Within the realm of early fault-tolerant quantum computing (EFTQC), quantum Krylov subspace diagonalization (QKSD) has emerged as a promising quantum algorithm for the approximate Hamiltonian diagonalization via projection onto the quantum Krylov subspace. However, the algorithm often requires solving an ill-conditioned generalized eigenvalue problem (GEVP) involving erroneous matrix pairs, which can significantly distort the solution. Since EFTQC assumes limited-scale error correction, finite sampling error becomes a dominant source of error in these matrices. This work focuses on quantifying sampling errors during the measurement of matrix element in the projected Hamiltonian examining two measurement approaches based on the Hamiltonian decompositions: the linear combination of unitaries and diagonalizable fragments. To reduce sampling error within a fixed budget of quantum circuit repetitions, we propose two measurement strategies: the shifting technique and coefficient splitting. The shifting technique eliminates redundant Hamiltonian components that annihilate either the bra or ket states, while coefficient splitting optimizes the measurement of common terms across different circuits. Numerical experiments with electronic structures of small molecules demonstrate the effectiveness of these strategies, reducing sampling costs by a factor of 20-500.
Evaluating Sample-Based Krylov Quantum Diagonalization for Heisenberg Models with Applications to Materials Science
Volkswagen AG（大众汽车）；Technical University of Munich（慕尼黑工业大学）；IBM Quantum（IBM量子）；RWTH Aachen（亚琛工业大学）；University of Ulm（乌尔姆大学）；IBM T.J. Watson Research Center（IBM T.J. 沃森研究中心）
The Lanczos algorithm is evaluated for solving the time-independent as well as the time-dependent Dirac equation with arbitrary electromagnetic fields. We demonstrate that the Lanczos algorithm can yield very precise eigenenergies and allows very precise time propagation of relativistic wave packets. The unboundedness of the Dirac Hamiltonian does not hinder the applicability of the Lanczos algorithm. As the Lanczos algorithm requires only matrix-vector products and inner products, which both can be efficiently parallelized, it is an ideal method for large-scale calculations. The excellent par…
Quantum Krylov subspace diagonalization (QKSD) algorithms provide a low-cost alternative to the conventional quantum phase estimation algorithm for estimating the ground and excited-state energies of a quantum many-body system. While QKSD algorithms typically rely on using the Hadamard test for estimating Krylov subspace matrix elements of the form, $\langle ϕ_i|e^{-i\hat{H}τ}|ϕ_j \rangle$, the associated quantum circuits require an ancilla qubit with controlled multi-qubit gates that can be quite costly for near-term quantum hardware. In this work, we show that a wide class of Hamiltonians re…
Integrable Floquet spin chains are known to host strong zero and $π$ modes which are boundary operators that respectively commute and anticommute with the Floquet unitary generating stroboscopic time-evolution, in addition to anticommuting with a discrete symmetry of the Floquet unitary. Thus the existence of strong modes imply a characteristic pairing structure of the full spectrum. Weak interactions modify the strong modes to almost strong modes that almost commute or anticommute with the Floquet unitary. Manifestations of strong and almost strong modes are presented in two different Krylov …
Green's Functions from Sample-based Krylov Quantum Diagonalization: An Impurity Solver for Dynamical Mean-Field Theory
Center for Computation and Technology, Louisiana State University（路易斯安那州立大学计算与技术中心）；Department of Physics, The Pennsylvania State University（宾夕法尼亚州立大学物理系）；Department of Physics and Astronomy, Louisiana State University（路易斯安那州立大学物理与天文学系）
We study the quantum dynamics of a simple translation invariant, center-of-mass (CoM) preserving model of interacting fermions in one dimension (1D), which arises in multiple experimentally realizable contexts. We show that this model naturally displays the phenomenology associated with fractonic systems, wherein single charges can only move by emitting dipoles. This allows us to demonstrate the rich Krylov fractured structure of this model, whose Hilbert space shatters into exponentially many dynamically disconnected subspaces. Focusing on exponentially large Krylov subspaces, we show that th…
