[Paper Review] Quantum algorithm for linear non-unitary dynamics with near-optimal dependence on all parameters

Dong An, Andrew M. Childs+1 more|arXiv (Cornell University)|Dec 6, 2023

Quantum Computing Algorithms and Architecture8 citations

TL;DR

The paper generalizes the linear combination of Hamiltonian simulation (LCHS) to express linear non-unitary evolution as a weighted sum of unitary evolutions, achieving near-optimal parameter scaling and optimal state preparation for solving linear ODEs on a quantum computer.

ABSTRACT

We introduce a family of identities that express general linear non-unitary evolution operators as a linear combination of unitary evolution operators, each solving a Hamiltonian simulation problem. This formulation can exponentially enhance the accuracy of the recently introduced linear combination of Hamiltonian simulation (LCHS) method [An, Liu, and Lin, Physical Review Letters, 2023]. For the first time, this approach enables quantum algorithms to solve linear differential equations with both optimal state preparation cost and near-optimal scaling in matrix queries on all parameters.

Motivation & Objective

• Motivate efficient quantum simulation of large-scale linear ODEs beyond Hamiltonian dynamics.
• Introduce a generalized LCHS framework that expresses non-unitary propagators as weighted integrals over unitary evolutions.
• Achieve near-optimal (poly-logarithmic) dependence on precision and optimal state preparation cost.
• Enable Gibbs state preparation and time-independent case improvements via the new LCHS kernel functions.

Proposed method

• Represent general linear non-unitary evolution as a kernel-weighted integral of unitary evolutions via a kernel f(k) and a decomposition A(t)=L(t)+iH(t).
• Prove a generalized LCHS identity: T e^{-∫ A(s) ds} = ∫ f(k)/(1−ik) T e^{-i∫(kL(s)+H(s)) ds} dk under mild analyticity, decay, and normalization conditions.
• Introduce a specific kernel family f(z)=1/(2π e^{-2^{β}} e^{(1+iz)^{β}}) with β∈(0,1) to obtain near-exponential decay in k, allowing smaller truncation K=O((log(1/ε))^{1/β}).
• Discretize the integral via composite Gaussian quadrature and simulate each unitary evolution with truncated Dyson series; combine via quantum LCU (linear combination of unitaries).
• Leverage improved kernel to attain near-optimal scaling in matrix queries and optimal state preparation cost across time-dependent and time-independent A(t).
• Extend to Gibbs state preparation and discuss a hybrid, near-term implementation path.

Experimental results

Research questions

• RQ1 How can non-unitary linear dynamics with A(t) and inhomogeneous term b(t) be represented as a sum over unitary evolutions?
• RQ2 What kernel functions enable near-exponential decay to improve precision scaling in LCHS-based quantum ODE algorithms?
• RQ3 What are the resulting query complexities for general time-dependent and time-independent linear ODEs when using the improved LCHS framework?
• RQ4 Can the improved LCHS framework support Gibbs state preparation and hybrid (near-term) implementations?
• RQ5 What are the open theoretical questions and potential extensions of the LCHS formula to broader stability or eigenvalue transformations?

Key findings

• A generalized LCHS formula expresses non-unitary propagators as a weighted integral of unitary evolutions with a kernel f(k).
• Using a kernel with near-exponential decay dramatically reduces the truncation parameter and improves precision scaling compared to the original LCHS.
• The improved LCHS algorithm solves linear ODEs with error ε using a matrix-query complexity that scales as Õ((||u0||+||b||_L1)/||u(T)|| · α_A T · (log(1/ε))^{1+1/β}) and optimal state preparation cost.
• For time-independent A, the complexity improves further to Õ((||u0||/||u(T)||) α_A T (log(1/ε))^{1/β}) with QSP/QSVT-based Hamiltonian simulation, approaching additive scaling with ε.
• The framework enables Gibbs state preparation with nearly linear dependence on γ and α_L and poly-log dependence on 1/ε.
• A hybrid implementation reduces ancilla requirements by sampling LCU terms and estimating observables via Hadamard tests and amplitude estimation.