Quaternion Optical Computing Chip (QOCC)

• Quaternion Optical Computing Chip (QOCC) is a silicon-photonic processor that implements quaternion algebra in a single optical pass, enabling efficient 3D and 4D data processing.
• It employs wavelength-division multiplexing, tunable micro-ring resonators, and photonic summation circuits to perform quaternion-valued linear algebra and convolutions in real-time.
• Key applications include 3D point cloud transformations, RGB chromatic rotations, and quaternion neural network inference, demonstrating high computational fidelity and scalability.

The Quaternion Optical Computing Chip (QOCC) is a silicon-photonic processor designed for high-throughput, energy-efficient computation of quaternion-valued linear algebra, convolution, and related transforms, targeting parallel high-dimensional data processing. The QOCC leverages the properties of quaternions—hypercomplex numbers well-suited to 3D and 4D data representations—by encoding their components across multiple optical wavelengths and manipulating them on-chip using wavelength-division multiplexing (WDM), thermally-tunable micro-ring resonator (MRR) arrays, and photonic summation circuits. By executing quaternion algebra in a single optical pass, the QOCC demonstrably reduces computational load and energy requirements compared to electronic digital implementations, achieving higher computational fidelity and scalability for applications such as point cloud transformation, chromatic rotation, and quaternion neural inference (Liu et al., 4 Jan 2026).

1. System Design and Photonic Architecture

The QOCC architecture integrates a number of specialized silicon-photonic components to enable programmable quaternion-valued computations. The WDM light source stage consists of three uncoherent distributed feedback (DFB) laser diodes operating at λ₁ ≈ 1550.8 nm, λ₂ ≈ 1552.4 nm, and λ₃ ≈ 1554.0 nm, which are combined by a dense wavelength-division multiplexer (DWDM). Each data vector to be processed is encoded as a sequence S = [⋯, xᵢ, yᵢ, zᵢ, ⋯] by a high-speed arbitrary waveform generator (AWG, 65 GSa/s), amplified and modulated (EA + Mach–Zehnder Modulator at 40 GHz) so that each quaternion component modulates a distinct wavelength.

The signals are synchronized via an 8 km optical time delay line (OTDL) introducing 108 ps symbol-offsets between adjacent wavelengths, aligned to a baud rate f_b = 9.2 GHz. The QOCC chip itself consists of three parallel photonic computation branches (corresponding to the quaternion output components) with each branch containing an array of three serially-connected, thermally-tunable add–drop MRRs. These implement programmable weights $w_{ij}$ for each wavelength channel via resonance detuning. After analog recombination, the signals are converted to electrical voltages using balanced photodetectors (BPDs). Final post-processing involves subsampling every third time-slot to reconstruct the valid quaternion result stream.

Fabrication details include a 220 nm-thick Si device layer (CUMEC CSiP180Al MPW), 500 nm × 220 nm Si strip waveguides with SiO₂ upper cladding, MRRs with radius ~10 μm (FSR ≈ 6 nm, Q ≈ 9,000), TE-polarized grating couplers (~3 dB loss), and etched isolation trenches (< 5 % thermal crosstalk).

2. Implementation of Quaternion Algebra

Quaternions are hypercomplex numbers of the form $q = a + b\,i + c\,j + d\,k$ with non-commutative multiplication rules: $$i^2 = j^2 = k^2 = i\,j\,k = -1; \quad i\,j = -j\,i = k,\; j\,k = -k\,j = i,\; k\,i = -i\,k = j.$$ Addition and multiplication are implemented in accordance with their algebraic definitions: $$q_1 + q_2 = (a_1 + a_2) + (b_1 + b_2)\,i + (c_1 + c_2)\,j + (d_1 + d_2)\,k,$$

$$q_1 q_2 = (a_1 a_2 - b_1 b_2 - c_1 c_2 - d_1 d_2) + (a_1 b_2 + b_1 a_2 + c_1 d_2 - d_1 c_2)\,i + (a_1 c_2 - b_1 d_2 + c_1 a_2 + d_1 b_2)\,j + (a_1 d_2 + b_1 c_2 - c_1 b_2 + d_1 a_2)\,k.$$

For 3D vector processing, input data p = (x, y, z) are modeled as “pure quaternions” (0 + x i + y j + z k); each coordinate is amplitude-modulated onto a distinct optical carrier. MRR detuning allows per-wavelength weighting in [−1, 1], enabling on-chip implementation of quaternion-valued matrix multiplications and rotations. Recombination in the electrical domain via incoherent detection delivers algebraic summation consistent with quaternionic addition. Arbitrary quaternion products or convolutions can be implemented in a single optical pass by appropriately configuring the 3×3 weighting matrix in the MRR arrays.

