Encoding electronic spectra using second-quantized entanglement networks of angular momenta

Abstract

According to the invention an apparatus is provided for encoding electronic spectra comprising a unit for generating eigenfunctions of quantum angular momenta to transform the Coulomb operator of electronic spectra into an isospectral diagonal form and a unit for decomposing said isospectral diagonal form into atomic orbital angular momentum blocks.

Description

[0001] ENCODING ELECTRONIC SPECTRA USING SECOND-QUANTIZED ENTANGLEMENT NETWORKS OF ANGULAR MOMENTA FIELD OF THE INVENTION

[0002] This invention relates to quantum computing and more particularly to encoding electronic spectra using second-quantized entanglement networks of angular momenta.

[0003] BACKGROUND

[0004] Vector-coupling and vector-uncoupling schemes in the quantum theory of angular momentum correspond to unitary Clebsch-Gordan transformations that operate on quantum angular momentum states and thereby control their degree of entanglement.

[0005] The addition of quantum angular momentum from this transformation is suitable for reducing the degree of entanglement of quantum angular momentum, leading to simple and effective calculations of the molecular integrals of solid harmonic Gaussian orbitals (SHGO).

[0006] Even with classical computers, the speed-up ratio in the evaluation of molecular nuclear Coulomb integrals with SHGOs can be up to four orders of magnitude for atomic orbitals with high angular momentum quantum number. This indicates that the less entanglement there is for a quantum system the easiest it is to simulate. It also demonstrates that molecular integrals with SHGOs are particularly well-suited for quantum computing.

[0007] High-efficiency quantum circuits previously developed for unitary and cascading Clebsch-Gordan transformations of angular momentum states can be applied to the differential and product rules of solid harmonics to efficiently compute two-electron Coulomb integrals ubiquitous in quantum chemistry. Combined with such quantum circuits and variational quantum eigensolver algorithms, the high computational efficiency of molecular integrals in solid harmonic bases unveiled in this paper may open an avenue for accelerating full quantum computational chemistry.

[0008] In the atomic theory of quantum mechanics, orbitals, with well-defined energy, angular momentum and magnetic momentum, are the building blocks of a system electronic structure. Modern computational chemistry methods, such as Hartree- Fock and density functional theory, use mathematical functions to represent the orbitals. These functions are typically a product of a radial component which defines the energy and an eigenfunction of the angular momentum operator for the angular part of the atomic orbitals. In physics and quantum mechanics, in particular, angular momentum plays a central role in systems with rotational symmetry.

[0009] Complicated calculations can be naturally factorized into several simpler parts using the angular momentum. In general, these simple factors may be divided into two types. The first type is invariant under rotation, mainly determined by the precise physical nature of the quantum system under consideration. In contrast, the second type depends solely on the system rotational properties and is relatively independent of its physical nature. Therefore, the second type of factors can be precisely expressed as a function of angular momenta, laying the foundation for a very general theory of angular momentum algebra.

[0010] The elegant computational methods derived from this theory were widely applied to many problems such as atomic, molecular, and nuclear spectroscopies, and nuclear reactions. Similar computational methods were developed for one-electron molecular integrals using solid harmonic Gaussian orbitals. In particular, recent developments have shown significant advantages in switching Cartesian Gaussian orbitals (CGOs) to solid harmonic Gaussian orbitals (SHGOs). Efficient calculations of molecular Coulomb integrals with SHGOs have not been reported previously. Additionally, evaluation of molecular integrals with SHGOs may be particularly well suited for implementation of quantum computing approaches to computational chemistry. Due to the non-zero Clebsch-Gordan coefficients of angular momentum vector-coupling and vector-uncoupling, there is some entanglement in molecular integrals with SHGOs. Quantum entanglement is a unique feature of quantum computing. The angular momentum entanglement of rotational degrees of freedom in molecular integrals is inherently quantum. Consequently, molecular integrals with SHGOs are well suited for quantum computing.

[0011] SUMMARY

[0012] The invention disclosed herein provides for a method and computer for encoding electronic spectra comprising a unit for generating eigenfunctions of quantum angular momenta to transform the Coulomb operator of electronic spectra into an isospectral diagonal form and a unit for decomposing said isospectral diagonal form into atomic orbital angular momentum blocks. According to implementations thereof the method or computer executes quantum computing simulations of molecules and condensed matters; and / or there is reduced entanglement between the quantum angular momenta at a same center in execution.

