Fast-forwardable Lindbladians imply quantum phase estimation

TL;DR

This work explicitly bridges the standard quantum limit-Heisenberg limit transition to the fast-forwarding of dissipative dynamics, and adopts the fast-forwarding algorithm for efficient Gibbs state preparation and demonstrates the counter-intuitive implication: the allowance of a quadratically accelerated decoherence effect under arbitrary Pauli noise.

Abstract

Quantum phase estimation (QPE) and Lindbladian dynamics are both foundational in quantum information science and central to quantum algorithm design. In this work, we bridge these two concepts: certain simple Lindbladian processes can be adapted to perform QPE-type tasks. However, unlike QPE, which achieves Heisenberg-limit scaling, these Lindbladian evolutions are restricted to standard quantum limit complexity. This indicates that, different from Hamiltonian dynamics, the natural dissipative evolution speed of such Lindbladians does not saturate the fundamental quantum limit, thereby suggesting the potential for quadratic fast-forwarding. We confirm this by presenting a quantum algorithm that simulates these Lindbladians for time $t$ within an error $\varepsilon$ using $\mathcal{O}\left(\sqrt{t\log(\varepsilon^{-1})}\right)$ cost, whose mechanism is fundamentally different from the fast-forwarding examples of Hamiltonian dynamics. As a bonus, this fast-forwarded simulation naturally serves as a new Heisenberg-limit QPE algorithm. Therefore, our work explicitly bridges the standard quantum limit-Heisenberg limit transition to the fast-forwarding of dissipative dynamics. We also adopt our fast-forwarding algorithm for efficient Gibbs state preparation and demonstrate the counter-intuitive implication: the allowance of a quadratically accelerated decoherence effect under arbitrary Pauli noise.

Figures (3)

• Figure 1: Comparison between Lindbladian fast-forwarding and Hamiltonian fast-forwarding. $H$ can be put in either Hamiltonian dynamics or the Lindbladian Eq. (\ref{['mainslme']}). Since Hamiltonian dynamics naturally achieves the Heisenberg limit, we can only expect the fast-forwarding for quite restricted Hamiltonians (green) whose special properties allow for surpassing the Heisenberg limit. In contrast, the natural evolution of the Lindbladian Eq. (\ref{['mainslme']}) only achieves the standard quantum limit, meaning improving to the Heisenberg limit is sufficient for fast-forwarding, which works for an arbitrary Hamiltonian (orange).
• Figure B.2: The quantum circuit of standard quantum phase estimation nielsen2010quantum.
• Figure E.3: The quantum circuit of our Lindbladian fast-forwarding algorithm for Eq. (\ref{['mainslme']}) in Theorem \ref{['the2']}. The fast-forwarding algorithm can also be used for QPE tasks with a Heisenberg-limit scaling (Theorem \ref{['the3']}).

Theorems & Definitions (11)

• Definition 1: Eigenvalue estimation
• Definition 2: Eigenstate preparation
• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Lemma 2
• proof
• Corollary 1