Search

The quantum phase estimation algorithm stands as the primary method for determining the ground state energy of a molecular electronic Hamiltonian on a quantum computer. In this context, the ability to initialize a classically tractable state that has a strong overlap with the desired ground state is critical as it directly affects the runtime of the algorithm. However, several numerical studies have shown that this overlap decays exponentially with system size. In this work, we demonstrate that this decay can be alleviated by optimizing the molecular orbital basis, for an initial state constructed from a single Slater determinant. We propose a practical method to achieve this optimization without knowledge of the true molecular ground state and test this method numerically. By comparing the resulting optimized orbitals to the natural orbitals, we find improved overlap. Specifically, for four iron-sulfur molecules, which are known to suffer from the mentioned decay, we show that our method yields one to two orders of magnitude improvement compared to localized molecular orbitals.

The computational cost of quantum algorithms for physics and chemistry is closely linked to the spectrum of the Hamiltonian, a property that manifests in the necessary rescaling of its eigenvalues. The typical approach of using the 1-norm as an upper bound to the spectral norm to rescale the Hamiltonian suits the most general case of bounded Hermitian operators but neglects the influence of symmetries commonly found in chemical systems. In this work, we introduce a hierarchy of symmetry-aware spectral bounds that provide a unified understanding of the performance of quantum phase estimation algorithms using block-encoded electronic structure Hamiltonians. We present a variational and numerically tractable method for computing these bounds, based on orbital optimization, to demonstrate that the computed bounds are smaller than conventional spectral bounds for a variety of molecular benchmark systems. We also highlight the unique analytical and numerical scaling behavior of these bounds in the thermodynamic and complete basis set limits. Our work shows that there is room for improvement in reducing the 1-norm, not yet achieved through methods like double factorization and tensor hypercontraction, but highlights potential challenges in improving the performance of current quantum algorithms beyond small constant factors through 1-norm reduction techniques alone., Comment: 11 pages, 3 figures, 1 table

Quantum phase estimation based on qubitization is the state-of-the-art fault-tolerant quantum algorithm for computing ground-state energies in chemical applications. In this context, the 1-norm of the Hamiltonian plays a fundamental role in determining the total number of required iterations and also the overall computational cost. In this work, we introduce the symmetry-compressed double factorization (SCDF) approach, which combines a compressed double factorization of the Hamiltonian with the symmetry shift technique, significantly reducing the 1-norm value. The effectiveness of this approach is demonstrated numerically by considering various benchmark systems, including the FeMoco molecule, cytochrome P450, and hydrogen chains of different sizes. To compare the efficiency of SCDF to other methods in absolute terms, we estimate Toffoli gate requirements, which dominate the execution time on fault-tolerant quantum computers. For the systems considered here, SCDF leads to a sizeable reduction of the Toffoli gate count in comparison to other variants of double factorization or even tensor hypercontraction, which is usually regarded as the most efficient approach for qubitization.

We present the first hardware implementation of electrostatic interaction energies using a trapped-ion quantum computer. As test system for our computation, we focus on the reduction of $\mathrm{NO}$ to $\mathrm{N}_2\mathrm{O}$ catalyzed by a nitric oxide reductase (NOR). The quantum computer is used to generate an approximate ground state within the NOR active space. To efficiently measure the necessary one-particle density matrices, we incorporate fermionic basis rotations into the quantum circuit without extending the circuit length, laying the groundwork for further efficient measurement routines using factorizations. Measurements in the computational basis are then used as inputs for computing the electrostatic interaction energies on a classical computer. Our experimental results strongly agree with classical noise-less simulations of the same circuits, finding electrostatic interaction energies within chemical accuracy despite hardware noise. This work shows that algorithms tailored to specific observables of interest, such as interaction energies, may require significantly fewer quantum resources than individual ground state energies would in the straightforward supermolecular approach.

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., Comment: 55 pages, 18 figures, 4 tables

The MultiExp Gaussian quadrature rule proposed by Gill and Chien has many compelling properties for the integration of radial integrands encountered in electronic structure, but thus far has only been computed up to N = 50, with some nontrivial holes in the existing tables for 27 <= N < 50. In this work, a simple recipe is developed to compute the MultiExp Gaussian quadrature for larger N at close to the double precision machine epsilon. Notably, this recipe uses only double precision operations, so no computer algebra systems or arbitrary precision libraries are needed. There are three primary outcomes: (A) The MultiExp tables for all N <= 1000 are presented in a supplement to this report - these might prove useful in building better medium and high quality density functional theory quadrature grids (B) these tables are compared to the known Gill and Chien tables with excellent agreement obtained modulo a noticeable problem with the weights of the largest N = 50 Gill and Chien table and (C) it may be the case that the Boley-Golub plus extremely high-order Gauss-Legendre quadrature approximation recipe developed in this work could be used to automatically derive other quadrature rules throughout electronic structure theory and beyond., Comment: 13 pages, 8 figures, 3 tables

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., Comment: 36 pages, 9 figures

Quantum computers promise to impact industrial applications, for which quantum chemical calculations are required, by virtue of their high accuracy. This perspective explores the challenges and opportunities of applying quantum computers to drug design, discusses where they could transform industrial research and elaborates on what is needed to reach this goal.

