# Collective unitary evolution with linear optics by Cartan decomposition

Wen-Qiang Liu1,2, Xin-Jie Zhou1, and Hai-Rui Wei1111Corresponding author:
<EMAIL_ADDRESS>1 School of Mathematics and Physics, University of Science
and Technology Beijing, Beijing 100083, China
2 Center for Quantum Technology Research and Key Laboratory of Advanced
Optoelectronic Quantum Architecture and Measurements (MOE), School of Physics,
Beijing Institute of Technology, Beijing 100081, China

###### Abstract

Unitary operation is an essential step for quantum information processing. We
first propose an iterative procedure for decomposing a general unitary
operation without resorting to controlled-NOT gate and single-qubit rotation
library. Based on the results of decomposition, we design two compact
architectures to deterministically implement arbitrary two-qubit polarization-
spatial and spatial-polarization collective unitary operations, respectively.
The involved linear optical elements are reduced from 25 to 20 and 21 to 20,
respectively. Moreover, the parameterized quantum computation can be flexibly
manipulated by wave plates and phase shifters. As an application, we construct
the specific quantum circuits to realize two-dimensional quantum walk and
quantum Fourier transformation. Our schemes are simple and feasible with the
current technology.

###### pacs:

03.67.-a, 03.67.Lx, 42.50.Ex

##  I Introduction

Linear optics provides an alternative physical platform for realizing varied
quantum technologies, ranging from quantum computing KLM ; LOQC ; Liu ,
quantum communication communication1 ; QSDC1 ; Teleportation3 ; QSDC0 ; QSDC ;
QSDC2 ; EPP14 ; qkd ; QSDC3 ; QSDC4 ; QSDC5  to quantum algorithms Shor1 ;
Equations  and quantum metrology Metrology ; Metrology1 . Photon is considered
as an outstanding candidate for complex quantum information processing (QIP)
because it naturally possesses various inherent qubit-like degrees of freedom
(DOFs), long coherence time, robustness against the decoherence, and the
availability of manipulation at single-photon level manipulate1 ; LOQC .
Small-scale quantum computing with linear optics can be implemented
probabilistically KLM . To date, ambitions to realize multi-qubit photonic
quantum computing on only one DOF are still obstructed since the physical
interaction between individual photons goes beyond those currently available.

QIP with multiple DOFs of photons is a promising research field for
implementing effectively large-scale quantum computation and quantum
communication, due to its appealing features, such as deterministic property
Deterministic ; Deterministic1 , high capacity of quantum channel capacity1 ,
low photon loss rate Low-loss , reduced quantum resources Lanyon , and
flexible interactions between different DOF qubits P-M-Cluster1 ; P-M-Cluster2
; P-M-Cluster3 ; hyperentanglement . Multiple DOFs of a single photon have
been widely used in linear optical QIP. For instance, the implementation of
spatial-polarization Deutsch–Jozsa algorithm have been reported by Scholz _et
al_. DJ1  in 2006 and Zhang _et al_. DJ2  in 2010. The creation of four-qubit
and six-qubit polarization-momentum entangled cluster states have been
experimentally demonstrated in recent years P-M-Cluster1 ; P-M-Cluster2 ; P-M-
Cluster3 . In 2010, Sheng _et al_. Sheng  first proposed hyperentanglement
Bell-state analysis and quantum teleportation in the polarization and spatial
DOFs. Subsequently, Wang _et al_. teleportation-2  experimentally demonstrated
the quantum teleportation in both polarization and orbital angular momentum
DOFs. In 2012, Abouraddy _et al_. orbital2  proposed universal polarization-
momentum optical entangled quantum gates and the generation of entangled
states. Later, a single-photon time-polarization controlled-phase gate was
also experimentally demonstrated CPF-time-polarization . In 2015, based on
cosine-sine decomposition (CSD) CSD , Goyal _et al_. quantum-walk  proposed an
interesting scheme to complete spatial-polarization quantum walk in single
photon level. Subsequently, the CSD algorithm was applied to realize arbitrary
discrete unitary spatial-polarization transformation by using linear optics
Dhand-Goyal . In 2019, arbitrary spatial-temporal collective unitary operation
was theoretically reported Su ; Kumar . In 2021, deterministic polarization-
orbital-angular-momentum Toffoli gate and Fredkin gate with a single photon
have been experimentally demonstrated hybrid-Toffoli ; hybrid-Fredkin . We
also note that multiple DOFs of photons have been widely used in entanglement
purification Deterministic ; Deterministic1  and concentration concentration ,
quantum secure direct communication (QSDC) QSDC6 ; QSDC7 , quantum key
distribution (QKD) qKD , and etc.

