The term potential energy was introduced by the 19th-century Scottish engineer and physicist William Rankine, although it has links to Greek philosopher Aristotle's concept of potentiality. The unit for energy in the International System of Units (SI) is the joule, which has the symbol J. Ĭommon types of potential energy include the gravitational potential energy of an object that depends on its mass and its distance from the center of mass of another object, the elastic potential energy of an extended spring, and the electric potential energy of an electric charge in an electric field. Our approach makes limited connectivity quantum computing architectures more useful for quantum chemistry simulations.In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. This operation reduces the depth of the quantum circuit as one does not need to create long-distance correlations. Our new algorithm abbreviated as permutation VQE (PermVQE) allows for efficient reordering of qubits, so that those qubits that are highly correlated are mapped to qubits that are closer on a given chip. Here we present a novel procedure to perform VQE calculations with lower depth circuits on quantum chips with limited connectivity, which is present on most quantum computing architectures. The difficulty is due to the limited number of qubits and/or massive quantum circuits coupled with the large number of variational parameters to optimize. Although successful implementations of the VQE on noisy intermediate-scale gate-based quantum computers have been presented across several “toy” molecular systems, demonstrating quantum chemistry advantage is challenging. The variational quantum eigensolver (VQE) is currently the leading approach to solve the molecular electronic Hamiltonian eigenvalue-eigenvector problem on near-term quantum computers, producing the ground and excited states of molecules and generating potential energy surfaces. It promises an exponential speedup over classical computing, as it reduces the scaling in the number of quantum particles as well as addresses the problem of memory storage required to represent the quantum wave function. Quantum chemistry is regarded to be one of the first disciplines that will be revolutionized by quantum computing. In particular, we demonstrate the beneficial effect of qubit permutations to build fermionic–adaptive derivative assembled pseudo-Trotter ansatz on a linear qubit connectivity architecture with nearly a twofold reduction of the number of controlled not gates. The main ideas can also be applied to simulate molecules with other ansatz as well as variational quantum algorithms beyond the VQE. The approach is designed for hardware-efficient ansatz of any qubit connectivity, and examples are demonstrated for linear and two-dimensional grid architectures. For representative molecular systems, LiH, H 2, ( H 2 ) 2, H 4 ≠, H 3 +, and N 2, we demonstrate that placing entangled qubits in close proximity leads to shallower depth circuits required to reach a given eigenvalue-eigenvector accuracy. Encoding strongly entangled spin-orbitals into proximal qubits on a quantum chip naturally reduces the circuit depth needed to prepare the ground state. The choice of permutations is based on mutual information, which is a measure of interaction between electrons and/or holes in spin-orbitals. Our approach, called “PermVQE,” adds an additional optimization loop to the VQE that permutes qubits in order to solve for the qubit Hamiltonian that maximally localizes correlations in the ground state. In this work, we propose an approach to reduce ansatz circuit depth. Increased depth can both degrade the accuracy of the results and reduce trainability. However, the circuit depth is expected to grow significantly with the problem size. The variational quantum eigensolver (VQE) is a method of choice to solve the electronic structure problem for molecules on near-term gate-based quantum computers.
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