Leveraging Quantum Annealing for Large MIMO Processing in Centralized Radio Access Networks

1 INTRODUCTION

• Large and Massive MIMO antenna arrays are being pursued to support many users sharing the wireless medium and meet increasing capacity and throughput demand.The paper situates this demand in ongoing 5G efforts.Source: §1 · p. 1
• Centralized RAN moves most physical-layer processing from APs to a data center, where processing from multiple APs can be aggregated.The architecture is presented as a promising, cost-effective approach for implementing 5G technologies.Source: §1 · p. 1
• Sphere Decoder-based maximum-likelihood detection can outperform linear filters but has computational requirements that grow exponentially with antenna count.Linear zero-forcing and MMSE filters are less computationally demanding but can suffer on poorly conditioned channels, often encountered when user and AP antenna counts are similar.Source: §1 · p. 1
• Wireless systems must complete decoding before feedback is needed; Wi-Fi allows tens of microseconds, while 4G LTE allows 3 ms and WCDMA 10 ms.These timing constraints help explain why current systems often use linear filters despite performance losses.Source: §1 · p. 1
• QuAMax applies quantum annealing to maximum-likelihood MIMO decoding in a centralized RAN, with the annealer co-located with data-center resources serving wireless APs.The paper also introduces Time-to-BER, a metric for evaluating the time needed to reach a target BER.Source: §1 · p. 2
• 10–20 µs of D-Wave 2000Q computation achieves BER 10^-6 and frame error rate 10^-4 for 48-user, 48-AP-antenna BPSK, while 14 × 14 QPSK needs 30–40 µs at 20 dB SNR.For the real-world 8 × 8 trace, QuAMax requires 2 µs for BPSK and 2–10 µs for QPSK.Source: §1 · p. 2

2.1 Primer: Maximum Likelihood Detection

• Maximum-likelihood detection searches transmitted-symbol combinations for the one minimizing mismatch with received symbols, yielding throughput-optimal decoding.The received vector is modeled as y = H v̄ + n, with H the wireless channel and n additive white Gaussian noise.Source: §2.1 · p. 2
• The Sphere Decoder limits its search to symbol combinations within a hypersphere, then uses QR decomposition to express detection as a tree search.The tree height is Nt and its branching factor is the constellation size |O|.Source: §2.1 · p. 2
• For 8 clients with 16-QAM, 15 with QPSK, or 30 with BPSK, Sphere Decoding visits close to 2,000 tree nodes, saturating the cited Skylake core i7 arithmetic subsystem.The described evaluation uses 50 subcarriers over a 20 MHz, 13 dB SNR Rayleigh channel; the search space grows exponentially despite Sphere Decoder reductions.Source: §2.1 · p. 2

2.2 Primer: Quantum Annealing

• Quantum annealing maps optimization objectives to an Ising model and seeks a low-energy solution through an annealing schedule that evolves signals A(t) and B(t).The final qubit values represent a candidate solution, ideally the objective's ground state.Source: §2.2 · pp. 2–3
• Because current imperfect annealers lack a first-principles predictive theory for schedule parameters, the paper uses empirical evaluation.It investigates problem-to-Ising mapping, annealing times of 1–300 µs, and pauses during annealing.Source: §2.2 · p. 3

3.1 QA Problem Formulation

• The Ising and QUBO formulations provide equivalent quadratic objectives, using spin or binary variables to represent the detection problem.In the Ising form, fields express spin preferences and couplings express preferred relations; QUBO variables are binary, with diagonal and off-diagonal terms corresponding to those parameters.Source: §3.1 · pp. 3–4

3.2.1 ML-to-QUBO problem reduction.

• QuAMax represents each transmitted symbol with log2(|O|) binary variables and chooses linear transforms so ML detection expands directly into a QUBO objective.The number of variables is Nt · log2(|O|); the paper gives modulation-specific transforms for BPSK, QPSK, and 16-QAM.Source: §3.2.1 · p. 4
• For 16-QAM, QuAMax retains its linear symbol transform and post-translates decoded bits into Gray-coded output, with analogous translation for higher-order QAM.A direct Gray-coded transform would introduce cubic and quartic terms requiring quadratization; the post-translation instead converts the receiver output to the transmitter’s Gray code.Source: §3.2.1 · pp. 4–5, Figure 2 · p. 5

3.2.2 ML-to-Ising problem reduction.

• QuAMax derives generalized Ising parameters from the channel H and received signal y, making conversion to the QA problem computationally insignificant.The Ising form follows by converting the QUBO; QPSK’s real and imaginary components have no coupler between them.Source: §3.2.2 · p. 5

