Many applications of quantum simulation require one to prepare and then characterize quantum states by efficiently estimating k-body reduced density matrices (k-RDMs), from which observables of interest may be obtained. For instance, the fermionic 2-RDM contains the energy, charge density, and energy gradients of an electronic system, while the qubit 2-RDM contains the spatial correlation functions of magnetic systems. Naive estimation of such RDMs requires repeated state preparations for each matrix element, which makes for prohibitively large computation times. However, commuting matrix elements may be measured simultaneously, allowing for a significant cost reduction. In this work, we design schemes for such a parallelization with near-optimal complexity in the system size N. We first describe a scheme to sample all elements of a qubit k-RDM using only O(3klogk-1N) unique measurement circuits, an exponential improvement over prior art. We then describe a scheme for sampling all elements of the fermionic 2-RDM using only O(N2) unique measurement circuits, each of which requires only a local O(N)-depth measurement circuit. We prove a lower bound of ω(ϵ-2Nk) on the number of state preparations, Clifford circuits, and measurement in the computational basis required to estimate all elements of a fermionic k-RDM, making our scheme for sampling the fermionic 2-RDM asymptotically optimal. We finally construct circuits to sample the expectation value of a linear combination of ω anticommuting two-body fermionic operators with only O(ω) gates on a linear array. These circuits allows for sampling any linear combination of fermionic 2-RDM elements in O(N4/ω) time, with a significantly lower measurement circuit complexity than prior art. Our results improve the viability of near-term quantum simulation of molecules and strongly correlated material systems.

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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.

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Variational quantum eigensolvers offer a small-scale testbed to demonstrate the performance of error mitigation techniques with low experimental overhead. We present successful error mitigation by applying the recently proposed symmetry verification technique to the experimental estimation of the ground-state energy and ground state of the hydrogen molecule. A finely adjustable exchange interaction between two qubits in a circuit QED processor efficiently prepares variational ansatz states in the single-excitation subspace respecting the parity symmetry of the qubit-mapped Hamiltonian. Symmetry verification improves the energy and state estimates by mitigating the effects of qubit relaxation and residual qubit excitation, which violate the symmetry. A full-density-matrix simulation matching the experiment dissects the contribution of these mechanisms from other calibrated error sources. Enforcing positivity of the measured density matrix via scalable convex optimization correlates the energy and state estimate improvements when using symmetry verification, with interesting implications for determining system properties beyond the ground-state energy.

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Modeling chemical reactions and complicated molecular systems has been proposed as the“killer application”of a future quantumcomputer. Accurate calculations of derivatives of molecular eigenenergies are essential toward this end, allowing for geometryoptimization, transition state searches, predictions of the response to an applied electric or magneticfield, and molecular dynamicssimulations. In this work, we survey methods to calculate energy derivatives, and present two new methods: one based onquantum phase estimation, the other on a low-order response approximation. We calculate asymptotic error bounds andapproximate computational scalings for the methods presented. Implementing these methods, we perform geometry optimizationon an experimental quantum processor, estimating the equilibrium bond length of the dihydrogen molecule to within 0:014 Å ofthe full configuration interaction value. Within the same experiment, we estimate the polarizability of the H2molecule,findingagreement at the equilibrium bond length to within 0:06 a.u. (2%relative error)@en

We investigate the performance of error mitigation via measurement of conserved symmetries on near-term devices. We present two protocols to measure conserved symmetries during the bulk of an experiment, and develop a third, zero-cost, post-processing protocol which is equivalent to a variant of the quantum subspace expansion. We develop methods for inserting global and local symmetries into quantum algorithms, and for adjusting natural symmetries of the problem to boost the mitigation of errors produced by different noise channels. We demonstrate these techniques on two- and four-qubit simulations of the hydrogen molecule (using a classical density-matrix simulator), finding up to an order of magnitude reduction of the error in obtaining the ground-state dissociation curve.

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