The EO-GRAPE and EO-DRLPE framework co-optimized the process fidelity and energetic cost (the time-integrated norm of the total Hamiltonian) of quantum gate synthesis, but only at fixed gate duration and only for gates drawn from the {CNOT, Hadamard, T} universal set. The open problem is to complete that picture along the two axes the source signposts for future work: co-optimize total evolution time, guided by quantum speed limits, together with fidelity and energetic cost; and discover universal gate sets whose energy-optimal pulse implementations, under realistic hardware restrictions on frequencies, bandwidths, and waveforms, can lower the energetic cost of compiling arbitrary quantum computations relative to the {CNOT, Hadamard, T} baseline.

Background

Quantum optimal control (QOC) shapes classical control fields that steer a quantum system along a desired evolution. For gate synthesis, a system evolves under a Hamiltonian H^(t)=H^D+∑kuk(t)H^k\hat H(t)=\hat H_D+\sum_k u_k(t)\hat H_k, combining a fixed drift term with tunable controls, and the goal is that the propagator at the end of the pulse reproduces a target unitary gate UU up to high process fidelity (the squared Hilbert-Schmidt overlap between implemented and target unitary, normalized by the dimension and insensitive to global phase, so that it equals 1 exactly when the implemented unitary matches the target up to a global phase). Gradient ascent pulse engineering (GRAPE) is the standard open-loop method: control amplitudes are parametrized on a discrete time grid and updated by gradients of the fidelity; Krotov-type schemes (an alternative gradient-based optimal-control family) and chopped random-basis (CRAB) optimization are alternative numerics, and deep reinforcement learning (DRL) offers a closed-loop, model-free alternative in which an agent learns pulses by interacting with a (simulated) quantum environment.

A line of work rooted in quantum thermodynamics asks what such control costs energetically. Deffner (2021) defined the energetic cost of a unitary UU implemented over duration τ\tau as E[U]=∫0τ∥H^(t)∥ dtE[U]=\int_0^\tau \|\hat H(t)\|\,dt, the time-integrated norm of the total Hamiltonian. Quantum speed limits (QSL) are bounds of Mandelstam-Tamm and Margolus-Levitin type on the minimal time required for a quantum system to evolve between distinguishable states or to implement a target transformation. Aifer and Deffner (2022) connected the two, deriving state-independent lower bounds on the energetic cost of quantum gates from QSL reasoning, which establishes theoretically that faster gates generically demand more control energy.

Against this backdrop, Fauquenot, Sarkar, and Feld (arXiv:2411.06556, published in Advanced Quantum Technologies, 2025) made the trade-off operational: they derived the gradient of E[U]E[U] for pulse engineering and introduced two optimizers that co-optimize fidelity and energetic cost, the open-loop EO-GRAPE (energy-optimized GRAPE) and the closed-loop EO-DRLPE (energy-optimized deep reinforcement learning for pulse engineering). Their findings include an empirical Pareto front (the set of non-dominated fidelity-energy pairs, where neither quantity can be improved without worsening the other) for gates from the universal gate set {CNOT, Hadamard, T} - roughly, a 10% reduction in energetic cost buys about 1% more infidelity in the low-infidelity regime - as well as a positive correlation between energetic cost and the length of the path the unitary traces on the Bloch sphere (the unit sphere of qubit states), supporting the view that the most energy-efficient implementation follows the geodesic (shortest path) at fixed duration. Both the Pareto front and the 10%/1% figure were obtained at fixed duration τ\tau and in terms of the duration-normalized energetic cost of the source's Eq. (19), ϕe=(∑nΔt ∥H^D+∑kuk(n)σ^k∥)/(τ ∥H^D+∑kσ^k∥)\phi_e=\big(\sum_n \Delta t\,\|\hat H_D+\sum_k u_k(n)\hat\sigma_k\|\big)\big/\big(\tau\,\|\hat H_D+\sum_k\hat\sigma_k\|\big) - the discretized time-integrated norm divided by a fixed normalization constant - with the norm evaluated through the Hilbert-Schmidt identity ∥A∥=Tr⁡(A∗A)\|A\|=\sqrt{\operatorname{Tr}(A^{*}A)} and the fidelity term taken as the normalized, global-phase-insensitive overlap ∣Tr⁡(UT†U(τ))/Tr⁡(UT†UT)∣2|\operatorname{Tr}(U_T^{\dagger}U(\tau))/\operatorname{Tr}(U_T^{\dagger}U_T)|^{2}.

