Source-linked AI summary
Polyhedral RIS Architecture with Quantum-Assisted Facet Selection for Uniform Coverage
Noha Hassan, Halim Yanikomeroglu
TL;DR
Planar panels lose gain toward grazing angles, while polyhedral RIS operation adds discrete facet selection alongside mounting uncertainty and UE mobility. The paper analyzes directional gain guarantees for a chamfered polyhedral RIS and formulates each facet-selection subproblem of the precoded max-min design as an exact QUBO. Simulated annealing is optimal in 95% of 150 instances with K = 6, with maximal gap of 0.04 dB; the polyhedral RIS keeps the higher minimum rate in every setting.
Problem
Planar panels lose gain toward grazing angles, while polyhedral RIS operation adds discrete facet selection alongside mounting uncertainty and UE mobility.
Source: §I · p. 1, §III · p. 2Method
The paper analyzes directional gain guarantees for a chamfered polyhedral RIS and formulates each facet-selection subproblem of the precoded max-min design as an exact QUBO.
Source: §I · p. 1, §III · p. 2, B. Facet Selection Algorithm Description · p. 3Results
Simulated annealing is optimal in 95% of 150 instances with K = 6, with maximal gap of 0.04 dB; the polyhedral RIS keeps the higher minimum rate in every setting.
Source: §V · p. 4Takeaways & Limitations
Facet activation can be re-solved as UE locations vary across coherence intervals, while phase and spacing parameters remain fixed during each interval.
Source: §I · p. 1, §III · p. 2The QUBO is exact for each facet-selection subproblem, but is not asserted to be equivalent to the full max-min optimization problem.
Source: B. Facet Selection Algorithm Description · p. 4
Abstract
from arXiv · showhide
Reconfigurable intelligent surfaces (RISs) direct coverage toward moving user equipment (UE), but practical deployments involve discrete phase control, uncertain facet positions, and UE mobility, which require the problem to be re-solved continually. We consider these challenges for the chamfered hexagonal polyhedral RIS, which provides nearly uniform coverage. With both incidence and departure angles, we bound in closed form the gain loss of its best facet relative to the best possible reflecting facet, include the known b-bit quantization loss, and extend the bound to joint random mounting errors of all facets. For a precoded multi-user downlink, each facet-selection subproblem of the max-min rate design is shown to be an exact quadratic unconstrained binary optimization (QUBO) problem, suited to repeated time-constrained solving. The deployment of the base station in relation to the RIS allows the polyhedral surface to bring down the percentage of dead zones in vertical planar panels from 77% to 0.3% and increase the minimum rate by 34% relative to the STAR panel. The simulated annealing algorithm finds the optimum solution in 95% of the cases. The quantum approximation algorithm fails to do so.
I. INTRODUCTION
The introduction motivates a polyhedral RIS as a way to address the grazing-angle loss of planar panels, while noting that facet activation adds a discrete design variable. It presents gain guarantees and QUBO-based facet selection as the paper’s responses to that challenge.
I. INTRODUCTION
- Facet activation is a new discrete design variable in the polyhedral RIS, whose many reflecting normals address the grazing-angle loss of planar panels.The introduction contrasts this geometry with single-sided panels serving one half-space and double-sided or STAR panels whose gain rolls off toward grazing angles.Source: §I · p. 1
- QPRO uses a hybrid quantum-classical strategy, refining the phase and spacing parameters classically.The formulation incorporates the discrete facet-selection problem into the optimizer.Source: §I · p. 1
- The paper derives a worst-case directional gain guarantee accounting for incidence, departure, and phase quantization, then extends it to joint random facet-mounting perturbations.The introduction describes these as guarantees for the underlying polyhedral geometry.Source: §I · p. 1
- With precoding and coupling, each facet-selection subproblem of the max-min design is an exact QUBO, evaluated against the exhaustive optimum using Monte Carlo simulations.The paper also compares noiseless QAOA and a real quantum computer with classical algorithms for feasibility and optimality gaps.Source: §I · p. 1
II. RELATED WORK
Prior RIS optimization work addresses beamforming and transmission/reflection control, but not the binary facet-selection variable unique to multifaceted surfaces. The paper therefore formulates facet activation as a QUBO rather than rounding a continuous relaxation.
II. RELATED WORK
- Alternating-optimization (AO) updates RIS beamformers based on phase shifts using semidefinite relaxation, successive convex approximation, or fractional programming; a greedy AO heuristic for large arrays lacks an optimality guarantee.D-STAR methods use AO to optimize two-sided panels’ transmission and reflection coefficients and show improved spectral efficiency while solving sum-rate optimization.Source: §II · p. 1
- Prior quantum-annealing and hybrid quantum methods address wireless or network optimization,, but do not optimize discrete facet selection.The related-work discussion says those studies had not considered multifaceted surfaces.Source: §II · p. 2
- Discrete phase-beamforming algorithms and known quantization-loss bounds,, motivate formulating facet activation as a QUBO rather than rounding a continuous relaxation.The paper identifies the binary coupled constrained combinatorial problem as unique to multi-facet polyhedron surfaces.Source: §II · p. 2
III. SYSTEM MODEL AND POLYHEDRAL ARCHITECTURE
The system uses a 20-facet chamfered hexagonal prism, with its bevel chosen to close the elevation coverage gap. The facet normals provide bounded directional coverage under nominal mounting, while mounting errors and UE motion enter the model’s operating conditions.
