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DOE OSTI2023

Time-warping invariant quantum recurrent neural networks via quantum-classical adaptive gating

Adaptive gating plays a key role in temporal data processing via classical recurrent neural networks (RNNs), as it facilitates retention of past information necessary to predict the future, providing a mechanism that preserves invariance to time warping transformations. This paper builds on quantum RNNs (QRNNs), a dynamic model with quantum memory, to introduce a novel class of temporal data processing quantum models that preserve invariance to time-warping transformations of the (classical) input-output sequences. The model, referred to as time warping-invariant QRNN (TWI-QRNN), augments a QRNN with a quantum–classical adaptive gating mechanism that chooses whether to apply a parameterized unitary transformation at each time step as a function of the past samples of the input sequence via a classical recurrent model. The TWI-QRNN model class is derived from first principles, and its capacity to successfully implement time-warping transformations is experimentally demonstrated on examples with classical or quantum dynamics.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

A universal variational quantum eigensolver for non-Hermitian systems

Abstract Many quantum algorithms are developed to evaluate eigenvalues for Hermitian matrices. However, few practical approach exists for the eigenanalysis of non-Hermintian ones, such as arising from modern power systems. The main difficulty lies in the fact that, as the eigenvector matrix of a general matrix can be non-unitary, solving a general eigenvalue problem is inherently incompatible with existing unitary-gate-based quantum methods. To fill this gap, this paper introduces a Variational Quantum Universal Eigensolver (VQUE), which is deployable on noisy intermediate scale quantum computers. Our new contributions include: (1) The first universal variational quantum algorithm capable of evaluating the eigenvalues of non-Hermitian matrices—Inspired by Schur’s triangularization theory, VQUE unitarizes the eigenvalue problem to a procedure of searching unitary transformation matrices via quantum devices; (2) A Quantum Process Snapshot technique is devised to make VQUE maintain the potential quantum advantage inherited from the original variational quantum eigensolver—With additional $$O(log_{2}{N})$$ O ( l o g 2 N ) quantum gates, this method efficiently identifies whether a unitary operator is triangular with respect to a given basis; (3) Successful deployment and validation of VQUE on a real noisy quantum computer, which demonstrates the algorithm’s feasibility. We also undertake a comprehensive parametric study to validate VQUE’s scalability, generality, and performance in realistic applications.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

Diagonalizing the Born–Oppenheimer Hamiltonian via Moyal perturbation theory, nonadiabatic corrections, and translational degrees of freedom

This article describes a method for calculating higher order or nonadiabatic corrections in Born–Oppenheimer theory and its interaction with the translational degrees of freedom. The method uses the Wigner–Weyl correspondence to map nuclear operators into functions on the classical phase space and the Moyal star product to represent operator multiplication on those functions. These are explained in the body of the paper. The result is a power series in κ 2 , where κ = (m/M)1/4 is the usual Born–Oppenheimer parameter. The lowest order term is the usual Born–Oppenheimer approximation, while higher order terms are nonadiabatic corrections. These are needed in calculations of electronic currents, momenta, and densities. The separation of nuclear and electronic degrees of freedom takes place in the context of the exact symmetries (for an isolated molecule) of translations and rotations, and these, especially translations, are explicitly incorporated into our discussion. This article presents an independent derivation of the Moyal expansion in molecular Born–Oppenheimer theory. Here, we show how electronic currents and momenta can be calculated within the framework of Moyal perturbation theory; we derive the transformation laws of the electronic Hamiltonian, the electronic eigenstates, and the derivative couplings under translations; we discuss in detail the rectilinear motion of the molecular center of mass in the Born–Oppenheimer representation; and we show how the elimination of the translational components of the derivative couplings leads to a unitary transformation that has the effect of exactly separating the translational degrees of freedom.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2025

