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arXiv:2606.01589

Towards Heisenberg Scaling: Measurement-Efficient Non-Orthogonal Quantum Eigensolver

Hang Ren, Yipei Zhang, Thilo Scharnhorst, K. Birgitta Whaley

quant-phSubmitted Jun 1, 2026• Announced Jun 2, 2026• 0 citations
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Abstract

The Non-Orthogonal Quantum Eigensolver (NOQE) provides an accurate framework for electronic-structure calculations, but the estimation of its Hamiltonian and overlap matrix elements relies on sampling and requires $O(1/\varepsilon^2)$ circuit repetitions to achieve additive precision $\varepsilon$. Here, we reformulate this matrix-element estimation step as a collection of amplitude-estimation tasks and integrate iterative quantum amplitude estimation into the NOQE workflow. The resulting protocol achieves near-Heisenberg query complexity $O(1/\varepsilon)$ for these estimation tasks, by replacing incoherent statistical averaging with coherent amplitude amplification. We present explicit circuit constructions and the corresponding implementation procedure. Numerical simulations for the electronic states of the hydrogen molecule show that the proposed method reaches chemical accuracy with substantially fewer total queries than the original sampling-based protocol. Overall, this work provides a measurement-efficient route to high-precision energy estimation and illustrates how sampling-limited quantum algorithms can be systematically reformulated to leverage quantum coherence and achieve lower measurement costs.

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Quantum subspace diagonalization methods are promising algorithms for quantum chemistry on near-term quantum computers. These methods can estimate low-lying energies of molecular systems using shallow quantum circuits, at the cost of many circuit repetitions to estimate the projected matrix elements. Errors in these matrix elements can be converted into much larger eigenvalue errors by an ill-conditioned overlap matrix. We study this bottleneck for the nonorthogonal quantum eigensolver (NOQE), which constructs a compact multireference subspace from dressed unrestricted Hartree-Fock states. We prove a finite-shot perturbation bound showing that, after overlap thresholding, the eigenvalue sensitivity is controlled by the condition number of the retained overlap matrix rather than by a worst-case dimension factor. With a scalable thresholding scheme, the upper bound on the per-matrix-element shot count required to reach a target accuracy scales as $\mathcal{O}(M)$, improving on the previously known $\mathcal{O}(M^3)$ bound, where $M$ is the number of reference states. Numerical experiments on hydrogen chains and rings suggest that, in practice, the measurement cost of structured NOQE instances can grow even more slowly than this linear bound.

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Efficient Measurement and Error Mitigation in Quantum Algorithms

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Measurement overhead and hardware noise present fundamental obstacles to the practical realization of quantum algorithms on near-term quantum devices. In many quantum applications, the dominant computational cost arises from repeated measurements needed to extract accurate information and from handling noise effects. This dissertation develops a unified framework for efficient measurement and error mitigation in quantum algorithms.In this work, measurement is treated as a controllable and designable component across hardware, algorithmic, and statistical levels, rather than as a fixed final step. By tailoring measurement schemes and estimators, we show that substantial reductions in measurement cost can be achieved without sacrificing accuracy. This approach is developed and validated across multiple settings, from hardware-level randomized measurement implementations to algorithm-level molecular energy estimation.At the hardware level, we investigate randomized measurement schemes based on photonic metasurfaces. We find that the device itself naturally realizes the required family of measurement bases without repeated active optical reconfiguration, providing an efficient platform for randomized measurements. We analyze the effect of realistic measurement noise and develop calibration-based error mitigation techniques that suppress noise-induced bias. Numerical studies show that this approach enables accurate estimation of state properties under realistic measurement noise.At the algorithmic level, we investigate measurement-efficient quantum eigen solvers through the Non-Orthogonal Quantum Eigensolver (NOQE). We develop a detailed implementation framework for NOQE, including circuit construction and simplification, analysis of measurement complexity, and resource estimation. Building on this understanding, we incorporate randomized measurements as an alternative to standard Hadamard-test-based schemes and develop shadow-tomography-enhanced variants of NOQE that significantly reduce circuit complexity and qubit requirements while maintaining favorable sample complexity. Shadow-based error mitigation methods, including shadow distillation, further improve robustness against noise.Finally, we investigate a more fundamental redesign of the NOQE measurement task by reformulating matrix-element estimation as a quantum amplitude estimation problem. By integrating iterative quantum amplitude estimation into the NOQE framework, we replace the original incoherent sampling-based estimation with coherent information accumulation through amplitude amplification and achieve near-Heisenberg scaling in measurement complexity, improving on the standard sampling-based methods. Numerical results for molecular energy estimation demonstrate that this approach reaches chemical accuracy with fewer queries than the original NOQE.Overall, this dissertation shows that measurement overhead is not a fixed limitation, but a resource bottleneck that can be identified and reduced at multiple levels in quantum applications. Across the settings studied here, improvements are achieved at different layers, including hardware-aware randomized measurements, algorithm-level redesign of measurement protocols, estimator-level error mitigation, and statistical improvements based on coherent information accumulation. Taken together, these results provide a unified framework for understanding and reducing measurement cost under realistic noise, and offer practical guidance for the design of quantum algorithms on near-term quantum devices.

