Balanced mode filtering for scalable Pauli-X gates in high-dimensional OAM space

Abstract

The orbital angular momentum (OAM) of photons provides a pivotal resource for carrying out high-dimensional quantum Pauli-X operations due to its unique discrete infinite orthogonality, which opens enticing perspectives for high-capacity quantum communication and multi-parallel quantum computation. However, the state-of-the-art Pauli-X gates that require cyclic transformation of OAM modes are mostly limited to four modes, face scalability challenges owing to the system redundancy and fidelity degradation induced by increased processing modes. Here, we propose a general strategy to design scalable quantum Pauli-X gates in high-dimensional OAM space by taking full advantage of the order-correlated spatial divergence property of OAM modes. Leveraging the mechanism of balance mode filtering, partitioned phase is embedded between two structurally symmetric Q-plate elements that enable bidirectional mode shifting to filter out the highest-order OAM mode for spatially selective phase modulation. Derived from radial matching between modes and partitioned phase profiles, this design of parallel mode transformation underpins the extension to higher dimensions by merely adjusting filter parameters. We demonstrate the five-mode cyclic transformation with mode purity and diffraction efficiency higher than 92% and 92.7%, respectively. By incorporating polarization-entangled photon sources, we further realize five-dimensional quantum Pauli-X gates encoded on OAM space and experimentally verify the entanglement of output mode states. Low modulation loss and efficient quantum state tomography (QST) techniques endow the reconstructed density matrices with fidelities exceeding 92.3% ± 0.1%, indicating the well-performed entanglement preservation of output states. These results confirm that our approach achieves scalable quantum Pauli-X gate operations through a simple balanced architecture with high coherence, consolidating a theoretical and experimental foundation for large-scale quantum computation encoded on high-dimensional OAM space.

Graphical abstract

Keywords

balanced mode filtering / scalable Pauli-X gates / high-dimensional OAM space

1 Introduction

High-dimensional photonic quantum logic gates enable large-alphabet quantum communication protocols and multi-parallel quantum computing by encoding a substantial amount of information on a single photon, which holds significant implications for constructing high-speed and large-capacity quantum information processing systems [1−3]. As a unique photon spatial structure, the orbital angular momentum (OAM) provides an intrinsic multi-mode state space to encode information [4−8], and thus has emerged as a promising candidate for high-dimensional quantum information [9−11]. The high-dimensional Pauli-X gate serves as a fundamental building block for quantum information processing based on OAM modes, because it can achieve arbitrary unitary transformations with the combination of high-dimensional Pauli-Z gate [12]. Manifesting as a phase modulator, the effect of high-dimensional Pauli-Z gate performing on a set of orthogonal OAM modes is to introduce mode-dependent phase delay, which is simply conducted by a Dove prism [13]. In contrast, the generalization of Pauli-X transformation makes it a cyclic ladder operator on a high-dimensional Hilbert space, necessitating periodic shifting of OAM modes in a clockwise sequence. The complex transformations impose stringent requirements on modulation strategy, presenting significant difficulty in implementing high-dimensional Pauli-X gates. Although techniques for the generation, shifting and measurement of multiple photonic qubits carrying OAM have gained progress, the general methods for efficient high-dimensional cyclic transformation still remain a challenge [14−21].

