# Unitarization of Pseudo-Unitary Quantum Circuits in the S-matrix Framework

Dennis Lima1 and Saif Al-Kuwari1
1 Qatar Center for Quantum Computing (QC2), Qatar
<EMAIL_ADDRESS>and<EMAIL_ADDRESS>
###### Abstract

Pseudo-unitary circuits are recurring in both S-matrix theory and analysis of
No-Go theorems. We propose a matrix and diagrammatic representation for the
operation that maps S-matrices to T-matrices and, consequently, a unitary
group to a pseudo-unitary one. We call this operation “partial inversion” and
show its diagrammatic representation in terms of permutations. We find the
expressions for the deformed metrics and deformed dot products that preserve
physical constraints after partial inversion. Subsequently, we define a
special set that allows for the simplification of expressions containing
infinities in matrix inversion. Finally, we propose a renormalized-growth
algorithm for the T-matrix as a possible application. The outcomes of our
study expand the methodological toolbox needed to build a family of pseudo-
unitary and inter-pseudo-unitary circuits with full diagrammatic
representation in three dimensions, so that they can be used to exploit
pseudo-unitary flexibilization of unitary No-Go Theorems and renormalized
circuits of large scattering lattices.

Keywords: pseudo-unitary, unitarization, S-matrix

##  1 Introduction

Over the years, pseudo-unitary quantum mechanics has rapidly grown from the
interest in pseudo-Hermitian Hamiltonians [1] in early 2000 to describe PT-
symmetric quantum systems [2], and recently to simulate and experiment pseudo-
unitary quantum cloning and quantum deletion in optical circuits [3, 4].
Furthermore, controlling Hilbert-space inner products [5] for fast quantum
algorithms in PT-symmetric systems [6, 7] is a direct implication of knowing
how to manipulate pseudo-unitary groups. A consistent and experiment-friendly
explanation of how pseudo-unitary algebras are built based on transformations
of Hilbert spaces is given by Karuvade _et al._ [5].

Another motivation to study pseudo-unitary circuits comes from the S-matrix
theory. The general solution $\psi(x)$ to the Schrödinger equation for a one-
dimensional wave scattering across a localized potential is equation (1), with
$\psi_{ab}(x)$ the solution between $x=a$ and $x=b$, the region where the
potential is non-zero.

$\displaystyle\psi(x)=\begin{cases}\psi_{00}e^{ikx}+\psi_{01}e^{-ikx},&\quad
x<a;\\\ \psi_{ab}(x),&\quad a<x<b;\\\ \psi_{10}e^{ikx}+\psi_{11}e^{-ikx}&\quad
x>b.\end{cases}$ (1)

$\displaystyle\begin{pmatrix}\psi_{01}\\\
\psi_{10}\end{pmatrix}=S\begin{pmatrix}\psi_{00}\\\ \psi_{11}\end{pmatrix}.$
(2)

If the S-matrix of a system is unitary (equation (3)), then the transfer
matrix (equation (4)) derived from it is known to be pseudo-unitary with
respect to either the $\sigma_{3}\otimes I_{m}$ metric [8] or the
$I_{m}\otimes\sigma_{3}$ metric [9], where $\sigma_{3}=|0\rangle\langle
0|-|1\rangle\langle 1|$ is a Pauli matrix and $I_{m}$ is the identity matrix
with dimensions $(m,m)$. In equations (3) and (4), $t$, $t^{\prime}$, $r$ and
$r^{\prime}$ are Hermitian matrices [8].

$\displaystyle S$ $\displaystyle=\begin{pmatrix}t&r\\\
-r^{\prime}&t^{\prime}\end{pmatrix},\quad\text{with}\quad
tt^{\prime}+rr^{\prime}=I_{m};$ (3) $\displaystyle T$
$\displaystyle=\begin{pmatrix}T^{++}&T^{+-}\\\
T^{-+}&T^{--}\end{pmatrix}=\begin{pmatrix}(t^{\prime})^{-1}&r(t^{\prime})^{-1}\\\
(t^{\prime})^{-1}r^{\prime}&(t^{\prime})^{-1}\end{pmatrix}.$ (4)

The S-matrix theory was initially proposed by Wheeler [10] to study subatomic
particles. It provides maximal analyticity, but cannot compute interparticle
forces [11]. In many-body systems of electrons, the S-matrix and, more often,
the T-matrix, are alternative formalisms to analyse critical behaviour,
conductance and localization in scattering processes [12].

Feynman diagrams [13] or, equivalently, quantum circuits, provide us with a
visual interpretation of both S and T matrices by rearranging the arrows that
represent states in a scattering process (figure 1) [14]. Lattices are grown
by juxtaposing these diagrams side by side. This ease of growing periodic
lattices with T-matrices may motivate the preference to use T-matrix formalism
for periodic potentials [15].

Figure 1: S-matrix and T-matrix scattering diagrams for two-component spinors.

Recent studies towards experimental control of pseudo-unitary operators focus
on PT-symmetric systems, but lack mentions of examples from the T-matrix
theory [3, 5]. As consequence, the pseudo-unitarity of the T-matrix formalism
is restricted to analytical studies for calculation of conductance, specially
in conductors like graphene [9].

The design of inter-unitary protocols has the advantage to allow for the use
of a matrix product in place of a tensor product for calculation of physical
properties like the Landauer conductance in equation (5). This possibility is
clarified by the diagrammatic representation of scattering lattices and the
change of direction of diagrammatic growth after solving a linear system of
equations [9].

$\displaystyle
G=\frac{4e^{2}}{h}\mathrm{Tr}\left\\{[(T^{--})^{\dagger}T^{--}]^{-1}\right\\}$
(5)

Although such protocols that take advantage of matrix products in different
pseudo-unitary groups have been used [9], the formal transformation that
allows it and its mathematical properties in the scope of pseudo-unitary
groups are not well understood.

We extend it and provide tools to build inter-unitary circuits, i.e., circuits
that make use of more than one pseudo-unitary group, by defining the matrix
representation for the transformation algorithm that performs a step-wise
matrix inversion necessary to convert between S-matrix and T-matrix
formalisms. We show that it contains a representation of the symmetric group,
can be used to transit between pseudo-unitary groups, leads to the definition
of a special set useful to manipulate singular matrices and is equipped with
the diagrammatic representation to further conceive a renormalized quantum
circuit growth.

###  1.1 Contributions

The contributions of this paper can be summarized as follows:

  1. 1.

proposition of partial inversion algorithm with an operator and diagrammatic
representations;

  2. 2.

proposition of a heuristic diagram that maps a family of pseudo-unitary groups
to one another through partial inversion;

  3. 3.

definition of the effect of partial inversion on dot product metric with
conservation of dot product;

  4. 4.

demonstration of the application of partial inversion to invert and invert
back matrices with zeros;

  5. 5.

demonstration of the application of partial inversion to grow the dimensions
of a circuit, while the dimension of the matrix representing the circuit is
kept constant.

###  1.2 Organization

The rest of the paper is organized as follows: In section 2, we motivate the
study of partial inversion in S-matrix theory with the effect of Legendre
transform in dot products. Subsequently, in section 3, we reformulate partial
inversion as an operator in terms of matrices and propose a diagrammatic
representation that is consistent with representations of the symmetric group.
We follow to defining heuristically the diagram that solves the deformed
algebra equation of non-commutativity of partial inversion and we show that it
maps a family of $SU(2,\eta)$ groups to one another. Next, we present a
unitary representation of a sequence of partial inversions and we explore how
partial inversion deforms the dot product by deforming the conservation metric
$g$. Finally, we discuss two potential applications in section 4. The first
one is the use of partial inversion to invert matrices forth and back even in
the presence of inverted zeros (non-finite matrices). The second one is the
use of partial inversion as a means to extend unitary circuits in two
dimensions without changing the dimensions of the matrix that represents the
circuit. We call this approach ”renormalized growth”. We conclude the paper in
section 5 with a few final remarks.

##  2 Methods

In this section we furnish the dot product conservation equation in a set of
Legendre-transformed functions as a motivational model. Subsequently, in the
light of the Gauss-Jordan elimination algorithm, we introduce a typical linear
equation solver in an indexed fashion. This solver, as the core algorithm of
this paper, is reformulated in section 3 as an operator given by a non-linear
composition of matrices. Next, its role in transforming between pseudo-unitary
groups and group-like objects is investigated.

###  2.1 Dot product conservation formula in sets of Legendre-transformed
functions

Legendre transforms are vastly used in Lagrangian Mechanics and Thermodynamics
[16] to design a formalism that is analytically appropriate for the variables
of a system. Consider $U=U(V_{1},V_{2},\cdots,V_{n})$ a function of $n$
variables $V_{i}$ and $U[1]=U-p_{1}V_{1}$ its Legendre transform with respect
to $V_{i}$.

$\displaystyle U$ $\displaystyle=\sum_{i=1}^{n}p_{i}V_{i}$ (6)
$\displaystyle\mathrm{d}U-p_{1}\mathrm{d}V_{1}$
$\displaystyle=\sum_{i=2}^{n}p_{i}\mathrm{d}V_{i}$ (7)
$\displaystyle\mathrm{d}U-p_{1}\mathrm{d}V_{1}-V_{1}\mathrm{d}p_{1}$
$\displaystyle=-V_{1}\mathrm{d}p_{1}+\sum_{i=2}^{n}p_{i}\mathrm{d}V_{i}$ (8)
$\displaystyle\mathrm{d}(U-p_{1}V_{1})$
$\displaystyle=-V_{1}\mathrm{d}p_{1}+\sum_{i=2}^{n}p_{i}\mathrm{d}V_{i}$ (9)
$\displaystyle\mathrm{d}U[1]$
$\displaystyle=-V_{1}\mathrm{d}p_{1}+\sum_{i=2}^{n}p_{i}\mathrm{d}V_{i}.$ (10)

In matrix formalism, equations (6) and (10) can be expressed as dot products
(equation (11)) between the vector $\mathbf{V}_{U}$ and its Legendre conjugate
$\mathbf{V}^{\star}_{U}$ under the metric $g_{U}$.