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Ground-state estimation lies at the heart of a broad range of quantum simulations. Most near-term approaches are cast as variational energy minimization and thus inherit the challenges of problem-specific energy landscapes. We develop the quantum amplitude-amplification eigensolver (QAAE), which departs from the variational paradigm and instead coherently drives a trial state toward the ground state via quantum amplitude amplification. Each amplitude-amplification round interleaves a reflection about the learned trial state with a controlled short-time evolution under a normalized Hamiltonian; an ancilla readout yields an amplitude-amplified pure target state that a state-learning step then re-encodes into an ansatz circuit for the next round -- without evaluating the energy gradients. Under standard assumptions (normalized $\hat{H}$, a nondegenerate ground-state, and a learning update), the ground-state overlap increases monotonically per round and the procedure converges; here, a per-round depth bound in terms of the ansatz depth and Hamiltonian-simulation cost establishes hardware compatibility. Cloud experiments on IBMQ processor verify our amplification mechanism on a two-level Hamiltonian and a two-qubit Ising model, and numerical benchmarks on $\mathrm{H}_2$, $\mathrm{LiH}$, and a $10$-qubit longitudinal-and-transverse-field Ising model show that QAAE integrates with chemistry-inspired and hardware-efficient circuits and can surpass gradient-based VQE in accuracy and stability. These results position QAAE as a variational-free and hardware-compatible route to ground-state estimation for near-term quantum simulation.

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Error-mitigated nonorthogonal quantum eigensolver via shadow tomography

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We present a shadow-tomography-enhanced nonorthogonal quantum eigensolver (NOQE) for more efficient and accurate electronic structure calculations on near-term quantum devices. By integrating shadow tomography into the NOQE, the measurement cost scales linearly rather than quadratically with the number of reference states, while also reducing the required qubits and circuit depth by half. This approach enables extraction of all matrix elements via randomized measurements and classical postprocessing. We analyze its sample complexity and show that, for small systems, it remains constant in the high-precision regime, while for larger systems, it scales linearly with the system size. We further apply shadow-based error mitigation to suppress noise-induced bias without increasing quantum resources. Demonstrations on the hydrogen molecule in the strongly correlated regime achieve chemical accuracy under realistic noise, showing that our method is both resource-efficient and noise-resilient for practical quantum chemistry simulations in the near term.

