Programmable photonic unitary circuits for light computing

3.1 Design strategy

In describing light behaviors in both isolated and open systems, a unitary matrix illustrates the evolutions of the couplings between optical elements, as shown in Eqs. (1) and (2). In this context, a major challenge in constructing higher-dimensional U _N matrices arises from their off-diagonal components, which necessitate the couplings between far-off elements. Because nearby interactions are dominant especially in the platforms for photonic integrated circuits whether exploiting the couplings through propagating [62] or evanescent [45] light, the basic strategy of realizing U _N is to develop the systematic assembly of nearby interactions to effectively achieve far-off couplings. Mathematically, this strategy is based on the conservation of the unitarity under the multiplications, as (VUW)^†(VUW) = I, where U, V, and W are unitary matrices. Therefore, a common objective in this method is to achieve the diagonalization through the multiplications, as VU _N W = D, where D is the diagonal matrix that can be obtained simply with isolated optical elements, and V and W represent unitary matrices achievable with nearby couplings. While the target unitary matrix can be reconstructed with the multiplication form U _N = V ^† DW ^†, the resulting matrix multiplication corresponds to the cascaded operations exerted on light, which are executed with a series of varying optical systems along the spatial or temporal axis.

As an example, we introduce the method of implementing an arbitrary U _N using a set of two-dimensional (2D) unitary matrices U ₂, which was firstly developed in Ref. [61]. For simplicity, we restrict our discussion to U ₂ that reflects only the nearest-neighbor (NN) couplings in one-dimensional (1D) systems, which cover conventional programmable photonic integrated circuits using multiple waveguides [25]. To apply the unitary multiplication, we develop the N-dimensional unitary matrix T _m that includes a nontrivial 2D unitary operation U ₂ at the components (m, m), (m, m + 1), (m + 1, m), and (m + 1, m + 1), while the rest components of T _m follow the N-dimensional identity matrix I _N . The role of T _m is to cancel out an off-diagonal component through the multiplications of T _m U and UT _m , which allow for nulling one of the mth or (m + 1)th row and the mth or (m + 1)th column components, respectively (Figure 2a).

Figure 2:

Universal unitaries composed of lower-dimensional units through nulling processes. (a) Example of nulling processes in the case of U ₄ (components with white squares). By multiplying T ₃ (red squares with SU(2) operations, blue squares with 1, and black squares with 0) on the right side of U ₄, one of the U ₄ T ₃ components becomes zero (black square) by designing the SU(2) operation to satisfy u ₁ s ₁ + u ₂ s ₂ = 0. While some of the components in U ₄ T ₃ are identical to those in U ₄, the other components are the SU(2)-transformed U ₄ components. The purple box in U ₄ denotes the SU(2)-transformed part. (b, c) The nulling processes of the (b) Reck [61] and (c) Clement [29] designs, and (d, e) their hardware implementations, respectively. The color of each arrow in (b, c) corresponds to the optical element of the same color in the circuits of (d, e). In (b–e), the nulling process is achieved sequentially, following the order of the orange, green, and purple arrows. The colored boxes in (d, e) represent the SU(2) building block described in Sections 3.2 and 3.3. In (d, e), i _n and o _n are the nth input and output signals, respectively (1 ≤ n ≤ 4).

To achieve the diagonalization with a set of T _m , N(N – 1)/2 nulling processes applied to either the upper or lower triangular off-diagonal components are necessary (Figure 2b and c), because a triangular unitary matrix is a diagonal matrix. Therefore, in the unitary matrix reconstruction U _N = V ^† DW ^†, V and W are constructed with a series of NN-coupling unitaries T _m . We note that the sequence of the nulling processes supports design freedom as long as a series of the cascaded nulling processes operate independently, that is, the protection of the nulled components during the subsequent processes. Representative examples include the Reck ([61], Figure 2b) and Clements ([29], Figure 2c) designs, which result in the optical hardware with different feature sizes and symmetry conditions (Figure 2d and e). Notably, owing to its structural symmetry, the Clements design achieves almost half the circuit depth and higher fidelity with error-tolerant interferences, compared to the Reck design.

The range of the unitary group accessible through nulling processes is determined by the universality of the unit unitary matrix T _m and the diagonal matrix D. Particularly, with the universal form of T _m and D, a universal set of U _N that covers random Haar matrices [63] can be achieved deterministically. Because the diagonal components of D have the unit modulus, the tailored phase evolutions in individual optical elements, such as optical waveguides or resonators with phase shifters, enables the universal form of D. Therefore, a major challenge in achieving programmable and universal U _N construction is the realization of the reconfigurable building block for T _m , which has to satisfy the complete coverage of the SU(2) group, at least, up to the global phase for nulling an arbitrary off-diagonal element during the diagonalization. In the following Sections 3.2 and 3.3, we explore the spatial- and temporal-domain realizations of T _m , which possess distinct characteristics in their design procedures and the performance figures of the resulting U _N .