Adding optical elements to the quantum circuit increases its overall
imperfections, which prevents the realization of quantum computation within
sufficient precision and also poses a challenge for the stability of the
circuits. It is therefore crucial to construct efficient quantum circuit with
significantly fewer linear optics. In this letter, we aim to reduce the cost
of the linear-optics-based arbitrary collective unitary operation using
polarization and spatial-mode DOFs of a photon. We first reconstruct a
universal unitary operation $U\in$ SU$(2^{n})$ by Cartan decomposition
technique. Based on the results of decomposition, we realize arbitrary two-
qubit linear optical collective unitary operation with the polarization-
spatial DOF and spatial-polarization DOF of a single photon, respectively. As
an application, we design the specific schemes for implementing two-
dimensional quantum walk and quantum Fourier transformation.

Our schemes have the following characteristics: (i) Our schemes are to reduce
the number of required linear optical elements and are not constructed in
terms of CNOT gates and single-qubit rotations. (ii) The parameterized quantum
computation can be easily manipulated by quarter-wave plates (QWPs), half-wave
plates (HWPs) and phase shifters (PSs). (iii) The number of required linear
optical elements for implementing two-qubit polarization-spatial (spatial-
polarization) collective unitary operation is reduced from 25 (21) to 20 (20).
(iv) Our schemes reduce the quantum resource cost, are more robustness against
the photonic dissipation, and feasible with the current technology.

##  II Arbitrary polarization-spatial collective unitary operation

Cartan decomposition Cartan ; Cartan1  derives from the Lie group and relies
on its Lie algebra, and it is a decomposition of the Lie group of unitary
evolutions. A Cartan decomposition of semi-simple real Lie algebra
$\mathfrak{g}$ is a vector space decomposition

$\displaystyle\mathfrak{g}=\mathfrak{l}\oplus\mathfrak{p},$ (1)

where Lie algebras $\mathfrak{l}$ and $\mathfrak{p}=\mathfrak{l}^{\bot}$
satisfy,

$\displaystyle[\mathfrak{l},\mathfrak{l}]\subseteq\mathfrak{l},\;\;\;[\mathfrak{l},\mathfrak{p}]\subseteq\mathfrak{p},\;\;\;[\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{l}.$
(2)

Here the Lie bracket for matrix algebras
$[\mathfrak{a},\mathfrak{b}]=\mathfrak{a}\mathfrak{b}-\mathfrak{b}\mathfrak{a}$.

According to the relationship between the Lie group $G$ and the Lie algebra
$\mathfrak{g}$, any given unitary transformation $U\in G$ (element of
$G=e^{\mathfrak{g}}$) can be decomposed as

$\displaystyle U=K_{1}\;A\;K_{2}.$ (3)

Here $K_{1},K_{2}\in e^{\mathfrak{l}}$ and $A\in e^{\mathfrak{h}}\subset
e^{\mathfrak{p}}$. $\mathfrak{h}$ is called Cartan subalgebra contained in
$\mathfrak{p}$, which is a maximal Abelian subalgebra.

Cartan decomposition is the most popular and powerful tool for constructing
efficient quantum circuit due to its flexible $\mathfrak{l}$ and
$\mathfrak{h}$. Such excellent property induces some particular decompositions
of $\text{SU}(2^{n})$ group for every $n$-qubit unitary evolution, such as
Khaneja and Glaser decomposition (KGD) Cartan1 , concurrence canonical
decomposition (CCD) CCD , odd-even decomposition (OED) OED , and quantum
Shannon decomposition (QSD) QSD . The best result is the QSD-based
construction Dhand-Goyal , which beats the QR-based one QR . In the following,
we simplify arbitrary polarization-spatial collective unitary operation by
employing new $\mathfrak{l}$ and $\mathfrak{h}$.

