Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2607.16866

Certified Bounds on the Measurement Cost of Molecular Hamiltonians

Federico Zahariev, Vanda Glezakou

quant-phphysics.chem-phSubmitted Jul 18, 2026• Announced Jul 21, 2026• 0 citations• Revised Oct 7, 2026 (v2)
v1 · Jul 18, 2026v2 · Oct 7, 2026
arXivHTMLPDF

Abstract

Quantum estimates of molecular energy require repeated state preparation and measurement. We bound the minimum repetition count for given measurements and processing rules from above with a practical schedule and from below with independently verified inequalities. For an H$_4$ chain at H-H distances of 0.90 and 1.80 Åwith 2 per cent uniform noise in the ground state, entangling Clifford measurements, which map Pauli products onto one another, cut leading-order repetitions by at least 65.9 and 61.0 per cent, respectively, relative to any schedule using only single-qubit Pauli measurements and Hamiltonian terms. With 69 unaltered measurement settings, each a basis change followed by qubit readout, processing further observables saves at least 12.04 and 6.70 per cent. For four lanthanide and actinide Hamiltonians with depolarized Hartree-Fock covariances, reducing the sum of absolute Hamiltonian coefficients does not proportionally reduce sampling. The bounds cover linear unbiased estimates from independent batches at small error.

Comments
Revised version. Heavy-element results recomputed from audited operators, text revised throughout, references updated. 36 pages, 2 figures, including Supplementary Information

Related papers

Quantum expectation value estimation by doubling the number of qubits

Hiroshi Yano, Masaya Kohda, Shoichiro Tsutsui +4 more

Expectation value estimation is ubiquitous in quantum algorithms. The expectation value of a Hamiltonian, which is essential in various practical applications, is often estimated by measuring a large number of Pauli strings on quantum computers and performing classical post-processing. In the case of $n$-qubit molecular Hamiltonians in quantum chemistry calculations, it is necessary to evaluate $O(n^4)$ Pauli strings, requiring a large number of measurements for accurate estimation. To reduce the measurement cost, we assess an existing idea that uses two copies of an $n$-qubit quantum state of interest and coherently measures them in the Bell basis, which enables the simultaneous estimation of the absolute values of expectation values of all the $n$-qubit Pauli strings. We numerically investigate the efficiency of energy estimation for molecular Hamiltonians of up to 12 qubits. The results show that, when the target precision is no smaller than tens of milli-Hartree, this method requires fewer measurements than conventional sampling methods. This suggests that the method may be useful for many applications that rely on expectation value estimation of Hamiltonians and other observables as well when moderate precision is sufficient.

arXiv:2412.14466•Dec 19, 2024•0 citations

Beyond Orbital Rotations: Correlation-Rank Limits and Clifford-Accessible Measurement, from Algebra and Global Optimization

Federico Zahariev, Vanda Glezakou

One should not assume that reconstructing an electronic observable with a smaller set of measurement settings will cut the number of repetitions in proportion. Each setting is simply a basis change and qubit readout, but the savings are not guaranteed to be commensurate. Take two electrons of opposite spin in two spatial orbitals: the orbital degree of freedom of each electron acts as a logical qubit and rotations are independent in the spin sectors. From the observable's one- and two-qubit coefficients we derive an exact floor for the setting count and a decomposition to meet it in no more than three settings. With fewer settings, the minimum error in the two-qubit correlation coefficients is set by the singular values left out. A case in point is an exchange with equal nonzero couplings on the three Pauli axes. This demands three orbital settings yet only one joint measurement. We offer a Clifford circuit for four physical qubits to effect the necessary mapping of Pauli products under conjugation. In the case of a one-parameter family of logical singlet-triplet mixtures, the orbital-to-Clifford shot cost ratio is three at maximum mixing and trends toward 3/2 near the logical singlet. Where spin sectors may be mixed, isotropic exchange needs one orbital setting. For spin-conserving Hamiltonians with $N$ spatial orbitals encoded via Jordan-Wigner, the $X$-rank, the binary rank of the $X$ parts of its strings, of any commuting Pauli string family is no more than $2(N-1)$. Searches with the Robust Adaptive Nature-inspired Global Explorer (RANGE) reach this limit for CH$_4$, with the rank of the Hamiltonian support vouching for the maximality of the search values. And for all nonidentity Pauli strings on $n$ qubits, semidefinite witnesses from the Lovász theta number give a lower bound on the Clifford setting count that agrees with the integer cover and counting bounds at $2^n+1$.

