Beyond-Diagonal RIS: Models & Architectures

• BD-RISs are advanced reconfigurable metasurfaces that use non-diagonal scattering matrices to enable controlled inter-element coupling.
• Their architecture ranges from single- to group- and fully-connected designs, unifying reflective, transmissive, and hybrid modes.
• Optimized design methodologies and hardware realizations in BD-RIS enhance system performance in communication, sensing, and power transfer.

Beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) are reconfigurable metasurfaces whose electromagnetic response is described by a non-diagonal scattering matrix rather than the diagonal phase-shift matrix of conventional RISs. By allowing controllable inter-element coupling through reconfigurable impedance or admittance networks, BD-RISs enlarge the set of attainable scattering matrices, unify reflective, transmissive, and hybrid modes, and subsume single-connected, group-connected, and fully-connected architectures within one multiport framework (Li et al., 2022, Ntougias et al., 7 May 2026).

1. Scattering-matrix formulation

A conventional diagonal RIS with $M$ elements is typically modeled by

$$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$

so each element only affects its own port and there is no inter-element coupling. BD-RIS generalizes this description by replacing the diagonal matrix with a general complex matrix, written as $\Phi\in\mathbb{C}^{M\times M}$, $\Theta\in\mathbb{C}^{N\times N}$, or $\Psi\in\mathbb{C}^{N_I\times N_I}$ depending on the paper, whose off-diagonal entries encode controlled coupling between ports (Ntougias et al., 7 May 2026, Li et al., 2022).

In multiport network form, one standard admittance-based relation is

$$\boldsymbol{\Theta}=(\mathbf{I}+Z_0\mathbf{Y})^{-1}(\mathbf{I}-Z_0\mathbf{Y}),$$

where $Z_0$ is the reference impedance and $\mathbf{Y}$ is the reciprocal $N$-port admittance matrix. Equivalent impedance-domain formulations are also used, such as

$$\bm{\Theta}=(\mathbf{Z}+Z_0\mathbf{I}_D)^{-1}(\mathbf{Z}-Z_0\mathbf{I}_D).$$

These relations make the circuit topology explicit: off-diagonal entries in $$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$0 or $$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$1 generate off-diagonal entries in the scattering matrix (Nerini et al., 7 Jan 2026, Sena et al., 2024).

For passive reciprocal BD-RIS implementations, symmetry and energy conservation are central. Several works impose

$$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$2

or the corresponding blockwise versions for group-connected structures. In hybrid reflecting/transmitting formulations, the reflective and transmissive matrices $$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$3 and $$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$4 satisfy

$$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$5

which is the matrix counterpart of per-cell power splitting in STAR-RIS-type designs (Li et al., 2022, Ming et al., 13 Apr 2025).

This formulation is not merely notational. It changes the attainable channel set: instead of element-wise phase control, BD-RIS performs a constrained multiport transformation of the incident field. A plausible implication is that BD-RIS should be viewed less as a bank of independent phase shifters and more as a programmable reciprocal microwave network.

2. Architectural taxonomy

The core architectural distinction in BD-RIS research is the connectivity pattern of the underlying impedance network. This pattern determines which entries of the scattering matrix can be nonzero and therefore which spatial transformations are physically realizable (Li et al., 2022, Li et al., 2022, Zhou et al., 2024).

Architecture	Matrix structure	Defining property
Single-connected	Diagonal	No inter-element interconnections
Group-connected	Block diagonal	Full coupling within each group only
Fully-connected	Full symmetric unitary	All elements interconnected
Dynamic group-connected	Permuted block-diagonal	Grouping adapts to CSI
Q-stem connected	Structured $$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$6	Non-stem ports connect only to stem ports

Single-connected RIS is the diagonal limit. Group-connected RIS partitions the surface into groups and uses block-diagonal scattering matrices, while fully-connected RIS allows dense coupling over the whole surface. In the hybrid transmitting/reflecting setting, these same connectivity choices apply to the reflective and transmissive matrices, yielding the cell-wise single-connected, group-connected, and fully-connected taxonomy (Li et al., 2022).

Dynamic grouping extends fixed group-connected BD-RIS by letting the partition itself depend on CSI. The resulting scattering matrix is permuted block-diagonal, and the stated motivation is precisely that fixed architectures “regardless of channel state information (CSI)” limit achievable performance. Simulation results in that work show that dynamically group-connected BD-RIS outperforms fixed group-connected architectures (Li et al., 2022).

