# Error-detected three-photon hyperparallel Toffoli gate with state-selective
reflection

Yi-Ming Wu, Gang Fan and Fang-Fang Du111Corresponding author<EMAIL_ADDRESS>Science and Technology on Electronic Test and Measurement Laboratory, North
University of China, Taiyuan 030051, China

###### Abstract

We present an error-detected hyperparallel Toffoli (hyper-Toffoli) gate for a
three-photon system based on the interface between polarized photon and
cavity-nitrogen-vacancy(NV) center system. This hyper-Toffoli gate can be used
to perform double Toffoli gate operations simultaneously on both the
polarization and spatial-mode degrees of freedom (DoFs) of a three-photon
system with a low decoherence, shorten operation time, and less quantum
resources required, in compared with those on two independent three-photon
systems in one DoF only. As the imperfect cavity-NV-center interactions are
transformed into the detectable failures rather than infidelity based on the
heralding mechanism of detectors, a near-unit fidelity of the quantum hyper-
Toffoli gate can be implemented. By recycling the procedures, the efficiency
of our protocol for the hyper-Toffoli gate is improved further. Meanwhile, the
evaluation of gate performance with achieved experiment parameters shows that
it is feasible with current experimental technology and provides a promising
building block for quantum compute.

Hyperparallel Toffoli gate, photon system, quantum information processing

##  I Introduction

Compared to the best-known classical information processing, quantum
information processing (QIP) Quantum2002 , adhered to the principles of
quantum mechanics, has great advantages in secure communication Error2019 ;
Quantum2020 ; Generic2020 ; Measurement2021 ; Quantum2021 ; Wang2021 ,
algorithm Polynomial-Time1999 ; Quantum1997 ; Cavity2018 , and machine
learning Entanglement2015 ; Quantum2018 ; Experimental2020 ; Machine2021 , due
to its great security and super-fast computing speed. Basic quantum logic
gates are the building block of quantum computing, so they are regarded as an
important unit of QIP. However, they still face the challenge of large-scale
integration in practice. Therefore, to simplify the constantly growing
complexity of qubits and decrease the overheads of resources, multi-qubit
gates become the central ingredients in scalable QIP. Generally, the simplest
universal multi-qubit gates is Toffoli or Fredkin gate Optimal2020 ; Low-
cost2020 . To construct a $n$-qubit Toffoli gate, $[(4^{n}-3n-1)/4]$ two-qubit
gates by Shende _et al._ Minimal2004  can be required. Ralph _et al._
introduced qudit to model Toffoli gate with polynomial two-qubit gates
Efficient2007 , where the qudit takes advantage of ancillary higher-
dimensional spaces. Based on multi-level physical systems, Radu _et al._
Generalized2009  presented the generalized $n$-qubit Toffoli gate using $n+1$
subsystems. Yu _et al._ deemed five two-qubit gates as optimal for
implementing the three-qubit Toffoli gate in theory Five2013 . Fiurášek _et
al._ Linear2006  proposed a linear-optics-based quantum Toffoli gate. Soeken
_et al._ Quantum2013  investigated the relationships between Pauli matrices
and quantum logical gates for simplifying the quantum circuits. Recently, some
physical architectures, including superconducting circuits Implementation2012
; Single-step2020 ; Machine-learning-based2020 , nitrogen-vacancy (NV) centers
Compact2013 , quantum dots (QDs) Scalable2014 ; Universal2014O ; Heralded2021
, and trapped ions, Realization2009  have been proposed to implement extensive
Toffoli gates in experiment and theory.

Less resources required from quantum gates is crucial in quantum computation,
originating from two strategies. One is to exploit a system with additional
qudit Fast2013 ; Simplifying2009 ; Scalable2020 , the other one is to encode
the quantum information in multiple degrees of freedom (DoFs) of a system in
universal computational tasks, i.e., hyperparallel quantum gates Universal2015
; Hyperparallel2016 ; Robust2019 ; Heralded2021O ; Universal2014L ;
Realization2021 ; Wei:16  simultaneously operating more than one independent
operations. The hyper-parallel quantum computing can simplify the quantum
circuit, improve the information-processing speed, reduce the quantum resource
consumption, and suppress the photonic dissipation noise. Recently,
implementing hyper-parallel photon-based controlled-NOT (CNOT) gate
Universal2015 ; Hyperparallel2016 ; Robust2019 ; Heralded2021O , hyper-
parallel photon-matter-based universal CNOT gate Universal2014L , and hyper-
parallel photon-based Toffoli gate Wei:16  have been proposed. Ru _et al._
Realization2021  have realized a deterministic quantum Toffoli gate on one
photon in orbital-angular-momentum and polarization DoFs. Therefore, enquiry
of the hyper-parallel multi-qubit Toffoli gate will be provided with great
significance in scalable hyper-parallel QIP.

Up to now, solid-state-spin systems play a promising platform for the
realization of quantum computing, as solid-state nature combined with
nanofabrication techniques provide a useful way to incorporate the electronic
spins into optical microcavities Room-temperature2006 ; Gigahertz2009 ;
Deterministric2010 ; Experimental2015 . One appealing type of solid-state-spin
system is the electron spin in a NV center High-filelity2011 ; Quantum2007 ;
Room2010 , which not only provides convenient ways of optical initialization,
single-qubit manipulation, and high-fidelity readout at room temperature, but
also processes a milliseconds coherence time of the electron spin in NV
center, resorting to spin echo techniques Decoherence-protected2012 ; Room2014
. In recent years, the NV centers have numerous applications, such as
distributed quantum computing Experimental2014 , the hybrid quantum gates
acting on the electron spin and the nearby nuclear spin Quantum2010 ,
geometric single-qubit gates Entangled2015 , CNOT gate Heralded2013 ;
Robust2019 , and universal gates on NV center electronic Universal2015  or
photonic qubits Hyperentanglement2013 ; Universal2014P  , and so on. Moreover,
NV-center-emission-based entanglement Universal2013  and hyperentanglement
purification Efficient2019  were also proposed recently.

In this paper, we propose an error-detected hyper-parallel Toffoli gate for a
three-photon system assisted by the state-selective reflection of one-sided
cavity-NV-center system. Our scheme has some characters. First, this hyper-
Toffoli gate is equivalent to perform double Toffoli gate operations
simultaneously on both the polarization and spatial DoFs of a three-photon
system with a low decoherence, short operation time, and less quantum
resources required, in compared with those on two independent three-photon
systems in one DoF only. Second, the fidelity of the quantum hyper-Toffoli
gate is unity in principle, as the inevitable imperfect performances can be
detected by single-photon detectors. Third, the strong coupling limitations
can be avoided, reducing the experimental conditions. Fourth, the success of
the error-detected scheme can be heralded by the single-photon detectors.
Fifth, the efficiency of the scheme can be further improved by repeating the
operation processes when the detectors are clicked.

##  II  Hyper-paralleled Quantum Toffoli Gate

Figure 1:  A cavity-NV center system consists of the negatively charged NV
center confined in an optical one-sided cavity. The optical transitions from
the spin-ground states $|\pm 1\rangle$ to the excited state $|A_{2}\rangle$
are coupled by the $|\sigma_{\mp}\rangle$ circularly polarized photons.