III. SYSTEM MODEL AND POLYHEDRAL ARCHITECTURE
- The chamfered hexagonal prism has 20 facets across vertical, upper- and lower-beveled, and top and bottom surfaces, with a 60° bevel tilt.Each facet has its own normal, elements, spacings, and independent phase vector.Source: §III · p. 2
- The half-power span exactly closes the elevation gap between the vertical and cap facets, avoiding the horizon dead zone that shallower or steeper chamfers would leave.This geometry is shown in Fig. 1.Source: §III · p. 2
- With 60° azimuth spacing, every direction is within approximately 33.7° of its best-aligned nominal facet normal.This covering-radius guarantee assumes nominal, unperturbed mounting; actual facet normals can deviate because of random mounting orientation errors.Source: §III · p. 2
- UE locations vary slowly relative to the coherence time, when phase and spacing remain fixed but facet activation can be re-solved; the BS illuminates from above and direct UE links are obstructed.The actual facet normal is modeled as its nominal normal plus a mounting error.Source: §III · p. 2
A. Channel Model
The channel model maps BS transmissions to UE receptions through active facets, with element responses shaped by incidence and departure angles. It also accounts for inter-facet coupling and specifies the direct-link propagation model.
A. Channel Model
- The RIS maps BS symbols to each UE through an effective channel dependent on facet activation; inactive facets do not re-radiate.The model identifies the BS-to-facet and facet-to-UE links used in this mapping.Source: A. Channel Model · p. 2
- Each element’s response combines incidence-dependent gain roll-off and controlled phase, and the facet-to-UE pattern depends on both incidence and departure angles.The pattern is zero beyond 90°, and facets not illuminated by the BS do not contribute.Source: A. Channel Model · p. 2
- Inter-facet scattering enters through a fixed-point channel relation; the model retains the first-order coupling term, with convergence guaranteed for every activation and phase configuration.The coupling strength is κc, and the channel is also expressed in block form over all facets.Source: A. Channel Model · p. 2
- The BS-to-facet direct-link entries are Rician, and the propagation model carries free-space path loss and incidence-dependent spatial phase.The same model applies to the BS-antenna and facet-to-UE links.Source: A. Channel Model · p. 2
B. Signal Model
The signal model defines each UE’s SINR from its received signal and interference, then derives its data rate from SINR. Facet alignment affects the rate through the element-pattern terms.
B. Signal Model
- UE k’s SINR accounts for its desired signal, interference from other UEs’ precoded streams, and additive noise.The interference reaches UE k through its own channel.Source: B. Signal Model · p. 3
- The data rate is Rk = log2(1 + SINRk).Source: B. Signal Model · p. 3
- Worsening facet alignment lowers the element-pattern term ρf,k and therefore the UE’s rate Rk.Source: B. Signal Model · p. 3
IV. PROBLEM FORMULATION AND ALGORITHM DESCRIPTION
The section bounds the directional gain of the best available facet against the ideal reflecting facet, accounting for facet coverage and phase quantization. It extends the guarantee to random mounting errors across all facets simultaneously.
IV. PROBLEM FORMULATION AND ALGORITHM DESCRIPTION
- For angle Ωk between BS and UE directions, any facet has ρf,k ≤ cos²(Ωk/2), while the best facet has ρf,k ≥ cos²(Ωk/2 + δc) when Ωk/2 + δc < 90°.The upper bound is attained only at the bisecting normal; the lower bound uses a facet within δc of that direction.Source: §IV · p. 3
- The worst-case gain-loss bound includes the known b-bit phase-quantization loss, which vanishes as b → ∞.Source: §IV · p. 3
- With independent isotropic mounting perturbations, replacing δc by δc + δϵ gives the margin for all directions simultaneously with probability at least 1 − ϵ.δϵ is the (1 − ϵ)^(1/F) quantile of each facet’s angular mounting error.Source: §IV · p. 3
A. Joint Optimization Problem
The joint design maximizes the minimum UE rate over phase, spacing, precoding, and facet activation, subject to hardware, power, service, and coupling constraints. Its iterative algorithm alternates facet-selection QUBO solving with phase and spacing refinement.
