Optimization of universal quantum gates with higher-dimensional spaces

Wen-Qiang Liu
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
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

Hai-Rui Wei
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China

Leong-Chuan Kwek
Centre for Quantum Technologies, National University of Singapore, Singapore 117543, Singapore
MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit,  Singapore UMI 3654, Singapore
National Institute of Education and Institute of Advanced Studies, Nanyang Technological University, Singapore 637616, Singapore

Minimizing the number of necessary two-qubit gates  is an important task in quantum information processing. By introducing non-computational quantum states, we construct effective circuits for the controlled-NOT (CNOT) gate and the $n$-qubit Toffoli gate with $(2n-3)$ qubit-qudit gates and $(2n-4)$ single-qudit gates. The cost of a three-qubit Toffoli gate is reduced from the minimum limit of five two-qubit gates to three nearest-neighbor qubit-qudit gates.  By operating with the spatial-mode degree of freedom, polarization CNOT and Toffoli gates are designed with linear optics. Our optical schemes can be achieved with a higher success probability and no extra auxiliary photons are needed in the scheme.

§ INTRODUCTION

Multi-qubit quantum gates have complex structures and play an important role in quantum computing [Barenco et al., 1995], quantum algorithms [Grover, 1997, Shor, 1999, Figgatt et al., 2017, Nam et al., 2020], cryptography [Gisin et al., 2002], etc. [Nielsen and Chuang, 2010]. The most popular paradigm for implementing a quantum gate is the quantum circuit model [Barenco et al., 1995]. However, when the cost (also called complexity) of the quantum circuit (measured by the number of two-qubit gates involved in a quantum circuit) is high, it is difficult to perform the experiments because of the low computing fidelity and limited coherence time. Moreover, the cost of a universal quantum circuit increases exponentially as the number of qubits increases. The theoretical lower bound for simulating an $n$-qubit universal quantum circuit is $(4^n-3n-1)/4$ controlled-NOT (CNOT) gates in qubit system [Shende et al., 2004]. Hence, it is crucial to find an effective method for building a universal quantum circuit in the simplest possible way.

Several matrix decomposition techniques have been introduced to optimize a large-scale quantum circuit [Khaneja and Glaser, 2001, Möttönen et al., 2004, Bullock and Brennen, 2004, Shende et al., 2006, D'Alessandro and Albertini, 2007, Di and Wei, 2013, Iten et al., 2016]. Two-qubit universal quantum circuits have also  been constructed with the lowest cost (resources) in qubit systems [Vatan and Williams, 2004, Vidal and Dawson, 2004, Shende et al., 2004, Shende et al., 2004]. However, there is still a gap between the current best result [Shende et al., 2006] and the theoretical lower bound [Shende et al., 2004] for a multi-qubit universal quantum circuit. Fortunately, Ralph et al. [Ralph et al., 2007] found that the quantum circuit may be optimized further by using higher-dimensional Hilbert spaces, and this proposal was later experimentally demonstrated in optical [Lanyon et al., 2009] and superconducting systems [Fedorov et al., 2012].
Following this, Liu et al. [Liu et al., 2012, Li et al., 2013] reduced the cost of the $n$-qubit universal circuit to $(5/16)\times 4^n-(5/4)\times 2^n+2n$ CNOT gates when $n$ was even and $(5/16)\times 4^n-2^n+2(n-1)$ CNOTs when $n$ was odd. Liu et al.  simplified a Fredkin gate from eight CNOTs to five CNOTs [Liu and Wei, 2020] or three qubit-qudit gates [Liu et al., 2020]. In addition, higher-dimensional systems have also been used in the optimization of quantum communication [Cerf et al., 2002, Malik et al., 2016, Zhang et al., 2017, Wang et al., 2017, Hu et al., 2018, Luo et al., 2019, Hu et al., 2020] and quantum metrology [D'ambrosio et al., 2013].

The Toffoli (controlled-NOT-NOT) gate, a three-qubit conditional operation, is one of the most popular universal multi-qubit quantum gates [Shi, 2002]. It is also an essential component in complex quantum algorithms [Grover, 1997, Shor, 1999, Figgatt et al., 2017, Nam et al., 2020], quantum error correction [Cory et al., 1998, Reed et al., 2012], and quantum fault tolerance [Paetznick and Reichardt, 2013, Guillaud and Mirrahimi, 2019]. In 1995, Barenco et al. [Barenco et al., 1995] proposed a concrete construction of a three-qubit Toffoli gate with five two-qubit entangled gates. When two-qubit gates are restricted to CNOT gates, the optimal cost of a Toffoli gate increases to six [Shende and Markov, 2008]. In 2013, Yu et al. [Yu et al., 2013, Yu and Ying, 2015] confirmed that the minimum resource for simulating a three-qubit Toffoli gate is five two-qubit gates. In 2020, Kiktenko et al. [Kiktenko et al., 2020] constructed a generalized $m$-qubit Toffoli gate with $(2m-3)$ CNOTs based on qudits. Independent of the standard decomposition-based approach, Toffoli gates have been implemented experimentally in superconducting circuits [Fedorov et al., 2012, Reed et al., 2012], linear optics [Lanyon et al., 2009, Mičuda et al., 2013, Lemr et al., 2015, Mičuda et al., 2015, Ru et al., 2021], trapped ions [Monz et al., 2009], atoms [Beterov et al., 2018, Levine et al., 2019], and quantum dots [Gullans and Petta, 2019].

In this paper, we propose a simplified CNOT and multi-qubit Toffoli gates by using higher-dimensional spaces.
Specifically, $(2n-3)$ qubit-qudit and  $(2n-4)$ single-qudit gates are required to implement an $n$-qubit Toffoli gate. The cost of the three-qubit Toffoli gate is reduced from the lower bound of five two-qubit gates to one CNOT and two partial-swap (P-SWAP) gates. In addition, using the spatial-mode degree of freedom (DOF) of the single photon, we design a feasible optical architecture for implementing CNOT and Toffoli gates with linear optics. Our proposals have several other advantages: (i) Our optical implementation of the CNOT gate does not require an extra entangled photon pair or a single photon, and the success probability of the gate is enhanced. (ii) The cost of the three-qubit Toffoli gate is reduced from the unconstructed lower bound of five two-qubit gates [Yu et al., 2013, Yu and Ying, 2015] three qubit-qudit gates. (iii) Linear optical Toffoli  gates can be constructed with a higher success probability than other existing optical schemes [Ralph et al., 2007, Lanyon et al., 2009, Fiurášek, 2006].

§ CONSTRUCTION OF CNOT AND TOFFOLI GATES WITH HIGHER-DIMENSIONAL SPACES

We focus on the simplification of controlled quantum gates with minimal resource overheads, and the main idea of our proposal is to extend temporarily the higher-dimensional subspaces on one of the controlled qubit carriers and then perform corresponding logical operations.

