Electronic structure calculations of molecular systems are among the most promising applications for fault-tolerant quantum computing (FTQC) in quantum chemistry and drug design. However, while recent algorithmic advancements such as qubitization and tensor hypercontraction (THC) have significantly reduced the complexity of such calculations, they do not yet achieve computational runtimes short enough to be practical for industrially relevant use cases. In this work, we combine several advances to electronic structure calculation for molecular systems, resulting in a 2-orders-of-magnitude speedup of estimated runtimes over prior-art algorithms run on comparable quantum devices. One of these advances is a novel framework for block-invariant symmetry-shifted tensor hypercontraction, with which we achieve the tightest Hamiltonian factorizations reported to date. We compile our algorithm for an active volume (AV) architecture, a technical layout that has recently been proposed for fusion-based photonic quantum hardware. AV compilation contributes towards a lower runtime of our computation by eliminating overheads stemming from connectivity issues in the underlying surface code. We present a detailed benchmark of our approach, focusing primarily on the computationally challenging benchmark molecule P450. Leveraging a number of hardware trade-offs in interleaving-based photonic FTQC, we estimate runtimes for the electronic structure calculation of P450 as a function of the device footprint.

The problems of quantum state preparation and matrix block-encoding are ubiquitous in quantum computing: they are crucial parts of various quantum algorithms for the purpose for initial state preparation as well as loading problem relevant data. We first present an algorithm based on QRS that prepares a quantum state |ψ_f⟩∝∑^N_x=1 f(x)|x⟩. When combined with efficient reference states the algorithm reduces the cost of quantum state preparation substantially, if certain criteria on f are met. When the preparation of the reference state is not the dominant cost, and the function f and relevant properties are efficiently computable or provided otherwise with cost o(N), the QRS-based method outperforms the generic state preparation algorithm, which has cost O(N). We demonstrate the detailed performance (in terms of the number of Toffoli gates) of the QRS-based algorithm for quantum states commonly appearing in quantum applications, e.g., those with coefficients that obey power law decay, Gaussian, and hyperbolic tangent, and compare it with other methods. Then, we adapt QRS techniques to the matrix block-encoding problem and introduce a QRS-based algorithm for block-encoding a given matrix A = ∑_ij A_ij |i⟩⟨ j|. We work out rescaling factors for different access models, which encode how the information about the matrix is provided to the quantum computer. We exemplify these results for a particular Toeplitz matrix with elements A_𝐢𝐣= 1/𝐢-𝐣^2, which appears in quantum chemistry, and PDE applications, e.g., when the Coulomb interaction is involved. Our work unifies, and in certain ways goes beyond, various quantum state preparation and matrix block-encoding methods in the literature, and gives detailed performance analysis of important examples that appear in quantum applications.

The efficient computation of observables beyond the total energy is a key challenge and opportunity for fault-tolerant quantum computing approaches in quantum chemistry. Here we consider the symmetry-adapted perturbation theory (SAPT) components of the interaction energy as a prototypical example of such an observable. We provide a guide for calculating this observable on a fault-tolerant quantum computer while optimizing the required computational resources. Specifically, we present a quantum algorithm that estimates interaction energies at the first-order SAPT level with a Heisenberg-limited scaling. To this end, we exploit a high-order tensor factorization and block encoding technique that efficiently represents each SAPT observable. To quantify the computational cost of our methodology, we provide resource estimates in terms of the required number of logical qubits and Toffoli gates to execute our algorithm for a range of benchmark molecules, also taking into account the cost of the eigenstate preparation and the cost of block encoding the SAPT observables. Finally, we perform the resource estimation for a heme and artemisinin complex as a representative large-scale system encountered in drug design, highlighting our algorithm's performance in this new benchmark study and discussing possible bottlenecks that may be improved in future work.

In this work, we introduce a new way to quantify information flow in quantum systems, especially for parameterized quantum circuits (PQCs). We use a graph representation of the circuits and propose a new distance metric using the mutual information between gate nodes. We then present an optimization procedure for variational algorithms using paths based on the distance measure. We explore the features of the algorithm by means of the variational quantum eigensolver, in which we compute the ground state energies of the Heisenberg model. In addition, we employ the method to solve a binary classification problem using variational quantum classification. From numerical simulations, we show that our method can be successfully used for optimizing the PQCs primarily used in near-term algorithms. We further note that information-flow based paths can be used to improve convergence of existing stochastic gradient based methods.