Real classical shadows

Efficiently learning expectation values of a quantum state using classical shadow tomography has become a fundamental task in quantum information theory. In a classical shadows protocol, one measures a state in a chosen basis $\mathcal{W}$ after it has evolved under a unitary transformation randomly sampled from a chosen distribution $\mathcal{U}$. In this work we study the case where $\mathcal{U}$ corresponds to either local or global orthogonal Clifford gates, and $\mathcal{W}$ consists of real-valued vectors. Our results show that for various situations of interest, this ‘real’ classical shadow protocol improves the sample complexity over the standard scheme based on general Clifford unitaries. For example, when one is interested in estimating the expectation values of arbitrary real-valued observables, global orthogonal Cliffords typically decrease the required number of samples by a factor of two. More dramatically, for k-local observables composed only of real-valued Pauli operators, sampling local orthogonal Cliffords leads to a reduction by an exponential-in-k factor in the sample complexity over local unitary Cliffords. Finally, we show that by measuring in a basis containing complex-valued vectors, orthogonal shadows can, in the limit of large system size, exactly reproduce the original unitary shadows protocol.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2024

Theory of quantum circuits with Abelian symmetries

Quantum circuits with gates (local unitaries) respecting a global symmetry have broad applications in quantum information science and related fields, such as condensed-matter theory and quantum thermodynamics. However, despite their widespread use, fundamental properties of such circuits are not well understood. Recently, it was found that generic unitaries respecting a global symmetry cannot be realized, even approximately, using gates that respect the same symmetry. This observation raises important open questions: What unitary transformations can be realized with k -local gates that respect a global symmetry? In other words, in the presence of a global symmetry, how does the locality of interactions constrain the possible time evolution of a composite system? In this work, we address these questions for the case of Abelian (commutative) symmetries and develop constructive methods for synthesizing circuits with such symmetries. Remarkably, as a corollary, we find that, while the locality of interactions still imposes additional constraints on realizable unitaries, certain restrictions observed in the case of non-Abelian symmetries do not apply to circuits with Abelian symmetries. For instance, in circuits with a general non-Abelian symmetry such as SU ( d ) , the unitary realized in a subspace with one irreducible representation (charge) of the symmetry dictates the realized unitaries in multiple other sectors with inequivalent representations of the symmetry. Furthermore, in certain sectors, rather than all unitaries respecting the symmetry, the realizable unitaries are the symplectic or orthogonal subgroups of this group. We prove that none of these restrictions appears in the case of Abelian symmetries. This result suggests that global non-Abelian symmetries may affect the thermalization of quantum systems in ways not possible under Abelian symmetries. Published by the American Physical Society 2024

Marvian, Iman↗
DOE OSTI2025

Quantum complexity in gravity, quantum field theory, and quantum information science

Quantum complexity quantifies the difficulty of preparing a state or implementing a unitary transformation with limited resources. Applications range from quantum computation to condensed matter physics and quantum gravity. Here, we seek to bridge the approaches of these fields, which define and study complexity using different frameworks and tools. We describe several definitions of complexity, along with their key properties. In quantum information theory, we focus on complexity growth in random quantum circuits. In quantum many-body systems and quantum field theory (QFT), we discuss a geometric definition of complexity in terms of geodesics on the unitary group. In dynamical systems, we explore a definition of complexity in terms of state or operator spreading, as well as concepts from tensor-networks. We also outline applications to simple quantum systems, quantum many-body models, and QFTs including conformal field theories (CFTs). Finally, we explain the proposed relationship between complexity and gravitational observables within the holographic anti-de Sitter (AdS)/CFT correspondence.

Baiguera, Stefano [Istituto Nazionale di Fisica Nu↗
DOE OSTI2021

Inversion of dynamical Bragg intensities to complex structure factors by iterated projections. For Ultramic. 2020. ("Pico" Festschrift, May 2021)

We discuss a method for recovering complex structure factors from many simultaneously excited Bragg beam intensities is described. The method is applied to simulated transmission electron diffraction data over a wide range of crystal thickness and beam energies. The method is based on iterated projections between structure and scattering matrices, which are related by a matrix unitary transformation, exponential, which we invert. The algorithm removes multiple-scattering perturbations from diffraction data and might be extended to other fields, including X-ray and neutron diffraction and cryo-electron microscopy. Because coherent multiple scattering involves interference between Bragg beams, the method also solves the phase problem. Unlike dynamical inversion from electron microscope images or ptychography data, the method, which starts with Bragg beam intensities, provides complex structure factors unaffected by focusing errors or resolution limitations imposed by lenses. We provide inversions from simulated data with 441 simultaneously excited Bragg beams over a range of thickness and beam energy. We discuss the retrieval of chirality information from enantiomorphs, the efficient incorporation of symmetry information using the irreducible representation of the group of structure matrices, and the effect of HOLZ lines to provide three-dimensional information.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2024

