## No file available

To read the file of this research,

you can request a copy directly from the authors.

Preprints and early-stage research may not have been peer reviewed yet.

Current quantum computers are limited in the number of qubits and coherence time, constraining the algorithms executable with sufficient fidelity. Variational quantum eigensolver (VQE) is an algorithm to find an approximate ground state of a quantum system and expected to work on even such a device. The deep VQE [K. Fujii, et al., arXiv:2007.10917] is an extension of the original VQE algorithm, which takes a divide-and-conquer approach to relax the hardware requirement. While the deep VQE is successfully applied for spin models and periodic material, its validity on a molecule, where the Hamiltonian is highly non-local in the qubit basis, is still unexplored. Here, we discuss the performance of the deep VQE algorithm applied to quantum chemistry problems. Specifically, we examine different subspace forming methods and compare their accuracy and complexity on a ten H-atom tree-like molecule as well as a 13 H-atom version. Additionally, we propose multiple methods to lower the number of qubits required to calculate the ground state of a molecule. We find that the deep VQE can simulate the electron-correlation energy of the ground-state to an error of below 1%, thus helping us to reach chemical accuracy in some cases. The accuracy differences and qubits reduction highlights the basis creation method's impact on the deep VQE.

To read the file of this research,

you can request a copy directly from the authors.

ResearchGate has not been able to resolve any citations for this publication.

We propose a divide-and-conquer method for the quantum-classical hybrid algorithm to solve larger problems with small-scale quantum computers. Specifically, we concatenate a variational quantum eigensolver (VQE) with a reduction in the system dimension, where the interactions between divided subsystems are taken as an effective Hamiltonian expanded by the reduced basis. Then the effective Hamiltonian is further solved by the VQE, which we call deep VQE. Deep VQE allows us to apply quantum-classical hybrid algorithms on small-scale quantum computers to large systems with strong intrasubsystem interactions and weak intersubsystem interactions, or strongly correlated spin models on large regular lattices. As proof-of-principle numerical demonstrations, we use the proposed method for quasi-one-dimensional models, including one-dimensionally coupled 12-qubit Heisenberg antiferromagnetic models on kagome lattices as well as two-dimensional Heisenberg antiferromagnetic models on square lattices. The largest problem size of 64 qubits is solved by simulating 20-qubit quantum computers with a reasonably good accuracy approximately a few %. The proposed scheme enables us to handle the problems of >1000 qubits by concatenating VQEs with a few tens of qubits. While it is unclear how accurate ground-state energy can be obtained for such a large system, our numerical results on a 64-qubit system suggest that deep VQE provides a good approximation (discrepancy within a few percent) and has room for further improvement. Therefore, deep VQE provides us a promising pathway to solve practically important problems on noisy intermediate-scale quantum computers.

Quantum computers are promising for simulations of chemical and physical systems, but the limited capabilities of today’s quantum processors permit only small, and often approximate, simulations. Here we present a method, classical entanglement forging, that harnesses classical resources to capture quantum correlations and double the size of the system that can be simulated on quantum hardware. Shifting some of the computation to classical postprocessing allows us to represent ten spin orbitals of the water molecule on five qubits of an IBM Quantum processor in the most accurate variational simulation of the H2O ground-state energy using quantum hardware to date. We discuss conditions for applicability of classical entanglement forging and present a roadmap for scaling to larger problems.

A programmable quantum device that has a large number of qubits without fault-tolerance has emerged recently. Variational quantum eigensolver (VQE) is one of the most promising ways to utilize the computational power of such devices to solve problems in condensed matter physics and quantum chemistry. As the size of the current quantum devices is still not large for rivaling classical computers at solving practical problems, Fujii et al. proposed a method called “Deep VQE”, which can provide the ground state of a given quantum system with the smaller number of qubits by combining the VQE and the technique of coarse graining [K. Fujii, K. Mitarai, W. Mizukami, and Y. O. Nakagawa, arXiv:2007.10917]. In this paper, we extend the original proposal of Deep VQE to obtain the excited states and apply it to quantum chemistry calculation of a periodic material, which is one of the most impactful applications of the VQE. We first propose a modified scheme to construct quantum states for coarse graining in Deep VQE to obtain the excited states. We also present a method to avoid a problem of meaningless eigenvalues in the original Deep VQE without restricting variational quantum states. Finally, we classically simulate our modified Deep VQE for quantum chemistry calculation of a periodic hydrogen chain as a typical periodic material. Our method reproduces the ground-state energy and the first-excited-state energy with the errors up to O(1)% despite the decrease in the number of qubits required for the calculation by two or four compared with the naive VQE. Our result will serve as a beacon for tackling quantum chemistry problems with classically-intractable sizes by smaller quantum devices in the near future.