3.2 Spatial domain implementation

Because T _m describes the evolution of photonic states in a two-level system up to the global phase, the corresponding operation is characterized by two degrees of freedom: the relative differences in the amplitudes and phases of the fields inside two elements [25]. As discussed in quantum computation [3], the universality of such an operation is guaranteed by a set of two arbitrary rotation operations among R _x , R _y , and R _z , which denote the rotations about x-, y-, and z-axis, respectively. These rotation operations, generally describing the θ-angle rotation about the vector n, can be expressed as the following propagator form:

where σ is the Pauli vector σ = x σ _x + y σ _y + z σ _z for the following Pauli matrices:

Regarding passive and active optical elements widely used in integrated photonics, the rotation operations that are practically more accessible are R _x and R _z . First, as shown in the form of σ _x in Eq. (4), which is the generator of R _x , the rotation R _x can be obtained with the in-phase coupling between two elements, such as two coupled waveguides composing a directional coupler. The amount of the R _x rotation is then determined by the magnitude of the coupling coefficient and the length of the coupler. Second, the rotation R _z is achieved with σ _z in Eq. (4), which is achieved with the decoupled optical elements with different on-site energies such as the propagation constants in waveguides and the resonance frequencies in resonators. To achieve arbitrary rotations of R _x and R _z , the listed optical parameters – coupling coefficient, coupler length, propagation constants, and resonance frequencies – need to be reconfigurable. It is worth mentioning that the realization of R _y , which requires the gauge field in the coupling due to the imaginary off-diagonal terms in σ _y , is not straightforward in the spatial-domain integrated photonic platforms. Instead, R _y can be obtained with a series of R _x and R _z , as R _y (θ) = R _x (3π/2)R _z (θ)R _x (π/2).

The SU(2) gate, which conducts an arbitrary U ₂ ∈ SU(2) matrix operation for the nulling processes in Figure 2, can be realized by cascading the constituent operations R _x and R _z , as U ₂ = Π _k R _n(k)(θ _k ), where n(k) denotes the rotation axis x or z, and k is the order index of the SU(2) operation. It is noted that the configuration of U ₂, which is determined by the number and order of the rotations, is not unique. Although the simplest form of U ₂ would be the straightforward multiplication of two orthogonal rotations, such as U ₂ = R _z (θ ₂)R _x (θ ₁) or R _z (θ ₂)R _y (θ ₁), one needs to consider the practical implementation of each configuration in terms of the operation fidelity, the method for reconfigurability, and the compatibility with fabrication techniques.

Regarding the difficulty in directly implementing R _y , the most straightforward architecture employs tunable directional coupling [64], [65], [66] (Figure 3a and b). The tunable directional coupler corresponds to a reconfigurable R _x rotation, which enables the following relation:

where γ is the rotation angle of 0 ≤ γ ≤ π about the x-axis. Although stable manipulation of the coupling is a critical engineering issue, as discussed in below, the mathematically straightforward structure of the unit and the resulting geometrical simplicity are clear advantages of this approach.

Figure 3:

Spatial implementation of universal unitary circuits. (a, b) SU(2) gate with the reconfigurable R _x rotation: (a) schematic of the gate, and (b) its real implementation [64]. (c, d) SU(2) gate with the fixed R _x rotations: (c) schematic of the gate, and (d) its real implementation [10]. (e) The PPC implementation using a set of SU(2) gates in (d). (f) Effect of hardware errors on pattern recognition [67] for 28 × 28 images from the MNIST dataset [68]. ‘EC’ denotes the error correction achieved by the self-configuration of Mach-Zehnder interferometers (MZIs) and MZI with an additional beam splitter (3-MZI). (g, h) The pruning of PPCs for noise-resilient unitary operations [69]. (g) The averages of the phase shift ξ = <θ _m,l > for the PPCs of 100 U ₃₂ realizations. (h) Comparison of the fidelities of the U ₁₂₈ PPCs in different groups: pruning body (red line), pruning tail (blue line), noisy body (orange line), and noisy tail (green error bars). The thicknesses of the colored lines and the error bars present the range of the fidelities between their maxima and minima. Panel (b) reprinted with permission from Ref. [64], Optica. Panels (d, e) reprinted with permission from Ref. [10], Springer Nature Limited. Panels (f) and (g, h) are adapted with permission from Refs. [67], [69], respectively, Springer Nature Limited.

