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Adaptive variational quantum algorithms arguably offer the best prospects for quantum advantage in the NISQ era. Since the inception of the first such algorithm, ADAPT-VQE, many improvements have appeared in the literature. We combine the key improvements along with a novel operator pool -- which we term Coupled Exchange Operator (CEO) pool -- to assess the cost of running state-of-the-art ADAPT-VQE on hardware in terms of measurement counts and circuit depth. We show a dramatic reduction of these quantum resources compared to the early versions of the algorithm. We also find that our state-of-the-art CEO-ADAPT-VQE outperforms UCCSD, the most widely regarded static VQE ansatz, in all relevant metrics.

Standing as one of the most significant barriers to reaching quantum advantage, state-preparation fidelities on noisy intermediate-scale quantum processors suffer from quantum-gate errors, which accumulate over time. A potential remedy is pulse-based state preparation. We numerically investigate the minimal evolution times (METs) attainable by optimizing (microwave and exchange) pulses on silicon hardware. We investigate two state preparation tasks. First, we consider the preparation of molecular ground states and find the METs for H$_2$, HeH$^+$, and LiH to be 2.4 ns, 4.4 ns, and 27.2 ns, respectively. Second, we consider transitions between arbitrary states and find the METs for transitions between arbitrary four-qubit states to be below 50 ns. For comparison, connecting arbitrary two-qubit states via one- and two-qubit gates on the same silicon processor requires approximately 200 ns. This comparison indicates that pulse-based state preparation is likely to utilize the coherence times of silicon hardware more efficiently than gate-based state preparation. Finally, we quantify the effect of silicon device parameters on the MET. We show that increasing the maximal exchange amplitude from 10 MHz to 1 GHz accelerates the METs, e.g., for H$_2$ from 84.3 ns to 2.4 ns. This demonstrates the importance of fast exchange. We also show that increasing the maximal amplitude of the microwave drive from 884 kHz to 56.6 MHz shortens state transitions, e.g., for two-qubit states from 1000 ns to 25 ns. Our results bound both the state-preparation times for general quantum algorithms and the execution times of variational quantum algorithms with silicon spin qubits., Comment: 9 + (7) pages, 6 figs, comments are welcomed

The cluster-based Mean Field method (cMF) and it's second order perturbative correction[1], was introduced by Jim\'enez-Hoyos and Scuseria to reduce the cost of modeling strongly correlated systems by dividing an active space up into small clusters, which are individually solved in the mean-field presence of each other. In that work, clusters with unpaired electrons are treated naturally, by allowing the $\alpha$ and $\beta$ orbitals to spin polarize. While that provided significant energetic stabilization, the resulting cMF wavefunction was spin-contaminated, making it difficult to use as a reference state for spin-pure post-cMF methods. In this work, we propose the Restricted Open-shell cMF (RO-cMF) method, extending the cMF approach to systems with open-shell clusters, while not permitting spin-polarization. While the resulting RO-cMF energies are necessarily higher in energy than the unrestricted orbital cMF, the new RO-cMF provides a simple reference state for post-cMF methods that recover the missing inter-cluster correlations. We provide a detailed explanation of the method, and report demonstrative calculations of exchange coupling constants for three systems: a di-iron complex, a di-chromium complex, and a dimerized organic radical. We also report the first perturbatively corrected RO-cMF-PT2 results as well.

In conventional variational quantum eigensolvers (VQEs), trial states are prepared by applying series of parameterized gates to a reference state, with the gate parameters being varied to minimize the energy of the target system. Recognizing that the gates are intermediates which are ultimately compiled into a set of control pulses to be applied to each qubit in the lab, the recently proposed ctrl-VQE algorithm takes the amplitudes, frequencies, and phases of the pulse as the variational parameters used to minimize the molecular energy. In this work, we explore how all three degrees of freedom interrelate with one another. To this end, we consider several distinct strategies to parameterize the control pulses, assessing each one through numerical simulations of a transmon-like device. For each parameterization, we contrast the pulse duration required to prepare a good ansatz, and the difficulty to optimize that ansatz from a well-defined initial state. We deduce several guiding heuristics to implement practical ctrl-VQE in hardware, which we anticipate will generalize for generic device architectures., Comment: 13 pages (10 of main text), 6 figures