quant-pharXiv:2607.16869•Jul 21, 2026•0 citations

Practical techniques for high-precision measurements on near-term quantum hardware and applications in molecular energy estimation

Keijo T. Korhonen, Hetta Vappula, Adam Glos +6 more

Achieving high-precision measurements on near-term quantum devices is critical for advancing quantum computing applications. Quantum computers suffer from high readout errors, making quantum simulations with high accuracy requirements particularly challenging. This paper implements practical techniques to reach accuracies essential for quantum chemistry by addressing key overheads and noise sources. Specifically, we leverage locally biased random measurements for reducing shot overhead, repeated settings with parallel quantum detector tomography for reducing circuit overhead and mitigating readout errors, and blended scheduling for mitigating time-dependent noise. We demonstrate these techniques via molecular energy estimation of the BODIPY molecule on a Hartree-Fock state on an IBM Eagle r3, obtaining a reduction in measurement errors by an order of magnitude from 1-5% to 0.16%. These strategies pave the way for more reliable quantum computations, particularly for applications requiring precise molecular energy calculations.

arXiv:2409.02575•npj Quantum Information•Jul 2, 2025•4 citations

Evaluating higher-order product formulae for molecular ground-state energy estimation

Hiromu Abe, Keita Kanno, Ryosuke Kimura +2 more

We evaluate deterministic higher-order product formulae for molecular ground-state energy estimation. Motivated by recent fault-tolerant architectures in which non-Clifford operations may be generated more locally and cheaply than in conventional assumptions, we re-examine such formulae as practical candidates for quantum chemistry. Using one-dimensional hydrogen chains from $\mathrm{H}_2$ to $\mathrm{H}_{15}$ as benchmarks, we estimate both the total gate count and the depth of $R_Z$-rotation layers required to reach a target energy error. To make this comparison feasible at larger system sizes, we use a perturbative method to estimate the eigenvalue error induced by each product formula and thereby evaluate the cost of the corresponding phase-estimation procedure. Among the previously considered formulae, the eighth-order construction introduced by Morales et al. [M. E. S. Morales et al., "Greatly improved higher-order product formulae for quantum simulation," arXiv:2210.15817v2 (2024)] minimizes both cost metrics in the benchmark at a chemically relevant target error. We also find that increasing the formal order does not automatically reduce the total cost: near chemical accuracy, the tenth-order formula introduced in the same work can be less efficient than the eighth-order one. Motivated by this observation, we construct a new fourth-order formula; it achieves the lowest total gate count among the formulae considered for all H-chain instances near chemical accuracy and over much of the 0.1-10 mHa target-error window for most instances, while also reducing the $R_Z$-layer depth. These results clarify how deterministic higher-order product formulae should be selected for molecular ground-state energy estimation.

quant-pharXiv:2605.30967•Jun 1, 2026•0 citations

Almost optimal measurement scheduling of molecular Hamiltonian via finite projective plane

Wataru Inoue, Koki Aoyama, Yusuke Teranishi +3 more

We propose an efficient and almost optimal scheme for measuring molecular Hamiltonians in quantum chemistry on quantum computers, which requires 2N2 distinct measurements in the leading order with N being the number of molecular orbitals. It achieves the state-of-the-art by improving a previous proposal by Bonet-Monroig [], which exhibits 103N2 scaling in the leading order. We develop a method based on a finite projective plane to construct sets of simultaneously measurable operators contained in molecular Hamiltonians. Each measurement only requires a depth- O(N) circuit consisting of O(N2) one- and two-qubit gates under the Jordan-Wigner and parity mapping, assuming the linear connectivity of qubits on quantum hardwares. We perform numerical simulation of our method for molecular Hamiltonians of hydrogen chains. We count the number of sets of simultaneously measurable operators generated by our method and estimate the number of measurement shots required to achieve the small standard deviation of the energy expectation value for precise quantum chemistry calculations. Because evaluating expectation values of molecular Hamiltonians is one of the major bottlenecks in the applications of quantum devices to quantum chemistry, our finding is expected to accelerate such applications. Published by the American Physical Society 2024

arXiv:2301.07335•Physical Review Research•Jan 25, 2024•3 citations

Reducing the sampling complexity of energy estimation in quantum many-body systems using empirical variance information