On the other hand, the most widely used configuration involves a fixed R _x (Figure 3c and d), highlighting the technical challenges in achieving reconfigurability of coupling coefficients. To adjust the coupling for controlling R _x (θ) without affecting R _z rotations, the optical modes within the elements must be perturbed symmetrically. Furthermore, the nonlinear relationships between material perturbations and modal profiles [70], [71], [72], as well as between the perturbed modal profiles and coupling coefficient [73], need to be mitigated. Therefore, to avoid the sensitivity and nonlinearity in performing reconfigurable R _x (θ), most conventional programmable photonic circuits (PPCs) have leveraged the relationship of R _x (π)R _y (ξ) = R _x (π/2)R _z (ξ)R _x (π/2), which leads to

As shown in Figure 3a and b, the corresponding SU(2) gate is composed of two identical passive Mach-Zehnder interferometers both for R _x (π/2), and two phase shifters of ξ and η for R _z (ξ) and R _z (η), respectively. When 0 ≤ ξ ≤ π and 0 ≤ η < 2π, the degrees of freedom provided by two rotations R _z (η) and the effective R _y (ξ) allow for the realization of universal SU(2) operations, completely covering the entire evolutions on the Bloch sphere. It is worth mentioning that although other configurations, such as U ₂ = R _z (η)R _x (3π/2)R _z (ξ)R _x (π/2) = R _z (η)R _y (ξ) that is a mathematically more concise form, are also allowed, the critical advantage of the configuration in Eq. (5) lies in the identical form of R _x (π/2), which results in the same fabrication conditions for the Mach–Zehnder interferometers.

To achieve the necessary modulation for the reconfigurable operations of the SU(2) gate, various modulation schemes have been examined. A conventional approach involves using thermo-optic modulation [10], [65], [74], [75], [76], [77], which possesses advantages of compact gate design due to substantial changes in the refractive index, and low insertion loss. However, this approach presents several challenges, such as limited operation bandwidth of a few MHz and high-power consumption. To address each issue, various alternative solutions have been successfully demonstrated. For example, electro-optical modulation using lithium niobite films offers from 1 to 100 GHz modulation for phase shifts [78], [79]. In terms of power consumption, piezo-optomechanical techniques enable μW-scale operations per modulator with operation speeds about 100 MHz [80], [81]. Owing to the sharply different features of each modulation technique, the method of modulation is a core factor in determining the performance and application target of the entire PPC (Figure 3e).

All modulation techniques and the passive elements used in the SU(2) gate inevitably involve noise and defects. Furthermore, such random errors scale with O(N ^1/2) due to the O(N) circuit depth. Therefore, addressing fidelity issues through error correction is critical, especially when conducting computing functionalities in larger N, even for stochastic operation such as deep learning. Furthermore, devising hardware error correction can also be employed in the in situ training of deep learning in terms of the suppression of cost functions [82], [83].

To address imperfections, for example in linear-optic elements such as beam splitters, various systematic calibration procedures have been developed. In these procedures, the underlying mechanisms are developed upon two notions: measurement-based reconfigurability and the utilization of auxiliary dimensions within system parameter spaces. First, error correction necessitates both signal detection and the consequent reconfiguration of systems. Although an individual correction of each element is expected to be ideal for high fidelity [28], [84], [85], [86], [87], such an ideal condition not only burdens the fabrication difficulty but also raises additional noises due to the necessity of local signal monitoring. Therefore, there have been substantial efforts on reducing the required number of internal detectors and tunable elements. The approach at the extreme is global optimization, for example, using gradient descent [88] or the in situ training with adjoint optimization, which requires only detecting signals at the system boundaries [82], [83]. The challenge of employing these techniques to error correction would be the difficulty in properly setting initial conditions to explore global optimum and the necessity of problem-specific training. Recently, error correction without internal detectors for sufficiently low local errors has been demonstrated using self-configuration method, which corrects the splitting error in each beam splitter [89], [90], [91].

Another approach – auxiliary dimensions in system parameter spaces – increases the upper bound of accuracy at a given level of device defects. At an elemental level, it was demonstrated that a beam splitter targeting 50:50 power division but with errors up to 85:15 can be adjusted by doubling the system dimension through additional phase shifters and beam splitters [28]. In a similar context, employing additional passive elements to each gate, such as a beam splitter and waveguide crossing, allows for avoiding forbidden regions in the Hilbert space of optical states [67]. This improved coverage enhances the accuracy of the circuit operations (Figure 3f) beyond previous works [89], [90], [91], achieving fault tolerant pattern recognition. The use of auxiliary dimension has also been employed in system levels, such as the realization of random Haar matrices using an additional beam splitter layer at the terminal port of the photonic circuit [92].

In a similar context, a method inspired by network science has recently been proposed, focusing on categorizing active optical elements based on their impacts on achieving reconfigurable universal unitary operations [69]. The work demonstrated that the building blocks of PPCs designed under the Clements approach [29] for achieving random Haar matrices follow the power law and the Pareto principle, stating that most of the significant rotation operations come from about 20 % of the building blocks – the ‘tail’ elements of the circuit. This analysis inspires the pruning of less important phase shifters in the overall circuit (Figure 3g), which enables the removal of tunable phase shifters that are usually the most expensive elements in PPCs in terms of noise and footprint engineering. As shown in Figure 3h, pruning the ‘body’ of the phase shifters – less important ones, constituting about 80 % of the circuit – can yield even better performance at a specific level of noise, aligning with the underlying mechanisms of hub nodes in network science [93]. Recently, the concept of pruning for photonic circuits has also been employed in relation to the training process [94], [95], achieving reduced power consumption and noise resilience.