§.§ Synthesis of a CNOT gate using qutrits

A CNOT gate with two P-SWAP gates using qutrits is shown in Fig. <ref>. The gate qubits are encoded on two computational states, $|0\rangle$ and $|1\rangle$. The single-qutrit $X_A$ gate provides a three-dimensional subspace on the control qubit. In the following, we describe the construction process of our protocol in detail.

Synthesis of a CNOT gate. The single-qutrit $X_A$ gate implements the transformation $|1\rangle \leftrightarrow |2\rangle$. The controlled node $\odot$  is turned on for the input $|0\rangle$ or $|1\rangle$. That is, a swap operation is applied to $c$ and $t$, if and only if, the control qubit $c$ is in the state $|0\rangle$ or $|1\rangle$. $H$ is a single-qubit Hardmard gate to achieve operations $|0\rangle \leftrightarrow \frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$ and $|1\rangle \leftrightarrow \frac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$. $\sigma_z$ completes $\sigma_z|0\rangle=|0\rangle$ and $\sigma_z|1\rangle=-|1\rangle$.

Suppose that the state of the system is initially

\begin{align}
\begin{split}             \label{eq1}
|\phi_0\rangle=&\alpha_1|0_{c}\rangle|0_{t}\rangle+\alpha_2|0_{c}\rangle|1_{t}\rangle \\&
\end{split}
\end{align}

where $\alpha_i$ ($i=1, 2, 3, 4$) are complex coefficients that satisfy the normalization condition $\sum^{4}_{i=1} |\alpha_{i}|^{2}=1$. Subscripts $c$ and $t$ denote the control and target qubits, respectively.

First, qubit $c$ undergoes a single-qutrit gate $X_A$, which introduces an ancillary state $|2\rangle$ on $c$ and completes the transformations $|1_{c}\rangle \stackrel{X_A}{\longleftrightarrow} |2_{c}\rangle$ and $|0_{c}\rangle \stackrel{X_A}{\longleftrightarrow} |0_{c}\rangle$. After the $X_A$ gate and a Hadamard ($H$) gate are applied to $c$ and $t$, the initial state $|\phi_0\rangle$ is changed to

\begin{align}
\begin{split}             \label{eq2}
|\phi_1\rangle=&\frac{1}{\sqrt{2}}[\alpha_1|0_{c}\rangle(|0_{t}\rangle+|1_{t}\rangle)\\&+\alpha_2|0_{c}\rangle(|0_{t}\rangle-|1_{t}\rangle) \\
\end{split}
\end{align}

Second, a P-SWAP gate is applied to $c$ and $t$, and it transforms $|\phi_1\rangle$ into

\begin{align}
\begin{split}             \label{eq3}
|\phi_2\rangle=&\frac{1}{\sqrt{2}}[\alpha_1(|0_{c}\rangle+|1_{c}\rangle)|0_{t}\rangle  \\&+\alpha_2(|0_{c}\rangle-|1_{c}\rangle)|0_{t}\rangle \\
\end{split}
\end{align}

Here, the P-SWAP gate performs a swap operation only between two computational states $|0\rangle$ and $|1\rangle$, that is,

\begin{eqnarray}              \label{eq4}
\begin{split}
&|00\rangle \xrightarrow{\text{P-SWAP}} |00\rangle, \;\; &|01\rangle \xrightarrow{\text{P-SWAP}} |10\rangle, \\
&|10\rangle \xrightarrow{\text{P-SWAP}} |01\rangle, \;\; &|11\rangle \xrightarrow{\text{P-SWAP}} |11\rangle, \\
&|20\rangle \xrightarrow{\text{P-SWAP}} |20\rangle, \;\; &|21\rangle \xrightarrow{\text{P-SWAP}} |21\rangle.
\end{split}
\end{eqnarray}

Third, a $\sigma_z$ operation acts on $t$ to change $|\phi_2\rangle$ to

\begin{align}
\begin{split}             \label{eq5}
|\phi_3\rangle=&\frac{1}{\sqrt{2}}[\alpha_1(|0_{c}\rangle+|1_{c}\rangle)|0_{t}\rangle\\&+\alpha_2(|0_{c}\rangle-|1_{c}\rangle)|0_{t}\rangle \\
\end{split}
\end{align}

Finally, after the P-SWAP gate, the $X_A$ gate and $H$ operation are applied to $c$ and $t$ again,  $|\phi_3\rangle$ is changed to

\begin{align}
\begin{split}             \label{eq6}
\end{split}
\end{align}

Note that Eq. (<ref>) is a CNOT gate, and such a construction can be achieved in linear optics with a high success probability and without additional photons (see Sec. <ref>).

§.§ Construction of Toffoli gates with higher-dimensional spaces

§.§.§ Synthesis of a three-qubit Toffoli gate using qutrits

Based on the designed CNOT and P-SWAP gates, the process for implementing a three-qubit Toffoli gate with four-dimensional space  is presented in Fig. <ref>.

Simplified synthesis of a three-qubit Toffoli gate with two P-SWAP and one CNOT gates. As shown in Fig. <ref>, the operations in the dotted rectangle are a CNOT gate.

Considering an arbitrary normalization three-qubit initial state

\begin{align}
\begin{split}             \label{eq7}
\\&+\alpha_3|0_{c_1}\rangle|1_{c_2}\rangle|0_{t}\rangle+\alpha_4|0_{c_1}\rangle|1_{c_2}\rangle|1_{t}\rangle\\
\\&+\alpha_7|1_{c_1}\rangle|1_{c_2}\rangle|0_{t}\rangle+\alpha_8|1_{c_1}\rangle|1_{c_2}\rangle|1_{t}\rangle.
\end{split}
\end{align}

First, the $X_A$ gate acts on $c_2$ to achieve $|1_{c_2}\rangle \stackrel{X_A}{\longleftrightarrow} |2_{c_2}\rangle$ and $|0_{c_2}\rangle \stackrel{X_A}{\longleftrightarrow} |0_{c_2}\rangle$. After the first P-SWAP gate is executed on $c_1$ and $c_2$, $|\psi_0\rangle$ becomes

\begin{align}              \label{eq8}
\begin{split}
\\&+\alpha_3|0_{c_1}\rangle|2_{c_2}\rangle|0_{t}\rangle+\alpha_4|0_{c_1}\rangle|2_{c_2}\rangle|1_{t}\rangle\\
\\&+\alpha_7|1_{c_1}\rangle|2_{c_2}\rangle|0_{t}\rangle+\alpha_8|1_{c_1}\rangle|2_{c_2}\rangle|1_{t}\rangle.
\end{split}
\end{align}

