BD-RIS: Beyond-Diagonal Intelligent Surfaces

• BD-RISs are reconfigurable intelligent surfaces that use non-diagonal multiport networks to enable inter-element coupling, unlike conventional independent phase shifts.
• They facilitate advanced amplitude-and-phase manipulation through tunable impedance/admittance components, supporting reflective, transmissive, and hybrid operating modes.
• They bridge microwave network theory and practical hardware constraints, improving spectral efficiency, wireless optimization, and enabling diverse architecture designs.

Beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) are reconfigurable intelligent surfaces in which the electromagnetic response is governed by a non-diagonal multiport network rather than by a diagonal per-element phase-shift model. In conventional diagonal RISs, each element is independently loaded and no controllable inter-element coupling is present. In BD-RISs, distinct elements can be interconnected through tunable impedance or admittance components, so incident energy can be redistributed across ports and re-radiated with richer amplitude-and-phase control. This places BD-RISs at the intersection of microwave network theory, circuit-constrained wave control, and wireless optimization, and it has motivated a family of architectures spanning reflective, transmissive, hybrid, multi-sector, group-connected, tree/forest-connected, fully-connected, planar-connected, lossy, and active designs (Li et al., 2022, Khan et al., 29 Dec 2025).

1. Multiport network formulation

A BD-RIS is naturally modeled as an $N$-port reciprocal network with admittance matrix $\mathbf{Y}\in\mathbb{C}^{N\times N}$ and scattering matrix $\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$. If $Y_n$ denotes the tunable admittance from element $n$ to ground and $Y_{n,m}=Y_{m,n}$ the tunable admittance interconnecting elements $n$ and $m$, then the network admittance entries are

$[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$

With reference impedance $Z_0$, typically $\mathbf{Y}\in\mathbb{C}^{N\times N}$0, the scattering-admittance relation is

$\mathbf{Y}\in\mathbb{C}^{N\times N}$1

Under the lossless reciprocal model used in several works, $\mathbf{Y}\in\mathbb{C}^{N\times N}$2, where $\mathbf{Y}\in\mathbb{C}^{N\times N}$3 is symmetric, and therefore

$\mathbf{Y}\in\mathbb{C}^{N\times N}$4

This makes explicit that BD-RIS control acts through a structured susceptance matrix rather than through elementwise phase shifts alone (Nerini et al., 7 Jan 2026).

The corresponding physical constraints are standard multiport constraints. Reciprocity implies $\mathbf{Y}\in\mathbb{C}^{N\times N}$5 or, equivalently, $\mathbf{Y}\in\mathbb{C}^{N\times N}$6. Passivity imposes $\mathbf{Y}\in\mathbb{C}^{N\times N}$7, and in the ideal lossless case $\mathbf{Y}\in\mathbb{C}^{N\times N}$8 is unitary. Several formulations are equally used in the literature: impedance-based forms $\mathbf{Y}\in\mathbb{C}^{N\times N}$9, admittance-based forms, and hybrid transmitting/reflecting scattering models. The common point is that BD-RIS departs from the diagonal operator $\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$0 by admitting structured off-diagonal couplings (Nerini et al., 2024, Khan et al., 29 Dec 2025).

For hybrid transmitting and reflecting implementations, the $\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$1-port formulation is especially important. The effective matrices toward the reflective and transmissive sides, $\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$2 and $\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$3, satisfy

$\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$4

in the lossless case. This expresses the power-conservation coupling between the two half-spaces and generalizes the per-element amplitude-splitting rule of STAR-RIS-like models (Li et al., 2022).

2. Architectural classes and operating modes

BD-RISs are classified both by connectivity and by electromagnetic operating mode. Connectivity determines which ports are coupled by tunable circuit components; operating mode determines whether the surface reflects, transmits, splits power between both, or partitions space into more than two sectors.