3.3 Temporal domain implementation

Despite the great success of spatial-domain PPCs especially in terms of fidelity [28], [31], [67], [69], [82], [86], [87], [88], [89], [90], [91], [92], [96], [97], scalability remains another critical challenge. To diagonalize a U _N matrix using SU(2) operations, N(N – 1)/2 nulling processes are necessary as described in Figure 2. Even when approximately N/2 operations are achieved in parallel, as is available in the Clements design [29], the circuit depth is proportional to N, which results in O(N ²) scalability in both device footprint and gate number regarding the channel number N. This poor scalability, for example, requiring over 10⁶ elements with footprints exceeding a few cm² for a deep learning model with only 1,000 neurons, also raises substantial fidelity issue due to imperfections in device fabrication and light sources.

One possible approach to overcoming this limitation is to exploit the mathematical similarity between the governing equations in different physical axes. For example, the spatial propagation of light along a waveguide and the temporal evolution inside a resonator can be described by the almost same equations of da/dx = –iβa and da/dt = +iωa [73], respectively, where a is a field amplitude, β is the wavenumber along the waveguide, and ω is the resonance frequency of the resonator. Therefore, one can replace the field evolution of light along the spatial domain with the evolution along an alternative physical axis – time or frequency – which allows for the substitution of the device footprint with the operation time or mode number. Similar approaches have been employed in examining synthetic-dimensional phenomena [13], [98], [99], [100], [101], [102], [103], supersymmetric transformation [104], crystal optics [39], [105], and disordered photonics [41], [43].

The proposal of programmable unitary circuits using temporal-axis computation, referred to as programmable photonic time circuits (PPTCs) [106], corresponds to the space-to-time replacement of wave evolutions to address the poor footprint scalability of PPCs. The underlying mechanism of PPTCs relies on tailoring optical spatial modes using time-domain modulation, which has been intensively studied in various platforms. For example, recent studies on photonic time crystals [39], [105] and time disorder [40], [41], [43] have focused on controlling directivity of light with broken continuous time translational symmetry while preserving spatial translational symmetry. The simultaneous breaking of spatial and time translational symmetry allows for advanced manipulation of optical modes, such as wavelength conversion [107], complete optical isolation [108], effective magnetic fields [109], and non-Hermitian band manipulation [110]. In line with these historical cornerstones, the goal of PPTCs is to achieve reconfigurable and universal unitary operations of optical spatial modes using broken spatial and time translational symmetries in integrated photonic platforms, by adjusting the design strategy of PPCs.

In PPTCs, the nulling process for diagonalization is identical to the spatial counterpart, while the spatial SU(2) gate is replaced with the temporal one. This SU(2) ‘time gate’ consists of two identical travelling-wave resonators with resonance frequency ω ₀ (Figure 4a) to describe two-level systems with the spinor state Ψ = [ψ _m , ψ _n ]^T, where ψ _m and ψ _n are the field amplitudes of the specific pseudospin modes in the neighboring mth and nth resonators, respectively. The resonators are coupled via a pair of single-mode waveguide loops with a decay rate 1/τ, which have been widely employed in topological photonics to achieve pseudogauge fields in coupling [34], [111], [112]. When preparing tunable phase shifters in the resonators and waveguide loops, the spinor state is governed by the following spinor Hamiltonian [106]:

where ξ _mn ^U(t) and ξ _mn ^L(t) are the time-varying phase shifts in the upper and lower loops, and Δω is the time-varying averaged resonance perturbation due to the tunable phase shifters in each resonator. Equation (7) shows that the full degrees of freedom for the rotation operations, R _x,y,z , are accessible with the SU(2) time gate. By imposing the digital modulation on tunable phase shifters, the rotation operations about an arbitrary axis on the xy-plane (Figure 4b) and about the z-axis (Figure 4c) can be realized, comprising a universal set for the unitary operation of a spinor state. Such a universal SU(2) operation facilitates the extension to universal and reconfigurable U _N operations following the method described in Section 3.1, as demonstrated in the realization of the quantum Fourier transform and random Haar matrices [106], and the emulation of light behaviors inside hyperbolic lattices [113]. Because the computation is achieved with light stored inside coupled resonators, the PPTC can be understood as compressing the spatial circuit depth from N to 1, by transferring the domain for necessary computations to the temporal axis. When considering the conventional integration level of cutting-edge photonic circuits – about 10⁴ gates within a cm²-scale chip [45], [111] – the potential to reduce spatial footprint using PPTCs sheds light on realizing very-large-scale integration for light-based matrix computations. The design strategy with tunable gauge fields in PPTCs has recently been extended to the implementation of integrated photonic platforms for reconfigurable matrix-valued gauge fields and the following programmable non-Abelian physics and braiding [114], which revealed novel topological phases in photonic lattices.