$\displaystyle U$ $\displaystyle=\mathbf{V}_{U}^{\star}g_{U}\mathbf{V}_{U},$
(11) $\displaystyle\text{with}\quad\mathbf{V}_{U}$
$\displaystyle=\begin{bmatrix}V_{1}\\\ V_{2}\\\ \vdots\\\
V_{n}\end{bmatrix},\quad\mathbf{V}_{U}^{\star}=\begin{bmatrix}p_{1}&p_{2}&\cdots&p_{n}\end{bmatrix}.$
(12)

Conservation of the dot product $U$ implies that $\mathrm{d}U$ in equation
(13) is zero.

$\displaystyle\mathrm{d}U=\mathbf{V}_{U}^{\star}g_{U}\mathrm{d}\mathbf{V}_{U}$
(13)

A Legendre transform over $U$ swaps entries between $\mathbf{V}_{U}$ and
$\mathbf{V}^{\star}_{U}$, also changing the metric $g_{U}$ and resulting in a
new dot product. The new metric (equation (14)) due to the Legendre transform
over the $i$th variable of the vector $\mathbf{V}_{U}$ is obtained by
extraction of the signs that accompany the products $p_{i}V_{i}$.

$\displaystyle g_{U[n]}=(I-2|n\rangle\langle n|)g_{U}$ (14)

The problem of a matrix representation of the transformation that maps a
unitary operator to an $\eta$-pseudo-unitary one is the focus of this study.
The reappearance of equation (14), the dot product metric, is for now the main
connection between the Legendre transform and the pseudo-unitary-metric
transform to be presented in section 3.3.

###  2.2 Step-wise linear equation solver

Gauss-Jordan elimination is a traditional, robust, and still largely used
matrix inversion algorithm due to its parallelization capability [17]. It
involves a parallel application of a sequence of row summations and row
multiplications by constants in both a sample matrix and the identity matrix
until the identity switches place with the sample-matrix. This way the matrix
inverse is found.

The Gauss-Jordan elimination algorithm is a type of linear equation solver for
a system of the form of equation (15), where $\mathbf{x}$ is usually a vector
of variables to be solved for and $\mathbf{v}$ is a vector of independent
variables or constants. They are related by the coefficient matrix $Z$,
whereas the solved equation (16) relates $\mathbf{v}$ to $\mathbf{x}$ by the
inverse of $Z$.

$\displaystyle\mathbf{v}$ $\displaystyle=Z\mathbf{x}$ (15)
$\displaystyle\mathbf{x}$ $\displaystyle=Z^{-1}\mathbf{v}$ (16)

The method for solving the system for only one entry of $\mathbf{x}$ involves
isolation of a variable in a row of the resulting system and its subsequent
substitution in other rows. This algorithm $\hat{Y}_{ik}$ is summarized by a
sequence of four transformations throughout a matrix $Z$ (equations (17)
through (20)), with $i,k$ fixed and $\sigma,\rho$ changing. The $i$th entry of
the output-vector is swapped with the $k$th entry of the input-vector in the
process.

$\displaystyle Z_{ik}$ $\displaystyle\to Z_{ik}^{-1};$ (17) $\displaystyle
Z_{i\sigma}$ $\displaystyle\to-Z_{ik}^{-1}Z_{i\sigma},\quad\text{for
all}\quad\sigma\neq k;$ (18) $\displaystyle Z_{\rho k}$ $\displaystyle\to
Z_{\rho k}Z_{ik}^{-1},\quad\text{for all}\quad\rho\neq i;$ (19) $\displaystyle
Z_{\rho\sigma}$ $\displaystyle\to Z_{\rho\sigma}-Z_{\rho
k}Z_{ik}^{-1}Z_{i\sigma},\quad\text{for all}\quad\sigma\neq k,\rho\neq i.$
(20)

This well-known procedure is the core algorithm to transit between a family of
pseudo-unitary groups. The primarily numerical approach of equations (17)
through (20) was reformulated in a semi-analytical fashion for clarity,
followed by a deeper study of its representations, potential applications and
experimental implications.

##  3 Results and Discussion

###  3.1 Correspondence between partial inversion, permutation and total
matrix inversion

We use $\hat{Y}_{ik}$ to represent an application of equations (17) through
(20), and adopt the name ”partial inversion” to refer to it throughout this
paper, since it can be used to invert matrices. Its matrix representation is
obtained from a sum of deformed detachments, as illustrated by the matrix
grids in equation (27) that represent the adding terms of equation (28) for
the case of $\hat{Y}_{22}$ over a square matrix of dimension $4$. The grids
take $\cdot$ for unchanged elements, $\bullet$ for non-zero changed elements
and empty entries for zeros.

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$\displaystyle\hat{Y}_{ik}Z$
$\displaystyle=Z+ZC_{kk}-C_{ii}Z+C_{ii}ZC_{kk}-ZC_{ik}^{\mathrm{T}}Z,$ (28)
$\displaystyle\text{with}\quad C_{ik}=\frac{|i\rangle\langle k|}{\langle
i|Z|k\rangle}.$

If $i=k$, equation (28) becomes (29), which reduces to the operator defined in
equation (30), with the appearance of equation (32), notoriously similar to
the Yang-Baxter equation from Braid Theory [18], except for the transposition.

$\displaystyle\hat{Y}_{ii}Z=Z+ZC_{ii}-C_{ii}Z+C_{ii}ZC_{ii}-ZC_{ii}^{\mathrm{T}}Z,$
(29) $\displaystyle\text{with}\quad C_{ii}=\frac{|i\rangle\langle i|}{\langle
i|Z|i\rangle}.$

$\displaystyle\hat{Y}_{ii}$
$\displaystyle=\hat{I}+\widehat{[(\quad),C_{ii}]}+\widehat{[(\quad),C_{ii}]}_{\star},$
(30) $\displaystyle\text{with}\quad C_{ii}$
$\displaystyle=\frac{|i\rangle\langle i|}{\langle i|(\quad)|i\rangle},$
$\displaystyle[A,B]$ $\displaystyle=AB-BA,$ (31) $\displaystyle[A,B]_{\star}$
$\displaystyle=BAB-AB^{\mathrm{T}}A.$ (32)

Automatically from the permutation of input and output vectors, any sequence
of $\hat{Y}_{ii}$ for all $i$ in a matrix leads to its total inverse. The
occurrence of permutations is exemplified in equation (33) for 2-vectors,
where $\pi_{1}$ is a permutation matrix.

$\displaystyle\begin{pmatrix}x\\\ y\end{pmatrix}$
$\displaystyle=Z\begin{pmatrix}a\\\ b\end{pmatrix},$ (33)
$\displaystyle\begin{pmatrix}x\\\ y\end{pmatrix}$
$\displaystyle=(Z\pi_{1})\begin{pmatrix}b\\\ a\end{pmatrix},$
$\displaystyle\begin{pmatrix}a\\\ y\end{pmatrix}$
$\displaystyle=\hat{Y}_{00}(Z)\begin{pmatrix}x\\\ b\end{pmatrix}.$

The partial inversion group $G(\hat{Y}_{ik},n)$ of $n$-dimensional square
matrices contains and is larger than the general linear group $GL(n)$, since
$G(\hat{Y}_{ik})$ contains the operation of total matrix inverse. The group
$G(\hat{Y}_{ik})$ also contains representations of the symmetric group $S(n)$
and the Unitary group $U(n)$ (section 3.4). The relations that define
generators of the symmetric group are defined in equations (34) through (36)),
with a chained cycle of indices, such as $\hat{Y}_{i,i+1}$,
$\hat{Y}_{i+2,i+1}$, $\hat{Y}_{i+2,i+3}$, $\cdots$, as generators. The
commutation in equation (36) expresses that permutations that are not chained
are independent of order.

$\displaystyle\hat{Y}_{ik}\hat{Y}_{ik}$ $\displaystyle=\hat{I},$ (34)
$\displaystyle\hat{Y}_{i-1,i}\hat{Y}_{i+1,i}\hat{Y}_{i-1,i}$
$\displaystyle=\hat{Y}_{i+1,i}\hat{Y}_{i-1,i}\hat{Y}_{i+1,i},$ (35)
$\displaystyle[\hat{Y}_{ik},\hat{Y}_{j\ell}]$
$\displaystyle=0\,\iff\,|i-j|>0,\,|k-\ell|>0.$ (36)

The correct partial inversion that maps an S-matrix to a T-matrix depends on
the conventions used for both S-matrix and T-matrix. From the first
presentation of $S$ in equation (3), other possible conventions use $r$ in
place of $t$, or $r^{\prime}$ instead of $-r^{\prime}$ (cf. [9]).

In terms of $\hat{Y}_{ik}$ over the transfer matrix $T$, the $SU(2)$ S-matrix
is read as equation (37).

$\displaystyle S$ $\displaystyle=\hat{Y}_{11}T$ (37)
$\displaystyle=T+TC_{11}-C_{11}T+C_{11}TC_{11}-TC_{11}^{\mathrm{T}}T$
$\displaystyle=\left[\begin{matrix}\frac{1}{t^{*}}&\frac{r}{t^{*}}\\\
\frac{r^{*}}{t^{*}}&\frac{1}{t^{*}}\end{matrix}\right]+\left[\begin{matrix}0&r\\\
0&1\end{matrix}\right]-\left[\begin{matrix}0&0\\\
r^{*}&1\end{matrix}\right]+\left[\begin{matrix}0&0\\\
0&t^{*}\end{matrix}\right]-\left[\begin{matrix}\frac{rr^{*}}{t^{*}}&\frac{r}{t^{*}}\\\
\frac{r^{*}}{t^{*}}&\frac{1}{t^{*}}\end{matrix}\right]$
$\displaystyle=\begin{pmatrix}t&r\\\ -r^{*}&t^{*}\end{pmatrix}.$

Equations (38) and (39) exemplify other applications of partial inversion on
the $SU(2)$ S-matrix. For higher dimensions, since the S and T-matrices are
composed by four blocks of matrices, equations (17) through (20) should be
used with the appropriate order of matrix multiplication, using the blocks in
place of the entry $Z$.

