[Paper Review] Single-ancilla ground state preparation via Lindbladians

Zhiyan Ding, Chi-Fang Chen+1 more|arXiv (Cornell University)|Aug 30, 2023

Quantum Computing Algorithms and ArchitectureComputer Science44 references3 citations

TL;DR

This paper proposes a fault-tolerant quantum algorithm for ground state preparation using a single ancilla qubit and a tailored Lindbladian dynamics, where the target ground state is the unique stationary state. The method enables efficient preparation even from initial states with zero overlap to the ground state, bypassing a key limitation of quantum phase estimation, and achieves near-optimal simulation cost via a discrete-time variant with polynomial scaling in system size and precision.

ABSTRACT

We design a quantum algorithm for ground state preparation in the early fault tolerant regime. As a Monte Carlo-style quantum algorithm, our method features a Lindbladian where the target state is stationary. The construction of this Lindbladian is algorithmic and should not be seen as a specific approximation to some weakly coupled system-bath dynamics in nature. Our algorithm can be implemented using just one ancilla qubit and efficiently simulated on a quantum computer. It can prepare the ground state even when the initial state has zero overlap with the ground state, bypassing the most significant limitation of methods like quantum phase estimation. As a variant, we also propose a discrete-time algorithm, demonstrating even better efficiency and providing a near-optimal simulation cost depending on the desired evolution time and precision. Numerical simulation using Ising and Hubbard models demonstrates the efficacy and applicability of our method.

Motivation & Objective

• To address the critical bottleneck of ground state preparation in early fault-tolerant quantum computing.
• To overcome the limitation of existing methods that require initial states with non-zero overlap to the ground state.
• To design a systematic, algorithmic Lindbladian generator that ensures the ground state is the unique stationary state, independent of physical system-bath coupling.
• To achieve efficient simulation with minimal qubit overhead, using only one ancilla qubit.
• To demonstrate near-optimal simulation cost through a discrete-time variant of the algorithm.

Proposed method

• The method employs a custom Lindbladian generator whose unique stationary state is the target ground state, constructed algorithmically rather than as a physical approximation.
• The Lindbladian dynamics are designed such that the ground state is asymptotically reached as the fixed point, regardless of initial state.
• The algorithm uses a single ancilla qubit to mediate the dissipative dynamics, enabling efficient implementation on a quantum computer.
• A discrete-time variant is introduced, which achieves better efficiency and near-optimal simulation cost scaling as O(T log(T/ϵ)/log log(T/ϵ)) with time T and precision ϵ.
• The method relies on a stochastic simulation scheme with time steps τ, where the evolution is approximated using random superoperators drawn from a distribution ΞA.
• Theoretical analysis establishes concentration of the stochastic trajectory around the deterministic dynamics, with error bounds depending on system size and simulation parameters.

Experimental results

Research questions

• RQ1 Can ground state preparation be achieved efficiently without requiring initial state overlap with the ground state?
• RQ2 Can a Lindbladian be systematically constructed such that the ground state is the unique stationary state, independent of physical bath assumptions?
• RQ3 What is the minimal qubit overhead required for fault-tolerant ground state preparation using dissipative dynamics?
• RQ4 How does the simulation cost scale with evolution time and precision in a fault-tolerant setting?
• RQ5 Can a discrete-time version of the Lindbladian algorithm achieve near-optimal complexity?

Key findings

• The algorithm prepares the ground state even when the initial state has zero overlap with the ground state, overcoming a major limitation of quantum phase estimation.
• The method uses only one ancilla qubit, significantly reducing qubit resource requirements compared to other approaches.
• The discrete-time variant achieves near-optimal simulation cost scaling as O(T log(T/ϵ)/log log(T/ϵ)) with respect to time T and precision ϵ.
• Numerical simulations on Ising and Hubbard models confirm the efficacy and applicability of the method across different quantum many-body systems.
• Theoretical analysis proves that the stochastic simulation trajectory concentrates around the deterministic dynamics with error bounds scaling as O(exp(CΞT)CΞ√(Tτ) + CFTτ), where CΞ and CF are system-dependent constants.
• The method demonstrates polynomial scaling in system size for mixing time under physically relevant assumptions, suggesting practical feasibility for realistic Hamiltonians.