Figure 4:

Temporal implementation of universal unitary circuits. (a) The SU(2) time gate between the mth and nth channels. Gray circles and curved triangles represent resonators and waveguide loops, respectively. Purple boxes are phase shifters. The upper right inset illustrates a PPTC comprised of multiple SU(2) time gates. (b, c) Temporal evolution of pseudospinor modes on the Bloch sphere for two operation modes: (b) even-parity mode for ξ _mn ^U(t) = ξ _mn ^L(t) = ξ and Δω(t) = 0 in (a), and (c) odd-parity mode for ξ _mn ^U(t) = –ξ _mn ^L(t) = π/2 and Δω(t) ≠ 0 in (a). B denotes the pseudospinor rotation axis. (d) PPTC fidelities with respect to the cutoff frequency ω _c of low-pass filtering. Circles and error bars represent the average and standard deviation of 20 realizations of random Haar matrices at each ω _c and triangles represent quantum Fourier transforms for N = 4 (two qubits; orange) and N = 8 (three qubits; blue). Panels (a–d) reprinted with permission from Ref. [106], APS.

Remaining issues in the practical implementation of PPTCs lie in the design of modulation signals and the limitations in modulation speed. Because Eq. (7) is nonlinear, the calculation of the required phase shifts generally does not yield an analytical solution. In Ref. [106], digital modulation is applied as an analogy of the discretized building blocks in spatial-domain PPCs. However, because the speed of temporal modulation is limited by the response time of materials, ideal discretization in the PPTC is actually unattainable, and therefore, the fidelity significantly degrades with slower modulation of phase shifts (Figure 4d). The solution to this issue can be found in digital signal processing techniques [106], such as predistortion [115], and analog computation [116], [117], [118], [119], [120].

A more critical issue would be the necessity of ultrahigh quality (Q-) factors that should guarantee the storage of light during modulation. Notably, because the SU(2) gate is executed sequentially in previous works [106], [113] due to the lack of synchronization between SU(2) time gates, the resulting scalability of the necessary time period scales as O(N ²). Therefore, given ultra-high quality factor of Q > 10⁸ achieved in all-waveguide resonators [121], which enforces the termination of the entire operations inside the PPTC within about 10 ns, the lower bound of the modulation speed becomes about (N ²/10) GHz for U(N) operations. The realization of parallel SU(2) operations, analogous to the spatial implementation [29], [61], can substantially mitigate this problem, facilitating O(N) scalability. Dynamical control of coupling quality factors [122] can also provide an optimum solution by restricting unwanted dissipation to the intrinsic loss of resonators.

4.1 Design strategy

Although the detailed design process described in this section is distinct from that in Section 3, the underlying philosophy is identical: constructing the entire U(N) group with N ² degrees of freedom from the layers of U(N) subsets that are physically allowed in terms of nearby interactions. The difference lies in the dimensionality of the nontrivial parts of U(N) layers. In Section 3, we introduced the use of two-dimensional SU(2) operations while the other channels experience U(1) evolution. In this section, we focused on the use of N-dimensional operations, which can be implemented through nearby interactions. Consider the realization of N-dimensional universal unitary operations U using the N-dimensional cascaded unitary unit cells V _m (m = 1, 2, …, M; Figure 5a), as U = V _M V _M–1, …, V ₂ V ₁, while V _m does not need to be universal. The first step in covering N ² degrees of freedom of U is to devise a photonic unit cell that provides V _m through nearby interactions, because achieving long-range interactions is challenging on conventional integrated platforms. By performing cascaded multiplications of V _m , we can then obtain the corresponding photonic hardware that approximates U through the substantially enhanced degrees of freedom.

Figure 5:

Universal unitaries composed of higher-dimensional cells through numerical optimization. (a) The circuit realizing universal unitary operations U. i _n and o _n are the nth input and output signals, respectively (1 ≤ n ≤ N). V _m is the reconfigurable N-dimensional unitary unit cell (1 ≤ m ≤ M) of which the operation is determined by nearby interactions accessible with integrated photonic platforms. (b, c) Detailed implementation of V _m cell. (b) V _m composed of the diagonal unitary operation D and the Fourier transform U _FT. The diagonal unitary matrix is realized with an array of phase shifters: φ _n for the phase shift applied to the nth channel. (c) V _m realized through the transfer matrix T, which depends on L real-valued parameters p _l (1 ≤ l ≤ L).