The cavity-NV center system is originated from a diamond NV center that is
coupled to an optical one-sided cavity, as shown in Fig. 1. The bottom mirror
of the optical cavity is partially reflection and the top one is 100%
reflection. The negatively charged NV center in the diamond lattice is
composed by a substitutional nitrogen atom, an adjacent vacancy and six
electrons. Here, the spin-ground state of the negatively charged NV center
owing to spin-spin interaction is split into $|0\rangle(m_{s}=0)$ and $|\pm
1\rangle(m_{s}=\pm 1)$ with a 2.87GHz zero-field splitting.
$|A_{2}\rangle=(|E_{-}\rangle|+1\rangle+|E_{+}\rangle|-1\rangle)/\sqrt{2}$ is
one of the six excited states of the negatively charged NV center Quantum2010
; Entangling2011 , taking into account spin-orbit and spin-spin interactions
at the same time, and is robust due to stable symmetric property.
$|E_{\pm}\rangle|(J_{s}=\pm 1)$ is orbit states of the cavity-NV center
system. The optical transition $|+1\rangle\leftrightarrow|A_{2}\rangle$
($|-1\rangle\leftrightarrow|A_{2}\rangle$) is driven by the absorption or
emission of a left-(right-) circularly polarized photon $|R\rangle$
($|L\rangle$), which can be improved by the NV center trapped in a frequency-
degenerate two-mode cavity. The above polarized states can be rewrote as
$|R\rangle=|0\rangle^{p}$, $|L\rangle=|1\rangle^{p}$ in the following
presentation. Under the rotating wave approximation, the Hamiltonian of the
cavity-NV-center system can be written as BOOK1997

$\displaystyle
H=\omega_{d}\hat{\sigma}_{+}\hat{\sigma}_{-}+\omega_{c}\hat{a}^{\dagger}\hat{a}+ig(\hat{\sigma}_{+}\hat{a}-\hat{a}^{\dagger}\hat{\sigma}_{-}),$
(1)

where $\omega_{d}$ and $\omega_{c}$ denote the frequencies of the negatively
charged NV center and the cavity field, respectively. $\hat{\sigma}_{+}$ and
$\hat{\sigma}_{-}$ are the lifting and lowering operators of the negatively
charged NV center. $\hat{a}$ and $\hat{a}^{\dagger}$ are the annihilation and
creation operators of the cavity filed. $g$ is the coupling strength between
the NV center and the one-sided cavity.

The Heisenberg-Langevin equations about the operators $\hat{a}$ and
$\hat{\sigma}_{-}$, and the input-output relationship of cavity filed could be
described as Deterministic2008

$\displaystyle\frac{d\hat{a}}{dt}$ $\displaystyle=$
$\displaystyle-\left[i\left(\omega_{c}-\omega_{p}\right)+\frac{\kappa}{2}\right]\hat{a}-g\hat{\sigma}_{-}$
$\displaystyle-\sqrt{\kappa}\hat{a}_{in}+\hat{H},$
$\displaystyle\frac{d\hat{\sigma}_{-}}{dt}$ $\displaystyle=$
$\displaystyle-\left[i\left(\omega_{d}-\omega_{p}\right)+\frac{\gamma}{2}\right]\hat{\sigma}_{-}-g\hat{\sigma}_{z}\hat{a}$
$\displaystyle+\sqrt{\gamma}\hat{\sigma}_{z}\hat{b}_{in}+\hat{G},$
$\displaystyle\hat{a}_{out}$ $\displaystyle=$
$\displaystyle\hat{a}_{in}+\sqrt{\kappa}\hat{a}.$ (2)

Here, $\omega_{p}$ represents the frequency of the input photon.
$\hat{\sigma}_{z}=\hat{\sigma}_{+}\hat{\sigma}_{-}-\hat{\sigma}_{-}\hat{\sigma}_{+}$
is the population operator. $\kappa$ and $\gamma$ are the decay rate of the
cavity field and the electron-spin state in the negatively charged NV center,
respectively. $\hat{H}$ and $\hat{G}$ are the noise operators. $\hat{a}_{in}$
and $\hat{a}_{out}$ are the input and output vacuum filed operators,
respectively. Under the weak excitation
$\left(\langle\hat{\sigma}_{z}\rangle\approx-1\right)$, the reflection
coefficient $r_{1}$ of the input photon interacting with the one-sided cavity-
NV center system can be described as Quantum-information2009

$\displaystyle
r_{1}=\frac{\left[i(\omega_{c}-\omega_{p})-\frac{\kappa}{2}\right]\left[i(\omega_{t}-\omega_{p})+\frac{\gamma}{2}\right]+g^{2}}{\left[i(\omega_{c}-\omega_{p})+\frac{\kappa}{2}\right]\left[i(\omega_{t}-\omega_{p})+\frac{\gamma}{2}\right]+g^{2}}.$
(3)

When the input photon encounters a cold cavity, that is, the coupling strength
$g$ = 0, the reflection coefficient $r_{0}$ can be changed into

$\displaystyle
r_{0}=\frac{i(\omega_{c}-\omega_{p})-\frac{\kappa}{2}}{i(\omega_{c}-\omega_{p})+\frac{\kappa}{2}}.$
(4)

Thus in the realistic condition, the state-selective reflection of circularly
polarized photon interacting with one-sided cavity-NV center system can be
summarized as

$\displaystyle|0\rangle^{p}|+1\rangle\rightarrow
r_{1}|0\rangle^{p}|+1\rangle,$
$\displaystyle|1\rangle^{p}|+1\rangle\rightarrow
r_{0}|1\rangle^{p}|+1\rangle,$
$\displaystyle|0\rangle^{p}|-1\rangle\rightarrow
r_{0}|0\rangle^{p}|-1\rangle,$
$\displaystyle|1\rangle^{p}|-1\rangle\rightarrow
r_{1}|1\rangle^{p}|-1\rangle.$ (5)

In this section, we will construct a near-unit fidelity and heralded hyper-
Toffoli gate based on the above state-selective reflection in Eq. (II). The
Toffoli gate completes the function that the target qubit is flipped only when
the two control qubits are both in the state $|1\rangle$, there is no
operation on the target qubit when the two control qubits is in the state
$|0\rangle$. The hyper-Toffoli gate simultaneously performs the Toffoli gate
operations on the spatial-polarization DoFs of the three-photon system.

The quantum circuit of the error-detected hyper-Toffoli gate can be divided
into two parts. One is used to perform the Toffoli gate operations on the
polarized DoF of a three-photon system (shown in Fig. 2) and the other is on
spatial DoF (shown in Fig. 3). Here, the building block1 in the green frame
interacts with the external cavity-NVi center systems, ($i$=1, 2). Suppose
that the initial states of NV1 and NV2 are both $|-1\rangle$ and the initial
polarized and spatial states of three photons are shown as follows

$\displaystyle|\phi\rangle_{a}^{p}=$
$\displaystyle\alpha_{1}|0\rangle_{a}^{p}+\alpha_{2}|1\rangle_{a}^{p},\quad|\phi\rangle_{a}^{s}$
$\displaystyle=\delta_{1}|0\rangle_{a}^{s}+\delta_{2}|1\rangle_{a}^{s},$
$\displaystyle|\phi\rangle_{b}^{p}=$
$\displaystyle\beta_{1}|0\rangle_{b}^{p}+\beta_{2}|1\rangle_{b}^{p},\quad\;|\phi\rangle_{b}^{s}$
$\displaystyle=\epsilon_{1}|0\rangle_{b}^{s}+\epsilon_{2}|1\rangle_{b}^{s},$
$\displaystyle|\phi\rangle_{c}^{p}=$
$\displaystyle\gamma_{1}|0\rangle_{c}^{p}+\gamma_{2}|1\rangle_{c}^{p},\quad\
|\phi\rangle_{c}^{s}$
$\displaystyle=\zeta_{1}|0\rangle_{c}^{s}+\zeta_{2}|1\rangle_{c}^{s}.$ (6)

The coefficients of the same type satisfy the normalized relationship, i.e.
$\lvert Y_{1}\rvert^{2}+\lvert Y_{2}\rvert^{2}=1$ ($Y=\alpha$, $\beta$,
$\gamma$, $\delta$, $\epsilon$, $\zeta$ ). We will present our proposals in
order of function.

