\rho)=\operatorname{tr}\rho$ for all $\rho$). It
is worth noting, however, that because the operator-sum (Kraus) representation
of a given channel is not uniqueEkert (2022), different channels can lead to
the same density matrix. Hence, the state-channel duality allows one to
ascertain whether a given channel can be physically implemented, but it does
not imply that this will be the only possible implementation of a given
density matrix.

Our suggestion is to apply this to high energy scattering processes by making
the reverse use of the duality: because through quantum state tomographyAshby-
Pickering, Barr, and Wierzchucka (2022); Barr (2022) it is possible to
reconstruct the density matrix of a given process, which will also
automatically fulfil the formal characteristics required for it to correspond
to a physical process, such as positivityMartens, Ralston, and Takaki (2017)
($\rho\geqslant 0$), we conclude that it is in principle possible to construct
a quantum channel connecting the initial and final state density matrices.

##  IV An example model: $\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow t\bar{t}$
electroweak process with polarised beams

In this section we present the process of electron-positron annihilation
producing a top quark and its anti-quark with polarised beams as an example
model to illustrate how state-channel duality may be applied to simulate
helicity states density matrices.

This process is particularly interesting because the leptonic decays of the
$t$ and $\bar{t}$ (anti-) quarks effectively induce a measurement of spin
along the axis of the emitted leptonFabbrichesi, Floreanini, and Panizzo
(2021), thus allowing access to the spin states of the $t\bar{t}$ system and
making the quantum-computing simulated density matrix amenable to experimental
verification. Hence, it allows us to test the assumption that colliders can be
identified as privileged observatories for quantum information processing (cf.
Section I, point (i)). Other Standard Model processes for which spin
tomography is possible include the decay of $W$ bosons (e.g. produced from
Higgs boson decays) to leptonsBarr (2022).

Figure 1: The $\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow t\bar{t}$ electroweak
process viewed in the centre-of-mass frame (top-left), corresponding lowest-
order Feynman diagram (top-right), and four of the $16$ possible helicity
combinations (adapted from ref.Thomson (2021))

###  IV.1 Notes on polarisation choices

Our example process $\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow t\bar{t}$
contains two spin-$\frac{1}{2}$ particles in the incoming state, and two
spin-$\frac{1}{2}$ particles in the outgoing state. Thus both the initial and
final state are naturally represented by two-qubit systems, and the Hilbert
state for the initial state (and similarly that for the final state) is
therefore of dimension $2^{2}=4$. The density operators on each of those
Hilbert spaces are $4\times 4$ matrices, and the full Choi matrix, which maps
the initial to the final state (eq. 6), is of dimension $16\times 16$.

To more simply illustrate the principles we consider a reduced subset of the
Choi matrix elements relating particular polarisation choices of the initial
and final states. We restrict our attention to those initial- and final-state
density matrices that can be represented by polarisation (Bloch) vectors that
point along the momentum direction of each particle, which is equivalent to
saying that the states that are diagonal in the helicity basis. Since the
density matrices are diagonal in the helicity basis the Choi matrix elements
representing them are also diagonal.

This restriction should not be interpreted as meaning that the other Choi
elements, those that are off-diagonal in the helicity basis, are necessarily
inaccessible. For the final state, techniques have been developed Afik and de
Nova (2021); Ashby-Pickering, Barr, and Wierzchucka (2022) that can
reconstruct the full bipartite density matrix for a pair of top quarks, so all
possible densities in the final-state Hilbert space can be accessed and
measured experimentally. For the initial state, technologies exist (see e.g.
ref.Sokolov and Ternov (1963); Sobloher (2012); Blondel  _et al._ (2019)) that
allow us to polarise high-energy electron beams in both the longitudinal
direction (i.e. with the spin polarisation vector (anti-)parallel to the beam)
and in the transverse direction, thus each beam having an arbitrary Bloch
vector. The beams are not entangled, so the density matrices of the bipartite
initial state which are experimentally accessible are those that can be
represented by the product state

$\rho=\rho_{A}\otimes\rho_{B}$ (3)

for some $\rho_{A}$ and $\rho_{B}$ representing the density matrices of the
two beams respectively.

In what follows, we restrict our attention to only those Choi matrix elements
that are diagonal in the helicity basis in order to more compactly illustrate
the technique. For compactness, we report these values in a $4\times 4$
matrix, with the initial state polarisations represented by rows and the
final-state polarisations represented by columns.

###  IV.2 Example model illustration

Following the traditional quantum field theoretical approach (cf. ref.Thomson
(2021) for full steps), the process cross-section $\sigma$ at leading order in
the electroweak coupling constants is proportional to

$\sigma\propto|\mathcal{M}|^{2}=\left|\mathcal{M}_{\gamma}+\mathcal{M}_{Z}\right|^{2}$
(4)

where $\mathcal{M}_{\gamma}$ and $\mathcal{M}_{Z}$ are the density matrices
related to the $Z$ boson and photon, $\gamma$, exchange Feynman diagrams (Fig.
1). Since our points are sufficiently illustrated by considering separately
the cases of the photon and $Z$ boson propagators, we neglect in what follows
the interference term
$2\left|\mathcal{M}_{\gamma}\mathcal{M}_{Z}{}^{*}\right|$. Adding that term is
a straight-forward modification, but it adds to the complexity of the
expression and is not required for our illustrative purposes.

