The problem of simulating sparse Hamiltonians on quantum computers is well studied. The evolution of a sparse N x N Hamiltonian H for time t can be simulated using O(||Ht||poly(log N)) operations, which is essentially optimal due to a no--fast-forwarding theorem. Here, we consider non-sparse Hamiltonians and show significant limitations on their simulation. We generalize the no--fast-forwarding theorem to dense Hamiltonians, ruling out generic simulations taking time o(||Ht||), even though ||H|| is not a unique measure of the size of a dense Hamiltonian $H$. We also present a stronger limitation ruling out the possibility of generic simulations taking time poly(||Ht||,log N), showing that known simulations based on discrete-time quantum walk cannot be dramatically improved in general. On the positive side, we show that some non-sparse Hamiltonians can be simulated efficiently, such as those with graphs of small arboricity.
Hybrid quantum-classical variational algorithms such as the variational quantum eigensolver (VQE) and the quantum approximate optimization algorithm (QAOA) are promising applications for noisy, intermediate-scale quantum (NISQ) computers. Both VQE and QAOA variationally extremize the expectation value of a Hamiltonian. All work to date on VQE and QAOA has been limited to Pauli representations of Hamiltonians. However, many cases exist in which a sparse representation of the Hamiltonian is known but there is no efficient Pauli representation. We extend VQE to general sparse Hamiltonians. We pro…
Quantum chemistry has been viewed as one of the potential early applications of quantum computing. Two techniques have been proposed for electronic structure calculations: (i) the variational quantum eigensolver and (ii) the phase-estimation algorithm. In both cases, the complexity of the problem increases for basis sets where either the Hamiltonian is not sparse, or it is sparse, but many orbitals are required to accurately describe the molecule of interest. In this work, we explore the possibility of mapping the molecular problem onto a sparse Hubbard-like Hamiltonian, which allows a Green's…
Estimation of a sparse multi-qubit Hamiltonian via compressed sensing
State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, University of Chinese Academy of Sciences（中国科学院大学数学科学学院）；College of Electronic and Information Engineering, Tongji University（同济大学电子与信息工程学院）；Shanghai Research Institute for Intelligent Autonomous Systems, Tongji University（同济大学上海智能无人驾驶系统研究院）；Institute for Advanced Study, Tongji University（同济大学高等研究院）；Department of Electrical and Electronic Engineering, University of Melbourne（墨尔本大学电气与电子工程系）；Laboratory of Quantum Information, University of Science and Technology of China（中国科学技术大学量子信息实验室）；CAS Center For Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China（中国科学技术大学中科院量子信息与量子物理卓越创新中心）；Hefei National Laboratory（合肥国家实验室）
We present a quantum algorithm for simulating the dynamics of Hamiltonians that are not necessarily sparse. Our algorithm is based on the input model where the entries of the Hamiltonian are stored in a data structure in a quantum random access memory (qRAM) which allows for the efficient preparation of states that encode the rows of the Hamiltonian. We use a linear combination of quantum walks to achieve poly-logarithmic dependence on precision. The time complexity of our algorithm, measured in terms of the circuit depth, is $O(t\sqrt{N}\|H\|\,\mathrm{polylog}(N, t\|H\|, 1/ε))$, where $t$ is …
Trans-dimensional Hamiltonian model selection and parameter estimation from sparse, noisy data
University of Chicago（芝加哥大学）；Argonne National Laboratory（阿贡国家实验室）；QuTech, Delft University of Technology（代尔夫特理工大学 QuTech）；Kavli Institute of Nanoscience, Delft University of Technology（代尔夫特理工大学 卡弗里纳米科学研究所）
Nearly optimal algorithms to learn sparse quantum Hamiltonians in physically motivated distances
Google Quantum AI（谷歌量子人工智能）；Scuola Normale Superiore（高等师范学院）；Centrum Wiskunde & Informatica (CWI)（数学与计算机科学中心）；QuSoft；Quriosity；Inria-Saclay（法国国家信息与自动化研究所萨克雷分中心）；ILLC, University of Amsterdam（阿姆斯特丹大学逻辑与计算中心）；NEST；Istituto Nanoscienze（纳米科学研究所）
Polynomial-time exact diagonalization via sparse guided eigenwalks
IBM Quantum, IBM T.J. Watson Research Center（IBM量子，IBM托马斯·J·沃森研究中心）；Department of Physics, Massachusetts Institute of Technology（麻省理工学院物理系）；NVIDIA Corporation（英伟达公司）；Center for Ultracold Atoms and Research Laboratory of Electronics, Massachusetts Institute of Technology（麻省理工学院超冷原子中心与电子研究实验室）
Heisenberg-limited Hamiltonian learning without short-time control
School of Computational Sciences, Korea Advanced Institute of Science and Technology（韩国科学技术院计算科学学院）；Institute of Computer Technology, Seoul National University（首尔大学计算机技术研究所）；Department of Physics, Korea Advanced Institute of Science and Technology（韩国科学技术院物理系）
We describe methods for simulating general second-quantized Hamiltonians using the compact encoding, in which qubit states encode only the occupied modes in physical occupation number basis states. These methods apply to second-quantized Hamiltonians composed of a constant number of interactions, i.e., linear combinations of ladder operator monomials of fixed form. Compact encoding leads to qubit requirements that are optimal up to logarithmic factors. We show how to use sparse Hamiltonian simulation methods for second-quantized Hamiltonians in compact encoding, give explicit implementations f…
Observation of Improved Accuracy over Classical Sparse Ground-State Solvers using a Quantum Computer
IBM Quantum（IBM量子）；T. J. Watson Research Center（T.J.沃森研究中心）；IBM Research Cambridge（IBM剑桥研究院）；IBM Research Silicon Valley（IBM硅谷研究院）；IBM Research Europe（IBM欧洲研究院）；Pritzker School of Molecular Engineering, The University of Chicago（芝加哥大学普里茨克分子工程学院）；IBM France Lab（IBM法国实验室）
We demonstrate a method that merges the quantum filter diagonalization (QFD) approach for hybrid quantum/classical solution of the time-independent electronic Schrödinger equation with a low-rank double factorization (DF) approach for the representation of the electronic Hamiltonian. In particular, we explore the use of sparse "compressed" double factorization (C-DF) truncation of the Hamiltonian within the time-propagation elements of QFD, while retaining a similarly compressed but numerically converged double-factorized representation of the Hamiltonian for the operator expectation values ne…
Evolutions of local Hamiltonians in short times are expected to remain local and thus limited. In this paper, we validate this intuition by proving some limitations on short-time evolutions of local time-dependent Hamiltonians. We show that the distribution of the measurement output of short-time (at most logarithmic) evolutions of local Hamiltonians are \emph{concentrated} and satisfy an \emph{isoperimetric inequality}. To showcase explicit applications of our results, we study the \textsc{MaxCut} problem and conclude that quantum annealing needs at least a run-time that scales logarithmicall…
For noisy intermediate-scale quantum (NISQ) devices only a moderate number of qubits with a limited coherence is available thus enabling only shallow circuits and a few time evolution steps in the currently performed quantum computations. Here, we present how to bypass this challenge in practical molecular chemistry simulations on NISQ devices by employing a classical-quantum hybrid algorithm allowing us to produce a sparse Hamiltonian which contains only $\mathcal{O}(n^2)$ terms in a Gaussian orbital basis when compared to the $\mathcal{O}(n^4)$ terms of a standard Hamiltonian, where $n$ is t…