Q-stem connected RIS provides an explicit performance-complexity interpolation. It uses $$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$7 stem ports connected to all other ports, while non-stem ports connect only to stem ports. The special cases are $$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$8 for single-connected RIS, $$\Phi_{\text{diag}}=\operatorname{diag}\big(e^{j\theta_1},\ldots,e^{j\theta_M}\big),$$9 for tree-connected RIS, and $\Phi\in\mathbb{C}^{M\times M}$0 for fully-connected RIS. Its circuit complexity is

$\Phi\in\mathbb{C}^{M\times M}$1

and for $\Phi\in\mathbb{C}^{M\times M}$2, $\Phi\in\mathbb{C}^{M\times M}$3, the architecture uses 484 admittances, compared with 2080 for the fully-connected case. In the reported simulations, $\Phi\in\mathbb{C}^{M\times M}$4 can match fully-connected performance, and $\Phi\in\mathbb{C}^{M\times M}$5 can attain the sum channel gain achieved by fully connected RIS, where $\Phi\in\mathbb{C}^{M\times M}$6 is the multiplexing gain (Zhou et al., 2024).

A common misconception is that BD-RIS is synonymous with fully-connected RIS. The literature instead treats fully-connected RIS as one endpoint of a broader design space that also includes diagonal, group-connected, dynamic group-connected, tree/forest-connected, stem-connected, and transmitting/reflecting realizations.

3. Design and optimization methodologies

The non-diagonal, symmetric, unitary, and sometimes hybrid constraints make BD-RIS design intrinsically non-convex. The dominant algorithmic pattern is therefore block-wise optimization: optimize the active precoder, the scattering matrix, and any auxiliary variables in alternating or nested steps (Li et al., 2022, Fidanovski et al., 24 Sep 2025, Ntougias et al., 7 May 2026).

A foundational formulation for RIS-aided MU-MISO systems maximizes the sum rate by jointly designing the transmit precoder and the BD-RIS matrix under architecture-specific constraints. That work applies fractional programming, specifically the Lagrangian dual transform and quadratic transform, and then uses block coordinate descent over the BS precoder and the BD-RIS matrices. For group-connected and fully-connected architectures, the passive subproblem is solved on a complex Stiefel manifold; for single-connected hybrid RIS, the elementwise amplitudes and phases are updated by scalar convex search and phase alignment (Li et al., 2022).

For reciprocal group-connected hybrid active/passive BD-RIS, the design separates into a spatial subproblem and a power subproblem. The spatial part is solved per group by Takagi factorization. With

$\Phi\in\mathbb{C}^{M\times M}$7

Takagi factorization gives

$\Phi\in\mathbb{C}^{M\times M}$8

This choice makes the per-group contribution achieve $\Phi\in\mathbb{C}^{M\times M}$9, and the amplification factors are then obtained in closed form by Cauchy–Schwarz under reflect-power constraints. The paper’s key observation is that the optimal $\Theta\in\mathbb{C}^{N\times N}$0 is independent of the amplification factors, which makes alternating optimization converge in one iteration (Ntougias et al., 7 May 2026).

Reciprocal BD-RIS sum-rate maximization has also been attacked directly by manifold optimization. In that formulation, symmetry is encouraged through a penalty term,

$\Theta\in\mathbb{C}^{N\times N}$1

while unitary constraints are enforced on the Stiefel manifold. The final reciprocal projection symmetrizes the iterate,

$\Theta\in\mathbb{C}^{N\times N}$2

and then applies SVD-based projection to obtain a symmetric unitary scattering matrix. Simulation results in that work show that the reciprocal design outperforms the cited state-of-the-art baselines in sum-rate maximization (Fidanovski et al., 24 Sep 2025).

Other design strategies exploit structure more explicitly. In CF-mMIMO SWIPT, a heuristic scattering design projects an “energy-focused” matrix onto the symmetric-unitary manifold; in Q-stem RIS, a low-complexity least-squares algorithm and an LS-based quasi-Newton algorithm are used to fit the structured susceptance matrix to an SVD-derived channel-matching target (Hua et al., 2024, Zhou et al., 2024). This suggests that BD-RIS optimization is increasingly driven by architecture-aware parameterizations rather than architecture-agnostic relaxation.

4. Hybrid, active, and hardware-realized BD-RIS

One major branch of BD-RIS research moves beyond purely reflective passive surfaces by combining transmission, reflection, and even active amplification. The resulting designs replace the simple “phase-only reflector” picture with hybrid electromagnetic networks capable of power splitting, amplifier sharing, and independent spatial control on multiple sides of the aperture (Ming et al., 13 Apr 2025, Ntougias et al., 7 May 2026).