Let us first consider the individually electromagnetic process associated with
$\mathcal{M}_{\gamma}$, which is given by

$\displaystyle\mathcal{M}_{\gamma}$ $\displaystyle=-\frac{e^{2}}{q^{2}}g_{\mu
v}\left[\bar{v}\left(p_{2}\right)\gamma^{\mu}u\left(p_{1}\right)\right]\left[\bar{u}\left(p_{3}\right)\gamma^{\nu}v\left(p_{4}\right)\right]$
(5) $\displaystyle=-\frac{e^{2}}{s}j_{e}\cdot j_{t},$

where $\bar{v}\left(p_{2}\right)$, $u\left(p_{1}\right)$,
$\bar{u}\left(p_{3}\right)$, $v\left(p_{4}\right)$ are the helicity spinors
associated with the initial-state positron, initial-state electron, final-
state top quark and final-state antitop quark respectively, $\gamma^{\mu}$ are
the Dirac–Pauli representation of the $\gamma$-matrices, $e^{2}/4\pi$ is the
electromagnetic fine-structure constant, $g_{\mu v}$ is the Minkowski metric,
$q$ is the four-momentum of the photon propagator, $j_{e}$ and $j_{t}$ are the
four-vector electron-postron and top-antitop currents respectively, and the
centre-of-mass energy $\sqrt{s}=2E_{\text{beam }}$ is equal to twice the beam
energy.

Because we do not wish to average over spin degrees of freedom, we need to
obtain and multiply together the electron-positron and top-antitop currents
for every one of the possible 16 helicity combinations (e.g.
$t_{\uparrow}\bar{t}_{\downarrow}=j_{t,RL}=\left(j_{t}^{0},j_{t}^{1},j_{t}^{2},j_{t}^{3}\right)$)
without using spin trace techniques. We thus construct the matrix

$\left|\mathcal{M}_{\gamma}\right|^{2}\propto\left[\begin{array}[]{cccc}\left|M_{RL\rightarrow
RL}\right|^{2}&\left|M_{RL\rightarrow RR}\right|^{2}&\left|M_{RL\rightarrow
LL}\right|^{2}&\left|M_{RL\rightarrow LR}\right|^{2}\\\ \left|M_{RR\rightarrow
RL}\right|^{2}&\left|M_{RR\rightarrow RR}\right|^{2}&\left|M_{RR\rightarrow
LL}\right|^{2}&\left|M_{RR\rightarrow LR}\right|^{2}\\\ \left|M_{LL\rightarrow
RL}\right|^{2}&\left|M_{LL\rightarrow RR}\right|^{2}&\left|M_{LL\rightarrow
LL}\right|^{2}&\left|M_{LL\rightarrow LR}\right|^{2}\\\ \left|M_{LR\rightarrow
RL}\right|^{2}&\left|M_{LR\rightarrow RR}\right|^{2}&\left|M_{LR\rightarrow
LL}\right|^{2}&\left|M_{LR\rightarrow
LR}\right|^{2}\end{array}\right]{\color[rgb]{0,0,1}\uwave{.}}$ (6)

Disregarding any pre-factors, we obtain the following matrix

$\left|\mathcal{M}_{\gamma}\right|^{2}\propto\left[\begin{array}[]{cccc}(1+\cos\theta)^{2}&0&0&(1-\cos\theta)^{2}\\\
(\sin\theta)^{2}&0&0&(\sin\theta)^{2}\\\
(\sin\theta)^{2}&0&0&(\sin\theta)^{2}\\\
(1-\cos\theta)^{2}&0&0&(1+\cos\theta)^{2}\end{array}\right]{\color[rgb]{0,0,1}\uwave{.}}$
(7)

On the other hand, in the limit where chiral and helicity states are
essentially the same (ultra-relativistic), $\mathcal{M}_{Z}$ can be directly
derived in a similar manner to (7) above, but introducing the couplings
between the $Z$ boson and the fermions involved, and noticing that some
helicity combinations give $zero$ matrix elements. One finds

$\left|\mathcal{M}_{Z}\right|^{2}\propto\left[\begin{array}[]{cccc}\left(c_{R}^{e}c_{R}^{t}\right)^{2}(1+\cos\theta)^{2}&0&0&\left(c_{R}^{e}c_{L}^{t}\right)^{2}(1-\cos\theta)^{2}\\\
0&0&0&0\\\ 0&0&0&0\\\
\left(c_{L}^{e}c_{R}^{t}\right)^{2}(1-\cos\theta)^{2}&0&0&\left(c_{L}^{e}c_{L}^{t}\right)^{2}(1+\cos\theta)^{2}\end{array}\right]$
(8)

where $c_{L}$ and $c_{R}$ are the coupling constants of the left-chiral and
right-chiral fermions respectively. These calculations already remind one of
the construction of a Choi matrix (eq. 2), and we can easily check that the
obtained matrices satisfy the conditions to be implemented through quantum
channels (cf. Section III). In fact, using the IBM Quantum Lab
(https://quantum-computing.ibm.com/), we constructed quantum channels that
were able to reproduce them, which is the subject of the next section.