Alexander Gresch, Uğur Tepe, Martin Kliesch

We consider the problem of estimating the energy of a quantum state preparation for a given Hamiltonian in Pauli decomposition. For various quantum algorithms, in particular in the context of quantum chemistry, it is crucial to have energy estimates with error bounds, as captured by guarantees on the problem's sampling complexity. In particular, when limited to Pauli basis measurements, the smallest sampling complexity guarantee comes from a simple single-shot estimator via a straightforward argument based on Hoeffding's inequality. In this work, we construct an adaptive estimator using the state's actual variance. Technically, our estimation method is based on the Empirical Bernstein stopping (EBS) algorithm and grouping schemes, and we provide a rigorous tail bound, which leverages the state's empirical variance. In a numerical benchmark of estimating ground-state energies of several Hamiltonians, we demonstrate that EBS consistently improves upon elementary readout guarantees up to one order of magnitude.

arXiv:2502.01730•Feb 3, 2025•1 citation
Quantum estimates of molecular energy require repeated state preparation and measurement. We bound the minimum repetition count for given measurements and processing rules from above with a practical schedule and from below with independently verified inequalities. For an H4_44​ chain at H-H distances of 0.90 and 1.80 Åwith 2 per cent uniform noise in the ground state, entangling Clifford measurements, which map Pauli products onto one another, cut leading-order repetitions by at least 65.9 and 61.0 per cent, respectively, relative to any schedule using only single-qubit Pauli measurements and Hamiltonian terms. With 69 unaltered measurement settings, each a basis change followed by qubit readout, processing further observables saves at least 12.04 and 6.70 per cent. For four lanthanide and actinide Hamiltonians with depolarized Hartree-Fock covariances, reducing the sum of absolute Hamiltonian coefficients does not proportionally reduce sampling. The bounds cover linear unbiased estimates from independent batches at small error.
Expectation value estimation is ubiquitous in quantum algorithms. The expectation value of a Hamiltonian, which is essential in various practical applications, is often estimated by measuring a large number of Pauli strings on quantum computers and performing classical post-processing. In the case of nnn-qubit molecular Hamiltonians in quantum chemistry calculations, it is necessary to evaluate O(n4)O(n^4)O(n4) Pauli strings, requiring a large number of measurements for accurate estimation. To reduce the measurement cost, we assess an existing idea that uses two copies of an nnn-qubit quantum state of interest and coherently measures them in the Bell basis, which enables the simultaneous estimation of the absolute values of expectation values of all the nnn-qubit Pauli strings. We numerically investigate the efficiency of energy estimation for molecular Hamiltonians of up to 12 qubits. The results show that, when the target precision is no smaller than tens of milli-Hartree, this method requires fewer measurements than conventional sampling methods. This suggests that the method may be useful for many applications that rely on expectation value estimation of Hamiltonians and other observables as well when moderate precision is sufficient.
One should not assume that reconstructing an electronic observable with a smaller set of measurement settings will cut the number of repetitions in proportion. Each setting is simply a basis change and qubit readout, but the savings are not guaranteed to be commensurate. Take two electrons of opposite spin in two spatial orbitals: the orbital degree of freedom of each electron acts as a logical qubit and rotations are independent in the spin sectors. From the observable's one- and two-qubit coefficients we derive an exact floor for the setting count and a decomposition to meet it in no more than three settings. With fewer settings, the minimum error in the two-qubit correlation coefficients is set by the singular values left out. A case in point is an exchange with equal nonzero couplings on the three Pauli axes. This demands three orbital settings yet only one joint measurement. We offer a Clifford circuit for four physical qubits to effect the necessary mapping of Pauli products under conjugation. In the case of a one-parameter family of logical singlet-triplet mixtures, the orbital-to-Clifford shot cost ratio is three at maximum mixing and trends toward 3/2 near the logical singlet. Where spin sectors may be mixed, isotropic exchange needs one orbital setting. For spin-conserving Hamiltonians with NNN spatial orbitals encoded via Jordan-Wigner, the XXX-rank, the binary rank of the XXX parts of its strings, of any commuting Pauli string family is no more than 2(N−1)2(N-1)2(N−1). Searches with the Robust Adaptive Nature-inspired Global Explorer (RANGE) reach this limit for CH4_44​, with the rank of the Hamiltonian support vouching for the maximality of the search values. And for all nonidentity Pauli strings on nnn qubits, semidefinite witnesses from the Lovász theta number give a lower bound on the Clifford setting count that agrees with the integer cover and counting bounds at 2n+12^n+12n+1.