To explore the possibilities of a near-term intermediate-scale quantum algorithm and long-term fault-tolerant quantum computing, a fast and versatile quantum circuit simulator is needed. Here, we introduce Qulacs, a fast simulator for quantum circuits intended for research purpose. We show the main concepts of Qulacs, explain how to use its features via examples, describe numerical techniques to speed-up simulation, and demonstrate its performance with numerical benchmarks.

Quantum error correction protects fragile quantum information by encoding it into a larger quantum system1,2. These extra degrees of freedom enable the detection and correction of errors, but also increase the control complexity of the encoded logical qubit. Fault-tolerant circuits contain the spread of errors while controlling the logical qubit, and are essential for realizing error suppression in practice3–6. Although fault-tolerant design works in principle, it has not previously been demonstrated in an error-corrected physical system with native noise characteristics. Here we experimentally demonstrate fault-tolerant circuits for the preparation, measurement, rotation and stabilizer measurement of a Bacon–Shor logical qubit using 13 trapped ion qubits. When we compare these fault-tolerant protocols to non-fault-tolerant protocols, we see significant reductions in the error rates of the logical primitives in the presence of noise. The result of fault-tolerant design is an average state preparation and measurement error of 0.6 per cent and a Clifford gate error of 0.3 per cent after offline error correction. In addition, we prepare magic states with fidelities that exceed the distillation threshold7, demonstrating all of the key single-qubit ingredients required for universal fault-tolerant control. These results demonstrate that fault-tolerant circuits enable highly accurate logical primitives in current quantum systems. With improved two-qubit gates and the use of intermediate measurements, a stabilized logical qubit can be achieved. Fault-tolerant circuits for the control of a logical qubit encoded in 13 trapped ion qubits through a Bacon–Shor quantum error correction code are demonstrated.

Recently, several quantum machine learning algorithms have been proposed that may offer quantum speed-ups over their classical counterparts. Most of these algorithms are either heuristic or assume that data can be accessed quantum-mechanically, making it unclear whether a quantum advantage can be proven without resorting to strong assumptions. Here we construct a classification problem with which we can rigorously show that heuristic quantum kernel methods can provide an end-to-end quantum speed-up with only classical access to data. To prove the quantum speed-up, we construct a family of datasets and show that no classical learner can classify the data inverse-polynomially better than random guessing, assuming the widely believed hardness of the discrete logarithm problem. Furthermore, we construct a family of parameterized unitary circuits, which can be efficiently implemented on a fault-tolerant quantum computer, and use them to map the data samples to a quantum feature space and estimate the kernel entries. The resulting quantum classifier achieves high accuracy and is robust against additive errors in the kernel entries that arise from finite sampling statistics. Many quantum machine learning algorithms have been proposed, but it is typically unknown whether they would outperform classical methods on practical devices. A specially constructed algorithm shows that a formal quantum advantage is possible.

We improve the quality of quantum circuits on superconducting
quantum computing systems, as measured by the
quantum volume, with a combination of dynamical decoupling,
compiler optimizations, shorter two-qubit gates, and excited state
promoted readout. This result shows that the path to larger
quantum volume systems requires the simultaneous increase of
coherence, control gate fidelities, measurement fidelities, and
smarter software which takes into account hardware details,
thereby demonstrating the need to continue to co-design the
software and hardware stack for the foreseeable future.