###  IV.3  CONSTRUCTING THE QUANTUM CHANNELS

When constructing a quantum channel (Fig. 2), we translate observable values
R, L to the Boolean computational basis given by {0,1}. We can characterise
each possible initial-final helicity state by a string of four digits, the
first two corresponding to the helicity of the initial-state electron and
positron, and the last two corresponding to the helicity of the final-state
quarks. The first application of the Hadamard gate (H) creates an equal
superposition of all the 16 possible combinations of initial-final helicity
states. Then through the quantum interference introduced by the action of
phase and C-NOT gates we enhance the probability amplitudes of selected
states. The final Hadamard gates ‘close’ the superposition and precede the
measurement gates. It is worth noting that quantum channels can approximate a
result with as much precision as needed, at the cost of increasing the number
of computational steps (cf. Solovay-Kitaev theorem Nielsen and Chuang (2010)).

Figure 2: Schematic representation of a quantum channel comprising $n$ qubits,
Hadamard (H), C-NOT ($+$), Phase (P), and measurement gates. Figure 3: Quantum
channel simulating $|\mathcal{M}_{\gamma}|^{2}$ (eq. 7) and related histograms
showing a parity preserving process (simulated using the IMB Quantum Lab
(https://quantum-computing.ibm.com).  Figure 4: Quantum channel simulating
$|\mathcal{M}_{Z}|^{2}$ (eq. 8) and related histograms showing a parity
violating process (simulated using the IMB Quantum Lab (https://quantum-
computing.ibm.com).

Considering the experimental measurements of the
$\mathrm{e}^{+}\mathrm{e}^{-}\rightarrow q\bar{q}$ (where $q$ and $\bar{q}$
are a quark and its anti-quark respectively) cross section from LEP close to
and above $Z$ resonance (from LEP and SLD Collaborations Phy (2006), we see
that the contribution to the cross section from the QED process alone (eq. 7)
presents an evident backward-forward symmetry in the angular distribution that
is lost above the $Z$ resonance as a consequence of the parity violation which
characterises the weak interaction (eq. 8), that is manifested via different
couplings for the left- and right-chiral fermions. Figures 3-4 illustrate a
simple example of how quantum circuits can be used to compute ansatz leading-
order matrix elements for $|\mathcal{M}_{\gamma}|^{2}$ and
$|\mathcal{M}_{Z}|^{2}$, which respect the observed angular distributions
(a)symmetries.

Consider, for example, how the histogram in Fig. 3 is associated with the
elements in matrix (7). A given combination of 0’s and 1’s determines a unique
matrix element according to matrix (6) (as an example, the top-left matrix
element $\left|M_{RL\rightarrow RL}\right|^{2}$ is associated with the
combination "0101"), while the associated probability amplitude gives the
corresponding value according to matrix (7). The parity preserving nature of
the electric interaction (eq. 7) can be seen in Fig. 3 by the fact that, by
substituting each 0 with a 1 and vice-versa in each string, the same graph
would be obtained. On the other hand, in the weak interaction (eq. 8) parity
violation manifests itself via the unequal values of the chiral couplings
$c_{L}$ and $c_{R}$. It follows that, by comparing matrix (8) with matrix (7),
we expect that matrix elements that had equal values in the former, will be
unequal in the latter. This is what we observe in Fig. 4, where we also notice
that substituting each 0 with a 1 and vice-versa in each string would not lead
to the same graph. Adding the interference term
$2\left|\mathcal{M}_{\gamma}\mathcal{M}_{Z}{}^{*}\right|$ would similarly lead
to parity violation at and above the $Z^{0}$ boson resonance.

In both cases the obtained probabilities are in agreement with the theoretical
predictions of eqs. (4-8). It is worth noting that, while the data we used to
construct the channels in the presented example model are not experimental,
there are no barriers to using experimental data in this context. Given the
quantity and high precision of data produced at the LHC and other colliders,
this provides an interesting research direction. The quantum circuits
presented could in fact be optimised by employing, for example, a hybrid
quantum-classical variational approach, where the effect of circuit gates is
modified to match a specific experimental distribution by tunable parameters
optimised by classical computing techniques (cf. Section VII for possible
interpretations of a mismatch between experimental and quantum channel
simulated distributions). Such techniques have been successfully applied to
data analysis at colliders: two recent proposals involve using unsupervised
training of generative models to generate synthetic data of high-energy
physics processes Delgado and Hamilton (2022), and the quantum variational
classifier method to recognise signatures from experimental data from LHCWu
_et al._ (2021). These methods share a common structure, as they consist in
repeatedly producing synthetic ansatz density matrices that are then compared
with experimental data and updated.