For $n=1$, the basis of the Lie algebra $\mathfrak{su}(2)$ can be spanned by

$\displaystyle\begin{split}\mathfrak{l}_{2^{1}}:=\text{span}\\{\sigma_{z}\\},\quad\mathfrak{p}_{2^{1}}:=\text{span}\\{\sigma_{x},\sigma_{y}\\}.\end{split}$
(4)

The Cartan subalgebra $\mathfrak{h}_{2^{1}}$ is selected as

$\displaystyle\begin{split}\mathfrak{h}_{2^{1}}&:=\text{span}\\{\sigma_{x}\\}.\end{split}$
(5)

Here $\sigma_{x}$, $\sigma_{y}$, and $\sigma_{z}$ represent the standard Pauli
spin matrices.

For $n=2$, $\mathfrak{su}(4)$ can be divided into the following form

$\displaystyle\begin{split}\mathfrak{l}_{2^{2}}&:=\text{span}\\{\mathfrak{su}(2)\otimes\mathbb{I}_{2},\mathfrak{u}(2)\otimes\sigma_{z}\\},\\\
\mathfrak{p}_{2^{2}}&:=\text{span}\\{\mathfrak{u}(2)\otimes\sigma_{x},\mathfrak{u}(2)\otimes\sigma_{y}\\}.\end{split}$
(6)

We select Cartan subalgebra $\mathfrak{h}_{2^{2}}$ as

$\displaystyle\begin{split}\mathfrak{h}_{2^{2}}&:=\text{span}\\{\mathbb{I}_{2}\otimes\mathfrak{h}_{2^{1}},\sigma_{z}\otimes\mathfrak{h}_{2^{1}}\\}.\end{split}$
(7)

Here $\mathfrak{u}(2):=\text{span}\\{\mathfrak{su}(2),\mathbb{I}_{2}\\}$,
$\mathbb{I}_{2}$ is a $2\times 2$ identity matrix.

For an $n$-qubit ($n\geq 2$) case, as shown in Fig. 1, the $n$th recurrence
step yields the $\mathfrak{su}(2^{n})$ Cartan decomposition

$\displaystyle\begin{split}\mathfrak{l}_{2^{n}}&:=\text{span}\\{\mathfrak{su}(2^{n-1})\otimes\mathbb{I}_{2},\mathfrak{u}(2^{n-1})\otimes\sigma_{z}\\},\\\
\mathfrak{p}_{2^{n}}&:=\text{span}\\{\mathfrak{u}(2^{n-1})\otimes\sigma_{x},\mathfrak{u}(2^{n-1})\otimes\sigma_{y}\\}.\end{split}$
(8)

Cartan subalgebra $\mathfrak{h}_{2^{n}}$ can be taken as

$\displaystyle\begin{split}\mathfrak{h}_{2^{n}}&:=\text{span}\\{\mathbb{I}_{2}\otimes\mathfrak{h}_{2^{n-1}},\sigma_{z}\otimes\mathfrak{h}_{2^{n-1}}\\}.\end{split}$
(9)

Here
$\mathfrak{u}(2^{n-1}):=\text{span}\\{\mathfrak{su}(2^{n-1}),\mathbb{I}_{2}^{\otimes{(n-1)}}\\}$.

Figure 1:  Schematic explanation for Cartan decomposition of a polarization-
spatial collective unitary operation by a recursive algorithm.

###  II.1 Implementation of arbitrary two-qubit polarization-spatial
collective unitary operation

Based on Eqs. (6-7), arbitrary two-qubit polarization-spatial collective
unitary operation $U_{\text{ps}}\in\text{SU}(4)$ group can be decomposed as

$\displaystyle\begin{split}U_{\text{ps}}&=\text{S}[\text{U}(2)\oplus\text{U}(2)]\cdot
A\cdot\text{S}[\text{U}(2)\oplus\text{U}(2)]\\\
&\overset{\text{def}}{=}(L_{1}\oplus L_{2})\cdot A\cdot(R_{1}\oplus
R_{2}),\end{split}$ (10)

in the basis $\\{|Ha_{1}\rangle$, $|Ha_{2}\rangle$, $|Va_{1}\rangle$,
$|Va_{2}\rangle\\}$. Here $H$ and $V$ represent the horizontal and vertical
polarization DOF of a single photon, respectively. $a_{1}$ and $a_{2}$
represent two spatial-mode DOF of a single photon. $L_{1}$ and $L_{2}$
($R_{1}$ and $R_{2}$) are the local polarization single-qubit gates acting on
each spatial-mode separately. $A$ is created by the Cartan subalgebra
$\mathfrak{h}_{2^{2}}$ with a block diagonal form

