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Constructing Noise-Robust Quantum Gates via Pontryagin's Maximum Principle

Joshua Hanson, Dennis Lucarelli

TL;DR

This paper formulates a general PMP-based framework for designing smooth, noise-robust quantum gates in arbitrary $SU(N)$ systems by augmenting the state with error-curves from a Magnus expansion. Robustness is enforced by driving all error curves $\Omega_k^{(i_1,\dots,i_k)}$ to zero at the final time, transforming gate synthesis into a constrained motion-planning problem on $\mathcal{M}=SU(N)\times\mathfrak{su}(N)^p$ and solving it via PMP on Lie groups. The authors illustrate the method with two single-qubit examples, achieving first-order and third-order robustness against disturbances and demonstrating that the costate encapsulates the necessary pulse-envelope information, enabling efficient reconstruction of optimal controls. The approach offers a general, architecture-agnostic path to smooth, implementable noise-robust gates, with extensions to leakage suppression and in situ recalibration discussed as future directions.

Abstract

Reliable quantum information technologies depend on precise actuation and techniques to mitigate the effects of undesired disturbances such as environmental noise and imperfect calibration. In this work, we present a general framework based in geometric optimal control theory to synthesize smooth control pulses for implementing arbitrary noise-robust quantum gates. The methodology applies to generic unitary quantum dynamics with any number of qubits or energy levels, any number of control fields, and any number of disturbances, extending existing dynamical decoupling approaches that are only applicable for limited gate sets or small systems affected by one or two disturbances. The noise-suppressing controls are computed via indirect trajectory optimization based on Pontryagin's maximum principle, eliminating the need to make heuristic structural assumptions on parameterized pulse envelopes.

Constructing Noise-Robust Quantum Gates via Pontryagin's Maximum Principle

TL;DR

This paper formulates a general PMP-based framework for designing smooth, noise-robust quantum gates in arbitrary systems by augmenting the state with error-curves from a Magnus expansion. Robustness is enforced by driving all error curves to zero at the final time, transforming gate synthesis into a constrained motion-planning problem on and solving it via PMP on Lie groups. The authors illustrate the method with two single-qubit examples, achieving first-order and third-order robustness against disturbances and demonstrating that the costate encapsulates the necessary pulse-envelope information, enabling efficient reconstruction of optimal controls. The approach offers a general, architecture-agnostic path to smooth, implementable noise-robust gates, with extensions to leakage suppression and in situ recalibration discussed as future directions.

Abstract

Reliable quantum information technologies depend on precise actuation and techniques to mitigate the effects of undesired disturbances such as environmental noise and imperfect calibration. In this work, we present a general framework based in geometric optimal control theory to synthesize smooth control pulses for implementing arbitrary noise-robust quantum gates. The methodology applies to generic unitary quantum dynamics with any number of qubits or energy levels, any number of control fields, and any number of disturbances, extending existing dynamical decoupling approaches that are only applicable for limited gate sets or small systems affected by one or two disturbances. The noise-suppressing controls are computed via indirect trajectory optimization based on Pontryagin's maximum principle, eliminating the need to make heuristic structural assumptions on parameterized pulse envelopes.
Paper Structure (9 sections, 1 theorem, 62 equations, 8 figures)

This paper contains 9 sections, 1 theorem, 62 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Formulating the control problem
  3. The augmented control system
  4. The motion planning problem
  5. Pontryagin's maximum principle
  6. Right-trivialized Hamiltonian dynamics
  7. Examples
  8. Numerical results
  9. Discussion

Key Result

Theorem 1

Consider a control system $\dot{x} = f(x,u)$, $x \in \mathcal{M}$ subject to initial and final endpoint constraints $x(0) \in \mathcal{S}_0$, $x(T) \in \mathcal{S}_1$. Suppose $u^* : [0,T] \to {\mathbb R}^m$ is an optimal control policy in the sense of minimizing eq:cost_functional and $x^* : [0,T]

Figures (8)

  • Figure 1: In-phase and quadrature components ($u_1$ and $u_2$) of a pulse envelope implementing a Hadamard gate that is first-order robust to dephasing and multiplicative control errors.
  • Figure 2: Trajectory of the ideal evolution operator implementing a Hadamard gate that is first-order robust to dephasing and multiplicative control errors.
  • Figure 3: First-order dephasing and amplitude error curves for the Hadamard gate.
  • Figure 4: Gate infidelity as a function of the two disturbance magnitudes $\epsilon_1$ and $\epsilon_2$ for the controls shown in Figure \ref{['fig:ex1_control']}. The vanishing noise limit is in the lower-left region.
  • Figure 5: In-phase pulse envelope implementing a $\sqrt{X}$ gate that is third-order robust to dephasing.
  • ...and 3 more figures

Theorems & Definitions (1)

  • Theorem 1: Pontryagin's maximum principle on manifolds