In this Section we have illustrated how state-channel duality can be applied
in the context of a simple $2\rightarrow 2$ particle process when considered
at the leading order in perturbation theory. In the following Sections we
describe how this could be generalised to additional processes and higher
orders in perturbation theory.

##  V  CONSTRUCTING A DICTIONARY

We suggest that some useful quantum channel operations, such as channel
concatenation and factorisation, which follow as corollaries of state-channel
dualityBergholm (2015), can be associated with a physical meaning in the field
of particle physics. For the quantum information nomenclature in this section,
we follow ref.Wilde (2017).

a. Serial concatenation submits the quantum state to subsequent quantum
channels. Let
$\mathcal{N}:\mathcal{B}(\mathcal{H}_{A})\rightarrow\mathcal{B}\left(\mathcal{H}_{B}\right)$
and
$\mathcal{M}:\mathcal{B}(\mathcal{H}_{B})\rightarrow\mathcal{B}\left(\mathcal{H}_{C}\right)$
correspond to the first and second quantum channels respectively, with the
respective Kraus operators being $\left\\{\mathrm{N}_{\mathrm{k}}\right\\}$
and $\left\\{\mathrm{M}_{\mathrm{k}}\right\\}$. The output of the first
channel $\mathcal{N}_{A\rightarrow
B}\left(\rho_{A}\right)\equiv\sum_{k}N_{k}\rho_{A}N_{k}^{\dagger}$ is
submitted to the second thus giving:

$\left(\mathcal{M}_{B\rightarrow C}\circ\mathcal{N}_{A\rightarrow
B}\right)\left(\rho_{A}\right)=\sum_{k}M_{k}\mathcal{N}_{A\rightarrow
B}(\rho)M_{k}^{\dagger}=\sum_{k,k^{\prime}}M_{k}N_{k^{\prime}}\rho_{A}N_{k^{\prime}}^{\dagger}M_{k}^{\dagger}$
(9)

In the context of particle physics, this is applicable to modelling amplitudes
for sequential processes such as cascade decays and jets (cf. eq. 1 in
ref.Knowles (1990), and Section VI below) (Fig. 5-a). Notice that the
dimensions of the final states Hilbert space need not match that of the
initial state: an appending channel can be used to encode the additional
degrees of freedom of decay products (cf. point d. below).

b. Parallel concatenation sends a system $A$ through a channel
$\mathcal{N}:\mathcal{B}(\mathcal{H}_{A})\rightarrow\mathcal{B}\left(\mathcal{H}_{C}\right)$
and a system $B$ through a channel
$\mathcal{M}:\mathcal{B}(\mathcal{H}_{B})\rightarrow\mathcal{B}\left(\mathcal{H}_{D}\right)$.
Thus, for an input density operator
$\rho_{AB}\in\mathcal{D}\left(\mathcal{H}_{A}\otimes\mathcal{H}_{B}\right)$,
we have:

$\left(\mathcal{N}_{A\rightarrow C}\otimes\mathcal{M}_{B\rightarrow
D}\right)\left(\rho_{AB}\right)=\sum_{k,k^{\prime}}\left(N_{k}\otimes
M_{k^{\prime}}\right)\left(\rho_{AB}\right)\left(N_{k}\otimes
M_{k^{\prime}}\right)^{\dagger}$ (10)

In the context of particle physics this is applicable where subsystems are
present and each subsystem undergoes such a process that can be regarded as
independent of the state of the other subsystem, i.e. it does not introduce
entanglement. As an example, consider parallel decays (Fig. 5-b).

c. A preparation channel prepares a quantum system $A$ in a given state
$\rho_{A}\in\mathcal{D}(\mathcal{H}_{A})$, and can correspond, for example, to
the state of one of the two incoming beams in Section IV. In Fig. 3-4 above,
the $q[0]$ and $q[1]$ registers, encoding the helicity states of the incoming
electron and positron respectively, are prepared in an equal superposition of
all initial helicity states by applying an Hadamard gate ($H$) on each
register (Fig. 5-c).

d. An appending channel is the parallel concatenation of the identity channel
and a preparation channel, and can correspond, for example, to the state of
second of the two incoming beams in Section IV (Fig. 5-d), as it can be
noticed by the fact that in the simulations leading to Figs. 3 & 4 above each
register was individually prepared and appended to each other.

Figure 5: Examples from the text of: (a) serial concatenation of channels, (b)
parallel concatenation of channels, (c) a preparation channel, (d) an
appending channels and their suggested interpretation in terms of particle
physics processes.

##  VI EXTENSIONS OF THE PRESENT WORK: higher-order corrections

While in the previous sections we applied state-channel duality to amplitudes
corresponding to tree-level Feynman diagrams, we expect that a mapping is also
possible at higher orders in perturbation theory, where quantum corrections
are included.