3.3 Embedding into QA hardware

• Because the ML-derived Ising problem is almost fully connected while Chimera is sparsely connected, QuAMax embeds logical variables as connected chains of physical qubits.The described embedding uses N(⌈N/4⌉ + 1) physical qubits for N logical variables; Table 2 summarizes requirements and feasibility on the available machine.Source: Table 2 · p. 5, §3.3 · p. 6, Figure 3 · p. 6
• Embedding adds a penalty coupling to keep each logical chain aligned, and QuAMax unembeds returned spins by majority vote when a chain disagrees.Tied votes are randomized; the resulting logical assignment is evaluated using the original Ising objective.Source: §3.3 · p. 6

4 IMPLEMENTATION

• A batch of Na anneals returns spin configurations, and the lowest-energy configuration is selected; physical parallelization can run multiple instances and reduce runtime.The reported parallelization factor is Pf ≃ Ntot/(N(⌈N/4⌉ + 1)); a 16-qubit problem using 80 physical qubits could run more than 20 times in parallel.Source: §4 · p. 6
• Figure 4 presents empirical results from six decoding problems to illustrate relationships among Ising energy, solution rank, and BER.The passage identifies the figure’s scope but does not supply its individual outcomes.Source: Figure 4 · p. 7
• Analog intrinsic control errors perturb programmed Ising coefficients, while excessive chain strength can obscure problem information at the device’s precision.The passage models coefficient perturbations as Gaussian and reports measured means and variances during the most delicate annealing phase.Source: §4 · pp. 6–7
• The implementation benchmarks chain strength, annealing time, pause settings, and improved coupling range because their net performance effects require empirical investigation.Improved range can strengthen couplers and reduce intrinsic control errors, but breaks Ising symmetry and prevents averaging over symmetric instances to mitigate leakage errors.Source: §4 · p. 7

5 EVALUATION

• The evaluation used the D-Wave 2000Q, equal client and AP antenna counts, and over 8 × 10^10 anneals; programming and readout overheads were excluded.Those overheads currently exceed pure computation time by several orders of magnitude but do not scale with problem size.Source: §5 · p. 7, §5.2 · p. 8

5.1 Understanding Empirical QA Results

• As modulation order increases and user number decreases, ground-state probability tends to fall while BER and FER rise, alongside a smaller relative Ising energy gap.The smaller gap is likely inversely correlated with the impact of intrinsic control errors on the problem instance.Source: §5.1 · p. 7

5.2.1 Metric: Time-to-Solution (TTS).

• TTS estimates the anneal time needed to find the ground state with a target success probability, conventionally P = 0.99.It uses the per-anneal ground-state probability and assumes independent, identically distributed anneals.Source: §5.2.1 · p. 8

5.2.2 Our Metrics: BER and Time-to-BER (TTB).

• TTB measures expected time to reach a target BER, accounting for low-error solutions beyond the ground state and possible channel-noise errors in the ground state.The metric uses the best-energy solution among a run’s anneals; TTB is computed from the anneal count needed to reach the target BER, adjusted by parallelization.Source: §5.2.2 · p. 8

5.3 Performance Under Annealer Noise

• The annealer-noise study uses random-phase channels, random transmitted bit strings, and BPSK, QPSK, and 16-QAM across varying user counts and AP antennas.It evaluates 780 problems per QA parameter setting and post-processes up to 50,000 anneals per run.Source: §5.3 · p. 8

5.3.2 Choosing whether to pause.

• Improved dynamic range is selected for relative robustness to |JF|, and Tp = 1 µs is chosen because longer pauses dominate anneal time.The TTS microbenchmarks found that extended dynamic range is less sensitive to |JF| and performs best at Ta = 1 µs across problem sizes.Source: §5.3.2 · p. 9
• Pausing achieves better BER-versus-time performance than not pausing, although each paused anneal takes twice as long when Ta = 1 µs.The comparison holds for both Opt and Fix parameter-selection strategies.Source: §5.3.2 · p. 9, Figure 8 · p. 9

5.3.3 QuAMax: End-to-End performance.

• For instances reaching target BER 10^-6 within 10 ms, Opt achieves superior BER within 1–100 µs, while QuAMax reaches an acceptable BER on problems beyond conventional decoder capability.Instances with TTB below the minimum anneal-plus-pause time require an amortized 2 µs through parallelization.Source: §5.3.3 · p. 10, Figure 10 · p. 10, Figure 13 · p. 10
• Tens of microseconds suffice for FER below 10^-3 in 60-user BPSK, 18-user QPSK, and four-user 16-QAM; performance shows low sensitivity across 50-byte to 1,500-byte frames.The FER comparison uses median Opt performance and mean Fix performance.Source: §5.3.3 · p. 10, Figure 11 · p. 10

