Energy shortcut of N-level quantum protocols by optimal control

1. Problem Statement

Quantum technologies (e.g., quantum computing, thermodynamics, and state engineering) rely heavily on controlling quantum systems. A standard method for accelerating quantum processes is Shortcut-to-Adiabaticity (STA), specifically the Counter-Diabatic (CD-STA) approach. CD-STA adds an auxiliary control field ($V_{CD}$) to the system Hamiltonian to force the system to follow an adiabatic trajectory at arbitrary speeds, thereby avoiding non-adiabatic transitions (quantum friction).

However, CD-STA suffers from a critical drawback: high energy cost. The additional drive required to strictly enforce the adiabatic trajectory at all times often demands significant power, leading to:

• High energy consumption for control devices.
• Unwanted dissipation and heating in the experimental setup.
• Increased work fluctuations.

The paper addresses the question: Can we achieve the same final quantum transformation as CD-STA but with a significantly lower energy cost? The authors argue that strictly following the adiabatic trajectory at every instant is an unnecessary constraint; only the initial and final states (or unitary transformation) matter.

2. Methodology

The authors introduce a new method named Quantum-Optimal-Shortcut-To-Energetics (QOSTE). The methodology combines Geometric Control Theory and Optimal Control Theory (OCT).

• Geometric Approach:

  • The problem is analyzed in a rotating frame defined by the original protocol Hamiltonian $H_0(t)$.
  • The energy cost of a control $V(t)$ is defined as the integral of the squared Frobenius norm: $C[V] = \int_0^{t_f} \|V(u)\|^2 du$.
  • Using the Cauchy-Schwarz inequality, the authors establish a lower bound for the energy cost: $C[V] \geq \frac{L^2}{t_f}$, where $L$ is the length of the trajectory in the unitary group $SU(N)$.
  • The minimum energy is achieved when the control drives the system along the geodesic (the shortest path) in the rotating frame.
  • The authors derive an explicit analytical solution for the optimal control $V_Q(t)$ that generates the target unitary transformation while following this geodesic.

• Optimal Control Theory (PMP & GRAPE):

  • The Pontryagin Maximum Principle (PMP) is used to rigorously prove that the geodesic solution minimizes the energy cost.
  • To address robustness (sensitivity to experimental uncertainties in Hamiltonian parameters or control amplitudes), the authors employ the GRAPE (Gradient Ascent Pulse Engineering) algorithm. This allows for the numerical optimization of controls that simultaneously maximize fidelity across a range of uncertainties and minimize energy.

• State-to-State vs. Unitary:

  • The method distinguishes between Unitary Transformations (mapping the entire basis) and State-to-State (STS) transfers (mapping only specific initial and final states). STS transfers are shown to be even less energetically demanding as they do not require controlling the relative phases of all basis states.

3. Key Contributions

1. Derivation of QOSTE: The paper provides the first analytical derivation of an energy-optimal control protocol ($V_Q$) for N-level quantum systems that achieves the same transformation as CD-STA but with minimal energy.
2. Geometric Insight: It establishes that the minimal energy cost is determined by the length of the geodesic in the rotating frame ($G_{0,t_f}$), whereas CD-STA is constrained by the longer adiabatic path length ($L_{0,t_f}$).
3. Scaling Law: The authors prove that for long control times ($t_f \to \infty$), the ratio of energy costs scales quadratically:
   $$\frac{C[V_{CD}]}{C[V_Q]} \propto t_f^2$$
   This indicates that CD-STA becomes exponentially more expensive than QOSTE as the operation time increases, contrary to intuition where slower operations usually cost less.
4. Robust QOSTE Design: By combining GRAPE with energy constraints, the authors design robust protocols that outperform standard CD-STA in both energy efficiency and robustness against parameter uncertainties.
5. Reduced Control Complexity: The method demonstrates that high-fidelity, low-energy transfers can be achieved with a reduced number of control fields (e.g., using only 3 controls for a 3-level system instead of the full set), which is crucial for practical hardware implementation.

4. Results and Benchmarks

The authors benchmarked QOSTE against CD-STA using two standard models:

• Landau-Zener Model (Qubit):

  • Scenario: Transferring a qubit from the ground to the excited state.
  • Result: For a specific non-adiabatic regime ($\omega t_f = 1$), the energy cost of CD-STA was $15.7\omega$, while QOSTE (Unitary) was $7.1\omega$, and QOSTE (STS) was only $2.8\omega$.
  • Robustness: While standard QOSTE was slightly less robust than CD-STA, the GRAPE-optimized robust QOSTE achieved significantly higher fidelity under control amplitude uncertainties ($\epsilon=0.15$) while maintaining a much lower energy cost than CD-STA.

• STIRAP (Three-Level System):

  • Scenario: Stimulated Raman Adiabatic Passage in a $\Lambda$-configuration.
  • Result: The CD drive required an energy cost of $0.4725 \Omega_0$. The QOSTE protocol reduced this to $0.01135 \Omega_0$, a reduction factor of ~42.
  • Control Reduction: Even when restricted to only 3 control fields (instead of the full set), the optimized QOSTE maintained an energy cost of $0.01138 \Omega_0$, nearly identical to the full-control theoretical minimum, proving the method's efficiency in resource-constrained scenarios.

5. Significance

• Energy Efficiency in Quantum Tech: As quantum technologies scale, the energy cost of control electronics and the associated cooling requirements become a bottleneck. QOSTE offers a fundamental reduction in this cost, addressing the "energy crisis" in quantum computing.
• Beyond Adiabaticity: The work challenges the dogma that adiabatic trajectories must be strictly followed to ensure robustness. It shows that optimal non-adiabatic paths can be both faster and more robust if designed correctly.
• Practical Implementation: The ability to design robust, low-energy protocols with a reduced number of control fields makes these methods highly applicable to current experimental setups (e.g., NMR, trapped ions, superconducting qubits) where control lines are limited.
• Theoretical Framework: The geometric interpretation of energy cost in the rotating frame provides a powerful tool for analyzing and designing future quantum control protocols, bridging the gap between quantum thermodynamics and optimal control theory.

In conclusion, the paper presents QOSTE as a superior alternative to CD-STA, offering a mathematically rigorous, energy-minimal, and robust framework for accelerating quantum processes in N-level systems.