Quantum simulation of chemistry and materials is predicted to be an important application for both near-term and fault-tolerant quantum devices. However, at present, developing and studying algorithms for these problems can be difficult due to the prohibitive amount of domain knowledge required in both the area of chemistry and quantum algorithms. To help bridge this gap and open the field to more researchers, we have developed the OpenFermion software package (www.openfermion.org). OpenFermion is an open-source software library written largely in Python under an Apache 2.0 license, aimed at enabling the simulation of fermionic and bosonic models and quantum chemistry problems on quantum hardware. Beginning with an interface to common electronic structure packages, it simplifies the translation between a molecular specification and a quantum circuit for solving or studying the electronic structure problem on a quantum computer, minimizing the amount of domain expertise required to enter the field. The package is designed to be extensible and robust, maintaining high software standards in documentation and testing. This release paper outlines the key motivations behind design choices in OpenFermion and discusses some basic OpenFermion functionality which we believe will aid the community in the development of better quantum algorithms and tools for this exciting area of research.

SciPy is an open-source scientific computing library for the Python programming language. Since its initial release in 2001, SciPy has become a de facto standard for leveraging scientific algorithms in Python, with over 600 unique code contributors, thousands of dependent packages, over 100,000 dependent repositories and millions of downloads per year. In this work, we provide an overview of the capabilities and development practices of SciPy 1.0 and highlight some recent technical developments. This Perspective describes the development and capabilities of SciPy 1.0, an open source scientific computing library for the Python programming language.

The variational quantum eigensolver (VQE), a variational algorithm to obtain an approximated ground state of a given Hamiltonian, is an appealing application of near-term quantum computers. To extend the framework to excited states, we here propose an algorithm, the subspace-search variational quantum eigensolver (SSVQE). This algorithm searches a low-energy subspace by supplying orthogonal input states to the variational ansatz and relies on the unitarity of transformations to ensure the orthogonality of the output states. The kth excited state is obtained as the highest-energy state in the low-energy subspace. The proposed algorithm consists only of two parameter optimization procedures and does not employ any ancilla qubits. The avoidance of the estimation of the inner product and the small number of procedures required are considerable improvements from the existing proposals for excited states, making our proposal an improved near-term quantum algorithm. We further generalize the SSVQE to obtain all excited states up to the kth by only a single optimization procedure. From numerical simulations, we verify the proposed algorithms. This work extends the applicable domain of the VQE to excited states and their related properties as a transition amplitude without sacrificing any of its feasibility. Moreover, the proposed variational subspace search, which generalizes the state search problem to the search of a unitary mapping to a specific subspace, itself would be useful for various quantum information processing methods such as finding a protected subspace or a good variational quantum error correction code.

The promise of quantum computers is that certain computational tasks might be executed exponentially faster on a quantum processor than on a classical processor¹. A fundamental challenge is to build a high-fidelity processor capable of running quantum algorithms in an exponentially large computational space. Here we report the use of a processor with programmable superconducting qubits2,3,4,5,6,7 to create quantum states on 53 qubits, corresponding to a computational state-space of dimension 2⁵³ (about 10¹⁶). Measurements from repeated experiments sample the resulting probability distribution, which we verify using classical simulations. Our Sycamore processor takes about 200 seconds to sample one instance of a quantum circuit a million times—our benchmarks currently indicate that the equivalent task for a state-of-the-art classical supercomputer would take approximately 10,000 years. This dramatic increase in speed compared to all known classical algorithms is an experimental realization of quantum supremacy8,9,10,11,12,13,14 for this specific computational task, heralding a much-anticipated computing paradigm.

Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away --- we should regard it as a significant step toward the more powerful quantum technologies of the future. Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.

