Protocols for quantum measurement are an essential part of quantum computing. Measurements are no longer confined to the final step of computation but are increasingly embedded within quantum circuits as integral components of noise-resilient algorithms. However, each observable typically requires a distinct measurement basis, often demanding a different circuit configuration. As the number of such configurations typically grows with the number of qubits, measurements constitute a major bottleneck. Focusing on electronic structure calculations in crystalline systems, we propose a measurement protocol that restricts the required measurement configurations to an absolute minimum of just three, independent of the number of qubits. This makes it one of the few known protocols that do not scale with qubit number. In particular, we derive the measurement protocol from the symmetries of tight-binding (TB) Hamiltonians and implement it within the Orthogonal-Ansatz Variational Quantum Eigensolver (OA-VQE) algorithm. We demonstrate its performance on three systems, namely a two-dimensional CuO$_2$ square lattice (3 qubits), bilayer graphene with hexagonal (Honeycomb) lattice (4 qubits) and three-dimensional diamond lattice (10 qubits). Beyond tight-binding systems, the protocol can be extended to enable efficient initial state preparation for many-body Hamiltonians, such as multi-orbital Hubbard models in a momentum space.
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Version 2 with added experimental realization, 13 pages, 9 figures, supplementary file available
Protocols for quantum measurement are an essential part of quantum computing. Measurements are no longer confined to the final step of computation but are increasingly embedded within quantum circuits as integral components of noise-resilient algorithms. However, each observable typically requires a distinct measurement basis, often demanding a different circuit configuration. As the number of such configurations typically grows with the number of qubits, measurements constitute a major bottleneck. Focusing on electronic structure calculations in crystalline systems, we propose a measurement protocol that restricts the required measurement configurations to an absolute minimum of just three, independent of the number of qubits. This makes it one of the few known protocols that do not scale with qubit number. In particular, we derive the measurement protocol from the symmetries of tight-binding (TB) Hamiltonians and implement it within the Orthogonal-Ansatz Variational Quantum Eigensolver (OA-VQE) algorithm. We demonstrate its performance on three systems, namely a two-dimensional CuO2 square lattice (3 qubits), bilayer graphene with hexagonal (Honeycomb) lattice (4 qubits) and three-dimensional diamond lattice (10 qubits). Beyond tight-binding systems, the protocol can be extended to enable efficient initial state preparation for many-body Hamiltonians, such as multi-orbital Hubbard models in a momentum space.
Abstract Protocols for quantum measurement are an essential part of quantum computing. Measurements are no longer confined to the final step of computation but are increasingly embedded within quantum circuits as integral components of noise-resilient algorithms. However, each observable typically requires a distinct measurement basis, often demanding a different circuit configuration. As the number of such configurations typically grows with the number of qubits, measurements constitute a major bottleneck. Focusing on electronic structure calculations in crystalline systems, we propose a measurement protocol that restricts the required measurement configurations to an absolute minimum of just three, independent of the number of qubits. This makes it one of the few known protocols that do not scale with qubit number. In particular, we derive the measurement protocol from the symmetries of tight-binding (TB) Hamiltonians and implement it within the Orthogonal-Ansatz variational quantum eigensolver algorithm. We demonstrate its performance on three systems, namely a two-dimensional CuO 2 square lattice (3 qubits), bilayer graphene with hexagonal (Honeycomb) lattice (4 qubits) and three-dimensional diamond lattice (10 qubits). Beyond TB systems, the protocol can be extended to enable efficient initial state preparation for many-body Hamiltonians, such as multi-orbital Hubbard models in a momentum space.
High Resolution Image Download MS PowerPoint Slide We propose a cost-efficient measurement scheme of the variational quantum eigensolver (VQE) for atomistic simulations of electronic structures based on a tight-binding (TB) theory. Leveraging the lattice geometry of a material domain, the sparse TB Hamiltonian is constructed in a bottom-up manner and is represented as a linear combination of the standard-basis (SB) operators. The cost function is evaluated with an extended version of the Bell measurement circuit that can simultaneously measure multiple SB operators and therefore reduces the number of circuits required by the evaluation process. The proposed VQE scheme is applied to find band gap energies of metal-halide-perovskite supercells that have finite dimensions with closed boundaries and are described with an sp 3 TB model. Experimental results confirm that the proposed scheme gives solutions that follow well the accurate ones but, more importantly, has computing efficiency that is obviously superior to the commutativity-based Pauli grouping methods. Extending the application scope of VQE to three-dimensional confined atomic structures, this work can serve as a practical guideline for handling TB simulations and, more generally, for calculations of sparse Hermitian matrices in the noise-intermediate-scale quantum devices.