A concrete hardware realization is the hybrid transmitting and reflecting BD-RIS built from two phase-reconfigurable antenna arrays interconnected by tunable two-port power splitters. At the cell level, the two-port scattering matrix is

$\Theta\in\mathbb{C}^{N\times N}$3

The reported prototype uses a $\Theta\in\mathbb{C}^{N\times N}$4 cell array, 2-bit phase-reconfigurable antennas, and a two-port power splitter whose power ratio of $\Theta\in\mathbb{C}^{N\times N}$5 over $\Theta\in\mathbb{C}^{N\times N}$6 is tunable from $\Theta\in\mathbb{C}^{N\times N}$7 dB to $\Theta\in\mathbb{C}^{N\times N}$8 dB. Each antenna has 200 MHz bandwidth at 2.4 GHz. Experiments show that the fabricated BD-RIS can realize beam steering in reflection and transmission mode, and that in hybrid mode it enables independent beam steering of the reflected and transmitted waves (Ming et al., 13 Apr 2025).

A different extension combines beyond-diagonal coupling with active amplification. In the family of hybrid BD-RIS architectures, the surface is partitioned into two reflecting subsurfaces, and each subsurface can be passive, fully-connected-active, or sub-connected-active. The active designs use reflect-type power amplifiers and closed-form per-group or per-cluster gains satisfying a reflect-power budget. In the SISO blocked-direct-path scenario considered, the proposed hybrid BD-RIS architectures attain the same or higher receive SNR than their diagonal counterparts while using significantly fewer reflect-type amplifiers (Ntougias et al., 7 May 2026).

The same study isolates three notable operating points. For active/passive hybrid BD-RIS, performance can be up to $\Theta\in\mathbb{C}^{N\times N}$9 dB below diagonal active/passive RIS in SISO when the active group size is moderate, but with fully-connected groups it gains $\Psi\in\mathbb{C}^{N_I\times N_I}$0 dB, consistent with the asymptotic BD-RIS power gain factor

$\Psi\in\mathbb{C}^{N_I\times N_I}$1

For fully-connected-active/sub-connected-active BD-RIS, the SNR tracks fully-active diagonal RIS within $\Psi\in\mathbb{C}^{N_I\times N_I}$2 dB across $\Psi\in\mathbb{C}^{N_I\times N_I}$3 while using $\Psi\in\mathbb{C}^{N_I\times N_I}$4 amplifiers. For sub-connected-active/sub-connected-active BD-RIS, the architecture uses only 2 amplifiers and, at $\Psi\in\mathbb{C}^{N_I\times N_I}$5, attains about 47 dB, which is $\Psi\in\mathbb{C}^{N_I\times N_I}$6 dB above the fully-active diagonal RIS in the reported SISO setting (Ntougias et al., 7 May 2026).

These results illustrate an important point: “beyond diagonal” is not limited to passive reciprocity. It also supports hybrid transmitting/reflecting and hybrid active/passive realizations, provided the physical constraints are built into the scattering-matrix model.

5. Performance in communication, sensing, and power-transfer systems

BD-RIS has been studied in a wide range of end-to-end systems, and the performance benefits are strongly architecture- and application-dependent. The literature consistently reports gains over diagonal RIS, but it also shows that the relevant figure of merit varies: SNR in SISO links, sum channel gain in multiuser broadcast channels, spectral efficiency and harvested energy in SWIPT, or CRB and SCNR in sensing-oriented systems (Hua et al., 2024, Sena et al., 2024, Wang et al., 2023, Chen et al., 30 Sep 2025).

In CF-mMIMO SWIPT, BD-RIS is used to enhance wireless power transfer to energy users while preserving acceptable information-user spectral efficiency. The analytical framework includes spatial correlation among BD-RIS elements and a non-linear energy harvesting circuit. With the heuristic symmetric-unitary scattering design, the reported gain is up to 12% average harvested energy compared to random BD-RIS, and the no-BD-RIS baseline fails to meet the energy users’ requirements in the stated setting (Hua et al., 2024).

In multi-band multi-cell MIMO, frequency dependence becomes a first-order issue. A practical frequency-dependent model built from resonant self- and inter-element impedances shows that BD-RIS behavior can vary substantially over 4–12 GHz. Fully-connected BD-RIS exhibits the highest reflection performance over most of the band, while group-connected BD-RIS provides a hardware-performance compromise and can be assigned to different carrier frequencies in multi-BS operation. The same study also shows the potential for harmful interference in the absence of synchronization between RISs and adjacent BSs (Sena et al., 2024).

Sensing-oriented systems leverage the extra matrix degrees of freedom even more directly. In hybrid reflecting/transmitting DFRC, BD-RIS enables full-space coverage and flexible architectures, and the joint optimization of waveform, BD-RIS matrices, and sensing receive filters maximizes the minimum SCNR subject to communication QoS. The reported simulations show that group-connected and fully-connected BD-RIS can achieve higher radar SCNR than STAR-RIS under the same communication requirement, while also improving communication and sensing compared with conventional RIS-aided DFRC (Wang et al., 2023).

