Introduction

Quantum computing, a burgeoning frontier technology in the field of information science, represents an efficient method of information processing that leverages the quantum entanglement and superposition effects of qubits1,2,3. Examples of physical qubit platforms include superconducting circuits4,5, trapped ions6,7, laser-cooled neutral atoms8,9,10, photonic circuits11,12, and spin-based solid-state platforms13,14 and so on. The photon platform stands out due to its extended decoherence time, low noise and crosstalk characteristics stemming from the weak interaction between photons and the environment15, as well as its exceptional capacity for long-distance quantum information exchange. It has manifested as the preferred option for constructing a large-scale, modular optical quantum computer that operates at room temperature.

Optical quantum logic gates serve as the cornerstone of optical quantum computing, manipulating qubit states through unitary transformations. With the extensive physical size resources and modulation flexibility of photons, some quantum gates have been verified as on various degrees of freedom (DOFs), including polarization16,17,18,19,20, orbital angular momentum (OAM)21,22,23, time bin24,25, frequency26,27,28, and others. Classical optical quantum logic gates typically rely on two-dimensional quantum systems, such as the Hadamard gate, Pauli-X gate, rotation gate16 and CNOT gate17. However, these gates are constrained by the limited information expression capacity of DOFs, challenged in satisfying the escalating demand for high-capacity and high-security information transmission. Consequently, there has been a growing interest in the utilization of high-dimensional quantum systems for quantum information encoding and processing29,30.

Photon-based high-dimensional OAM encoding has enabled the successful implementation of several high-dimensional quantum logic gates, including the Toffoli gate31, Swap gate32, and four-dimensional X gate33. High-dimensional quantum steering has been experimentally demonstrated using entangled states with OAM coding34,35. These accomplishments robustly illustrate that photon high-dimensional OAM coding exhibits exceptionally high information capacity, high security36 and robust anti-interference properties37. Additionally, hybrid encoding that combines multiple degrees of freedom within a single photon further increases the capacity potential of photon channels38,39,40,41,42,43, offering new possibilities for more efficient information transmission.

The double Feynman gate, known for its versatility, demonstrates significant potential across various scientific fields. In quantum-dot cellular automata (QCA), it serves as an efficient multiplexer44, improving core logic circuits by reducing delay time and quantum cost45. When implemented with silicon microring resonators, it provides enhanced flexibility and scalability for integration into photonic circuits46. In genetic engineering, E. coli cells have utilized neural network-based designs to develop a three-input, three-output reversible gate47, contributing to advancements in biocomputing research.

In this paper, we mainly focus on the coding principle and experimental implementation of the single-photon double Feynman gate. We design a modular three-qubit double Feynman gate through the integration of polarization and four-dimensional OAM spatial mode hybrid coding. By manipulating the polarization state of photon and the sequence of OAM modes, our experimental design offers flexible control over the coding method and information capacity, accommodating various application scenarios. This contributes to the advancement of high-dimensional quantum logic gates.

Experimental setup

The three-qubit double Feynman gate is shown in Fig. 1a. Its operation can be expressed as

$$U_{|A,B,C\rangle } = \left| {A,B \oplus A,C \oplus A} \right\rangle .$$
(1)
Fig. 1
Fig. 1
Full size image

Double Feynman gate with polarization and four-dimensional OAM hybrid DOFs encoding at single photon level.

The input value is \(A,B,C\), where \(A,B,C\) are three qubits respectively, and after passing through the double Feynman gate, the output value is \(P,Q,R\). The outputs are defined by \(P = A,Q = B \oplus A,R = C \oplus A\) and \(\oplus\) denotes addition modulo two.

We implement a double Feynman gate using polarization and four-dimensional OAM space hybrid coding at the single photon level. The photon’s horizontal \(|H\rangle\) and vertical \(|V\rangle\) polarization states are used as the first control qubit, while the four-dimensional OAM space serves as the second control qubit and the target qubit. The specific coding method is shown in Fig. 1b.