Abstract Variational Quantum Eigensolver (VQE) and its extension, Variational Quantum Deflation (VQD), have emerged as promising algorithms for computing ground and excited state energy eigenvalues of Hamiltonians, particularly in quantum physics, chemistry, and materials science. Recently, the VQE and VQD algorithms were implemented to calculate the electronic band structure described by the tight-binding Hamiltonian. Despite their success for simple models, scalability remains a challenge due to the need for a large number of circuit executions, limited qubit coherence times, and low gate fidelities. In this work, we implemented the VQE and VQD algorithms for tight-binding models of diamond and zincblende crystal structures (Sn, C, Si, Ge, AlP, AlAs, AlSb, and GaP) using a quantum computer simulator. We investigate the scalability challenges, focusing on the significant overhead caused by a large number of quantum circuit executions required to find the energy eigenvalues. Our analysis highlights the substantial amount of both measurement and optimization overheads in VQD, particularly as the number of excited states increases. Based on our findings, we critically discuss the implications of implementing VQD on current noisy intermediate-scale quantum (NISQ) devices.
Solving the electronic structure problem using the Variational Quantum Eigensolver (VQE) technique involves the measurement of the Hamiltonian expectation value. The 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(N4) with the size of the system, N. To optimize the 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 commuting within the corresponding single-qubit subspaces or qubit-wise commuting. We found that the qubit-wise commutativity between the Hamiltonian terms can be expressed as a graph and the problem of the optimal grouping is equivalent to 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 less compared to the total number of terms in the considered Hamiltonians.
Variational quantum algorithms are promising applications of noisy intermediate-scale quantum (NISQ) computers. These algorithms consist of a number of separate prepare-and-measure experiments that estimate terms in a Hamiltonian. The number of terms can become overwhelmingly large for problems at the scale of NISQ hardware that may soon be available. We use unitary partitioning (developed independently by Izmaylov et al. [J. Chem. Theory Comput. 16, 190 (2020)]) to define variational quantum eigensolver procedures in which additional unitary operations are appended to the ansatz preparation to reduce the number of terms. This approach may be scaled to use all coherent resources available after ansatz preparation. We also study the use of asymmetric qubitization to implement the additional coherent operations with lower circuit depth. Using this technique, we find a constant factor speedup for lattice and random Pauli Hamiltonians. For electronic structure Hamiltonians, we prove that linear term reduction with respect to the number of orbitals, which has been previously observed in numerical studies, is always achievable. For systems represented on 10--30 qubits, we find that there is a reduction in the number of terms by approximately an order of magnitude. Applied to the plane-wave dual-basis representation of fermionic Hamiltonians, however, unitary partitioning offers only a constant factor reduction. Finally, we show that noncontextual Hamiltonians may be reduced to effective commuting Hamiltonians using unitary partitioning.
Abstract Variational quantum algorithms such as the variational quantum eigensolver (VQE) and variational quantum deflation (VQD) have emerged as leading candidates for computing electronic structure on near-term quantum hardware. For periodic solids, however, their reach has been limited not by the quantum algorithm itself but by the cost of constructing a faithful second-quantized Hamiltonian for each material, typically via density-functional theory (DFT) followed by Wannierization or by hand-fitted empirical tight-binding parameters. SlaKoNet, a recently developed method, significantly reduces the computational cost of electronic Hamiltonian construction with a small accuracy tradeoff. It integrates modern deep-learning methods and the Slater-Koster tight-binding formalism to fit thousands of onsite and distance-dependent hopping and overlap tight-binding parameters across 65 elements of the periodic table. After fitting, these parameters can be used to deterministically construct an electronic Hamiltonian for any crystal made up of any of the 65 elements in the training set, and the crystal's band structure can subsequently be computed. In this work, we combine a SlaKoNet model trained on the JARVIS-TBmBJ dataset with a Qiskit-based variational quantum deflation algorithm, replacing the traditional and costly Hamiltonian construction step with a universal neural Hamiltonian generator. Within the 65-element set, the resulting SlaKoNet-VQD workflow is structure-agnostic, fully differentiable, fast, and suited to high-throughput screening of band structures. We benchmark the workflow on silicon, recovering the full eight-band band structure along the standard high-symmetry k-path with a mean absolute deviation of 1.78 meV from exact diagonalization on a 3-qubit statevector simulator. We further compute the full band structure for five additional conventional superconductors, Al, Ta, Nb, V, and ZrN, achieving a mean absolute deviation within approximately 1.8 meV, and demonstrate full execution of SlaKoNet-VQD on IBM Quantum hardware for a representative k-point ground-state calculation on aluminum with a mean absolute error of roughly 0.37 eV. Beyond the single-particle level, we promote the same SlaKoNet Hamiltonian to a correlated Hubbard model and solve it with dynamical mean-field theory, recovering the weakening of correlations across the group-5 metals and strong quasiparticle renormalization in the cuprate La₂CuO₄, and we identify the resulting impurity problem as the natural target for a quantum solver. The pipeline opens a path to high-throughput benchmarking of variational quantum algorithms across the periodic table and to gradient-based co-optimization of ansatz and Hamiltonian for materials discovery. An interactive web app is available at https://atomgpt.org/quantum.