A transmitter-side ISAC formulation places BD-RIS adjacent to the transmitter and jointly optimizes active beamforming and the BD-RIS scattering matrix to minimize the trace of the CRB and maximize the communication sum rate. In that setting, numerical results show the appealing capability of transmitter-side BD-RIS-aided ISAC over conventional diagonal RIS-aided ISAC in enhancing both sensing and communication performance, and the proposed AO-PSCA algorithm also reduces the computational complexity relative to the classic iterative baseline (Chen et al., 30 Sep 2025).

Taken together, these studies suggest that the main system-level value of BD-RIS is not a single universal gain figure. Rather, it is the ability to tailor the passive electromagnetic transformation to the relevant task: coherent amplification in SISO, richer effective channel shaping in multiuser MIMO, power focusing in SWIPT, and joint beampattern/sensing control in ISAC and DFRC.

6. Practical constraints, misconceptions, and open problems

The gap between abstract scattering-matrix design and realizable hardware is especially visible in BD-RIS. Loss, fabrication constraints, reciprocity, and control overhead all modify which architectures remain attractive once the idealized lossless model is relaxed (Peng et al., 28 Apr 2025, Nerini et al., 7 Jan 2026, Fidanovski et al., 24 Sep 2025, Hua et al., 2024, Sena et al., 2024).

Loss is a decisive example. In the lossy BD-RIS model based on practical tunable admittances, each admittance satisfies

$\Psi\in\mathbb{C}^{N_I\times N_I}$7

so the feasible admittances lie on a circle, and with bounded capacitance they occupy only an arc of that circle. Under this model, all BD-RIS architectures still outperform D-RIS in the presence of losses, but the lossless ranking is no longer universal: group-connected BD-RIS can outperform fully- and tree-connected BD-RISs in SISO systems with relatively high losses, whereas the opposite always holds true in the lossless case (Peng et al., 28 Apr 2025).

Fabrication imposes another nontrivial filter. Graph-theoretic analysis of planar-connected RISs characterizes which BD-RIS interconnection graphs can be realized on double-layer PCBs. Forest-connected RIS is always planar-connected; group-connected RIS is planar-connected if and only if group size $\Psi\in\mathbb{C}^{N_I\times N_I}$8; $\Psi\in\mathbb{C}^{N_I\times N_I}$9-stem-connected RIS is planar-connected if and only if $$\boldsymbol{\Theta}=(\mathbf{I}+Z_0\mathbf{Y})^{-1}(\mathbf{I}-Z_0\mathbf{Y}),$$0; $$\boldsymbol{\Theta}=(\mathbf{I}+Z_0\mathbf{Y})^{-1}(\mathbf{I}-Z_0\mathbf{Y}),$$1-band-connected RIS is planar-connected if and only if $$\boldsymbol{\Theta}=(\mathbf{I}+Z_0\mathbf{Y})^{-1}(\mathbf{I}-Z_0\mathbf{Y}),$$2; and fully-connected RIS is not planar-connected. The same work shows that planar graphs have at most $$\boldsymbol{\Theta}=(\mathbf{I}+Z_0\mathbf{Y})^{-1}(\mathbf{I}-Z_0\mathbf{Y}),$$3 interconnections, and maximal-planar-connected RIS therefore has $$\boldsymbol{\Theta}=(\mathbf{I}+Z_0\mathbf{Y})^{-1}(\mathbf{I}-Z_0\mathbf{Y}),$$4 inter-element admittances and $$\boldsymbol{\Theta}=(\mathbf{I}+Z_0\mathbf{Y})^{-1}(\mathbf{I}-Z_0\mathbf{Y}),$$5 total tunable admittances (Nerini et al., 7 Jan 2026).

A separate misconception is that reciprocity necessarily sacrifices performance. Reciprocal BD-RIS constrains the scattering matrix to be symmetric and unitary, but a direct sum-rate formulation on the Stiefel manifold shows that reciprocal BD-RIS can outperform the cited state-of-the-art design baselines in multi-user MISO settings. In that sense, symmetry is not merely a hardware burden; it can also be an algorithmically manageable physical prior (Fidanovski et al., 24 Sep 2025).

The open problems identified across the literature are therefore practical as much as mathematical. Hardware implementation of large symmetric-unitary scattering matrices remains challenging; joint optimization of scattering matrices, grouping, AP or BS mode selection, power control, and transmit precoding remains high-dimensional; channel estimation and robustness under imperfect CSI remain underdeveloped; and synchronization becomes critical in multi-band or multi-cell deployments because uncoordinated BD-RIS operation can create harmful interference (Hua et al., 2024, Sena et al., 2024). This suggests that the next stage of BD-RIS research will depend not only on better optimization but also on architecture-aware control, frequency-selective modeling, and fabrication-constrained circuit synthesis.