Class	Structural form	Circuit-complexity trend
Single-connected	Diagonal $\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$5 or one tunable load per element	$\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$6
Group-connected	Permuted block-diagonal $\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$7	$\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$8 for group size $\boldsymbol{\Theta}\in\mathbb{C}^{N\times N}$9
Fully-connected	Dense $Y_n$0	$Y_n$1
Tree-/forest-connected	Sparse graph with no cycles	Reduced hardware overhead
Q-stem-connected	Non-stem ports connect only to $Y_n$2 stems	Linear in $Y_n$3 for fixed $Y_n$4

Group-connected BD-RIS partitions the $Y_n$5 elements into disjoint groups and fully interconnects elements only within each group. In permutation form,

$Y_n$6

with each block symmetric and unitary in the lossless reciprocal case. Single-connected and fully-connected architectures are recovered as the special cases $Y_n$7 and $Y_n$8, respectively. This block structure is central to both fixed grouping and CSI-adaptive dynamic grouping (Nerini et al., 2024, Li et al., 2022).

Tree-connected and forest-connected BD-RISs enforce sparse interconnection graphs. They reduce circuit and calibration complexity while retaining non-diagonal behavior. A related sparse family is the Q-stem-connected RIS, in which a set of $Y_n$9 stem ports is allowed to connect to all ports, while non-stem ports do not interconnect among themselves. This topology interpolates between single-connected ($n$0), tree-connected ($n$1), and fully-connected ($n$2) designs, and its number of independent tunable admittance components is

$n$3

This suggests an explicit architecture-level continuum between linear- and quadratic-complexity designs (Zhou et al., 2024).

Operating modes add a second axis of classification. Reflective BD-RIS re-radiates into the same half-space; transmissive BD-RIS re-radiates into the opposite half-space; hybrid BD-RIS performs both simultaneously under the energy-splitting constraint above. Multi-sector BD-RIS generalizes hybrid operation further by dividing space into $n$4 sectors with directional antennas arranged as a polygon prism and a multi-port group-connected impedance network. In that setting, the sector blocks $n$5 satisfy

$n$6

which enforces passivity across sectors while enabling highly directional full-space coverage (Li et al., 2022).

3. Graph-theoretic synthesis and planar realizability

A BD-RIS connectivity pattern can be represented by a graph $n$7, where vertices correspond to RIS elements and edges to inter-element tunable admittances. Off-diagonal nonzeros of $n$8 or $n$9 coincide with graph edges. This graph-theoretic view is not merely descriptive: it determines whether a given architecture can be routed on a double-layer printed circuit board with one signal layer and one ground plane (Nerini et al., 7 Jan 2026).

A planar-connected RIS is defined as a BD-RIS whose interconnection graph is planar for any $Y_{n,m}=Y_{m,n}$0. Ground connections do not belong to the graph because they terminate on the dedicated ground layer through vias and do not induce interconnection crossings. The decisive constraint is that inter-element traces must be routed on a single signal layer without crossings. Kuratowski’s theorem gives the exact forbidden subgraph characterization, and for a simple planar graph with $Y_{n,m}=Y_{m,n}$1,

$Y_{n,m}=Y_{m,n}$2

Hence the architecture-side degrees of freedom

$Y_{n,m}=Y_{m,n}$3

satisfy

$Y_{n,m}=Y_{m,n}$4

on two layers. By contrast, a fully-connected reciprocal BD-RIS has

$Y_{n,m}=Y_{m,n}$5

so planarity forces a transition from quadratic to linear scaling (Nerini et al., 7 Jan 2026).

The maximal-planar-connected RIS saturates the planar bound $Y_{n,m}=Y_{m,n}$6. Three explicit classes were identified as maximal-planar-connected examples: a 3-band-connected RIS with $Y_{n,m}=Y_{m,n}$7, a central-plus-2-band design with one central vertex connected to all others and $Y_{n,m}=Y_{m,n}$8, and a two-centrals-plus-1-band design with two universal vertices plus nearest-neighbor links. These are the most flexible architectures implementable on two PCB layers under the paper’s definitions (Nerini et al., 7 Jan 2026).

