Dong An

The well-conditioned multi-product formula (MPF), proposed by [Low, Kliuchnikov, and Wiebe, 2019], is a simple high-order time-independent Hamiltonian simulation algorithm that implements a linear combination of standard product formulas of low order. While the MPF aims to simultaneously exploit commutator scaling among Hamiltonians and achieve near-optimal time and precision dependence, its lack of a rigorous error bound on the nested commutators renders its practical advantage ambiguous. In this work, we conduct a rigorous complexity analysis of the well-conditioned MPF, demonstrating explicit commutator scaling and near-optimal time and precision dependence at the same time. Using our improved complexity analysis, we present several applications of practical interest where the MPF based on a second-order product formula can achieve a polynomial speedup in both system size and evolution time, as well as an exponential speedup in precision, compared to second-order and even higher-order product formulas. Compared to post-Trotter methods, the MPF based on a second-order product formula can achieve polynomially better scaling in system size, with only poly-logarithmic overhead in evolution time and precision.

We extend the findings of (Quantum \textbf{9}, 1887 (2025)), which demonstrated that the discrete adiabatic quantum linear system solver exhibits constant factors approximately 1,200 times smaller in practice than previously estimated by worst-case bounds and about an order of magnitude more efficient than using a randomised approach from [arXiv:2305.11352]. In the present work, we introduce a comparison between the adiabatic-based quantum walk method and the more recent ``shortcut'' quantum linear system solver proposed in [arXiv:2406.12086], which also achieves the asymptotically optimal scaling $O(\kappa\log(1/\varepsilon))$, but with a better constant factor guarantee than the quantum walk method, especially when the solution norm is known. Specifically, we conduct a comprehensive numerical analysis contrasting the two methods in two regimes: when the norm of the solution is unknown and when it is known. Our results indicate that in cases where we know \emph{a priori} $\|x\|$, the shortcut method achieves a lower total cost than QW, providing a potential practical application for problems where the norm of the solution is available.

Solving linear ordinary differential equations (ODE) is one of the most promising applications for quantum computers to demonstrate exponential advantages. The challenge of designing a quantum ODE algorithm is how to embed non-unitary dynamics into intrinsically unitary quantum circuits. In this work, we propose a new quantum algorithm for solving ODEs by harnessing open quantum systems. Specifically, we propose a novel technique called non-diagonal density matrix encoding, which leverages the inherent non-unitary dynamics of Lindbladians to encode general linear ODEs into the non-diagonal blocks of density matrices. This framework enables us to design quantum algorithms with both theoretical simplicity and high performance. Combined with the state-of-the-art quantum Lindbladian simulation algorithms, our algorithm can outperform all existing quantum ODE algorithms and achieve near-optimal dependence on all parameters under a plausible input model. We also give applications of our algorithm including the Gibbs state preparations and the partition function estimations.

We investigate Lindbladian fast-forwarding and its applications to estimating Gibbs state properties. Fast-forwarding refers to the ability to simulate a system of time $t$ using significantly fewer than $t$ queries or circuit depth. While various Hamiltonian systems are known to circumvent the no fast-forwarding theorem, analogous results for dissipative dynamics, governed by Lindbladians, remain largely unexplored. We first present a quantum algorithm for simulating purely dissipative Lindbladians with unitary jump operators, achieving additive query complexity $ \mathcal{O}\left(t + \frac{\log(\varepsilon^{-1})}{\log\log(\varepsilon^{-1})}\right)$ up to error~$\varepsilon$, improving previous algorithms. When the jump operators have certain structures (i.e., block-diagonal Paulis), the algorithm can be modified to achieve exponential fast-forwarding, attaining circuit depth $\mathcal{O}\left(\log\left(t + \frac{\log(\varepsilon^{-1})}{\log\log(\varepsilon^{-1})}\right)\right)$, while preserving query complexity. Using these fast-forwarding techniques, we develop a quantum algorithm for estimating Gibbs state properties of the form $\langle \psi_1 | e^{-\beta(H + I)} | \psi_2 \rangle$, up to additive error $\epsilon$, with $H$ the Hamiltonian and $\beta$ the inverse temperature. For input states exhibiting certain coherence conditions ---e.g.,~$\langle 0|^{\otimes n} e^{-\beta(H + I)} |+\rangle^{\otimes n}$---our method achieves exponential improvement in complexity (measured by circuit depth), $\mathcal{O} (2^{-n/2} \epsilon^{-1} \log \beta ),$ compared to the quantum singular value transformation-based approach, with complexity $\tilde{\mathcal{O}} (\epsilon^{-1} \sqrt{\beta} )$. We show how to apply this exponential improvement to applications such as the ground state overlap testing and amplitude estimation. For general $| \psi_1 \rangle$ and $| \psi_2 \rangle$, we also show how the level of improvement is changed with the coherence resource in $| \psi_1 \rangle$ and $| \psi_2 \rangle$.