$\displaystyle\begin{split}A&=e^{i(\alpha\mathbb{I}_{2}\otimes\sigma_{x}+\beta\sigma_{z}\otimes\sigma_{x})}.\end{split}$
(11)

We denote $\theta_{1}=\alpha+\beta$ and $\theta_{2}=\alpha-\beta$. In order to
realize the polarization-spatial operation $A$ efficiently, we further
decompose $A$ as

$\displaystyle\begin{split}A=&\left(\begin{array}[]{cccc}0&0&i&0\\\ 0&1&0&0\\\
i&0&0&0\\\ 0&0&0&-1\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\
0&0&1&0\\\ 0&0&0&1\\\ \end{array}\right)\\\
&\cdot\left(\begin{array}[]{cccc}i\cos\theta_{1}&0&i\sin\theta_{1}&0\\\
0&i\cos\theta_{2}&0&i\sin\theta_{2}\\\ i\sin\theta_{1}&0&-i\cos\theta_{1}&0\\\
0&i\sin\theta_{2}&0&-i\cos\theta_{2}\\\ \end{array}\right)\\\
&\cdot\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}0&0&-1&0\\\ 0&-i&0&0\\\
1&0&0&0\\\ 0&0&0&-i\\\ \end{array}\right).\end{split}$ (12)

In Eq. (12), the second and the fourth matrices represent a CNOT gate in the
basis $\\{|Ha_{1}\rangle$, $|Ha_{2}\rangle$, $|Va_{1}\rangle$,
$|Va_{2}\rangle\\}$, which can be implemented conveniently by using a
polarizing beam splitter (PBS), because the PBS transmits the $H$-polarized
photon and reflects the $V$-polarized photon. The third matrix can be
completed by two HWPs rotated to angles $\frac{\theta_{1}}{2}$ and
$\frac{\theta_{2}}{2}$ acting on spatial modes $a_{1}$ and $a_{2}$,
separately. The first and the last matrices can be absorbed into the single-
qubit operations $L_{1}$, $L_{2}$, $R_{1}$, and $R_{2}$, i.e., $i\sigma_{x}$
is absorbed into $L_{1}$, $\sigma_{z}$ is absorbed into $L_{2}$,
$-i\sigma_{y}$ is absorbed into $R_{1}$, and $-i\mathbb{I}_{2}$ is absorbed
into $R_{2}$.

Putting all the pieces together, as shown in Fig. 2, one can find that 20
linear optical elements are sufficient to realize an arbitrary two-qubit
polarization-spatial collective unitary operation. Each of $L_{1}$, $L_{2}$,
$R_{1}$, and $R_{2}$ in Fig. 2 can be efficiently achieved by employing at
most two QWPs, one HWP and one PS single-qubit1 ; single-qubit2 ; single-
qubit3 . The effects of optical elements PS, HWP, and QWP oriented to angle
$\theta$ can be expressed by single-qubit1

$\displaystyle\begin{split}&U_{\text{PS}^{\theta}}=e^{i\theta}\mathbb{I}_{2},\\\
&U_{\text{HWP}^{\theta}}=e^{i\frac{\pi}{2}}\left(\begin{array}[]{cc}\cos
2\theta&\sin 2\theta\\\ \sin 2\theta&-\cos 2\theta\\\ \end{array}\right),\\\
&U_{\text{QWP}^{\theta}}=\frac{1}{\sqrt{2}}\left(\begin{array}[]{cc}1+i\cos
2\theta&i\sin 2\theta\\\ i\sin 2\theta&1-i\cos 2\theta\\\
\end{array}\right).\end{split}$ (13)

Figure 2: Implementing arbitrary two-qubit polarization-spatial collective
unitary operation with up to 20 linear optics. $a_{1}$ and $a_{2}$ represent
two different spatial modes of a single photon. PBS is the polarizing beam
splitter. HWPθ denotes a half-wave plate rotated to angle $\theta$. $L_{1}$
and $R_{1}$ ($L_{2}$ and $R_{2}$) are polarization single-qubit gates acting
on spatial-mode $a_{1}$ ($a_{2}$).