$\displaystyle A$
$\displaystyle=\hat{Y}_{01}\hat{Y}_{00}S=\frac{1}{r}\begin{pmatrix}-1&t^{*}\\\
-t&1\end{pmatrix},$ (38) $\displaystyle B$
$\displaystyle=\hat{Y}_{01}S=\frac{1}{r}\begin{pmatrix}-t&1\\\
-1&t^{*}\end{pmatrix}.$ (39)

An example closer to experimental realization is illustrated by the matrix
representation of the half-wave plate, a gadget that rotates the polarization
of the electromagnetic oscillation planes of a photon by an angle $2\theta$.
This gadget, represented by equation (40), reproduces the Hadamard gate in
polarization space at $\theta=\pi/8$ [19]. Equation (41) shows an example of a
$\sigma_{3}$-pseudo-unitary version of it.

$\displaystyle C(\theta)$
$\displaystyle=\begin{bmatrix}\cos(2\theta)&\sin(2\theta)\\\
\sin(2\theta)&-\cos(2\theta)\end{bmatrix},$ (40) $\displaystyle
C^{\prime}(\theta)=\hat{Y}_{11}C(\theta)$
$\displaystyle=\begin{bmatrix}\sec(2\theta)&-\tan(2\theta)\\\
\tan(2\theta)&-\sec(2\theta)\end{bmatrix}.$ (41)

Similarly to how the $X,Y,Z$ gates are generators of the the Pauli group as
both unitary and Hermitian matrices, $W$ in equation (42) is the general
solution to the constraints for a $\sigma_{3}$-pseudo-Hermitian and
$\sigma_{3}$-pseudo-unitary matrix.

$\displaystyle
W(z,\theta)=\begin{pmatrix}\sqrt{1+z^{2}}&z\mathrm{e}^{\mathrm{i}\theta}\\\
-z\mathrm{e}^{\mathrm{i}\theta}&-\sqrt{1+z^{2}}\end{pmatrix},\quad\mathrm{with}\quad
z,\,\theta\in\mathbb{R}.$ (42)

###  3.2 Diagrammatic representation

Since permutations have a plane diagrammatic representation [18], a natural
extension to partial inversions as permutations through graph time requires an
extra dimension. In the proposed representation, we omit the straight
(unaltered) strands common to permutation and braid diagrams to ease the
tridimensional visualization. The bottom tile is the initial matrix, with
permutations growing upwards.

---
$\displaystyle\leftrightarrow$ $\displaystyle\begin{pmatrix}x\\\
y\end{pmatrix}=M\begin{pmatrix}a\\\ b\end{pmatrix}.$ (44)

$\displaystyle\begin{tabular}[]{c}\includegraphics{tow12}\end{tabular}=M\hat{\pi}_{1},$
(46)

$\displaystyle\begin{tabular}[]{c}\includegraphics{tow32}\end{tabular}=\hat{\pi}_{1}M,$
(48)

$\displaystyle\begin{tabular}[]{c}\includegraphics{back}\end{tabular}=\hat{Y}_{00}M,$
(50)

---
$\displaystyle=\hat{Y}_{11}M,$ (52)
---
$\displaystyle=\hat{Y}_{01}M,$ (54)
---
$\displaystyle=\hat{Y}_{10}M.$ (56)

Each pair of parallel or crossing planes is a commutative pair, therefore they
can be applied simultaneously.

$\displaystyle\begin{tabular}[]{c}\includegraphics{tow11}\end{tabular}=\begin{tabular}[]{c}\includegraphics{tow12}\end{tabular}$
(59)
$\displaystyle\leftrightarrow\quad\hat{Y}_{11}\hat{Y}_{00}\hat{Y}_{01}\hat{Y}_{00}\hat{Y}_{10}M=M\hat{\pi}_{1},$
(60)

$\displaystyle\begin{tabular}[]{c}\includegraphics{tow31}\end{tabular}=\begin{tabular}[]{c}\includegraphics{tow32}\end{tabular}$
(63)
$\displaystyle\leftrightarrow\quad\hat{Y}_{11}\hat{Y}_{00}\hat{Y}_{10}\hat{Y}_{11}\hat{Y}_{01}M=\hat{\pi}_{1}M,$
(64)

$\displaystyle\begin{tabular}[]{c}\includegraphics{tow21}\end{tabular}=\begin{tabular}[]{c}\includegraphics{tow22}\end{tabular}$
(67)
$\displaystyle\leftrightarrow\quad\hat{Y}_{11}\hat{Y}_{00}\hat{Y}_{10}\hat{Y}_{01}M=\hat{\pi}_{1}M\hat{\pi}_{1},$
(68)

$\displaystyle\begin{tabular}[]{c}\includegraphics{tow41}\end{tabular}=\begin{tabular}[]{c}\includegraphics{tow42}\end{tabular}=\begin{tabular}[]{c}\includegraphics{tow53}\end{tabular}$
(72)
$\displaystyle\leftrightarrow\quad\hat{\pi}_{1}\hat{Y}_{11}(\hat{\pi}_{1}M)=[\hat{Y}_{00}(M\hat{\pi}_{1})]\hat{\pi}_{1}=\hat{Y}_{01}M,$
(73)

$\displaystyle\begin{tabular}[]{c}\includegraphics{tow51}\end{tabular}=\begin{tabular}[]{c}\includegraphics{tow52}\end{tabular}=\begin{tabular}[]{c}\includegraphics{tow43}\end{tabular}$
(77)
$\displaystyle\leftrightarrow\quad\hat{\pi}_{1}\hat{Y}_{00}(\hat{\pi}_{1}M)=[\hat{Y}_{11}(M\hat{\pi}_{1})]\hat{\pi}_{1}=\hat{Y}_{10}M.$
(78)

###  3.3 Deformed algebra of pseudo-unitary metric operator

As exemplified in the S-matrix case, the application of partial inversion to a
matrix entry of a matrix representation of an $\eta_{0}$-pseudo-unitary group
maps it to a different, $\eta_{1}$-pseudo-unitary group. This is a direct
consequence of the non-commutativity of some partial inversions with a
similarity transformation $\hat{\eta}$. Equation (79) fully describes the
problem of defining $\eta_{1}$ in terms of $\eta_{0}$ for an
$\eta_{0}$-pseudo-unitary matrix $Z$. In contrast, the sequence of partial-
inversions that amounts to total inversion $\hat{y}_{1}$ commutes with
similarity transformations (equation (80)). Here
$\hat{\kappa}_{1}Z=Z^{\dagger}$.

$\displaystyle\hat{Y}_{ik}\hat{y}_{1}\hat{\kappa}_{1}\hat{\eta}_{0}U$
$\displaystyle=\hat{y}_{1}\hat{\kappa}_{1}\hat{\eta}_{1}\hat{Y}_{ik}U$ (79)
$\displaystyle\leftrightarrow\quad\hat{Y}_{ik}Z$
$\displaystyle=\hat{\eta_{1}}^{-1}\hat{\kappa}_{1}\hat{y}_{1}\hat{Y}_{ik}\hat{y}_{1}\hat{\kappa}_{1}\hat{\eta}_{0}Z.$

$\displaystyle\hat{y}_{1}\hat{\eta}Z$ $\displaystyle=\hat{\eta}\hat{y}_{1}Z$
(80) $\displaystyle\leftrightarrow\quad(\eta^{-1}Z\eta)^{-1}$
$\displaystyle=\eta^{-1}Z^{-1}\eta.$

The non-commutativity relation can be recast as a deformed commutator, taking
$\hat{\eta}Z=\eta^{-1}Z\eta$ as a metric operator acting over an arbitrary
matrix $Z$. Consider the deformed commutator in equation (81), with
$\hat{\eta}_{1}=\hat{\Omega}\hat{\eta}_{0}$ the new pseudo-unitary metric
operator obtained after a partial inversion. $\hat{\Omega}$ is the deformation
metric operator.

$\displaystyle[\hat{Y}_{ik},\hat{\eta}_{0}]_{\Omega}=\hat{Y}_{ik}\hat{\eta}_{0}-\hat{\Omega}\hat{\eta}_{0}\hat{Y}_{ik}=0.$
(81)

Opening the terms of equation (81) to seek the general solution for
$\hat{\eta}_{1}$ in terms of $\eta_{0}$ and $(i,k)$ results in a non-
analytical and hardly tractable equation. Fortunately, numerical evaluations
based on representations of the dihedral group are easily obtained following
the occurrence of $\hat{\eta}_{1}=\hat{\sigma}_{3}$ for the T-matrix from an
$SU(2)$ S-matrix.

We use as representations of the dihedral group $\hat{\tau}_{i}$ for
transposition around the $i$th axis (also diagonal), $\hat{\kappa}_{i}$ for
complex conjugation with transposition and $\hat{y}_{i}$ for inversion along
the $i$th diagonal. They obey equations (82) through (84) (in place of
$\hat{a}$).