[0013] BRIEF DESCRIPTION OF THE DRAWINGS

[0014] The drawings have not necessarily been drawn to scale. Similarly, some components and / or operations can be separated into different blocks or combined into a single block for the purposes of discussion of some of the implementations of the present technology. Moreover, while the technology is amenable to various modifications and alternative forms, specific implementations have been shown by way of example in the drawings and are described in detail below. The intention, however, is not to limit the technology to the particular implementations described. On the contrary, the technology is intended to cover all modifications, equivalents, and alternatives falling within the scope of the technology as defined by the appended claims.

[0015] Figure 1 shows a computational speed-up ratio in the calculation of nuclear attraction Coulomb integrals as a function of highest orbital angular momentum number.

[0016] DETAILED DESCRIPTION Developed herein is an approach for efficient evaluation of molecular integrals with SHGOs using a combination of vector-coupling and vector-uncoupling schemes of angular momenta. We demonstrate that the highly efficient calculation of molecular Coulomb integrals with SHGOs arises not only from simpler mathematical expressions but also from their quantum nature (the entanglement degree of orbital angular momentum states). In fact, the vector-coupling and vector- uncoupling schemes, which provide natural means to efficiently evaluate molecular integrals, refer to a unitary transformation, also known as the Clebsch-Gordan transformation . This transformation performs quantum angular momentum coupling and uncoupling, naturally preserving entanglement, a physical resource central to quantum information and quantum computing. This transformation is also applied to construct efficient quantum circuits for practical and efficient quantum algorithms. As a result, efficient evaluation of molecular integrals exploiting entanglement of orbital angular momenta may open a new vista to full quantum computational chemistry for accelerating novel materials and high -potency drug discovery. Note that variational quantum eigensolver algorithms have been developed and applied to the calculations of the ground state and excited energies of molecules

[0017] Taking the overlap and nuclear Coulomb attraction integrals as paradigms in this paper, we show a detailed derivation of efficient-to-evaluate expressions for molecular integrals using SHGOs and relevant vector-coupling and vector-uncoupling schemes of angular momenta. In general, SHGOs are defined as where a is the orbital atomic center, α the Gaussian exponent, N (la, α ) the normalization constant, and are solid harmonics. Solid harmonics are related to the spherical harmonics as where ( ) are the spherical harmonics with the phase convention of Condon and Shortley, I and m are the orbital angular momentum and magnetic quantum numbers, respectively. With the Hobson theorem of solid harmonics, SHGOs can be further simplified using solid harmonic derivatives with respect to the orbital atomic center,

[0018] Without the normalization constants, the general expression for a molecular overlap integral is of the form

[0019] Using the Gaussian product rule and integration yield overlap integrals given by

[0020] The overlap integral in Eq. (6) can be evaluated by applying the differential and product rules of solid harmonic derivative9. This approach is similar to the vector-uncoupling scheme of angular momenta because there are no terms related to the la+ lbquantum number. However, this is suboptimal in terms of computational efficiency, with too many harmonic derivatives acting on different atomic centers, resulting in deep angular momentum entanglement of the atomic orbitals. Our new approach can eliminate the harmonics derivatives with the addition of angular momentum . With this addition, we shift orbital atomic centers to the same center P of Gaussian and obtain the following equation

[0021] Here, the vector-coupling coefficients of angular momenta are given by and are also Clebsch-Gordan transformation coefficients. Therefore, a two-center overlap integral is transformed into a single -center integral,

[0022] With the orthonormality of solid harmonics, we obtain

[0023]

[0024] The summation in the second line of Eq. (10) is independent of orbitals and determined only by the rotational properties of angular momenta for an overlap integral. The last line in Eq. (9) is an elegant solution to demonstrate that quantum angular momenta of the same atomic center can not entangle with each other due to their orthonormality. It indicates that if a quantum system does not have a lot of entanglement, it will be much easier to simulate. As a result, in the same way, the nuclear Coulomb attraction integrals can be directly calculated by simply introducing a nuclear Coulomb operator, located at the center c , into the integral of the last line in Eq. (9),

[0025] The last line of Eq. (11) clearly expresses the nuclear Coulomb attraction integral as the interaction between Gaussian distributed angular momenta at the center and the nuclear Coulomb potential at the center. It can be further simplified using solid harmonics derivatives to

[0026]

[0027] Compared with all available integrals of the same type with either SHGOs or CGOs, Eq. (12) is the simplest expression for general nuclear Coulomb attraction integrals. The final expression combines Eq. (11) and Eq. (12) and contains three factors.

