The rapid progress of noisy intermediate-scale quantum (NISQ) computing underscores the need to test and evaluate new devices and applications. Quantum chemistry is a key application area for these devices, and therefore serves as an important benchmark for current and future quantum computer performance. Previous benchmarks in this field have focused on variational methods for computing ground and excited states of various molecules, including a benchmarking suite focused on performance of computing ground states for alkali-hydrides under an array of error mitigation methods. Here, we outline state of the art methods to reach chemical accuracy in hybrid quantum-classical electronic structure calculations of alkali hydride molecules on NISQ devices from IBM. We demonstrate how to extend the reach of variational eigensolvers with new symmetry preserving Ansätze. Next, we outline how to use quantum imaginary time evolution and Lanczos as a complementary method to variational techniques, highlighting the advantages of each approach. Finally, we demonstrate a new error mitigation method which uses systematic error cancellation via hidden inverse gate constructions, improving the performance of typical variational algorithms. These results show that electronic structure calculations have advanced rapidly, to routine chemical accuracy for simple molecules, from their inception on quantum computers a few short years ago, and they point to further rapid progress to larger molecules as the power of NISQ devices grows.
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）
We show in full generality that time-correlation function of a physical observable analytically continued to imaginary time is a tau-function of integrable Toda hierarchy. Using this relation we show that the singularity along the imaginary axis, which is a generic behavior for quantum non-integrable many-body system, is due to delocalization in Krylov space.
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…
Imaginary-time evolution (ITE) on a quantum computer is a promising formalism for obtaining the ground state of a quantum system. As a kind of it, the probabilistic ITE (PITE) takes advantage of measurements to implement the nonunitary operations. We propose a new approach of PITE which requires only a single ancillary qubit. Under a practical approximation, the circuit is constructed from the forward and backward real-time evolution (RTE) gates as black boxes, generated by the original many-qubit Hamiltonian. All the efficient unitary quantum algorithms for RTE proposed so far and those in th…
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…
Accurately treating electron correlation in the wavefunction is a key challenge for both classical and quantum computational chemistry. Classical methods have been developed which explicitly account for this correlation by incorporating inter-electronic distances into the wavefunction. The transcorrelated method transfers this explicit correlation from the wavefunction to a transformed, non-Hermitian Hamiltonian, whose right-hand eigenvectors become easier to obtain than those of the original Hamiltonian. In this work, we show that the transcorrelated method can reduce the resources required t…
Hardware-Efficient Ground-State Preparation using Variational Imaginary-Time Majorana Evolution
IQM Quantum Computers GmbH（IQM量子计算机有限公司）；RPTU Kaiserslautern-Landau（莱布尼茨技术大学凯泽斯劳滕-兰道分校）；Universidad Autónoma de Madrid（马德里自治大学）；Technical University of Munich（慕尼黑工业大学）
We introduce a novel open-source software package QForte, a comprehensive development tool for new quantum simulation algorithms. QForte incorporates functionality for handling molecular Hamiltonians, fermionic encoding, ansatz construction, time evolution, and state-vector simulation, requiring only a classical electronic structure package as a dependency. QForte also contains black-box implementations of a wide variety of quantum algorithms including (but not limited to): variational and projective quantum eigensolvers, adaptive eigensolvers, quantum imaginary time evolution, quantum Krylov …
Quantum computers have been widely speculated to offer significant advantages in obtaining the ground state of difficult Hamiltonian in chemistry and physics. In this work, we first propose a Lyapunov control-inspired strategy to accelerate the well-established imaginary-time method for ground-state preparation. We also dig for the source of acceleration of the imaginary-time process under Lyapunov control with theoretical understanding and dynamic process visualization. To make the method accessible in the noisy intermediate-scale quantum era, we further propose a variational form of the algo…
Evolution in imaginary time is a prominent technique for finding the ground state of quantum many-body systems, and the heart of a number of numerical methods that have been used with great success in quantum chemistry, condensed matter and nuclear physics. We propose an algorithm to implement imaginary time propagation on a quantum computer. Our algorithm is devised in the context of an efficient encoding into an optimized gate, drawing on the underlying characteristics of the quantum device, of a unitary operation in an extended Hilbert space. However, we proved that for simple problems it c…
Convergence and efficiency proof of quantum imaginary time evolution for bounded order systems
Northeastern University - London（伦敦东北大学）；Khoury College of Computer Sciences, Northeastern University（东北大学科里计算机科学学院）；The Cyprus Institute（塞浦路斯研究所）；Deutsches Elektronen-Synchrotron DESY（德国电子同步加速器中心）
We propose a quantum-classical hybrid algorithm of the power method, here dubbed as quantum power method, to evaluate $\hat{\cal H}^n |ψ\rangle$ with quantum computers, where $n$ is a nonnegative integer, $\hat{\cal H}$ is a time-independent Hamiltonian of interest, and $|ψ\rangle$ is a quantum state. We show that the number of gates required for approximating $\hat{\cal H}^n$ scales linearly in the power and the number of qubits, making it a promising application for near term quantum computers. Using numerical simulation, we show that the power method can control systematic errors in approxi…