$\displaystyle\hat{a}_{i}\hat{a}_{i+1}$ $\displaystyle=\hat{a}_{i+2},$ (82)
$\displaystyle\hat{a}_{i}\hat{a}_{i}$ $\displaystyle=\hat{I},$ (83)
$\displaystyle[\hat{a}_{i},\hat{a}_{j}]$ $\displaystyle=0.$ (84)

Although Pauli matrices in equation (85) anti-commute (equation (86)), their
similarity transformations
$\hat{\sigma}_{i}(\quad)=\sigma_{i}(\quad)\sigma_{i}$ do commute and form a
dihedral group (equation (87)).

$\displaystyle\sigma_{1}=\begin{pmatrix}0&1\\\
1&0\end{pmatrix},\quad\sigma_{2}=\begin{pmatrix}0&-i\\\
i&0\end{pmatrix},\quad\sigma_{3}=\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}.$
(85)

$\displaystyle[\sigma_{i},\sigma_{i+1}]$
$\displaystyle=2\mathrm{i}\sigma_{i+2},$ (86)
$\displaystyle\sigma_{i}\sigma_{i+1}Z\sigma_{i+1}\sigma_{i}$
$\displaystyle=\mathrm{i}\sigma_{i+2}Z(-\mathrm{i})\sigma_{i+2}=\sigma_{i+2}Z\sigma_{i+2}.$
(87)

Transposition along the main diagonal is denoted by an operator whose matrix
entries are orthonormal functions $F_{\mu\nu}(\theta)$ under integration,
therefore $\int\mathrm{d}\theta
F_{\mu\nu}F_{\lambda\rho}=\delta_{\mu\lambda}\delta_{\nu\rho}$, serving the
purpose of permuting entries around the main diagonal. The three
representations of $D_{2}(n)$ (equations (88) through (90), equation (91), and
$\hat{\kappa}_{i}$ as for $\hat{\tau}_{i}$ with a complex conjugation
operator) have the common element $\hat{\tau}_{3}$, i.e.,
$D_{2}(\hat{\tau})\cap D_{2}(\hat{y})\cap D_{2}(\hat{\sigma})\cap
D_{2}(\hat{\kappa})=\hat{\tau_{3}}$.

$\displaystyle\hat{\tau}_{1}$
$\displaystyle=\int\mathrm{d}\theta\begin{pmatrix}F_{00}&F_{01}\\\
F_{10}&F_{11}\end{pmatrix}\left(\,\right)\begin{pmatrix}F_{00}&F_{01}\\\
F_{10}&F_{11}\end{pmatrix},$ (88) $\displaystyle\hat{\tau}_{2}$
$\displaystyle=\hat{\tau_{3}}\hat{\tau_{1}},$ (89)
$\displaystyle\hat{\tau}_{3}$ $\displaystyle=\begin{pmatrix}0&1\\\
1&0\end{pmatrix}(\,)\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},$ (90)

$\displaystyle\hat{y}_{1}=\prod_{i=0}^{N-1}\hat{Y}_{ii},\quad\hat{y}_{2}=\prod_{i=0}^{N-1}\hat{Y}_{i,N-i-1},\quad\hat{\tau_{3}}=\hat{y}_{1}\hat{y}_{2}.$
(91)

As consequence, any product of them can exchange $\hat{\tau}_{3}$ to change
axes (e.g., equation (92)).

$\displaystyle\hat{\sigma}_{2}\hat{\tau}_{1}=\hat{\sigma}_{2}\hat{\tau}_{3}\hat{\tau}_{2}=\hat{\sigma}_{3}\hat{\tau}_{2},\quad\hat{\sigma}_{1}=\hat{\tau}_{3}.$
(92)

Running a routine in python to check if any combination of the above-mentioned
metric operators is a candidate for the metric operator of a pseudo-unitary
group, taking the $SU(2)$ S-matrix as starting point, one finds that this
example of S-matrix itself pertains to three special pseudo-unitary groups,
and so do other groups and group-like structures derived from it by partial
inversion. It is, therefore, convenient to use equation (93) to represent the
group of a metric vector $\vec{\alpha}$ as the intersection of the groups of
each vector entry. In our case, the metric vectors are 3-vectors, and only
two-dimensional matrices are studied.

$\displaystyle SU(2,\vec{\hat{\alpha}})=SU(2,\hat{\alpha}^{(0)})\cap
SU(2,\hat{\alpha}^{(1)})\cap SU(2,\hat{\alpha}^{(2)}).$ (93)

Figure 2:  Graph that defines the relation between $U(2^{n},I^{\otimes n})$
and $U(2^{n},\sigma_{3}^{\otimes n})$ (magenta link between nodes of
respective metrics), and defines $SU(2,\eta)$ metric-deformation operators in
$\hat{\Omega}(i,k,\nu,\rho)$ (full black and red edges) as links between
metric vectors (nodes $\vec{\hat{\alpha}}$, $\vec{\hat{\phi}}$).
$\vec{\hat{\phi}}$ and $\vec{\hat{\epsilon}}$ are linked by powers, not
$\hat{\Omega}$ (dashed edges). The magenta edge marks a sequence of equal
indices as the first $2^{n-1}$ odious numbers in the odious series. Black
edges mark $i=k$, and red edges mark $i\neq k$, with $i,k$ the partial
inversion indices. The nodes of pseudo-unitary metrics are linked to the nodes
of pseudo-special-unitary metrics by dotted edges to represent their relation
with simpler and more symmetric sets of metrics.

In figure 2, the upper region of the graph with two nodes linked by a magenta
edge marks a map between pseudo-unitary groups of higher dimensions that are
grown by tensor product from elements of $U(2,\hat{I})$ and
$U(2,\hat{\sigma}_{3})$, respectively. These groups were numerically found to
be mapped to each other by a sequence of partial inversions with the first
$2^{n-1}$ odious numbers as indices, where $2^{n}$ is the dimension of the
group element.

Odious numbers are integers whose binary representations have odd numbers of
”1”s. They are object of study of Number Theory and are of rare occurrence in
mathematical problems, along with their reciprocal set of evil numbers, the
integers with even number of ”1”s [20]. The first ten numbers of the odious
sequence $\mathbb{O}$ are given in equation (94).

$\displaystyle\mathbb{O}=\\{1,2,4,7,8,11,13,14,16,19,\cdots\\}$ (94)

The first three examples of the map between the two pseudo-unitary groups in
figure 2 are given in equations (95) through (97). The matrix indices are
conveyed to start from $0$.

$\displaystyle U(2,\hat{\sigma}_{3})$
$\displaystyle=\hat{Y}_{11}U(2,\hat{I}_{2})$ (95) $\displaystyle
U(4,\hat{\sigma}_{3}^{\otimes 2})$
$\displaystyle=\hat{Y}_{22}\hat{Y}_{11}U(4,\hat{I}_{2}^{\otimes 2})$ (96)
$\displaystyle U(8,\hat{\sigma}_{3}^{\otimes 3})$
$\displaystyle=\hat{Y}_{77}\hat{Y}_{44}\hat{Y}_{22}\hat{Y}_{11}U(8,\hat{I}_{2}^{\otimes
3})$ (97)

Partial inversions do not always produce new groups. In figure 2,
$\vec{\hat{\alpha}}_{i}$ are metric vectors of pseudo-unitary groups,
$\vec{\hat{\phi}}_{i}$ are group-like structures that obey the pseudo-unitary
inversion constraint, but are made only of odd powers. Their even powers do
form pseudo-unitary groups and are marked on the diagram by
$\vec{\hat{\epsilon}}_{i}$ and $\vec{\hat{\alpha}}_{1}$ nodes.

$\displaystyle\vec{\hat{\alpha}}_{1}$ $\displaystyle=\begin{pmatrix}\hat{I}\\\
\hat{\sigma}_{3}\hat{\tau}_{1}\hat{\kappa}_{2}\\\
\hat{y}_{1}\hat{\tau}_{2}\hat{\sigma}_{3}\end{pmatrix},\quad\vec{\hat{\alpha}}_{2}=\begin{pmatrix}\hat{\sigma}_{3}\\\
\hat{\sigma}_{3}\hat{\tau}_{2}\\\
\hat{y}_{1}\hat{\tau}_{1}\hat{\kappa}_{2}\end{pmatrix},$ (98)
$\displaystyle\vec{\hat{\phi}}_{3}$
$\displaystyle=\begin{pmatrix}-\hat{\tau}_{2}\\\ \hat{\sigma}_{3}\\\
-\hat{y}_{1}\hat{\tau}_{1}\hat{\kappa}_{2}\hat{\sigma}_{3}\end{pmatrix},\quad\vec{\hat{\phi}}_{4}=\begin{pmatrix}-\hat{\sigma_{3}}\hat{\tau}_{1}\hat{\kappa}_{2}\\\
\hat{I}\\\ -\hat{y}_{1}\hat{\sigma}_{3}\hat{\tau}_{2}\end{pmatrix},$ (99)
$\displaystyle\vec{\hat{\phi}}_{5}$
$\displaystyle=\begin{pmatrix}-\hat{\sigma_{3}}\hat{\tau}_{1}\\\
-\hat{\sigma}_{3}\\\
\hat{y}_{1}\hat{\tau}_{1}\hat{\kappa}_{1}\end{pmatrix},\quad\vec{\hat{\phi}}_{6}=\begin{pmatrix}-\hat{\sigma_{3}}\\\
\hat{\tau}_{1}\\\
-\hat{y}_{1}\hat{\sigma}_{2}\hat{\tau}_{1}\hat{\kappa}_{2}\end{pmatrix},$
(100) $\displaystyle\vec{\hat{\epsilon}}_{5}$
$\displaystyle=\begin{pmatrix}\hat{\sigma_{3}}\\\
\hat{\tau}_{1}\hat{\sigma}_{3}\\\
\hat{y}_{1}\hat{\tau}_{1}\hat{\kappa}_{1}\end{pmatrix},\quad\vec{\hat{\epsilon}}_{6}=\begin{pmatrix}\hat{\sigma_{3}}\\\
\hat{\tau}_{1}\\\
\hat{y}_{1}\hat{\sigma}_{2}\hat{\tau}_{1}\hat{\kappa}_{2}\end{pmatrix}.$ (101)

In addition to equations (98) through (101), $\vec{\hat{\epsilon}}_{3}$ is
highly constrained and contains all possible combinations of $D_{2}$
generators as vector entries. It is worth noting that all entries containing
$\hat{y}_{1}$ express a symmetry property (equation (102)). For example, if
$\hat{\eta}=\hat{a}\hat{y}_{1}$, then

$\displaystyle\hat{y}_{1}Z$
$\displaystyle=\hat{\eta}\hat{\kappa}_{1}Z=\hat{y}_{1}\hat{a}\hat{\kappa}_{1}Z\quad\iff\quad
Z=\hat{a}\hat{\kappa}_{1}Z.$ (102)

This heuristic graph in figure 2 allows us to define the deformation matrix
$\hat{\Omega}$ that maps one metric vector to another, instead of trying to
solve equation (81). It depends both on the partial inversion indices $i,k$
and on the graph coordinates $\nu,\rho$. $\nu$ can be interpreted as the
distance from the unitary group (defined within $\vec{\hat{\alpha}}_{1}$),
whereas $\rho$ is a rolling number, and the superscript notation $[\rho]$ is
used to roll matrix entries along the main diagonal.