[0028] The first one contains solid harmonics and is independent of Gaussian exponents. The second one involves solid harmonics dependent on Gaussian exponents25. The third one is the Boys function dependent on Gaussian exponents. Table 1 shows a comparison of the computational cost of evaluating the nuclear Coulomb attraction integral with CGOs and SHGOs26.27dominant computational cost with CGOs scales as L7P2(L is the highest angular momentum number and P is the number of primitive Gaussians). For SHGO with only the vector-uncoupling scheme of angular momentum, as implemented in the current ParaGauss package, a similar computational cost of L6P2is also found16. The main computational cost comes from the vector-uncoupling scheme of angular momentum. This relatively high cost is the result of three-fold summations over angular momentum / and magnetic quantum m, respectively12. The interaction of two orbital angular momenta from different centers with a nuclear Coulomb potential can generate a lot of quantum entanglement under the vector-uncoupling scheme.

[0029] For a realistic comparison of computational speed-up, Eqs. (11) and (12) were programmed in Fortran 90 and implemented for nuclear Coulomb attraction integrals within the ParaGauss package. This package evaluates the nuclear Coulomb attraction integral using the three-fold nested product rule of solid harmonics, with a main computational cost about L6P2. This rule mainly applies to the vector-uncoupling scheme of angular momenta. As a result, the vector- uncoupling scheme of angular momenta is not suitable for multi-center molecular integrals. This can be the main reason why two-electron Coulomb integrals with SHGOs have not been implemented in this quantum chemistry package. Figure 1 shows the computational overall speed- up ratio achieved with the ParaGauss package using SHGOs. About three-orders of magnitude increase in the computational speed-up ratio can be achieved, depending on the quantum number of angular momenta. For calculations of CGOs using l=7 (h orbital) as the highest angular momentum number and P=10 for the number of primitive Gaussians, the speed-up ratio is roughly about 14,600 times, based on an estimate from Table 1. This represents a computational efficiency gain of four orders of magnitude.

[0030] Table 1. Cost associated with computing nuclear attraction integrals

[0031] L is the highest orbital angular msmeatum number of the Gaussian crbstafe and P the number of primitive Gaussiaa functions.

[0032] For a general two-electron Coulomb integral, there are four Gaussian exponents α , β , δ , and X located at four atomic orbital centers respectively. We can transform this two-electron four-center Coulomb integral into a two -center one with the centres located at in terms of four Gaussian distributed angular momenta based on Eq. (7). Here, we will focus on the potential of angular momenta located at the center Q . Based on the general three-dimensional Green’s function for Poisson’s equation and Hobson theorem23, the Green’s function can be expressed as, where the angular momentum potential components are transformed into harmonic derivatives using the Hobson theorem. Then, the general solution of Poisson’s equation is, where is fer the radial integration range from to infinity and s for the radial integration range from zero to shows that the two -electron Coulomb integral is actually related to the orbital angular momentum interaction in the electronic Coulomb potential by the general solution of Poisson’s equation in spherical coordinates. The detailed derivation is given in the Supplement Materials. There are two of the two-electron Coulomb integral. Their general expressions, independent of the original angular momentum numbers of the Gaussian orbitals, are given by

[0033] wher s an integer becaus must be even due to the Wign

[0034] Since the Wigner 3-j symbol lias several other constraint conditions on the quantum numbers of the three angular momenta such as the infinite sum over the angular momentum in Eq. (13) becomes finite, due to natural truncation. Moreover, symmetry conditions for the Wigner 3-j symbol also reduce the number of nonzero terms in Eq. (15), resulting in significant computational saving. More importantly, calculations are all exact, with no approximation to justify truncation of the series expansion necessary . With the solid harmonics product rule using the vector-coupling scheme, the main computational cost of two-electron Coulomb integrals is generally about L6P4, but there are additional computational savings. First, the three-fold product rule applied to this integral is a nested loop, halving the computational cost. Second, the constraint conditions of the Wigner 3-j symbol also apply to integrals. Third, all terms associated with solid harmonics vanish if their angular momentum quantum numbers are less than that of the solid harmonic derivative which translates again in tremendous reduction of computational cost. In comparison with the dominant computational cost such as sociated with two-electron Coulomb integrals with CGOs27, the speed-up ratio may increase by several orders of magnitude. Like for nuclear Coulomb attraction integrals, the computational speedup in evaluating two -electron Coulomb integrals also comes from the absence of entanglement between the quantum angular momenta at the same center. Note that the two -electron Coulomb integrals include terms related to multipole solid harmonic derivatives, which can be attributed to the entanglement of angular momenta from different atomic centers. In fact, solid harmonic derivatives and the product rule are all based on a unitary Clebsch-Gordan transformation for the quantum angular momentum. The efficient quantum circuit for this transformation has been already developed to convert angular momentum states. The quantum circuit for the cascading Clebsch-Gordan transform may apply to the product rule of solid harmonic derivatives. The quantum circuit for Clebsch-Gordan transform only requires three qubits, two controlled operation X gates, and one doubly controlled rotational gate. Therefore, an efficient quantum circuit with only a few qubits and gates for Clebsch-Gordan transform would be able to evaluate the two-electron Coulomb integral, with an estimated computational cost of about LP4.