The first controlled qubit is encoded as

$$|V\rangle \to |0\rangle ,\quad |H\rangle \to |1\rangle ,$$
(2)

while the second controlled qubit and the target qubit are jointly encoded as

$$\begin{gathered} \left| {l = - 2} \right\rangle \to \left| {01} \right\rangle ,\;\left| {l = 0} \right\rangle \to \left| {10} \right\rangle , \hfill \\ \left| {l = - 1} \right\rangle \to \left| {00} \right\rangle ,\;\left| {l = + 1} \right\rangle \to \left| {11} \right\rangle \hfill \\ \end{gathered}$$
(3)

in a four-dimensional Hilbert space spanned by OAM quantum states.

The initial quantum state is prepared as the first stage in the experiment, as shown in Fig. 2. The Gaussian beam, pumped by a 405 nm continuous-wave diode laser, passes through a half-wave plate (HWP) and a quarter-wave plate (QWP) and is then adjusted to a horizontally polarized state. The beam is focused by a lens onto a 5 mm-long collinear periodically poled potassium titanyl phosphate (PPKTP) crystal. This crystal responds only to horizontally polarized light and generates 810 nm polarization-correlated photon pairs via type-II spontaneous parametric down-conversion. The 405 nm pump light is effectively filtered out using high-pass filters, leaving a Gaussian beam for the preparation of the OAM state.

Fig. 2
Fig. 2
Full size image

Experimental setup for the double Feynman gate. A 405 nm diode laser pumps a type-II PPKTP crystal to generate orthogonally polarized, correlated photon pairs. The vertically polarized idler photon is used to herald the horizontally polarized signal photon. SLM1, mirror and a set of plates are employed to modulate the transverse and polarization modes of the signal photon independently. Blocks (a)–(c) are designed to implement the interchange operation of the double Feynman gate in the OAM-encoded Hilbert space. Finally, the two photon paths are recombined at the PBS2, and the photon is projectively detected via SLM2 and SMF. (The digital drawings in this figure were created by Yanbing Zhu using SOLIDWORKS® Premium 2019 SP0.0, while device annotations were added with Microsoft Visio® 2021. For further details, refer to the official websites of SOLIDWORKS® 2019 (https://www.solidworks.com/) and Microsoft Visio® 2021 (https://www.microsoft.com/en-us/microsoft-365/visio/).)

A polarization beam splitter (PBS) separates the polarization-correlated photon pairs. The idler photon, with vertical polarization, is coupled into a single-mode fiber (SMF) and detected by a single-photon avalanche detector (SPAD) used for heralding. The other signal photon with horizontal polarization, after passing through the polarization-maintaining fiber (PMF), is coupled into Spatial Light Modulator 1 (SLM 1). This transforms the signal photon from a spatial Gaussian mode into an OAM quantum state \(\left| {\varphi_{1} } \right\rangle = \alpha \left| {01} \right\rangle + \beta \left| {00} \right\rangle + \gamma \left| {10} \right\rangle + \delta \left| {11} \right\rangle\), where the complex amplitude satisfies the normalization condition, i.e., \(\left| \alpha \right|^{2} + \left| \beta \right|^{2} + \left| \gamma \right|^{2} + \left| \delta \right|^{2} = 1\). The control qubit of the signal photon is then adjusted using the QWP and HWP to prepare arbitrary polarization quantum state.