We propose the regularized compressed double factorization (RC-DF) method to classically compute compressed representations of molecular Hamiltonians that enable efficient simulation with noisy intermediate scale (NISQ) and error corrected quantum algorithms. We find that already for small systems with 12 to 20 qubits, the resulting NISQ measurement scheme reduces the number of measurement bases by roughly a factor of three and the shot count to reach chemical accuracy by a factor of three to six compared to truncated double factorization (DF) and we see order of magnitude improvements over Pauli grouping schemes. We demonstrate the scalability of our approach by performing RC-DF on the CpdI species of cytochrome P450 with 58 orbitals and find that using the resulting compressed Hamiltonian cuts the run time of qubitization and truncated DF based error corrected algorithms almost in half and even outperforms the lambda parameters achievable with tensor hypercontraction (THC) while at the same time reducing the CCSD(T) energy error heuristic by an order of magnitude.

We propose a class of randomized quantum Krylov diagonalization (rQKD) algorithms capable of solving the eigenstate estimation problem with modest quantum resource requirements. Compared to previous real-time evolution quantum Krylov subspace methods, our approach expresses the time evolution operator, $e^{-i\hat{H} \tau}$, as a linear combination of unitaries and subsequently uses a stochastic sampling procedure to reduce circuit depth requirements. While our methodology applies to any Hamiltonian with fast-forwardable subcomponents, we focus on its application to the explicitly double-factorized electronic-structure Hamiltonian. To demonstrate the potential of the proposed rQKD algorithm, we provide numerical benchmarks for a variety of molecular systems with circuit-based statevector simulators, achieving ground state energy errors of less than 1~kcal~mol$^{-1}$ with circuit depths orders of magnitude shallower than those required for low-rank deterministic Trotter-Suzuki decompositions.

Efficient representations of the Hamiltonian such as double factorization drastically reduce circuit depth or number of repetitions in error corrected and noisy intermediate scale quantum (NISQ) algorithms for chemistry. We report a Lagrangian-based approach for evaluating relaxed one- and two-particle reduced density matrices from double factorized Hamiltonians, unlocking efficiency improvements in computing the nuclear gradient and related derivative properties. We demonstrate the accuracy and feasibility of our Lagrangian-based approach to recover all off-diagonal density matrix elements in classically-simulated examples with up to 327 quantum and 18470 total atoms in QM/MM simulations, with modest-sized quantum active spaces. We show this in the context of the variational quantum eigensolver (VQE) in case studies such as transition state optimization, ab initio molecular dynamics simulation and energy minimization of large molecular systems., Comment: 22 pages, 5 figures

The computation of interaction energies on noisy intermediate-scale quantum (NISQ) computers appears to be challenging with straightforward application of existing quantum algorithms. For example, use of the standard supermolecular method with the variational quantum eigensolver (VQE) would require extremely precise resolution of the total energies of the fragments to provide for accurate subtraction to the interaction energy. Here we present a symmetry-adapted perturbation theory (SAPT) method that may provide interaction energies with high quantum resource efficiency. Of particular note, we present a quantum extended random-phase approximation (ERPA) treatment of the SAPT second-order induction and dispersion terms, including exchange counterparts. Together with previous work on first-order terms, this provides a recipe for complete SAPT(VQE) interaction energies up to second order. The SAPT interaction energy terms are computed as first-level observables with no subtraction of monomer energies invoked, and the only quantum observations needed are the the VQE one- and two-particle density matrices. We find empirically that SAPT(VQE) can provide accurate interaction energies even with coarsely optimized, low circuit depth wavefunctions from the quantum computer, simulated through ideal statevectors. The errors on the total interaction energy are orders of magnitude lower than the corresponding VQE total energy errors of the monomer wavefunctions., Comment: 38 pages, 6 primary figures

We develop a quartic-scaling implementation of coupled-cluster singles and doubles based on low-rank tensor hypercontraction (THC) factorizations of both the electron repulsion integrals (ERIs) and the doubles amplitudes. This extends our rank-reduced coupled-cluster method to incorporate higher-order tensor factorizations. The THC factorization of the doubles amplitudes accounts for most of the gain in computational efficiency as it is sufficient, in conjunction with a Cholesky decomposition of the ERIs, to reduce the computational complexity of most contributions to the CCSD amplitude equations. Further THC factorization of the ERIs reduces the complexity of certain terms arising from nested commutators between the doubles excitation operator and the two-electron operator. We implement this new algorithm using graphical processing units (GPUs) and demonstrate that it enables CCSD calculations for molecules with 250 atoms and 2500 basis functions using a single computer node. Further, we show that the new method computes correlation energies with comparable accuracy to the underlying RR-CCSD method.