###  II.2 Application: the polarization-spatial quantum walk and quantum
Fourier transformation

Quantum walk and quantum Fourier transformation are both used as important
benchmarks for realizing QIP tasks. Two-dimensional quantum walk and quantum
Fourier transformation matrices are described by Walk ; book

$\displaystyle U_{w}=\frac{1}{2}\left(\begin{array}[]{cccc}-1&1&1&1\\\
1&-1&1&1\\\ 1&1&-1&1\\\ 1&1&1&-1\\\ \end{array}\right),$ (18) $\displaystyle
U_{F}=\frac{1}{2}\left(\begin{array}[]{cccc}1&1&1&1\\\ 1&i&-1&-i\\\
1&-1&1&-1\\\ 1&-i&-1&i\\\ \end{array}\right).$ (23)

Based on Cartan decomposition described in Eqs. (6-7), $U_{w}$ can be
decomposed as

$\displaystyle\begin{split}U_{w}=&\left(\begin{array}[]{cccc}-1&0&-1&0\\\
0&-1&0&-1\\\ 1&0&-1&0\\\ 0&-1&0&1\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}i&0&0&0\\\ 0&0&i&0\\\
0&i&0&0\\\ 0&0&0&-i\\\ \end{array}\right)\\\
&\cdot\frac{i}{2}\left(\begin{array}[]{cccc}-1&0&1&0\\\ 0&1&0&1\\\ 1&0&1&0\\\
0&1&0&-1\\\ \end{array}\right).\end{split}$ (24)

Here the second matrix $A_{w}$ in Eq. (24) can be further factorized as

$\displaystyle\begin{split}A_{w}=&\left(\begin{array}[]{cccc}0&1&0&0\\\
1&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}0&0&i&0\\\ 0&i&0&0\\\
i&0&0&0\\\ 0&0&0&-i\\\ \end{array}\right)\\\
&\cdot\left(\begin{array}[]{cccc}0&1&0&0\\\ 1&0&0&0\\\ 0&0&1&0\\\ 0&0&0&1\\\
\end{array}\right).\end{split}$ (25)

Based on Eqs. (24-25), we design a quantum circuit to implement a
deterministic two-dimensional quantum walk in the basis $\\{|Ha_{1}\rangle$,
$|Ha_{2}\rangle$, $|Va_{1}\rangle$, $|Va_{2}\rangle\\}$ with 11 linear optics,
see Fig. 3.

Figure 3:  The scheme for implementing a two-dimensional polarization-spatial
quantum walk.

Similarly, two-qubit quantum Fourier transformation can be decomposed as

$\displaystyle\begin{split}U_{F}=&\left(\begin{array}[]{cccc}-1&0&-1&0\\\
0&-1&0&-1\\\ -1&0&1&0\\\ 0&-1&0&1\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}i&0&0&0\\\ 0&0&i&0\\\
0&i&0&0\\\ 0&0&0&-i\\\ \end{array}\right)\\\
&\cdot\frac{i}{2}\left(\begin{array}[]{cccc}1&0&1&0\\\ 0&1&0&1\\\ 1&0&-1&0\\\
0&-i&0&i\\\ \end{array}\right).\end{split}$ (26)

Based on Eq. (26), Fig. 4 shows a specific setup for implementing two-qubit
polarization-spatial quantum Fourier transform with 12 linear optics.

Figure 4:  The scheme for implementing a two-qubit polarization-spatial
quantum Fourier transformation.

##  III Arbitrary spatial-polarization collective unitary operation

We note that Cartan decomposition is not unique, and the form of decomposition
depends on the span of the Lie subalgebra and the Cartan subalgebra. We next
investigate the decomposition for arbitrary spatial-polarization collective
unitary operation.

For $n=1$, the Cartan decomposition of the Lie algebra $\mathfrak{su}(2)$ can
still be taken the form

$\displaystyle\begin{split}\tilde{\mathfrak{l}}_{2^{1}}:=\text{span}\\{\sigma_{z}\\},\quad\tilde{\mathfrak{p}}_{2^{1}}:=\text{span}\\{\sigma_{x},\sigma_{y}\\}.\end{split}$
(27)

The Cartan subalgebra $\tilde{\mathfrak{h}}_{2^{1}}$ is taken as

$\displaystyle\begin{split}\tilde{\mathfrak{h}}_{2^{1}}&:=\text{span}\\{\sigma_{x}\\}.\end{split}$
(28)

