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Variational Quantum Eigensolvers (VQEs) represent a promising approach to computing molecular ground states and energies on modern quantum computers. These approaches use a classical computer to optimize the parameters of a trial wave function, while the quantum computer simulates the energy by preparing and measuring a set of bitstring observations, referred to as shots, over which an expected value is computed. Although more shots improve the accuracy of the expected ground state, it also increases the simulation cost. Hence, we propose modifications to the standard Bayesian optimization algorithm to leverage few-shot circuit observations to solve VQEs with fewer quantum resources. We demonstrate the effectiveness of our proposed approach, Bayesian optimization with priors on surface topology (BOPT), by comparing optimizers for molecular systems and demonstrate how current quantum hardware can aid in finding ground state energies., Comment: 28 pages, 9 figures

Quantum computers currently rely on a hybrid quantum-classical approach known as Variational Quantum Algorithms (VQAs) to solve problems. Still, there are several challenges with VQAs on the classical computing side: it corresponds to a black-box optimization problem that is generally non-convex, the observations from the quantum hardware are noisy, and the quantum computing time is expensive. The first point is inherent to the problem structure; as a result, it requires the classical part of VQAs to be solved using global optimization strategies. However, there is a trade-off between cost and accuracy; typically, quantum computers return a set of bit strings, where each bitstring is referred to as a shot. The probabilistic nature of quantum computing (QC) necessitates many shots to measure the circuit accurately. Since QC time is charged per shot, reducing the number of shots yields cheaper and less accurate observations. Recently, there has been increasing interest in using basic Bayesian optimization (BO) methods to globally optimize quantum circuit parameters. This work proposes two modifications to the basic BO framework to provide a shot-efficient optimization strategy for VQAs. Specifically, we provide a means to place a prior on the periodicity of the rotation angles and a framework to place a topological prior using few-shot quantum circuit observations. We demonstrate the effectiveness of our proposed approach through an ablation study, showing that using both proposed features statistically outperforms a standard BO implementation within VQAs for computational chemistry simulations., Comment: 8 pages, 5 figures

Variational quantum eigensolvers are touted as a near-term algorithm capable of impacting many applications. However, the potential has not yet been realized, with few claims of quantum advantage and high resource estimates, especially due to the need for optimization in the presence of noise. Finding algorithms and methods to improve convergence is important to accelerate the capabilities of near-term hardware for VQE or more broad applications of hybrid methods in which optimization is required. To this goal, we look to use modern approaches developed in circuit simulations and stochastic classical optimization, which can be combined to form a surrogate optimization approach to quantum circuits. Using an approximate (classical CPU/GPU) state vector simulator as a surrogate model, we efficiently calculate an approximate Hessian, passed as an input for a quantum processing unit or exact circuit simulator. This method will lend itself well to parallelization across quantum processing units. We demonstrate the capabilities of such an approach with and without sampling noise and a proof-of-principle demonstration on a quantum processing unit utilizing 40 qubits., Comment: 7 pages, + appendix

The unitary coupled cluster (UCC) ansatz is a promising tool for achieving high-precision results using the variational quantum eigensolver (VQE) algorithm in the NISQ era. However, results on quantum hardware are thus far very limited and simulations have only accessed small system sizes. We advance the state of the art of UCC simulations by utilizing an efficient sparse wavefunction circuit solver and studying systems up to 64 qubits. Here we report results obtained using this solver that demonstrate the power of the UCC ansatz and address pressing questions about optimal initial parameterizations and circuit construction, among others. Our approach enables meaningful benchmarking of the UCC ansatz, a crucial step in assessing the utility of VQE for achieving quantum advantage.

It is crucial to reduce the resources required to run quantum algorithms and simulate physical systems on quantum computers due to coherence time limitations. With regards to Hamiltonian simulation, a significant effort has focused on building efficient algorithms using various factorizations and truncations, typically derived from the Hamiltonian alone. We introduce a new paradigm for improving Hamiltonian simulation and reducing the cost of ground state problems based on ideas recently developed for classical chemistry simulations. The key idea is that one can find efficient ways to reduce resources needed by quantum algorithms by making use of two key pieces of information: the Hamiltonian operator and an approximate ground state wavefunction. We refer to our algorithm as the $\textit{Self-consistent Quantum Iteratively Sparsified Hamiltonian}$ (SQuISH). By performing our scheme iteratively, one can drive SQuISH to create an accurate wavefunction using a truncated, resource-efficient Hamiltonian. Utilizing this truncated Hamiltonian provides an approach to reduce the gate complexity of ground state calculations on quantum hardware. As proof of principle, we implement SQuISH using configuration interaction for small molecules and coupled cluster for larger systems. Through our combination of approaches, we demonstrate how SQuISH performs on a range of systems, the largest of which would require more than 200 qubits to run on quantum hardware. Though our demonstrations are on a series of electronic structure problems, our approach is relatively generic and hence likely to benefit additional applications where the size of the problem Hamiltonian creates a computational bottleneck., Comment: 15 pages, 11 figures

Compact representations of fermionic Hamiltonians are necessary to perform calculations on quantum computers that lack error-correction. A fermionic system is typically defined within a subspace of fixed particle number and spin while unnecessary states are projected out of the Hilbert space. We provide a bijective mapping using combinatoric ranking to bijectively map fermion basis states to qubit basis states and express operators in the standard spin representation. We then evaluate compact mapping using the Variational Quantum Eigensolver (VQE) with the unitary coupled cluster singles and doubles excitations (UCCSD) ansatz in the compact representation. Compactness is beneficial when the orbital filling is well away from half, and we show at 30 spin orbital $H_{2}$ calculation with only 8 qubits. We find that the gate depth needed to prepare the compact wavefunction is not much greater than the full configuration space in practice. A notable observation regards the number of calls to the optimizer needed for the compact simulation compared to the full simulation. We find that the compact representation converges faster than the full representation using the ADAM optimizer in all cases. Our analysis demonstrates the effect of compact mapping in practice., Comment: 10 pages, 4 figures

Reference arXiv:1904.03196 recently introduced an algorithm (QPS) for simulating parton showers with intermediate flavor states using polynomial resources on a digital quantum computer. We make use of a new quantum hardware capability called dynamical quantum computing to improve the scaling of this algorithm to significantly improve the method precision. In particular, we modify the quantum parton shower circuit to incorporate mid-circuit qubit measurements, resets, and quantum operations conditioned on classical information. This reduces the computational depth from $\mathcal{O}(N^5\log_2(N)^2)$ to $\mathcal{O}(N^3\log_2(N)^2)$ and the qubit requirements are reduced from $\mathcal{O}(N\log_2(N))$ to $\mathcal{O}(N)$. Using "matrix product state" statevector simulators, we demonstrate that the improved algorithm yields expected results for 2, 3, 4, and 5-steps of the algorithm. We compare absolute costs with the original QPS algorithm, and show that dynamical quantum computing can significantly reduce costs in the class of digital quantum algorithms representing quantum walks (which includes the QPS)., Comment: Updated to journal version