We develop analytical gradients of ground- and excited-state energies with respect to system parameters including the nuclear coordinates for the hybrid quantum/classical multistate contracted variational quantum eigensolver (MC-VQE) applied to fermionic systems. We show how the resulting response contributions to the gradient can be evaluated with a quantum effort similar to that of obtaining the VQE energy and independent of the total number of derivative parameters (e.g. number of nuclear coordinates) by adopting a Lagrangian formalism for the evaluation of the total derivative. We also demonstrate that large-step-size finite-difference treatment of directional derivatives in concert with the parameter shift rule can significantly mitigate the complexity of dealing with the quantum parameter Hessian when solving the quantum response equations. This enables the computation of analytical derivative properties of systems with hundreds of atoms, while solving an active space of their most strongly correlated orbitals on a quantum computer. We numerically demonstrate the exactness the analytical gradients and discuss the magnitude of the quantum response contributions., Comment: 23 pages, 5 figures

We explore the use of symmetry-adapted perturbation theory (SAPT) as a simple and efficient means to compute interaction energies between large molecular systems with a hybrid method combing NISQ-era quantum and classical computers. From the one- and two-particle reduced density matrices of the monomer wavefunctions obtained by the variational quantum eigensolver (VQE), we compute SAPT contributions to the interaction energy [SAPT(VQE)]. At first order, this energy yields the electrostatic and exchange contributions for non-covalently bound systems. We empirically find from ideal statevector simulations that the SAPT(VQE) interaction energy components display orders of magnitude lower absolute errors than the corresponding VQE total energies. Therefore, even with coarsely optimized low-depth VQE wavefunctions, we still obtain sub kcal/mol accuracy in the SAPT interaction energies. In SAPT(VQE), the quantum requirements, such as qubit count and circuit depth, are lowered by performing computations on the separate molecular systems. Furthermore, active spaces allow for large systems containing thousands of orbitals to be reduced to a small enough orbital set to perform the quantum portions of the computations. We benchmark SAPT(VQE) (with the VQE component simulated by ideal state-vector simulators) against a handful of small multi-reference dimer systems and the iron center containing human cancer-relevant protein lysine-specific demethylase 5 (KDM5A)., Comment: 16 pages, 4 figures, plus supplemental data

We demonstrate a method that merges the quantum filter diagonalization (QFD) approach for hybrid quantum/classical solution of the time-independent electronic Schr\"odinger 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 needed in the QFD quantum matrix elements. Together with significant circuit reduction optimizations and number-preserving post-selection/echo-sequencing error mitigation strategies, the method is found to provide accurate predictions for low-lying eigenspectra in a number of representative molecular systems, while requiring reasonably short circuit depths and modest measurement costs. The method is demonstrated by experiments on noise-free simulators, decoherence- and shot-noise including simulators, and real quantum hardware.

We propose VQE circuit fabrics with advantageous properties for the simulation of strongly correlated ground and excited states of molecules and materials under the Jordan-Wigner mapping that can be implemented linearly locally and preserve all relevant quantum numbers: the number of spin up ($\alpha$) and down ($\beta$) electrons and the total spin squared. We demonstrate that our entangler circuits are expressive already at low depth and parameter count, appear to become universal, and may be trainable without having to cross regions of vanishing gradient, when the number of parameters becomes sufficiently large and when these parameters are suitably initialized. One particularly appealing construction achieves this with just orbital rotations and pair exchange gates. We derive optimal four-term parameter shift rules for and provide explicit decompositions of our quantum number preserving gates and perform numerical demonstrations on highly correlated molecules on up to 20 qubits., Comment: 26 pages, 10 figures

In this work, we present an efficient rank-compression approach for the classical simulation of Kraus decoherence channels in noisy quantum circuits. The approximation is achieved through iterative compression of the density matrix based on its leading eigenbasis during each simulation step without the need to store, manipulate, or diagonalize the full matrix. We implement this algorithm in an in-house simulator, and show that the low rank algorithm speeds up simulations by more than two orders of magnitude over an existing implementation of full rank simulator, and with negligible error in the target noise and final observables. Finally, we demonstrate the utility of the low rank method as applied to representative problems of interest by using the algorithm to speed-up noisy simulations of Grover's search algorithm and quantum chemistry solvers.