$\displaystyle\hat{\Omega}(i,k,\nu,\rho)=\begin{cases}P^{[\rho]}(\hat{\sigma}_{3},\hat{\kappa}^{\nu}\hat{\kappa}_{1}),\quad\text{if}\quad
i=k;\\\
P^{[\rho]}(\hat{\kappa}^{\nu}\hat{\kappa}_{2},-\hat{\sigma}_{3}),\quad\text{otherwise.}\end{cases}$
(103)

with $\hat{\kappa}Z=Z^{*}$ the complex conjugation operator over the matrix
$Z$, $\hat{\kappa}^{2}=\hat{I}$, and

$\displaystyle P^{[0]}(\hat{a},\hat{b})$
$\displaystyle=\begin{pmatrix}\hat{b}&0&0\\\ 0&\hat{a}\hat{b}&0\\\
0&0&\hat{a}\end{pmatrix},$ (104) $\displaystyle P^{[1]}(\hat{a},\hat{b})$
$\displaystyle=\begin{pmatrix}\hat{a}&0&0\\\ 0&\hat{b}&0\\\
0&0&\hat{a}\hat{b}\end{pmatrix},$ $\displaystyle P^{[2]}(\hat{a},\hat{b})$
$\displaystyle=\begin{pmatrix}\hat{a}\hat{b}&0&0\\\ 0&\hat{a}&0\\\
0&0&\hat{b}\end{pmatrix}.$

As a first example, equations (105) show the effect of partial inversion on
the metric vector from the S-matrix group $SU(2,\vec{\hat{\alpha}}_{1})$ to
the T-matrix group $SU(2,\vec{\hat{\alpha}}_{2})$.

$\displaystyle\hat{Y}_{11}S=T\quad\iff\quad\hat{\Omega}(1,1,0,1)\vec{\hat{\alpha}}_{1}$
$\displaystyle=\vec{\hat{\alpha}}_{2},$ (105) $\displaystyle
P^{[1]}(\hat{\sigma}_{3},\hat{\kappa}_{1})\vec{\hat{\alpha}}_{1}$
$\displaystyle=\vec{\hat{\alpha}}_{2},$
$\displaystyle\begin{pmatrix}\hat{\sigma}_{3}&&\\\ &\hat{\kappa}_{1}&\\\
&&\hat{\kappa}_{1}\hat{\sigma}_{3}\end{pmatrix}\begin{pmatrix}\hat{I}\\\
\hat{\sigma}_{3}\hat{\tau}_{1}\hat{\kappa}_{2}\\\
\hat{y}_{1}\hat{\tau_{2}}\hat{\sigma}_{3}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\hat{\sigma}_{3}\\\
\hat{\sigma}_{3}\hat{\tau}_{2}\\\
\hat{y}_{1}\hat{\tau}_{1}\hat{\kappa}_{2}\end{pmatrix}.$

###  3.4 Unitary representation of a sequence of partial inversions

A powerful consequence of figure 2 is that, for $Z\in SU(2,\hat{I}_{2})$ there
special sequences of partial inversions (around the hexagon graph in figure 2)
with $SU(2)$ representations, and similarly for the $\sigma_{3}$-pseudo-
special unitary group, as shown in equation (106).

$\displaystyle Z\in$ $\displaystyle
SU(2)\iff\hat{Y}_{f,f^{\prime}}\hat{Y}_{ee}\hat{Y}_{d,d^{\prime}}\hat{Y}_{cc}\hat{Y}_{b,b^{\prime}}\hat{Y}_{aa}Z\in
SU(2),$ (106) $\displaystyle\text{with}\quad
a,b,b^{\prime},c,d,d^{\prime},e,f,f^{\prime}\in\\{0,1\\},$
$\displaystyle\text{and with}\quad b^{\prime}\neq b,\,d^{\prime}\neq
d,\,f^{\prime}\neq f.$

Equation (106) means that it is possible to define a $SU(2)$ matrix
representation for sequences of partial inversions that obey that constraint.
The example in equations (107) through (109) emphasizes that such
representation depends on the entries of the target matrix $Z$, but a
universal representation can still be obtained as long as $Z$ is a general,
symbolic $SU(2)$ matrix.

$\displaystyle
Z^{\prime}=\hat{Y}_{01}\hat{Y}_{00}\hat{Y}_{10}\hat{Y}_{00}\hat{Y}_{01}\hat{Y}_{11}Z$
(107)
$\displaystyle\hat{Y}_{01}\hat{Y}_{00}\hat{Y}_{10}\hat{Y}_{00}\hat{Y}_{01}\hat{Y}_{11}$
$\displaystyle=Z^{\prime}Z^{-1}$ (108)
$\displaystyle=\begin{pmatrix}(t^{*})^{2}+(r^{*})^{2}&-(rt^{*}+r^{*}t)\\\
rt^{*}+r^{*}t&t^{2}+r^{2}\end{pmatrix},$

$\displaystyle\text{with}\quad Z=\begin{pmatrix}t&r\\\
-r^{*}&t^{*}\end{pmatrix}.$ (109)

In this example, the unitary representation is simply obtained by implementing
the transformations $t\to(t^{*})^{2}+(r^{*})^{2}$ and $r\to-(rt^{*}+r^{*}t)$.

###  3.5 Dot-product-conservation metric

The pseudo-unitary metric ($\eta$) also defines a pseudo-unitary dot-product
[21]. One can define a conservative dot-product (from a previously
conservative one) straightforwardly from the change of sign due to swaps of
vector entries between input and output vector in the partial inversion
(equation (110)).

$\displaystyle\psi_{1}^{*}\psi_{1}+\psi_{2}^{*}\psi_{2}$
$\displaystyle=\psi_{3}^{*}\psi_{3}+\psi_{4}^{*}\psi_{4}\quad$ (110)
$\displaystyle\leftrightarrow\quad\psi_{1}^{*}\psi_{1}-\psi_{3}^{*}\psi_{3}$
$\displaystyle=-\psi_{2}^{*}\psi_{2}+\psi_{4}^{*}\psi_{4}.$

The final input ($g_{\text{in}}^{\prime}$) and output
($g_{\text{out}}^{\prime}$) dot-product conservation metrics depend explicitly
on the partial inversion indices $i,k$ and the initial input and output
metrics $g_{\text{in}},g_{\text{out}}$ through equation (111).

$\displaystyle\text{If}\,g_{\alpha}(i,i)=g_{\alpha}(k,k):$ (111)
$\displaystyle\quad\quad g_{\text{in}}^{\prime}(i,k)=\left(I-2|k\rangle\langle
k|\right)g_{\text{in}},$ $\displaystyle\quad\quad
g_{\text{out}}^{\prime}(i,k)=\left(I-2|i\rangle\langle
i|\right)g_{\text{out}},$ else: $\displaystyle\quad\quad
g_{\alpha}^{\prime}=g_{\alpha}.$

Let us start an example from the S-matrix, whose vector space obeys the dot-
product conservation law under the identity metric $I$, i.e.
$g_{\text{in}}=g_{\text{out}}=I$, (equations (112) through (114), shortened
from the previously defined $\hat{I}\in\vec{\hat{\alpha}}_{1}$).

$\displaystyle S$ $\displaystyle\in U(n,\hat{\eta}=\hat{I}).$ (112)
$\displaystyle\begin{bmatrix}\psi_{01}\\\ \psi_{10}\end{bmatrix}$
$\displaystyle=S\begin{bmatrix}\psi_{00}\\\ \psi_{11}\end{bmatrix}.$ (113)
$\displaystyle\begin{bmatrix}\psi_{01}^{*}&\psi_{10}^{*}\end{bmatrix}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\begin{bmatrix}\psi_{01}\\\ \psi_{10}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}\psi_{00}^{*}&\psi_{11}^{*}\end{bmatrix}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\begin{bmatrix}\psi_{00}\\\ \psi_{11}\end{bmatrix}.$ (114)

An application of $\hat{Y}_{11}$ introduces the new pseudo-unitary metric
coincidentally as the new dot-product-conservation metric
($\eta=g_{\text{in}}=g_{\text{out}}=\sigma_{3}$, shortened from
$\sigma_{3}\in\vec{\hat{\alpha}}_{2}$). Since equations (111) apply and $i=k$,
then $g_{\text{in}}^{\prime}=g_{\text{out}}^{\prime}$, i.e., the metrics of
both input and output vectors are equal.