Conformal truncation is a powerful numerical method for solving generic strongly-coupled quantum field theories based on purely field-theoretic technics without introducing lattice regularization. We discuss possible speedups for performing those computations using quantum devices, with the help of near-term and future quantum algorithms. We show that this construction is very similar to quantum simulation problems appearing in quantum chemistry (which are widely investigated in quantum information science), and the renormalization group theory provides a field theory interpretation of conform…
Nonorthogonal variational quantum simulation for quantum chemistry
Hefei National Research Center for Physical Sciences at the Microscale and School of Physical Sciences, University of Science and Technology of China（合肥微尺度物质科学国家研究中心和中国科学技术大学物理学院）；Shanghai Research Center for Quantum Science and CAS Center for Excellence in Quantum Information and Quantum Physics, University of Science and Technology of China（上海量子科学研究中心和中国科学院量子信息重点实验室（中国科学技术大学））；Hefei National Laboratory, University of Science and Technology of China（合肥国家实验室（中国科学技术大学））；Key Laboratory of Theoretical and Computational Photochemistry, Ministry of Education, College of Chemistry, Beijing Normal University（教育部理论化学与计算化学重点实验室和北京师范大学化学学院）；Center on Frontiers of Computing Studies, School of Computer Science, Peking University（前沿计算研究中心和北京大学生命科学学院）
The generating function of a Hamiltonian $H$ is defined as $F(t)=\langle e^{-itH}\rangle$, where $t$ is the time and where the expectation value is taken on a given initial quantum state. This function gives access to the different moments of the Hamiltonian $\langle H^{K}\rangle$ at various orders $K$. The real and imaginary parts of $F(t)$ can be respectively evaluated on quantum computers using one extra ancillary qubit with a set of measurement for each value of the time $t$. The low cost in terms of qubits renders it very attractive in the near term period where the number of qubits is li…
In this work we intend to study a class of time-dependent quantum systems with non-Hermitian Hamiltonians, particularly those whose Hermitian counterpart are important for the comprehension of posed problems in quantum optics and quantum chemistry, which consists of an oscillator with time-dependent mass and frequency under the action of a time-dependent imaginary potential. The propagator for a general time-dependence of the parameters and the wave-functions are obtained explicitly for constant frequency and mass and a linear time-dependence in the potential. The wave-functions are used to ob…
An Analytic Theory of Quantum Imaginary Time Evolution
Department of Computer Science, University of Pittsburgh, Pittsburgh, PA 15260, USA（计算机科学系，匹兹堡大学）；Ming Hsieh Department of Electrical and Computer Engineering, University of Southern California, Los Angeles, CA 90089, USA（明希德电气与计算机工程系，南加州大学）；Department of Physics and Astronomy, University of Southern California, Los Angeles, CA 90089, USA（物理与天文学系，南加州大学）
Variational quantum algorithms are a promising hybrid framework for solving chemistry and physics problems with broad applicability to optimization as well. They are particularly well suited for noisy intermediate scale quantum (NISQ) computers. In this paper, we describe the unitary block optimization scheme (UBOS) and apply it to two variational quantum algorithms: the variational quantum eigensolver (VQE) and variational time evolution. The goal of VQE is to optimize a classically intractable parameterized quantum wave function to target a physical state of a Hamiltonian or solve an optimiz…
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）
Optimization algorithms play a central role in chemistry since optimization is the computational keystone of most molecular and electronic structure calculations. Herein, we introduce the iterative power algorithm (IPA) for global optimization and a formal proof of convergence for both discrete and continuous global search problems, which is essential for applications in chemistry such as molecular geometry optimization. IPA implements the power iteration method in quantics tensor train (QTT) representations. Analogous to the imaginary time propagation method with infinite mass, IPA starts wit…
Finding the ground state of a Hamiltonian system is of great significance in many-body quantum physics and quantum chemistry. We propose an improved iterative quantum algorithm to prepare the ground state of a Hamiltonian. The crucial point is to optimize a cost function on the state space via the quantum gradient descent (QGD) implemented on quantum devices. We provide practical guideline on the selection of the learning rate in QGD by finding a fundamental upper bound and establishing a relationship between our algorithm and the first-order approximation of the imaginary time evolution. Furt…
Adaptive time Compressed QITE (ACQ) and its geometrical interpretation
Universidad CEU Cardenal Herrera（塞乌卡迪纳尔埃雷拉大学）；Universitat Politècnica de València（瓦伦西亚理工大学）；Universitat de València-CSIC（瓦伦西亚大学-西班牙国家研究委员会）；The Barcelona Institute of Science and Technology（巴塞罗那科学与技术研究所）；Griffith University（格里菲斯大学）；VTT Technical Research Centre of Finland（芬兰技术研究中心）；Leibniz Institute of Hannover（汉诺威莱布尼茨研究所）；TU Berlin（柏林工业大学）
Google Brain（谷歌大脑）；Institute for Quantum Computing, University of Waterloo（滑铁卢大学量子计算研究所）；UC Berkeley（加州大学伯克利分校）；Stanford University（斯坦福大学）；Princeton University（普林斯顿大学）；University of Waterloo（滑铁卢大学）
The dual tasks of quantum Hamiltonian learning and quantum Gibbs sampling are relevant to many important problems in physics and chemistry. In the low temperature regime, algorithms for these tasks often suffer from intractabilities, for example from poor sample- or time-complexity. With the aim of addressing such intractabilities, we introduce a generalization of quantum natural gradient descent to parameterized mixed states, as well as provide a robust first-order approximating algorithm, Quantum-Probabilistic Mirror Descent. We prove data sample efficiency for the dual tasks using tools fro…