Two design freedoms were deliberately held fixed in that work and are what this problem concerns. First, the gate duration τ\tau was fixed: the Pareto front was mapped only in the (fidelity, energy) plane, even though QSL theory says duration and energetic cost are intertwined. Second, the gate set was prescribed: a universal gate set is a finite collection of gates from which any quantum computation can be compiled up to approximation, and {CNOT, Hadamard, T} is one such set, but there is no reason it is energetically optimal for given hardware.

Problem Statement

The problem, as signposted by Fauquenot-Sarkar-Feld, is to complete energy-efficient quantum gate synthesis along the two axes their paper signposts as future work: minimizing the energetic cost of implementing arbitrary quantum computations at target fidelity within a pragmatic pulse-level compiler, with gate duration (axis A) and gate-set choice (axis B) treated as free design variables. The source states the two axes as separate future-work items rather than as a conjunctive completion requirement.

(A) Time-energy-fidelity co-optimization. For a quantum system with drift Hamiltonian H^D\hat H_D and a fixed set of control Hamiltonians H^k\hat H_k driven by bounded control amplitudes uk(t)u_k(t), characterize and attain the joint trade-off surface among process fidelity FF, energetic cost E[U]=∫0τ∥H^(t)∥ dtE[U]=\int_0^\tau\|\hat H(t)\|\,dt, and total gate duration τ\tau. Existing results hold τ\tau fixed and map only the fidelity-energy Pareto front, and they do so in terms of the duration-normalized cost ϕe\phi_e rather than the raw integral E[U]E[U]; once τ\tau is itself a design variable, the two differ by a factor proportional to τ\tau - ϕe\phi_e is a time-average, amplitude-like quantity, while the raw E[U]E[U] additionally accumulates the drift contribution roughly linearly in τ\tau - so a trade-off surface over varying τ\tau is defined only relative to an explicitly chosen energetic-cost functional and norm convention. Two sub-questions jointly constitute this characterization: how does the minimal attainable energetic cost at fixed target fidelity behave as τ\tau approaches the quantum-speed-limited minimum for a given system and gate; and can a single open-loop or closed-loop optimizer (extending EO-GRAPE or EO-DRLPE) co-optimize all three objectives, using quantum speed limits to guide the choice of τ\tau?

(B) Energetically informed gate-set selection. Discover universal gate sets - alternatives to {CNOT, Hadamard, T} - whose members admit energy-optimal pulse implementations under a specific set of hardware restrictions (such as a specific waveform, set of frequencies, or bandwidth), and determine whether compiling arbitrary quantum computations in the discovered set can lower total energetic cost relative to the {CNOT, Hadamard, T} baseline at comparable fidelity.

