The measurement cost for estimating expectation values of Hamiltonians is a central bottleneck in variational quantum algorithms. Grouping strategies significantly reduce this cost, with overlapping techniques being the state of the art in the field. Overlapping grouping methods require i) a non-overlapping grouping of the Hamiltonian, typically obtained from the Sorted Insertion (SI) algorithm as initialization, and ii) the construction of covariance dictionaries from approximate wavefunctions to guide the optimization. It was recently shown that different initializations can potentially reduce measurement costs for overlapping methods. Motivated by these findings, we introduce variance-aware SI (VarSI), a family of covariance-informed non-overlapping Pauli grouping heuristics to reduce measurement counts. VarSI grouping leverages the covariance dictionaries, already required by overlapping methods, to construct better non-overlapping groups. We propose three variants: a global greedy grouping insertion rule, a variance-informed SI analog, and a local refinement step initialized from SI or our variance-informed variant. We showcase the use of groupings generated by our VarSI heuristic algorithms to initialize overlapping methods using the iterative coefficient-splitting (ICS) algorithm. Molecular benchmarks with 130 Hamiltonians demonstrate consistent, non-overlapping measurement improvements over SI of 38\% and enhanced downstream ICS results when initialized from VarSI groups. We find that the initializations considered here achieve up to 70\% measurement reductions for ICS, compared to the standard SI initialization with mean reductions of 9--15.3\% depending on qubit mappings and covariance dictionaries used. These results show that non-overlapping grouping remains a consequential design step even when the final estimator uses overlapping fragments.
The measurement cost for estimating expectation values of Hamiltonians is a central bottleneck in variational quantum algorithms. Grouping strategies significantly reduce this cost, with overlapping techniques being the state of the art in the field. Overlapping grouping methods require i) a non-overlapping grouping of the Hamiltonian, typically obtained from the Sorted Insertion (SI) algorithm as initialization, and ii) the construction of covariance dictionaries from approximate wavefunctions to guide the optimization. It was recently shown that different initializations can potentially reduce measurement costs for overlapping methods. Motivated by these findings, we introduce variance-aware SI (VarSI), a family of covariance-informed non-overlapping Pauli grouping heuristics to reduce measurement counts. VarSI grouping leverages the covariance dictionaries, already required by overlapping methods, to construct better non-overlapping groups. We propose three variants: a global greedy grouping insertion rule, a variance-informed SI analog, and a local refinement step initialized from SI or our variance-informed variant. We showcase the use of groupings generated by our VarSI heuristic algorithms to initialize overlapping methods using the iterative coefficient-splitting (ICS) algorithm. Molecular benchmarks with 130 Hamiltonians demonstrate consistent, non-overlapping measurement improvements over SI of 38% and enhanced downstream ICS results when initialized from VarSI groups. We find that the initializations considered here achieve up to 70% measurement reductions for ICS, compared to the standard SI initialization with mean reductions of 9--15.3% depending on qubit mappings and covariance dictionaries used. These results show that non-overlapping grouping remains a consequential design step even when the final estimator uses overlapping fragments.
Repeated measurements can dominate the resources required for quantum energy estimation, making the choice of which Pauli observables to measure together a central optimization problem for variational quantum algorithms. We formulate fully commuting measurement grouping as a classical binary optimization problem based on clique selection and construct non-overlapping groups using mixed-integer linear programming (MILP). For a benchmark of molecular Hamiltonians, groupings optimized with approximate covariances reduce the non-overlapping measurement requirement ε2M by 51.8% on average relative to sorted insertion (SI). Using the MILP groups to initialize iterative coefficient splitting (ICS), denoted MILP-ICS, yields an average 24.3% reduction relative to ICS initialized from SI (SI-ICS). The optimized groups also transfer across nearby molecular geometries while preserving substantial measurement savings. We further introduce O-clique, which directly optimizes overlapping commuting supports through candidate-clique selection and coefficient profiles. Although O-clique provides only modest additional reductions beyond MILP-ICS, it reduces the measurement requirement by 27.3% on average relative to SI-ICS with 100 iterations, despite using only five final coefficient refinement iterations, indicating improved support quality. Finally, we extend the comparison to Fermi--Hubbard, Kitaev--Heisenberg--Γ, and XYZ lattice Hamiltonians, where MILP-based groupings substantially outperform the corresponding SI-based strategies. Together, these results show that variance-informed optimization of measurement-group structure can substantially reduce sampling costs across molecular and lattice Hamiltonians, with optimized non-overlapping groups providing strong, transferable initializations and direct overlapping optimization offering further gains.