$\displaystyle T=\hat{Y}_{11}S\in U(n,\hat{\eta}=\hat{\sigma}_{3}).$ (115)

$\displaystyle\begin{bmatrix}\psi_{01}\\\ \psi_{11}\end{bmatrix}$
$\displaystyle=T\begin{bmatrix}\psi_{00}\\\ \psi_{10}\end{bmatrix}.$ (116)

$\displaystyle\begin{bmatrix}\psi_{01}^{*}&\psi_{11}^{*}\end{bmatrix}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\begin{bmatrix}\psi_{01}\\\
\psi_{11}\end{bmatrix}=\begin{bmatrix}\psi_{00}^{*}&\psi_{10}^{*}\end{bmatrix}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\begin{bmatrix}\psi_{00}\\\ \psi_{10}\end{bmatrix}.$ (117)

A subsequent application of $\hat{Y}_{01}$ leaves the dot-product conservation
metrics unchanged, since
$(g_{\alpha}^{\prime}(T))_{00}\neq(g_{\alpha}^{\prime}(T))_{11}$. The pseudo-
unitary metric vector for $A$ is $\vec{\hat{\alpha}}_{3}$, here shortened by
the entry $\hat{\tau}_{2}$ of transposition around the anti-diagonal followed
by product by $-1$.

$\displaystyle A=\hat{Y}_{01}T\in U\left(n,\hat{\eta}=-\hat{\tau}_{2}\right).$
(118)

$\displaystyle\begin{pmatrix}\psi_{10}\\\
\psi_{11}\end{pmatrix}=A\begin{pmatrix}\psi_{00}\\\ \psi_{01}\end{pmatrix}.$
(119)

$\displaystyle\begin{bmatrix}\psi_{10}^{*}&\psi_{11}^{*}\end{bmatrix}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\begin{bmatrix}\psi_{10}\\\
\psi_{11}\end{bmatrix}=\begin{bmatrix}\psi_{00}^{*}&\psi_{01}^{*}\end{bmatrix}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\begin{pmatrix}1&0\\\ 0&-1\end{pmatrix}\begin{pmatrix}1&0\\\
0&1\end{pmatrix}\begin{bmatrix}\psi_{00}\\\ \psi_{01}\end{bmatrix}.$ (120)

An example in which both the input and output dot-product conservation metrics
are different is given by the operator $B$, whose group-like object is fully
defined by node $\vec{\hat{\phi}}_{6}$ in figure 2.

$\displaystyle B=\hat{Y}_{01}S\in U\left(n,\hat{\eta}=-\hat{\tau}_{1}\right).$
(121) $\displaystyle\begin{bmatrix}\psi_{11}\\\
\psi_{10}\end{bmatrix}=B\begin{bmatrix}\psi_{00}\\\ \psi_{01}\end{bmatrix}.$
(122)
$\displaystyle\begin{bmatrix}\psi_{11}^{*}&\psi_{10}^{*}\end{bmatrix}\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}\begin{bmatrix}\psi_{11}\\\
\psi_{10}\end{bmatrix}=\begin{bmatrix}\psi_{00}^{*}&\psi_{01}^{*}\end{bmatrix}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\begin{bmatrix}\psi_{00}\\\ \psi_{01}\end{bmatrix}.$ (123)

Successive application of $B$ could lead to an ill-defined dot-product
conservation metric for medial terms. This ambiguity is suppressed by
alternating $B$ with its counterpart $\hat{Y}_{10}S$, whose vector space takes
the same dot-product conservation metrics as $B$ with opposite signs. Both
belong to the same group-like object $U(2,\vec{\hat{\phi}}_{6})$ that squares
to the group $U(2,\vec{\hat{\epsilon}}_{6})$.

##  4 Applications

###  4.1 Invertibility through singular matrices

The fact that the matrix inversion algorithm through partial inversion inverts
diagonal entries may pose a problem for matrices with zeros along the main
diagonal, even if they are invertible. In this section, we propose a solution
to this issue by manipulating zeros as variables in a special set defined
here. We proceed by analyzing the conditions that make hypothesis $H$ true,
with a finite matrix being just a matrix without infinities.

$H$: The partial inversion algorithm applied successively along the main
diagonal of a matrix produces the finite inverse matrix for any finite
invertible matrix.

If hypothesis $H$ is true, then invertible matrices such as permutation
matrices can lead us to define a formalism to manipulate zeros and infinities
in order to obtain a finite matrix. We take as an example the simplest of
permutation matrices, $\pi_{1}$, without simplifications other than
distributivity and associativity, putting symbols $o$ in place of zeros
(equations (124) through (127)).

$\displaystyle\gamma=\hat{Y}_{11}\hat{Y}_{00}\pi$
$\displaystyle=\hat{Y}_{11}\hat{Y}_{00}\begin{pmatrix}o&1\\\ 1&o\end{pmatrix}$
(124) $\displaystyle=\hat{Y}_{11}\begin{pmatrix}\frac{1}{o}&-\frac{1}{o}\\\
\frac{1}{o}&o-\frac{1}{o}\end{pmatrix}$ (125)
$\displaystyle=\hat{Y}_{11}\begin{pmatrix}\frac{1}{o}&-\frac{1}{o}\\\
\frac{1}{o}&\frac{o^{2}-1}{o}\end{pmatrix}$ (126)
$\displaystyle=\begin{pmatrix}\frac{1}{o}+\frac{1}{o^{2}}\left(\frac{o}{o^{2}-1}\right)&-\frac{1}{o}\left(\frac{o}{o^{2}-1}\right)\\\
-\frac{1}{o}\left(\frac{o}{o^{2}-1}\right)&\frac{o}{o^{2}-1}\end{pmatrix}.$
(127)

Let $o\in\mathbb{W}$ be the basis of the set $\mathbb{W}$, with
$\mathbb{M}=\mathbb{R}\cup\mathbb{W}$ a set that extends the real set
$\mathbb{R}$, and consider equations (128) through (131), with
$\\{N,M,p\\}\in\mathbb{R},\quad p\neq 0$.

$\displaystyle N+Mo^{p}$ $\displaystyle\in\mathbb{M};$ (128) $\displaystyle
Real(N+Mo^{p})$ $\displaystyle=\lim_{x\to 0}(N+Mx^{p})\in\mathbb{R};$ (129)
$\displaystyle Mem(N+0)$ $\displaystyle=N+Mo^{p}\in\mathbb{M};$ (130)
$\displaystyle mem(N+Mo^{p})$ $\displaystyle=Mo^{p}\in\mathbb{W}.$ (131)

In $\mathbb{W}$, differently from the real set, zeros are conveniently
quantifiable, or equivalently, in a more computational terminology, their
occurrences consume memory. It follows that equations (132) through (134) are
true in $\mathbb{W}$.

$\displaystyle o$ $\displaystyle\neq\sum^{N}_{i}o_{i}=No,$ (132)
$\displaystyle o$ $\displaystyle\neq\prod^{N}_{i}o_{i}=o^{N},$ (133)
$\displaystyle o+N$ $\displaystyle\neq N.$ (134)

Since $Real(\pi_{1})=Real(\pi_{1})^{-1}$ in real space, then the realization
of the inverse in $\mathbb{W}$ should preserve the equality if the hypothesis
$H$ is true, i.e., $Real(Mem(\pi))=Real(Mem(\pi)^{-1})$. With this assumption,
equation (127) implies equations (135) through (138), with $\gamma_{ij}$ the
entries of the matrix $\gamma$.

$\displaystyle Real(\gamma_{00})$ $\displaystyle=0,$ (135) $\displaystyle
Real(\gamma_{01})$ $\displaystyle=1,$ (136) $\displaystyle Real(\gamma_{10})$
$\displaystyle=1,$ (137) $\displaystyle Real(\gamma_{11})$ $\displaystyle=0.$
(138)

In order to investigate possible ambiguities during simplification, let us
assume the results $a,b,c,d$ of equations (139) through (142) are unknown, so
that we can seek their solutions in $\mathbb{M}$ that allow $H$ to be true.

$\displaystyle\frac{o}{o}$ $\displaystyle=a,$ (139) $\displaystyle 1-1$
$\displaystyle=b,$ (140) $\displaystyle o-o$ $\displaystyle=c,$ (141)
$\displaystyle\frac{1}{o}-\frac{1}{o}$ $\displaystyle=d.$ (142)

For $\gamma_{11}$, notice that $Real(\gamma_{11})=\frac{0}{0-1}=0$. For
$\gamma_{01}$ and $\gamma_{10}$, one can put the numerator $o$ in evidence and
apply $o/o=a$.

$\displaystyle-\frac{1}{o}\left(\frac{o}{o^{2}-1}\right)$
$\displaystyle=-\frac{o}{o}\left(\frac{1}{o^{2}-1}\right)$ (143)
$\displaystyle=\frac{-a}{o^{2}-1}.$ (144)

Upon realization in equation (145), and considering constraints defined in
equations (136) and (137), equation (144) is true iff $a=1$.

$\displaystyle
Real\left(\frac{-a}{o^{2}-1}\right)=\frac{-a}{0-1}=\frac{a}{1}.$ (145)

For $\gamma_{00}$, and considering we already found $o/o=1$, let us evidence
$\frac{1}{o}$ for the whole expression, put the numerator $o$ in evidence and
simplify the sum of fractions into one fraction, in equations (146) through
(150).