The leading platform for processing structured photons in in high-dimensions is Mach−Zehnder interferometers (MZIs) assembled with spiral phase plates, reflection mirrors and OAM beam splitter (OAM-BS) modules, which is built from beam splitters and Dove prims. Leveraging mode-dependent phase delay imparted during multiple reflections, this configuration separates even- and odd-parity OAM modes at distinct output ports through interference [22−24]. Subsequent mode-shifting operations along independent spatial paths then enable cyclic transformation across four OAM modes. On this basis, a four-dimensional OAM quantum Pauli-X gate has been verified by introducing entangled quantum light sources and phase-locking devices. But this approach requires splitting and recombine the light beam within the interferometric setup, which is cumbersome and difficult to maintain aligned. Furthermore, an N-mode transformation typically entails the characterization of ~N² elements, limiting the scalability to a large extent. With the advantages of strong manipulation and system compactness, the multi-plane light conversion (MPLC) technology has garnered rapid progress in multiple OAM mode conversion [25−27]. Utilizing only a few optimized phase masks separated by free-space propagation, the general cyclic transformations on OAM modes and even Pauli-X gates on single photons have been demonstrated. This scheme effectively minimizes the system redundancy in that the enhancement of mode modulation capability is finalized by progressively embedding phase patterns within a static hardware structure, thereby presenting potential scalability for high-dimensional quantum logic operations [28−31]. Nevertheless, the continuous stepwise diffraction modulation of multi-layered phase masks induces exponential-accumulated energy loss, which severely degrades the process fidelity of mode unitary transformation. Despite these strategies experimentally feasible, the quantum Pauli-X gates are limited to four modes thus far due to the escalating resource overhead and fidelity degradation with increasing mode count, still facing the impediments of scalability [32].

In accordance with the mechanism of balanced mode filtering [33−35], we developed a general solution for OAM mode cyclic transformation that enables the extension in mode bases and verified a five-dimensional quantum Pauli-X gate based on this framework. By virtue of the mode shift conversion of a liquid crystal Q-plate (Q-plate), the highest-order OAM mode in the inputs is filtered through the proper transformation to Gaussian mode while other incident modes become non-Gaussian profiles. Another structurally symmetric Q-plate element that conveys reverse mode shift is deployed for balancing. Introducing the partitioned phase masks among them, the radially separated modes are capable of getting access to spatially selective phase modulation in parallel, facilitating the achievement of cyclic mode transforming operation. Profiting from the radial matching between the OAM mode and partitioned phase profiles, our strategy can be generalized to higher dimensions by only adjusting the parameters of balanced filter, without any additional optical components. In a proof-of-concept experiment, we demonstrated a successful classical cyclic transformation of five OAM modes with the mode purity and efficiency higher than 92% and 92.7%, respectively. Adopting the polarization-entangled photon source and quantum state tomography technique in this transformation, a five-dimensional quantum Pauli-X gate was also realized, leading to the remarkable fidelities exceeding 92.3% ± 0.1%. Our results display great robustness and scalability of mode basis, and is conductive to the development of spatial mode quantum computer [36, 37].

2 Principle and method

The high-dimensional Pauli-X gate is one of the most important unitary operations on a single qubit and corresponds to a cyclic transformation. On a certain d-dimensional space, the effect of Pauli-X gate encoded on OAM mode bases can be mathematically expressed as

(1)\hat{X}|l⟩=|{\left(l+1\right)}_{\mathrm{m}\mathrm{o}\mathrm{d}\left(d\right)}⟩,

where l ∈ {0, 1, …, d−1} refers to different OAM modes in d-dimensional Hilbert space and (l + 1)_mod(d) denotes the value modulo the number of modes d. This simply results in each mode being transformed to its nearest neighbor in a clockwise direction with the last mode |d−1〉 and getting back to the first one |0〉.

