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Quantum Energetic Advantage before Computational Advantage in Boson Sampling

Ariane Soret, Nessim Dridi, Stephen C. Wein, Valérian Giesz, Shane Mansfield, Pierre-Emmanuel Emeriau

arXiv:2601.08068v1quant-ph

TL;DR

The paper investigates whether quantum computers can use less energy than classical systems for Boson Sampling before achieving a runtime advantage. It applies the Metric–Noise–Resource framework to a realistic photonic architecture, modeling hardware resources, loss, and distinguishability. The analysis finds an energetic advantage before computational advantage and identifies a near-term architecture and noise budget for observing it.

  • Problem

    Full-stack evidence remains limited on when quantum energetic advantages arise for concrete tasks, particularly while classical algorithms remain faster.

  • Method

    The study specializes the Metric–Noise–Resource framework to photonic Boson Sampling, connecting control parameters, noise processes, resources, and a classical-simulation-based performance metric.

  • Results

    Quantum energetic advantage emerges before computational advantage, with photonic devices consuming less energy per sample even while classical algorithms remain faster.

  • Takeaways & Limitations

    Energetic efficiency is an independent, experimentally accessible benchmark for quantum advantage, supported by a realistic noise and loss budget for near-term observation.

Abstract

from arXiv · show

Understanding the energetic efficiency of quantum computers is essential for assessing their scalability and for determining whether quantum technologies can outperform classical computation beyond runtime alone. In this work, we analyze the energy required to solve the Boson Sampling problem, a paradigmatic task for quantum advantage, using a realistic photonic quantum computing architecture. Using the Metric-Noise-Resource methodology, we establish a quantitative connection between experimental control parameters, dominant noise processes, and energetic resources through a performance metric tailored to Boson Sampling. We estimate the energy cost per sample and identify operating regimes that optimize energetic efficiency. By comparing the energy consumption of quantum and state-of-the-art classical implementations, we demonstrate the existence of a quantum energetic advantage -- defined as a lower energy cost per sample compared to the best-known classical implementation -- that emerges before the onset of computational advantage, even in regimes where classical algorithms remain faster. Finally, we propose an experimentally feasible Boson Sampling architecture, including a complete noise and loss budget, that enables a near-term observation of quantum energetic advantage.

I. INTRODUCTION

The paper asks whether quantum energetic advantage can arise before computational advantage and studies this question through a full-stack Boson Sampling analysis. It develops an MNR-based photonic model, compares quantum and classical energy per sample, and identifies experimentally relevant conditions for observing the effect.

  • Energetic advantage asks whether quantum hardware uses less energy for a task, independently of whether it is faster than classical hardware.
  • Full-stack hardware-level analyses connecting energy consumption to concrete quantum algorithms remain scarce, especially before computational advantage.
  • Boson Sampling is used because sampling is native to photonic hardware, requires minimal algorithmic overhead, and can be repeated at high rates.
  • The study specializes the MNR framework to photonic Boson Sampling and defines a metric reflecting classical simulation complexity under loss and partial distinguishability.
  • The analysis finds quantum energetic advantage before computational advantage, with quantum devices using less energy per sample while classical algorithms remain faster.
  • A detailed noise and loss budget identifies a realistic near-term regime for experimentally observing quantum energetic advantage.

II. METRIC-NOISE-RESOURCE

The MNR framework links adjustable control parameters, task performance, noise, and resource consumption. It formulates optimization as minimizing resources at a fixed target metric or maximizing efficiency under that constraint.

  • MNR relates a computation’s performance metric to physical noise processes and the resources required to control or compensate for them.
  • MNR components: Control parameters include adjustable hardware and software variables that influence performance and resource consumption.
  • MNR components: The metric quantifies computation quality or success, with higher values indicating better performance.
  • MNR components: Noise represents physical processes that reduce performance, while mitigation can lower noise at increased resource cost.
  • MNR components: Resources include physical and computational requirements such as energy, gate operations, and classical control overhead.
  • Optimization: The methodology expresses metric, noise, and resources through control parameters and optimizes resource use at a fixed target M0.
  • Optimization: Alternatively, MNR maximizes efficiency, defined as performance divided by resource consumption, under the same target constraint.

III. MNR FOR BOSON SAMPLING

The Boson Sampling MNR model combines photonic hardware resources with loss and distinguishability noise, using a classical-simulation-based metric to characterize task difficulty. It relates system size, temperature, transmission, and cryogenic infrastructure to performance and energy cost.