Consider the Feynman path integral formulation Feynman (1948), which relates
the quantum field theoretic picture (RHS) to a quantum mechanical amplitude
(LHS)

$\left\langle\Phi\left(t_{f}\right)\mid\Phi\left(t_{i}\right)\right\rangle=\int\mathcal{D}[\Phi]\mathrm{e}^{i\mathcal{S}[\Phi]}$
(11)

where $\mathcal{S}$[$\Phi$] is the action, functionally dependent on field
configurations $\Phi$, and $\mathcal{D}[\Phi]$ is the measure of integration.
Notice that the LHS of eq. 11 above is a quantum mechanical amplitude, which
makes it suitable for the application of state-channel duality. We hence
expect, within the limits of applicability of eq. 11, to be able to find an
equivalent unitary operator (or collection thereof) corresponding to a quantum
channel.

We can identify such operator(s) by considering the equivalent S-matrix
formulation. Following the exposition in the classical ref.Peskin and
Schroeder (2019), according to this formulation the transition amplitude
between input and output states of definite momentum is given by

${}_{\text{out
}}\left\langle\mathbf{p}_{1}\mathbf{p}_{2}\cdots\mid\mathbf{k}_{\mathcal{A}}\mathbf{k}_{\mathcal{B}}\right\rangle_{\text{in
}}\equiv\left\langle\mathbf{p}_{1}\mathbf{p}_{2}\cdots|S|\mathbf{k}_{\mathcal{A}}\mathbf{k}_{\mathcal{B}}\right\rangle$
(12)

where $S$ is a unitary operator, i.e. $S^{\dagger}S=1$. It is worth noting
that in quantum field theory, $S$ is also limiting unitary, which means that
it can be decomposed in such a way that ‘in and out states are related by the
limit of a sequence of unitary operators’Peskin and Schroeder (2019).

Following a standard procedure (cf. ref.Peskin and Schroeder (2019), Section
4.5, and specifically eq. 4.73 therein), it is possible to see that this
condition remains valid at any order in perturbation theory. $S$ can in fact
be decomposed into

$S=\mathbf{1}+iT$ (13)

where the $T$ matrix encodes the interactions and is proportional to the
unitary operator

$U\left(t,t_{0}\right)=T\left\\{\exp\left[-i\int_{t_{0}}^{t}dt^{\prime}H_{I}\left(t^{\prime}\right)\right]\right\\}$
(14)

where $H_{I}$ is the interacting Hamiltonian proportional to a small coupling
constant. Higher-order corrections in perturbation theory are calculated
through the power series expansion of the $T$ operator, and amplitudes at each
order tolerate an error that is proportional to the related power in the
coupling constant.

Because of the unitarity of the $S$ matrix, the sum of all the relevant
contributions associated with it (eq. 12 corresponds to the sum over all
connected, amputated Feynman diagrams with the specified number of fields and
vertices) up to a given order in perturbation theory has the form of a quantum
mechanical amplitude, and can thus be interpreted as a quantum channel. Also
notice that because it includes all the possible processes, this is the
relevant amplitude when performing unitarity tests (cf. section VII below for
discussion).

Let us illustrate the above points by considering the radiative corrections in
Fig. 6.

Figure 6: Feynman diagrams displaying vertex, external leg and vacuum
polarisation corrections respectively (adapted from ref.Peskin and Schroeder
(2019)). Figure 7: Bremsstrahlung diagrams (adapted from ref.Peskin and
Schroeder (2019))

While we refer to the classical ref Peskin and Schroeder (2019) (Sections
7.2-3) for a detailed calculation of each contribution, for the purpose of the
present argument it suffices to notice that the inclusion of all the relevant
corrections (fig. 6-7) preserves the quantum-mechanical amplitude nature of
the S-matrix (cf. eq. 12), since ultraviolet divergences cancel out of
expressions for observable quantities such as cross sections, and the infrared
divergences associated to the diagrams in Fig. 6 are cancelled out by adding
the contributions due to the soft Bremsstrahlung (fig. 7), thus allowing for
the application of state-channel duality to the amplitude corresponding to the
sum of all the relevant corrections.

Additionally, there exist cases when the application of state-channel duality
to specific corrections is also possible. As an example, in cases where the
correction corresponds to the inclusion of an additional final-state particle,
and calculations allow for the factorisation of the tree-level result, the
channel composition rules from Section V can applied. As a concrete example,
consider the amplitude associated to the Bremsstrahlung process (fig. 7):