WFOT: A Wave Function Overlap Tool between Single- and Multi-Reference Electronic Structure Methods for Spectroscopy Simulation

We report the development of a novel diagnostic tool, named wave function overlap tool (WFOT), designed to evaluate the overlap between wave functions computed at single-reference [i.e., time-dependent density functional theory or configuration interaction singles (CIS)] and multireference (i.e., CASSCF/CASPT2) electronic structure levels of theory. It relies on truncating the single- and multireference WFs to CIS-like expansions spanning the same configurational space and maximizing the molecular orbital overlap by means of a unitary transformation. To demonstrate the functionality of the tool, we calculate the transient spectrum of acetylacetone by evaluating excited state absorption signals with multireference quality on top of single-reference on-the-fly dynamics simulations. Semiautomatic spectra generation is facilitated by interfacing the tool with the COBRAMM package, which also allows one to use WFOT with several quantum chemistry codes such as Gaussian, NWChem, and OpenMolcas. Finally, other exciting possibilities for the utilization of the code beyond the simulation of transient absorption spectroscopy are eventually discussed.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2021

Hamiltonian systems, Toda lattices, solitons, Lax pairs on weighted Z -graded graphs

In this study, we consider discrete one-dimensional nonlinear equations and present the procedure of lifting them to Z -graded graphs. We identify conditions that allow one to lift one-dimensional solutions to solutions on graphs. In particular, we prove the existence of solitons for static potentials on graded fractal graphs. We also show that even for a simple example of a topologically interesting graph, the corresponding non-trivial Lax pairs and associated unitary transformations do not lift to a Lax pair on the Z -graded graph.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Complex Fermi–Löwdin orbital self-interaction correction

Here, this paper introduces the use of complex Fermi orbital descriptors (FODs) in the Fermi–Löwdin self-interaction-corrected density functional theory (FLOSIC). With complex FODs, the Fermi–Löwdin orbitals (FLOs) that are used to evaluate the SIC correction to the total energy become complex. Complex FLO-SIC (cFLOSIC) calculations based on the local spin density approximation produce total energies that are generally lower than the corresponding energies found with FLOSIC restricted to real orbitals (rFLOSIC). The cFLOSIC results are qualitatively similar to earlier Perdew–Zunger SIC (PZ-SIC) calculations using complex orbitals. The energy lowering stems from the exchange–correlation part of the self-interaction correction. The Hartree part of the correction is more negative in rFLOSIC. The energy difference between real and complex solutions is greater for more strongly hybridized FLOs in atoms and for FLOs corresponding to double and triple bonds in molecules. The case of N2 is examined in detail to show the differences between the real and complex FLOs. We show that the complex triple-bond orbitals are simple, and physically appealing combinations of π and σ g orbitals that have not been discussed before. Consideration of complex FODs, and resulting unitary transformations, underscores the fact that FLO centroids are not necessarily good guesses for FOD positions in a FLOSIC calculation.

37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
DOE OSTI2023

Geometric Event-Based Quantum Mechanics

In this work, we propose a special relativistic framework for quantum mechanics. It is based on introducing a Hilbert space for events. Events are taken as primitive notions (as customary in relativity), whereas quantum systems (e.g. fields and particles) are emergent in the form of joint probability amplitudes for position and time of events. Textbook relativistic quantum mechanics and quantum field theory can be recovered by dividing the event Hilbert spaces into space and time (a foliation) and then conditioning the event states onto the time part. Our theory satisfies the full Lorentz symmetry as a ‘geometric’ unitary transformation, and possesses relativistic observables for space (location of an event) and time (position in time of an event).