arXiv:2504.16008•Physical Review Research•Dec 30, 2025•1 citation
The Non-Orthogonal Quantum Eigensolver (NOQE) provides an accurate framework for electronic-structure calculations, but the estimation of its Hamiltonian and overlap matrix elements relies on sampling and requires O(1/ε2)O(1/\varepsilon^2)O(1/ε2) circuit repetitions to achieve additive precision ε\varepsilonε. Here, we reformulate this matrix-element estimation step as a collection of amplitude-estimation tasks and integrate iterative quantum amplitude estimation into the NOQE workflow. The resulting protocol achieves near-Heisenberg query complexity O(1/ε)O(1/\varepsilon)O(1/ε) for these estimation tasks, by replacing incoherent statistical averaging with coherent amplitude amplification. We present explicit circuit constructions and the corresponding implementation procedure. Numerical simulations for the electronic states of the hydrogen molecule show that the proposed method reaches chemical accuracy with substantially fewer total queries than the original sampling-based protocol. Overall, this work provides a measurement-efficient route to high-precision energy estimation and illustrates how sampling-limited quantum algorithms can be systematically reformulated to leverage quantum coherence and achieve lower measurement costs.
Quantum subspace diagonalization methods are promising algorithms for quantum chemistry on near-term quantum computers. These methods can estimate low-lying energies of molecular systems using shallow quantum circuits, at the cost of many circuit repetitions to estimate the projected matrix elements. Errors in these matrix elements can be converted into much larger eigenvalue errors by an ill-conditioned overlap matrix. We study this bottleneck for the nonorthogonal quantum eigensolver (NOQE), which constructs a compact multireference subspace from dressed unrestricted Hartree-Fock states. We prove a finite-shot perturbation bound showing that, after overlap thresholding, the eigenvalue sensitivity is controlled by the condition number of the retained overlap matrix rather than by a worst-case dimension factor. With a scalable thresholding scheme, the upper bound on the per-matrix-element shot count required to reach a target accuracy scales as O(M)\mathcal{O}(M)O(M), improving on the previously known O(M3)\mathcal{O}(M^3)O(M3) bound, where MMM is the number of reference states. Numerical experiments on hydrogen chains and rings suggest that, in practice, the measurement cost of structured NOQE instances can grow even more slowly than this linear bound.
Measurement overhead and hardware noise present fundamental obstacles to the practical realization of quantum algorithms on near-term quantum devices. In many quantum applications, the dominant computational cost arises from repeated measurements needed to extract accurate information and from handling noise effects. This dissertation develops a unified framework for efficient measurement and error mitigation in quantum algorithms.In this work, measurement is treated as a controllable and designable component across hardware, algorithmic, and statistical levels, rather than as a fixed final step. By tailoring measurement schemes and estimators, we show that substantial reductions in measurement cost can be achieved without sacrificing accuracy. This approach is developed and validated across multiple settings, from hardware-level randomized measurement implementations to algorithm-level molecular energy estimation.At the hardware level, we investigate randomized measurement schemes based on photonic metasurfaces. We find that the device itself naturally realizes the required family of measurement bases without repeated active optical reconfiguration, providing an efficient platform for randomized measurements. We analyze the effect of realistic measurement noise and develop calibration-based error mitigation techniques that suppress noise-induced bias. Numerical studies show that this approach enables accurate estimation of state properties under realistic measurement noise.At the algorithmic level, we investigate measurement-efficient quantum eigen solvers through the Non-Orthogonal Quantum Eigensolver (NOQE). We develop a detailed implementation framework for NOQE, including circuit construction and simplification, analysis of measurement complexity, and resource estimation. Building on this understanding, we incorporate randomized measurements as an alternative to standard Hadamard-test-based schemes and develop shadow-tomography-enhanced variants of NOQE that significantly reduce circuit complexity and qubit requirements while maintaining favorable sample complexity. Shadow-based error mitigation methods, including shadow distillation, further improve robustness against noise.Finally, we investigate a more fundamental redesign of the NOQE measurement task by reformulating matrix-element estimation as a quantum amplitude estimation problem. By integrating iterative quantum amplitude estimation into the NOQE framework, we replace the original incoherent sampling-based estimation with coherent information accumulation through amplitude amplification and achieve near-Heisenberg scaling in measurement complexity, improving on the standard sampling-based methods. Numerical results for