Graph-theoretic criteria also delimit common sparse architectures. Forest- and tree-connected RISs are always planar. Group-connected RIS is planar if and only if the group size $Y_{n,m}=Y_{m,n}$9. Q-stem-connected RIS is planar if and only if $n$0. Q-band-connected RIS is planar if and only if $n$1. This clarifies a frequent source of confusion: a sparse architecture may still be non-planar once routed physically, and practical implementability depends on the graph, not merely on the nominal count of nonzero couplings (Nerini et al., 7 Jan 2026).

4. Optimization frameworks and channel parametrization

The communication-theoretic role of a BD-RIS is to parameterize an effective cascaded channel. In a standard downlink MISO setting with transmitter–RIS channel $n$2, RIS–user channel $n$3, and precoder $n$4, the effective user channel is

$n$5

and the signal-to-interference-plus-noise ratio is optimized jointly over $n$6 and the structured $n$7. The canonical sum-rate problem maximizes $n$8 subject to transmitter power, reciprocity, passivity or losslessness, and architecture-imposed sparsity in $n$9 or block structure in $m$0 (Nerini et al., 7 Jan 2026, Li et al., 2022).

Several algorithmic lines have emerged. For grouped architectures, static grouping is formulated as a bi-level design: an offline upper-level optimization selects a permutation $m$1 from long-term channel statistics, and an online lower-level problem configures the intra-group scattering matrices per realization. In the single-user case the lower-level objective reduces to maximizing $m$2; in the multi-user case it becomes $m$3, followed by zero-forcing beamforming. The grouping search space is combinatorial, so a local-search heuristic based on pairwise swaps between groups is used; it is monotone and terminates because the number of groupings is finite (Nerini et al., 2024).

For reciprocal BD-RIS, symmetry-preserving manifold formulations are prominent. One representative method optimizes a symmetric scattering matrix through a Stiefel-manifold formulation with a symmetry penalty term $m$4, Riemannian conjugate-gradient ascent, Armijo line search, QR-based retraction, and a final projection that symmetrizes the iterate and then projects it to a unitary matrix through singular value decomposition. This directly enforces physically feasible reciprocal, lossless designs rather than optimizing an unconstrained matrix and correcting it afterward (Fidanovski et al., 24 Sep 2025).

Lossy BD-RIS requires a different treatment because each tunable admittance lies on a circle in the complex plane: $m$5 The feasible set is therefore an arc rather than an unrestricted imaginary axis, and amplitude-phase coupling becomes intrinsic. For SISO power maximization, a custom majorization-minimization plus alternating direction method of multipliers scheme is developed; for MU-MISO sum-rate maximization, the same lossy constraints are embedded in a fractional-programming and block-coordinate-descent framework (Peng et al., 28 Apr 2025).

A conceptually distinct line reinterprets BD-RIS through a physics-compliant diagonal representation. Any control circuit is decomposed into a static linear circuit and a set of individually tunable loads, so the overall channel is a cascade of radio environment, static linear circuit, and diagonal tunable loads. The resulting representation retains the physical BD-RIS effect while making the tunable stage diagonal. A plausible implication is that a large class of algorithms developed for physics-compliant diagonal RIS can be reused directly for BD-RIS system-level estimation and optimization, provided the static cascade is identified correctly (Hougne, 2024).

5. Applications and empirical performance

In multi-user communication, BD-RIS consistently enlarges the design space relative to diagonal RIS. For static grouping in correlated channels, optimized grouping produced up to $m$6 single-user received-power gain and up to $m$7 multi-user sum-rate gain over adjacent grouping when $m$8, $m$9, and $[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$0. The same study emphasized that optimized groups tend to spread elements spatially across the surface, thereby aggregating less-correlated elements in each group (Nerini et al., 2024).

