Variational quantum eigensolver~(VQE) typically optimizes variational parameters in a quantum circuit to prepare eigenstates for a quantum system. Its applications to many problems may involve a group of Hamiltonians, e.g., Hamiltonian of a molecule is a function of nuclear configurations. In this paper, we incorporate derivatives of Hamiltonian into VQE and develop some hybrid quantum-classical algorithms, which explores both Hamiltonian and wavefunction spaces for optimization. Aiming for solving quantum chemistry problems more efficiently, we first propose mutual gradient descent algorithm for geometry optimization by updating parameters of Hamiltonian and wavefunction alternatively, which shows a rapid convergence towards equilibrium structures of molecules. We then establish differential equations that governs how optimized variational parameters of wavefunction change with intrinsic parameters of the Hamiltonian, which can speed up calculation of energy potential surface. Our studies suggest a direction of hybrid quantum-classical algorithm for solving quantum systems more efficiently by considering spaces of both Hamiltonian and wavefunction.
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…
Traditional methods in quantum chemistry rely on Hartree-Fock-based Slater-determinant (SD) representations, whose underlying zeroth-order picture assumes separability by particle. Here, we explore a radically different approach, based on separability by Cartesian component, rather than by particle [J. Chem. Phys., 2018, 148, 104101]. The approach appears to be very well suited for 3D grid-based methods in quantum chemistry, and thereby also for so-called "first-quantized" quantum computing. We first present an overview of the approach as implemented on classical computers, including numerical…
Quantum annealing devices have been subject to various analyses in order to classify their usefulness for practical applications. While it has been successfully proven that such systems can in general be used for solving combinatorial optimization problems, they have not been used to solve chemistry applications. In this paper we apply a mapping, put forward by Xia et al. (The Journal of Physical Chemistry B 122.13 (2017): 3384-3395.), from a quantum chemistry Hamiltonian to an Ising spin glass formulation and find the ground state energy with a quantum annealer. Additionally we investigate th…
InstaDeep（英斯塔迪普）；University of Oxford（牛津大学）；Russian Quantum Center（俄罗斯量子中心）
In recent years, neural network quantum states (NNQS) have emerged as powerful tools for the study of quantum many-body systems. Electronic structure calculations are one such canonical many-body problem that have attracted significant research efforts spanning multiple decades, whilst only recently being attempted with NNQS. However, the complex non-local interactions and high sample complexity are significant challenges that call for bespoke solutions. Here, we parameterise the electronic wavefunction with a novel autoregressive neural network (ARN) that permits highly efficient and scalable…
In this work we investigate methods to improve the efficiency and scalability of quantum algorithms for quantum chemistry applications. We propose a transformation of the electronic structure Hamiltonian in the second quantization framework into the particle-hole (p/h) picture, which offers a better starting point for the expansion of the trial wavefunction. The state of the molecular system at study is parametrized in a way to efficiently explore the sector of the molecular Fock space that contains the desired solution. To this end, we explore several trial wavefunctions to identify the most …
A new approach to find exact solutions to one--dimensional quantum mechanical systems is devised. The scheme is based on the introduction of a potential function for the wavefunction, and the equation it satisfies. We recover known solutions as well as to get new ones for both free and interacting particles with wavefunctions having vanishing and non--vanishing Bohm potentials. For most of the potentials, no solutions to the Schrödinger equation produce a vanishing Bohm potential. A (large but) restricted family of potentials allows the existence of particular solutions for which the Bohm pote…
Quantum ground-state problems are computationally hard problems; for general many-body Hamiltonians, there is no classical or quantum algorithm known to be able to solve them efficiently. Nevertheless, if a trial wavefunction approximating the ground state is available, as often happens for many problems in physics and chemistry, a quantum computer could employ this trial wavefunction to project the ground state by means of the phase estimation algorithm (PEA). We performed an experimental realization of this idea by implementing a variational-wavefunction approach to solve the ground-state pr…
Quantum chemistry often considers atoms and molecules with non-zero spin. In such cases, the need for proper spin functions results in the theory of configuration state functions. Here, we consider the construction of such wavefunctions using the symmetric group and more specifically Young projectors. We discuss the formalism and detail an example to illustrate the theory. Additionally, we consider the pros and cons of specific implementations of spin symmetry in quantum simulation.