molecular energy estimation demonstrate that this approach reaches chemical accuracy with fewer queries than the original NOQE.Overall, this dissertation shows that measurement overhead is not a fixed limitation, but a resource bottleneck that can be identified and reduced at multiple levels in quantum applications. Across the settings studied here, improvements are achieved at different layers, including hardware-aware randomized measurements, algorithm-level redesign of measurement protocols, estimator-level error mitigation, and statistical improvements based on coherent information accumulation. Taken together, these results provide a unified framework for understanding and reducing measurement cost under realistic noise, and offer practical guidance for the design of quantum algorithms on near-term quantum devices.
Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. This strategy doubles the size of the state preparation circuits but not their depth, while adding a small number of additional two-qubit gates relative to standard variational quantum eigensolver. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H4 and the π-system of hexatriene (C6H8).
Variational algorithms for strongly correlated chemical and materials systems are one of the most promising applications of near-term quantum computers. We present an extension to the variational quantum eigensolver that approximates the ground state of a system by solving a generalized eigenvalue problem in a subspace spanned by a collection of parametrized quantum states. This allows for the systematic improvement of a logical wavefunction ansatz without a significant increase in circuit complexity. To minimize the circuit complexity of this approach, we propose a strategy for efficiently measuring the Hamiltonian and overlap matrix elements between states parametrized by circuits that commute with the total particle number operator. This strategy doubles the size of the state preparation circuits but not their depth, while adding a small number of additional two-qubit gates relative to standard variational quantum eigensolver. We also propose a classical Monte Carlo scheme to estimate the uncertainty in the ground state energy caused by a finite number of measurements of the matrix elements. We explain how this Monte Carlo procedure can be extended to adaptively schedule the required measurements, reducing the number of circuit executions necessary for a given accuracy. We apply these ideas to two model strongly correlated systems, a square configuration of H 4 and the π -system of hexatriene (C 6 H 8 ).
Ground-state estimation lies at the heart of a broad range of quantum simulations. Most near-term approaches are cast as variational energy minimization and thus inherit the challenges of problem-specific energy landscapes. We develop the quantum amplitude-amplification eigensolver (QAAE), which departs from the variational paradigm and instead coherently drives a trial state toward the ground state via quantum amplitude amplification. Each amplitude-amplification round interleaves a reflection about the learned trial state with a controlled short-time evolution under a normalized Hamiltonian; an ancilla readout yields an amplitude-amplified pure target state that a state-learning step then re-encodes into an ansatz circuit for the next round -- without evaluating the energy gradients. Under standard assumptions (normalized H^\hat{H}H^, a nondegenerate ground-state, and a learning update), the ground-state overlap increases monotonically per round and the procedure converges; here, a per-round depth bound in terms of the ansatz depth and Hamiltonian-simulation cost establishes hardware compatibility. Cloud experiments on IBMQ processor verify our amplification mechanism on a two-level Hamiltonian and a two-qubit Ising model, and numerical benchmarks on H2\mathrm{H}_2H2​, LiH\mathrm{LiH}LiH, and a 101010-qubit longitudinal-and-transverse-field Ising model show that QAAE integrates with chemistry-inspired and hardware-efficient circuits and can surpass gradient-based VQE in accuracy and stability. These results position QAAE as a variational-free and hardware-compatible route to ground-state estimation for near-term quantum simulation.
We present a shadow-tomography-enhanced nonorthogonal quantum eigensolver (NOQE) for more efficient and accurate electronic structure calculations on near-term quantum devices. By integrating shadow tomography into the NOQE, the measurement cost scales linearly rather than quadratically with the number of reference states, while also reducing the required qubits and circuit depth by half. This approach enables extraction of all matrix elements via randomized measurements and classical postprocessing. We analyze its sample complexity and show that, for small systems, it remains constant in the high-precision regime, while for larger systems, it scales linearly with the system size. We further apply shadow-based error mitigation to suppress noise-induced bias without increasing quantum resources. Demonstrations on the hydrogen molecule in the strongly correlated regime achieve chemical accuracy under realistic noise, showing that our method is both resource-efficient and noise-resilient for practical quantum chemistry simulations in the near term.