The Krylov subspace method is a standard approach to approximate quantum evolution, allowing to treat systems with large Hilbert spaces. Although its application is general, and suitable for many-body systems, estimation of the committed error is involved. This makes it difficult to automate its use. In this paper, we solve this problem by realizing that such error can be regarded as a Loschmidt echo in a tight-binding Hamiltonian. We show that the different time-regimes of the approximation can be understood using simple physical ideas. More importantly, we obtain computationally cheap error …
Quantum subspace diagonalization (QSD) algorithms have emerged as a competitive family of algorithms that avoid many of the optimization pitfalls associated with parameterized quantum circuit algorithms. While the vast majority of the QSD algorithms have focused on solving the eigenpair problem for ground, excited-state, and thermal observable estimation, there has been a lot less work in considering QSD algorithms for the problem of quantum dynamical simulation. In this work, we propose several quantum Krylov fast-forwarding (QKFF) algorithms capable of predicting long-time dynamics well beyo…
Quantum chemistry with provable convergence via randomized sample-based Krylov quantum diagonalization
IBM Quantum, IBM Research Europe - Zurich（IBM量子，IBM欧洲苏黎世研究实验室）；Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院物理研究所）；Institute for Theoretical Physics, ETH Zürich（苏黎世联邦理工学院理论物理研究所）；The Hartree Centre, STFC（哈特里中心，STFC）；Yusuf Hamied Department of Chemistry, University of Cambridge（剑桥大学尤素福·哈米德化学系）；Max Planck Institute for Solid State Research（马克斯·普朗克固体研究所）；IBM T.J. Watson Research Center（IBM托马斯·J·沃森研究中心）
The Gradient Ascent Pulse Engineering (GRAPE) is a celebrated control algorithm with excellent converging rates, owing to a piece-wise-constant ansatz for the control function that allows for cheap objective gradients. However, the computational effort involved in the exact simulation of quantum dynamics quickly becomes a bottleneck limiting the control of large systems. In this paper, we propose a modified version of GRAPE that uses Krylov approximations to deal efficiently with high-dimensional state spaces. Even though the number of parameters required by an arbitrary control task scales li…
We propose an improved scheme to do the time dependent variational principle (TDVP) in finite matrix product states (MPS) for two-dimensional systems or one-dimensional systems with long range interactions. We present a method to represent the time-evolving state in a MPS with its basis enriched by state-averaging with global Krylov vectors. We show that the projection error is significantly reduced so that precise time evolution can still be obtained even if a larger time step is used. Combined with the one-site TDVP, our approach provides a way to dynamically increase the bond dimension whil…
Krylov Complexity Under Hamiltonian Deformations and Toda Flows
University of Luxembourg（卢森堡大学）；Mie University（三重大学）；Kanazawa University（金泽大学）；Vrije Universiteit Brussel (VUB)（布鲁塞尔自由大学）；Yukawa Institute for Theoretical Physics, Kyoto University（京都大学汤川理论物理研究所）；RIKEN Center for Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS)（理化学研究所跨学科理论与数学科学中心）；Donostia International Physics Center（圣塞巴斯蒂安国际物理中心）
Quantum Chaos and Spread of States in Krylov Subspace: A Topical Review
Asia Pacific Center for Theoretical Physics（亚太理论物理中心）；Pohang University of Science and Technology（浦项科技大学）；Instituto de Física Teórica UAM/CSIC（马德里自治大学/西班牙国家科学研究委员会理论物理研究所）
Technical University of Munich（慕尼黑工业大学）；TUM School of Natural Sciences, Physics Department（TUM自然科学学院物理系）；Walther-Meißner-Institut, Bayerische Akademie der Wissenschaften（瓦尔特·迈斯纳研究所，巴伐利亚科学院）；Munich Center for Quantum Science and Technology (MCQST)（慕尼黑量子科学与技术中心）；Department of Physics and Materials Science, University of Luxembourg（卢森堡大学物理与材料科学系）；Donostia International Physics Center（圣塞巴斯蒂安国际物理中心）；Department of Applied Mathematics and Theoretical Physics, University of Cambridge（剑桥大学应用数学与理论物理系）