  • Boson Sampling task: Boson Sampling samples photon outputs from an m×m interferometer given n single-photon inputs, with output probabilities determined by matrix permanents.
  • Boson Sampling task: Calculating permanents is #P-hard, making exact Boson Sampling classically intractable as system size grows, while approximate sampling is believed efficiently solvable only quantumly.
  • Hardware model: The modeled setup uses single-photon sources, demultiplexers, an m×m interferometer, and m detectors, with sources and detectors in a cryostat at temperature T.
  • Hardware model: The architecture uses quantum-dot photon sources, active demultiplexing, fiber delays, a 51-mode interferometer, and SNSPD detection to estimate full-stack energy per sample.
  • Control parameters: The control parameters are input-photon number, mode number, and cryostat temperature, with m = ⌈2.1n⌉ sufficient beyond the collision-free regime.
  • Resources: Cryogenic cooling dominates current photonic-platform power because static heat loads exceed dynamic dissipation, while laser, monitoring, and classical overhead contribute a fixed cost Pfix.
  • Resources: The model sets total cryogenic requirements through detector count and cryostat capacity, with m = ⌈2.1n⌉ relating modes to input photons.
  • Noise: The main noise sources are photon loss and distinguishability, with the analysis initially assuming a fixed number l of lost photons.

A. Classical vs. quantum comparison

The comparison evaluates quantum and classical energy cost per Boson Sampling sample as an independent dimension of quantum advantage. It accounts for photon-loss fluctuations when comparing average sample energies.

  • Quantum energetic advantage is defined independently of runtime advantage: a device may use less energy per sample while classical algorithms remain faster.
  • The comparison targets the quantum energy cost per sample, EQsample, against the classical cost, ECsample, for the same Boson Sampling task.
  • The analysis optimizes over l lost photons for l ∈ [1, 25] before averaging over transmitted-photon fluctuations.

1. Classical Energetic Cost per Sample

The classical baseline uses the Clifford–Clifford exact Boson Sampling algorithm and conservatively estimates its energy from a lower bound on floating-point operations and efficient supercomputing hardware.

  • The Clifford–Clifford algorithm has time complexity O(n2n + mn2), providing the classical simulation baseline.
  • Computing permanent-expansion coefficients for a j × j matrix requires j2j floating-point operations.
  • Forming an m-element array by summing j terms m times requires mj floating-point operations.
  • Repeating these steps over j ∈ {2, ..., n} yields a lower bound L(m, n) on the algorithm’s floating-point operations.
  • The classical energy per sample is obtained by dividing L(m, n) by hardware efficiency ηe in Flops per watt, with n replaced by the performance metric M for partially distinguishable photons.
  • The estimate favors classical simulation by using a lower operation bound, energy-efficient hardware, and no memory, communication, or scheduling overheads.

2. Quantum Energetic Cost per Sample

The quantum sample cost combines total device power with sample-generation time, which is determined by the rate at which the photonic system emits the required photons.

  • Quantum energy per sample equals total power consumption from Eq. (3) multiplied by the time required to produce one sample.
  • The quantum energy and time results are plotted against the average performance metric, with quantum points colored by end-to-end transmission efficiency.
  • At larger input-photon numbers, fixed per-component losses degrade the metric when transmission is not improved.
  • The sample-generation time is the time needed to emit n photons and is obtained from the sample generation rate rsample.
  • With one source per cryostat, the generation rate is limited by the maximum number of photons sent by one source for given n and m.

3. Quantum Energetic Advantage in Boson Sampling

The study compares realistic quantum and classical hardware assumptions across energy and runtime. It finds an energy-advantage window before the computational-advantage threshold, while transmission requirements and optical loss constrain scaling.

  • Hardware assumptions: The classical baseline uses JEDI, with ηe = 72.733 GFlops/W, while the quantum model uses an AttocubeCMC cryostat requiring 1,396 kW at 3 K.
  • Hardware assumptions: The hardware assumptions include smax = 26, implying n̄ ≤ 12, and a single-photon generation rate rSPS = 1 GHz.
  • Energetic scaling: Classical energy per sample scales exponentially with M0, whereas quantum energy per sample has more favorable scaling.
  • Energetic scaling: For fluctuating transmitted-photon numbers, the average classical energy is normalized over N = 10,000 samples and compared with quantum energy per sample.
  • Advantage regimes: The energy-advantage threshold is M*e = 15: above this average metric, the quantum computer consumes less energy than the classical one.
  • Scaling constraints: Increasing the metric requires higher transmission and larger, better-performing photonic chips, while optical loss eventually makes the metric drop at larger input-photon counts.
  • Scaling constraints: The runtime comparison uses classical performance Rmax = 4.5 PFLOP/s and quantum sample-generation rates.
  • Advantage regimes: Quantum computational advantage begins beyond M*c = 18, while genuine quantum energy advantage occurs for 15 ≤ M0 ≤ 18 despite slower quantum sampling.

B. Experimental feasibility

The analysis shows that reaching energetic or computational advantage requires jointly increasing system size and transmission efficiency despite transmission losses in larger optical circuits. A near-term architecture is identified for energetic advantage, with modest improvements making the target experimentally attainable.

