Extension of Exactly-Solvable Hamiltonians Using Symmetries of Lie Algebras.

Exactly solvable Hamiltonians that can be diagonalized by using relatively simple unitary transformations are of great use in quantum computing. They can be employed for the decomposition of interacting Hamiltonians either in Trotter-Suzuki approximations of the evolution operator for the quantum phase estimation algorithm or in the quantum measurement problem for the variational quantum eigensolver. One of the typical forms of exactly solvable Hamiltonians is a linear combination of operators forming a modestly sized Lie algebra. Very frequently, such linear combinations represent noninteracting Hamiltonians and thus are of limited interest for describing interacting cases. Here, we propose an extension in which the coefficients in these combinations are substituted by polynomials of the Lie algebra symmetries. This substitution results in a more general class of solvable Hamiltonians, and for qubit algebras, it is related to the recently proposed noncontextual Pauli Hamiltonians. In fermionic problems, this substitution leads to Hamiltonians with eigenstates that are single Slater determinants but with different sets of single-particle states for different eigenstates. The new class of solvable Hamiltonians can be measured efficiently using quantum circuits with gates that depend on the result of a midcircuit measurement of the symmetries.

Block-Invariant Symmetry Shift: Preprocessing Technique for Second-Quantized Hamiltonians to Improve Their Decompositions to Linear Combination of Unitaries.

Computational cost of energy estimation for molecular electronic Hamiltonians via quantum phase estimation (QPE) grows with the difference between the largest and smallest eigenvalues of the Hamiltonian. In this work, we propose a preprocessing procedure that reduces the norm of the Hamiltonian without changing its eigenspectrum for the target states of a particular symmetry. The new procedure, block-invariant symmetry shift (BLISS), builds an operator T̂ such that the cost of implementing H^-T^ is reduced compared to that of H, yet H^-T^ acts on the subspaces of interest the same way as H does. BLISS performance is demonstrated for a linear combination of unitaries (LCU)-based QPE approaches on a set of small molecules. Using the number of electrons as the symmetry specifying the target set of states, BLISS provided a factor of 2 reduction of 1-norm for several LCU decompositions compared to their unshifted versions.

Growth Reduction of Similarity-Transformed Electronic Hamiltonians in Qubit Space.

Accurately solving the electronic structure problem through the variational quantum eigensolver (VQE) is hindered by the available quantum resources of current and near-term devices. One approach to relieving the circuit depth requirements for VQE is to "pre-process" the electronic Hamiltonian by a similarity transformation incorporating some degree of electronic correlation, with the remaining correlation left to be addressed by the circuit ansatz. This often comes at the price of a substantial increase in the number of terms to measure in the similarity-transformed Hamiltonian. In this work, we propose an efficient approach to sampling elements from the complete Pauli group for N qubits which minimizes the onset of new terms in the transformed Hamiltonian while facilitating substantial energy lowering. We benchmark the growth-mitigating generator selection technique for ground state energy estimations applied to models of the H4, N2, and H2O molecular systems. It is found that utilizing a selection procedure which obtains the growth-minimizing generator from the set of operators with the maximal energy gradient is the most competitive approach to reducing the onset of Hamiltonian terms while achieving systematic energy lowering of the reference state.

A Hybrid Quantum-Classical Algorithm for Multichannel Quantum Scattering of Atoms and Molecules.

We propose a hybrid quantum-classical algorithm for solving the time-independent Schrödinger equation for atomic and molecular collisions. The algorithm is based on the S-matrix version of the Kohn variational principle, which computes the fundamental scattering S-matrix by inverting the Hamiltonian matrix expressed in the basis of square-integrable functions. The computational bottleneck of the classical algorithm─symmetric matrix inversion─is addressed here using the variational quantum linear solver (VQLS), a recently developed noisy intermediate-scale quantum (NISQ) algorithm for solving systems of linear equations. We apply our algorithm to single- and multichannel quantum scattering problems, obtaining accurate vibrational relaxation probabilities in collinear atom-molecule collisions. We also show how the algorithm could be scaled up to simulate collisions of large polyatomic molecules. Our results demonstrate that it is possible to calculate scattering cross sections and rates for complex molecular collisions on NISQ quantum processors, opening up the possibility of scalable digital quantum computation of gas-phase bimolecular collisions and reactions of relevance to astrochemistry and ultracold chemistry.

Measurement Optimization Techniques for Excited Electronic States in Near-Term Quantum Computing Algorithms.