5.4 Performance under AWGN Noise

• As SNR rises, TTB improves at a fixed user count; at fixed SNR, TTB degrades gracefully as user count increases across modulations.At 10 dB, the energy gap between the lowest and second-lowest energy solutions narrows to 3%.Source: §5.4 · p. 10
• 10–1000 times faster: QuAMax reaches zero-forcing’s BER or better in BPSK and QPSK at bad SNR scenarios.The zero-forcing processing-time estimates use single-core BigStation results; the paper notes that more cores reduce time but not BER.Source: §5.4 · p. 11, Figure 14 · p. 11

5.5 Trace-Driven Channel Performance

• 10 µs for QPSK and amortized 2 µs for BPSK: QuAMax achieves 10^-6 BER and 10^-4 FER on real 8×8 channel traces.The trace is measured at 2.4 GHz; the BPSK result uses multiple problem instances operating in parallel.Source: §5.5 · p. 11

6 RELATED WORK

• Prior quantum annealing applications span planning and scheduling, databases, fault diagnostics, machine learning, finance, data analysis, chemistry, and space sciences.Work on fully connected graph optimization found QA performance could match highly optimized simulated annealing on Intel processors.Source: §6 · p. 11
• QAOA and QA both use QUBO or Ising formulations for unconstrained problems, while QAOA requires digital gate-model hardware.The paper notes that QAOA can, in principle, simulate QA and outperform it.Source: §6 · p. 11
• Sphere decoders implemented in ASICs can fall short as antenna count or modulation rises, while extreme parallelization or GPUs might resolve the issue.The passage cautions that practical constraints may eventually limit classical-platform performance gains.Source: §6 · p. 11

7 DISCUSSION

• The D-Wave 2000Q draws 16 kW, primarily for cryogenic refrigeration, and operating expenses are about USD $17,000 yearly.Liquid nitrogen is required one to two times a month.Source: §7 · p. 12
• 30–50 ms preprocessing, 6–8 ms programming, and 0.125 ms readout per anneal mean QuAMax cannot be deployed today within wireless processing times.The paper says integration efforts could reduce these overhead times by several orders of magnitude; end-to-end evaluation on more advanced hardware remains future work.Source: §7 · p. 12

8 CONCLUSION

• QuAMax is the first design, implementation, and experimental evaluation of a quantum-computing solver for ML MIMO decoding, establishing a baseline for future integrated systems.The authors say quantum computation should be considered for future high-capacity wireless networks once integration is optimized.Source: §8 · p. 12
• The paper identifies further optimization of |JF|, Ta, and sp, plus reverse annealing, as possible ways to close the gap to Opt performance.Each chain in the anticipated architecture would require N/12 + 1 qubits.Source: §8 · p. 12
• Future annealer improvements are expected to support chips with tens of thousands of highly connected qubits and order-of-magnitude reductions in overhead times.The conclusion also anticipates qubits with twice the Chimera degree and count, longer-range couplings, and improved parallelization opportunities.Source: §8 · p. 12
• Gate-model systems offer additional controls, but currently cannot support algorithms decoding more than 4×4 BPSK.The authors identify MIMO ML decoding on gate-model QPUs as future work.Source: §8 · p. 12

• 2 × 2 BPSK maximum-likelihood detection is transformed into QUBO form by expanding its norm and expressing symbols with binary variables.Source: A QUBO FORMS · p. 14
• For BPSK, each symbol vi ∈ {−1, 1} maps to qi ∈ {0, 1} through vi = 2qi − 1, with 0 representing −1 and 1 representing 1.Source: A QUBO FORMS · p. 14
• The resulting detector minimizes Q11q1 + Q22q2 + Q12q1q2 over q1 and q2, with Q12 determined by products of real and imaginary channel coefficients.Source: A QUBO FORMS · p. 14

• Embedding maps the Ising problem to an equivalent one with the same ground state that satisfies Chimera graph constraints.Source: B EMBEDDED ISING · p. 15
• Each logical variable si is associated with a chain of ⌈N/4⌉ + 1 qubits, and |JF| penalizes disagreement so the chain qubits take the same value.Larger |JF| makes this agreement more likely.Source: B EMBEDDED ISING · p. 15
• Because hardware caps the maximum negative energy at −1, |JF| renormalizes objective terms by |JF|^-1, while each linear term fi is additionally divided by chain length.Source: B EMBEDDED ISING · p. 15
• Duplicating variables ensures that a pair of qubits in two chains can be connected by a physical Chimera coupler.Source: B EMBEDDED ISING · p. 15

• The section presents Ising parameters for 16-QAM.Source: C 16-QAM ISING MODEL PARAMETERS · p. 15
• Because each symbol's real and imaginary terms are independent, the coupler strength between s4n−3, s4n−2 and s4n−1, s4n is 0.Source: C 16-QAM ISING MODEL PARAMETERS · p. 15
• For other si and sj, the section gives 16-QAM Ising coupler strengths in cases indexed by i and j.Source: C 16-QAM ISING MODEL PARAMETERS · p. 15