Hybrid transmitting-and-reflecting BD-RIS extends these gains to full-space operation. In a dynamically group-connected architecture, CSI-adaptive grouping outperformed fixed group-connected structures across transmit powers. For $[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$1, the sum-rate improvement was about $[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$2 at $[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$3 and $[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$4 at $[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$5; for $[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$6, the improvements were about $[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$7 at $[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$8 and $[\mathbf{Y}]_{n,m}= \begin{cases} -Y_{n,m}, & n\neq m,\[4pt] Y_n+\sum_{k\neq n}Y_{n,k}, & n=m. \end{cases}$9 at $Z_0$0. With $Z_0$1 and $Z_0$2, the dynamically group-connected design nearly matched fully-connected performance with substantially fewer active interconnections (Li et al., 2022).

BD-RIS has also been integrated into dual-function radar-communication and broader ISAC formulations. In BD-RIS-aided DFRC, hybrid reflecting/transmitting group-connected matrices $Z_0$3 and $Z_0$4 are jointly optimized with the transmit waveform and sensing filters to maximize the minimum signal-to-clutter-plus-noise ratio subject to communication quality-of-service constraints. The reported trends show that CW-GC and CW-FC outperform the STAR-RIS-like CW-SC baseline in both sensing and communication, with the advantage increasing under stricter QoS constraints (Wang et al., 2023). A transmitter-side ISAC architecture places BD-RIS at the transmitter to reduce the need for large-scale fully digital RF chains while jointly minimizing the trace of the Cramér–Rao bound and maximizing sum rate; it is presented as a low-complexity projection-based alternative to classical iterative solvers (Chen et al., 30 Sep 2025).

For wireless power transfer and SWIPT, BD-RIS has been embedded into cell-free massive MIMO systems with simultaneous information and energy users. In that setting, a heuristic symmetric-unitary scattering-matrix design yielded up to $Z_0$5 higher average sum harvested energy than random BD-RIS design, while the analytical framework quantified how RIS spatial correlation enters both spectral efficiency and fourth-order harvested-energy moments (Hua et al., 2024).

Physical-layer security is another area where non-diagonal coupling materially changes the feasible set. In a MIMO wiretap channel assisted by BD-RIS, a penalty product Riemannian conjugate gradient descent method optimized the transmit beamforming and the group-connected or fully-connected reciprocal BD-RIS jointly. The reported gains include secrecy rate $Z_0$6 bps/Hz for fully-connected BD-RIS at $Z_0$7 or $Z_0$8 dBW, corresponding to $Z_0$9 over diagonal RIS and $\mathbf{Y}\in\mathbb{C}^{N\times N}$00 over the alternating-optimization baseline. The robustness study under imperfect CSI also showed slower secrecy-rate degradation than diagonal RIS and random fully-connected RIS (Xiong et al., 19 Nov 2025).

Wideband and multi-band behavior introduces a different set of phenomena. A practical frequency-dependent model for fully- and group-connected BD-RIS reported that a fully-connected RIS with $\mathbf{Y}\in\mathbb{C}^{N\times N}$01 achieved maximal received power of approximately $\mathbf{Y}\in\mathbb{C}^{N\times N}$02 mW at approximately $\mathbf{Y}\in\mathbb{C}^{N\times N}$03 GHz and sustained at least $\mathbf{Y}\in\mathbb{C}^{N\times N}$04 of that value from approximately $\mathbf{Y}\in\mathbb{C}^{N\times N}$05 to $\mathbf{Y}\in\mathbb{C}^{N\times N}$06 GHz. The same study also showed that lack of synchronization between RIS and adjacent base stations can create harmful interference, with a spectral-efficiency gap of approximately $\mathbf{Y}\in\mathbb{C}^{N\times N}$07 bits/s/Hz in one tested setting (Sena et al., 2024).