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Existence of a robust optimal control process for efficient measurements in a two-qubit system

The verification of quantum entanglement is essential for quality control in quantum communication. In this work we propose an efficient protocol to directly verify the two-qubit entanglement of a known target state through a single-expectation-value measurement. Our method provides exact entanglement quantification using the concurrence measure without performing quantum state tomography. We prove the existence of a unitary transformation that drives a known initial state of a two-qubit system to a designated final state, where the trace over a chosen observable directly yields the concurrence of the initial state. Furthermore, we implement an optimal control process of that transformation and demonstrate its effectiveness through numerical simulations. We also show that this process is robust to environmental noise. Our approach offers advantages in directly verifying entanglement with low circuit depth, making it suitable for industrial-scale quality control of entanglement generation. Our results presented here provide mathematical justification for our earlier computational experiments.

Rodriguez, Ricardo [McPherson Coll.] (ORCID:000000↗
DOE OSTI2026

Computational schemes for the Magnus expansion of the in-medium similarity renormalization group

The in-medium similarity renormalization group (IMSRG) is a popular many-body method used for computations of nuclei. It solves the many-body Schrödinger equation through a continuous unitary transformation of the many-body Hamiltonian. The IMSRG transformation is typically truncated at the normal-ordered two-body level, the IMSRG(2), but recently several approaches have been developed to capture the effects of normal-ordered three-body operators, the IMSRG(3). In particular, a factorized approximation to the IMSRG(3) proposes to capture the leading effects of three-body operators at the same computational cost as the IMSRG(2) approximation. This approach often employs an approximate scheme for solving the IMSRG equations, the so-called hunter-gatherer scheme. In this work, I study the uncertainty associated with this scheme. I find that the hunter-gatherer scheme differs by up to 7MeV for ground-state energies and 0.5MeV for excitation energies from standard IMSRG(2) approaches. These differences are in some cases comparable to the expected size of IMSRG(3) corrections.

39 ≤ A ≤ 58↗
DOE OSTI2021

Fermi-Löwdin-orbital self-interaction correction using the optimized-effective-potential method within the Krieger-Li-Iafrate approximation

Perdew-Zunger self-interaction correction (PZ-SIC) offers a route to remove self interaction errors on an orbital-by-orbital basis. A recent formulation of PZ-SIC by Pederson, Ruzsinszky and Perdew proposes restricting the unitary transformation to localized orbitals called Fermi-Löwdin orbitals. This formulation, called the FLOSIC method, simplifies PZ-SIC calculations and was implemented self-consistently using a Jacobi-like (FLOSIC-Jacobi) iteration scheme. In this work we implement the FLOSIC approach using the Krieger-Li-Iafrate (KLI) approximation to the optimized effective potential (OEP). We compare the results of present FLOSIC-KLI approach with FLOSIC-Jacobi scheme for atomic energies, atomization energies, ionization energies, barrier heights, polarizability of chains of hydrogen molecules etc. to validate the FLOSIC-KLI approach. The FLOSICKLI approach, which is within the realm of Kohn-Sham theory, predicts smaller energy gaps between frontier orbitals due to the lowering of eigenvalues of the lowest unoccupied orbitals. Results show that atomic energies, atomization energies, ionization energy as an absolute of highest occupied orbital eigenvalue, and polarizability of chains of hydrogen molecules between the two methods agree within 2%. Lastly the FLOSIC-KLI approach is used to determine the vertical ionization energies of water clusters.

74 ATOMIC AND MOLECULAR PHYSICS↗
DOE OSTI2021

Iterated projected position algorithm for constructing exponentially localized generalized Wannier functions for periodic and nonperiodic insulators in two dimensions and higher

Localized bases play an important role in understanding electronic structure. In periodic insulators, a natural choice of localized basis is given by the Wannier functions which depend on a choice of unitary transform known as a gauge transformation. Over the past few decades, there have been many works that have focused on optimizing the choice of the gauge so that the corresponding Wannier functions are maximally localized or reflect some symmetry of the underlying system. In this work, we consider fully nonperiodic materials where the usual Wannier functions are not well defined and gauge optimization is impractical. To tackle the problem of calculating exponentially localized generalized Wannier functions in both periodic and nonperiodic systems, we discuss the ‘iterated projected position (IPP)” algorithm. Here, the IPP algorithm is based on matrix diagonalization and therefore unlike optimization-based approaches, it does not require initialization and cannot get stuck at a local minimum. Furthermore, the IPP algorithm is guaranteed by a rigorous analysis to produce exponentially localized functions under certain mild assumptions. We numerically demonstrate that the IPP algorithm can be used to calculate exponentially localized bases for the Haldane model, the Kane-Mele model (in both Z 2 invariant even and Z 2 invariant odd phases), and the p x + ip y model on a quasicrystal lattice.