In this approach toward universal unitaries, the first example of V _m is the unit cell that conducts Fourier-transform operations in the spatial domain. Various photonic systems can operate as gates for the Fourier transform of optical information, which is a subclass of N-dimensional unitary operations deterministically accessible with nearby interactions between optical elements. A representative example is a diffractive optical system, where light propagation in the Fraunhofer regime leads to the Fourier transform of the transverse profile of an optical field [38]. As a theoretical background, it was demonstrated that a reconfigurable y-axis rotation operation on the Bloch sphere of an arbitrary pair of channels can be achieved by using an N-channel discrete Fourier transform in conjunction with an N-channel diagonal unitary operation [57]. Because the diagonal unitary operation permits tunable z-axis rotations for any pair of channels, the combination of a set of Fourier-transforming (U _FT) and diagonal (D) unitary operations enables the configuration of the universal SU(2) operation for each pair of channels, as well as the subsequent N-dimensional unitary operations as demonstrated in Section 3 (Figure 5b).

It is important to note that the Fourier transform is not the only high-dimensional configuration suitable for constructing universal unitaries. Any form of reflectionless transfer matrix can serve as the unit cell for composing universal U _N (Figure 5c). Although such transfer matrices in integrated platforms are typically characterized by nearby interactions, long-range interactions can also be achievable using specific platforms. A particularly interesting example involves employing the concept of synthetic dimensions [13], [98], [99], [100], [101], [102], [103] to construct the unit transfer matrix using higher-order couplings. This approach also replaces spatial degrees of freedom with the frequency axis for enhanced scalability in footprint and element number, as similar to the approach in Section 3.3.

In the following Sections 4.2 and 4.3, we explore the realizations of V _m using the Fourier transform in the spatial domain and the transfer matrix in the frequency synthetic-dimensional domain, discussing their theoretical background, numerical assessment, and practical implementation.

4.2 Spatial Fourier implementation

In integrated photonics, which leverages nearby interactions between guided lights in 2D platforms, the Fourier transform is implemented upon the concept of the fractional Fourier transform (FrFT) [123], a generalization of the standard Fourier transform. Because the Fourier transform operator and its generalization, the FrFT operator, are both unitary, their physical implementations rely on exploring Hermitian Hamiltonians of which the propagators correspond to these operators. Consider the following operator representation of the 1D Fourier transform:

where the conventional expression of the Fourier transform of f(x) is obtained by applying F to f(x), and then, replacing x at the right side of the equation with the wavenumber k. The underlying concept of the FrFT involves interpreting the Fourier transform operator F as F = F(π/4), as described in below. As shown in Ref. [123], the operator F satisfies the following eigenvalue equation:

where H _n (x) is the Hermite polynomial of order n. Notably, the eigenmodes in Eq. (9) are familiar in quantum mechanics, because they are the eigenmodes of the harmonic oscillator Hamiltonian:

while the eigenvalue of the Hamiltonian is 2n. Regarding that F is a unitary operator, the Fourier transform can be considered the unitary propagator of the harmonic oscillator Hamiltonian according to the Stone’s theorem [2]. From Eq. (1), the propagator applied to the nth eigenmode is exp(–iHξ) = exp(–2inξ), which leads to exp(–inπ/2) at ξ = π/4. Because this relationship is established for all the eigenmodes independent of n, the Hamiltonian of Eq. (10) provides the Fourier transform of an arbitrary state ψ(ξ ₀) exactly at ψ(ξ ₀ + π/4) from Eq. (9), allowing for introducing the relation F = F(π/4) (Figure 6a). A generalized propagator F(ξ), which leads to exp(–iHξ) for the Hamiltonian H in Eq. (10), then satisfies:

Figure 6:

Spatial Fourier implementation of universal unitary circuits. (a) Schematic for illustrating the Fourier transform achieved through wave evolution in a harmonic oscillator potential. The arrows connecting colored circles represent the phase evolution of each eigenmode through the propagator exp(–2inξ). (b) Pictorial view of FrFT in phase space, where Z = 2ξ in Eq. (11). (c) A schematic of an engineered waveguide array for an optical harmonic oscillator potential. (d) Schematic for the proposed decomposition of universal linear N × N matrices with the complex diagonal matrices D ^(m) and the DFrFT matrix F. The diagonal matrices are further decomposed into a layer of phase shifters Φ^(m) and a layer of amplitude modulators A^(m). Schematic of a photonic-device architecture is shown in the bottom figure. (e, f) Errors L in reproducing universal linear matrices for (e) N = 4 and (f) N = 6, while varying the total number of layers M. Panels (b, c) reprinted with permission from Ref. [124], Springer Nature Limited. Panels (d–f) reprinted with permission from Ref. [125], APS.

This unitary propagator F(ξ) corresponds to the FrFT operator, which includes the Fourier transform as the special case when F(ξ = π/4). The graphical illustration of Eq. (11) in phase space is shown in Figure 6b.

Equations (10) and (11) demonstrate that the optical implementation of a harmonic oscillator potential enables light-based computing of both the Fourier transform and FrFT. According to the mathematical similarity between the Schrödinger equation and the optical paraxial wave equation [6], [7], the harmonic oscillator potential can be readily obtained by engineering an effective refractive index profile [126], [127], [128]. Because a higher refractive index corresponds to a lower quantum potential, the index profile of optical harmonic oscillators is shaped like a convex lens, with a higher index around the center.