For $n=2$, in order to realize the spatial-polarization collective unitary
operation, $\mathfrak{su}(4)$ can be factorized as

$\displaystyle\begin{split}\tilde{\mathfrak{l}}_{2^{2}}&:=\text{span}\\{\mathbb{I}_{2}\otimes\mathfrak{su}(2),\sigma_{z}\otimes\mathfrak{u}(2)\\},\\\
\tilde{\mathfrak{p}}_{2^{2}}&:=\text{span}\\{\sigma_{x}\otimes\mathfrak{u}(2),\sigma_{y}\otimes\mathfrak{u}(2)\\},\end{split}$
(29)

with Cartan subalgebra

$\displaystyle\begin{split}\tilde{\mathfrak{h}}_{2^{2}}&:=\text{span}\\{\sigma_{x}\otimes\sigma_{y},\sigma_{y}\otimes\sigma_{x}\\}.\end{split}$
(30)

For an $n$-qubit ($n>2$) case, as shown in Fig. 5, the factorization of the
$\mathfrak{su}(2^{n})$ can be expressed by

$\displaystyle\begin{split}\tilde{\mathfrak{l}}_{2^{n}}&:=\text{span}\\{\mathbb{I}_{2}\otimes\mathfrak{su}(2^{n-1}),\sigma_{z}\otimes\mathfrak{u}(2^{n-1})\\},\\\
\tilde{\mathfrak{p}}_{2^{n}}&:=\text{span}\\{\sigma_{x}\otimes\mathfrak{u}(2^{n-1}),\sigma_{y}\otimes\mathfrak{u}(2^{n-1})\\},\end{split}$
(31)

with Cartan subalgebra

$\displaystyle\begin{split}\tilde{\mathfrak{h}}_{2^{n}}&:=\text{span}\\{\tilde{\mathfrak{h}}_{2^{n-1}}\otimes\mathbb{I}_{2},\tilde{\mathfrak{h}}_{2^{n-1}}\otimes\sigma_{z}\\}.\end{split}$
(32)

Figure 5:  Schematic explanation for Cartan decomposition of a spatial-
polarization collective unitary operation by a recursive algorithm.

###  III.1 Implementation of arbitrary two-qubit spatial-polarization
collective unitary operation

Based on Eqs. (29-30), one can see that arbitrary two-qubit collective unitary
operation $U_{\text{sp}}\in\text{SU}(4)$ group in the basis
$\\{|a_{1}H\rangle$, $|a_{1}V\rangle$, $|a_{2}H\rangle$, $|a_{2}V\rangle\\}$
can be factorized as

$\displaystyle\begin{split}U_{\text{sp}}&=\text{S}[\text{U}(2)\oplus\text{U}(2)]\cdot\widetilde{A}\cdot\text{S}[\text{U}(2)\oplus\text{U}(2)]\\\
&\overset{\text{def}}{=}\left(\begin{array}[]{cc}\widetilde{L}_{1}&0\\\
0&\widetilde{L}_{2}\\\
\end{array}\right)\cdot\widetilde{A}\cdot\left(\begin{array}[]{cc}\widetilde{R}_{1}&0\\\
0&\widetilde{R}_{2}\\\ \end{array}\right).\end{split}$ (33)

Here $\widetilde{L}_{1}$ ($\widetilde{L}_{2}$) and $\widetilde{R}_{1}$
($\widetilde{R}_{2}$) are arbitrary single-qubit polarization gates acting on
the spatial-mode $a_{1}$ ($a_{2}$). $\widetilde{A}$ can be generated by
$\tilde{\mathfrak{h}}_{2^{2}}$ with the following form

$\displaystyle\begin{split}\widetilde{A}&=e^{i(\alpha\sigma_{x}\otimes\sigma_{y}+\beta\sigma_{y}\otimes\sigma_{x})}.\end{split}$
(34)

$\widetilde{A}$ can be further factorized as

$\displaystyle\begin{split}\widetilde{A}=&\left(\begin{array}[]{cccc}-i&0&0&0\\\
0&i&0&0\\\ 0&0&-i&0\\\ 0&0&0&i\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ \end{array}\right)\\\
&\cdot\left(\begin{array}[]{cccc}i\cos\theta_{2}&i\sin\theta_{2}&0&0\\\
i\sin\theta_{2}&-i\cos\theta_{2}&0&0\\\ 0&0&i\cos\theta_{1}&i\sin\theta_{1}\\\
0&0&i\sin\theta_{1}&-i\cos\theta_{1}\\\ \end{array}\right)\\\
&\cdot\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
\end{array}\right).\end{split}$ (35)