In this paper we design and use two Deep Learning models to generate the ground and excited wavefunctions of different Hamiltonians suitable for the study the vibrations of molecular systems. The generated neural networks are trained with Hamiltonians that have analytical solutions, and ask the network to generalize these solutions to more complex Hamiltonian functions. This approach allows to reproduce the excited vibrational wavefunctions of different molecular potentials. All methodologies used here are data-driven, therefore they do not assume any information about the underlying physical …
Quantum mechanics is an outstandingly successful description of nature, underpinning fields from biology through chemistry to physics. At its heart is the quantum wavefunction, the central tool for describing quantum systems. Yet it is still unclear what the wavefunction actually is: does it merely represent our limited knowledge of a system, or is it an element of reality? Recent no-go theorems argued that if there was any underlying reality to start with, the wavefunction must be real. However, that conclusion relied on debatable assumptions, without which a partial knowledge interpretation …
Is there a single linearly evolving Wave Function of the Universe that is able to lead to all the nonlinearities we see around us? This proposal seems a priori highly implausible. I claim that instead, in the real Universe, generically only local wave functions exist. Non-local wave functions occur for carefully engineered contexts such as Bell experiments, but there is no single wave function for a cat or similar macroscopic objects such as a brain. Contextual wave function collapse leads to a defensible version of the Copenhagen interpretation, where classical macro levels provide the contex…
Spin vs. position conjugation in quantum simulations with atoms: application to quantum chemistry
Quantum Technology Centre, M.V. Lomonosov Moscow State University（莫斯科国立大学量子技术中心）；St. Petersburg State University（圣彼得堡国立大学）；Russian Quantum Center（俄罗斯量子中心）；Centre for Interdisciplinary Basic Research, HSE University（高等经济学院跨学科基础研究系）；Department of Optics, Palacký University（帕拉茨基大学光学系）；Department of Physics, Old Dominion University（老道明大学物理系）
We propose a method to obtain Floquet states---also known as light-induced states---and their quasi-energies from real-time wavefunctions without solving the Floquet equation. This is useful for the analysis of various phenomena in time-dependent quantum dynamics if the Hamiltonian is not strictly periodic, as in short laser pulses, for instance. There, the population of the Floquet states depends on the pulse form and is automatically contained in the real-time wavefunction but not in the standard Floquet approach. Several examples in the area of intense laser-atom interaction are exemplarily…
In the first days of quantum mechanics Dirac pointed out an analogy between the time-dependent coefficients of an expansion of the Schrödinger equation and the classical position and momentum variables solving Hamilton's equations. Here it is shown that the analogy can be made an equivalence in that, in principle, systems of classical oscillators can be constructed whose position and momenta variables form time-dependent amplitudes which are identical to the complex quantum amplitudes of the coupled wavefunction of an N-level quantum system with real coupling matrix elements. Hence classical m…
Ab initio electronic excited state calculations are necessary for the quantitative study of photochemical reactions, but their accurate computation on classical computers is plagued by prohibitive scaling. The Variational Quantum Deflation (VQD) is an extension of the Variational Quantum Eigensolver (VQE) for calculating electronic excited state energies, and has the potential to address some of these scaling challenges using quantum computers. However, quantum computers available in the near term can only support a limited number of circuit operations, so reducing the quantum computational co…
A unitary coupled-cluster (UCC) form for the wavefunction in the variational quantum eigensolver has been suggested as a systematic way to go beyond the mean-field approximation and include electron correlation in solving quantum chemistry problems on a quantum computer. Although being exact in the limit of including all possible coupled-cluster excitations, practically, the accuracy of this approach depends on how many and what kind of terms are included in the wavefunction parametrization. Another difficulty of UCC is a growth of the number of simultaneously entangled qubits even at the fixe…
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）