$\displaystyle\gamma_{00}=\frac{1}{o}+\frac{1}{o^{2}}\left(\frac{o}{o^{2}-1}\right)$
$\displaystyle=\frac{1}{o}\left[1+\frac{1}{o}\left(\frac{o}{o^{2}-1}\right)\right]$
(146)
$\displaystyle=\frac{1}{o}\left[1+\frac{o}{o}\left(\frac{1}{o^{2}-1}\right)\right]$
(147) $\displaystyle=\frac{1}{o}\left[1+\frac{1}{o^{2}-1}\right]$ (148)
$\displaystyle=\frac{1}{o}\left[\frac{o^{2}-1+1}{o^{2}-1}\right]$ (149)
$\displaystyle=\frac{o^{2}-1+1}{o^{3}-o}.$ (150)

Dividing both the numerator and denominator of equation (150) by $o$
(equations (151) through (154)), the denominator becomes realizable to $-1$,
so the constraint defined in equation (135) implies the numerator must be
realizable to zero, leading to equation (154).

$\displaystyle\gamma_{00}$
$\displaystyle=\frac{o-\frac{1}{o}+\frac{1}{o}}{o^{2}-1}$ (151)
$\displaystyle=\frac{o+\frac{1}{o}(1-1)}{o^{2}-1}$ (152)
$\displaystyle=\frac{o+\frac{1}{o}b}{o^{2}-1}.$ (153) $\displaystyle
Real\left(o+\frac{b}{o}\right)$ $\displaystyle=0\quad\iff\quad
Real\left(\frac{b}{o}\right)=0.$ (154)

Finally, $Real(\frac{b}{o})=0$ requires $b=Mo^{n}$, with $M,n$ real numbers,
$M\neq 0$, and $n>1$ .

Once $1-1$ is defined, we can multiply both sides by $o$ and find the values
of $c$ and $d$.

$\displaystyle c=o(1-1)$ $\displaystyle=oMo^{n}$ (155) $\displaystyle=o-o$
$\displaystyle=Mo^{n+1}.$

and

$\displaystyle d=\frac{1}{o}(1-1)$ $\displaystyle=\frac{1}{o}Mo^{n}$ (156)
$\displaystyle=\frac{1}{o}-\frac{1}{o}$ $\displaystyle=Mo^{n-1}.$

Repeating the previous steps for other powers, we obtain equation (157). The
conditions on $n$ to be a sufficiently large positive is to make $Mo^{n+p}$
realizable to $0$ for any power $p$, including negatives and $p=0$, but
excepting $p=-n$, to be discussed later.

$\displaystyle o^{p}-o^{p}$ $\displaystyle=Mo^{n+p},$ (157) with
$\displaystyle n>|p|,\quad n>0,\quad\\{n,p\\}\in\mathbb{R}.$

It may be intuitive to try to conceive $1-1=o$ as true, however, it implies
$o^{-1}-o^{-1}=1$, and recurrently leads to $o^{p}-o^{p}=o^{p+1}$ and to
$o=2$, which is obviously false under realization. Therefore the definition of
the exponent $n$ makes the algorithm realizable without inconsistencies.

The reversibility of this model can be attested even in non-invertible
matrices (e.g., equations (158) through (162), using the entry-wise algorithm
defined in equations (17) through (20)). In this case, non-invertibility of
$M$ means that no matrix can multiply $M$ and result in an identity matrix,
even though the partial inversion algorithm can be used to invert it
reversibly in $\mathbb{W}$.

$\displaystyle X$ $\displaystyle=\begin{pmatrix}o&o\\\ o&o\end{pmatrix}$ (158)
$\displaystyle\hat{Y}_{11}\hat{Y}_{00}\hat{Y}_{11}\hat{Y}_{00}X$
$\displaystyle=\hat{Y}_{11}\hat{Y}_{00}\hat{Y}_{11}\begin{pmatrix}o^{-1}&-1\\\
1&o-\frac{o^{2}}{o}\end{pmatrix}$ (159)
$\displaystyle=\hat{Y}_{11}\hat{Y}_{00}\hat{Y}_{11}\begin{pmatrix}o^{-1}&-1\\\
1&o-o\end{pmatrix}$ (160)
$\displaystyle=\hat{Y}_{11}\hat{Y}_{00}\begin{pmatrix}\frac{1}{o}+\frac{1}{o-o}&-\frac{1}{o-o}\\\
-\frac{1}{o-o}&\frac{1}{o-o}\end{pmatrix}$ (161)
$\displaystyle=\hat{Y}_{11}\hat{Y}_{00}\begin{pmatrix}\frac{1-1+1}{Mo^{n+1}}&\frac{-1}{Mo^{n+1}}\\\
\frac{-1}{Mo^{n+1}}&\frac{1}{Mo^{n+1}}\end{pmatrix}.$ (162)

Use of $1-1=Mo^{n}$ implies two simplification routes for $1-1+1$, that give
different results in the memory set, but the same result in $\mathbb{R}$. By
considering $(1-1)+1=Mo^{n}+1$ at the last equation of 162:

$\displaystyle\hat{Y}_{11}$
$\displaystyle\hat{Y}_{00}\begin{pmatrix}\frac{1-1+1}{Mo^{n+1}}&\frac{-1}{Mo^{n+1}}\\\
\frac{-1}{Mo^{n+1}}&\frac{1}{Mo^{n+1}}\end{pmatrix}=\hat{Y}_{11}\hat{Y}_{00}\begin{pmatrix}\frac{Mo^{n}+1}{Mo^{n+1}}&\frac{-1}{Mo^{n+1}}\\\
\frac{-1}{Mo^{n+1}}&\frac{1}{Mo^{n+1}}\end{pmatrix}$ (163)
$\displaystyle=\hat{Y}_{11}\begin{pmatrix}\frac{Mo^{n+1}}{Mo^{n}+1}&\frac{1}{Mo^{n}+1}\\\
\frac{-1}{Mo^{n}+1}&\frac{1}{Mo^{n+1}}-\frac{1}{(Mo^{n+1})^{2}}\frac{Mo^{n+1}}{Mo^{n}+1}\end{pmatrix}$
(164)
$\displaystyle=\hat{Y}_{11}\begin{pmatrix}\frac{Mo^{n+1}}{Mo^{n}+1}&\frac{1}{Mo^{n}+1}\\\
\frac{-1}{Mo^{n}+1}&\frac{Mo^{n}+1-1}{(Mo^{n+1})(Mo^{n}+1)}\end{pmatrix}$
(165)
$\displaystyle=\hat{Y}_{11}\begin{pmatrix}\frac{Mo^{n+1}}{Mo^{n}+1}&\frac{1}{Mo^{n}+1}\\\
\frac{-1}{Mo^{n}+1}&\frac{2}{o(Mo^{n}+1)}\end{pmatrix}$ (166)
$\displaystyle=\begin{pmatrix}\frac{Mo^{n+1}}{Mo^{n}+1}+\frac{1}{(Mo^{n}+1)^{2}}\frac{o(Mo^{n}+1)}{2}&\frac{o(Mo^{n}+1)}{2}\frac{1}{Mo^{n}+1}\\\
\frac{1}{Mo^{n}+1}\frac{o(Mo^{n}+1)}{2}&\frac{o(Mo^{n}+1)}{2}\end{pmatrix}$
(167)
$\displaystyle=\begin{pmatrix}\frac{Mo^{n+1}+o}{2(Mo^{n}+1)}&\frac{o}{2}\\\
\frac{o}{2}&\frac{Mo^{n+1}+o}{2}\end{pmatrix}.$ (168)

Since elements $X$ in $\mathbb{W}$ retain information about occurrence of some
operations, $Mem(X)\neq(Mem(X)^{-1})^{-1}$, however the equation
$Real(Mem(X))=Real((Mem(X)^{-1})^{-1})$ even for non-invertible matrices, as
can be seen from the last equation in equations (168).

Starting from the last equation of equations (162) by considering the
alternative simplification $(1+1)-1=1$, the realization of the result is
consistently the realization of the initial matrix (equations (169) through
(173)).

$\displaystyle\hat{Y}_{11}\hat{Y}_{00}$
$\displaystyle\begin{pmatrix}\frac{1-1+1}{Mo^{n+1}}&\frac{-1}{Mo^{n+1}}\\\
\frac{-1}{Mo^{n+1}}&\frac{1}{Mo^{n+1}}\end{pmatrix}=\hat{Y}_{11}\hat{Y}_{00}\begin{pmatrix}\frac{1}{Mo^{n+1}}&\frac{-1}{Mo^{n+1}}\\\
\frac{-1}{Mo^{n+1}}&\frac{1}{Mo^{n+1}}\end{pmatrix}$ (169)
$\displaystyle=\hat{Y}_{11}\begin{pmatrix}Mo^{n+1}&1\\\
-1&\frac{1}{Mo^{n+1}}-\frac{1}{Mo^{n+1}}\end{pmatrix}$ (170)
$\displaystyle=\hat{Y}_{11}\begin{pmatrix}Mo^{n+1}&1\\\
-1&\frac{Mo^{n}}{Mo^{n+1}}\end{pmatrix}$ (171)
$\displaystyle=\hat{Y}_{11}\begin{pmatrix}Mo^{n+1}&1\\\
-1&\frac{1}{o}\end{pmatrix}$ (172)
$\displaystyle=\begin{pmatrix}Mo^{n+1}+o&o\\\ o&o\end{pmatrix}.$ (173)

The actual values and nature of $M,n$ were vaguely defined earlier. In
addition to equations (139) throught (142), equations (174) through (178) give
the last axiom to sustain the set $\mathbb{W}$, the axiom of total
realizability. Differently from the treatment we gave up to now, it starts
with proposing that each application of $1-1=Mo^{n}$ results in an independent
and unique pair of variables $M,n$, i.e., the generation of
$Mo^{n}\in\mathbb{W}$ is not necessarily copiable.