Achieving high-precision measurements on near-term quantum devices is critical for advancing quantum computing applications. Quantum computers suffer from high readout errors, making quantum simulations with high accuracy requirements particularly challenging. This paper implements practical techniques to reach accuracies essential for quantum chemistry by addressing key overheads and noise sources. Specifically, we leverage locally biased random measurements for reducing shot overhead, repeated settings with parallel quantum detector tomography for reducing circuit overhead and mitigating readout errors, and blended scheduling for mitigating time-dependent noise. We demonstrate these techniques via molecular energy estimation of the BODIPY molecule on a Hartree-Fock state on an IBM Eagle r3, obtaining a reduction in measurement errors by an order of magnitude from 1-5% to 0.16%. These strategies pave the way for more reliable quantum computations, particularly for applications requiring precise molecular energy calculations.
We evaluate deterministic higher-order product formulae for molecular ground-state energy estimation. Motivated by recent fault-tolerant architectures in which non-Clifford operations may be generated more locally and cheaply than in conventional assumptions, we re-examine such formulae as practical candidates for quantum chemistry. Using one-dimensional hydrogen chains from H2\mathrm{H}_2H2​ to H15\mathrm{H}_{15}H15​ as benchmarks, we estimate both the total gate count and the depth of RZR_ZRZ​-rotation layers required to reach a target energy error. To make this comparison feasible at larger system sizes, we use a perturbative method to estimate the eigenvalue error induced by each product formula and thereby evaluate the cost of the corresponding phase-estimation procedure. Among the previously considered formulae, the eighth-order construction introduced by Morales et al. [M. E. S. Morales et al., "Greatly improved higher-order product formulae for quantum simulation," arXiv:2210.15817v2 (2024)] minimizes both cost metrics in the benchmark at a chemically relevant target error. We also find that increasing the formal order does not automatically reduce the total cost: near chemical accuracy, the tenth-order formula introduced in the same work can be less efficient than the eighth-order one. Motivated by this observation, we construct a new fourth-order formula; it achieves the lowest total gate count among the formulae considered for all H-chain instances near chemical accuracy and over much of the 0.1-10 mHa target-error window for most instances, while also reducing the RZR_ZRZ​-layer depth. These results clarify how deterministic higher-order product formulae should be selected for molecular ground-state energy estimation.
We propose an efficient and almost optimal scheme for measuring molecular Hamiltonians in quantum chemistry on quantum computers, which requires 2N2 distinct measurements in the leading order with N being the number of molecular orbitals. It achieves the state-of-the-art by improving a previous proposal by Bonet-Monroig [], which exhibits 103N2 scaling in the leading order. We develop a method based on a finite projective plane to construct sets of simultaneously measurable operators contained in molecular Hamiltonians. Each measurement only requires a depth- O(N) circuit consisting of O(N2) one- and two-qubit gates under the Jordan-Wigner and parity mapping, assuming the linear connectivity of qubits on quantum hardwares. We perform numerical simulation of our method for molecular Hamiltonians of hydrogen chains. We count the number of sets of simultaneously measurable operators generated by our method and estimate the number of measurement shots required to achieve the small standard deviation of the energy expectation value for precise quantum chemistry calculations. Because evaluating expectation values of molecular Hamiltonians is one of the major bottlenecks in the applications of quantum devices to quantum chemistry, our finding is expected to accelerate such applications. Published by the American Physical Society 2024
We consider the problem of estimating the energy of a quantum state preparation for a given Hamiltonian in Pauli decomposition. For various quantum algorithms, in particular in the context of quantum chemistry, it is crucial to have energy estimates with error bounds, as captured by guarantees on the problem's sampling complexity. In particular, when limited to Pauli basis measurements, the smallest sampling complexity guarantee comes from a simple single-shot estimator via a straightforward argument based on Hoeffding's inequality. In this work, we construct an adaptive estimator using the state's actual variance. Technically, our estimation method is based on the Empirical Bernstein stopping (EBS) algorithm and grouping schemes, and we provide a rigorous tail bound, which leverages the state's empirical variance. In a numerical benchmark of estimating ground-state energies of several Hamiltonians, we demonstrate that EBS consistently improves upon elementary readout guarantees up to one order of magnitude.