The variational quantum eigensolver (VQE) remains one of the most popular near-term quantum algorithms for solving the electronic structure problem. Yet, for its practicality, the main challenge to overcome is improving the quantum measurement efficiency. Numerous quantum measurement techniques have been developed recently, but it is unclear how these state-of-the-art measurement techniques will perform in extensions of VQE for obtaining excited electronic states. Assessing the measurement techniques' performance in the excited state VQE is crucial because the measurement requirements in these extensions are typically much greater than in the ground state VQE, as one must measure the expectation value of multiple observables in addition to that of the electronic Hamiltonian. Here, we adapt various measurement techniques to two widely used excited state VQE algorithms: multistate contraction and quantum subspace expansion. Then, the measurement requirements of each measurement technique are numerically compared. We find that the best methods for multistate contraction are ones utilizing Hamiltonian data and wave function information to minimize the number of measurements. In contrast, randomized measurement techniques are more appropriate for quantum subspace expansion, with many more observables of vastly different energy scales to measure. Nevertheless, when the best possible measurement technique for each excited state VQE algorithm is considered, significantly fewer measurements are required in multistate contraction than in quantum subspace expansion.

Quantum Kinetic Rates within the Nonequilibrium Steady State.

The nonequilibrium steady state (NESS) of a quantum network is central to a host of physical and biological scenarios. Examples include natural processes such as vision and photosynthesis as well as technical devices such as photocells, both activated by incoherent light (e.g., sunlight) and leading to quantum transport. Assessing time scales of the relevant chemical processes in the steady state is thus of utmost interest and is our goal in this paper. Here, a completely general approach to defining components of a quantum network in the NESS and obtaining rates of processes between these components is provided. Quantum effects are explicitly included throughout, both in (a) defining network components via projection operators and (b) determining the role of coherences in rate processes. As examples, the methodology is applied to model cases, two versions of the V-level system, and to the spin-boson model, wherein the roles of the environment and of internal system properties in determining the rates are examined. In addition, the role of Markovian vs non-Markovian contributions is quantified, exposing conditions under which NESS rates can be obtained by perturbing the nonequilibrium steady state.

Deterministic improvements of quantum measurements with grouping of compatible operators, non-local transformations, and covariance estimates.

Obtaining the expectation value of an observable on a quantum computer is a crucial step in the variational quantum algorithms. For complicated observables such as molecular electronic Hamiltonians, one of the strategies is to present the observable as a linear combination of measurable fragments. The main problem of this approach is a large number of measurements required for accurate estimation of the observable's expectation value. We consider three previously studied directions that minimize the number of measurements: (1) grouping commuting operators using the greedy approach, (2) involving non-local unitary transformations for measuring, and (3) taking advantage of compatibility of some Pauli products with several measurable groups. The last direction gives rise to a general framework that not only provides improvements over previous methods but also connects measurement grouping approaches with recent advances in techniques of shadow tomography. Following this direction, we develop two measurement schemes that achieve a severalfold reduction in the number of measurements for a set of model molecules compared to previous state-of-the-art methods.

Improving Quantum Measurements by Introducing "Ghost" Pauli Products.

Reducing the number of measurements required to estimate the expectation value of an observable is crucial for the variational quantum eigensolver to become competitive with state-of-the-art classical algorithms. To measure complicated observables such as a molecular electronic Hamiltonian, one of the common strategies is to partition the observable into linear combinations (fragments) of mutually commutative Pauli products. The total number of measurements for obtaining the expectation value is then proportional to the sum of variances of individual fragments. We propose a method that lowers individual fragment variances by modifying the fragments without changing the total observable expectation value. Our approach is based on adding Pauli products ("ghosts") that are compatible with members of multiple fragments. The total expectation value does not change because a sum of coefficients for each "ghost" Pauli product introduced to several fragments is zero. Yet, these additions change individual fragment variances because of the non-vanishing contributions of "ghost" Pauli products within each fragment. The proposed algorithm minimizes individual fragment variances using a classically efficient approximation of the quantum wavefunction for variance estimations. Numerical tests on a few molecular electronic Hamiltonian expectation values show several-fold reductions in the number of measurements in the "ghost" Pauli algorithm compared to those in the other recently developed techniques.

Fidelity Overhead for Nonlocal Measurements in Variational Quantum Algorithms.

