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Papers by Mohammad Haidar

Physical Review A, 2020

The largest hyperfine interaction coefficients in the hydrogen molecular ion HD + , i.e. the elec... more The largest hyperfine interaction coefficients in the hydrogen molecular ion HD + , i.e. the electronproton and electron-deuteron spin-spin scalar interactions, are calculated with estimated uncertainties slightly below 1 ppm. The (Zα) 2 EF relativistic correction, for which a detailed derivation is presented, QED corrections up to the order α 3 ln 2 (α) along with an estimate of higher-order terms, and nuclear structure corrections are taken into account. Improved results are also given for the electron-proton interaction coefficient in H + 2 , in excellent agreement with RF spectroscopy experiments. In HD + , a 4σ difference is found in the hyperfine splitting of the (v, L) = (0, 3) → (9, 3) two-photon transition that was recently measured with high precision. The origin of this discrepancy is unknown.

arXiv (Cornell University), Dec 23, 2022

The Trotterized Unitary Coupled Cluster Single and Double (UCCSD) ansatz has recently attracted i... more The Trotterized Unitary Coupled Cluster Single and Double (UCCSD) ansatz has recently attracted interest due to its use in Variation Quantum Eigensolver (VQE) molecular simulations on quantum computers. However, when the size of molecules increases, UCCSD becomes less interesting as it cannot achieve sufficient accuracy. Including higher-order excitations is therefore mandatory to recover the UCC's missing correlation effects. In this Letter, we extend the Trotterized UCC approach via the addition of (true) Triple T excitations introducing UCCSDT. We also include both spin and orbital symmetries. Indeed, in practice, these later help to reduce unnecessarily circuit excitations and thus accelerate the optimization process enabling to tackle larger molecules. Our initial numerical tests (12-14 qubits) show that UCCSDT improves the overall accuracy by at least two-orders of magnitudes with respect to standard UCCSD. Over-1

arXiv (Cornell University), Jun 17, 2022

Quantum Chemistry (QC) is one of the most promising applications of Quantum Computing. However, p... more Quantum Chemistry (QC) is one of the most promising applications of Quantum Computing. However, present quantum processing units (QPUs) are still subject to large errors. Therefore, noisy intermediate-scale quantum (NISQ) hardware is limited in terms of qubit counts/circuit depths. Variational Quantum Eigensolver (VQE) algorithms can potentially overcome such issues. Here, we introduce the OpenVQE open-source QC package. It provides tools for using and developing chemicallyinspired adaptive methods derived from Unitary Coupled Cluster (UCC). It facilitates the development and testing of VQE algorithms and is able to use the Atos Quantum Learning Machine (QLM), a general quantum programming framework enabling to write/optimize/simulate quantum computing programs. We present a specific, freely available QLM open-source module, myQLM-fermion. We review its key tools for facilitating QC computations (fermionic second quantization, fermion-spin transforms, etc.). OpenVQE largely extends the QLM's QC capabilities by providing: (i) the functions to generate the different types of excitations beyond the commonly used UCCSD ansatz; (ii) a new Python implementation of the "adaptive derivative assembled pseudo-Trotter method" (ADAPT-VQE). Interoperability with other major quantum programming frameworks is ensured thanks to the myQLMinterop package, which allows users to build their own code and easily execute it on existing QPUs. The combined OpenVQE/myQLM-fermion libraries facilitate the implementation, testing and development of variational quantum algorithms, while offering access to large molecules as the noiseless Schrödinger-style dense simulator can reach up to 41 qubits for any circuit. Extensive benchmarks are provided for molecules associated to qubit counts ranging from 4 up to 24. We focus on reaching chemical accuracy, reducing the number of circuit gates and optimizing parameters and operators between "fixed-length" UCC and ADAPT-VQE ansätze.

Physical Review A

We consider higher-order corrections to hyperfine coefficients related to the spin-orbit and spin... more We consider higher-order corrections to hyperfine coefficients related to the spin-orbit and spinspin tensor interactions in hydrogen molecular ions. The mα 7 ln(α)-order radiative correction is derived in the NRQED framework. We present complete numerical calculations, including as well the mα 6-order relativistic correction, for the case of H + 2. The theoretical uncertainty is reduced by more than one order of magnitude with respect to the Breit-Pauli level, down to a few ppm. We also compare our results with available rf spectroscopy data.