The Adaptive Derivative-Assembled Pseudo-Trotter Variational Quantum Eigensolver (ADAPT-VQE) has emerged as a pivotal promising approach for electronic structure challenges in quantum chemistry with noisy quantum devices. Nevertheless, to surmount existing technological constraints, this study endeavors to enhance ADAPT-VQE's efficacy. Leveraging insights from electronic structure theory, we concentrate on optimizing state preparation without added computational burden and guiding ansatz expansion to yield more concise wavefunctions with expedited convergence toward exact solutions. These advancements culminate in shallower circuits and, as demonstrated, reduced measurement requirements. This research delineates these enhancements and assesses their performance across mono, di, and tridimensional arrangements of H4 models, as well as in the water molecule. Ultimately, this work attests to the viability of physically-motivated strategies in fortifying ADAPT-VQE's efficiency, marking a significant stride in quantum chemistry simulations.

The task of computing wavefunctions that are accurate, yet simple enough mathematical objects to use for reasoning has long been a challenge in quantum chemistry. The difficulty in drawing physical conclusions from a wavefunction is often related to the generally large number of configurations with similar weights. In Tensor Product Selected CI, we use a locally correlated tensor product state basis, which has the effect of concentrating the weight of a state onto a smaller number of physically interpretable degrees of freedom. In this paper, we apply TPSCI to a series of three molecular systems ranging in separability, one of which is the first application of TPSCI to an open-shell bimetallic system. For each of these systems, we obtain accurate solutions to large active spaces, and analyze the resulting wavefunctions through a series of different approaches including (i) direct inspection of the TPS basis coefficients, (ii) construction of Bloch effective Hamiltonians, and (iii) computation of cluster correlation functions.

Adaptive protocols enable the construction of more efficient state preparation circuits in variational quantum algorithms (VQAs) by utilizing data obtained from the quantum processor during the execution of the algorithm. This idea originated with ADAPT-VQE, an algorithm that iteratively grows the state preparation circuit operator by operator, with each new operator accompanied by a new variational parameter, and where all parameters acquired thus far are optimized in each iteration. In ADAPT-VQE and other adaptive VQAs that followed it, it has been shown that initializing parameters to their optimal values from the previous iteration speeds up convergence and avoids shallow local traps in the parameter landscape. However, no other data from the optimization performed at one iteration is carried over to the next. In this work, we propose an improved quasi-Newton optimization protocol specifically tailored to adaptive VQAs. The distinctive feature in our proposal is that approximate second derivatives of the cost function are recycled across iterations in addition to parameter values. We implement a quasi-Newton optimizer where an approximation to the inverse Hessian matrix is continuously built and grown across the iterations of an adaptive VQA. The resulting algorithm has the flavor of a continuous optimization where the dimension of the search space is augmented when the gradient norm falls below a given threshold. We show that this inter-optimization exchange of second-order information leads the Hessian in the state of the optimizer to better approximate the exact Hessian. As a result, our method achieves a superlinear convergence rate even in situations where the typical quasi-Newton optimizer converges only linearly. Our protocol decreases the measurement costs in implementing adaptive VQAs on quantum hardware as well as the runtime of their classical simulation.

ADAPT-VQE is one of the leading VQE algorithms which circumvents the choice-of-ansatz conundrum by iteratively growing compact and arbitrarily accurate problem-tailored ans\"atze. However, for hardware-efficient operator pools, the gradient-measurement step of the algorithm requires the estimation of $O(N^8)$ observables, which may represent a bottleneck for relevant system sizes on real devices. We present an efficient strategy for measuring the pool gradients based on simultaneously measuring commuting observables. We argue that our approach is relatively robust to shot-noise effects, and show that measuring the pool gradients is in fact only $O(N)$ times as expensive as a naive VQE iteration. Our proposed measurement strategy significantly ameliorates the measurement overhead of ADAPT-VQE and brings us one step closer to practical implementations on real devices., Comment: 8 pages, 2 figures