References

  1. Sebastiaan Fauquenot, Aritra Sarkar, Sebastian Feld, "Open and Closed Loop Approaches for Energy Efficient Quantum Optimal Control", Advanced Quantum Technologies 8, 2400690 (2025), DOI 10.1002/qute.202400690, https://doi.org/10.1002/qute.202400690; preprint arXiv:2411.06556 (2024), posted under the title "EO-GRAPE and EO-DRLPE: Open and Closed Loop Approaches for Energy Efficient Quantum Optimal Control", https://arxiv.org/abs/2411.06556
  2. Sebastian Deffner, "Energetic cost of Hamiltonian quantum gates", Europhysics Letters 134, 40002 (2021), DOI 10.1209/0295-5075/134/40002, https://doi.org/10.1209/0295-5075/134/40002
  3. Maxwell Aifer, Sebastian Deffner, "From quantum speed limits to energy-efficient quantum gates", New Journal of Physics 24, 055002 (2022), DOI 10.1088/1367-2630/ac6821, https://doi.org/10.1088/1367-2630/ac6821
  4. Stefanie Günther, N. Anders Petersson, "A practical approach to determine minimal quantum gate durations using amplitude-bounded quantum controls", AVS Quantum Science 5, 043802 (2023), DOI 10.1116/5.0173373, https://doi.org/10.1116/5.0173373; preprint arXiv:2307.13168, https://arxiv.org/abs/2307.13168
  5. Denis Janković, Saba Taherpour, Paul-Louis Etienney, Paul-Antoine Hervieux, Christoph Wolf, "Probing the Weak-Driving Quantum Speed Limit via Drift-Aware Shooting Methods", arXiv:2606.22212 (2026), https://arxiv.org/abs/2606.22212
  6. Francesco Preti, Tommaso Calarco, Felix Motzoi, "Continuous quantum gate sets and pulse class meta-optimization", arXiv:2203.13594 (2022), https://arxiv.org/abs/2203.13594
  7. Fernando Gago-Encinas, Monika Leibscher, Christiane P. Koch, "Graph test of controllability in qubit arrays: a systematic way to determine the minimum number of external controls", Quantum Science and Technology 8, 045002 (2023), DOI 10.1088/2058-9565/ace1a4, https://doi.org/10.1088/2058-9565/ace1a4; preprint arXiv:2212.04828, https://arxiv.org/abs/2212.04828
  8. Sibasish Mishra, Aritra Sarkar, Sebastian Feld, "EQISA: Energy-efficient Quantum Instruction Set Architecture using Sparse Dictionary Learning", arXiv:2603.20646 (2026), https://arxiv.org/abs/2603.20646
  9. Aritra Sarkar, Akash Kundu, Matthew Steinberg, Sibasish Mishra, Sebastiaan Fauquenot, Tamal Acharya, Jarosław A. Miszczak, Sebastian Feld, "YAQQ: yet another quantum quantizer design space exploration of quantum gate sets using novelty search", New Journal of Physics 28 (2026), DOI 10.1088/1367-2630/ae5a54, https://doi.org/10.1088/1367-2630/ae5a54
  10. Tan Van Vu, Tomotaka Kuwahara, Keiji Saito, "Fidelity-dissipation relations in quantum gates", Physical Review Research 6, 033225 (2024), DOI 10.1103/PhysRevResearch.6.033225, https://doi.org/10.1103/PhysRevResearch.6.033225
  11. Tan Van Vu, Keiji Saito, "Finite-Time Quantum Landauer Principle and Quantum Coherence", Physical Review Letters 128, 010602 (2022), DOI 10.1103/PhysRevLett.128.010602, https://doi.org/10.1103/PhysRevLett.128.010602
  12. Yonggang Peng, Zhenbo Ni, Yujun Zheng, "Energetic cost as a consequence of parallel transporting speed limit", Physica Scripta 99, 095125 (2024), DOI 10.1088/1402-4896/ad6e2a, https://doi.org/10.1088/1402-4896/ad6e2a
  13. Daoyi Dong, Chuan-Cun Shu, Jiangchao Chen, Xi Xing, Hailan Ma, Yu Guo, Herschel Rabitz, "Learning Control of Quantum Systems Using Frequency-Domain Optimization Algorithms", IEEE Transactions on Control Systems Technology 29(4), 1791-1798 (2021), DOI 10.1109/TCST.2020.3018500, https://doi.org/10.1109/TCST.2020.3018500
  14. Aritra Sarkar, "DeComp2: Description Complexity aware Decomposition", arXiv:2607.23990 (2026), https://arxiv.org/abs/2607.23990