$iM=\bar{u}\left(p^{\prime}\right)\left[M_{0}\left(p^{\prime},p\right)\right]u(p)\cdot\left[e\left(\frac{p^{\prime}\cdot\varepsilon^{*}}{p^{\prime}\cdot
k}-\frac{p\cdot\varepsilon^{*}}{p\cdot k}\right)\right],$ (15)

which is the amplitude for tree-level elastic scattering multiplied by an
extra factor accounting for the emission of the photon (where $k$ and
$\varepsilon^{*}$ are the photon momentum and polarisation vector
respectively) (cf. Section 6.1 of ref.Peskin and Schroeder (2019), and
specifically eq. 6.22 therein). On the quantum computing side, this could be
modelled as a quantum channel corresponding to the amplitude for tree-level
elastic scattering, followed by the application, through serial concatenation,
of an ‘amplitude damping’ channel, that accounts for the extra factor. This
operation effectively embeds the density matrix associated to the lower-order
Feynman diagram into that of the higher-order one by rescaling the relevant
degrees of freedom (thus ‘damping it’) to account for the additional final-
state degrees of freedom. Notice this preserves the unitarity of the overall
process once the degrees of freedom of the additional final-state photon are
also taken into account, and has elsewherePreskill (2018) been used to model
spontaneous emission processes.

##  VII FINAL REMARKS

State-channel duality may prove to be a useful instrument for foundational
tests of quantum mechanics at colliders as it allows us to tackle rarely
explored features of high energy physics systems by analysing their
corresponding quantum channels. By directing our attention not only to a given
density matrix, but also to the corresponding quantum channels, we may be able
to systematically search for properties such as entanglement, rotation
invariant quantities and symmetries. As an example, separability
considerations related to state-channel dualityAntipin (2020) could be applied
to probe the quantum mechanics features in particle physics problems, even at
leading order (cf., e.g., ref.Eckstein and Horodecki (2022)). In fact, since
_"bipartite states are well classified according to correlations therein
Werner (1989); Horodecki  _et al._ (2009); Luo (2008), we may classify
channels by translating the classifications in bipartite states with the
mediation of channel-state duality and investigate the conditions for channels
in order to generate corresponding states with certain correlation
structures."_Jiang, Luo, and Fu (2013).

In addition, it is worth noting that effective field theories that respect the
limiting unitarity condition outlined in Section VI can be modelled as quantum
channels though ‘oracle operators’Nielsen and Chuang (2010) connecting input
to output states. This may have an application, where physics beyond the
Standard Model is expected to be observed, as experimentally probing all
matrix elements through polarisation choices (cf. Section IV-A) would allow
one to construct a theory-independent quantum channel, from which it could
nonetheless be possible to extract information on the underlying theory such
as chirality structure and various forms of symmetry violation. In fact, since
state-channel duality establishes a mapping between the quantum channel and
the S-matrix, and not directly the Hamiltonian, the quantum channel
formulation can be agnostic to the underlying full particle physics
Hamiltonian while outputting a result that is nonetheless compatible with the
underlying dynamics.

Finally, notice that when an experimentally measured density matrix cannot be
turned into a quantum channel, this might be taken to suggest that it cannot
be produced by unitary dynamics alone and, if the traditional quantum
mechanics picture is complete and the perturbative approach applicable at
given energy level, then additional interactions (e.g. with a subsystem) may
be present.

For these reasons, we are hopeful the presented framework will prove to be a
useful tool for the particle physics community, and provide an interesting
addition to the quantum computing methods applied to particle physics problems
Delgado and Hamilton (2022); Wu  _et al._ (2021); Bepari  _et al._ (2022).

##  VIII  ACKNOWLEDGEMENTS

CA is particularly grateful to Malcolm Fairbairn for helpful comments and
discussion. We would like to thank the members of the Oxford ATLAS ‘SM &
beyond’ group, the participants to the conferences ‘Foundational Tests of
Quantum Mechanics at the LHC’ (Merton College, Oxford, 20-22 March 2023) and
‘Quantum Computing for High-Energy Physics’ (Durham, 19-20 September 2023) and
particularly Michał Eckstein, for enlightening discussion. We would also like
to thank two anonymous referees for valuable comments and suggestions that
helped us improve the present work. We acknowledge the use of IBM Quantum
services for this work. The views expressed are those of the authors and do
not reflect the official policy or position of IBM or the IBM Quantum team.
AJB is funded through STFC grants ST/R002444/1 and ST/S000933/1, by the
University of Oxford and by Merton College, Oxford. CA is funded through a
King’s College London NMES studentship. For the purpose of Open Access, the
authors have applied a CC BY public copyright licence to any Author Accepted
Manuscript (AAM) version arising from this submission.

##  Appendix A Choice of basis example: helicity

As an example of choice of basis, the particle (eq. 1) and antiparticle (eq.
2) right ($R$) and left ($L$) handed chiral Dirac spinors satisfying