As for the experimental setup as shown in Fig. 2. When the input polarization state of the first control qubit is \(|H\rangle\), the photon passes through PBS1 and Blocks (a)–(c) to perform the interchange operation. Block (a) depicts an embedded polarization-correlated Sagnac interferometer. The two HWPs are set to angles of \({\pi \mathord{\left/ {\vphantom {\pi 8}} \right. \kern-0pt} 8}\) and \({{3\pi } \mathord{\left/ {\vphantom {{3\pi } 8}} \right. \kern-0pt} 8}\), respectively. An internally mounted Dove prism, with an angle of \({\pi \mathord{\left/ {\vphantom {\pi 4}} \right. \kern-0pt} 4}\) relative to the optical axis, is positioned within the setup. The manipulation of the OAM state \(\left| \ell \right\rangle\) follows the rule:

$$\left| \ell \right\rangle \mathop{\longrightarrow}\limits^{Dove}ie^{i2\alpha \ell } \left| { - \ell } \right\rangle ,$$
(4)

where \(\alpha\) is the angle of rotation of the Dove prism. This configuration effectively causes the relative angles between the clockwise and counterclockwise paths of the interferometer to rotate the phases of incident photon with odd or even OAM modal values by angle of \(\pi\) or \(2\pi\). According to the above settings, suppose the input state is \(|H\rangle |l\rangle\). After passing through Block (a), the quantum state becomes

$$i\frac{1}{2}(e^{i\pi l} + 1)|H\rangle | - l\rangle + i\frac{1}{2}(e^{i\pi l} - 1)|V\rangle | - l\rangle .$$
(5)

The result \(i|H\rangle | - l\rangle\) is obtained by substituting Eq. (5) when the value of OAM mode l is even; in this case, only horizontal polarization is present. Similarly, when l is odd, the polarization state of the photon corresponds to vertical polarization. Hence, the Sagnac interferometer functions as an OAM parity sorter, which separates the photon based on the parity of their OAM mode values.

The Mach–Zehnder interferometer as shown in Block (b). After passing through the PBS, if the OAM mode number l is odd, the corresponding photon is reflected. Conversely, if the OAM mode number l is even, the corresponding photon is transmitted. In the reflected path, after passing through the PBS, three mirrors in sequence, and a QWP, the state is given as

$$- \frac{1}{2}(e^{i\pi l} - 1)|V\rangle | - l\rangle .$$
(6)

In the PBS transmission direction, A Spiral Phase Plate (SPP) is used to interchange the even modes. After passing through two mirrors sequentially, and an HWP, we can get the state

$$- i\frac{1}{2}(e^{i\pi l} + 1)|H\rangle | - l\rangle .$$
(7)

Finally, the two photon paths are recombined at the PBS. To compensate for the phasing caused by the total number of reflections on the relative phases between the odd and even modes, the transmissive and reflected paths in Block (b) are further augmented with HWP and QWP, at \({\pi \mathord{\left/ {\vphantom {\pi 2}} \right. \kern-0pt} 2},0\) angles relative to its optical axis respectively. To ensure that the optical path of both paths is the same-that is, no extra phase is added-the PZT in the reflected path is coupled to a motorized translation stage. Block (c) illustrates an OAM combiner, which is structurally the inverse of the OAM parity sorter. The inverted-rotated Dove prism flattens the phase of the incident photon, allowing for the coherent reassembly of the separated even and odd OAM modes.

$$\begin{gathered} \left| {100} \right\rangle \mathop{\longrightarrow}\limits^{(a)} - i\left| {011} \right\rangle \mathop{\longrightarrow}\limits^{(b)}i\left| {000} \right\rangle \mathop{\longrightarrow}\limits^{(c)}i\left| {111} \right\rangle , \hfill \\ \left| {101} \right\rangle \mathop{\longrightarrow}\limits^{(a)}i\left| {101} \right\rangle \mathop{\longrightarrow}\limits^{(b)} - i\left| {110} \right\rangle \mathop{\longrightarrow}\limits^{(c)}i\left| {110} \right\rangle , \hfill \\ \left| {110} \right\rangle \mathop{\longrightarrow}\limits^{(a)}i\left| {110} \right\rangle \mathop{\longrightarrow}\limits^{(b)} - i\left| {101} \right\rangle \mathop{\longrightarrow}\limits^{(c)}i\left| {101} \right\rangle , \hfill \\ \left| {111} \right\rangle \mathop{\longrightarrow}\limits^{(a)} - i\left| {000} \right\rangle \mathop{\longrightarrow}\limits^{(b)}i\left| {011} \right\rangle \mathop{\longrightarrow}\limits^{(c)}i\left| {100} \right\rangle . \hfill \\ \end{gathered}$$
(8)

