Measurement Optimization in the Variational Quantum Eigensolver Using a Minimum Clique Cover (1907.03358v4)

Abstract: Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves measurement of the Hamiltonian expectation value. Current hardware can perform only projective single-qubit measurements, and thus, the Hamiltonian expectation value is obtained by measuring parts of the Hamiltonian rather than the full Hamiltonian. This restriction makes the measurement process inefficient because the number of terms in the Hamiltonian grows as $O(N^4)$ with the size of the system, $N$. To optimize VQE measurement one can try to group as many Hamiltonian terms as possible for their simultaneous measurement. Single-qubit measurements allow one to group only the terms that commute within corresponding single-qubit subspaces or qubit-wise commuting. We found that qubit-wise commutativity between the Hamiltonian terms can be expressed as a graph and the problem of the optimal grouping is equivalent of finding a minimum clique cover (MCC) for the Hamiltonian graph. The MCC problem is NP-hard but there exist several polynomial heuristic algorithms to solve it approximately. Several of these heuristics were tested in this work for a set of molecular electronic Hamiltonians. On average, grouping qubit-wise commuting terms reduced the number of operators to measure three times compared to the total number of terms in the considered Hamiltonians.

• The paper demonstrates that grouping Hamiltonian terms via a minimum clique cover heuristic cuts the measurement count nearly threefold.
• It leverages graph theory to map qubit commutativity into cliques and applies several polynomial-time heuristics including LF, GC, and SL.
• Results indicate enhanced computational feasibility in VQE for electronic structure problems, paving the way for advanced quantum algorithms.

Measurement Optimization in the Variational Quantum Eigensolver Using a Minimum Clique Cover

This paper explores an important aspect of the Variational Quantum Eigensolver (VQE), a vital quantum algorithm suited for solving electronic structure problems on quantum computers, specifically regarding the optimization of the Hamiltonian measurement process. The discussion revolves around the inherent limitations presented by current quantum hardware, which predominantly supports projective single-qubit measurements. This constraint complicates the measurement process because it necessitates the separation of the Hamiltonian into multiple terms, leading to inefficiencies that scale with the system size, N, as $O(N^4)$.

The paper focuses on reducing the computational demand of VQE by optimizing how these Hamiltonian terms are grouped for simultaneous measurement. The central concept is to leverage qubit-wise commutativity between terms in a Hamiltonian, framing the problem as one involving graph theory, specifically known as the minimum clique cover (MCC) problem. Here, the Hamiltonian terms are nodes in a graph, with edges indicating commutativity. The goal is to find the smallest number of cliques (fully connected subgraphs), allowing the grouping of terms for concurrent measurement.

Addressing the MCC problem involves solving NP-hard challenges, requiring efficient approximation techniques. Several polynomial-time heuristics were tested, including graph-coloring algorithms (which map the MCC problem onto a complementary graph) and direct clique search and removal strategies. The heuristics deployed included Greedy Coloring (GC), Largest First (LF), Smallest Last (SL), and others, each varying by strategy for ordering and traversing graph vertices for optimal results.

Notably, the paper demonstrates that by using these heuristic methods, the number of individual measurements required can be reduced approximately by a factor of three, compared to measuring each term separately. Among the heuristics, LF exhibited superior performance in minimizing the number of measurement groups required and computational feasibility for larger Hamiltonians. This finding is significant as it lays a foundation for efficient quantum computations in electronic structure problems, which are central to advancements in materials science, chemistry, and related fields.

The results indicate that further improvements could focus on algorithms that incorporate additional degrees of freedom provided by multi-qubit transformations to potentially further reduce the number of measurement groups. The paper opens avenues for further exploration of efficient measurement techniques, contributing to the pragmatic execution of quantum chemical calculations on near-term quantum devices.

This research highlights important implications for both theoretical developments and practical applications in quantum computing, suggesting that while optimizing measurement remains a substantial challenge, viable strategies already exist to make quantum computations more tractable within current technological constraints. Future research could address integrating these techniques with other quantum algorithms or exploring their applications to more complex system models.