In a recent paper [JCTC, 2020, 16, 6098], we introduced a new approach for accurately approximating full CI ground states in large electronic active-spaces, called Tensor Product Selected CI (TPSCI). In TPSCI, a large orbital active space is first partitioned into disjoint sets (clusters) for which the exact, local many-body eigenstates are obtained. Tensor products of these locally correlated many-body states are taken as the basis for the full, global Hilbert space. By folding correlation into the basis states themselves, the low-energy eigenstates become increasingly sparse, creating a more compact selected CI expansion. While we demonstrated that this approach can improve accuracy for a variety of systems, there is even greater potential for applications to excited states, particularly those which have some excited state character. In this paper, we report on the accuracy of TPSCI for excited states, including a far more efficient implementation in the Julia programming language. In traditional SCI methods that use a Slater determinant basis, accurate excitation energies are obtained only after a linear extrapolation and at a large computational cost. We find that TPSCI with perturbative corrections provides accurate excitation energies for several excited states of various polycyclic aromatic hydrocarbons (PAH) with respect to the extrapolated result (i.e. near exact result). Further, we use TPSCI to report highly accurate estimates of the lowest 31 eigenstates for a tetracene tetramer system with an active space of 40 electrons in 40 orbitals, giving direct access to the initial bright states and the resulting 18 doubly excited (biexcitonic) states.

Accurate modeling of the response of molecular systems to an external electromagnetic field is challenging on classical computers, especially in the regime of strong electronic correlation. In this paper, we develop a quantum linear response (qLR) theory to calculate molecular response properties on near-term quantum computers. Inspired by the recently developed variants of the quantum counterpart of equation of motion (qEOM) theory, the qLR formalism employs "killer condition" satisfying excitation operator manifolds that offers a number of theoretical advantages along with reduced quantum resource requirements. We also used the qEOM framework in this work to calculate state-specific response properties. Further, through noise-less quantum simulations, we show that response properties calculated using the qLR approach are more accurate than the ones obtained from the classical coupled-cluster based linear response models due to the improved quality of the ground-state wavefunction obtained using the ADAPT-VQE algorithm.

Adaptive quantum variational algorithms are particularly promising for simulating strongly correlated systems on near-term quantum hardware, but they are not yet viable due, in large part, to the severe coherence time limitations on current devices. In this work, we introduce an algorithm called TETRIS-ADAPT-VQE, which iteratively builds up variational ans\"atze a few operators at a time in a way dictated by the problem being simulated. This algorithm is a modified version of the ADAPT-VQE algorithm in which the one-operator-at-a-time rule is lifted to allow for the addition of multiple operators with disjoint supports in each iteration. TETRIS-ADAPT-VQE results in denser but significantly shallower circuits, without increasing the number of CNOT gates or variational parameters. Its advantage over the original algorithm in terms of circuit depths increases with the system size. Moreover, the expensive step of measuring the energy gradient with respect to each candidate unitary at each iteration is performed only a fraction of the time compared to ADAPT-VQE. These improvements bring us closer to the goal of demonstrating a practical quantum advantage on quantum hardware., Comment: 10 pages, 7 figures

The recent quantum information boom has effected a resurgence of interest in unitary coupled cluster (UCC) theory. Our group's interest in local energy landscapes of unitary ans\"atze prompted us to investigate the classical approach of truncating the Taylor series expansion (instead of a perturbative expansion) of UCCSD energy at second-order. This amounts to an approach where electron correlation energy is estimated by taking a single Newton-Raphson step from Hartree-Fock toward UCCSD. Such an approach has been explored previously, but the accuracy was not extensively studied. In this paper, we investigate the performance and observe similar pathologies to linearized coupled cluster with singles and doubles. We introduce the use of derivatives of order three or greater to help partially recover the variational lower bound of true UCCSD, restricting these derivatives to those of the "unmixed" category in order to simplify the model. By testing the approach on several potential energy surfaces and reaction energies, we find this "diagonal" approximation to higher order terms to be effective at reducing sensitivity near singularities for strongly correlated regimes, while not significantly diminishing the accuracy of weakly correlated systems., Comment: 11 pages, 7 figures (including SI)