The second and the fourth matrices in Eq. (35) denote a spatial-polarization
CNOT gate which can be easily realized by a PBS. The third matrix can be
achieved by two HWPs with the angles $\frac{\theta_{2}}{2}$ and
$\frac{\theta_{1}}{2}$ acting on spatial-modes $a_{1}$ and $a_{2}$,
respectively. The first matrix will be absorbed into $\widetilde{L}_{1}$ and
$\widetilde{L}_{2}$. Each of single-qubit operations $\widetilde{L}_{1}$,
$\widetilde{L}_{2}$, $\widetilde{R}_{1}$, and $\widetilde{R}_{2}$, can be
completed by at most two QWPs, one HWP and one PS single-qubit1 ; single-
qubit2 ; single-qubit3 . Figure 6 presents the setup to implement arbitrary
two-qubit spatial-polarization collective unitary operation with at most 20
linear optics.

Figure 6: Implementing arbitrary two-qubit spatial-polarization collective
unitary operation with up to 20 linear optics.

###  III.2 Application: the spatial-polarization quantum walk and quantum
Fourier transformation

In order to realize spatial-polarization quantum walk, based on Eqs. (29-30),
two-dimensional quantum walk can be redecomposed as

$\displaystyle\begin{split}U_{w}=&\left(\begin{array}[]{cccc}-1&-1&0&0\\\
1&-1&0&0\\\ 0&0&-1&-1\\\ 0&0&-1&1\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&-1&0\\\
0&1&0&0\\\ 0&0&0&1\\\ \end{array}\right)\\\
&\cdot\frac{1}{2}\left(\begin{array}[]{cccc}1&-1&0&0\\\ -1&-1&0&0\\\
0&0&1&1\\\ 0&0&1&-1\\\ \end{array}\right).\end{split}$ (36)

The second matrix $\widetilde{A}_{w}$ in Eq. (36) can be further decomposed as

$\displaystyle\begin{split}\widetilde{A}_{w}=&\left(\begin{array}[]{cccc}-i&0&0&0\\\
0&i&0&0\\\ 0&0&-i&0\\\ 0&0&0&i\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ \end{array}\right)\\\
&\cdot\left(\begin{array}[]{cccc}0&i&0&0\\\ i&0&0&0\\\ 0&0&i&0\\\ 0&0&0&-i\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ \end{array}\right).\end{split}$ (37)

Based on Eqs. (36-37), Fig. 7 shows the setup to implement a two-dimensional
spatial-polarization quantum walk with 12 optical elements.

Figure 7:  The scheme for implementing a two-dimensional spatial-polarization
quantum walk.

Similarly, we redecompose two-qubit quantum Fourier transformation matrix in
Eq. (38).

$\displaystyle\begin{split}U_{F}=&\text{diag}\bigg{\\{}\left(\begin{array}[]{cc}i\sin\frac{\pi}{8}-\cos\frac{\pi}{8}&-\sin\frac{\pi}{8}-i\cos\frac{\pi}{8}\\\
\cos\frac{\pi}{8}+i\sin\frac{\pi}{8}&\sin\frac{\pi}{8}-i\cos\frac{\pi}{8}\\\
\end{array}\right),\\\
&\left(\begin{array}[]{cc}\cos\frac{\pi}{8}-i\sin\frac{\pi}{8}&-\sin\frac{\pi}{8}-i\cos\frac{\pi}{8}\\\
-\cos\frac{\pi}{8}-i\sin\frac{\pi}{8}&\sin\frac{\pi}{8}-i\cos\frac{\pi}{8}\\\
\end{array}\right)\bigg{\\}}\\\
&\cdot\left(\begin{array}[]{cccc}\cos\frac{3\pi}{8}&0&0&\sin\frac{3\pi}{8}\\\
0&\cos\frac{\pi}{8}&-\sin\frac{\pi}{8}&0\\\
0&\sin\frac{\pi}{8}&\cos\frac{\pi}{8}&0\\\
-\sin\frac{3\pi}{8}&0&0&\cos\frac{3\pi}{8}\\\ \end{array}\right)\\\
&\cdot\frac{1}{2}\left(\begin{array}[]{cccc}-i&\frac{i-1}{\sqrt{2}}&0&0\\\
i&\frac{i-1}{\sqrt{2}}&0&0\\\ 0&0&1&-\frac{i+1}{\sqrt{2}}\\\
0&0&-1&-\frac{i+1}{\sqrt{2}}\\\ \end{array}\right).\end{split}$ (38)