The cyclic operator on OAM modes requires independent helical phase modulation to reconfigure the order of output mode. Meanwhile, thanks to the inherent property of OAM modes that the divergence is positively correlated with the order, it spurs the insight to apply parallel helical phase modulation to spatially separated OAM modes by introducing a partitioned phase modulation. However, utilizing this approach for direct multi-mode cyclic transformation requires precise segmentation into multiple sub-regions, which inevitably induces mode-field crosstalk in the phase modulation process. To tackle this, the proposed mechanism of balanced mode filtering embeds two Q-plates sandwiching a partitioned phase, and achieves symmetric mode shifting on each side to reduce the number of regions, as shown in Fig. 1. For the input OAM modes {|−4〉, |−2〉, |0〉, |+2〉, |+4〉}, the Q-plate1 conducts the mode shifting with a format of |l〉→|l + 2q〉 through its intrinsic spin−orbit coupling [38−41], converting input modes into Gaussian and non-Gaussian intensity profiles. Here, the parameter q is a structural constant whose value is assigned to −2 according to the predetermined mode transformation protocol. Arising from the incomplete overlap of Gaussian mode |0〉 and non-Gaussian mode |!0〉 attributed to mode divergence, each of them undergoes tailored helical phase conversion after a partitioned phase modulation designed with incorporating only dual spatial regions is performed. Specifically, the mode |0〉 undergoes modulation by the phase profiles φ_in in the inner region while the other |!0〉 OAM modes are imposed with the spiral phase φ_out in the complementary outer region. In this case, φ_in = exp(−i8θ) and φ_out = exp(i2θ), where θ is the azimuthal angle. In our arrangement, an OAM mode filter is constituted collectively by Q-plate1 and the partitioned phase, selecting the highest-order mode to be modulated by the center phase region, and transmitting other modes with phase in the peripheral region. For the restoration to the original mode basis, Q-plate2 executes inverse mode conversion with the same format of |l〉→|l + 2q〉 (q = 2) through its symmetric structure of optical axis orientation, thereby enabling cyclic transformation and converting the order of output modes to {|−2〉, |0〉, |+2〉, |+4〉, |−4〉}.

To achieve mode shifting, Q-plate elements that possess the spin−orbit coupling property have been employed, which means imparting the conjugate phase to different spin states [42, 43]. The optical axis direction of the Q-plate is specified as

(2)\beta \left(r,\varphi \right)=q\varphi +{\varphi }_0,

where (r, ϕ) denotes the polar coordinates, ϕ is also the azimuthal angle, ϕ₀ is the initial orientation of the optical axis, which is set as zero, and as mentioned above, q is a constant, indicating the spatial rotation ratio of the optical axis. According to the modulation principle of Jones matrix with the homogeneous birefringent phase retardation setting as π, the Q-plate elements exhibit the independent phase responses E_LCP → exp(i2qϕ)∙E_RCP and E_RCP → exp(−i2qϕ)∙E_LCP for the left-handed polarized (LCP) and right-handed polarized (RCP) light, respectively, which establishes the foundation for the effect of mode shifting.

Following this regulation principle, we only need to map the helical phase to the azimuth of optical axes, which is half of the resulting phase. In the optical axis distribution map, the size of the Q-plate element is 9 mm × 9 mm with a sampling resolution of 768 × 768 pixels in total, and the optical axis in each pixel is modulated by writing a nanograting. Except for the azimuthal of the optical axis, the phase retardance also affects birefringent modification of the element, which depends on the refractive indices and the writing depth. The effective refractive indices of the nanogrooves for ordinary n_o and extraordinary n_e can be written as

(3)\begin{array}{rl}& n_{\mathrm{o}}=\sqrt{f\times n_1^2+\left(1-f\right)n_2^2},\\ & n_{\mathrm{e}}=\sqrt{\frac{n_1^2{n}_2^2}{f\times n_2^2+\left(1-f\right)n_1^2}},\end{array}

where f ≈ 0.15 is the filling factor, n₁ and n₂ are the refractive indices of the layers that constitute the nanograting. The value of n_e − n_o is approximately −3 × 10⁻³ based on known parameters. The phase retardation of the element calculated using the formula θ = 2π∙(n_e − n_o)∙h/λ of the element is chosen as π at a wavelength of 1550 nm for the phase modulation, and thus the writing depth h is about 250 μm.

