Trade-off between complexity and energy in quantum phase estimation

TL;DR

The paper introduces a universal framework to study the trade-off between implementation energy and gate complexity in quantum procedures, focusing on sequential quantum phase estimation (QPE). By linking the complexity $C$ and per-gate energy $E$ to a total resource $R = C E$, and incorporating gate-imperfection via an error parameter $\epsilon$, the authors derive a non-monotonic resource landscape with a potential sweet spot for energy–complexity co-optimisation. They quantify QPE performance through the quantum Fisher information $F_N$, show how noise in gate implementation degrades $F_N$, and provide a concrete first-principles optical model that yields explicit scaling laws for the optimal number of steps $N_{opt}$, energy per gate, and total resource $R$, including state-preparation and measurement costs. The results highlight practical guidelines for energy benchmarking in quantum sensing and metrology and suggest that combining energy-aware analyses with existing resource theories can inform the sustainable deployment of near-term quantum technologies.

Abstract

Driven by the desire to make quantum technologies more sustainable, in this work we introduce a framework for analysing the interplay between complexity and energy cost of quantum procedures. In particular, we study 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 repetitively until the probe is measured and the outcome leads to an estimate on the phase. We establish a trade-off relation between the implementation energy of the channel and the number of times it is applied (complexity), while reaching a desired estimation precision. The principles of our analysis can be adapted to optimise the energy consumption in other quantum protocols and devices.

Figures (7)

• Figure 1: A qualitative plot of the key variables characterising a quantum protocol as in Eq. \ref{['noisyRes']}. A vanishing error $\epsilon=0$ corresponds to the ideal limit, while competition takes place between (a) growing complexity $C(\epsilon)$ and (b) decreasing energy cost $E(\epsilon)$ per gate as the error level $\epsilon$ increases. (c) The resulting total resource cost $R(\epsilon)$ may (solid curve) or may not (dashed curve) have a global minimum at finite $\epsilon$.
• Figure 2: (a) Plots of $F_N/N$ for different values of $\kappa$ and fixed $\phi=0.5$. Observe the non-monotonic behaviour predicted by Eq. \ref{['QFIapproxNichol']}: the ratio grows linearly with $N$ first, reaches its maximum and decays exponentially afterwards. As indicated by the vertical dotted lines, the optimal step $N_{\textrm{opt}}$ is well approximated by $-[2\log(\lambda_\perp)]^{-1}$; (b) The resulting complexity \ref{['QPEcomplexity']} is plotted against $1/\kappa$ (representing the implementation error $\epsilon$) for $\delta^2=10^{-4}$. Observe the growing pattern as anticipated by Figure \ref{['figure:resource']}(a). The inset exhibits the zoom-in at small $1/\kappa$ for $\phi=0.25$. Each teeth of the zigzag pattern corresponds to a region that applies the same number of complete rounds $Q_N$. The dotted line corresponds to the raw complexity computed by Eq. \ref{['rawcomplexity']}. Notice that the raw complexity vanishes in the ideal limit, and approximates the true complexity better as $1/\kappa$ grows.
• Figure 3: (a) Plots of the total resource cost \ref{['ResourceQPE2']}, with $\delta^2=10^{-4}$ and the implementation error $\epsilon$ represented by $1/\bar{m}$. The vertical and horizontal dotted lines locate the saturation point \ref{['optimalpoint']} where the plateau starts. This matches the behaviour indicated by the dashed curve in Figure \ref{['figure:resource']}(c). Solid and dotted lines in the inset are the corresponding true and raw complexities \ref{['approxQFIComp']}, respectively; (b) The energy plot in (a) is repeated for various $\delta^2$ and fixed $g=2.5$. The blue solid curve depicts evolution of the saturation point as both $\bar{m}$ and $\delta^2$ vary.
• Figure 4: Plots of the total resource cost from gate implementation and state preparation, with $g=2.5$, $\xi=0.2$, $\delta^2=10^{-4}$ and the error represented by $1/\bar{m}$. Solid lines are computed by Eq.\ref{['totalcostext']}, while dashed ones \ref{['QPEresourceT']} approximate the cost through the raw complexity. Contrary to the plateau in Figure \ref{['plot:QPEEnergy']}, the total resource cost ends up increasing with larger implementation error as shown in the inset, since a larger error, or smaller $\bar{m}$, leads to more rounds (see Eq. \ref{['optimalstepapprox']}) and hence a larger cooling cost. This corresponds to the behaviour indicated by the solid curve in Figure \ref{['figure:resource']}(c). However, the increment is negligible as the cost of cooling is much smaller than that of gate implementation.
• Figure 5: Plots of the total resource cost with gate implementation and measurement taken into account. The fixed parameters are $g=2.5$, $\delta^2=10^{-4}$. Solid and dashed lines correspond to when the total resource cost and the number of gates are minimised, respectively amounting to $R_{N_R}$ and $R_{N_C}$; see Eqs. \ref{['totalcostext']}--\ref{['tworesources']}, with $E_{\textrm{ext}}$ given by Eq. \ref{['mmtperround']}. The two costs coincide when they involve the same number of full rounds, since then their external costs, $E_{\textrm{ext}}\cdot (Q_N+1)$, are the same and so minimising the total energy cost and only the cost of gate implementation are equivalent. Otherwise, a gap opens up between $R_{N_C}$ and $R_{N_R}$ when the numbers of full rounds for each differ. This happens when the external cost of an extra round outweighs the cost of the gates it saves for complexity minimisation. The larger and the smaller $E_{\textrm{ext}}$ and $\bar{m}$ are, respectively, the larger the gap grows between the two costs and the choice regarding which quantity to optimise becomes more important.