Second, a CNOT gate is applied to $c_1$ and $t$ (which can be achieved by the circuit in the
dotted rectangle), resulting in

\begin{align}              \label{eq9}
\begin{split}
\\&+\alpha_3|0_{c_1}\rangle|2_{c_2}\rangle|0_{t}\rangle+\alpha_4|0_{c_1}\rangle|2_{c_2}\rangle|1_{t}\rangle\\
\\&+\alpha_7|1_{c_1}\rangle|2_{c_2}\rangle|1_{t}\rangle+\alpha_8|1_{c_1}\rangle|2_{c_2}\rangle|0_{t}\rangle.
\end{split}
\end{align}

Finally, the P-SWAP and $X_A$ gates are applied again. The two operations induce $|\psi_2\rangle$ as the final state

\begin{align}              \label{eq10}
\begin{split}
\\&+\alpha_3|0_{c_1}\rangle|1_{c_2}\rangle|0_{t}\rangle+\alpha_4|0_{c_1}\rangle|1_{c_2}\rangle|1_{t}\rangle\\
\\&+\alpha_7|1_{c_1}\rangle|1_{c_2}\rangle|1_{t}\rangle+\alpha_8|1_{c_1}\rangle|1_{c_2}\rangle|0_{t}\rangle.
\end{split}
\end{align}

From Eqs. (<ref>-<ref>), one can see that a three-qubit Toffoli gate can be simulated using three nearest-neighbor  qubit-qudit gates and two single-qutrit gates.  The cost of this construction outperforms the previous synthesis ones that require six CNOT gates [Shende and Markov, 2008] or five two-qubit gates [Barenco et al., 1995].

§.§.§ Synthesis of $n$-control-qubit Toffoli gate using qudits

Using a higher-dimensional space, the method can be applied to any multi-qubit Toffoli gate. As shown in Fig. <ref>, an $n$-control-qubit Toffoli gate is constructed with ($2n-1$) qubit-qudit and ($2n-2$) single-qudit gates, which flips the target qubit states $|0\rangle$ and $|1\rangle$ if and only if the $n$ control-qubits are all $|1\rangle$. Here, single-qudit gates $X_a$, $X_b$, $\cdots$, $X_n$ create multi-level qudits on $c_n$ and complete transformations $|0_{c_n}\rangle\leftrightarrow |2_{c_n}\rangle$, $|1_{c_n}\rangle\leftrightarrow |3_{c_n}\rangle$, $|0_{c_n}\rangle\leftrightarrow |4_{c_n}\rangle$, $\cdots$, $|0_{c_n}\rangle\leftrightarrow |n_{c_n}\rangle$ when $n$ is even or $|0_{c_n}\rangle\leftrightarrow |2_{c_n}\rangle$, $|1_{c_n}\rangle\leftrightarrow |3_{c_n}\rangle$, $|0_{c_n}\rangle\leftrightarrow |4_{c_n}\rangle$, $\cdots$, $|1_{c_n}\rangle\leftrightarrow |n_{c_n}\rangle$ when $n$ is odd. These single-qudit gates can temporarily expand the two-dimensional space of $c_n$ to an ($n$+1)-dimensional subspace. All CNOT and P-SWAP gates act on computational states $|0\rangle$  and $|1\rangle$. The synthesis requires $O(n)$  qubit-qudit gates, which is considerably less than the previous result, which required $O(n^2)$ CNOT gates [Barenco et al., 1995]. The low-cost advantage is more evident in our scheme as the number of qubits increases.

Synthesis of an $n$-control-qubit Toffoli gate in a higher-dimensional space. Single-qudit gates $X_a$, $X_b$, $\cdots$, $X_n$ complete operations $|0_{c_n}\rangle\leftrightarrow |2_{c_n}\rangle$, $|1_{c_n}\rangle\leftrightarrow |3_{c_n}\rangle$, $\cdots$, $|0_{c_n}\rangle\leftrightarrow |n_{c_n}\rangle$ when $n$ is even or $|0_{c_n}\rangle\leftrightarrow |2_{c_n}\rangle$, $|1_{c_n}\rangle\leftrightarrow |3_{c_n}\rangle$, $\cdots$, $|1_{c_n}\rangle\leftrightarrow |n_{c_n}\rangle$ when $n$ is odd, respectively.

§ IMPLEMENTATION OF LINEAR OPTICAL CNOT  AND TOFFOLI GATES

In the previous section, we proposed the simulation of CNOT and Toffoli gates based on P-SWAP gates and auxiliary higher-dimensional spaces. In an optical system, two computational states can be encoded on the polarization DOF of a single photon in the spatial-mode $i$, that is, $|0\rangle \equiv |H\rangle_i$ and $|1\rangle \equiv |V\rangle_i$. Here, $H$ and $V$ represent the horizontal and vertical polarized components, respectively. The higher-dimensional state can be encoded on the $V$-polarized component in a new spatial-mode $i'$, that is, $|2\rangle \equiv |V\rangle_{i'}$. The qutrit operation $X_A$ can be achieved by employing a polarizing beam splitter (PBS), which reflects the $V$-polarized component and transmits the $H$-polarized component, respectively. Before describing the implementation of the CNOT gate, we first detail the step-by-step construction of the P-SWAP gate with linear optical elements.

As shown in Fig. <ref>, the injected photon 1 is divided into $H$-polarized component and $V$-polarized component by a PBS. The $H$-polarized component passes into the spatial-mode $1_{in}$, which is encoded on $|H\rangle_{1_{in}}\equiv |0\rangle$  (and $V$-polarized component in the spatial-mode $1_{in}$ is encoded on $|V\rangle_{1_{in}}\equiv |1\rangle$), while the $V$-polarized component is reflected into another spatial-mode $1^{'}_{in}$, which is encoded on $|V\rangle_{1^{'}_{in}} \equiv |2\rangle$. The photon 2  from the spatial-mode $2_{in}$ is encoded on $|H\rangle_{2_{in}} \equiv |0\rangle$ and $|V\rangle_{2_{in}} \equiv |1\rangle$. A general injected photon state can be considered as

\begin{align}              \label{eq11}
\begin{split}
\end{split}
\end{align}

First, PBS$_1$ and PBS$_2$ transmit the $H$-photons into modes 1 and 3 to interact with half-wave plates HWP$^{45^\circ}$ and HWP$^{22.5^\circ}$ and reflect the $V$-photons into modes 2 and 4 to interact with HWP$^{45^\circ}$ and HWP$^{67.5^\circ}$. Here, HWP$^{45^\circ}$ is a half-wave plate set to 45 degrees and achieves the qubit-flip operation $|H\rangle\leftrightarrow|V\rangle$. HWP$^{22.5^\circ}$ completes the transformations

\begin{align}                  \label{eq12}
\begin{split}
&|H\rangle\stackrel{\text{HWP$^{22.5^\circ}$}}{\longleftrightarrow}\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle),  \\
\end{split}
\end{align}