We develop a quantum filter diagonalization method (QFD) that lies somewhere between the variational quantum eigensolver (VQE) and the phase estimation algorithm (PEA) in terms of required quantum circuit resources and conceptual simplicity. QFD uses a set of of time-propagated guess states as a variational basis for approximate diagonalization of a sparse Pauli Hamiltonian. The variational coefficients of the basis functions are determined by the Rayleigh-Ritz procedure by classically solving a generalized eigenvalue problem in the space of time-propagated guess states. The matrix elements of the subspace Hamiltonian and subspace metric matrix are each determined in quantum circuits by a one-ancilla extended swap test, i.e., statistical convergence of a one-ancilla PEA circuit. These matrix elements can be determined by many parallel quantum circuit evaluations, and the final Ritz estimates for the eigenvectors can conceptually be prepared as a linear combination over separate quantum state preparation circuits. The QFD method naturally provides for the computation of ground-state, excited-state, and transition expectation values. We numerically demonstrate the potential of the method by classical simulations of the QFD algorithm for an N=8 octamer of BChl-a chromophores represented by an 8-qubit ab initio exciton model (AIEM) Hamiltonian. Using only a handful of time-displacement points and a coarse, variational Trotter expansion of the time propagation operators, the QFD method recovers an accurate prediction of the absorption spectrum., Comment: 10 pages, 3 figures

The maturation of analytical derivative theory over the past few decades has enabled classical electronic structure theory to provide accurate and efficient predictions of a wide variety of observable properties. However, classical implementations of analytical derivative theory take advantage of explicit computational access to the approximate electronic wavefunctions in question, which is not possible for the emerging case of hybrid quantum/classical methods. Here, we develop an efficient Lagrangian-based approach for analytical first derivatives of hybrid quantum/classical methods using only observable quantities from the quantum portion of the algorithm. Specifically, we construct the key first-derivative property of the nuclear energy gradient for the recently-developed multistate, contracted variant of the variational quantum eigensolver (MC-VQE) within the context of the ab initio exciton model (AIEM). We show that a clean separation between the quantum and classical parts of the problem is enabled by the definition of an appropriate set of relaxed density matrices, and show how the wavefunction response equations in the quantum part of the algorithm (coupled-perturbed MC-VQE or CP-MC-VQE equations) are decoupled from the wavefunction response equations and and gradient perturbations in the classical part of the algorithm. We explore the magnitudes of the Hellmann-Feynman and response contributions to the gradients in quantum circuit simulations of MC-VQE+AIEM and demonstrate a quantum circuit simulator implementation of adiabatic excited state dynamics with MC-VQE+AIEM., Comment: 26 pages, 3 figures, 4 tables

The optimization of circuit parameters of variational quantum algorithms such as the variational quantum eigensolver (VQE) or the quantum approximate optimization algorithm (QAOA) is a key challenge for the practical deployment of near-term quantum computing algorithms. Here, we develop a hybrid quantum/classical optimization procedure inspired by the Jacobi diagonalization algorithm for classical eigendecomposition, and combined with Anderson acceleration. In the first stage, analytical tomography fittings are performed for a local cluster of circuit parameters via sampling of the observable objective function at quadrature points in the circuit angles. Classical optimization is used to determine the optimal circuit parameters within the cluster, with the other circuit parameters frozen. Different clusters of circuit parameters are then optimized in "sweeps,'' leading to a monotonically-convergent fixed-point procedure. In the second stage, the iterative history of the fixed-point Jacobi procedure is used to accelerate the convergence by applying Anderson acceleration/Pulay's direct inversion of the iterative subspace (DIIS). This Jacobi+Anderson method is numerically tested using a quantum circuit simulator (without noise) for a representative test case from the multistate, contracted variant of the variational quantum eigensolver (MC-VQE), and is found to be competitive with and often faster than Powell's method and L-BFGS., Comment: 31 pages, 16 figures

We develop an extension of the variational quantum eigensolver (VQE) algorithm - multistate, contracted VQE (MC-VQE) - that allows for the efficient computation of the transition energies between the ground state and several low-lying excited states of a molecule, as well as the oscillator strengths associated with these transitions. We numerically simulate MC-VQE by computing the absorption spectrum of an ab initio exciton model of an 18-chromophore light-harvesting complex from purple photosynthetic bacteria., Comment: Updated with addl references and SI case study of H-aggregate BChl-a stack

Configuration-space matrix elements of N-body potentials arise naturally and ubiquitously in the Ritz-Galerkin solution of many-body quantum problems. For the common specialization of local, finite-range potentials, we develop the eXact Tensor HyperContraction (X-THC) method, which provides a quantized renormalization of the coordinate-space form of the N-body potential, allowing for a highly separable tensor factorization of the configuration-space matrix elements. This representation allows for substantial computational savings in chemical, atomic, and nuclear physics simulations, particularly with respect to difficult "exchange-like" contractions., Comment: Third version of the manuscript after referee's comments. In press in PRL. Main text: 4 pages, 2 figures, 1 table; Supplemental material (also included): 14 pages, 2 figures, 2 tables