In recent years, quantum algorithms have been proposed which use quantum phase estimation (QPE) coherently as a subroutine without measurement. In order to do this effectively, the routine must be able to distinguish eigenstates with success probability close to unity. In this paper, we provide the first systematic comparison between two approaches towards maximizing this success probability, one using the quantum singular value transform and the other leveraging window functions, which have been previously studied as priors of the phase value distribution. We find that the quantum singular value transform is significantly outclassed by the window function approach, with the latter able to achieve between 3 and 5 orders of magnitude improvement in the success probability with approximately 1/4 the query cost. Our circuit simulation results indicate that QPE is not a domain which benefits from the integration of QSVT and we show that the use of the Kaiser window function is currently the most practical choice for realizing QPE with high success probability.

Over the past three decades significant reductions have been made to the cost of estimating ground-state energies of molecular Hamiltonians with quantum computers. However, comparatively little attention has been paid to estimating the expectation values of other observables with respect to said ground states, which is important for many industrial applications. In this work we present a novel expectation value estimation (EVE) quantum algorithm which can be applied to estimate the expectation values of arbitrary observables with respect to any of the system's eigenstates. In particular, we consider two variants of EVE: std-EVE, based on standard quantum phase estimation, and QSP-EVE, which utilizes quantum signal processing (QSP) techniques. We provide rigorous error analysis for both both variants and minimize the number of individual phase factors for QSPEVE. These error analyses enable us to produce constant-factor quantum resource estimates for both std-EVE and QSP-EVE across a variety of molecular systems and observables. For the systems considered, we show that QSP-EVE reduces (Toffoli) gate counts by up to three orders of magnitude and reduces qubit width by up to 25% compared to std-EVE. While estimated resource counts remain far too high for the first generations of fault-tolerant quantum computers, our estimates mark a first of their kind for both the application of expectation value estimation and modern QSP-based techniques.

Quantum computing is believed to be particularly useful for the simulation of chemistry and materials, among the various applications. In recent years, there have been significant advancements in the development of near-term quantum algorithms for quantum simulation, including VQE and many of its variants. However, for such algorithms to be useful, they need to overcome several critical barriers including the inability to prepare high-quality approximations of the ground state. Current challenges to state preparation, including barren plateaus and the high-dimensionality of the optimization landscape, make state preparation through ansatz optimization unreliable. In this work, we introduce the method of ground state boosting, which uses a limited-depth quantum circuit to reliably increase the overlap with the ground state. This circuit, which we call a booster, can be used to augment an ansatz from VQE or be used as a stand-alone state preparation method. The booster converts circuit depth into ground state overlap in a controllable manner. We numerically demonstrate the capabilities of boosters by simulating the performance of a particular type of booster, namely the Gaussian booster, for preparing the ground state of N2 molecular system. Beyond ground state preparation as a direct objective, many quantum algorithms, such as quantum phase estimation, rely on high-quality state preparation as a subroutine. Therefore, we foresee ground state boosting and similar methods as becoming essential algorithmic components as the field transitions into using early fault-tolerant quantum computers.

Parameterized quantum circuits (PQCs) are a central component of many variational quantum algorithms, yet there is a lack of understanding of how their parameterization impacts algorithm performance. We initiate this discussion by using principal bundles to geometrically characterize two-qubit PQCs. On the base manifold, we use the Mannoury-Fubini-Study metric to find a simple equation relating the Ricci scalar (geometry) and concurrence (entanglement). By calculating the Ricci scalar during a variational quantum eigensolver (VQE) optimization process, this offers us a new perspective to how and why Quantum Natural Gradient outperforms the standard gradient descent. We argue that the key to the Quantum Natural Gradient's superior performance is its ability to find regions of high negative curvature early in the optimization process. These regions of high negative curvature appear to be important in accelerating the optimization process.

We present a review of the Unitary Coupled Cluster (UCC) ansatz and related ansätze which are used to variationally solve the electronic structure problem on quantum computers. A brief history of coupled cluster (CC) methods is provided, followed by a broad discussion of the formulation of CC theory. This includes touching on the merits and difficulties of the method and several variants, UCC among them, in the classical context, to motivate their applications on quantum computers. In the core of the text, the UCC ansatz and its implementation on a quantum computer are discussed at length, in addition to a discussion on several derived and related ansätze specific to quantum computing. The review concludes with a unified perspective on the discussed ansätze, attempting to bring them under a common framework, as well as with a reflection upon open problems within the field.

A universal fault-tolerant quantum computer that can solve efficiently problems such as integer factorization and unstructured database search requires millions of qubits with low error rates and long coherence times. While the experimental advancement towards realizing such devices will potentially take decades of research, noisy intermediate-scale quantum (NISQ) computers already exist. These computers are composed of hundreds of noisy qubits, i.e. qubits that are not error-corrected, and therefore perform imperfect operations in a limited coherence time. In the search for quantum advantage with these devices, algorithms have been proposed for applications in various disciplines spanning physics, machine learning, quantum chemistry and combinatorial optimization. The goal of such algorithms is to leverage the limited available resources to perform classically challenging tasks. In this review, we provide a thorough summary of NISQ computational paradigms and algorithms. We discuss the key structure of these algorithms, their limitations, and advantages. We additionally provide a comprehensive overview of various benchmarking and software tools useful for programming and testing NISQ devices.