Physical Review A, 2020

A complete effective Hamiltonian for relativistic corrections at orders mα 6 and mα 6 (m/M) in a ... more A complete effective Hamiltonian for relativistic corrections at orders mα 6 and mα 6 (m/M) in a one-electron molecular system is derived from the NRQED Lagrangian. It includes spin-independent corrections to the energy levels and spin-spin scalar interactions contributing to the hyperfine splitting, both of which had been studied previously. In addition, corrections to electron spin-orbit and spin-spin tensor interactions are newly obtained. This allows improving the hyperfine structure theory in the hydrogen molecular ions. Improved values of the spin-orbit hyperfine coefficient are calculated for a few transitions of current experimental interest. I. INTRODUCTION High-resolution spectroscopy of the hydrogen molecular ions H + 2 and HD + may contribute significantly to the determination of fundamental constants such as the proton-electron mass ratio m p /m e [1]. A pure rotational transition in HD + has recently been measured with a relative uncertainty of 1.3 × 10 −11 [2]. The experimental accuracy of ro-vibrational transition frequencies is expected to reach a few parts per trillion in the near future using spectroscopy in the Lamb-Dicke regime [2-4] or in a Doppler-free geometry [5, 6]. While information on fundamental constants is obtained from comparison of spin-averaged transition frequencies with theoretical predictions, the hyperfine splitting of ro-vibrational lines also allows for precise tests of theory. So far, the hyperfine structure of H + 2 and HD + has been calculated within the Breit-Pauli approximation [7, 8], taking into account the anomalous magnetic moment of the electron. All terms at orders mα 4 and mα 5 are included, so that the theoretical accuracy of the hyperfine coefficients is of order α 2 ∼ 5 × 10 −5. Higherorder corrections to the largest coefficients, i.e. the spin-spin Fermi contact interaction, were later calculated in [9, 10], which allowed to get excellent agreement with available RF spectroscopy data in H + 2 [11] at the level of ∼ 1 ppm. The following step to improve the hyperfine structure theory is to evaluate higher-order corrections to the next largest coefficients, i.e. the electron spin-orbit and spin-spin tensor interaction, starting with relativistic corrections at the mα 6 order. With this aim, we derive in the present work the complete effective Hamiltonian for the hydrogen molecular ions at the mα 6 and mα 6 (m/M) orders, following the NRQED approach [12-14]. Then, we use it to calculate numerically the corrections to the electron spin-orbit interaction for a few transitions studied in ongoing experiments. The paper is organized as follows: in Secs. II and III, we recall the expression of the NRQED Lagrangian and associated interaction vertices. We then systematically derive the effective potentials, which are organized in three categories: tree-level interactions involving the exchange of a Coulomb or transverse photon (Sec. IV), terms due to retardation in the transverse photon exchange (Sec. V), and finally those coming from a seagull diagram with simultaneous exchange of two photons (Sec. VI). In Sec. VII, we collect our results to write the total effective Hamiltonian, separating the different types of interactions: spinindependent, electronic spin-orbit, spin-spin scalar and tensor interactions. Finally, in Sec. VIII we present numerical calculations of the spin-orbit interaction coefficient. II. NRQED LAGRANGIAN Natural (Lorenz-Heaviside) units (h = c = 1) are used throughout. We assume that e is the electron's charge and thus is negative, the elementary charge is then denoted by |e|.

Physical Review A, 2020

The NRQED approach is applied to the calculation of relativistic corrections to the fine and hype... more The NRQED approach is applied to the calculation of relativistic corrections to the fine and hyperfine structure of hydrogenlike atoms at orders mα 6 and mα 6 (m/M). Results are found to be in agreement with those of the relativistic theory. This confirms that the derived NRQED effective potentials are correct, and may be used for studying more complex atoms or molecules. Furthermore, we verify the equivalence between different forms of the NRQED Lagrangian used in the literature.

Recent Progress in Few-Body Physics, 2020

Although they do not lend themselves to analytical resolution, three-body atomic or molecular sys... more Although they do not lend themselves to analytical resolution, three-body atomic or molecular systems are still simple enough to allow for very precise theoretical predictions of their energy levels, which makes them attractive candidates for fundamental tests and determination of fundamental physical constants. Focusing on the hydrogen molecular ions (H + 2 , HD + , D + 2), we outline the methods which have been used to improve the theoretical accuracy by several orders of magnitude over the last two decades. The three-body Schrödinger equation can be solved with extreme precision by variational methods with trial functions involving exponentials of interparticle distances. Quantum electrodynamics (QED) corrections are evaluated in the framework of nonrelativistic QED (NRQED). The current status of theory and possibilities of further improvement are briefly sketched.