Python‐based simulations of chemistry framework (P y SCF) is a general‐purpose electronic structure platform designed from the ground up to emphasize code simplicity, so as to facilitate new method development and enable flexible computational workflows. The package provides a wide range of tools to support simulations of finite‐size systems, extended systems with periodic boundary conditions, low‐dimensional periodic systems, and custom Hamiltonians, using mean‐field and post‐mean‐field methods with standard Gaussian basis functions. To ensure ease of extensibility, P y SCF uses the Python language to implement almost all of its features, while computationally critical paths are implemented with heavily optimized C routines. Using this combined Python/C implementation, the package is as efficient as the best existing C or Fortran‐based quantum chemistry programs. In this paper, we document the capabilities and design philosophy of the current version of the P y SCF package. WIREs Comput Mol Sci 2018, 8:e1340. doi: 10.1002/wcms.1340
This article is categorized under: Structure and Mechanism > Computational Materials Science
Electronic Structure Theory > Ab Initio Electronic Structure Methods
Software > Quantum Chemistry

Dendrimers are nano-sized, radially symmetric molecules with well-defined, homogeneous, and monodisperse structure that has a typically symmetric core, an inner shell, and an outer shell. Their three traditional macromolecular architectural classes are broadly recognized to generate rather polydisperse products of different molecular weights. A variety of dendrimers exist, and each has biological properties such as polyvalency, self-assembling, electrostatic interactions, chemical stability, low cytotoxicity, and solubility. These varied characteristics make dendrimers a good choice in the medical field, and this review covers their diverse applications.

Quantum computers promise to efficiently solve important problems that
are intractable on a conventional computer. For quantum systems, where
the dimension of the problem space grows exponentially, finding the
eigenvalues of certain operators is one such intractable problem and
remains a fundamental challenge. The quantum phase estimation algorithm
can efficiently find the eigenvalue of a given eigenvector but requires
fully coherent evolution. We present an alternative approach that
greatly reduces the requirements for coherent evolution and we combine
this method with a new approach to state preparation based on ans\"atze
and classical optimization. We have implemented the algorithm by
combining a small-scale photonic quantum processor with a conventional
computer. We experimentally demonstrate the feasibility of this approach
with an example from quantum chemistry: calculating the ground state
molecular energy for He-H+, to within chemical accuracy. The proposed
approach, by drastically reducing the coherence time requirements,
enhances the potential of the quantum resources available today and in
the near future.

Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging and complex landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chemistry, such as the electronic structure of molecules. In the past two decades, significant advances have been made in developing algorithms and physical hardware for quantum computing, heralding a revolution in simulation of quantum systems. This Review provides an overview of the algorithms and results that are relevant for quantum chemistry. The intended audience is both quantum chemists who seek to learn more about quantum computing and quantum computing researchers who would like to explore applications in quantum chemistry.

An n-qubit quantum circuit performs a unitary operation on an exponentially large, 2ⁿ-dimensional, Hilbert space, which is a major source of quantum speed-ups. We develop a new “Quantum singular value transformation” algorithm that can directly harness the advantages of exponential dimensionality by applying polynomial transformations to the singular values of a block of a unitary operator. The transformations are realized by quantum circuits with a very simple structure - typically using only a constant number of ancilla qubits - leading to optimal algorithms with appealing constant factors. We show that our framework allows describing many quantum algorithms on a high level, and enables remarkably concise proofs for many prominent quantum algorithms, ranging from optimal Hamiltonian simulation to various quantum machine learning applications. We also devise a new singular vector transformation algorithm, describe how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum, and show how to efficiently implement principal component regression. Finally, we also prove a quantum lower bound on spectral transformations.

Variational quantum eigensolver (VQE) is an efficient computational method promising chemical accuracy in electronic structure calculations on a universal-gate quantum computer. However, such a simple task as computing the electronic energy of a hydrogen molecular cation, H+2 , is not possible for a general VQE protocol because the calculation will invariably collapse to a lower energy of the corresponding neutral form, H2. The origin of the problem is that VQE effectively performs an unconstrained energy optimization in the Fock space of the original electronic problem. We show how this can be avoided by introducing necessary constraints directing VQE to the electronic state of interest. The proposed constrained VQE can find an electronic state with a certain number of electrons, spin, or any other property. Moreover, the new algorithm naturally removes unphysical kinks in potential energy surfaces, which frequently appeared in the regular VQE and required significant additional quantum resources for their removal. We demonstrate performance of the constrained VQE by simulating potential energy surfaces of various states of H2 and H2O on Rigetti Computing Inc’s 19Q-Acorn quantum processor.