Figure 2: Schematic diagram for the hyper-parallel Toffoli gate for a three-
photon system in polarized DoF assisted by two NV-cavity systems. CPBSj ($j$ =
1, 2, 3, 4) represents a circularly polarized beam splitter, which completes
the reflection of state $|0\rangle^{p}$ and the transmission of state
$|1\rangle^{p}$. Hk ($k$ = 1, 2, 3, 4, 5) is a half-wave plate inclined at
$22.5^{\circ}$, which performs the polarized Hadamard operation, i.e.,
$|0\rangle^{p}\rightarrow(|0\rangle^{p}+|1\rangle^{p})/\sqrt{2},|1\rangle^{p}\rightarrow(|0\rangle^{p}-|1\rangle^{p})/\sqrt{2}$.
X performs a qubit-flip operation on the polarized DoF of a photon
$[|0\rangle^{p}\leftrightarrow|1\rangle^{p}]$ by tilting the half-wave plate
at $45^{\circ}$. VBS1 is an adjustable beam splitter with transmission
coefficient $(r_{1}-r_{0})/2$ and reflection coefficient
$\sqrt{1-[(r_{1}-r_{0})/2]^{2}}$. Similarly, the transmission and reflection
coefficient of VBS2 are $[(r_{1}-r_{0})/2]^{3}$ and
$\sqrt{1-[(r_{1}-r_{0})/2]^{6}}$, respectively. D represents a single-photon
detector. DL is a delay line, which makes the two wave packets confluent at
the same time.

In the first part shown in Fig. 2, the Toffoli gate operations are performed
on the polarized DoF of three-photon system $abc$, unaffecting the spatial
states of the three photons. First, let the photons $a$ and $b$ simultaneously
enter into path $|0\rangle_{a}^{s}$ (or $|1\rangle_{a}^{s}$) and
$|0\rangle_{b}^{s}$ (or $|1\rangle_{b}^{s}$) from opposite directions,
respectively, i.e., photon $a$ interacts with the block1-NV2, meanwhile,
photon $b$ interacts with the block1-NV1. In particular, before the two
photons $a$ and $b$ pass through block1-NVi (i=1, 2), the Hadamard operations
He, i.e.,
$|+1\rangle\leftrightarrow(|+1\rangle+|-1\rangle)/\sqrt{2},|-1\rangle\leftrightarrow(|+1\rangle-|-1\rangle)/\sqrt{2}$,
are performed on the electron-spin state in NVi (i=1, 2) center by using a
$\pi/2$ femtosecond-level optical pulse Picosecond2018 . In the block1, the
polarized state $|0\rangle^{p}$ is reflected by the first circularly
polarizing beam splitter (CPBS), then it interacts with H1, NVi, H1 and X in
sequence, while the polarized state $|1\rangle^{p}$ is transmitted to a delay
line (DL) and an adjustable beam splitter (VBS). After that, the two optical
wave packets $|1\rangle^{p}$ and $|0\rangle^{p}$ converge on another CPBS2.
When all detectors Ds in the block1 are no response, the whole system is
transmitted from $|\phi_{0}\rangle$ into $|\phi_{1}\rangle$. Here,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|\phi_{0}\rangle$ $\displaystyle=$
$\displaystyle|\phi\rangle_{a}^{p}\otimes|\phi\rangle_{a}^{s}\otimes|\phi\rangle_{b}^{p}\otimes|\phi\rangle_{b}^{s}\otimes|\phi\rangle_{c}^{p}\otimes|\phi\rangle_{c}^{s}$
$\displaystyle\otimes|-1\rangle_{1}|-1\rangle_{2},$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|\phi_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{r_{1}-r_{0}}{2\sqrt{2}}[|+1\rangle_{2}(\alpha_{1}|0\rangle_{a}^{p}+\alpha_{2}|1\rangle_{a}^{p})$
(7)
$\displaystyle+|-1\rangle_{2}(\alpha_{1}|0\rangle_{a}^{p}-\alpha_{2}|1\rangle_{a}^{p})]$
$\displaystyle\frac{r_{1}-r_{0}}{2\sqrt{2}}[|+1\rangle_{1}(\beta_{1}|0\rangle_{b}^{p}+\beta_{2}|1\rangle_{b}^{p})$
$\displaystyle+|-1\rangle_{1}(\beta_{1}|0\rangle_{b}^{p}-\beta_{2}|1\rangle_{b}^{p})]$
$\displaystyle\otimes|\phi\rangle_{a}^{s}|\phi\rangle_{b}^{s}|\phi\rangle_{c}^{p}|\phi\rangle_{c}^{s}.$

From Eq. (II), the function of block1-NVi is equal to performing the
controlled-phase-flip (CPF) gate operation on the polarized state
$|0\rangle^{p}$ of photon $a$ or $b$ when the electron-spin state are
$|-1\rangle$ in NVi ($i$=1, 2) centers without affecting its spatial state.
When the single-photon detector D in the block1 on the spatial mode
$|0\rangle_{a}^{s}$ ($|1\rangle_{a}^{s}$, $|0\rangle_{b}^{s}$, or
$|1\rangle_{b}^{s}$) responses, it means that the errors originated from the
imperfect photon-spin interaction are filtered by CPBS2 and heralded by the D
in the block1. That is, the block1 has an error-detected function. After
photon $a$ passes through block1-NV2 and photon $b$ passes through block1-NV1,
the electron-spin state in NVi ($i$=1, 2) centers perform the Hadamard
operation He again. The quantum state $|\phi_{1}\rangle$ is changed into
$|\phi_{1^{\prime}}\rangle$

$\displaystyle|\phi_{1^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{2}\left(\alpha_{1}|0\rangle_{a}^{p}|+1\rangle_{2}+\alpha_{2}|1\rangle_{a}^{p}|-1\rangle_{2}\right)$
(8)
$\displaystyle\left(\beta_{1}|0\rangle_{b}^{p}|+1\rangle_{1}+\beta_{2}|1\rangle_{b}^{p}|-1\rangle_{1}\right)$
$\displaystyle\otimes|\phi\rangle_{a}^{s}|\phi\rangle_{b}^{s}|\phi\rangle_{c}^{p}|\phi\rangle_{c}^{s}.$

Second, the photon $c$ is put on the quantum circuit. After the CPBS3, the
polarized state $|1\rangle_{c}^{p}$ passes through H2, block1-NV1, H3,
block1-NV2, H4, block1-NV1 and H5 in sequence, meanwhile, the polarized state
$|0\rangle_{c}^{p}$ passes through DL and VBS2 with transmission coefficient
$[(r_{1}-r_{0})/2]^{3}$ and reflection coefficient
$\sqrt{1-[(r_{1}-r_{0})/2]^{6}}$. Then the two wave packets cross at another
CPBS4 simultaneously by DL. When all detectors Ds in two block1s do not
response, the above operations change the quantum state
$|\phi_{1^{\prime}}\rangle$ into