Physical Review A, 2020

The largest hyperfine interaction coefficients in the hydrogen molecular ion HD + , i.e. the elec... more The largest hyperfine interaction coefficients in the hydrogen molecular ion HD + , i.e. the electronproton and electron-deuteron spin-spin scalar interactions, are calculated with estimated uncertainties slightly below 1 ppm. The (Zα) 2 EF relativistic correction, for which a detailed derivation is presented, QED corrections up to the order α 3 ln 2 (α) along with an estimate of higher-order terms, and nuclear structure corrections are taken into account. Improved results are also given for the electron-proton interaction coefficient in H + 2 , in excellent agreement with RF spectroscopy experiments. In HD + , a 4σ difference is found in the hyperfine splitting of the (v, L) = (0, 3) → (9, 3) two-photon transition that was recently measured with high precision. The origin of this discrepancy is unknown.

arXiv (Cornell University), Dec 23, 2022

The Trotterized Unitary Coupled Cluster Single and Double (UCCSD) ansatz has recently attracted i... more The Trotterized Unitary Coupled Cluster Single and Double (UCCSD) ansatz has recently attracted interest due to its use in Variation Quantum Eigensolver (VQE) molecular simulations on quantum computers. However, when the size of molecules increases, UCCSD becomes less interesting as it cannot achieve sufficient accuracy. Including higher-order excitations is therefore mandatory to recover the UCC's missing correlation effects. In this Letter, we extend the Trotterized UCC approach via the addition of (true) Triple T excitations introducing UCCSDT. We also include both spin and orbital symmetries. Indeed, in practice, these later help to reduce unnecessarily circuit excitations and thus accelerate the optimization process enabling to tackle larger molecules. Our initial numerical tests (12-14 qubits) show that UCCSDT improves the overall accuracy by at least two-orders of magnitudes with respect to standard UCCSD. Over-1

arXiv (Cornell University), Jun 17, 2022

Quantum Chemistry (QC) is one of the most promising applications of Quantum Computing. However, p... more Quantum Chemistry (QC) is one of the most promising applications of Quantum Computing. However, present quantum processing units (QPUs) are still subject to large errors. Therefore, noisy intermediate-scale quantum (NISQ) hardware is limited in terms of qubit counts/circuit depths. Variational Quantum Eigensolver (VQE) algorithms can potentially overcome such issues. Here, we introduce the OpenVQE open-source QC package. It provides tools for using and developing chemicallyinspired adaptive methods derived from Unitary Coupled Cluster (UCC). It facilitates the development and testing of VQE algorithms and is able to use the Atos Quantum Learning Machine (QLM), a general quantum programming framework enabling to write/optimize/simulate quantum computing programs. We present a specific, freely available QLM open-source module, myQLM-fermion. We review its key tools for facilitating QC computations (fermionic second quantization, fermion-spin transforms, etc.). OpenVQE largely extends the QLM's QC capabilities by providing: (i) the functions to generate the different types of excitations beyond the commonly used UCCSD ansatz; (ii) a new Python implementation of the "adaptive derivative assembled pseudo-Trotter method" (ADAPT-VQE). Interoperability with other major quantum programming frameworks is ensured thanks to the myQLMinterop package, which allows users to build their own code and easily execute it on existing QPUs. The combined OpenVQE/myQLM-fermion libraries facilitate the implementation, testing and development of variational quantum algorithms, while offering access to large molecules as the noiseless Schrödinger-style dense simulator can reach up to 41 qubits for any circuit. Extensive benchmarks are provided for molecules associated to qubit counts ranging from 4 up to 24. We focus on reaching chemical accuracy, reducing the number of circuit gates and optimizing parameters and operators between "fixed-length" UCC and ADAPT-VQE ansätze.

Physical Review A

We consider higher-order corrections to hyperfine coefficients related to the spin-orbit and spin... more We consider higher-order corrections to hyperfine coefficients related to the spin-orbit and spinspin tensor interactions in hydrogen molecular ions. The mα 7 ln(α)-order radiative correction is derived in the NRQED framework. We present complete numerical calculations, including as well the mα 6-order relativistic correction, for the case of H + 2. The theoretical uncertainty is reduced by more than one order of magnitude with respect to the Breit-Pauli level, down to a few ppm. We also compare our results with available rf spectroscopy data.

