Contextual Subspace Variational Quantum Eigensolver

William M. Kirby1, Andrew Tranter1,2, and Peter J. Love1,3

1Department of Physics and Astronomy, Tufts University, Medford, MA 02155
2Cambridge Quantum Computing, 9a Bridge Street Cambridge, CB2 1UB United Kingdom
3Computational Science Initiative, Brookhaven National Laboratory, Upton, NY 11973

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We describe the $\textit{contextual subspace variational quantum eigensolver}$ (CS-VQE), a hybrid quantum-classical algorithm for approximating the ground state energy of a Hamiltonian. The approximation to the ground state energy is obtained as the sum of two contributions. The first contribution comes from a noncontextual approximation to the Hamiltonian, and is computed classically. The second contribution is obtained by using the variational quantum eigensolver (VQE) technique to compute a contextual correction on a quantum processor. In general the VQE computation of the contextual correction uses fewer qubits and measurements than the VQE computation of the original problem. Varying the number of qubits used for the contextual correction adjusts the quality of the approximation. We simulate CS-VQE on tapered Hamiltonians for small molecules, and find that the number of qubits required to reach chemical accuracy can be reduced by more than a factor of two. The number of terms required to compute the contextual correction can be reduced by more than a factor of ten, without the use of other measurement reduction schemes. This indicates that CS-VQE is a promising approach for eigenvalue computations on noisy intermediate-scale quantum devices.

The variational quantum eigensolver (VQE) is a quantum simulation algorithm that estimates the ground state energy of a system, given its Hamiltonian. The quantum computer is used to prepare a guess or “ansatz” for the ground state, and to evaluate its energy. A classical computer is then used to vary the ansatz, and this whole process is repeated, ideally until the energy approaches its global minimum, the ground state energy.
Contextuality is a feature of quantum mechanics that does not appear in classical physics. A system is contextual when one cannot model its observables as having preexisting values before measurement. Applied to VQE, contextuality is a property that the set of measurements involved in evaluating energies may or may not possess. When the set of measurements is noncontextual, it can be described by a classical statistical model, but when it is contextual, such models are generally ruled out.
In this work, we showed how to take a VQE instance and partition it into a noncontextual part and a remaining part that in general is contextual. The noncontextual part can be simulated classically, and the contextual part, which we can think of as encoding the “intrinsically quantum part” of the original problem, is simulated using VQE. We call this algorithm contextual subspace VQE or CS-VQE, and it is an example of a genuinely hybrid quantum-classical algorithm where part of the solution is obtained using a classical computer and part is obtained using a quantum computer.
Since the contextual part is only a subset of the original problem, the VQE algorithm it requires uses fewer qubits and measurements than the original problem, in general. We can vary the size of the contextual part to trade off use of more qubits and measurements for better accuracy in the overall approximation. We tested this for electronic structure Hamiltonians of various atoms and small molecules: in some cases we reached useful accuracy using fewer than half as many qubits as standard VQE, and in nearly all cases at least one qubit was saved. In summary, by using contextuality to isolate the “intrinsically quantum part” of a VQE instance, we can save quantum resources while still taking advantage of those that are available on our quantum computer.

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[1] Kishor Bharti, Alba Cervera-Lierta, Thi Ha Kyaw, Tobias Haug, Sumner Alperin-Lea, Abhinav Anand, Matthias Degroote, Hermanni Heimonen, Jakob S. Kottmann, Tim Menke, Wai-Keong Mok, Sukin Sim, Leong-Chuan Kwek, and Alán Aspuru-Guzik, "Noisy intermediate-scale quantum (NISQ) algorithms", arXiv:2101.08448.

[2] Maxwell D. Radin and Peter Johnson, "Classically-Boosted Variational Quantum Eigensolver", arXiv:2106.04755.

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