Based on the operating principles of the double Feynman gate, Eq. (8) provides the unitary transformation for the four quantum states in the Blocks (a)-(c), as described in the experimental setup. When the input polarization state of the first control qubit is \(|V\rangle\), the OAM modes of photon is reflected by PBS1 without any transformation operation, as shown in Eq. (9), and are eventually merged with the path transmitted through Blocks (a)-(c) at PBS2. The signal photon following the double Feynman gate

$$\begin{gathered} \left| {000} \right\rangle \mathop{\longrightarrow}\limits^{{}}\left| {000} \right\rangle ,\left| {001} \right\rangle \mathop{\longrightarrow}\limits^{{}}\left| {001} \right\rangle , \hfill \\ \left| {010} \right\rangle \mathop{\longrightarrow}\limits^{{}}\left| {010} \right\rangle ,\left| {011} \right\rangle \mathop{\longrightarrow}\limits^{{}}\left| {011} \right\rangle . \hfill \\ \end{gathered}$$
(9)

is analyzed by a detection Block consisting of SLM2 and a SMF. In this setup, the SLM2 flattens the phase of the incoming photon, transforming it into a Gaussian mode which can be efficiently coupled to the SMF. In this case, both the eigenstate modes and superposition OAM modes of single photon can be examined. The signal photon passes through the SPAD, and coincidence detection is performed with the idler heralding photon.

Measurement scheme and experimental results

To evaluate the quality of the double Feynman gate, we define the conversion efficiency \(P_{ij}\) of photon, which is obtained from the crosstalk measurements between the input and output modes in the computational basis. As shown in Eq. (10), \(N_{ij}\) is the coincidence count of the input mode i transformed into the output mode j in the crosstalk matrix.

$$P_{ij} = \frac{{N_{ij} }}{{\sum\nolimits_{ij} {N_{ij} } }}.$$
(10)

The coincidence counts of all output states resulting from the four possible OAM input eigenstates \(\left| {00} \right\rangle ,\left| {01} \right\rangle ,\left| {10} \right\rangle\) and \(\left| {11} \right\rangle\) are measured when the controlled qubits are in various quantum states. The experimental and theoretical results for different input states of the logic gate are presented in Fig. 3a. By comparing each set of experimental values (blue bars) and theoretical values (black wire frames), it is evident that the experimental values are lower than the theoretical values due to experimental limitations. The experimental results of the logic gate align with the theoretically predicted target state. Using \(\left| H \right\rangle\) and \(\left| V \right\rangle\) as the computational basis states of the controlled qubit, the average conversion rates \(P_{ij}\) are around 91.9% and 93.5%, respectively in that order.

Fig. 3
Fig. 3
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The conversion rate truth table of the double Feynman gate operation at the computational basis. In the 3D histogram, the conversion rate of each quantum state is represented collectively by a blue bar and a black wire-frame outline. The blue bar signifies the experimental measurement data, whereas the black wire-frame outline represents the theoretical prediction data. (a) The controlled qubits are sequentially positioned at \(\left| V \right\rangle\) and \(\left| H \right\rangle\). (b) The controlled qubit is located at \({{\left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right)} \mathord{\left/ {\vphantom {{\left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right)} {\sqrt 2 }}} \right. \kern-0pt} {\sqrt 2 }}\).