Harrow, Hassidim, and Lloyd showed that for a suitably specified $N \times N$
matrix $A$ and $N$-dimensional vector $\vec{b}$, there is a quantum algorithm
that outputs a quantum state proportional to the solution of the linear system
of equations $A\vec{x}=\vec{b}$. If $A$ is sparse and well-conditioned, their
algorithm runs in time $\mathrm{poly}(\log N, 1/\epsilon)$, where $\epsilon$ is
the desired precision in the output state. We improve this to an algorithm
whose running time is polynomial in $\log(1/\epsilon)$, exponentially improving
the dependence on precision while keeping essentially the same dependence on
other parameters. Our algorithm is based on a general technique for
implementing any operator with a suitable Fourier or Chebyshev series
representation. This allows us to bypass the quantum phase estimation
algorithm, whose dependence on $\epsilon$ is prohibitive.

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time by at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and which have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.

We investigate tree tensor network states for quantum chemistry. Tree tensor network states represent one of the simplest generalizations of matrix product states and the density matrix renormalization group. While matrix product states encode a one-dimensional entanglement structure, tree tensor network states encode a tree entanglement structure, allowing for a more flexible description of general molecules. We describe an optimal tree tensor network state algorithm for quantum chemistry. We introduce the concept of half-renormalization which greatly improves the efficiency of the calculations. Using our efficient formulation we demonstrate the strengths and weaknesses of tree tensor network states versus matrix product states. We carry out benchmark calculations both on tree systems (hydrogen trees and π-conjugated dendrimers) as well as non-tree molecules (hydrogen chains, nitrogen dimer, and chromium dimer). In general, tree tensor network states require much fewer renormalized states to achieve the same accuracy as matrix product states. In non-tree molecules, whether this translates into a computational savings is system dependent, due to the higher prefactor and computational scaling associated with tree algorithms. In tree like molecules, tree network states are easily superior to matrix product states. As an illustration, our largest dendrimer calculation with tree tensor network states correlates 110 electrons in 110 active orbitals.

Research work conducted in the field of fragmentation methods, a route to accurate calculations on large systems, is presented. The restricted variational space (RVS) analysis and the constrained space orbital variations (CSOV) method improve on the KM scheme by employing fully antisymmetrized intermediate wave functions. Wu et al. developed a density-based energy decomposition (EDA), in which the energies of the intermediate states are calculated using the densities of the fragments, rather than their wave functions. In 2008, Xie et al. modified the formulation of X-Pol to obtain rigorously analytic gradients. In 2002, Inadomi et al. proposed the FMO-MO method, which according to the classification of fragment methods as discussed above, belongs to a different category from the rest of the FMO methods. Bettens et al., have developed an energy based fragmentation method based on the idea of isodesmic reactions.

Solving linear systems of equations is a common problem that arises both on its own and as a subroutine in more complex problems: given a matrix A and a vector b(-->), find a vector x(-->) such that Ax(-->) = b(-->). We consider the case where one does not need to know the solution x(-->) itself, but rather an approximation of the expectation value of some operator associated with x(-->), e.g., x(-->)(dagger) Mx(-->) for some matrix M. In this case, when A is sparse, N x N and has condition number kappa, the fastest known classical algorithms can find x(-->) and estimate x(-->)(dagger) Mx(-->) in time scaling roughly as N square root(kappa). Here, we exhibit a quantum algorithm for estimating x(-->)(dagger) Mx(-->) whose runtime is a polynomial of log(N) and kappa. Indeed, for small values of kappa [i.e., poly log(N)], we prove (using some common complexity-theoretic assumptions) that any classical algorithm for this problem generically requires exponentially more time than our quantum algorithm.

- Y Zhang
- L Cincio
- C F A Negre
- P Czarnik
- P Coles
- P M Anisimov
- S M Mniszewski
- S Tretiak
- P A Dub

Y. Zhang, L. Cincio, C. F. A. Negre, P. Czarnik, P. Coles,
P. M. Anisimov, S. M. Mniszewski, S. Tretiak, and P. A.
Dub, Variational Quantum Eigensolver with Reduced
Circuit Complexity, arXiv:2106.07619 [quant-ph] (2021).