$\displaystyle Real(N)$ $\displaystyle=Real(N),$ (174) $\displaystyle Real(N)$
$\displaystyle=Real\left(N+\sum^{P}_{i=0}(1-1)_{i}\right),$ (175)
$\displaystyle Real(N)$
$\displaystyle=Real\left(N+\sum_{i=0}^{P}M_{i}o^{n_{i}}\right),$ (176)
$\displaystyle Real(N-N)$ $\displaystyle=Real(N(1-1))$ (177)
$\displaystyle=Real(N(M_{t}o^{n_{t}}))$
$\displaystyle=Real\left(\sum_{i=0}^{P}M_{i}o^{n_{i}}\right),$

$\displaystyle
Real(N)=Real\left(\sum_{i=0}^{P}\frac{M_{i}}{M_{t}}o^{n_{i}-n_{t}}\right).$
(178)

Equations (174) through (178) pose the problem that either $Mo^{n}$ is ill-
defined and inconsistent with $o$-number-processing, or $M,n$ are variables
that are constrained by real equations upon application of the realization
function $Real$.

The definition of $o$ and the series of statements that support it in this
paper are meant to allow for the simplification of expressions in the partial
inversion algorithm when it requires a bulky manipulation of zero-expressions,
and thus they were, from the start, required to be in agreement with the
equivalent equations in the real set. In other words, the proposed definitions
may be conveniently defined as true in a computer program to serve their
simplification purpose, although their consistency with set-theoretical
definitions was not confronted.

Having the memory set $\mathbb{W}$ well defined, partial inversion allows us
to study singular matrices and seek finite representations of them.

###  4.2 Renormalized-growth approach to renormalization

The S-matrix theory interpretation of pseudo-unitary groups allows us to look
at partial inversion as a means to change the direction of the graph flow.
This is particularly important in growing diagrams using operators that grow a
diagram perpendicularly. This is the case of Chalker and Coddington’s $M$ and
$M^{\prime}$ [14]. In equation (179) we defined a matrix
$A=\hat{Y}_{01}\hat{Y}_{11}S$ (in place of $M^{\prime}$) that grows a
scattering diagram perpendicularly to a transfer matrix $T$ by matrix product.
We use the subscript notation (a,b) to indicate the dimensions of the circuit
represented by the matrix.

$\displaystyle T_{(4,4)}=(\hat{Y}_{01}[A_{(2,2)}^{3}])^{3}.$ (179)

The more general case, for any positive integers $n,m$, is given in equation
(180), and illustrated in figure 3.

$\displaystyle T_{(2n,m+1)}$
$\displaystyle=(\hat{Y}_{01}[(\hat{Y}_{01}T)^{2n-1}])^{m}$ (180)
$\displaystyle=\hat{Y}_{01}[(\hat{Y}_{01}[T^{m}])^{2n-1}].$

The requirement of the odd number $2n-1$ is due to the fact that $A^{2n}$ is
diagonal. Therefore, a partial inversion at indices $(0,1)$ would invert a
zero, leading to a matrix with infinities. It is treatable using the special
set of simplification rules defined in the previous section, but it was
avoided in this example for simplicity.

Figure 3: Equality of two renormalized-growth paths shows agreement with
graphical interpretation of scattering lattice.

---

$=$ .

The renormalized matrix for a diagram of dimensions $(4,4)$ is equation (181).
It leads reasonably to the renormalization equations (182) through (185).

$\displaystyle
T_{(4,4)}=\begin{bmatrix}\frac{3r^{2}(r^{*})^{2}+rr^{*}}{(2rr^{*}-1)(t^{*})^{3}}&\frac{[r^{2}(r^{*})^{2}+tt^{*}+2]r^{3}}{(2rr^{*}-1)(t^{*})^{3}}\\\
\frac{[r^{2}(r^{*})^{2}+tt^{*}+2](r^{*})^{3}}{(2rr^{*}-1)(t^{*})^{3}}&\frac{3r^{2}(r^{*})^{2}-3rr^{*}}{(2rr^{*}-1)(t^{*})^{3}}\end{bmatrix}$
(181)

$\displaystyle T^{--}$
$\displaystyle=\frac{3r^{2}(r^{*})^{2}-3rr^{*}}{(2rr^{*}-1)(t^{*})^{3}},$
(182) $\displaystyle T^{++}$
$\displaystyle=\frac{3r^{2}(r^{*})^{2}+rr^{*}}{(2rr^{*}-1)(t^{*})^{3}},$ (183)
$\displaystyle T^{+-}$
$\displaystyle=\frac{(r^{2}(r^{*})^{2}+tt^{*}+2)r^{3}}{(2rr^{*}-1)(t^{*})^{3}}$
(184) $\displaystyle T^{-+}$
$\displaystyle=\frac{(r^{2}(r^{*})^{2}+tt^{*}+2)(r^{*})^{3}}{(2rr^{*}-1)(t^{*})^{3}}$
(185)

The graphical interpretation of Scattering Theory gives the partial inversion
representation a physical meaning by changing the diagrammatic time flow from
linear (in S-matrix formalism) to circular (in T-matrix formalism), and any
other configuration in between. The diagrammatic time flow is accompanied by
the charge ordering of the scattering process, with space-time coordinates and
other properties of the wavefunctions as generalized charges. By choosing a
preferential direction for diagrammatic time flow, i.e., a partial inversion
of the formalism, one chooses for which charge the matrix of the formalism is
a translation operator of, with the S-matrix as translation operator of time
levels, and T-matrix as translation operator of space levels.

The restrictions imposed on unitary systems by No-Go Theorems (no-cloning, no-
deleting, no-masking) are largely studied in pseudo-unitary systems due to the
possibility of flexibilizing these theorems with a set of $\eta$-orthogonal
states [4, 22]. These models focus on pseudo-unitary systems alone without
realizing the transition from unitary systems to $\eta$-pseudo-unitary ones
and vice-versa. Since $\eta$-orthogonal cloning was proved possible [4], in
order to exploit it in unitary systems, a pair of desunitarization and
reunitarization protocols is necessary.

The closure property of Lie groups does not allow a unitary gate to generate a
different $\eta$-pseudo-unitary one. This implies that no observable is
related to the conversion between $\eta$-pseudo-unitary groups, and a gate
representation is not required for the conversion. Instead, the operation of
partial inversion as defined in this paper allows for the interpretation of
the conversion in an abstract way, so that an $\eta$-pseudo-unitary circuit
can exist as an abstraction of a unitary circuit. In particular, we can create
a circuit such that it starts as unitary, gets converted into an
$\eta$-pseudo-unitary circuit, then extended with $\eta$-pseudo-unitary gates,
and finally converted back into a unitary circuit. After the restoration of
unitarity, the $\eta$-pseudo-unitary part becomes the application of a
parameterized general unitary gate.

A possible advantage of using an $\eta$-pseudo-unitary circuit in place of a
unitary one is purely syntactic, by building a $\sigma_{3}$-pseudo-unitary
circuit from compositions of $W$ in equation (42), and then using partial
inversion to obtain its unitary representation. This means that with an
appropriate desunitarization-reunitarization protocol in terms of partial
inversion, an $\eta$-pseudo-unitary circuit is an alternative way of modelling
unitary circuits.

In the light of the Tensor Renormalization Group, the graphical process of
partial inversion is regarded as rewiring [23]. As was shown in the
Renormalized-Growth algorithm, a special sequence of partial inversions and
matrix powers allows for the manipulation of corner states of a scattering
lattice. This algorithm is an example of a Tensor Renormalization procedure
that makes constant use of rewiring to re-scale the circuit for use in lower
dimensions. The experimental implications of this algorithm can be estimated
by calculation of Landauer conductance. Figure 4, with $t=\cos(\theta)$ and
$r=\sin(\theta)$ shows that increasing the dimensions of the scattering
diagram increases the scattering and reduces the transmission along different
values of $\theta$.

Figure 4: Landauer conductance of real T-matrix for diagrammatic dimensions
from $(2,2)$ to $(10,10)$ after application of the renormalized-growth
algorithm.

##  5 Conclusion

In this paper, we proposed a matrix formalism for the transformation in matrix
space that results from permuting input and output vector entries. We call
this operation partial inversion, since it can be used to invert matrices
step-wise. As we discussed, partial inversion affects the dot product in a
similar way to the Legendre transform that served as inspiration, with the
difference that, in partial inversion, a metric is defined such that the new
dot product is always a conserved quantity, regardless of its physical
significance.

We used permutation planes in a 3D grid as diagram representation of partial
inversion in a quantum circuit, and defined some of its properties. This
representation eases the further illustration of inter-pseudo-unitary
circuits, i.e., circuits that transit between $\eta$-pseudo-unitary groups.

We defined the rules that allow for the application of partial inversion even
on entries with zeros by defining a special set $\mathbb{M}$ that treats zero
as a numerical basis.

S-matrix theory provides us with examples to analyze metric changes in pseudo-
unitarity groups due to partial inversion, as it is already used to map
S-matrices to T-matrices; however without a proper name other than solving a
system of equations. We explored the fact that this transformation changes the
geometric growth of the circuit to propose a renormalized-growth algorithm.
For example, if a matrix $T$ grows a circuit along the $x$ axis, by matrix
product, then we can use partial inversion to find the $A$ matrix that will
continue the growth by matrix product along a perpendicular $y$ axis. The
final matrix is a representation of a larger two-dimensional circuit without
changing the matrix dimensions.

We found the $(2,2)$-dimensional $T$ matrix that represents a
$(4,4)$-dimensional system symbolically as a test of the proposed renormalized
growth. Our results can potentially be used to devise other inter-unitary
circuits that can be conservatively transformed into unitary circuits,
expanding the possibilities for quantum computing algorithms.

##  6 Acknowledgements

The first author is thankful to the fruitful discussions with prof. Dmitry
Melnikov at the International Institute of Physics (IIP) in the Federal
University of Rio Grande do Norte (UFRN), Brazil. This article was partially
funded by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior -
Brasil (CAPES) via affiliation of the first author to the Graduate Physics
Program of UFRN, and by QC2.

##  7 Data availability statement

The data that support the findings of this study are available upon request
from the authors.

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