$\displaystyle|\phi_{2}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[\alpha_{1}|0\rangle_{a}^{p}|+1\rangle_{2}(\beta_{1}\gamma_{1}|0\rangle_{b}^{p}|0\rangle_{c}^{p}|+1\rangle_{1}$
(9)
$\displaystyle+\beta_{1}\gamma_{2}|0\rangle_{b}^{p}|1\rangle_{c}^{p}|+1\rangle_{1}+\beta_{2}\gamma_{1}|1\rangle_{b}^{p}|0\rangle_{c}^{p}|-1\rangle_{1}$
$\displaystyle+\beta_{2}\gamma_{2}|1\rangle_{b}^{p}|1\rangle_{c}^{p}|-1\rangle_{1})$
$\displaystyle+\alpha_{2}|1\rangle_{a}^{p}|-1\rangle_{2}(\beta_{1}\gamma_{1}|0\rangle_{b}^{p}|0\rangle_{c}^{p}|+1\rangle_{1}$
$\displaystyle+\beta_{1}\gamma_{2}|0\rangle_{b}^{p}|1\rangle_{c}^{p}|+1\rangle_{1}+\beta_{2}\gamma_{1}|1\rangle_{b}^{p}|0\rangle_{c}^{p}|-1\rangle_{1}$
$\displaystyle-\beta_{2}\gamma_{2}|1\rangle_{b}^{p}|1\rangle_{c}^{p}|-1\rangle_{1})]\otimes|\phi\rangle_{a}^{s}|\phi\rangle_{b}^{s}|\phi\rangle_{c}^{s}.$

Third, we apply Hadamard operations He on the two electron-spin states in NV1
and NV2, respectively, and measure them with the basis
$\left\\{|+1\rangle,|-1\rangle\right\\}$. If the measurement results of the
two-electron-spin states are $|+1\rangle_{1}$ and $|+1\rangle_{2}$, The state
of the system is changed from $|\phi_{2}\rangle$ to

$\displaystyle|\phi_{3}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\alpha_{1}\beta_{1}|0\rangle_{a}^{p}|0\rangle_{b}^{p}+\alpha_{1}\beta_{2}|0\rangle_{a}^{p}|1\rangle_{b}^{p}$
(10)
$\displaystyle+\alpha_{2}\beta_{1}|1\rangle_{a}^{p}|0\rangle_{b}^{p})(\gamma_{1}|0\rangle_{c}^{p}+\gamma_{2}|1\rangle_{c}^{p})$
$\displaystyle+(\alpha_{2}\beta_{2}|1\rangle_{a}^{p}|1\rangle_{b}^{p})(\gamma_{1}|0\rangle_{c}^{p}-\gamma_{2}|1\rangle_{c}^{p})]$
$\displaystyle\otimes|\phi\rangle_{a}^{s}|\phi\rangle_{b}^{s}|\phi\rangle_{c}^{s},$

which is the result of the controlled-controlled-phase-flip (C2PF) gate on the
polarized DoF of a three-photon system without affecting its spatial state.
That is, when the two electron-spin states in NV1 and NV2 centers, and the
polarized states of two photons $a$ and $b$ are $|+1\rangle_{1}|+1\rangle_{2}$
and $|1\rangle_{a}^{p}|1\rangle_{b}^{p}$, the CPF gate operation on the
polarized state of photon $c$. However, the measurement results of the two
electron-spin states are other possible outcomes, the corresponding feed-
forward single-qubit operations are shown in Table 1. Here,
$\sigma_{z}^{p}=|0\rangle^{p}\langle 0|-|1\rangle^{p}\langle 1|$ completes the
phase-flip operation on polarized state of the photon, and $I$ means keeping
its original state. Therefore, the success probability of a C2PF gate on the
polarized DoF of a three-photon system is 100% in principle. Finally, before
and after the C2PF gate, Hadamard operation performed on the polarized DoF of
the target qubit $c$, i.e., $U^{p}_{Toffoli}=(H^{p}\otimes
I_{4})U^{p}_{C^{2}PF}(I_{4}\otimes H^{p})$, the Toffoli gate on the polarized
DoF of the three-photon system is finished without impacting on the spatial
DoF of the three-photon system. Here,

$\displaystyle|\phi_{4}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\alpha_{1}\beta_{1}|0\rangle_{a}^{p}|0\rangle_{b}^{p}+\alpha_{1}\beta_{2}|0\rangle_{a}^{p}|1\rangle_{b}^{p}$
(11)
$\displaystyle+\alpha_{2}\beta_{1}|1\rangle_{a}^{p}|0\rangle_{b}^{p})(\gamma_{1}|0\rangle_{c}^{p}+\gamma_{2}|1\rangle_{c}^{p})$
$\displaystyle+(\alpha_{2}\beta_{2}|1\rangle_{a}^{p}|1\rangle_{b}^{p})(\gamma_{1}|1\rangle_{c}^{p}+\gamma_{2}|0\rangle_{c}^{p})]$
$\displaystyle\otimes|\phi\rangle_{a}^{s}|\phi\rangle_{b}^{s}|\phi\rangle_{c}^{s}.$

Table 1: The measurement results of the two-electron-spin states in NV1 and NV2, and the corresponding feed-forward single-qubit operations on polarized DoF of three photons. measurement results | single-qubit operations
---|---
$|+1\rangle_{1}|+1\rangle_{2}$ | $I_{a}\otimes I_{b}\otimes I_{c}$
$|+1\rangle_{1}|-1\rangle_{2}$ | $(\sigma_{z}^{p})_{a}\otimes I_{b}\otimes I_{c}$
$|-1\rangle_{1}|+1\rangle_{2}$ | $I_{a}\otimes(\sigma_{z}^{p})_{b}\otimes I_{c}$
$|-1\rangle_{1}|-1\rangle_{2}$ | $(\sigma_{z}^{p})_{a}\otimes(\sigma_{z}^{p})_{b}\otimes I_{c}$

In next part shown in Fig. 3, the Toffoli gate operations, are performed on
the spatial DoF of three-photon, by utilizing two cavity-NV systems and five
block2 to finish system $abc$, without changing the polarized states of three
photons. Suppose that the states of the three photons are the same as ones
described in Eq. (II), and meanwhile the states of NV3 and NV4 are
$|+1\rangle_{3}$ and $|-1\rangle_{4}$, respectively.

Figure 3: Schematic diagram for the hyper-parallel Toffoli gate for a three-
photon system in spatial DoF assisted by two NV-microcavity systems. BS1 and
BS4 denote 50:50 beam splitter which completes the Hadamard operation of the
photon in spatial DoF, i.e.,
$|1\rangle_{c}^{s}\leftrightarrow(|0\rangle_{d}^{s}+|1\rangle_{d}^{s})/\sqrt{2}$.
BS2 and BS3 complete the operation, i.e.,
$|0\rangle_{d}^{s}\leftrightarrow(|0\rangle_{d}^{s}+|1\rangle_{d}^{s})/\sqrt{2},|1\rangle_{d}^{s}\leftrightarrow(|0\rangle_{d}^{s}-|1\rangle_{d}^{s})/\sqrt{2}$.