HWP$^{67.5^\circ}$ results in

\begin{align}                  \label{eq13}
\begin{split}
&|H\rangle\stackrel{\text{HWP$^{67.5^\circ}$}}{\longleftrightarrow}\frac{1}{\sqrt{2}}(-|H\rangle+|V\rangle), \\
\end{split}
\end{align}

The above operations, PBS$_1$ $\rightarrow$ HWP$^{45^\circ}$ (HWP$^{45^\circ}$) and PBS$_2$ $\rightarrow$ HWP$^{22.5^\circ}$ (HWP$^{67.5^\circ}$) cause $|\varphi_0\rangle$ to become

\begin{align}              \label{eq14}
\begin{split}
\end{split}
\end{align}

Implementation of a linear optical P-SWAP gate. The half-wave plate HWP$^{45^\circ}$ realizes the qubit-flip $|H\rangle\leftrightarrow|V\rangle$. HWP$^{22.5^\circ}$ completes the operations $|H\rangle\leftrightarrow\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle)$ and $|V\rangle\leftrightarrow\frac{1}{\sqrt{2}}(|H\rangle-|V\rangle)$, and HWP$^{67.5^\circ}$ completes $|H\rangle\leftrightarrow\frac{1}{\sqrt{2}}(-|H\rangle+|V\rangle)$ and $|V\rangle\leftrightarrow\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle)$. BS is a balanced beam splitter to realize $|V\rangle_{1^{'}_{in}} \leftrightarrow\frac{1}{\sqrt{2}}(|V\rangle_{1^{'}}+|V\rangle_{1^{''}})$.

Photons in mode $1'_{in}$ are then split into modes $1'$ and $1''$ by a balanced polarization beam splitter (BS), i.e., $|V\rangle_{1^{'}_{in}} \xrightarrow{\text{BS}}(|V\rangle_{1^{'}}+|V\rangle_{1^{''}})/\sqrt{2}$.  Photons emitted from modes 2 and 4 (1 and 3) are split into modes 5 and 6 (7 and 8) by PBS$_3$ (PBS$_4$) and followed  by HWP$^{22.5^\circ}$ (HWP$^{67.5^\circ}$). These elements change $|\varphi_1\rangle$ as

\begin{align}              \label{eq15}
\begin{split}
\end{split}
\end{align}

Finally, PBS$_5$ (PBS$_6$) induces photons into modes 9 and 10 (11 and 12). Photons in modes 10 and 12 will undergo HWP$^{45^\circ}$. Thus, the state of the system evolves as

\begin{align}              \label{eq16}
\begin{split}
\end{split}
\end{align}

Coincident expectation outgoing values for six logic basis inputs. $\langle n_{H_{i}/V_{i}}n_{H_{j}/V_{j}}\rangle$ is the number of $H/V$-photons detected in modes $i$ and $j$.

$\langle n_{H_{9}}n_{H_{12}}\rangle$      $\langle n_{V_{9}}n_{H_{12}} \rangle$          $\langle n_{H_{9}}n_{V_{12}} \rangle$         $\langle n_{V_{9}}n_{V_{12}} \rangle$      $\langle n_{V_{1'}}n_{H_{12}} \rangle$      $\langle n_{V_{1'}}n_{V_{12}} \rangle$

$\langle n_{H_{10}}n_{H_{12}}\rangle$      $\langle n_{V_{10}}n_{H_{12}} \rangle$          $\langle n_{H_{10}}n_{V_{12}} \rangle$         $\langle n_{V_{10}}n_{V_{12}} \rangle$      $\langle n_{V_{1'}}n_{H_{12}} \rangle$      $\langle n_{V_{1'}}n_{V_{12}} \rangle$

$\langle n_{H_{9}}n_{H_{11}}\rangle$      $\langle n_{V_{9}}n_{H_{11}} \rangle$          $\langle n_{H_{9}}n_{V_{11}} \rangle$         $\langle n_{V_{9}}n_{V_{11}} \rangle$      $-\langle n_{V_{1'}}n_{H_{11}} \rangle$      $-\langle n_{V_{1'}}n_{V_{11}} \rangle$

$\langle n_{H_{10}}n_{H_{11}}\rangle$      $\langle n_{V_{10}}n_{H_{11}} \rangle$          $\langle n_{H_{10}}n_{V_{11}} \rangle$         $\langle n_{V_{10}}n_{V_{11}} \rangle$      $-\langle n_{V_{1'}}n_{H_{11}} \rangle$      $-\langle n_{V_{1'}}n_{V_{11}} \rangle$

$H_{1_{in}}H_{2_{in}}$       1/8       0       0        0       0        0

$H_{1_{in}}V_{2_{in}}$         0       1/8      0      0       0        0

$V_{1_{in}}H_{2_{in}}$        0         0       1/8      0       0        0

$V_{1_{in}}V_{2_{in}}$        0        0         0       1/8      0        0

$V_{1'_{in}}H_{2_{in}}$      0         0       0      0        1/8        0

$V_{1'_{in}}V_{2_{in}}$       0         0       0       0        0        1/8

As shown in Table <ref>, there are four desired coincidence outcomes in the construction of the post-selection P-SWAP gate. (i) When the detected output modes are 9, $1'$, and 12, the state $|\varphi_3\rangle$ will collapse into

\begin{align}              \label{eq17}
\begin{split}
\end{split}
\end{align}

(ii) When the detected output modes are 10, $1'$, and 12, the state $|\varphi_3\rangle$ will collapse into

\begin{align}              \label{eq18}
\begin{split}
\end{split}
\end{align}

(iii) When the detected output modes are 9, $1'$, and 11, the state $|\varphi_3\rangle$ will collapse into

\begin{align}              \label{eq19}
\begin{split}
\end{split}
\end{align}

(iv) When the detected output modes are 10, $1'$, and 11, the state $|\varphi_3\rangle$ will collapse into

\begin{align}              \label{eq20}
\begin{split}
\end{split}
\end{align}

According to Eq. (<ref>) to Eq. (<ref>), one can see that when the outing components emitting from modes 9, 1', and 12 (or from 10, 1', and 12), the P-SWAP gate is completed. When the outing components emitting from 9, 1', and 11 (or from 10, 1' and 11), the gate can also be completed by setting a feed-forward phase shifter $P_\pi$ in mode $1'$ which achieves the transformations $-\alpha_5|V\rangle_{1^{'}}|H\rangle_{11}\xrightarrow{P_\pi} \alpha_5|V\rangle_{1^{'}}|H\rangle_{11}$ and $-\alpha_6|V\rangle_{1^{'}}|V\rangle_{11} \xrightarrow{P_\pi}  \alpha_6|V\rangle_{1^{'}}|V\rangle_{11}$. Based on the spatial modes of the outing photons, the classical feed-forward operations can work correctly.  Therefore, the quantum circuit shown in Fig. <ref> completes a linear optical P-SWAP gate in the coincidence basis with a success probability of 1/2. The success (or the output modes) of the scheme can be heralded by using the success instances in the post-selection in the applications.