Because quantum simulation of molecular systems is expected to provide the strongest advantage over classical computing methods for systems exhibiting strong electron correlation, it is critical that the performance of VQEs be assessed for strongly correlated systems. For classical simulation, strong correlation often results in symmetry-breaking of the Hartree-Fock reference, leading to L\"owdin's well-known ``symmetry dilemma'' whereby accuracy in the energy can be increased by breaking spin or spatial symmetries. Here, we explore the impact of symmetry breaking on the performance of ADAPT-VQE using two strongly correlated systems: (i) the ``fermionized" anisotropic Heisenberg model, where the anisotropy parameter controls the correlation in the system, and (ii) symmetrically-stretched linear \ce{H4}, where correlation increases with increasing H-H separation. In both of these cases, increasing the level of correlation of the system leads to spontaneous symmetry breaking (parity and $\hat{S}^{2}$, respectively) of the mean-field solutions. We analyze the role that symmetry breaking in the reference states and orbital mappings of the fermionic Hamiltonians have on the compactness and performance of ADAPT-VQE. We observe that improving the energy of the reference states by breaking symmetry has a deleterious effect on ADAPT-VQE by increasing the length of the ansatz necessary for energy convergence and exacerbating the problem of ``gradient troughs"., Comment: 16 pages, 5 figures

Adaptive variational quantum simulation algorithms use information from the quantum computer to dynamically create optimal trial wavefunctions for a given problem Hamiltonian. A key ingredient in these algorithms is a predefined operator pool from which trial wavefunctions are constructed. Finding suitable pools is critical for the efficiency of the algorithm as the problem size increases. Here, we present a technique called operator pool tiling that facilitates the construction of problem-tailored pools for arbitrarily large problem instances. By first performing an ADAPT-VQE calculation on a smaller instance of the problem using a large, but computationally inefficient operator pool, we extract the most relevant operators and use them to design more efficient pools for larger instances. We demonstrate the method here on strongly correlated quantum spin models in one and two dimensions, finding that ADAPT automatically finds a highly effective ansatz for these systems. Given that many problems, such as those arising in condensed matter physics, have a naturally repeating lattice structure, we expect the pool tiling method to be a widely applicable technique apt for such systems., Comment: 9 pages, 4 figures

Near-term quantum computers are expected to facilitate material and chemical research through accurate molecular simulations. Several developments have already shown that accurate ground-state energies for small molecules can be evaluated on present-day quantum devices. Although electronically excited states play a vital role in chemical processes and applications, the search for a reliable and practical approach for routine excited-state calculations on near-term quantum devices is ongoing. Inspired by excited-state methods developed for the unitary coupled-cluster theory in quantum chemistry, we present an equation-of-motion-based method to compute excitation energies following the variational quantum eigensolver algorithm for ground-state calculations on a quantum computer. We perform numerical simulations on H$_2$, H$_4$, H$_2$O, and LiH molecules to test our quantum self-consistent equation-of-motion (q-sc-EOM) method and compare it to other current state-of-the-art methods. q-sc-EOM makes use of self-consistent operators to satisfy the vacuum annihilation condition, a critical property for accurate calculations. It provides real and size-intensive energy differences corresponding to vertical excitation energies, ionization potentials and electron affinities. We also find that q-sc-EOM is more suitable for implementation on NISQ devices as it is expected to be more resilient to noise compared with the currently available methods.

Size extensivity, defined as the correct scaling of energy with system size, is a desirable property for any many-body method. Traditional CI methods are not size extensive hence the error increases as the system gets larger. Coupled electron pair approximation (CEPA) methods can be constructed as simple extensions of truncated configuration interaction (CI) that ensures size extensivity. One of the major issues with the CEPA and its variants is that singularities arise in the amplitude equations when the system starts to be strongly correlated. In this work, we extend the traditional Slater determinant-based coupled electron pair approaches like CEPA-0, averaged coupled-pair functional (ACPF) and average quadratic coupled-cluster (AQCC) to a new formulation based on tensor product states (TPS). We show that a TPS basis can often be chosen such that it removes the singularities that commonly destroy the accuracy of CEPA-based methods. A suitable TPS representation can be formed by partitioning the system into separate disjoint clusters and forming the final wavefunction as the tensor product of the many body states of these clusters. We demonstrate the application of these methods on simple bond breaking systems such as CH$_4$ and F$_2$ where determinant based CEPA methods fail. We further apply the TPS-CEPA approach to stillbene isomerization and few planar $\pi-$conjugated systems. Overall the results show that the TPS-CEPA method can remove the singularities and provide improved numerical results compared to common electronic structure methods.