First, after the Hadamard operations He are performed on the two electron-spin
states in NV3 and NV4, let the photons $a$ and $b$ simultaneously enter into
path $|0\rangle_{a}^{s}$ (or $|1\rangle_{a}^{s}$) and $|0\rangle_{b}^{s}$ (or
$|1\rangle_{b}^{s}$) from opposite directions, respectively. In detail, photon
$a$ from the spatial path $|1\rangle_{a}^{s}$ interacts with the block2-NV4,
meanwhile, photon $b$ from the spatial path $|1\rangle_{b}^{s}$ interacts with
the block2-NV3. Both photon $a$ from the spatial path $|0\rangle_{a}^{s}$ and
photon $b$ from the spatial path $|0\rangle_{b}^{s}$ pass through a VBS1 and a
DL in sequence. VBS1 is used to adjust the transmission coefficient. DL is
used to make the two paths $|0\rangle_{a}^{s}$ and $|1\rangle_{a}^{s}$, and
$|0\rangle_{b}^{s}$ and $|1\rangle_{b}^{s}$ of two photons $a$ and $b$ arrive
simultaneously. In the block2-NV3, the polarized state $|0\rangle^{p}$ of
photon $b$ is reflected by CPBS1, then it sequentially passes through H1, NV3,
and H1, while the polarized state $|1\rangle^{p}$ is transmitted to
sequentially pass through X1, H1, NV3, H1, and X2. After that, the two optical
wave packets converge on another CPBS2. So does the photon $a$ in the
block2-NV4. When all detectors Ds in the block2 and VBS1s are no response, the
state of the whole system is transmitted from $|\psi_{0}\rangle$ into
$|\psi_{1}\rangle$. Here,

$\displaystyle|\psi_{0}\rangle$ $\displaystyle=$
$\displaystyle|\phi\rangle_{a}^{p}\otimes|\phi\rangle_{a}^{s}\otimes|\phi\rangle_{b}^{p}\otimes|\phi\rangle_{b}^{s}\otimes|\phi\rangle_{c}^{p}\otimes|\phi\rangle_{c}^{s}$
$\displaystyle\otimes|+1\rangle_{3}\otimes|-1\rangle_{4},$
$\displaystyle|\psi_{1}\rangle$ $\displaystyle=$
$\displaystyle\frac{r_{1}-r_{0}}{2\sqrt{2}}[|+1\rangle_{4}(\delta_{1}|0\rangle_{a}+\delta_{2}|1\rangle_{a})$
(12)
$\displaystyle+|-1\rangle_{4}(-\delta_{1}|0\rangle_{a}+\delta_{2}|1\rangle_{a})]$
$\displaystyle\frac{r_{1}-r_{0}}{2\sqrt{2}}[|+1\rangle_{3}(\epsilon_{1}|0\rangle_{b}^{s}+\epsilon_{2}|1\rangle_{b}^{s})$
$\displaystyle+|-1\rangle_{3}(\epsilon_{1}|0\rangle_{b}^{s}-\epsilon_{2}|1\rangle_{b}^{s})]$
$\displaystyle\otimes|\phi\rangle_{a}^{p}|\phi\rangle_{b}^{p}|\phi\rangle_{c}^{p}|\phi\rangle_{c}^{s}.$

From Eq. (II), the function of block2-NVj ($j$ = 3, 4) is equal to performing
the CPF gate operation on the spatial state $|0\rangle^{s}$ of photon $a$ or
$b$ when the electron-spin state $|-1\rangle$ in NVi ($j$ = 3, 4) center,
without affecting its polarized state. Similarly, when the detector D in the
block2 on the spatial mode $|1\rangle_{a}^{s}$ ($|1\rangle_{b}^{s}$) or VBS1
on the spatial mode $|0\rangle_{a}^{s}$ ($|0\rangle_{b}^{s}$) responses, the
errors caused by the imperfect photon scattering are heralded by the D. That
is, the block2 also has an error-detected function. After the photons $a$ from
the spatial path $|1\rangle_{a}^{s}$ and $b$ from the spatial path
$|1\rangle_{b}^{s}$ interact with block2-NV4 and block2-NV3, respectively, the
two electron-spin states in NV3 and NV4 centers perform the Hadamard
operations He again. The quautum state $|\psi_{1}\rangle$ is changed into
$|\psi_{1^{\prime}}\rangle$

$\displaystyle|\psi_{1^{\prime}}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{2}(\delta_{1}|0\rangle_{a}^{s}|-1\rangle_{4}+\delta_{2}|1\rangle_{a}^{s}|+1\rangle_{4})$
(13)
$\displaystyle\otimes(\epsilon_{1}|0\rangle_{b}^{s}|+1\rangle_{3}+\epsilon_{2}|1\rangle_{b}^{s}|-1\rangle_{3})$
$\displaystyle\otimes|\phi\rangle_{a}^{p}|\phi\rangle_{b}^{p}|\phi\rangle_{c}^{p}|\phi\rangle_{c}^{s}.$

Second, the photon $c$ is put the quantum circuit. The photon $c$ from the
spatial state $|0\rangle_{c}^{s}$ passes through a VBS1 and a DL in sequence.
After the photon $c$ from the spatial path $|1\rangle_{c}^{s}$ passes through
BS1, it completes the transformation, i.e.,
$|1\rangle_{c}^{s}\leftrightarrow(|0\rangle_{d}^{s}+|1\rangle_{d}^{s})/\sqrt{2}$.
Subsequently, the photon $c$ from the spatial path $|1\rangle_{d}^{s}$ passes
through block2-NV3, BS2, block2-NV4, BS3, DL, VBS1 in sequence, while the
photon $c$ from the spatial path $|0\rangle_{d}^{s}$ passes through VBS1, DL,
BS2, VBS1, DL, BS3, block2-NV3 in sequence. Then the photon $c$ from the
spatial path $|0\rangle_{d}^{s}$ and $|1\rangle_{d}^{s}$ converge at BS4.
Here, BSk(k=2,3) is used to completes the Hadamard operation on the spatial
DoF of the photon $c$, i.e.,
$|0\rangle_{d}^{s}\leftrightarrow(|0\rangle_{d}^{s}+|1\rangle_{d}^{s})/\sqrt{2}$,
and
$|1\rangle_{d}^{s}\leftrightarrow(|0\rangle_{d}^{s}-|1\rangle_{d}^{s})/\sqrt{2}$.
DLs are used to make two spatial paths $|0\rangle_{d}^{s}$ and
$|1\rangle_{d}^{s}$ of the photon $c$ converge at simultaneously BS2, BS3, and
BS4, respectively. DLs from the spatial paths $|0\rangle_{a}^{s}$, and
$|0\rangle_{b}^{s}$, $|0\rangle_{c}^{s}$ make the photons $a$, $b$, and $c$
arrive simultaneously, respectively. When all detectors Ds in three block2s ,
VBS1s and VBS2s do not response, the above operations change the quautum state
$|\psi_{1^{\prime}}\rangle$ into