Implementation of a linear optical CNOT gate.  The operation in the dotted rectangle is a P-SWAP gate, as shown in Fig. <ref>. The numbers represent input-output modes.

As shown in Fig. <ref>, a P-SWAP-based CNOT gate can be realized in the coincidence basis with linear optical elements. PBS plays a role in the qutrit $X_A$ to provide an additional spatial mode. The operation in the black dotted rectangle corresponds to a P-SWAP gate in Fig. <ref>.

First, the initial state of the two injected  photons is

\begin{align}
\begin{split}             \label{eq21}
\end{split}
\end{align}

Second, photons 1 and 2 execute a PBS$_1$ and an HWP$^{22.5^\circ}$, respectively, to pass through the first P-SWAP gate. After the photons interact with the first P-SWAP gate, the outing photons emitted from modes 9, $1'$, and 12, and 10, $1'$, and 12 (without feed-forward operations) will be led to the next HWP$^{0^\circ}$ and the rightmost P-SWAP gate, and then, $|\chi_0\rangle$ is changed to $|\chi_{i,12}\rangle$, $i\in\{9,10\}$. Here,

\begin{align}
\begin{split}             \label{eq22}
\end{split}
\end{align}

Third, HWP$^{0^\circ}$ acts on mode 12 to complete $|H\rangle_{12} \rightarrow |H\rangle_{12}$ and $|V\rangle_{12} \rightarrow -|V\rangle_{12}$. The second P-SWAP gate produces eight desired outcomes of the system, that is,

(i) when the photons emitted from modes  9, $1’$, and 11, the state $|\chi_{i,12}\rangle$ will collapse into

\begin{align}
\begin{split}             \label{eq23}
\end{split}
\end{align}

(ii) When the photons emitted from modes  10, $1’$, and 11, the state $|\chi_{i,12}\rangle$ will collapse into

\begin{align}
\begin{split}             \label{eq24}
\end{split}
\end{align}

(iii) When the photons emitted from modes  9, $1’$, and 12, the state $|\chi_{i,12}\rangle$ will collapse into

\begin{align}
\begin{split}             \label{eq25}
\end{split}
\end{align}

(iv) When the photons emitted from modes  10, $1’$, and 12, the state $|\chi_{i,12}\rangle$ will collapse into

\begin{align}
\begin{split}             \label{eq26}
\end{split}
\end{align}

Finally, modes 9 (10) and $1'$ are combined into the same mode by PBS$_2$, and HWP$^{22.5^\circ}$ is applied to mode 12 (11) again. After PBS$_2$ and HWP$^{22.5^\circ}$, we obtain $|\chi^+_{9,12}\rangle'$, $|\chi^+_{10,12}\rangle'$, $|\chi^-_{9,11}\rangle'$, and  $|\chi^-_{10,11}\rangle'$. That is, $|\chi^\pm_{i,j}\rangle'$ with $i\in\{9,10\}$ and $j\in\{11,12\}$ is described by

\begin{align}
\begin{split}             \label{eq27}
\\
\end{split}
\end{align}

The minus signs of $|\chi^-_{9,11}\rangle'$ and $|\chi^-_{10,11}\rangle'$ can be corrected by setting the HWP$^{0^\circ}$ to modes 9 and 10 when two outgoing photons are at output ports 9 and 11, or 10 and 11. That is, after the feed-forward operations are only applied to the rightmost P-SWAP gate, an optical post-selection CNOT gate with a success probability of $1/8=1/4\times1/2$ is completed. Remarkably, additional entangled photon pairs or single photons are necessary for previous schemes [Pittman et al., 2001, Gasparoni et al., 2004, Bao et al., 2007, Zeuner et al., 2018], but are not required for our CNOT gate. In addition, the success probability of the gate is improved on the results without an auxiliary photon [Langford et al., 2005, Kiesel et al., 2005, Okamoto et al., 2005].

As shown in Fig. <ref>, after three input photons are injected into modes 1, 2, and 3, the initial state is given by

\begin{align}
\begin{split}             \label{eq28}
\end{split}
\end{align}

Optical implementation of a three-photon Toffoli gate.

First, after photons go through the PBS$_1$ and the leftmost P-SWAP gate, when we choose the output modes 9, $1'$ and 12, and 10, $1'$ and 12 (9, $1'$ and 11, and 10, $1'$ and 11 with the classical feed-forward operations), we can obtain four desired outcomes,

\begin{align}
\begin{split}             \label{eq29}
\end{split}
\end{align}

\begin{align}
\begin{split}             \label{eq30}
\end{split}
\end{align}

\begin{align}
\begin{split}             \label{eq31}
\end{split}
\end{align}

\begin{align}
\begin{split}             \label{eq32}
\end{split}
\end{align}

Second, the CNOT gate acts on photon 1 and photon 3, which can yield 16 desired outcomes $|\Xi_{i,j,k}\rangle$ (two-fold). Here, $|\Xi_{i,j,k}\rangle$ with $i\in\{9, 10\}, j\in\{9, 10\}$ and $k\in\{11, 12\},$  is described by

\begin{align}
\begin{split}             \label{eq33}
\end{split}
\end{align}

Third, after the rightmost P-SWAP gate acts on photon 1 and photon 2, when we choose the output modes 9, $1’$ and 12 (10, 1’ and 12), we can obtain 32 desired outcomes $|\Xi_{12,j,k}\rangle$ (eight-fold). Here, $|\Xi_{12,j,k}\rangle$ with $j\in\{9, 10\}$ and $k\in\{11, 12\}$,  is described by

\begin{align}
\begin{split}             \label{eq34}
\end{split}
\end{align}

Finally, photons in modes 9 (10) and $1’$ can be combined into the same mode by a PBS$_2$. Therefore, the linear optical Toffoli gate can been achieved without auxiliary photons.  Our proposal has a higher success probability ($1/64=1/2\times1/8\times1/4$) than the simplified CNOT-based one (1/72) [Ralph et al., 2007, Lanyon et al., 2009] and the one without a decomposition-based approach (1/133)  [Fiurášek, 2006]. In addition, optical single-qudit operation ensembles $X_a$, $X_b$, $\cdots$, $X_n$ can be achieved by employing a sequence of PBSs, and the linear optical $n$-control-photon Toffoli gate can be implemented in principle (see Fig. <ref>).

Implementation of an ($n+1$)-photon Toffoli gate.

