![]() Therefore, with an n-qubit state we can efficiently evaluate the expectation value of this 2 n × 2 n Hamiltonian. A quantum device can efficiently evaluate the expectation value of a tensor product of an arbitrary number of simple Pauli operators 23. Thus, the evaluation of ‹ › reduces to the sum of a polynomial number of expectation values of simple Pauli operators for a quantum state | ψ›, multiplied by some real constants. This class of Hamiltonians encompasses a wide range of physical systems, including the electronic structure Hamiltonian of quantum chemistry, the quantum Ising Model, the Heisenberg Model 20, 21, matrices that are well approximated as a sum of n-fold tensor products 22, 23, and more generally any k-sparse Hamiltonian without evident tensor product structure (see Supplementary Methods for details). We consider Hamiltonians that can be written as a polynomial number of terms, with respect to the system size. By using a variational algorithm, this approach reduces the requirement for coherent evolution of the quantum state, making more efficient use of quantum resources, and may offer an alternative route to practical quantum-enhanced computation. The QPU has been experimentally implemented using integrated photonics technology with a spontaneous parametric downconversion single-photon source and combined with an optimization algorithm run on a classical processing unit (CPU), which variationally computes the eigenvalues and eigenvectors of. We have developed a reconfigurable quantum processing unit (QPU), which efficiently calculates the expectation value of a Hamiltonian ( ), providing an exponential speedup over exact diagonalization, the only known exact solution to the problem on a traditional computer. Here we introduce an alternative to QPE that significantly reduces the requirements for coherent evolution. The time the quantum computer must remain coherent is determined by the necessity of O( p −1) successive applications of, each of which can require on the order of millions or billions of quantum gates for practical applications 17, 19, as compared to the tens to hundreds of gates achievable in the short term. In the standard formulation of QPE, one assumes the eigenvector | ψ› of a Hermitian operator is given as input and the problem is to determine the corresponding eigenvalue λ. The QPE algorithm offers an exponential speedup over classical methods and requires a number of quantum operations O( p −1) to obtain an estimate with precision p (refs 13, 14, 15, 16, 17, 18). Quantum approaches to finding eigenvalues have previously relied on the quantum phase estimation (QPE) algorithm. Recent developments in the field of quantum computation offer a way forward for determining efficient solutions of many instances of large eigenvalue problems that are classically intractable 6, 7, 8, 9, 10, 11, 12. Beyond chemistry, the solution of large eigenvalue problems 3 would have applications ranging from determining the results of internet search engines 4 to designing new materials and drugs 5. Many approximate methods 2 have been developed to treat these systems, but efficient, exact methods for large chemical problems remain out of reach for classical computers. However, because the dimension of the problem grows exponentially with the size of the physical system under consideration, exact treatment of these problems remains classically infeasible for compounds with more than 2–3 atoms 1. In chemistry, the properties of atoms and molecules can be determined by solving the Schrödinger equation. The proposed approach drastically reduces the coherence time requirements, enhancing the potential of quantum resources available today and in the near future. We experimentally demonstrate the feasibility of this approach with an example from quantum chemistry-calculating the ground-state molecular energy for He–H +. We implement the algorithm by combining a highly reconfigurable photonic quantum processor with a conventional computer. Here we present an alternative approach that greatly reduces the requirements for coherent evolution and combine this method with a new approach to state preparation based on ansätze and classical optimization. The quantum phase estimation algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution. For quantum systems, where the physical dimension grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer.
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