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!|\psi_{2}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\delta_{1}\epsilon_{1}|0\rangle_{a}^{s}|0\rangle_{b}^{s}+\delta_{1}\epsilon_{2}|0\rangle_{a}^{s}|1\rangle_{b}^{s}$
(14)
$\displaystyle+\delta_{2}\epsilon_{1}|1\rangle_{a}^{s}|0\rangle_{b}^{s})(\zeta_{1}|0\rangle_{c}^{s}+\zeta_{2}|1\rangle_{c}^{s})$
$\displaystyle+(\delta_{2}\epsilon_{2}|1\rangle_{a}^{s}|1\rangle_{b}^{s})(\zeta_{1}|0\rangle_{c}^{s}-\zeta_{2}|1\rangle_{c}^{s})]$
$\displaystyle\otimes|\phi\rangle_{a}^{p}|\phi\rangle_{b}^{p}|\phi\rangle_{c}^{p}.$

Third, we apply Hadamard operations He on the two electron-spin states in NV3
and NV4 centers, respectively, and measure them with the basis
$\left\\{|+1\rangle,|-1\rangle\right\\}$. If the measurement results of the
two electron-spin states are $|+1\rangle_{1}$ and $|+1\rangle_{2}$, The state
of the system is changed from $|\psi_{2}\rangle$ to

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|\psi_{3}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\delta_{1}\epsilon_{1}|0\rangle_{a}^{s}|0\rangle_{b}^{s}+\delta_{1}\epsilon_{2}|0\rangle_{a}^{s}|1\rangle_{b}^{s}$
(15)
$\displaystyle+\delta_{2}\epsilon_{1}|1\rangle_{a}^{s}|0\rangle_{b}^{s})(\zeta_{1}|0\rangle_{c}^{s}+\zeta_{2}|1\rangle_{c}^{s})$
$\displaystyle+\delta_{2}\epsilon_{2}|1\rangle_{a}^{s}|1\rangle_{b}^{s}(\zeta_{1}|0\rangle_{c}^{s}-\zeta_{2}|1\rangle_{c}^{s})]$
$\displaystyle\otimes|\phi\rangle_{a}^{p}|\phi\rangle_{b}^{p}|\phi\rangle_{c}^{p},$

which is the result of the C2PF gate on the spatial DoF of the three-photon
system without affecting its polarized state. That is, when the two electron-
spin states in NV3 and NV4, and the polarized states of two photons $ab$ are
$|+1\rangle_{3}|+1\rangle_{4}$ and $|1\rangle_{a}^{s}|1\rangle_{b}^{s}$, the
CPF gate operation on the spatial state of photon $c$. However, the
measurement results of the two-electron-spin states are other possible
outcomes, the corresponding feed-forward single-qubit operations are shown in
Table 2. Here, phase shifter $e^{i\pi}$ ,which completes the transformation
$|0\rangle^{p}\rightarrow-|0\rangle^{p}$ and
$|1\rangle^{p}\rightarrow-|1\rangle^{p}$, is performed on spatial state, and
$I$ means keeping its original state. Therefore, the success probability of a
C2PF gate on the spatial DoF of a three-photon system is 100% in principle.
Finally, before and after the C2PF gate, Hadamard operation is performed on
the spatial DoF of the target qubit $c$, i.e., $U^{s}_{Toffoli}=(H^{s}\otimes
I_{4})U^{s}_{C^{2}PF}(I_{4}\otimes H^{s})$, the Toffoli gate on the spatial
DoF of the three-photon system is finished without impacting on the polarized
DoF of the three-photon system. Here,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!|\psi_{4}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\delta_{1}\epsilon_{1}|0\rangle_{a}^{s}|0\rangle_{b}^{s}+\delta_{1}\epsilon_{2}|0\rangle_{a}^{s}|1\rangle_{b}^{s}$
(16)
$\displaystyle+\delta_{2}\epsilon_{1}|1\rangle_{a}^{s}|0\rangle_{b}^{s})(\zeta_{1}|0\rangle_{c}^{s}+\zeta_{2}|1\rangle_{c}^{s})$
$\displaystyle+\delta_{2}\epsilon_{2}|1\rangle_{a}^{s}|1\rangle_{b}^{s}(\zeta_{2}|0\rangle_{c}^{s}+\zeta_{1}|1\rangle_{c}^{s})]$
$\displaystyle\otimes|\phi\rangle_{a}^{p}|\phi\rangle_{b}^{p}|\phi\rangle_{c}^{p}.$

Finally, we can combine the above two parts in Fig. 2 and 3 to construct an
error-detected hyper-parallel Toffoli gate with state-selective reflection, as
the Toffoli gates of the three-photon system in the polarized and spatial DoFs
are independent of each other. The hybrid quantum CPF gate operations are the
key elements of this hyper-Toffoli gate. The above two Tables 1 and 2 are also
valid when the four electron spins of the NV-cavity systems are in other
states. Thus, our error-detected hyper-parallel Toffoli gate is very flexible.
Here,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!|\Phi\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\alpha_{1}\beta_{1}|0\rangle_{a}^{p}|0\rangle_{b}^{p}+\alpha_{1}\beta_{2}|0\rangle_{a}^{p}|1\rangle_{b}^{p}$
(17)
$\displaystyle+\alpha_{2}\beta_{1}|1\rangle_{a}^{p}|0\rangle_{b}^{p})(\gamma_{1}|0\rangle_{c}^{p}+\gamma_{2}|1\rangle_{c}^{p})$
$\displaystyle+(\alpha_{2}\beta_{2}|1\rangle_{a}^{p}|1\rangle_{b}^{p})(\gamma_{1}|1\rangle_{c}^{p}+\gamma_{2}|0\rangle_{c}^{p})]$
$\displaystyle\otimes(\frac{r_{1}-r_{0}}{2})^{5}[(\delta_{1}\epsilon_{1}|0\rangle_{a}^{s}|0\rangle_{b}^{s}+\delta_{1}\epsilon_{2}|0\rangle_{a}^{s}|1\rangle_{b}^{s}$
$\displaystyle+\delta_{2}\epsilon_{1}|1\rangle_{a}^{s}|0\rangle_{b}^{s})(\zeta_{1}|0\rangle_{c}^{s}+\zeta_{2}|1\rangle_{c}^{s})$
$\displaystyle+(\delta_{2}\epsilon_{2}|1\rangle_{a}^{s}|1\rangle_{b}^{s})(\zeta_{1}|1\rangle_{c}^{s}+\zeta_{2}|0\rangle_{c}^{s})].$

As the incomplete and imperfect cavity-NV-center interactions are transformed
into the detectable failure rather than infidelity based on the novel
heralding mechanism of detectors in the block1 and block2, our hyper-Toffoli
gate of the three-photon system is error-detected. The efficiencies of the two
blocks can be further improved by repeating the operation processes when the
detectors in blocks are clicked. Now, we have obtained the result of the
error-detected hyper-parallel Toffoli gate with state-selective reflection of
one-sided cavity-NV center system. That is, when the polarized state of two
photons $ab$ are $|1\rangle_{a}^{p}|1\rangle_{b}^{p}$, the polarized states of
the photon $c$ is flipped; when the spatial state of two photons $ab$ are
$|1\rangle_{a}^{s}|1\rangle_{b}^{s}$, the spatial states of the photon $c$ is
flipped. No operation is performed on the photon $c$ when the polarized state
of two photons $ab$ are $|0\rangle_{a}^{p}|0\rangle_{b}^{p}$
($|0\rangle_{a}^{p}|1\rangle_{b}^{p}$ or $|1\rangle_{a}^{p}|0\rangle_{b}^{p}$)
and the spatial state of two photons $ab$ are
$|0\rangle_{a}^{s}|0\rangle_{b}^{s}$ ($|0\rangle_{a}^{s}|1\rangle_{b}^{s}$ or
$|1\rangle_{a}^{s}|0\rangle_{b}^{s}$).