§ DISCUSSION AND CONCLUSION

The optimal cost of a Toffoli gate is six CNOT gates using the standard decomposition-based approach [Shende and Markov, 2008]. The theoretical lower bound of a Toffoli gate is five two-qubit gates [Yu et al., 2013]. Ralph et al. [Ralph et al., 2007] first reduced the cost of a Toffoli gate to three CNOT gates by introducing auxiliary Hilbert space. Inspired by traditional all CNOT gate synthesis [Ralph et al., 2007, Lanyon et al., 2009],  the quantum circuit we designed to implement the Toffoli gate possesses the higher success probability than the protocols in Refs. [Ralph et al., 2007, Lanyon et al., 2009] with the same cost in terms of required qubit-qudit gates. The required qubit-qudit entangled  gates are all nearest neighbors in our construction of the three-qubit Toffoli gate. Note that the nearest-neighbor quantum gate where each qubit interacts only with its nearest neighbors requires less resource overhead than the long-range one. For example, a long-range CNOT gate acting on the first qubit and the third qubit is constructed by four nearest-neighbor CNOT gates [Viamontes et al., 2009]. In addition, ($2n-3$)  qubit-qudit gates can simulate an $n$-qubit Toffoli gate in higher-dimensional spaces, which considerably improves on the previous result with $O(n^2)$ [Barenco et al., 1995].

Linear optics has inherent probability characteristics for the implementation of controlled quantum gates. With the help of an additional entangled photon pair [Gasparoni et al., 2004, Bao et al., 2007] or a single photon  [Zeuner et al., 2018], optical CNOT gate with a success probability of 1/4  or 1/8 can be realistically implemented. Without auxiliary photons, CNOT gate with a success probability of 1/9 has been experimentally demonstrated in linear optics [Langford et al., 2005, Kiesel et al., 2005, Okamoto et al., 2005]. Remarkably, the success probability of our P-SWAP-based CNOT gate is enhanced to 1/8 without additional photons. Moreover, the success probability of our P-SWAP-based Toffoli gate (1/64) is higher than the CNOT-based protocols (1/72) [Ralph et al., 2007, Lanyon et al., 2009] and it is also higher than the no-decomposition-based one (1/133) [Fiurášek, 2006].

The multi-level system is essential to realize our schemes. In optical system, we can encode polarization DOF of photons as two computational qubits and spatial-mode DOF as the qudit (extra level).
We can also encode these levels on orbital angular momentum of photons. Besides, diamond nitrogen-vacancy defect center [Xiang et al., 2013, Waldherr et al., 2014] and superconducting system [Peterer et al., 2015, Bækkegaard et al., 2019]  can also provide available multiple levels to implement the universal quantum gates due to their long coherence time and flexible manipulation.

In summary, by introducing higher-dimensional spaces, we proposed simplified CNOT and Toffoli gates.
A three-qubit Toffoli gate can be simulated with two P-SWAP, one CNOT, and two single-qutrit gates. $(2n-3)$ qubit-qudit gates  and $(2n-4)$ single-qudit gates are sufficient for constructing an $n$-qubit Toffoli gate.
Following the simplified synthesis, as a feasible example, linear optics architectures for implementing CNOT and Toffoli gates were designed with a higher success probability.

§ ACKNOWLEDGMENTS

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 Grants No. FRF-BR-19-011A3 and No. 230201506500024, and a grant from the China Scholarship Council.  L.-C.K. is supported by the Ministry of Education and the National Research Foundation Singapore.

[Barenco et al., 1995]
Adriano Barenco, Charles H Bennett, Richard Cleve, David P DiVincenzo, Norman
Margolus, Peter Shor, Tycho Sleator, John A Smolin, and Harald Weinfurter.
Elementary gates for quantum computation.
Physical Review A, 520 (5):0 3457, 1995.

[Grover, 1997]
Lov K Grover.
Quantum mechanics helps in searching for a needle in a haystack.
Physical Review Letters, 790 (2):0 325, 1997.

[Shor, 1999]
Peter W Shor.
Polynomial-time algorithms for prime factorization and discrete
logarithms on a quantum computer.
SIAM Review, 410 (2):0 303–332, 1999.

[Figgatt et al., 2017]
Caroline Figgatt, Dmitri Maslov, KA Landsman, Norbert Matthias Linke, Shantanu
Debnath, and C Monroe.
Complete 3-qubit Grover search on a programmable quantum computer.
Nature Communications, 80 (1):0 1918, 2017.

[Nam et al., 2020]
Yunseong Nam, Yuan Su, and Dmitri Maslov.
Approximate quantum Fourier transform with $o(n\log(n))$ T gates.
npj Quantum Information, 60 (1):0 26, 2020.

[Gisin et al., 2002]
Nicolas Gisin, Grégoire Ribordy, Wolfgang Tittel, and Hugo Zbinden.
Quantum cryptography.
Reviews of Modern Physics, 740 (1):0 145,

[Nielsen and Chuang, 2010]
Michael A Nielsen and Isaac L Chuang.
Quantum Computation and Quantum Information: 10th Anniversary
Cambridge University Press, 2010.

[Shende et al., 2004]
Vivek V Shende, Igor L Markov, and Stephen S Bullock.
Minimal universal two-qubit controlled-NOT-based circuits.
Physical Review A, 690 (6):0 062321,

[Khaneja and Glaser, 2001]
Navin Khaneja and Steffen J Glaser.
Cartan decomposition of SU($2^{n}$) and control of spin systems.
Chemical Physics, 2670 (1-3):0 11–23, 2001.

[Möttönen et al., 2004]
Mikko Möttönen, Juha J Vartiainen, Ville Bergholm, and Martti M
Quantum circuits for general multiqubit gates.
Physical Review Letters, 930 (13):0 130502,

[Bullock and Brennen, 2004]
Stephen S Bullock and Gavin K Brennen.
Canonical decompositions of $n$-qubit quantum computations and
Journal of Mathematical Physics, 450 (6):0
2447–2467, 2004.

[Shende et al., 2006]
Vivek V Shende, Stephen S Bullock, and Igor L Markov.
Synthesis of quantum-logic circuits.
IEEE Transactions on Computer-Aided Design of Integrated
Circuits and Systems, 250 (6):0 1000–1010, 2006.

[D'Alessandro and Albertini, 2007]
Domenico D'Alessandro and Francesca Albertini.
Quantum symmetries and Cartan decompositions in arbitrary
Journal of Physics A: Mathematical and Theoretical,
400 (10):0 2439, 2007.

[Di and Wei, 2013]
Yao Min Di and Hai Rui Wei.
Synthesis of multivalued quantum logic circuits by elementary gates.
Physical Review A, 870 (1):0 012325, 2013.