Grouping-based measurement strategies are widely used to reduce measurement complexity in near-term quantum algorithms. While these schemes have typically produced disjoint groups, recently this has been relaxed in what is known as overlapped grouping or coefficient splitting where operators may appear in more than one compatible group. In recent work, it has been numerically shown that this strategy can reduce the variance of energy estimates on small benchmark problems, motivating both the application and further analysis of the method. Here we prove that overlapped grouping for energy estimation can lead to a maximal variance reduction that is linear in the number of Hamiltonian terms. We introduce a new algorithm which we call repacking to transform existing groups into overlapped groups, and we show this repacking procedure iteratively reduces variance under mild assumptions. We also perform numerical simulations with Hamiltonians up to 44 qubits and 575⋅103 terms, assessing overlapped grouping at scale on problems of practical importance. Our numerics show that the variance reduction relative to state-of-the-art (disjoint) grouping increases linearly with the problem size, suggesting that overlapped grouping methods can be a powerful strategy for quantum energy estimation at the scale of Megaquop computers and beyond.
Accurately estimating expectation values of quantum observables with as few measurements as possible is crucial to many quantum computing applications. We introduce a framework that covers many of existing measurement strategies and introduce heuristics that can be used to enhance randomized schemes, including those based on Pauli grouping with inverse probability weighting and variants of the classical shadow algorithm. We show how to maximize information gain from such schemes, while carefully optimizing the distribution of possible measurements, and show that simple grouping algorithms can get close to, and in some cases exceed, state-of-the-art accuracy for unbiased estimation of expectation values on a standard quantum chemistry benchmark. We show how these randomized methods may be compared to more recent measurement schemes, such as shadow grouping, derandomized shadow, and overlapped grouping measurement, we show how the same strategies can be used to augment these schemes, and we demonstrate that we can reduce measurement costs by up to a factor of two by allowing Clifford measurement circuits for otherwise Clifford-less methods.
Obtaining the expectation value of an observable on a quantum computer is a crucial step in the variational quantum algorithms. For complicated observables such as molecular electronic Hamiltonians, one of the strategies is to present the observable as a linear combination of measurable fragments. The main problem of this approach is a large number of measurements required for accurate estimation of the observable's expectation value. We consider three previously studied directions that minimize the number of measurements: (1) grouping commuting operators using the greedy approach, (2) involving non-local unitary transformations for measuring, and (3) taking advantage of compatibility of some Pauli products with several measurable groups. The last direction gives rise to a general framework that not only provides improvements over previous methods but also connects measurement grouping approaches with recent advances in techniques of shadow tomography. Following this direction, we develop two measurement schemes that achieve a severalfold reduction in the number of measurements for a set of model molecules compared to previous state-of-the-art methods.
Abstract The adaptive variational quantum eigensolver (ADAPT-VQE) is a promising approach for quantum simulation in the noisy intermediate-scale quantum era, offering advantages over traditional VQE methods by reducing circuit depth and mitigating challenges in classical optimisation. Although ADAPT-VQE can require fewer total measurements than fixed ansätze for matched precision, the combined cost of repeated energy and gradient evaluations still leads to a substantial measurement overhead in practice, motivating strategies to reduce it further. In this work, we study two complementary strategies to reduce this cost using only standard projective Pauli measurements. First, we apply variance-based shot allocation (variance-minimised shot assignment and variance-preserved shot reduction (VPSR)) to both Hamiltonian energy and operator-gradient measurements, extending techniques previously studied only for energy estimation in standard VQE. VPSR achieves shot reductions of up to 43% for H 2 and 51% for LiH relative to uniform allocation while maintaining chemical accuracy. Second, we introduce a Pauli-measurement reuse protocol that stores outcomes from VQE optimisation and reuses them for gradient evaluation in the next ADAPT iteration. Across six molecular benchmarks (4–14 qubits), measurement reuse yields only marginal savings confined to the smallest systems and is effectively negligible for larger sizes, pivot-based grouping alone provides the dominant reduction and should be considered the practical baseline. These results characterise the interplay between shot allocation, measurement grouping, and cross-iteration reuse, offering practical guidance on when each strategy is most beneficial.
Quantum algorithms designed for realistic quantum many-body systems, such as chemistry and materials, usually require a large number of measurements of the Hamiltonian. Exploiting different ideas, such as importance sampling, observable compatibility, or classical shadows of quantum states, different advanced measurement schemes have been proposed to greatly reduce the large measurement cost. Yet, the underline cost reduction mechanisms seem distinct from each other, and how to systematically find the optimal scheme remains a critical challenge. Here, we address this challenge by proposing a unified framework of quantum measurements, incorporating advanced measurement methods as special cases. Our framework allows us to introduce a general scheme – overlapped grouping measurement, which simultaneously exploits the advantages of most existing methods. An intuitive understanding of the scheme is to partition the measurements into overlapped groups with each one consisting of compatible measurements. We provide explicit grouping strategies and numerically verify its performance for different molecular Hamiltonians with up to 16 qubits. Our numerical result shows significant improvements over existing schemes. Our work paves the way for efficient quantum measurement and fast quantum processing with current and near-term quantum devices.