To test the performance of double Feynman gate when the control qubit is in the superposition state, Fig. 3b displays the average \(P_{ij}\) corresponding to the target state, with the controlled qubit is in the \({{\left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right)} \mathord{\left/ {\vphantom {{\left( {\left| 0 \right\rangle + \left| 1 \right\rangle } \right)} {\sqrt 2 }}} \right. \kern-0pt} {\sqrt 2 }}\) state. The output average conversion rate is about 94.0%, which is consistent with expectations and confirms the dependability of this gate in the experiments. The imperfect phase modulation of the photonic OAM modes during projection measurements is the primary factor limiting the input–output conversion rate \(P_{ij}\) in this method. The decline in efficiency can also be attributed to the interferometer cascade and optical misalignment. For instance, spatial misalignment leads to a reduced coupling efficiency of this specific mode to the single-photon detector, which subsequently decreases the detection probability of this gate when the controlled state is at \(\left| H \right\rangle\) and the OAM state is at \(\left| {01} \right\rangle\).

We use a set of waveplates and an SLM2 in the experimental optical setup to perform quantum state tomography (QST) in the OAM Hilbert space. To demonstrate the effect of the QST reconstruction of the density matrix \(\rho\), we measured the fidelity \(F(\rho_{e} ,\rho_{t} ) = (Tr[(\sqrt {\rho_{t} } \rho_{e} \sqrt {\rho_{t} } )^{{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0pt} 2}}} ])^{2}\) between the experimental state \(\rho_{e}\) and the theoretical expectation state \(\rho_{t}\), where the theoretical expectation state \(\rho_{t} = U\rho_{i} U^{\dag }\) is calculated by the unitary transformation, and \(\rho_{i}\) is the input state. The fidelity of the different separable target states for single photon is shown in Table 1, where the first column corresponds to the state of the controlled qubit and the first row corresponds to the state of the OAM qubit. The average fidelity of the controlled qubit being in the single quantum state and the diagonal superposition state is nearly 92.7% and 94.0%, respectively.

Table 1 Fidelity of detecting photon in OAM eigenstate states through the double Feynman gate.
Full size table

In Table 2, the difference from Table 1 is that the four states \(\left| {\Psi_{ + } } \right\rangle\), \(\left| {\Psi_{ - } } \right\rangle\), \(\left| {\Phi_{ + } } \right\rangle\) and \(\left| {\Phi_{ - } } \right\rangle\) in the first row are a set of Bell-entangled target states \(B = \{ \left| {x_{{\left| {01} \right\rangle ,\left| {10} \right\rangle }}^{ + } } \right\rangle ,\left| {x_{{\left| {01} \right\rangle ,\left| {10} \right\rangle }}^{ - } } \right\rangle ,\left| {x_{{\left| {00} \right\rangle ,\left| {11} \right\rangle }}^{ + } } \right\rangle ,\left| {x_{{\left| {00} \right\rangle ,\left| {11} \right\rangle }}^{ - } } \right\rangle \}\), obtained by normalizing the pure states according to the \(\left| {x_{{\ell_{1} ,\ell_{2} }}^{ \pm } } \right\rangle = \frac{1}{\sqrt 2 }(|\ell_{1} \rangle \pm |\ell_{2} \rangle )\) defined by Pauli matrix \(\sigma_{x}\). Analyzing the data in the table, we can find that the fidelity of the photon in the Bell-entangled states is approximately 97.8% on average. The average fidelity of the probed target states,

Table 2 Fidelity of photon probing Bell entangled states through the double Feynman gate.
Full size table

excluding the impact of some experimental errors, is (92.7 ± 0.15)%. This indicates that the current logic gate can carry out its intended operation correctly.

Conclusion and discussion

In this work, we display a high-dimensional double Feynman gate using single-photon hybrid DOFs. Utilizing the single-photon polarization DOF and the four-dimensional OAM space to encode the control qubit and the target qubit respectively, a three-qubit double Feynman gate is realized, eliminating the need for additional auxiliary photon and probability post-selection. It sufficiency supports some moderately complex quantum applications, including quantum encryption, quantum simulation, and certain specific quantum algorithms, as well as to implement of basic quantum algorithms and quantum information processing tasks.