Table 2: The measurement results of the two-electron-spin states in NV3 and NV4, and the corresponding feed-forward single-qubit operations on polarized DoF of three photons. measurement results | single-qubit operations
---|---
$|+1\rangle_{3}|+1\rangle_{4}$ | $I_{a}\otimes I_{b}\otimes I_{c}$
$|+1\rangle_{3}|-1\rangle_{4}$ | $e^{i\pi}|0\rangle_{a}^{s}\otimes I_{b}\otimes I_{c}$
$|-1\rangle_{3}|+1\rangle_{4}$ | $I_{a}\otimes e^{i\pi}|1\rangle_{b}^{s}\otimes I_{c}$
$|-1\rangle_{3}|-1\rangle_{4}$ | $e^{i\pi}|0\rangle_{a}^{s}\otimes e^{i\pi}|1\rangle_{b}^{s}\otimes I_{c}$

##  III Discussion and conclusion

In this work, we have designed an error-detected quantum circuit for
implementing the hyper-parallel Toffoli gate on a three-photon system in both
the polarization and spatial DoFs, assisted by state-selective reflection of
one-sided cavity-NV-center system. The hyper-Toffoli gate can perform double
Toffoli gate operations on only one three-photon system in two DoFs with four
one-sided cavity-NV center systems, which can save the resource depletion and
decrease the photonic dispassion noise in QIP. The interaction of the photons
is accomplished by the photon-electron-spin interaction in the NV center,
which can be greatly improve by coupled to an optical cavity, fiber-based
microcavity or microring resonator either in strong coupling regime and in
weak coupling regime in experiment Room-temperature2006 ; Gigahertz2009 ;
Deterministric2010 ; Experimental2015 ; Coupling2013 ; Integrated2012 ;
Strain2014 . The identically optical transition of the four uncorrelated NV
centers, can be achieved by applying controlled external electric fields
Observation2009 ; Electrical2011 . Our scheme nearly immunes to spectral
diffusion and charge fluctuation because of the narrow linewidth of the state
$|A_{2}\rangle$ Entangled2015 . Many techniques have been explored to reduce
and eliminate the spectral-diffusion influence High-filelity2011 ; Quantum2007
; Room2010 ; Optically2013 .

We can also construct three types of flexible hybrid hyper-Toffoli gate
assisted by the state-selective reflection of one-sided cavity-NV-center
system. The quantum circuit of the hybrid hyper-Toffoli gates in Fig. 4 is
similar to the one of the hyper-Toffoli gate in Fig. 2 and Fig. 3, by
replacing H and block1 with BS and block2, respectively. As shown in Fig. 4,
we construct a hybrid hyper-Toffoli gate, that is, the polarized (spatial)
states of two photons $a$ and $b$ to control the spatial (polarized) state of
photon $c$, respectively. In detail, if all Ds can not click of Fig. 4, the
quantum circuit finishes the polarized states of two photons $a$ and $b$ to
control the spatial state of photon $c$. Here,

$\displaystyle|\phi_{h}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\alpha_{1}\beta_{1}|0\rangle_{a}^{p}|0\rangle_{b}^{p}+\alpha_{1}\beta_{2}|0\rangle_{a}^{p}|1\rangle_{b}^{p}$
(18)
$\displaystyle+\alpha_{2}\beta_{1}|1\rangle_{a}^{p}|0\rangle_{b}^{p})(\zeta_{1}|0\rangle_{c}^{s}+\zeta_{2}|1\rangle_{c}^{s})$
$\displaystyle+(\alpha_{2}\beta_{2}|1\rangle_{a}^{p}|1\rangle_{b}^{p})(\zeta_{1}|1\rangle_{c}^{s}+\zeta_{2}|0\rangle_{c}^{s})]$
$\displaystyle\otimes|\phi\rangle_{a}^{s}|\phi\rangle_{b}^{s}|\phi\rangle_{c}^{p}.$

If all Ds in Fig. 4 can not click, the quantum circuit completes that the
spatial states of two photons $a$ and $b$ control the polarized state of
photon $c$. Here,

$\displaystyle|\phi_{t}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\delta_{1}\epsilon_{1}|0\rangle_{a}^{s}|0\rangle_{b}^{s}+\delta_{1}\epsilon_{2}|0\rangle_{a}^{s}|1\rangle_{b}^{s}$
(19)
$\displaystyle+\delta_{2}\epsilon_{1}|1\rangle_{a}^{s}|0\rangle_{b}^{s})(\gamma_{1}|0\rangle_{c}^{p}+\gamma_{2}|1\rangle_{c}^{p})$
$\displaystyle+(\delta_{2}\epsilon_{2}|1\rangle_{a}^{s}|1\rangle_{b}^{s})(\gamma_{1}|1\rangle_{c}^{p}+\gamma_{2}|0\rangle_{c}^{p})]$
$\displaystyle\otimes|\phi\rangle_{a}^{p}|\phi\rangle_{b}^{p}|\phi\rangle_{c}^{s}.$

Obviously, we can unite the two parts to achieve a hybrid hyper-paralleled
Toffoli gate with state-selective reflection. Here,

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!|\Phi_{1}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\alpha_{1}\beta_{1}|0\rangle_{a}^{p}|0\rangle_{b}^{p}+\alpha_{1}\beta_{2}|0\rangle_{a}^{p}|1\rangle_{b}^{p}$
(20)
$\displaystyle+\alpha_{2}\beta_{1}|1\rangle_{a}^{p}|0\rangle_{b}^{p})(\zeta_{1}|0\rangle_{c}^{s}+\zeta_{2}|1\rangle_{c}^{s})$
$\displaystyle+(\alpha_{2}\beta_{2}|1\rangle_{a}^{p}|1\rangle_{b}^{p})(\zeta_{1}|1\rangle_{c}^{s}+\zeta_{2}|0\rangle_{c}^{s})]$
$\displaystyle\otimes(\frac{r_{1}-r_{0}}{2})^{5}[(\delta_{1}\epsilon_{1}|0\rangle_{a}^{s}|0\rangle_{b}^{s}+\delta_{1}\epsilon_{2}|0\rangle_{a}^{s}|1\rangle_{b}^{s}$
$\displaystyle+\delta_{2}\epsilon_{1}|1\rangle_{a}^{s}|0\rangle_{b}^{s})(\gamma_{1}|0\rangle_{c}^{p}+\gamma_{2}|1\rangle_{c}^{p})$
$\displaystyle+(\delta_{2}\epsilon_{2}|1\rangle_{a}^{s}|1\rangle_{b}^{s})(\gamma_{1}|1\rangle_{c}^{p}+\gamma_{2}|0\rangle_{c}^{p})].$

Similarly, we can also construct other types of flexible hybrid hyper-Toffoli
gate assisted by the state-selective reflection of one-sided cavity-NV-center
system. For example, the polarized (spatial) states of photon $a$ and the
spatial (polarized) states of photon $b$ control the spatial (polarized) state
of photon $c$, respectively. Here,

