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. 2022 Oct 11;23(6):bbac437. doi: 10.1093/bib/bbac437

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© The Author(s) 2022. Published by Oxford University Press.

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (https://creativecommons.org/licenses/by-nc/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is properly cited. For commercial re-use, please contact journals.permissions@oup.com

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(A) Scheme of VQE. Starting from the initialized state where all the qubits are set to (), a trial state (commonly called with the German word ansatz) is prepared using a quantum circuit (the ansatz circuit) whose operations are parametrized by a set of variational parameters (e.g. single-qubit rotations, each parameterized by some angle ). The choice of the ansatz circuit determines which subspace of the general Hilbert space can be spanned by the variational procedure. The electronic structure Hamiltonian can be written as a sum of appropriately weighted products of single-qubit operations, i.e. , where is, for example, , implying rotation on the first and second qubits, rotation on the third, identity on the fourth, in a four-qubit register. Each term is evaluated separately, and post-rotations can be needed to rotate the single qubits to the measurement basis (in the example, the first two qubits need to be rotated in order to measure ). Thanks to these operations, the expectation value of each product in the Hamiltonian can be written in terms of the probabilities of each possible outcome, implying that, for each global cycle, a sufficiently large number of state preparation and measurement cycles needs to be performed (Hamiltonian averaging). The measurements (i.e. the probabilities of each possible outcome) are passed to the training/optimization subroutine on the classical computer, whose task is to calculate and provide a new set of based on the previous history of values. The cycle is repeated until convergence. (B) The ansatz circuit for the Quantum Approximate Optimization Algorithm (QAOA) used in the QuASer (see main text). The QUBO problem of finding the optimal solution of a quadratic Hamiltonian is recast into a quantum annealing form , where is the ‘simple’ mixing Hamiltonian whose ground state is the equal superposition of all basis states and is obtained after the application of Hadamard gates to all the qubits. is the cost Hamiltonian and is defined in such a way that its ground state corresponds to the QUBO solution. The adiabatic evolution that would lead from to is approximated by the sequential application of and where the s and , the ‘time steps’ of the evolution, are the variational parameters (i.e. the set) to be optimized by the classical subroutine.

Inline graphic — (A) Scheme of VQE. Starting from the initialized state where all the qubits are set to (), a trial state (commonly called with the German word ansatz) is prepared using a quantum circuit (the ansatz circuit) whose operations are parametrized by a set of variational parameters (e.g. single-qubit rotations, each parameterized by some angle ). The choice of the ansatz circuit determines which subspace of the general Hilbert space can be spanned by the variational procedure. The electronic structure Hamiltonian can be written as a sum of appropriately weighted products of single-qubit operations, i.e. , where is, for example, , implying rotation on the first and second qubits, rotation on the third, identity on the fourth, in a four-qubit register. Each term is evaluated separately, and post-rotations can be needed to rotate the single qubits to the measurement basis (in the example, the first two qubits need to be rotated in order to measure ). Thanks to these operations, the expectation value of each product in the Hamiltonian can be written in terms of the probabilities of each possible outcome, implying that, for each global cycle, a sufficiently large number of state preparation and measurement cycles needs to be performed (Hamiltonian averaging). The measurements (i.e. the probabilities of each possible outcome) are passed to the training/optimization subroutine on the classical computer, whose task is to calculate and provide a new set of based on the previous history of values. The cycle is repeated until convergence. (B) The ansatz circuit for the Quantum Approximate Optimization Algorithm (QAOA) used in the QuASer (see main text). The QUBO problem of finding the optimal solution of a quadratic Hamiltonian is recast into a quantum annealing form , where is the ‘simple’ mixing Hamiltonian whose ground state is the equal superposition of all basis states and is obtained after the application of Hadamard gates to all the qubits. is the cost Hamiltonian and is defined in such a way that its ground state corresponds to the QUBO solution. The adiabatic evolution that would lead from to is approximated by the sequential application of and where the s and , the ‘time steps’ of the evolution, are the variational parameters (i.e. the set) to be optimized by the classical subroutine.