In our proof-of-concept demonstration, we performed the reciprocal opposing shifting effect of OAM modes through the same Q-plate element (Q-plate1/Q-plate2) with the value q = 2, whose surface characterization is shown in Fig. 2(a). To achieve this, a half-wave plate (HWP) is placed in front of the Q-plate1 to flip the chirality of the incident polarization state of the beam, yielding phase responses equivalent to those of another Q-plate element with its value q = −2. The simulated and measured cross-polarized detection images are displayed in Figs. 2(b) and (c), which exhibit a high degree of consistency. Figures 2(d) and (e) illustrate the equivalent optical axis profiles of the elements HWP-Q-plate1 combination and Q-plate2, respectively. The red bars indicate the orientation of the local optical axis, which changes periodically along the azimuthal directions. It can also be seen that the equivalent optical axis profiles of the element combination are almost locally symmetric to the optical axis profile of Q-plate2, ensuring the conjugate phase modulation during the OAM mode cyclic transformation. Concurrently, the theoretical conversion process of each OAM mode is depicted in Fig. 2(f). As described above, the OAM mode |+4〉 is transformed to the fundamental Gaussian mode |0〉 first and then transformed to |−8〉 by the center phase mask of the partitioned phase loaded on a spatial light modulator (SLM). After that, the mode is transformed back to |−4〉 through the modulation of Q-plate2. Subsequently, the other four OAM modes |−4〉, |−2〉, |0〉, |+2〉 were transformed into |−2〉, |0〉, |+2〉, |+4〉 by leveraging the identical three-layer balanced architecture.

3 Results

To verify the proposed strategy’s feasibility in cyclic transformation of OAM modes, we constructed a classical measurement apparatus combined with the balanced mode filtering device, and the details of this setup are outlined in supplementary material Note S1. Figure 3(a) shows the selected input OAM modes with orders {|−4〉, |−2〉, |0〉, |+2〉, |+4〉}, created by the combination of several waveplates and a Q−P (with q-values of 1 or 2). The order of modes can be detected by utilizing a cylindrical lens (C-lens); specifically, the number of dark diffraction fringes indicates the absolute value of mode order, and the tilt direction of the fringes represents the sign of the order. Under the modulation of the three-layer balanced architecture (including HWP-Q-plate1 combination, partitioned phase loaded on SLM and Q-plate2), the intensity and C-lens detection results of output OAM modes were measured, as shown in Fig. 3(b). Additionally, the optical evolution process of the input OAM modes passing through each component of the architecture was recorded in supplementary material Note S2. From these results, it can be concluded that the five input OAM modes have been successfully transformed in a clockwise direction to {|−2〉, |0〉, |+2〉, |+4〉, |−4〉}, which is in good agreement with the theoretical expectations. Notably, the intensity outcomes of the cyclic transformation exhibit slight annular discontinuity and non-uniformity. This mainly stems from the disruption to the partitioned phase modulation caused by deviations from the ideal mode states during the mode conversion process, along with the increase in the beam waist size. Misalignment and deviations in polarization states also contribute to some extent, though these effects are potentially mitigable through precise adjustments of the optical path. Nevertheless, the obtained results validate the effectiveness of our scheme for OAM mode cyclic transformation. Further analysis of the effect of misalignment and polarization state deviations can be found in supplementary material Notes S3 and S4.

For further quantitative analysis, we measured the purities of the output transformed OAM modes and the diffraction efficiency under each input mode, as presented in Figs. 3(c) and (d). The mode purity is obtained by projecting the generated light field onto the set of orthogonal OAM modes,by utilizing Q-plate elements with different q-values and then calculating the normalized ratio of the total in-plane optical field energy under a single projection to that across all projections; the detailed theoretical expressions are shown in supplementary material Note S5. As can be seen, the successfully converted mode purities are higher than 92% for all five OAM modes, implying effective cyclic transformation between them. In addition, the measured diffraction efficiency is defined as the optical power ratio (E) of the total output light intensity (I_out) to the input intensity (I_in), which can be written as E = I_out/I_in and representing the probability that the optical-field energy is effectively utilized for each selected input OAM mode. The results clearly demonstrate that the cyclic transformation operation has excellent performance in energy conversion, with the efficiency exceeding 92.7%.