[Iten et al., 2016]
Raban Iten, Roger Colbeck, Ivan Kukuljan, Jonathan Home, and Matthias
Quantum circuits for isometries.
Physical Review A, 930 (3):0 032318, 2016.

[Vatan and Williams, 2004]
Farrokh Vatan and Colin Williams.
Optimal quantum circuits for general two-qubit gates.
Physical Review A, 690 (3):0 032315, 2004.

[Vidal and Dawson, 2004]
Guifre Vidal and Christopher M Dawson.
Universal quantum circuit for two-qubit transformations with three
controlled-NOT gates.
Physical Review A, 690 (1):0 010301(R), 2004.

[Shende et al., 2004]
Vivek V Shende, Stephen S Bullock, and Igor L Markov.
Recognizing small-circuit structure in two-qubit operators.
Physical Review A, 700 (1):0 012310,

[Ralph et al., 2007]
T C Ralph, K J Resch, and Alexei Gilchrist.
Efficient Toffoli gates using qudits.
Physical Review A, 750 (2):0 022313, 2007.

[Lanyon et al., 2009]
Benjamin P Lanyon, Marco Barbieri, Marcelo P Almeida, Thomas Jennewein,
Timothy C Ralph, Kevin J Resch, Geoff J Pryde, Jeremy L O’brien, Alexei
Gilchrist, and Andrew G White.
Simplifying quantum logic using higher-dimensional Hilbert spaces.
Nature Physics, 50 (2):0 134–140, 2009.

[Fedorov et al., 2012]
Arkady Fedorov, Lars Steffen, Matthias Baur, Marcus P da Silva, and Andreas
Implementation of a Toffoli gate with superconducting circuits.
Nature, 4810 (7380):0 170–172, 2012.

[Liu et al., 2012]
Kai Liu, Wen Dong Li, Wen Zhao Zhang, Peng Shi, Chun Nian Ren, and Yong Jian
Optimizing quantum circuits using higher-dimensional Hilbert
Acta Physica Sinica, 610 (12):0 120301, 2012.

[Li et al., 2013]
Wen Dong Li, Yong Jian Gu, Kai Liu, Yuan Harng Lee, and Yao Zhong Zhang.
Efficient universal quantum computation with auxiliary Hilbert
Physical Review A, 880 (3):0 034303, 2013.

[Liu and Wei, 2020]
Wen Qiang Liu and Hai Rui Wei.
Optimal synthesis of the Fredkin gate in a multilevel system.
New Journal of Physics, 220 (6):0 063026,

[Liu et al., 2020]
Wen Qiang Liu, Hai Rui Wei, and Leong Chuan Kwek.
Low-cost Fredkin gate with auxiliary space.
Physical Review Applied, 140 (5):0 054057,

[Cerf et al., 2002]
Nicolas J Cerf, Mohamed Bourennane, Anders Karlsson, and Nicolas Gisin.
Security of quantum key distribution using $d$-level systems.
Physical Review Letters, 880 (12):0 127902,

[Malik et al., 2016]
Mehul Malik, Manuel Erhard, Marcus Huber, Mario Krenn, Robert Fickler, and
Anton Zeilinger.
Multi-photon entanglement in high dimensions.
Nature Photonics, 100 (4):0 248–252, 2016.

[Zhang et al., 2017]
Yingwen Zhang, Megan Agnew, Thomas Roger, Filippus S Roux, Thomas Konrad,
Daniele Faccio, Jonathan Leach, and Andrew Forbes.
Simultaneous entanglement swapping of multiple orbital angular
momentum states of light.
Nature Communications, 80 (1):0 632, 2017.

[Wang et al., 2017]
Feiran Wang, Manuel Erhard, Amin Babazadeh, Mehul Malik, Mario Krenn, and Anton
Generation of the complete four-dimensional Bell basis.
Optica, 40 (12):0 1462–1467, 2017.

[Hu et al., 2018]
Xiao Min Hu, Yu Guo, Bi Heng Liu, Yun Feng Huang, Chuan Feng Li, and Guang Can
Beating the channel capacity limit for superdense coding with
entangled ququarts.
Science Advances, 40 (7):0 eaat9304, 2018.

[Luo et al., 2019]
Yi Han Luo, Han Sen Zhong, Manuel Erhard, Xi Lin Wang, Li Chao Peng, Mario
Krenn, Xiao Jiang, Li Li, Nai Le Liu, Chao Yang Lu, A Zeilinger, and J W Pan.
Quantum teleportation in high dimensions.
Physical Review Letters, 1230 (7):0 070505,

[Hu et al., 2020]
Xiao Min Hu, Chao Zhang, Bi Heng Liu, Yu Cai, Xiang Jun Ye, Yu Guo, Wen Bo
Xing, Cen Xiao Huang, Yun Feng Huang, Chuan Feng Li, and Guang Can Guo.
Experimental high-dimensional quantum teleportation.
Physical Review Letters, 1250 (23):0 230501,

[D'ambrosio et al., 2013]
Vincenzo D'ambrosio, Nicolò Spagnolo, Lorenzo Del Re, Sergei Slussarenko,
Ying Li, Leong Chuan Kwek, Lorenzo Marrucci, Stephen P Walborn, Leandro
Aolita, and Fabio Sciarrino.
Photonic polarization gears for ultra-sensitive angular measurements.
Nature Communications, 40 (9):0 2432, 2013.

[Shi, 2002]
Yaoyun Shi.
Both Toffoli and controlled-NOT need little help to do universal
quantum computation.
Quantum Information and Computation, 30 (1):0
84–92, 2002.

[Cory et al., 1998]
David G Cory, MD Price, W Maas, Emanuel Knill, Raymond Laflamme, Wojciech H
Zurek, Timothy F Havel, and Shyamal S Somaroo.
Experimental quantum error correction.
Physical Review Letters, 810 (10):0 2152,

[Reed et al., 2012]
Matthew D Reed, Leonardo DiCarlo, Simon E Nigg, Luyan Sun, Luigi Frunzio,
Steven M Girvin, and Robert J Schoelkopf.
Realization of three-qubit quantum error correction with
superconducting circuits.
Nature, 4820 (7385):0 382–385, 2012.

[Paetznick and Reichardt, 2013]
Adam Paetznick and Ben W Reichardt.
Universal fault-tolerant quantum computation with only transversal
gates and error correction.
Physical Review Letters, 1110 (9):0 090505,

[Guillaud and Mirrahimi, 2019]
Jérémie Guillaud and Mazyar Mirrahimi.
Repetition cat qubits for fault-tolerant quantum computation.
Physical Review X, 90 (4):0 041053, 2019.

[Shende and Markov, 2008]
Vivek V Shende and Igor L Markov.
On the CNOT-cost of Toffoli gates.
Quantum Information and Computation, 90 (5):0
461–486, 2008.