Hardware demonstrations have begun to close the gap between scattering-matrix abstractions and realizable designs. A $\mathbf{Y}\in\mathbb{C}^{N\times N}$08 hybrid transmitting-and-reflecting BD-RIS prototype at $\mathbf{Y}\in\mathbb{C}^{N\times N}$09 GHz used two phase-reconfigurable antenna arrays interconnected by tunable two-port power splitters. Each splitter controlled the power ratio of $\mathbf{Y}\in\mathbb{C}^{N\times N}$10 over $\mathbf{Y}\in\mathbb{C}^{N\times N}$11 from $\mathbf{Y}\in\mathbb{C}^{N\times N}$12 dB to $\mathbf{Y}\in\mathbb{C}^{N\times N}$13 dB, and experiments verified independent beam steering of reflected and transmitted waves in hybrid mode (Ming et al., 13 Apr 2025). Active BD-RIS pushes further by embedding amplification into the beyond-diagonal network. In one reported comparison, to achieve the same spectral efficiency, the number of elements required by active BD-RIS was less than half of that required by active diagonal RIS (Shen et al., 14 Mar 2026).

6. Practical constraints, misconceptions, and open directions

A first practical constraint is loss. Real tunable components exhibit finite $\mathbf{Y}\in\mathbb{C}^{N\times N}$14, parasitic resistance and inductance, limited tuning ranges, and architecture-dependent insertion loss. This matters because the lossless ordering of architectures does not necessarily survive under realistic hardware. In particular, one study showed that in lossy SISO systems with relatively high losses, group-connected BD-RIS can outperform fully- and tree-connected BD-RISs, whereas the opposite always holds in the lossless case. This directly contradicts the common simplification that more couplings are always better (Peng et al., 28 Apr 2025).

A second misconception is that the only meaningful representation of BD-RIS at system level must be non-diagonal. The physics-compliant diagonal representation shows that the tunable part can always be separated into individually tunable loads after absorbing the static coupling circuitry into a fixed cascade. This does not remove the physical beyond-diagonal behavior; it changes the parametrization used for estimation and optimization. A plausible implication is that disagreements between “diagonal” and “beyond-diagonal” system models can sometimes be about representation rather than about physics (Hougne, 2024).

A third conceptual pitfall is to conflate architecture-side and link-side degrees of freedom. The planar-connected RIS work explicitly distinguishes $\mathbf{Y}\in\mathbb{C}^{N\times N}$15, the number of independent tunable circuit components, from the system-level degree of freedom $\mathbf{Y}\in\mathbb{C}^{N\times N}$16 of the wireless link. Under double-layer planarity,

$\mathbf{Y}\in\mathbb{C}^{N\times N}$17

yet certain maximal-planar-connected examples are stated to be optimal for systems with $\mathbf{Y}\in\mathbb{C}^{N\times N}$18. This separation is important because the rank or stream dimension of the communication problem does not equal the number of tunable impedances (Nerini et al., 7 Jan 2026).

The dominant implementation burdens remain fabrication, routing, calibration, CSI acquisition, and control overhead. Fully-connected architectures maximize flexibility but require $\mathbf{Y}\in\mathbb{C}^{N\times N}$19 tunables, substantial control signaling, and often multilayer PCB routing. Planar-connected, group-connected, tree-connected, and Q-stem-connected designs exist precisely to trade flexibility for realizability. The broader literature identifies further challenges in wideband modeling, frequency dispersion, mmWave/THz CSI acquisition, synchronization in multi-cell settings, and physically compliant integration of losses, quantization, and mutual coupling (Khan et al., 29 Dec 2025).

Current research directions therefore span several layers of abstraction. Circuit and architecture work is exploring selective multilayer bridges, dual-polarized elements, nonuniform interconnection density, and active amplification (Nerini et al., 7 Jan 2026, Shen et al., 14 Mar 2026). System-level work is extending BD-RIS to ISAC, SWIPT, multi-sector full-space coverage, and transmitter-side deployments (Li et al., 2022, Hua et al., 2024, Chen et al., 30 Sep 2025). Algorithmic work is moving toward robust manifold optimization, topology-aware channel estimation, and data-driven or hybrid quantum-classical control loops for fast beam prediction (Fidanovski et al., 24 Sep 2025, Khan et al., 29 Dec 2025). Taken together, these developments indicate that BD-RIS is best understood not as a single architecture, but as a circuit-constrained design space for programmable, non-diagonal electromagnetic operators.