$\gamma^{5}u_{R}=+u_{R},\;\;\;\;\;\;\gamma^{5}u_{L}=-u_{L},$ (16)

and

$\gamma^{5}v_{R}=-v_{R},\;\;\;\;\;\;\gamma^{5}v_{L}=+v_{L},$ (17)

being eigenstates of the unitary matrix

$\gamma^{5}\equiv
i\gamma^{0}\gamma^{1}\gamma^{2}\gamma^{3}=\left(\begin{array}[]{llll}0&0&1&0\\\
0&0&0&1\\\ 1&0&0&0\\\
0&1&0&0\end{array}\right)=\left(\begin{array}[]{ll}0&I\\\
I&0\end{array}\right)$ (18)

provide a good alternative for representing relativistic spin-$\frac{1}{2}$
fermions, as any such state can be prepared on a quantum circuit. We note that
generally qubit quantum computers implement unitary operations, e.g. within
IBM Quantum labIBM , any unitary matrix can be turned into a unitary gate with
UnitaryGate() (cf., as a possible protocol, initialising eigenstates of
$\gamma^{5}$, introducing relative phase differences though phase gates
corresponding to relative amplitudes, applying a depolarising channel to set
all amplitudes to zero, thus obtaining a unitary quantum channel that, when
run in reverse, outputs the desired state; cf., for example, ref.Bergholm
(2015)).

However, because chirality does not have a straightforward physical
interpretation, helicity spinors, which we used in the example model presented
in Section IV,

$u_{\uparrow}(p)=N\left(\begin{array}[]{c}c\\\ se^{i\phi}\\\
\frac{\mathrm{p}}{E+m}c\\\
\frac{\mathrm{p}}{E+m}se^{i\phi}\end{array}\right),u_{\downarrow}(p)=N\left(\begin{array}[]{c}-s\\\
ce^{i\phi}\\\ \frac{\mathrm{p}}{E+m}s\\\
-\frac{\mathrm{p}}{E+m}ce^{i\phi}\end{array}\right),$ (19)

$v_{\uparrow}(p)=N\left(\begin{array}[]{c}\frac{\mathrm{p}}{E+m}s\\\
-\frac{\mathrm{p}}{E+m}ce^{i\phi}\\\ -s\\\
ce^{i\phi}\end{array}\right),v_{\downarrow}(p)=N\left(\begin{array}[]{c}\frac{\mathrm{p}}{E+m}c\\\
\frac{\mathrm{p}}{E+m}se^{i\phi}\\\ c\\\ se^{i\phi}\end{array}\right),$ (20)

(where $p$ is particle’s 3-momentum, $N=\sqrt{E+m}$ is a normalization factor
with $E$ and $m$ being particle’s energy and mass respectively,
$s=\sin\frac{\theta}{2}$, $c=\cos\frac{\theta}{2}$ with azimuthal angle
$\theta$ defined as in Fig.1 and polar angle $\phi$), provide an even more
useful basis. The helicity operator commutes with the Dirac Hamiltonian, has
the straightforward physical interpretation of normalised spin along particle
flight direction, and in the ultra-relativistic (i.e. massless) limit it is
equivalent to chirality.