[Yu et al., 2013]
Nengkun Yu, Runyao Duan, and Mingsheng Ying.
Five two-qubit gates are necessary for implementing the Toffoli
Physical Review A, 880 (1):0 010304(R), 2013.

[Yu and Ying, 2015]
Nengkun Yu and Mingsheng Ying.
Optimal simulation of Deutsch gates and the Fredkin gate.
Physical Review A, 910 (3):0 032302, 2015.

[Kiktenko et al., 2020]
E O Kiktenko, A S Nikolaeva, Peng Xu, G V Shlyapnikov, and A K Fedorov.
Scalable quantum computing with qudits on a graph.
Physical Review A, 1010 (2):0 022304, 2020.

[Mičuda et al., 2013]
M Mičuda, Michal Sedlak, Ivo Straka, Martina Miková, M Dušek,
M Ježek, and J Fiurášek.
Efficient experimental estimation of fidelity of linear optical
quantum Toffoli gate.
Physical Review Letters, 1110 (16):0 160407,

[Lemr et al., 2015]
Karel Lemr, Karol Bartkiewicz, Antonín Černoch, Miloslav
Dušek, and Jan Soubusta.
Experimental implementation of optimal linear-optical
controlled-unitary gates.
Physical Review Letters, 1140 (15):0 153602,

[Mičuda et al., 2015]
M Mičuda, M Miková, I Straka, M Sedlák, M Dušek,
M Ježek, and J Fiurášek.
Tomographic characterization of a linear optical quantum Toffoli
Physical Review A, 920 (3):0 032312, 2015.

[Ru et al., 2021]
Shihao Ru, Yunlong Wang, Min An, Feiran Wang, Pei Zhang, and Fuli Li.
Realization of deterministic quantum Toffoli gate with a single
Physical Review A, 1030 (2):0 022606, 2021.

[Monz et al., 2009]
Thomas Monz, Kihwan Kim, Wolfgang Hänsel, M Riebe, AS Villar, Philipp
Schindler, Michael Chwalla, Markus Hennrich, and Rainer Blatt.
Realization of the quantum Toffoli gate with trapped ions.
Physical Review Letters, 1020 (4):0 040501,

[Beterov et al., 2018]
I I Beterov, I N Ashkarin, E A Yakshina, D B Tretyakov, V M Entin, I I
Ryabtsev, P Cheinet, P Pillet, and M Saffman.
Fast three-qubit Toffoli quantum gate based on three-body
Förster resonances in Rydberg atoms.
Physical Review A, 980 (4):0 042704, 2018.

[Levine et al., 2019]
Harry Levine, Alexander Keesling, Giulia Semeghini, Ahmed Omran, Tout T Wang,
Sepehr Ebadi, Hannes Bernien, Markus Greiner, Vladan Vuletić, Hannes
Pichler, and M D Lukin.
Parallel implementation of high-fidelity multiqubit gates with
neutral atoms.
Physical Review Letters, 1230 (17):0 170503,

[Gullans and Petta, 2019]
M J Gullans and J R Petta.
Protocol for a resonantly driven three-qubit Toffoli gate with
silicon spin qubits.
Physical Review B, 1000 (8):0 085419, 2019.

[Fiurášek, 2006]
Jaromír Fiurášek.
Linear-optics quantum Toffoli and Fredkin gates.
Physical Review A, 730 (6):0 062313, 2006.

[Pittman et al., 2001]
T B Pittman, B C Jacobs, and J D Franson.
Probabilistic quantum logic operations using polarizing beam
Physical Review A, 640 (6):0 062311, 2001.

[Gasparoni et al., 2004]
Sara Gasparoni, Jian Wei Pan, Philip Walther, Terry Rudolph, and Anton
Realization of a photonic CNOT gate sufficient for quantum
Physical Review Letters, 930 (2):0 020504,

[Bao et al., 2007]
Xiao Hui Bao, Teng Yun Chen, Qiang Zhang, Jian Yang, Han Zhang, Tao Yang, and
Jian Wei Pan.
Optical nondestructive controlled-NOT gate without using entangled
Physical Review Letters, 980 (17):0 170502,

[Zeuner et al., 2018]
Jonas Zeuner, Aditya N Sharma, Max Tillmann, René Heilmann, Markus
Gräfe, Amir Moqanaki, Alexander Szameit, and Philip Walther.
Integrated-optics heralded controlled-NOT gate for
polarization-encoded qubits.
npj Quantum Information, 40 (1):0 13, 2018.

[Langford et al., 2005]
Nathan K Langford, T J Weinhold, R Prevedel, K J Resch, Alexei Gilchrist, J L
O’Brien, G J Pryde, and A G White.
Demonstration of a simple entangling optical gate and its use in
Bell-state analysis.
Physical Review Letters, 950 (21):0 210504,

[Kiesel et al., 2005]
Nikolai Kiesel, Christian Schmid, Ulrich Weber, Rupert Ursin, and Harald
Linear optics controlled-phase gate made simple.
Physical Review Letters, 950 (21):0 210505,

[Okamoto et al., 2005]
Ryo Okamoto, Holger F. Hofmann, Shigeki Takeuchi, and Keiji Sasaki.
Demonstration of an optical quantum controlled-NOT gate without
path interference.
Physical Review Letters, 950 (21):0 210506,

[Viamontes et al., 2009]
George F Viamontes, Igor L Markov, and John P Hayes.
Quantum circuit simulation.
Springer Science, 2009.

[Xiang et al., 2013]
Ze Liang Xiang, Sahel Ashhab, J Q You, and Franco Nori.
Hybrid quantum circuits: Superconducting circuits interacting with
other quantum systems.
Reviews of Modern Physics, 850 (2):0 623,

[Waldherr et al., 2014]
Gerald Waldherr, Y Wang, S Zaiser, M Jamali, T Schulte-Herbrüggen, H Abe,
T Ohshima, J Isoya, J F Du, P Neumann, and J Wrachtrup.
Quantum error correction in a solid-state hybrid spin register.
Nature, 5060 (7487):0 204–207, 2014.

[Peterer et al., 2015]
Michael J Peterer, Samuel J Bader, Xiaoyue Jin, Fei Yan, Archana Kamal,
Theodore J Gudmundsen, Peter J Leek, Terry P Orlando, William D Oliver, and
Simon Gustavsson.
Coherence and decay of higher energy levels of a superconducting
transmon qubit.
Physical Review Letters, 1140 (1):0 010501,

[Bækkegaard et al., 2019]
Thomas Bækkegaard, L B Kristensen, Niels J S Loft, Christian Kraglund
Andersen, David Petrosyan, and Nikolaj T Zinner.
Realization of efficient quantum gates with a superconducting
qubit-qutrit circuit.
Scientific Reports, 90 (1):0 13389, 2019.