Variational Quantum Algorithms (VQAs) are promising candidates for finding practical applications of near to mid-term quantum computers. There has been an increasing effort to study the intricacies of VQAs, such as the presence or absence of barren plateaus and the design of good quantum circuit ans\"atze. Many of these studies can be linked to the loss landscape that is optimized as part of the algorithm, and there is high demand for quality software tools for flexibly studying these loss landscapes. In our work, we collect a variety of techniques that have been used to visualize the training of deep artificial neural networks and apply them to visualize the high-dimensional loss landscapes of VQAs. We review and apply the techniques to three types of VQAs: the Quantum Approximate Optimization Algorithm, the Quantum Circuit Born Machine, and the Variational Quantum Eigensolver. Additionally, we investigate the impact of noise due to finite sampling in the estimation of loss functions. For each case, we demonstrate how our visualization techniques can verify observations from past studies and provide new insights. This work is accompanied by the release of the open-source Python package $\textit{orqviz}$, which provides code to compute and flexibly plot 1D and 2D scans, Principal Component Analysis scans, Hessians, and the Nudged Elastic Band algorithm. $\textit{orqviz}$ enables flexible visual analysis of high-dimensional VQA landscapes and can be found at: $\textbf{github.com/zapatacomputing/orqviz}$.

When compared to fault-tolerant quantum computational strategies, variational quantum algorithms stand as one of the candidates with the potential of achieving quantum advantage for real-world applications in the near term. However, the optimization of the circuit parameters remains arduous and is impeded by many obstacles such as the presence of barren plateaus, many local minima in the optimization landscape, and limited quantum resources. A non-random initialization of the parameters seems to be key to the success of the parametrized quantum circuits (PQC) training. Drawing and extending ideas from the field of meta-learning, we address this parameter initialization task with the help of machine learning and propose FLIP: a FLexible Initializer for arbitrarily-sized Parametrized quantum circuits. FLIP can be applied to any family of PQCs, and instead of relying on a generic set of initial parameters, it is tailored to learn the structure of successful parameters from a family of related problems which are used as the training set. The flexibility advocated to FLIP hinges in the possibility of predicting the initialization of parameters in quantum circuits with a larger number of parameters from those used in the training phase. This is a critical feature lacking in other meta-learning parameter initializing strategies proposed to date. We illustrate the advantage of using FLIP in three scenarios: a family of problems with proven barren plateaus, PQC training to solve max-cut problem instances, and PQC training for finding the ground state energies of 1D Fermi-Hubbard models.

Variational hybrid quantum-classical algorithms are powerful tools to maximize the use of Noisy Intermediate Scale Quantum devices. While past studies have developed powerful and expressive ansatze, their near-term applications have been limited by the difficulty of optimizing in the vast parameter space. In this work, we propose a heuristic optimization strategy for such ansatze used in variational quantum algorithms, which we call "Parameter-Efficient Circuit Training" (PECT). Instead of optimizing all of the ansatz parameters at once, PECT launches a sequence of variational algorithms, in which each iteration of the algorithm activates and optimizes a subset of the total parameter set. To update the parameter subset between iterations, we adapt the dynamic sparse reparameterization scheme by Mostafa et al. (arXiv:1902.05967). We demonstrate PECT for the Variational Quantum Eigensolver, in which we benchmark unitary coupled-cluster ansatze including UCCSD and k-UpCCGSD, as well as the low-depth circuit ansatz (LDCA), to estimate ground state energies of molecular systems. We additionally use a layerwise variant of PECT to optimize a hardware-efficient circuit for the Sycamore processor to estimate the ground state energy densities of the one-dimensional Fermi-Hubbard model. From our numerical data, we find that PECT can enable optimizations of certain ansatze that were previously difficult to converge and more generally can improve the performance of variational algorithms by reducing the optimization runtime and/or the depth of circuits that encode the solution candidate(s).