Many classical optimization problems can be mapped to finding the ground states of diagonal Ising Hamiltonians, for which variational quantum algorithms such as the Quantum Approximate Optimization Algorithm (QAOA) provide heuristic methods. Because the solutions of such classical optimization problems are necessarily product states, it is unclear how entanglement affects their performance. An Adaptive Derivative-Assembled Problem-Tailored (ADAPT) variation of QAOA improves the convergence rate by allowing entangling operations in the mixer layers whereas it requires fewer CNOT gates in the entire circuit. In this work, we study the entanglement generated during the execution of ADAPT-QAOA. Through simulations of the weighted Max-Cut problem, we show that ADAPT-QAOA exhibits substantial flexibility in entangling and disentangling qubits. By incrementally restricting this flexibility, we find that a larger amount of entanglement entropy at earlier stages coincides with faster convergence at later stages. In contrast, while the standard QAOA quickly generates entanglement within a few layers, it cannot remove excess entanglement efficiently. Our results demonstrate that the role of entanglement in quantum optimization is subtle and provide guidance for building favorable features into quantum optimization algorithms., Comment: 15 pages, 7 figures. Comments are welcome

Variational quantum eigensolvers (VQEs) represent a powerful class of hybrid quantum-classical algorithms for computing molecular energies. Various numerical issues exist for these methods, however, including barren plateaus and large numbers of local minima. In this work, we consider Adaptive, Problem-Tailored (ADAPT)-VQE ans\"atze, and examine how they are impacted by these local minima. We find that while ADAPT-VQE does not remove local minima, the gradient-informed, one-operator-at-a-time circuit construction seems to accomplish two things: First, it provides an initialization strategy that is dramatically better than random initialization, and which is applicable in situations where chemical intuition cannot help with initialization, i.e., when Hartree-Fock is a poor approximation to the ground state. Second, even if an ADAPT-VQE iteration converges to a local trap at one step, it can still "burrow" toward the exact solution by adding more operators, which preferentially deepens the occupied trap. This same mechanism helps highlight a surprising feature of ADAPT-VQE: It should not suffer optimization problems due to "barren plateaus". Even if barren plateaus appear in the parameter landscape, our analysis and simulations reveal that ADAPT-VQE avoids such regions by design.

The preparation of Gibbs thermal states is an important task in quantum computation with applications in quantum simulation, quantum optimization, and quantum machine learning. However, many algorithms for preparing Gibbs states rely on quantum subroutines which are difficult to implement on near-term hardware. Here, we address this by (i) introducing an objective function that, unlike the free energy, is easily measured, and (ii) using dynamically generated, problem-tailored ans\"atze. This allows for arbitrarily accurate Gibbs state preparation using low-depth circuits. To verify the effectiveness of our approach, we numerically demonstrate that our algorithm can prepare high-fidelity Gibbs states across a broad range of temperatures and for a variety of Hamiltonians., Comment: 7 pages, 3 figures

Quantum simulation on NISQ devices is severely limited by short coherence times. A variational pulse-shaping algorithm known as ctrl-VQE was recently proposed to address this issue by eliminating the need for parameterized quantum circuits, which lead to long state preparation times. Here, we find the shortest possible pulses for ctrl-VQE to prepare target molecular wavefunctions for a given device Hamiltonian describing coupled transmon qubits. We find that the time-optimal pulses that do this have a bang-bang form consistent with Pontryagin's maximum principle. We further investigate how the minimal state preparation time is impacted by truncating the transmons to two versus more levels. We find that leakage outside the computational subspace (something that is usually considered problematic) speeds up the state preparation, further reducing device coherence-time demands. This speedup is due to an enlarged solution space of target wavefunctions and to the appearance of additional channels connecting initial and target states.