A. Joint Optimization Problem
- The joint objective optimizes phase, spacing, precoding, and facet activation for max-min UE rate.Source: A. Joint Optimization Problem · p. 3
- The constraints impose discrete phase levels, element-spacing and facet-aperture budgets, transmit-power and minimum-rate limits, active-facet caps, and coupling limits.Together, phase-dependent rates, binary activation, and distance-dependent coupling make the problem non-convex.Source: A. Joint Optimization Problem · p. 3
- Each iteration uses RZF precoding and solves a facet-selection QUBO, then refines phase and spacing with facet activations fixed.The listed QUBO solvers are simulated annealing and QAOA.Source: A. Joint Optimization Problem · p. 3
B. Facet Selection Algorithm Description
Facet selection separates classical refinement of phase and spacing from discrete activation, which is re-solved as user equipment moves. The activation step is formulated as a QUBO whose minimizers satisfy the activation constraints and maximize the weighted objective over feasible activations.
B. Facet Selection Algorithm Description
- Discrete activation is re-solved once per coherence interval, while phase, spacing, and precoding are refined classically in a two-stage procedure.The classical updates use regularized zero-forcing precoding and are retained when the minimum rate increases.Source: B. Facet Selection Algorithm Description · p. 3
- The discrete step minimizes a QUBO built from the weighted quadratic objective, with slack bits encoding every allowed active-facet count from 0 to Fmax.The slack-bit weights are [1, 2, . . . , 2S−2, Fmax − 2S−1 + 1].Source: B. Facet Selection Algorithm Description · p. 3
- Every minimizer of the penalized QUBO satisfies the activation constraints and maximizes the weighted objective over feasible activations.A feasible activation can use slack with zero penalty; the proof bounds the objective range to separate infeasible solutions.Source: B. Facet Selection Algorithm Description · p. 3
- The algorithm terminates after finitely many accepted updates, and one simulated-annealing sweep takes O((F + S)^2) time.For Fmax = 8, the QUBO has 24 variables; each QUBO has F + S variables.Source: B. Facet Selection Algorithm Description · p. 4
V. EXPERIMENTAL RESULTS
Experiments compare the polyhedral RIS with planar designs across coverage, rate, solver, and scaling tests. The polyhedron sharply reduces dead zones in the grazing-incidence setup, while simulated annealing often matches the exhaustive activation optimum; the tested quantum hardware does not find it.
V. EXPERIMENTAL RESULTS
- 0.3% of UEs are in a dead zone with the polyhedron, versus 77.1% with STAR; SA-QUBO achieves 0.477 bps/Hz, 34% above the STAR panel.Spacing alone gives 0.467 bps/Hz; the STAR panel re-tilted for the BS gives 0.564 bps/Hz.Source: §V · p. 4
- At grazing incidence from the BS, vertical panels leave 87.7% (single-sided) and 77.1% (STAR) of UEs in a dead zone, versus 0.3% for the polyhedron; its gain does not exceed the bisector bound by more than 3.8 dB.Source: §V · p. 4
- The polyhedral RIS has a higher minimum rate than the double-sided planar RIS across every tested Rician-factor, UE-count, and transmission-power setting.The comparison is reported in Table III for polyhedral SA-QUBO with Fmax = 8.Source: §V · p. 4, Table III · p. 4
- Simulated annealing is optimal in 95% of 150 instances, compared with 83% for greedy activation; their maximum gaps are 0.04 dB and 2.91 dB, respectively.All 9,330 QUBOs solved by SA were feasible, and no repairs were needed.Source: §V · p. 4, Table IV · p. 4
- For prism orders n = 6 to 12, exhaustive search takes 0.4–81 s per instance, while SA takes at most 0.43 s.SA solves 26 of 33 instances optimally, compared with 22 of 33 for greedy activation.Source: §V · p. 4, Table V · p. 5
- No IBM-hardware sample reaches the exhaustive optimum; 13.5% of samples are feasible, versus 33% for noiseless QAOA and 5% for uniform bits.The best feasible hardware sample is 3.6 dB from the exhaustive optimum in minimum SINR.Source: §V · p. 5, Figure 4 · p. 5
VI. CONCLUSION
The chamfered hexagonal prism bounds gain loss relative to the best reflecting facet in closed form, including facet-mounting perturbations. It nearly eliminates vertical-panel dead zones, improves minimum rate over the STAR panel, and yields exact QUBO facet-selection subproblems for which simulated annealing often finds the optimum.
VI. CONCLUSION
- 34% more minimum rate than the STAR panel is achieved when the base station is placed over the RIS.The polyhedron also almost eliminates the dead zone of vertical planar panels.Source: §VI · p. 5
- Closed-form bounds characterize gain loss relative to the best reflecting facet, including probabilistic facet-mounting perturbations.The reported loss degrades gracefully with phase quantization and mounting imperfections.Source: §VI · p. 5
- Exact QUBO facet-selection subproblems are solved optimally by simulated annealing in 95% of cases, while QAOA does not yet achieve this.Source: §VI · p. 5
- The facet-selection solution is intended for iterative re-solving as UEs move through coherence intervals.Source: §VI · p. 5