3. Data Encoding, Pipeline, and Signal Flow

The QOCC’s optical information pipeline utilizes wavelength-division multiplexing and time-interleaving for simultaneous manipulation of high-dimensional, multi-component data. Three wavelength channels each encode one component of the input quaternion per symbol period, with the sequence S = [x₁, y₁, z₁, x₂, y₂, z₂, ...] modulated at a symbol rate of 9.2 Gbaud, giving one quaternion per 108 ps interval.

Upon entering the QOCC, the three channels are temporally aligned by the OTDL and then split into three compute branches (corresponding to output quaternion components), each passing through three tunable MRRs for programmable weight application. The recombined and photodetected outputs yield continuous electrical signals corresponding to weighted quaternion sums. The output is digitized (80 GSa/s oscilloscope), and every third symbol is sampled to reconstruct the valid high-dimensional result sequence.

4. Application Benchmarks and Experimental Performance

4.1 3D Point Cloud Processing

In transformation tests with point clouds (“SJTU,” 2,500 points, and “Dog,” 10,000 points), the QOCC applied quaternion rotations (e.g., $q=1 -0.4\,i -0.1\,j +0.3\,k$; $q=1 + 0\,i +0\,j -1\,k$) to entire datasets using the 3-wavelength scheme. Observed RMSE was consistently < 0.035, with ~5-bit positioning precision. The QOCC executed full transforms in a single light pass, whereas classic electronic implementations require multiple sequential real-valued MACs (16 per q⋅p⋅q⁻¹), resulting in a reduction of 2/3 of operations.

4.2 RGB Chromatic Transformation

For space transformations of color images (pixels encoded as $c = 0 + R\,i + G\,j + B\,k$), the QOCC performed two types of unit-quaternion rotations ($q_{120°} = 0.5 + 0.5\,i + 0.5\,j + 0.5\,k$, $q_{-120°}= 0.5 -0.5\,i -0.5\,j -0.5\,k$) on 512×512 images. Each result maintained RMSE < 0.035 relative to ideal electronic calculations, with waveforms indicating close visual correspondence with perfect color rotations.

4.3 Quaternion Convolutional Neural Network (QCNN) for CIFAR-5

The QOCC implemented a four-layer QCNN for color image recognition on CIFAR-5, with 32×32×3 RGB images regarded as quaternion matrices. Each QCNN layer used 2×2 quaternion kernels, mapped to the MRRs' weight matrix. After end-to-end training (Adam optimizer, backprop), the optical QCNN achieved ∼80.2% test accuracy (mathematical ideal: ∼82%), compared to a standard CNN baseline of ∼78%. Cross-entropy loss for the QCNN reached 0.14 (vs. 0.47 for the standard CNN). Optical error was quantified as RMSE ∼0.035 with σ = 0.0212 (≈ 5 bits).

4.4 Throughput and Energy Efficiency

The QOCC achieved a measured computational throughput of 331.2 GOPS (9.2 Gbaud × 3 branches × 4 modules × 3 wavelengths). The estimated energy per quaternion convolution (QC) operation was ≈11.6 pJ (108 ps × 3 mW MRR heating per device × 9 MRRs × 4 modules). This demonstrates a scaling advantage over traditional digital optical and electronic processors.

5. Comparison with Electronic Computing Architectures

The QOCC offers significant improvements in parallelism, computational load, and energy when compared to conventional SIMD/DRAM electronic systems. For 3D rotations and color-space transforms, the QOCC achieves optical RMSE < 0.035, compared to ideal numerics on electronic platforms (machine precision, Δ=0). In computational load, typical electronic quaternion multiplication for rigid-body transforms requires ~24 real-valued operations per point (for q p q⁻¹), whereas the QOCC requires a single light-pass MAC, yielding a two-thirds reduction in sequential operations. Throughput measurements (QOCC: 331.2 GOPS) exceed typical electronic limits by up to 30× (∼10 GOPS). Energy per MAC is 11.6 pJ for QOCC, compared to electronic figures in the pJ–nJ range.

Comparison Metric	Electronic System	QOCC
Fidelity (RMSE)	0 (machine precision)	< 0.035
Throughput (OPS)	∼10 GOPS	331.2 GOPS
Energy per QC operation	pJ–nJ	∼11.6 pJ
Computational Load	24 MACs/q p q⁻¹	1 light-pass MAC

6. Significance and Outlook

The QOCC enables direct, high-fidelity hardware implementation of quaternion-valued signal processing and computation, addressing the inefficiency of real- and complex-valued hardware for high-dimensional data. Its integration of WDM, time-interleaving, and programmable photonic weighting within a compact silicon-photonic platform provides a scalable approach to parallelizing quaternion algebra for data analytics, artificial intelligence, and multidimensional physical simulations. Achieving one-pass quaternion computation positions the QOCC as a candidate architecture for next-generation high-dimensional optical processors, overcoming throughput and energy bottlenecks that limit electronic platforms for processing tasks requiring native quaternion representations (Liu et al., 4 Jan 2026).