For a Bose condensate in a double-well potential or with two Josephson-coupled internal states, the condensate wavefunction is a superposition. Here we consider coupling two such Bose condensates, and suggest the existence of a joint condensate wavefunction, which is in general a superposition of all products of the bases condensate wavefunctions of the two condensates. The corresponding many-body state is a product of such superposed wavefunctions, with appropriate symmetrization. These states may be potentially useful for quantum computation. There may be robustness and stability due to macr…
It is shown that it is possible to construct the quantum wave functions for non-separable but integrable two-dimensional Hamiltonian systems, by solving suitable Dirichlet boundary values problems inside and outside the regions spanned by particular families of classical trajectories, in one-to-one correspondence with the quantum state. The method is applied both to the Schrodinger equation, and to the quantum Hamilton-Jacobi equation. The boundary values are obtained by integrating the one-dim equations on the caustics arcs enveloping the classical trajectories. This approach gives the same r…
Two-photon photoassociation of hot magnesium atoms by femtosecond laser pulses, creating electronically excited magnesium dimer molecules, is studied from first principles, combining \textit{ab initio} quantum chemistry and molecular quantum dynamics. This theoretical framework allows for rationalizing the generation of molecular rovibrational coherence from thermally hot atoms [L. Rybak \textit{et al.}, Phys. Rev. Lett. {\bf 107}, 273001 (2011)]. Random phase thermal wave functions are employed to model the thermal ensemble of hot colliding atoms. Comparing two different choices of basis func…
Point-group symmetry analysis of many-electron wavefunctions on a quantum computer
JSR Corporation（杰士创株式会社）；Keio University（庆应义塾大学）；Deloitte Tohmatsu LLC（德勤都马茨有限责任公司）；TCG Centres for Research and Education in Science and Technology（TCG科学与技术研究与教育中心）；Mitsubishi Chemical Corporation（三菱化学株式会社）；IBM Quantum, IBM Research–Tokyo（IBM量子，IBM研究-东京）
Quantum chemistry provides a target for quantum simulation of considerable scientific interest and industrial importance. The majority of algorithms to date have been based on a second-quantized representation of the electronic structure Hamiltonian - necessitating qubit requirements that scale linearly with the number of orbitals. The scaling of the number of gates for such methods, while polynomial, presents some serious experimental challenges. However, because the number of electrons is a good quantum number for the electronic structure problem it is unnecessary to store the full Fock spac…
Digital quantum computers provide a computational framework for solving the Schrödinger equation for a variety of many-particle systems. Quantum computing algorithms for the quantum simulation of these systems have recently witnessed remarkable growth, notwithstanding the limitations of existing quantum hardware, especially as a tool for electronic structure computations in molecules. In this review, we provide a self-contained introduction to emerging algorithms for the simulation of Hamiltonian dynamics and eigenstates, with emphasis on their applications to the electronic structure in molec…
Excited states of molecules lie in the heart of photochemistry and chemical reactions. The recent development in quantum computational chemistry leads to inventions of a variety of algorithms that calculate the excited states of molecules on near-term quantum computers, but they require more computational burdens than the algorithms for calculating the ground states. In this study, we propose a scheme of supervised quantum machine learning which predicts the excited-state properties of molecules only from their ground state wavefunction resulting in reducing the computational cost for calculat…
A balanced description of both static and dynamic correlations in electronic systems with nearly degenerate low-lying states presents a challenge for multi-configurational methods on classical computers. We present here a quantum algorithm utilizing the action of correlating cluster operators to provide high-quality wavefunction ansätze employing a non-orthogonal multireference basis that captures a significant portion of the exact wavefunction in a highly compact manner, and that allows computation of the resulting energies and wavefunctions at polynomial cost with a quantum computer. This en…
The speculation that gravity is the key to solving the quantum measurement problem has been alive for decades, without any convincing demonstration of a solution. One necessary factor in the relevant proposals is that the gravitational energy of mutual interaction, which scales quadratically with the mass, facilitates the spontaneous collapse of the wavefunctions in spatially separated superpositions. Relying on a simple physical input from electrodynamics, supported by robust first principle calculations, we show that the speculations connecting gravity and the hypothetical spontaneous collap…