## References

  * Feynman (1982) R. P. Feynman, “Simulating physics with computers,” International Journal of Theoretical Physics 21, 467–488 (1982).
  * Nielsen and Chuang (2010) M. Nielsen and I. Chuang,  _Quantum computation and quantum information_ (Cambridge university press, 2010).
  * Barr, Caban, and Rembieliński (2022) A. J. Barr, P. Caban, and J. Rembieliński, “Bell-type inequalities for systems of relativistic vector bosons,” (2022), arXiv:2204.11063 [quant-ph] .
  * Fabbrichesi, Floreanini, and Panizzo (2021) M. Fabbrichesi, R. Floreanini, and G. Panizzo, “Testing Bell Inequalities at the LHC with Top-Quark Pairs,” Physical Review Letters 127 (2021), 10.1103/physrevlett.127.161801.
  * Afik and de Nova (2021) Y. Afik and J. R. M. de Nova, “Entanglement and quantum tomography with top quarks at the LHC,” The European Physical Journal Plus 136 (2021), 10.1140/epjp/s13360-021-01902-1.
  * Ashby-Pickering, Barr, and Wierzchucka (2022) R. Ashby-Pickering, A. J. Barr, and A. Wierzchucka, “Quantum state tomography, entanglement detection and bell violation prospects in weak decays of massive particles,” (2022), arXiv:2209.13990 [quant-ph] .
  * Barr (2022) A. J. Barr, “Testing Bell inequalities in Higgs boson decays,” Physics Letters B 825, 136866 (2022).
  * Möttönen  _et al._ (2004) M. Möttönen, J. J. Vartiainen, V. Bergholm, and M. M. Salomaa, “Quantum circuits for general multiqubit gates,” Physical Review Letters 93 (2004), 10.1103/physrevlett.93.130502.
  * Delgado and Hamilton (2022) A. Delgado and K. E. Hamilton, “Unsupervised Quantum Circuit Learning in High Energy Physics,” (2022), arXiv:2203.03578 [quant-ph] .
  * Wu  _et al._ (2021) S. L. Wu _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 48, 125003 (2021).
  * Gustafson  _et al._ (2022) G. Gustafson, S. Prestel, M. Spannowsky, and S. Williams, “Collider events on a quantum computer,” (2022), arXiv:2207.10694 [hep-ph] .
  * Jiang, Luo, and Fu (2013) M. Jiang, S. Luo, and S. Fu, “Channel-state duality,” Phys. Rev. A 87, 022310 (2013).
  * Klco, Roggero, and Savage (2022) N. Klco, A. Roggero, and M. J. Savage, “Standard model physics and the digital quantum revolution: thoughts about the interface,” Reports on Progress in Physics 85, 064301 (2022).
  * Bauer  _et al._ (2023a) C. W. Bauer, Z. Davoudi, N. Klco, and M. J. Savage, “Quantum simulation of fundamental particles and forces,” Nature Rev. Phys. 5, 420–432 (2023a).
  * Bauer  _et al._ (2023b) C. W. Bauer _et al._ , “Quantum simulation for high-energy physics,” PRX Quantum 4, 027001 (2023b).
  * Bañuls  _et al._ (2020) M. C. Bañuls _et al._ , “Simulating lattice gauge theories within quantum technologies,” The European Physical Journal D 74 (2020), 10.1140/epjd/e2020-100571-8.
  * Bertlmann and Krammer (2008) R. A. Bertlmann and P. Krammer, “Bloch vectors for qudits,” Journal of Physics A: Mathematical and Theoretical 41, 235303 (2008).
  * Dogra, Dorai, and Arvind (2018) S. Dogra, K. Dorai, and Arvind, “Majorana representation, qutrit Hilbert space and NMR implementation of qutrit gates,” Journal of Physics B: Atomic, Molecular and Optical Physics 51, 045505 (2018).
  * Weyl (2014) H. Weyl,  _The theory of groups and Quantum Mechanics_ (Martino Publishing, 2014).
  * Gell-Mann (1962) M. Gell-Mann, “Symmetries of baryons and mesons,” Phys. Rev. 125, 1067–1084 (1962).
  * Ekert (2022) A. Ekert, “’Introduction to quantum information’ course page,” Available at https://zhenyucai.com/post/intro_to_qi/ (2022).
  * Martens, Ralston, and Takaki (2017) J. C. Martens, J. P. Ralston, and J. D. T. Takaki, “Quantum tomography for collider physics: illustrations with lepton-pair production,” The European Physical Journal C 78 (2017), 10.1140/epjc/s10052-017-5455-8.
  * Thomson (2021) M. Thomson,  _Modern Particle Physics_ (Cambridge University Press, 2021).
  * Sokolov and Ternov (1963) A. A. Sokolov and I. M. Ternov, “On polarization and spin effects in the theory of synchrotron radiation,” Dokl. Akad. Nauk SSSR 153, 1052–1054 (1963).
  * Sobloher (2012) B. Sobloher, “Polarisation and polarimetry at HERA,” (2012), arXiv:1201.3836 [physics.ins-det] .
  * Blondel  _et al._ (2019) A. Blondel _et al._ , “Polarization and centre-of-mass energy calibration at FCC-ee,” (2019), arXiv:1909.12245 [physics.acc-ph] .
  * Phy (2006) “Precision electroweak measurements on the Z resonance,” Physics Reports 427, 257–454 (2006).
  * Bergholm (2015) V. Bergholm, “A diagrammatic approach to map-state and channel-state dualities,” (2015), arXiv:1509.08339 [quant-ph] .
  * Wilde (2017) M. Wilde,  _Quantum Information theory_ (Cambridge University Press, 2017).
  * Knowles (1990) I. Knowles, “A linear algorithm for calculating spin correlations in hadronic collisions,” Computer Physics Communications 58, 271–284 (1990).
  * Feynman (1948) R. P. Feynman, “Space-time approach to non-relativistic quantum mechanics,” Rev. Mod. Phys. 20, 367–387 (1948).
  * Peskin and Schroeder (2019) M. E. Peskin and D. V. Schroeder,  _An introduction to quantum field theory_ (CRC Press, 2019).
  * Preskill (2018) J. Preskill, “Lecture notes for PH219/CS219: Quantum information chapter 3, California Institute of Technology,”  (2018).
  * Antipin (2020) K. V. Antipin, “Channel-state duality and the separability problem,” (2020), arXiv:1909.13309 [quant-ph] .
  * Eckstein and Horodecki (2022) M. Eckstein and P. Horodecki, “Probing the limits of quantum theory with quantum information at subnuclear scales,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 478 (2022), 10.1098/rspa.2021.0806.
  * Werner (1989) R. F. Werner, “Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model,” Phys. Rev. A 40, 4277–4281 (1989).
  * Horodecki  _et al._ (2009) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, “Quantum entanglement,” Reviews of Modern Physics 81, 865–942 (2009).
  * Luo (2008) S. Luo, “Using measurement-induced disturbance to characterize correlations as classical or quantum,” Phys. Rev. A 77, 022301 (2008).
  * Bepari  _et al._ (2022) K. Bepari, S. Malik, M. Spannowsky, and S. Williams, “Quantum walk approach to simulating parton showers,” Phys. Rev. D 106, 056002 (2022).
  * (40) https://quantum-computing.ibm.com/.