Physical Review A, 2020

A complete effective Hamiltonian for relativistic corrections at orders mα 6 and mα 6 (m/M) in a ... more A complete effective Hamiltonian for relativistic corrections at orders mα 6 and mα 6 (m/M) in a one-electron molecular system is derived from the NRQED Lagrangian. It includes spin-independent corrections to the energy levels and spin-spin scalar interactions contributing to the hyperfine splitting, both of which had been studied previously. In addition, corrections to electron spin-orbit and spin-spin tensor interactions are newly obtained. This allows improving the hyperfine structure theory in the hydrogen molecular ions. Improved values of the spin-orbit hyperfine coefficient are calculated for a few transitions of current experimental interest. I. INTRODUCTION High-resolution spectroscopy of the hydrogen molecular ions H + 2 and HD + may contribute significantly to the determination of fundamental constants such as the proton-electron mass ratio m p /m e [1]. A pure rotational transition in HD + has recently been measured with a relative uncertainty of 1.3 × 10 −11 [2]. The experimental accuracy of ro-vibrational transition frequencies is expected to reach a few parts per trillion in the near future using spectroscopy in the Lamb-Dicke regime [2-4] or in a Doppler-free geometry [5, 6]. While information on fundamental constants is obtained from comparison of spin-averaged transition frequencies with theoretical predictions, the hyperfine splitting of ro-vibrational lines also allows for precise tests of theory. So far, the hyperfine structure of H + 2 and HD + has been calculated within the Breit-Pauli approximation [7, 8], taking into account the anomalous magnetic moment of the electron. All terms at orders mα 4 and mα 5 are included, so that the theoretical accuracy of the hyperfine coefficients is of order α 2 ∼ 5 × 10 −5. Higherorder corrections to the largest coefficients, i.e. the spin-spin Fermi contact interaction, were later calculated in [9, 10], which allowed to get excellent agreement with available RF spectroscopy data in H + 2 [11] at the level of ∼ 1 ppm. The following step to improve the hyperfine structure theory is to evaluate higher-order corrections to the next largest coefficients, i.e. the electron spin-orbit and spin-spin tensor interaction, starting with relativistic corrections at the mα 6 order. With this aim, we derive in the present work the complete effective Hamiltonian for the hydrogen molecular ions at the mα 6 and mα 6 (m/M) orders, following the NRQED approach [12-14]. Then, we use it to calculate numerically the corrections to the electron spin-orbit interaction for a few transitions studied in ongoing experiments. The paper is organized as follows: in Secs. II and III, we recall the expression of the NRQED Lagrangian and associated interaction vertices. We then systematically derive the effective potentials, which are organized in three categories: tree-level interactions involving the exchange of a Coulomb or transverse photon (Sec. IV), terms due to retardation in the transverse photon exchange (Sec. V), and finally those coming from a seagull diagram with simultaneous exchange of two photons (Sec. VI). In Sec. VII, we collect our results to write the total effective Hamiltonian, separating the different types of interactions: spinindependent, electronic spin-orbit, spin-spin scalar and tensor interactions. Finally, in Sec. VIII we present numerical calculations of the spin-orbit interaction coefficient. II. NRQED LAGRANGIAN Natural (Lorenz-Heaviside) units (h = c = 1) are used throughout. We assume that e is the electron's charge and thus is negative, the elementary charge is then denoted by |e|.

Physical Review A, 2020

The NRQED approach is applied to the calculation of relativistic corrections to the fine and hype... more The NRQED approach is applied to the calculation of relativistic corrections to the fine and hyperfine structure of hydrogenlike atoms at orders mα 6 and mα 6 (m/M). Results are found to be in agreement with those of the relativistic theory. This confirms that the derived NRQED effective potentials are correct, and may be used for studying more complex atoms or molecules. Furthermore, we verify the equivalence between different forms of the NRQED Lagrangian used in the literature.

Recent Progress in Few-Body Physics, 2020

Although they do not lend themselves to analytical resolution, three-body atomic or molecular sys... more Although they do not lend themselves to analytical resolution, three-body atomic or molecular systems are still simple enough to allow for very precise theoretical predictions of their energy levels, which makes them attractive candidates for fundamental tests and determination of fundamental physical constants. Focusing on the hydrogen molecular ions (H + 2 , HD + , D + 2), we outline the methods which have been used to improve the theoretical accuracy by several orders of magnitude over the last two decades. The three-body Schrödinger equation can be solved with extreme precision by variational methods with trial functions involving exponentials of interparticle distances. Quantum electrodynamics (QED) corrections are evaluated in the framework of nonrelativistic QED (NRQED). The current status of theory and possibilities of further improvement are briefly sketched.