Quantum computing is moving beyond its early stage and seeking for commercial applications in chemical and biomedical sciences. In the current noisy intermediate-scale quantum computing era, quantum resource is too scarce to support these explorations. Therefore, it is valuable to emulate quantum computing on classical computers for developing quantum algorithms and validating quantum hardware. However, existing simulators mostly suffer from the memory bottleneck so developing the approaches for large-scale quantum chemistry calculations remains challenging. Here we demonstrate a high-performa…
Over the last few decades, quantum chemistry has progressed through the development of computational methods based on modern digital computers. However, these methods can hardly fulfill the exponentially-growing resource requirements when applied to large quantum systems. As pointed out by Feynman, this restriction is intrinsic to all computational models based on classical physics. Recently, the rapid advancement of trapped-ion technologies has opened new possibilities for quantum control and quantum simulations. Here, we present an efficient toolkit that exploits both the internal and motion…
Last years witnessed a remarkable interest in application of quantum computing for solving problems in quantum chemistry more efficiently than classical computers allow. Very recently, even first proof-of-principle experimental realizations have been reported. However, so far only the non-relativistic regime (i.e. Schroedinger equation) has been explored, while it is well known that relativistic effects can be very important in chemistry. In this letter we present the first quantum algorithm for relativistic computations of molecular energies. We show how to efficiently solve the eigenproblem …
Quantum algorithms for quantum chemistry map the dynamics of electrons in a molecule to the dynamics of a coupled spin system. To reach chemical accuracy for interesting molecules, a large number of quantum gates must be applied which implies the need for quantum error correction and fault-tolerant quantum computation. Arbitrary fault-tolerant operations can be constructed from a small, universal set of fault-tolerant operations by gate compilation. Quantum chemistry algorithms are compiled by decomposing the dynamics of the coupled spin-system using a Trotter formula, synthesizing the decompo…
One of the most promising suggested applications of quantum computing is solving classically intractable chemistry problems. This may help to answer unresolved questions about phenomena like: high temperature superconductivity, solid-state physics, transition metal catalysis, or certain biochemical reactions. In turn, this increased understanding may help us to refine, and perhaps even one day design, new compounds of scientific and industrial importance. However, building a sufficiently large quantum computer will be a difficult scientific challenge. As a result, developments that enable thes…
We propose a modification of the Variational Quantum Eigensolver algorithm for electronic structure optimization using quantum computers, named non-unitary Variational Quantum Eigensolver (nu-VQE), in which a non-unitary operator is combined with the original system Hamiltonian leading to a new variational problem with a simplified wavefunction Ansatz. In the present work, we use, as non-unitary operator, the Jastrow factor, inspired from classical Quantum Monte Carlo techniques for simulation of strongly correlated electrons. The method is applied to prototypical molecular Hamiltonians for wh…
We introduce novel algorithms for the quantum simulation of molecular systems which are asymptotically more efficient than those based on the Trotter-Suzuki decomposition. We present the first application of a recently developed technique for simulating Hamiltonian evolution using a truncated Taylor series to obtain logarithmic scaling with the inverse of the desired precision, an exponential improvement over all prior methods. The two algorithms developed in this work rely on a second quantized encoding of the wavefunction in which the state of an $N$ spin-orbital system is encoded in ${\cal …
A method is proposed for extracting ionization amplitudes from the solution of the time-dependent Schrödinger equation (TDSE) describing a system in a time-dependent external field. The method is a hybrid of two earlier developed methods, the time-dependent surface flux (t-SURFF) method and the method using the propagated wavepacket as the source term in a time-independent driven Schrödinger equation with the field-free Hamiltonian. It is demonstrated that the method combines the advantages of the parent ones and allows the extraction of ionization amplitudes by solving the TDSE within a small…