$\displaystyle|\Phi_{2}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\alpha_{1}\epsilon_{1}|0\rangle_{a}^{p}|0\rangle_{b}^{s}+\alpha_{1}\epsilon_{2}|0\rangle_{a}^{p}|1\rangle_{b}^{s}$
$\displaystyle+\alpha_{2}\epsilon_{1}|1\rangle_{a}^{p}|0\rangle_{b}^{s})(\zeta_{1}|0\rangle_{c}^{s}+\zeta_{2}|1\rangle_{c}^{s})$
$\displaystyle+(\alpha_{2}\epsilon_{2}|1\rangle_{a}^{p}|1\rangle_{b}^{s})(\zeta_{1}|1\rangle_{c}^{s}+\zeta_{2}|0\rangle_{c}^{s})]$
$\displaystyle\otimes(\frac{r_{1}-r_{0}}{2})^{5}[(\delta_{1}\beta_{1}|0\rangle_{a}^{s}|0\rangle_{b}^{p}+\delta_{1}\beta_{2}|0\rangle_{a}^{s}|1\rangle_{b}^{p}$
$\displaystyle+\delta_{2}\beta_{1}|1\rangle_{a}^{s}|0\rangle_{b}^{p})(\gamma_{1}|0\rangle_{c}^{p}+\gamma_{2}|1\rangle_{c}^{p})$
$\displaystyle+(\delta_{2}\beta_{2}|1\rangle_{a}^{s}|1\rangle_{b}^{p})(\gamma_{1}|1\rangle_{c}^{p}+\gamma_{2}|0\rangle_{c}^{p})],$
$\displaystyle|\Phi_{3}\rangle$ $\displaystyle=$
$\displaystyle(\frac{r_{1}-r_{0}}{2})^{5}[(\alpha_{1}\epsilon_{1}|0\rangle_{a}^{p}|0\rangle_{b}^{s}+\alpha_{1}\epsilon_{2}|0\rangle_{a}^{p}|1\rangle_{b}^{s}$
(21)
$\displaystyle+\alpha_{2}\epsilon_{1}|1\rangle_{a}^{p}|0\rangle_{b}^{s})(\gamma_{1}|0\rangle_{c}^{p}+\gamma_{2}|1\rangle_{c}^{p})$
$\displaystyle+(\alpha_{2}\beta_{2}|1\rangle_{a}^{p}|1\rangle_{b}^{p})(\gamma_{1}|1\rangle_{c}^{p}+\gamma_{2}|0\rangle_{c}^{p})]$
$\displaystyle\otimes(\frac{r_{1}-r_{0}}{2})^{5}[(\delta_{1}\beta_{1}|0\rangle_{a}^{s}|0\rangle_{b}^{p}+\delta_{1}\beta_{2}|0\rangle_{a}^{s}|1\rangle_{b}^{p}$
$\displaystyle+\delta_{2}\beta_{1}|1\rangle_{a}^{s}|0\rangle_{b}^{p})(\zeta_{1}|0\rangle_{c}^{s}+\zeta_{2}|1\rangle_{c}^{s})$
$\displaystyle+(\delta_{2}\epsilon_{2}|1\rangle_{a}^{s}|1\rangle_{b}^{s})(\zeta_{1}|1\rangle_{c}^{s}+\zeta_{2}|0\rangle_{c}^{s})].$

Figure 4: Schematic diagram of a hybrid hyper-Toffoli gate.  Table 3: The efficiency $\eta_{T}$ of the hyper-Toffoli gates and the probability $\eta_{D}$ of D triggered in the block1 or block2 vs the ratio of $g/\kappa$ when $\omega_{c}=\omega_{p}=\omega_{c}$ at $\gamma=0.01\kappa$.  |  $g/\kappa=0.5$ |  $g/\kappa=1.5$ |  $g/\kappa=2.4$
---|---|---|---
$\eta_{T}$ | 90.53$\%$ | 98.90$\%$ | 99.57$\%$
$\eta_{D}$ | 0.01$\%$ | $0$ | $0$

In the above protocols, we have taken into account practical experiment
environment, including the cavity leakage rate and the coupling strength. The
probability $\eta_{D}^{p}$ of D triggered in the block1 and the one
$\eta_{D}^{s}$ in the block2 is equal to
$\eta_{D}^{p}=\eta_{D}^{s}=\eta_{D}=(\frac{r_{1}+r_{0}}{2})^{2}$, in Fig. 5,
which are related to the coupling strength $g$, cavity decay and leakage rate
$\kappa,\gamma$ at $\gamma=0.01\kappa$. Obviously, the probability $\eta_{D}$
shown in Table 3 exists zero values even in the weak coupling rate, which
results from the destructive interference. Therefore, the fidelity of the
hyper-Toffoli gates is unity in principle without strong couple limitation
between photon and NV center. These positive characteristics may make our
projects conducive to increase the information processing capability and
decrease the complexity of large-scale integration. The efficiency of the
hyper-Toffoli gates is $\eta_{T}=(\frac{r_{1}-r_{0}}{2})^{10}$, the ratio of
the number of output photons to the input. The relation between $\eta_{T}$ and
$g/\kappa$ is shown in Fig. 5 and the partial detailed data is shown in Table
3. One can see that our schemes can work not only in strong coupling rate but
also in the weak coupling rate. To achieve a higher efficiency $\eta_{T}$,
$\gamma/\kappa$ should be as small as possible, which can be realized by
improving the design and manufacturing process of cavity. Besides, the
efficiency of our scheme can also reduced from non-ideal single-photon
sources, imperfect linear-optical elements (BSs and CPBSs), and invalid dark
detectors, which will be improved with the further development of the current
technology.

Figure 5:  The efficiency $\eta_{T}$ of the hyper-Toffoli gates and the
probability $\eta_{D}$ of D triggered in the block1 vs the ratio of $g/\kappa$
when $\omega_{c}=\omega_{p}=\omega_{c}$ at $\gamma=0.01\kappa$.

In summary, we have proposed an error-detected hyper-Toffoli gate for the
three-photon system based on the interface between polarized photon and
cavity-NV-center system. This hyper-Toffoli gate is equivalent to perform
double Toffoli gate operations simultaneously on both the polarization and
spatial DoFs of the three-photon system with a low decoherence, short
operation time, and less quantum resources required, in compared with those on
two independent three-photon systems in one DoF only. Based on the heralding
mechanism of detectors, the imperfect interactions between the cavity-NV
center and the photon can be translated into judging the response status of
the detectors, so the fidelity of our proposal is near unity. The efficiency
of our optimal protocol for the hyper-Toffoli gate can be improved further by
repeating the operation processes when the detectors are clicked. We can also
construct three types of hybrid hyper-Toffoli gate assisted by the state-
selective reflection of one-sided cavity-NV-center system, as the quantum
circuit of the hyper-Toffoli gate is flexible and adjustable. Besides, the
modular circuits effectively reduce the complexity of Toffoli gate, and beyond
that, two photons enter the systems simultaneously from opposite directions
can effectively reduce the overall running time. Moreover, the evaluation of
gate performance with achieved experiment parameters, even in the weak
coupling, shows that it is feasible with current experimental technology and
provides a promising building block for quantum compute.

## Acknowledgments

This work was supported in part by the Natural Science Foundation of China
under Contract 61901420; in part by the Shanxi Province Science Foundation for
Youths under Contract 201901D211235; in part by the Scientific and
Technological Innovation Programs of Higher Education Institutions in Shanxi
under Contract 2019L0507; in part by the Shanxi ”1331 Project” Key Subjects
Construction.

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