Furthermore, to verify the functionality of our proposed balanced mode filtering mechanism in the quantum regime, we constructed a high-dimensional OAM mode quantum Pauli-X gate by incorporating a polarization-entangled photon source and a detection system to the original framework. In the schematics of the experimental setup as illustrated in Fig. 4(a), a pump laser with a central wavelength of 780 nm passes through the periodically poled lithium niobite waveguide (PPLN) to generate the entangled photon pairs at 1550 nm by spontaneous parametric down-conversion (SPDC). The photon pairs counts and polarization interference visibility measurements are performed to characterize the quality of the photon source, and its detailed performance is shown in supplementary material Note S6. Through the SPDC process, the produced photon pairs possess mutually orthogonal polarization states. Due to phase matching conditions, the signal photons and idler photons, presenting polarization entanglement, are emitted along non-collinear wave vectors and are spatially separated into different ports by the reflection from the prism. Prior to this, the BPF in each path is utilized to filter the continuous waves and photons at other wavelengths. The signal photons are then guided to the optical components: HWP, QWP and Q-P, for the generation of the prepared OAM mode quantum state. We employ the modulation of the HWP-Q-plate1 and Q-plate2 to shift the states, and load the SLM with designed partitioned phase to implement the Pauli-X gate. In this process, the QWPs are used to convert the polarization state of OAM modes between |R〉/|L〉 and |H〉/|V〉, and the lens pairs control the diffraction length of different mode states. On the other path, the idler photons are directly coupled into the detection system consisting of the polarization controller (the element combination of a QWP, HWP and LP with various optical axis orientations for orthogonal polarization projections), a set of single photon detectors (SPDs) and a time-correlated single photon counting (TCSPC) equipment.

The quantum entanglement property is preserved after the cyclic transformation of OAM mode quantum states, and we perform quantum state tomography (QST) to reconstruct the density matrices by recording the coincidence counts between the idler and signal photons. Here, instead of complex measurements on quantum superposition states, we convert the generated five types of states to the polarization entangled state, respectively. For each specific state, the cascaded polarization control wave plates with distinct optical axis orientations constitute 16 orthogonal projection operators, and the photon counts obtained undergo a series of matrix transformations to yield the corresponding density matrices, which is described in supplementary material Note S7. By utilizing the proper Q-Ps to convert wavefronts from helical to Gaussian, the signal photons are transferred to zero-order photons and coupled into a single-mode fiber for analysis. To avoid decoherence induced by path differences between the signal photons and idler photons, a 5-meter-long optical fiber is added to the idler path to compensate for phase delays. The SPDs operate in continuous mode to detect photons, in which the counting time is set as 0.1s, while the dead time and quantum efficiency are configured as 20 μs and 10%, respectively, to adjust the number of detected photons. For an integration time of 1s, ~100 coincidence counts are measured without any projections. From a total of 16 measurements performed for each polarization-entangled state after the quantum Pauli-X gate operation, we recovered the density matrices and measured the fidelities of these states, as shown in Figs. 4(b1)–(b5). It is clear to see that reconstructed entangled states are close to the theoretical Bell states for the original photon source with fidelities of 0.983 ± 0.001, 0.923 ± 0.001, 0.938 ± 0.001, 0.951 ± 0.001, 0.960 ± 0.001, respectively. where we define the fidelity between the recovered ρ_e and theoretical density matrices ρ_t by F (ρ_e, ρ_t) = Tr ((ρ_e^1/2∙ρ_t∙ρ_e^1/2)^1/2). These results are in good agreement with theory, indicating that the mode crosstalk and wavefront distortion are effectively suppressed in high-dimensional quantum operations based on OAM. Additionally, the quantum coherence is well-preserved, highlighting the system stability of our proposed balanced mode filtering architecture. Furthermore, the gate operation error confined to within 7.7% can potentially be corrected via high-dimensional quantum error correction codes, thereby enabling the future construction of more complex quantum circuits, such as cascaded quantum logic gate structures on high-dimensional platforms.