In integrated photonics, achieving a continuous variation of the refractive index could be challenging because the unit elements, such as optical waveguides or resonators, are usually discretized with fixed geometrical and material specifications. Therefore, a common approach lies in realizing the discrete fractional Fourier transform (DFrFT) [129], which corresponds to the generalization of a discrete Fourier transform. Because harmonic oscillators can also be effectively reproduced in discretized systems, such as waveguide arrays with detuned coupling or propagation vectors [130], [131], [132], the DFrFT and its special case of the discrete Fourier transform can be implemented by tailoring the evolution parameter ξ within optical harmonic oscillators (Figure 6c) [124].

The DFrFT on integrated photonic platforms has been intensively studied to realize light-based computing by performing universal unitary operations [133], [134], convolution operations [135], and arbitrary linear operations [125] (Figure 6d–f). Although it has been demonstrated that universal unitary operations can be obtained through alternating multiplications of diagonal unitary matrices and Fourier-transforming (U _FT) matrices [57], designing the diagonal matrices and determining the number of unit cells (M in Figures 5a and 6e and f) still requires numerical assessments. Recently, a more general form, involving the DFrFT instead of the discrete Fourier transform, was used to obtain universal unitary operations [133]. In this approach, numerical optimization with the Levenberg-Marquardt algorithm [136], [137] was employed to inversely design diagonal unitary matrices, which correspond to the phase-shifting layers (D in Figure 5b). The result demonstrates that (N + 1) phase-shifting layers are sufficient to reproduce universal unitary operations approximately, resulting in O(N ²) scalability both for the footprint and element number, similar to the SU(2)-based design in Section 3.2. Although the design has yet to demonstrate superior scalability, its more simplified geometry without Mach–Zehnder interferometers could be advantageous especially in large-scale realizations. Furthermore, because the design is defect-resilient, maintaining the universality with one faulted phase shifter at each layer, an extended comparison with the SU(2)-based design will be necessary in future studies. It is worth mentioning that alternating multiplications of the Fourier transform, a complex-valued diagonal matrix, and the inverse Fourier transform allow for the construction of circulant matrices [138], enabling the realization of an arbitrary complex matrix [139]. In this context, the approach in Ref. [133] has recently been generalized to universal linear operations using the DFrFT [125], which can be directly applied to weight matrices in photonic deep learning and application devices, such as real-time optical direction-finding sensors [140]. In terms of resolving scalability, the utilization of the FrFT in the time domain [141] for constructing unitary operations can be a topic of future studies, as the correspondence of the relationship between PPCs in Section 3.2 and PPTCs in Section 3.3.

4.3 Synthetic-dimensional implementation

Despite the simplified and defect-resilient design using the spatial-domain DFrFT, the overall scalability of the footprint and element number remains O(N ²), which hinders large-scale implementations required for deep-learning applications. Similar to the utilization of the temporal axis in Ref. [106], employing computation on alternative physical axes instead of the spatial domain also became necessary in unitary operations using higher-dimensional unit cells. One approach is to utilize the synthetic frequency dimension [100], [142].

The concept of the synthetic dimension allows for exploring light behaviors in higher-dimensional spaces beyond the spatial dimension by exploiting internal degrees of freedom of photons or imposing extended system parameters on the governing Hamiltonian [100]. Figure 7a illustrates an example of the first approach: the use of multimode optical resonances within a single ring resonator, which extends a zero-dimensional (0D) photonic structure to an effective 1D or higher-dimensional system. The key mechanism relies on the analogy between signal transport along a spatial axis and coupling across multiple optical modes, enabling the frequency axis, which represents resonance modes, to be considered as a synthetic dimension. Because resonance modes are initially orthogonal to each other in both the spatial and temporal domains, such coupling can be achieved through spatio-temporal modulation to break this orthogonality, for example, by finite-space modulation for broken spatial orthogonality [107], and at the same time, by designed temporal modulation to control frequency-domain coupling distributions [98], [143], [144], [145]. In this field, electro-optical modulators (EOM) have been widely employed for their sufficiently large modulation speed and depth, which determine the accessible range of the free-spectral range (FSR), the accessible orders of neighboring couplings, and the coupling strength. The synthetic dimension has been widely applied in topological photonics, permitting the observation of three- [13] or even four-dimensional [142] topological phenomena in 2D integrated photonic platforms, with the experimental verification of the concept [101], [146].