75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
DOE OSTI2024

Simplified projection on total spin zero for state preparation on quantum computers

Here, we introduce a simple algorithm for projecting on J = 0 states of a many-body system by performing a series of rotations to remove states with angular momentum projections greater than zero. Existing methods rely on unitary evolution with the two-body operator J 2 , which when expressed in the computational basis contains many complicated Pauli strings requiring Trotterization and leading to very deep quantum circuits. Our approach performs the necessary projections using the one-body operators J x and J z . By leveraging the method of Cartan decomposition, the unitary transformations that perform the projection can be parametrized as a product of a small number of two-qubit rotations, with angles determined by an efficient classical optimization. Given the reduced complexity in terms of gates, this approach can be used to prepare approximate ground states of even-even nuclei by projecting onto the J = 0 component of deformed Hartree-Fock states. We estimate the resource requirements in terms of the universal gate set {H,S, CNOT ,T} and briefly discuss a variant of the algorithm that projects onto J = 1/2 states of a system with an odd number of fermions.

73 NUCLEAR PHYSICS AND RADIATION PHYSICS↗
DOE OSTI2025

N -way parametric frequency beamsplitter for quantum photonics

Optical networks are the leading platform for the transfer of information due to their low loss and ability to scale to many information channels using optical frequency modes. To fully leverage the quantum properties of light in this platform, it is desired to manipulate higher-dimensional superpositions by orchestrating linear, beamsplitter-type interactions between several channels simultaneously. We propose a method of achieving simultaneous, all-to-all coupling between N optical frequency modes via N -way Bragg-scattering four-wave mixing. By exploiting the frequency degree of freedom, additional modes can be multiplexed in an interaction medium of fixed volume and loss while avoiding the introduction of excess noise. We generalize the theory of the frequency-encoded two-mode interaction to N modes under this four-wave-mixing approach, finding that arbitrary unitary transformations are possible through successive four-wave-mixing pump configurations. We experimentally verify the quantum nature of this scheme by demonstrating three-way multiphoton interference. Two input photons are shared among three frequency modes and display interference differing from that of two classical (coherent-state) inputs. These results show the potential of our approach for the scalability of photonic quantum information processing to general N -mode systems in the frequency domain. Published by the American Physical Society 2025

Oliver, Richard (ORCID:0000000207760657)↗
DOE OSTI2023

Equivalence between Fermion-to-Qubit Mappings in two Spatial Dimensions

We argue that all locality-preserving mappings between fermionic observables and Pauli matrices on a two-dimensional lattice can be generated from the exact bosonization in Chen et al., whose gauge constraints project onto the subspace of the toric code with emergent fermions. Starting from the exact bosonization and applying Clifford finite-depth generalized local unitary transformation, we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of Majorana fermions). In particular, we discover a new supercompact encoding using 1.25 qubits per fermion on the square lattice. We prove the existence of finite-depth quantum circuits to obtain fermion-to-qubit mappings with qubit-fermion ratios r = 1 + 1/2k for positive integers k, utilizing the trivialness of quantum cellular automata in two spatial dimensions. Also, we provide direct constructions of fermion-to-qubit mappings with ratios arbitrarily close to 1. When the ratio reaches 1, the fermion-to-qubit mapping reduces to the one-dimensional Jordan-Wigner transformation along a certain path in the two-dimensional lattice. Finally, we explicitly demonstrate that the Bravyi-Kitaev superfast simulation, the Verstraete-Cirac auxiliary method, Kitaev’s exactly solved model, the Majorana loop stabilizer codes, and the compact fermion-to-qubit mapping can all be obtained from the exact bosonization.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