Abhinav Anand &,1, ∗ Philipp Schleich &,2, 3, 4, † Sumner Alperin-Lea &,1 Phillip W. K. Jensen &,1 Sukin Sim,5 Manuel Díaz-Tinoco,1, 6 Jakob S. Kottmann,1, 2 Matthias Degroote,1, 2 Artur F. Izmaylov,1, 6, ‡ and Alán Aspuru-Guzik1, 2, 4, 7, § Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Canada. Department of Computer Science, University of Toronto, Canada. Applied and Computational Mathematics, Department of Mathematics, RWTH Aachen University, Aachen, Germany. Vector Institute for Artificial Intelligence, Toronto, Canada. Department of Chemistry and Chemical Biology, Harvard University, USA. Department of Physical and Environmental Sciences, University of Toronto Scarborough, Canada. Canadian Institute for Advanced Research (CIFAR) Lebovic Fellow, Toronto, Canada (Dated: December 3, 2021)

With the increasing size of quantum processors, submodules that constitute the processor hardware will become too large to accurately simulate on a classical computer. Therefore, one would soon have to fabricate and test each new design primitive and parameter choice in time-consuming coordination between design, fabrication, and experimental validation. Here we show how one can design and test the performance of next-generation quantum hardware-by using existing quantum computers. Focusing on superconducting transmon processors as a prominent hardware platform, we compute the static and dynamic properties of individual and coupled transmons. We show how the energy spectra of transmons can be obtained by variational hybrid quantum-classical algorithms that are well suited for near-term noisy quantum computers. In addition, single-and two-qubit gate simulations are demonstrated via Suzuki-Trotter decomposition. Our methods pave a promising way towards designing candidate quantum processors when the demands of calculating submodule properties exceed the capabilities of classical computing resources.

The increasing integration of software and automation in modern chemical laboratories prompts special emphasis on two important skills in the chemistry classroom. First, students need to learn the technical skills involved in modern scientific computing and automation. Second, applying these techniques in practice requires effective collaboration in teams. This work aims at developing a teaching module to help students gain both skills. In particular, we describe a modular and collaborative approach for introducing undergraduate students to scientific computing in the context of automated and autonomous chemical laboratories. Using online collaboration tools, students work in parallel teams to develop central components of an automated computer vision system that monitors color changes in ongoing chemical reactions. These components include three different aspects: image capture, communication, and data visualization. The image capture team collects and stores the images of the chemical reaction, the communication team processes the images, and the visualization team develops the tools for analyzing the processed image data. Using this educational framework, students built an open-source Python tool called AutoVis that enables the automated tracking of color and intensity changes in a liquid. The software is tested by simulating chemical reactions with dilute solutions of food coloring in water. It is shown that the system reliably tracks color and intensity, providing feedback to the experimentalist and enabling further computational analysis. Over the course of the project, students gain proficiency in scientific computing using Python and collaborate on software development using GitHub. In this way, they learn the role of software in chemical laboratories of the 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.

Parameterized quantum circuits (PQCs) play an essential role in the performance of many variational quantum algorithms. One challenge in implementing such algorithms is choosing an effective circuit that well represents the solution space while maintaining a low circuit depth and parameter count. To characterize and identify expressible, yet compact, circuits, several descriptors are proposed, including expressibility and entangling capability, that are statistically estimated from classical simulations. These descriptors are computed for different circuit structures, varying the qubit connectivity and selection of gates. From these simulations, circuit fragments that perform well with respect to the descriptors are identified. In particular, a substantial improvement in performance of two‐qubit gates in a ring or all‐to‐all connected arrangement, compared to that of those on a line, is observed. Furthermore, improvement in both descriptors is achieved by sequences of controlled X‐rotation gates compared to sequences of controlled Z‐rotation gates. In addition, it is investigated how expressibility “saturates” with increased circuit depth, finding that the rate and saturated value appear to be distinguishing features of a PQC. While the correlation between each descriptor and algorithm performance remains to be investigated, methods and results from this study can be useful for algorithm development and design of experiments.

Quantum simulations are bound to be one of the main applications of near-term quantum computers. Quantum chemistry and condensed matter physics are expected to benefit from these technological developments. Several quantum simulation methods are known to prepare a state on a quantum computer and measure the desired observables. The most resource economic procedure is the variational quantum eigensolver (VQE), which has traditionally employed unitary coupled cluster as the ansatz to approximate ground states of many-body fermionic Hamiltonians. A significant caveat of the method is that the initial state of the procedure is a single reference product state from a classical Hartree–Fock calculation with no pairing correlations, hence it cannot represent superconducting states. In this work, we propose to improve the method by initializing the algorithm with a more general fermionic Gaussian state, an idea borrowed from the field of nuclear physics. We show how this Gaussian reference state can be prepared with a linear-depth circuit of quantum matchgates. By augmenting the set of available gates with nearest-neighbor phase coupling, we generate a low-depth circuit ansatz that can accurately prepare the ground state of correlated fermionic systems. This extends the range of applicability of the VQE to systems with strong pairing correlations such as superconductors, atomic nuclei, and topological materials.