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arxiv: 2511.05458 · v3 · submitted 2025-11-07 · 🪐 quant-ph

Trade-off between complexity and energy in quantum phase estimation

Yukuan Tao , Madalin Guta , Gerardo Adesso This is my paper

Pith reviewed 2026-05-17 23:30 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum phase estimationenergy costcomplexity trade-offquantum metrologysequential protocolsquantum sensing
0
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The pith

Quantum phase estimation requires balancing total energy cost against the number of channel applications to hit a target precision.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework that quantifies the energetic cost of applying a quantum channel repeatedly in a sequential phase estimation protocol. It derives an explicit trade-off showing that fewer channel applications force higher total energy expenditure to maintain the same estimation accuracy. A co-optimization point is identified where both energy and the number of applications reach their joint minimum for the desired precision. This relation provides a concrete way to evaluate efficiency limits in quantum sensing tasks.

Core claim

In the sequential quantum phase estimation protocol, a trade-off relation exists between the total energy cost of the protocol and the number of times the channel is applied, for any fixed estimation precision, with a sweet spot where the two quantities are co-optimised.

What carries the argument

The trade-off relation that connects total energy cost to the count of sequential channel applications needed for a chosen phase precision.

If this is right

  • For any target precision, lowering the number of channel applications necessarily raises the total energy cost.
  • A single operating point simultaneously minimises the combined energy and complexity burden.
  • The same analysis method can be used to assess energy requirements in other sequential quantum protocols.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Designers of quantum sensors could use the trade-off curve to select the best number of repetitions for given hardware energy limits.
  • The relation may help compare sequential protocols against parallel or continuous-time alternatives in metrology.
  • Hardware-specific tests could reveal whether the assumed energy model needs adjustment for real devices such as trapped ions or photons.

Load-bearing premise

The framework relies on a specific model of energetic cost for quantum operations and channel applications that is assumed to capture the dominant physical contributions in the sequential protocol.

What would settle it

An experiment that records energy consumption versus number of channel uses in a real quantum phase estimation setup and checks whether the measured points lie on or violate the predicted trade-off curve.

Figures

Figures reproduced from arXiv: 2511.05458 by Gerardo Adesso, Madalin Guta, Yukuan Tao.

Figure 1
Figure 1. Figure 1: FIG. 1. A qualitative plot of the key variables characterising a quantum protocol as in Eq. (3). A vanishing error [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) Plots of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Plots of the total resource cost (18), with [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Plots of the total resource cost with gate implementation and [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Full circuit diagram of the QPE protocol for which a trade-off between complexity and energy is established in this paper. [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Plots of [PITH_FULL_IMAGE:figures/full_fig_p014_7.png] view at source ↗
read the original abstract

Quantifying the energetic cost of implementing quantum operations is essential for assessing the efficiency and scalability of quantum sensing and information-processing technologies. Here, we introduce a framework for analysing the interplay between complexity and energy cost of quantum processes. In particular, we apply our framework to a sequential quantum phase estimation protocol, where a phase of physical significance is encoded in a quantum channel. The channel is applied to a probe state repeatedly until the probe is measured and the outcome leads to an estimate on the phase. We establish a trade-off relation between the total energy cost of the protocol and the number of times the channel is applied (complexity), while reaching a desired estimation precision. A sweet spot is located where the two quantities are co-optimised. The principles of our analysis can be adapted to benchmark the energetic requirements in other quantum protocols and devices.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a framework for quantifying the interplay between complexity (measured by the number of sequential channel applications) and energetic cost in quantum protocols. Applied to a sequential quantum phase estimation protocol, it derives a trade-off relation between the total energy cost E_total and the number of channel uses N required to achieve a target estimation precision, and identifies a co-optimized 'sweet spot' N* that balances the two quantities.

Significance. If the postulated energy-cost model is shown to be the dominant contribution under standard physical implementations, the trade-off and sweet-spot result could offer a practical benchmark for assessing energetic scalability in quantum sensing. The framework's adaptability to other protocols is noted as a potential strength, though its impact hinges on the physical justification of the cost functional.

major comments (1)
  1. [Framework and energy-cost definition (near the protocol description)] The central trade-off E_total(N) and the existence of a co-optimized N* rest on the assumption that each channel application incurs a fixed, state-independent energy cost. No derivation from a microscopic time-dependent control Hamiltonian is provided to show that this cost remains constant when the probe evolves under repeated applications; if the integrated energy depends on accumulated phase or control schedule, the functional form of the trade-off and the location of the sweet spot would change. This assumption is load-bearing for the main claim.
minor comments (1)
  1. [Abstract and introduction] The abstract and framework description would benefit from an explicit statement of the precision metric (e.g., variance or Holevo bound) used to fix the target accuracy when plotting or deriving the E_total(N) curve.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address the major comment in detail below and have made revisions to clarify the energy cost model as suggested.

read point-by-point responses

  1. Referee: [Framework and energy-cost definition (near the protocol description)] The central trade-off E_total(N) and the existence of a co-optimized N* rest on the assumption that each channel application incurs a fixed, state-independent energy cost. No derivation from a microscopic time-dependent control Hamiltonian is provided to show that this cost remains constant when the probe evolves under repeated applications; if the integrated energy depends on accumulated phase or control schedule, the functional form of the trade-off and the location of the sweet spot would change. This assumption is load-bearing for the main claim.

    Authors: The referee correctly identifies that our trade-off relation relies on modeling the energy cost as constant per channel application. This is a deliberate choice in the framework to focus on the interplay between complexity (N) and energy, treating the per-use cost as an effective parameter that can be calibrated for different physical systems. We do not claim a universal microscopic derivation because the appropriate Hamiltonian depends on the implementation details, which are beyond the scope of the general framework. However, we acknowledge the importance of this point and have added text in the revised manuscript (near the protocol description) explaining that the assumption holds when the control fields are adjusted to compensate for state-dependent effects or when the energy is primarily dissipated in a state-independent manner, such as in the driving of the channel. We also discuss that if the cost varies with the phase, the sweet spot N* would need to be determined numerically for the specific case, but the existence of a trade-off remains. This revision clarifies the scope of our results without altering the main claims. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation of trade-off relation

full rationale

The paper introduces an explicit model for energetic cost of operations and channel applications, then derives the trade-off E_total(N) versus N for fixed precision by applying that model to the sequential protocol. No equation or step reduces the claimed relation to a self-definition, a fitted parameter renamed as prediction, or a self-citation chain. The cost functional is stated as an assumption rather than derived from a microscopic Hamiltonian, but this is an external modeling choice, not a circular reduction of the result to its inputs. The derivation therefore remains self-contained once the cost model is granted.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review provides no explicit free parameters, axioms, or invented entities; ledger left empty.

pith-pipeline@v0.9.0 · 5435 in / 917 out tokens · 23092 ms · 2026-05-17T23:30:56.713560+00:00 · methodology

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Reference graph

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