低温下近似量子配分函数的高效算法

论量子配分函数的复杂性

On the complexity of quantum partition functions

本文研究量子多体系统配分函数与自由能近似的计算复杂性，提出稠密 2-局域哈密顿量的经典 poly(n) 时间算法，并证明自由能近似与 QMA 见证态计数等量子近似计数问题多项式时间等价。

Simpler (classical) and faster (quantum) algorithms for Gibbs partition functions

We present classical and quantum algorithms for approximating partition functions of classical Hamiltonians at a given temperature. Our work has two main contributions: first, we modify the classical algorithm of Štefankovič, Vempala and Vigoda (\emph{J.~ACM}, 56(3), 2009) to improve its sample complexity; second, we quantize this new algorithm, improving upon the previously fastest quantum algorithm for this problem, due to Harrow and Wei (SODA 2020). The conventional approach to estimating partition functions requires approximating the means of Gibbs distributions at a set of inverse tempera…

当少即是多：用块结构近似量子几何张量

When Less is More: Approximating the Quantum Geometric Tensor with Block Structures

提出一种按网络层划分的块对角量子几何张量近似方法，在保留关键曲率的同时消除跨层噪声，显著提升了神经量子态优化的收敛速度、能量和稳定性。

分块展开用于近似张量网络收缩

Partitioned Expansions for Approximate Tensor Network Contractions

分块展开方法通过将张量网络分割为更易计算的部分，提高了近似收缩的精度，尤其在BP失效时仍能保持有效性。

Minimizing State Preparations in Variational Quantum Eigensolver by Partitioning into Commuting Families

Variational quantum eigensolver (VQE) is a promising algorithm suitable for near-term quantum machines. VQE aims to approximate the lowest eigenvalue of an exponentially sized matrix in polynomial time. It minimizes quantum resource requirements both by co-processing with a classical processor and by structuring computation into many subproblems. Each quantum subproblem involves a separate state preparation terminated by the measurement of one Pauli string. However, the number of such Pauli strings scales as $N^4$ for typical problems of interest--a daunting growth rate that poses a serious li…

Approximating Gibbs states of local Hamiltonians efficiently with PEPS

We analyze the error of approximating Gibbs states of local quantum spin Hamiltonians on lattices with Projected Entangled Pair States (PEPS) as a function of the bond dimension ($D$), temperature ($β^{-1}$), and system size ($N$). First, we introduce a compression method in which the bond dimension scales as $D=e^{O(\log^2(N/ε))}$ if $β<O(\log (N))$. Second, building on the work of Hastings [Phys. Rev. B 73, 085115 (2006)], we derive a polynomial scaling relation, $D=\left(N/ε\right)^{O(β)}$. This implies that the manifold of PEPS forms an efficient representation of Gibbs states of local qua…

Partition Function Estimation: Quantum and Quantum-Inspired Algorithms

We present two algorithms, one quantum and one classical, for estimating partition functions of quantum spin Hamiltonians. The former is a DQC1 (Deterministic quantum computation with one clean qubit) algorithm, and the first such for complex temperatures. The latter, for real temperatures, achieves performance comparable to a state-of-the-art DQC1 algorithm [Chowdhury et al. Phys. Rev. A 103, 032422 (2021)]. Both our algorithms take as input the Hamiltonian decomposed as a linear combination Pauli operators. We show this decomposition to be DQC1-hard for a given Hamiltonian, providing new ins…

Approximating quantum thermodynamic properties using DFT

The fabrication, utilisation, and efficiency of quantum technologies rely on a good understanding of quantum thermodynamic properties. Many-body systems are often used as hardware for these quantum devices, but interactions between particles make the complexity of related calculations grow exponentially with the system size. Here we explore and systematically compare `simple' and `hybrid' approximations to the average work and entropy variation built on static density functional theory concepts. These approximations are computationally cheap and could be applied to large systems. We exemplify …

Tunneling catastrophes of the partition function

We use path-integrals to derive a general expression for the semiclassical approximation to the partition function of a one-dimensional quantum-mechanical system. Our expression depends solely on ordinary integrals which involve the potential. For high temperatures, the semiclassical expression is dominated by single closed paths. As we lower the temperature, new closed paths may appear, including tunneling paths. The transition from single to multiple-path regime corresponds to well-defined catastrophes. Tunneling sets in whenever they occur. Our formula fully accounts for this feature.

Quantum Force Induced on a Partition Wall in a Harmonic Potential

Boundary effects in quantum mechanics are examined by considering a partition wall inserted at the centre of a harmonic oscillator system. We put an equal number of particles on both sides of the impenetrable wall keeping the system under finite temperatures. When the wall admits distinct boundary conditions on the two sides, then a net force is induced on the wall. We study the temperature behaviour of the induced force both analytically and numerically under the combination of the Dirichlet and the Neumann conditions, and determine its scaling property for two statistical cases of the partic…