Selective decoupling in multi-level quantum systems by the SU(2) sign anomaly

1. Problem Statement

• Context: Dynamical Decoupling (DD) is a standard technique in quantum technologies used to mitigate noise by applying sequences of pulses to reverse the sign of specific Hamiltonian terms. While well-established for two-level systems (qubits), extending these strategies to multilevel systems (qudits, such as qutrits) is challenging.
• The Challenge: In multilevel systems, the goal is often to selectively decouple specific transitions (e.g., between ground state $|g\rangle$ and excited state $|e\rangle$) while preserving others (e.g., between $|e\rangle$ and $|f\rangle$).
• Limitation of Standard Approaches: Traditional geometric strategies require control Hamiltonians oriented in specific directions to address the "logic" basis directly. However, in many real-world quantum hardware platforms, such direct control is unavailable or limited.
• Objective: The authors aim to develop a flexible DD protocol that can selectively suppress unwanted couplings (off-diagonal elements) in a multilevel system without requiring direct control over the specific transition being decoupled.

2. Methodology

The authors propose a novel protocol leveraging the SU(2) sign anomaly inherent in multilevel systems.

• System Model:
  • A 3-Level System (3LS) with logic basis $\{|g\rangle, |e\rangle, |f\rangle\}$.
  • The target is to block-diagonalize the effective Hamiltonian, decoupling the subspace $\{|g\rangle\}$ from $\{|e\rangle, |f\rangle\}$, while preserving the internal dynamics of the $\{|e\rangle, |f\rangle\}$ subspace.
• The Core Mechanism (SU(2) Sign Anomaly):
  • Instead of applying pulses to the $|g\rangle \leftrightarrow |e\rangle$ transition (which is hard to control), the protocol applies $2\pi$ pulses to a different transition: the $|e\rangle \leftrightarrow |f\rangle$ subspace.
  • The rotation operator is $R = e^{-i\pi\sigma_{ef}}$, where $\sigma_{ef}$ is an SU(2) generator in the $\{|e\rangle, |f\rangle\}$ subspace.
  • The Anomaly: A $2\pi$ rotation in SU(2) introduces a phase factor of $e^{\pm i\pi} = -1$ for the rotated subspace relative to the unrotated state $|g\rangle$. This effectively flips the sign of the off-diagonal coupling terms ($g-e$) in the Hamiltonian when viewed in the rotating frame, while leaving other terms invariant.
• Sequence Design:
  • The total evolution time $T_f$ is divided into $n+1$ intervals separated by $n$ instantaneous $2\pi$ pulses.
  • The evolution alternates between free evolution under the original Hamiltonian $H$ and the rotated Hamiltonian $H_R = RHR^\dagger$.
• Analytical Framework:
  • The authors use the Magnus Expansion to derive the effective Hamiltonian ($H_{eff}$) of the sequence.
  • They analyze the expansion order by order ($\Omega_1, \Omega_2, \Omega_3$) to determine the optimal time intervals ($\delta_i$) that minimize unwanted couplings.
  • The condition for perfect decoupling is that the effective Hamiltonian approaches the target $H_T = \frac{1}{2}(H + H_R)$.

3. Key Results

• First-Order Decoupling:
  • To cancel the first-order term (average Hamiltonian), the sum of time intervals for even and odd steps must be equal ($\sum \delta_{even} = \sum \delta_{odd} = 1/2$).
• Second-Order Analysis ($n=2$ pulses):
  • For a 2-pulse sequence, the authors derived an exact solution: $\delta_0 = \delta_2 = 1/4$ and $\delta_1 = 1/2$. This sequence successfully cancels the unwanted coupling to the first order.
• Higher-Order Analysis ($n=4$ pulses):
  • For $n > 2$, the system of equations required to cancel all terms up to the third order of the Magnus expansion becomes over-constrained.
  • Key Finding: There is no real physical solution that exactly reproduces the target Hamiltonian for $n=4$ while satisfying all constraints simultaneously.
  • Relaxed Solution: By relaxing the requirement to preserve the full target Hamiltonian and focusing solely on vanishing the unwanted couplings, a class of solutions exists.
  • Uhrig Sequence Suitability: The authors discovered that the Uhrig Dynamical Decoupling (UDD) sequence, originally designed for pure dephasing in qubits, is suitable for this problem when implemented as $2\pi$ pulses. The UDD sequence effectively cancels the coupling between the $\{|g\rangle\}$ and $\{|e\rangle, |f\rangle\}$ subspaces.
• Approximation: While an exact solution for arbitrary $T_f$ does not exist, sequences can be engineered to achieve the desired decoupling up to order $T_f^2$.

4. Significance and Contributions

• New Control Paradigm: The paper introduces a strategy that bypasses the need for direct control over the transition to be decoupled. By exploiting the topological sign change of a $2\pi$ pulse in a different subspace, it achieves selective decoupling.
• Experimental Flexibility: This approach is highly relevant for experimental platforms where control is restricted to specific transitions (e.g., superconducting circuits or trapped ions with limited addressing capabilities).
• Generalization: While demonstrated on a 3-level system (qutrit), the framework is generalizable to $N$-level systems using the sign anomaly of SU($M$) subalgebras to suppress leakage from specific subspaces.
• Practical Applications:
  • Quantum Networking: Controlling internode interactions.
  • Decoherence Suppression: Actively suppressing noise affecting specific transitions.
  • Quantum Sensing: Enhancing sensing capabilities by decoupling specific environmental modes.
  • Ultrastrong Coupling: Modulating couplings in quantum architectures operating in the ultrastrong coupling regime.

5. Conclusion

The authors successfully demonstrate that the SU(2) sign anomaly can be harnessed to create a flexible, selective dynamical decoupling protocol for multilevel quantum systems. Although exact cancellation of all higher-order errors is mathematically impossible for certain pulse counts, the method allows for highly effective suppression of unwanted couplings (up to second order in time) using standard pulse sequences like Uhrig's, adapted for $2\pi$ rotations. This significantly expands the toolbox for quantum control in complex, multilevel hardware.