4 Discussion

In this study, our system can be scaled to higher dimensions without changing the modulation architecture in principle. For the incident beam with (2l/k+1) OAM modes {|−l〉, |−l+k〉, …, |l−k〉, |l〉}, the radial partitioned phase drives the mode modification for parallel conversion, in which the phase profile of the inner region is generalized as φ_in = exp(−i∙2l∙θ) and the complementary outer region is filled with the phase φ_out = exp(i∙k∙θ). The cyclic transformation can be achieved by reconfiguring the phase masks of two sub-regions and by adjusting the parameters of Q-plate1 and Q-plate2 whose values are defined as q = \mpl/2 to satisfy the center filtering condition for OAM modes. While the variables are theoretically unbounded, the maximum number of supportable OAM modes N_max fundamentally depends on the beam waist of incident mode. To be more specific, the excessively large waist radius of high-order OAM modes impedes spatial overlap with finite inscription areas and thus hinders effective modulation. Moreover, spatially separated modulation also induces deviations from the ideal mode state due to insufficient diffraction distance in the mode transformation process, thereby disrupting the subsequent spatially separated modulation. In this case, we progressively increased the order of incident OAM modes at identical intervals and calculated the corresponding state fidelity, in the same manner as the previous experimental demonstration. The results indicate that the maximum supportable mode order is almost l = 8, and thus N_max = 17 under the condition of F > 90%, as shown in supplementary material Note S8.

As for the extensibility of photonic logic operations, this scheme can be developed into an enhanced architecture that cascades n radial partitioned phase profiles within symmetric Q-plate devices for the implementation of Pauli-Xⁿ gate, that is n integer power of Pauli-X gate, transforming each mode to its n-th nearest neighbor mode. As shown in Figs. 5(a) and (b), we experimentally demonstrated the three-mode cyclic transformation for Pauli-X² gate with the re-established balanced mode filtering frame by insetting two identical localized phase masks inside of a pair of inverse Q-plate elements.

The results from C-Lens detection align with theoretical predictions, confirming the input modes {|−2〉, |0〉, |+2〉} are converted to their second-nearest position {|+2〉, |−2〉, |0〉} in a clockwise direction. Furthermore, slight spot distortions arising from non-uniform optical field propagation had a negligible impact on the outcomes. Compared with prior methods, our approach enables the augmentation of logic gate operation functions by increasing the number of partitioned phase modulation in a relatively compact manner. Moreover, by adjusting the number of cascaded phases, our approach facilitates the function switching and reconfiguration between different Pauli-Xⁿ gates with different exponents in high-dimensional OAM mode space. Besides, through employing advanced nanofabrication techniques and two-photon 3D polymerization lithography methods to create metasurface elements that are capable of classifying polarization and controlling phase of light with high efficiency, it is promising to replace the SLM and Q-plate devices in our architecture with a miniature platform to perform quantum logic operations on more degrees, which makes it an appropriate candidate for integrated quantum computing.

5 Conclusion

In summary, we have introduced a balanced mode filtering architecture for the scalable implementation of quantum Pauli-X gate operation in high-dimensional OAM space, breaking the number of mode processing limitations that exists in conventional transformation methods. Driven by the embedded partitioned phase mask between a pair of structurally symmetric Q-plate elements, the highest-order OAM mode of the inputs is filtered out as Gaussian intensity profiles and subjected to selective phase modulation from distinct regions with other converted non-Gaussian profiles, enabling the synergistic periodic movement among different OAM modes. Benefitting from the radial matching between the OAM mode and partitioned phase modulation, it is possible to extend to higher dimensions just by tweaking the relevant parameters of the filter. Given this concept, the classical five-modes cyclic transformation is successfully demonstrated with mode purity and efficiency higher than 92% and 92.7%, respectively. By adopting the polarization-entangled photon source, our scheme is straightforwardly utilized for conducting five-dimensional quantum Pauli-X gate, and the output mode states present well-performed entanglement maintain owing to the high fidelities of reconstructed density matrices exceeding 92.3% ± 0.1% measured by QST. Our achievement underscores the effectiveness of the proposed strategy for the high-dimensional quantum Pauli-X logic gate operation on OAM modes in the general manner and shows the potential in significant scalability, which holds great promise for the practical deployment of high-dimensional spatial mode quantum information processing.

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