Figure 7:

Synthetic-dimensional implementation of universal unitary circuits. (a) Dynamically modulated ring resonator and its band structure for comprising the synthetic frequency dimension. The resonator is modulated via the multiple harmonic signal V _M applied to a part of the ring (left), which can create nearest-neighbor (κ ₁) or long-range (κ ₂) coupling (right). ‘EOM’ denotes an electro-optic phase modulator. (b) The entire system for generating universal unitary operations and the subsequent arbitrary linear transformations. The input is encoded in five-dimensional frequency modes. (c) A unit cell design. The dielectric property of the ring resonator is controlled through the multiple harmonic EOM signal, which allows for the reconfigurable unitary transfer matrix applied to the input profile {s _m ⁺}. The purple rings represent the internal auxiliary rings for creating frequency boundaries. (d, e) The errors as functions of the number of the unit cells N _r and the number of modulation harmonics N _f in obtaining a five-dimensional (d) permutation matrix and (e) Vandermonde matrix. Panels (a) reprinted with permission from Ref. [146], Springer Nature Limited. Panels (b–e) reprinted with permission from Ref. [147], Springer Nature Limited.

Employing the extended dimension from the concept of the frequency-synthetic dimension, O(N)-scalable implementation of universal unitary operations has been recently demonstrated (Figure 7b) [147], which can be generalized to arbitrary linear transformations. In the proposed system, superior scalability is achieved by compressing the spatial circuit width from N to 1, which is obtained by replacing N-channel operations with a higher-dimensional N-element lattice unit cell developed in the 0D resonator. Each basis vector for matrix operations corresponds to the resonance mode, ω _m = ω ₀ + mΩ_R, where m is an integer, ω ₀ is the reference frequency, and Ω_R denotes the FSR. To configure the lattice, the EOM modulates the permittivity in the finite region of the resonator through a combination of multiple harmonics, as Δε = Σ _l δε _l cos(lΩ_R t + θ _l ) (1 ≤ l ≤ N _f), where δε _l and θ _l are the modulation depth and phase for the modulation frequency lΩ _R , respectively. We note that l values greater than 1 enable long-range couplings, which are one of the critical advantages of utilizing synthetic dimensions compared to the intricate design required for spatial long-range coupling [148]. Such long-range couplings allow for the efficient access to off-diagonal elements in matrix operations, providing higher-dimensional unit cells with excellent expressivity for constructing universal unitaries. The dimensionality N of unitary operations is defined by establishing spectral boundaries that block undesired couplings into too higher- or lower-modes, which can be realized with inner auxiliary rings. The rings split the modes outside the spectral boundaries through the coupling process, which deviates eigenfrequencies, and thus, results in decoupling (Figure 7c) [101].

To fulfill the necessary degrees of freedom required for unitary operations, the entire system consists of an array of N _r unit cells, each performing an N-dimensional unitary transfer matrix T (Figures 5c and 7b). The unit cell is composed of a synthetic-dimensional ring resonator side-coupled to an external waveguide. Across each unit cell, the waveguide supports the N × 1 input mode vector s ⁺ = […, s _–1 ⁺, s ₀ ⁺, s ₁ ⁺, …]^T and the N × 1 output mode vector s ^– = […, s _–1 ^–, s ₀ ^–, s ₁ ^–, …]^T, where s _m ⁺ and s _m ^– are the input and output modes, respectively, which are coupled with the resonance mode at ω _m . The unit cell then conducts a unitary operation of s ^– = Ts ⁺. Notably, the circuit depth N _r is determined by the entire degrees of freedom in multiple harmonic modulations 2N _f from δε _l and θ _l , and the dimensionality N, enforcing the condition of 2N _f N _r ≥ N ² for universality. As similar to the optimization process in Section 4.2, this synthetic-dimensional resonator array is inversely designed using a gradient-based procedure: the limited-memory Broyden–Fletcher–Goldfarb–Shanno algorithm [149]. In contrast to the spatial- or temporal-domain realizations treated in Sections 3.1, 3.2, and 4.2, the unitary transformations in this system are non-reciprocal due to the spatio-temporal modulations [108], [150]. An arbitrary linear transformation was also examined by exploiting the sub-dimensional space of the system for unitary operations [151].

When compared with the PPTC [106], which also achieves O(N) scalability through the transfer of the circuit depth into the temporal axis, the synthetic-dimensional approach that transfers the circuit width to the frequency axis allows for a more mitigated condition on computing time [147], because each resonator needs to store light during 1/N part of the entire computation. Nonetheless, we note that the synthetic-dimensional system requires equally-spaced multi-wavelength sources and detectors due to its operation principles, which could compose a challenge in the extension to large-scale integrated systems. For example, because the number of modulation frequencies, N _f, should be comparable to N for universal unitaries within O(N) spatial footprint, the system requires both high modulation frequencies and small FSRs. Although lithium-niobate phase modulators offer modulation bandwidths of about 100 GHz [78], the realization of N = 1,000 synthetic-dimensional systems necessitates FSRs of the order of 100 MHz. Therefore, the recent experimental demonstration of synthetic-dimensional systems has mainly implemented in fiber optic platforms, which allow for meter-scale optical resonators [101], [144]. Recent achievement of realizing meter-scale delay lines on a mm²-scale photonic chip can be applied to synthetic-dimensional computing in integrated photonic platforms [152].