The second matrix $\widetilde{A}_{F}$ in Eq. (38) can be further factorized as

$\displaystyle\begin{split}\widetilde{A}_{F}=&\left(\begin{array}[]{cccc}-i&0&0&0\\\
0&i&0&0\\\ 0&0&-i&0\\\ 0&0&0&i\\\
\end{array}\right)\cdot\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&0\\\
1&0&0&0\\\ 0&0&0&1\\\ \end{array}\right)\\\
&\cdot\left(\begin{array}[]{cccc}i\cos\frac{\pi}{8}&i\sin\frac{\pi}{8}&0&0\\\
i\sin\frac{\pi}{8}&-i\cos\frac{\pi}{8}&0&0\\\
0&0&i\cos\frac{3\pi}{8}&i\sin\frac{3\pi}{8}\\\
0&0&i\sin\frac{3\pi}{8}&-i\cos\frac{3\pi}{8}\\\ \end{array}\right)\\\
&\cdot\left(\begin{array}[]{cccc}0&0&1&0\\\ 0&1&0&0\\\ 1&0&0&0\\\ 0&0&0&1\\\
\end{array}\right).\end{split}$ (39)

Figure 8 shows the setup to implement a two-qubit spatial-polarization quantum
Fourier transformation with 19 linear optical elements.

Figure 8: The scheme for implementing a two-qubit spatial-polarization quantum
Fourier transformation. Figure 9: CSD-based arbitrary two-qubit spatial-
polarization collective unitary operation Dhand-Goyal . Beam splitter
BS${}^{\frac{\pi}{4}}$ completes operations
$|a_{1}\rangle\leftrightarrow(|a_{1}\rangle-|a_{2}\rangle)/\sqrt{2}$ and
$|a_{2}\rangle\leftrightarrow(|a_{1}\rangle+|a_{2}\rangle)/\sqrt{2}$.  Figure
10: CSD-SWAP-based arbitrary two-qubit polarization-spatial collective unitary
operation Dhand-Goyal . Figure 11:  Implementation of arbitrary unitary
operation with spatial modes $m$=4 and polarization modes $n=2$. $L^{(2)}_{1}$
($L^{(2)}_{2}$) and $R^{(2)}_{1}$ ($R^{(2)}_{2}$) are two-qubit gates acting
on spatial-modes $a_{1}$ and $a_{2}$ ($a_{3}$ and $a_{4}$), which can be
realized by Fig. 6.

##  IV Conclusion

In this letter, by devising Cartan decompositions in detail, we have presented
two deterministic optical architectures to implement arbitrary two-qubit
collective unitary operation in polarization-spatial DOF and spatial-
polarization DOF of a single photon, respectively. The parameterized quantum
computation can be flexibly controlled by adjusting the angles of HWP, QWP,
and PS. The number of required linear optical elements for two-qubit spatial-
polarization scheme is reduced from 21 Dhand-Goyal  (the CSD-based circuit,
see Fig. 9) to 20, and the polarization-spatial one is reduced from 25 Dhand-
Goyal  (the CSD-SWAP-based circuit, see Fig. 10) to 20. Our method is also
suitable for more spatial modes ($m>2$). For example, based on the iterative
spatial-polarization Cartan decomposition in Fig. 5, arbitrary collective
unitary operation with spatial modes $m$=4 and polarization modes $n=2$ can be
realized with 48 linear optical elements (see Fig. 11), which improves on the
CSD-based one with 74 optical elements Dhand-Goyal .

As an application, we designed four compact schemes to implement two-
dimensional quantum walk and quantum Fourier transformation. Our perspective
may have potential advantages in simplifying quantum computing and quantum
communication using multiple DOFs encoding.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
under Grant No. 11604012, the Fundamental Research Funds for the Central
Universities under Grant No. FRF-TP-19-011A3, and a grant from the China
Scholarship Council.

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