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Quantum Approximate Optimization Algorithm for MIMO with Quantized b-bit Beamforming

Nikos A Mitsiou, Ioannis Krikidis, George K Karagiannidis

TL;DR

This work tackles the NP-hard problem of optimizing $b$-bit quantized beamforming in MIMO systems by blending the Quantum Approximate Optimization Algorithm (QAOA) with alternating optimization (AO) and a warm-start variant. It formalizes a cost Hamiltonian that maps beamforming phase shifts to quantum rotation gates, derives the corresponding quantum circuits, and presents both a $b=2$ treatment and a general $b$-bit framework with a qubit-efficient encoding. Numerical results show the proposed AO-QAOA methods outperform classical baselines such as quantized SVD and convex AO-based schemes at low bit-depth, with WS-QAOA notably improving convergence by mitigating barren plateaus. The study highlights a practical quantum-classical pathway for low-complexity beamforming and points to future extensions in multi-user MIMO, sensing, RIS, and scalable circuit techniques like circuit knitting to accommodate larger systems.

Abstract

Multiple-input multiple-output (MIMO) is critical for 6G communication, offering improved spectral efficiency and reliability. However, conventional fully digital designs face significant challenges due to high hardware complexity and power consumption. Low-bit MIMO architectures, such as those employing b-bit quantized phase shifters, provide a cost-effective alternative but introduce NP-hard combinatorial problems in the pre- and post-coding design. This paper explores the use of the Quantum Approximate Optimization Algorithm (QAOA) and alternating optimization to address the problem of b-bit quantized phase shifters both at the transmitter and the receiver. We demonstrate that the structure of this quantized beamforming problem aligns naturally with hybrid-classical methods like QAOA, as the phase shifts used in beamforming can be directly mapped to rotation gates in a quantum circuit. Notably, this paper is the first to show that theoretical connection. Then, the Hamiltonian derivation analysis for the b-bit case is presented, which could have applications in different fields, such as integrated sensing and communication, and emerging quantum algorithms such as quantum machine learning. In addition, a warm-start QAOA approach is studied which improves computational efficiency. Numerical results highlight the effectiveness of the proposed methods in achieving an improved quantized beamforming gain over their classical optimization benchmarks from the literature.

Quantum Approximate Optimization Algorithm for MIMO with Quantized b-bit Beamforming

TL;DR

This work tackles the NP-hard problem of optimizing -bit quantized beamforming in MIMO systems by blending the Quantum Approximate Optimization Algorithm (QAOA) with alternating optimization (AO) and a warm-start variant. It formalizes a cost Hamiltonian that maps beamforming phase shifts to quantum rotation gates, derives the corresponding quantum circuits, and presents both a treatment and a general -bit framework with a qubit-efficient encoding. Numerical results show the proposed AO-QAOA methods outperform classical baselines such as quantized SVD and convex AO-based schemes at low bit-depth, with WS-QAOA notably improving convergence by mitigating barren plateaus. The study highlights a practical quantum-classical pathway for low-complexity beamforming and points to future extensions in multi-user MIMO, sensing, RIS, and scalable circuit techniques like circuit knitting to accommodate larger systems.

Abstract

Multiple-input multiple-output (MIMO) is critical for 6G communication, offering improved spectral efficiency and reliability. However, conventional fully digital designs face significant challenges due to high hardware complexity and power consumption. Low-bit MIMO architectures, such as those employing b-bit quantized phase shifters, provide a cost-effective alternative but introduce NP-hard combinatorial problems in the pre- and post-coding design. This paper explores the use of the Quantum Approximate Optimization Algorithm (QAOA) and alternating optimization to address the problem of b-bit quantized phase shifters both at the transmitter and the receiver. We demonstrate that the structure of this quantized beamforming problem aligns naturally with hybrid-classical methods like QAOA, as the phase shifts used in beamforming can be directly mapped to rotation gates in a quantum circuit. Notably, this paper is the first to show that theoretical connection. Then, the Hamiltonian derivation analysis for the b-bit case is presented, which could have applications in different fields, such as integrated sensing and communication, and emerging quantum algorithms such as quantum machine learning. In addition, a warm-start QAOA approach is studied which improves computational efficiency. Numerical results highlight the effectiveness of the proposed methods in achieving an improved quantized beamforming gain over their classical optimization benchmarks from the literature.
Paper Structure (18 sections, 1 theorem, 57 equations, 6 figures, 2 algorithms)

This paper contains 18 sections, 1 theorem, 57 equations, 6 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Motivation & Contribution
  4. Preliminaries on QAOA
  5. The QAOA method
  6. Warm-start QAOA (WS-QAOA)
  7. System Model
  8. QAOA-based beamforming optimization, $b = 2$
  9. Optimization of $\mathbf{f}$
  10. Optimization of $\mathbf{g}$
  11. QAOA-based $b$-bit beamforming optimization
  12. An illustrative example
  13. The General case
  14. The WS-QAOA approach
  15. Complexity analysis
  16. ...and 3 more sections

Key Result

Lemma 1

For any $b\in \mathbb{N}$, there is a subset of exactly $b$ angles, $\{\theta_0,\!\theta_1,\!\dots,\!\theta_{b-1}\}\!\subset\!\{\phi_0,\dots,\phi_{2^b-1}\},$ such that every$\phi_i$ can be written as $\phi_i = \sum_{j=0}^{b-1} x_j\,\theta_j$, where each $x_j\in \{0,1\}$. Equivalently, the map $\,(\p

Figures (6)

  • Figure 1: The 3-qubit rotation gate.
  • Figure 2: Average $\rho(\mathbf{f}^*, \mathbf{g}^*)$ vs $N_\mathrm{T}\times N_\mathrm{R}$, $b=2$.
  • Figure 3: Average $\rho(\mathbf{f}^*, \mathbf{g}^*)$ vs $N_\mathrm{T}\times N_\mathrm{R}$, $b=3$.
  • Figure 4: Average $\rho(\mathbf{f}^*, \mathbf{g}^*)$ vs $b$, $N_\mathrm{T}\times N_\mathrm{R}=3\times3$.
  • Figure 5: Different QAOA landscapes for a $5\times5$ MIMO scenario with $b = 2$.
  • ...and 1 more figures

Theorems & Definitions (1)

  • Lemma 1