  • Higher system size and transmission efficiency are simultaneously required to access the energetic and computational advantage regimes.Transmission nevertheless decreases as the optical circuit grows.
  • Table I compares state-of-the-art parameters with modest improvements required for energetic and computational advantage demonstrations.The table computes transmission η for m = 51 in the energetic columns and m = 59 in the computational column.
  • The transmission model includes two coupling interfaces, producing a factor of 2 before c_coup.
  • M*_e = 15 is the average metric required for energy advantage.The stated target can be reached with an end-to-end transmission of 60%, 51 modes, two sources in two cryostats, and 24 input photons.
  • The proposed energetic-advantage setup is not yet achievable with current state-of-the-art devices, but higher-efficiency demultiplexers could bridge the gap.The computational-advantage target requires 59 detectors, 3 DMX-12 units, 28 photons, 95% indistinguishability, and 65% end-to-end efficiency.
  • Quantum energy advantage is presented as an experimentally attainable goal in the near future.

C. Energetic comparison between photonic and superconducting modalities

The paper compares photonic Boson Sampling with superconducting Random Circuit Sampling as a hardware-level energetic benchmark, not as formally equivalent tasks. This comparison is simplified by the reported constant energy cost per superconducting sample.

  • Random Circuit Sampling is the superconducting counterpart to photonic Boson Sampling, with samples generated natively by gate-based quantum computers.Classical computation of its output distribution becomes exponentially hard with the number of qubits and gates.
  • The comparison is an order-of-magnitude hardware-level benchmark rather than a strict equivalence between the two tasks.The tasks use distinct physical models, noise mechanisms, and complexity assumptions, and no formal mapping between their hardness metrics is known.
  • Both tasks can be compared using the absolute runtime of the best-known classical algorithm for producing a sample.Dependence on the chosen classical strategy is intrinsic to quantum advantage demonstrations.
  • The Sycamore experiment reports an essentially constant energetic cost per RCS sample, independent of qubit number or circuit depth.All samples are generated within a single cryostat and the dynamic heat load is negligible.

V. CONCLUSIONS

The paper develops a full-stack MNR analysis linking photonic Boson Sampling controls, noise, and energy consumption. It finds energetic advantage before runtime advantage and translates the result into experimentally relevant requirements, while detailing modeling assumptions and distinguishability effects.

  • The full-stack resource model combines cryogenic power, laser power, and control-electronics consumption with the Metric–Noise–Resource methodology.
  • The analysis connects experimental control parameters, noise processes, and energy consumption for a concrete quantum computational task.
  • Photonic Boson Sampling can have a more favorable energy-cost-per-sample scaling than classical simulation before achieving computational advantage in runtime.
  • Classical energy costs are conservatively estimated using energy-efficient supercomputers, a lower bound on floating-point operations, and a fast classical simulation algorithm.Under these assumptions, classical energy consumption is likely underestimated.
  • The analysis provides a detailed noise and loss budget to translate energetic advantage into experimentally relevant requirements.
  • The indistinguishability model accounts for temperature-dependent broadening caused by thermal acoustic phonons and additional effects between distant sources.The effective indistinguishability subtracts a fixed 5% for slow spectral fluctuations.
  • For l = 12 lost photons, the optimization reaches the energy-advantage threshold at M0 = 17, with the lowest reported k_eff equal to 14.The choice l = 12 permits optimization while reaching the threshold.

C. Experimental feasibility

The proposed experimental configurations target the energy-advantage threshold by balancing photon number, transmission efficiency, and sampling fluctuations. At η = 60% with 29 input photons, the classical energy per sample is one order of magnitude larger than the quantum cost.

  • Thresholds: Me = 17 and Mc = 21 define the energy and computational advantage thresholds at l = 12 lost photons.The optimization requires 60% efficiency for Me and 65% for Mc, with a 3 K cryostat and indistinguishability I = 95% in both cases.
  • Thresholds: The proposed threshold setup uses η = 60%, 61 modes, and n = 29 photons to realize Me.The corresponding computational-advantage configuration uses η = 65%, 72 modes, and n = 34 photons to realize Mc.
  • Experimental variability: Photon-transmission fluctuations follow a binomial distribution and induce fluctuations in the performance metric M.The analysis compares average quantum and classical energy per sample rather than only the fraction of samples exceeding a threshold.
  • Energy comparison: At η = 60%, 61 modes, and 29 input photons, classical energy averages 1.3 × 10−7 Wh per sample versus 1.4 × 10−8 Wh quantum.The metric is centered around Me = 17, and the classical energy is one order of magnitude larger.
  • Experimental design: The feasibility analysis proposes an experiment at M*e = 15, where average classical and quantum energy per sample are equal.Reaching this threshold requires 28 photons at 56% transmission, while a 24-photon implementation requires 60% transmission and two sources.
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