Measuring quantum observables by grouping terms that can be rotated to sums of only products of Pauli z operators (Ising form) is proven to be efficient in near term quantum computing algorithms. This approach requires extra unitary transformations to rotate the state of interest so that the measurement of a fragment's Ising form would be equivalent to the measurement of the fragment for the unrotated state. These extra rotations allow one to perform a fewer number of measurements by grouping more terms into the measurable fragments with a lower overall estimator variance. However, previous estimations of the number of measurements did not take into account nonunit fidelity of quantum gates implementing the additional transformations. Through a circuit fidelity reduction, additional transformations introduce extra uncertainty and increase the needed number of measurements. Here we consider a simple model for errors introduced by additional gates needed in schemes involving groupings of commuting Pauli products. For a set of molecular electronic Hamiltonians, we confirm that the numbers of measurements in schemes using nonlocal qubit rotations are still lower than those in their local qubit rotation counterparts, even after accounting for uncertainties introduced by additional gates.

Controlling energy conservation in quantum dynamics with independently moving basis functions: Application to multi-configuration Ehrenfest.

Application of the time-dependent variational principle to a linear combination of frozen-width Gaussians describing the nuclear wavefunction provides a formalism where the total energy is conserved. The computational downside of this formalism is that trajectories of individual Gaussians are solutions of a coupled system of differential equations, limiting implementation to serial propagation algorithms. To allow for parallelization and acceleration of the computation, independent trajectories based on simplified equations of motion were suggested. Unfortunately, within practical realizations involving finite Gaussian bases, this simplification leads to breaking the energy conservation. We offer a solution for this problem by using Lagrange multipliers to ensure the energy and norm conservation regardless of basis function trajectories or basis completeness. We illustrate our approach within the multi-configurational Ehrenfest method considering a linear vibronic coupling model.

A quantum computing view on unitary coupled cluster theory.

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.

Computational approaches to efficient generation of the stationary state for incoherent light excitation.

Light harvesting processes are often computationally studied from a time-dependent viewpoint, in line with ultrafast coherent spectroscopy experiments. Yet, natural processes take place in the presence of incoherent light, which induces a stationary state. Such stationary states can be described using the eigenbasis of the molecular Hamiltonian, but for realistic systems, a full diagonalization is prohibitively expensive. We propose three efficient computational approaches to obtain the stationary state that circumvents system Hamiltonian diagonalization. The connection between the incoherent perturbations, decoherence, and Kraus operators is established.

Unitary Transformation of the Electronic Hamiltonian with an Exact Quadratic Truncation of the Baker-Campbell-Hausdorff Expansion.

The application of current and near-term quantum hardware to the electronic structure problem is highly limited by qubit counts, coherence times, and gate fidelities. To address these restrictions within the variational quantum eigensolver (VQE) framework, many recent contributions have suggested dressing the electronic Hamiltonian to include a part of electron correlation, leaving the rest to VQE state preparation. We present a new dressing scheme that combines the preservation of the Hamiltonian hermiticity and an exact quadratic truncation of the Baker-Campbell-Hausdorff expansion. The new transformation is constructed as the exponent of an involutory linear combination (ILC) of anti-commuting Pauli products. It incorporates important strong correlation effects in the dressed Hamiltonian and can be viewed as a classical preprocessing step to alleviate the resource requirements of the subsequent VQE application. The assessment of the new computational scheme for the electronic structure of the LiH, H2O, and N2 molecules shows a significant increase in efficiency compared to the conventional qubit coupled cluster dressings.

On the order problem in construction of unitary operators for the variational quantum eigensolver.

One of the main challenges in the variational quantum eigensolver (VQE) framework is construction of the unitary transformation. The dimensionality of the space for unitary rotations of N qubits is 4N- 1, which makes the choice of a polynomial subset of generators an exponentially difficult process. Moreover, due to non-commutativity of generators, the order in which they are used strongly affects results. Choosing the optimal order in a particular subset of generators requires testing the factorial number of combinations. We propose an approach based on the Lie algebra-Lie group connection and corresponding closure relations that systematically eliminates the order problem.

Measurement optimization in the variational quantum eigensolver using a minimum clique cover.

Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves the measurement of the Hamiltonian expectation value. The current hardware can perform only projective single-qubit measurements, and thus, the Hamiltonian expectation value is obtained by measuring parts of the Hamiltonian rather than the full Hamiltonian. This restriction makes the measurement process inefficient because the number of terms in the Hamiltonian grows as O(N4) with the size of the system, N. To optimize the VQE measurement, one can try to group as many Hamiltonian terms as possible for their simultaneous measurement. Single-qubit measurements allow one to group only the terms commuting within the corresponding single-qubit subspaces or qubit-wise commuting. We found that the qubit-wise commutativity between the Hamiltonian terms can be expressed as a graph and the problem of the optimal grouping is equivalent to finding a minimum clique cover (MCC) for the Hamiltonian graph. The MCC problem is NP-hard, but there exist several polynomial heuristic algorithms to solve it approximately. Several of these heuristics were tested in this work for a set of molecular electronic Hamiltonians. On average, grouping qubit-wise commuting terms reduced the number of operators to measure three times less compared to the total number of terms in the considered Hamiltonians.

Measuring All Compatible Operators in One Series of Single-Qubit Measurements Using Unitary Transformations.

The Variational Quantum Eigensolver approach to the electronic structure problem on a quantum computer involves measurement of the Hamiltonian expectation value. Formally, quantum mechanics allows one to measure all mutually commuting or compatible operators simultaneously. Unfortunately, the current hardware permits measuring only a much more limited subset of operators that share a common tensor product eigen-basis. We introduce unitary transformations that transform any fully commuting group of operators to a group that can be measured on current hardware. These unitary operations can be encoded as a sequence of Clifford gates and let us not only measure much larger groups of terms but also to obtain these groups efficiently on a classical computer. The problem of finding the minimum number of fully commuting groups of terms covering the whole Hamiltonian is found to be equivalent to the minimum clique cover problem for a graph representing Hamiltonian terms as vertices and commutativity between them as edges. Tested on a set of molecular electronic Hamiltonians with up to 50 thousand terms, the introduced technique allows for the reduction of the number of separately measurable operator groups down to few hundreds, thus achieving up to 2 orders of magnitude reduction. Based on the test set results, the obtained gain scales at least linearly with the number of qubits.

Iterative Qubit Coupled Cluster Approach with Efficient Screening of Generators.

An iterative version of the qubit coupled cluster (QCC) method [I. G. Ryabinkin et al., J. Chem. Theory Comput. 2019, 14, 6317] is proposed. The new method seeks to find ground electronic energies of molecules on noisy intermediate-scale quantum devices. Each iteration involves a canonical transformation of the Hamiltonian and employs constant-size quantum circuits at the expense of increasing the Hamiltonian size. We numerically studied the convergence of the method on ground-state calculations for LiH, H2O, and N2 molecules and found that the exact ground-state energies can be systematically approached only if the generators of the QCC ansatz are sampled from a specific set of operators. We report an algorithm for constructing this set that scales linearly with the size of the Hamiltonian.

Unitary Partitioning Approach to the Measurement Problem in the Variational Quantum Eigensolver Method.

To obtain estimates of electronic energies, the Variational Quantum Eigensolver (VQE) technique performs separate measurements for multiple parts of the system Hamiltonian. Current quantum hardware is restricted to projective single-qubit measurements, and, thus, only parts of the Hamiltonian that form mutually qubit-wise commuting groups can be measured simultaneously. The number of such groups in the electronic structure Hamiltonians grows as N4, where N is the number of qubits, thereby putting serious restrictions on the size of the systems that can be studied. Using a partitioning of the system Hamiltonian as a linear combination of unitary operators, we found a circuit formulation of the VQE algorithm that allows one to measure a group of fully anticommuting terms of the Hamiltonian in a single series of single-qubit measurements. Numerical comparison of the unitary partitioning to previously used grouping of Hamiltonian terms based on their qubit-wise commutativity is consistent with an N-fold reduction in the number of measurable groups.

Exact and approximate symmetry projectors for the electronic structure problem on a quantum computer.

Solving the electronic structure problem on a universal-gate quantum computer within the variational quantum eigensolver (VQE) methodology requires constraining the search procedure to a subspace defined by relevant physical symmetries. Ignoring symmetries results in convergence to the lowest eigenstate of the Fock space for the second quantized electronic Hamiltonian. Moreover, this eigenstate can be symmetry broken due to limitations of the wavefunction ansatz. To address this VQE problem, we introduce and assess methods of exact and approximate projectors to irreducible eigensubspaces of available physical symmetries. Feasibility of symmetry projectors in the VQE framework is discussed, and their efficiency is compared with symmetry constraint optimization procedures. Generally, projectors introduce a higher number of terms for VQE measurement compared to the constraint approach. On the other hand, the projection formalism improves accuracy of the variational wavefunction ansatz without introducing additional unitary transformations, which is beneficial for reducing depths of quantum circuits.