[0035] In the previous description, for the purposes of explanation, numerous specific details are set forth in order to provide a thorough understanding of implementations of the present technology. It will be apparent, however, to one skilled in the art that implementations of the present technology can be practiced without some of these specific details.

[0036] The design techniques introduced here can be implemented as special -purpose hardware (for example, circuitry), as programmable circuitry appropriately programmed with software and / or firmware, or as a combination of special-purpose and programmable circuitry. Hence, implementations can include a machine-readable medium having stored thereon instructions which can be used to program a computer (or other electronic devices) to perform a process. The machine-readable medium can include, but is not limited to, floppy diskettes, optical disks, compact disc read-only memories (CD-ROMs), magneto-optical disks, ROMs, random access memories (RAMs), erasable programmable read-only memories (EPROMs), electrically erasable programmable read-only memories (EEPROMs), magnetic or optical cards, flash memory, or other type of media / machine -readable medium suitable for storing electronic instructions.

[0037] The technology herein can be seen to be particularly suitable for execution with quantum programing. Quantum programming is a process of designing or assembling sequences of instructions, called quantum circuits, using gates, switches, and operators to manipulate a quantum system for a desired outcome or results of a given experiment. Quantum circuit algorithms can be implemented on integrated circuits, for instance the special purpose hardware or programable circuitry stated above; conducted with instrumentation, or written in a programming language for use with a quantum computer or a quantum processor.

[0038] With quantum processor based systems, quantum programming languages help express quantum algorithms using high-level constructs.

[0039] Quantum computers, such as those based on the KLM protocol, a linear optical quantum computing (LOQC) model, may use quantum algorithms (circuits) implemented with electronics, integrated circuits, instrumentation, sensors, and / or by other physical means.

[0040] The phrases “in some implementations,” “according to some implementations,” “in the implementations shown,” “in other implementations,” and the like generally mean the particular feature, structure, or characteristic following the phrase is included in at least one implementation of the present technology, andean be included in more than one implementation. In addition, such phrases do not necessarily refer to the same implementations or different implementations. ...

[0041] Unless the context clearly requires otherwise, throughout the description and the claims, the words “comprise,” “comprising,” and the like are to be construed in an inclusive sense, as opposed to an exclusive or exhaustive sense; that is to say, in the sense of “including, but not limited to.” As used herein, the terms “connected,” “coupled,” or any variant thereof, means any connection or coupling, either direct or indirect, between two or more elements; the coupling of connection between the elements can be physical, logical, or a combination thereof. Additionally, the words “herein,” “above,” “below,” and words of similar import, when used in this application, shall refer to this application as a whole and not to any particular portions of this application. Where the context permits, words in the above Detailed Description using the singular or plural number may also include the plural or singular number respectively. The word “or,” in reference to a list of two or more items, covers all of the following interpretations of the word: any of the items in the list, all of the items in the list, and any combination of the items in the list.

[0042] The above detailed description of implementations of the system is not intended to be exhaustive or to limit the system to the precise form disclosed above. While specific implementations of, and examples for, the system are described above for illustrative purposes, various equivalent modifications are possible within the scope of the system, as those skilled in the relevant art will recognize

[0043] These and other changes can be made to the invention in light of the above Detailed Description. While the above description describes certain implementations of the technology, and describes the best mode contemplated, no matter how detailed the above appears in text, the invention can be practiced in many ways. Details of the system may vary considerably in its implementation details, while still being encompassed by the technology disclosed herein. As noted above, particular terminology used when describing certain features or aspects of the technology should not be taken to imply that the terminology is being redefined herein to be restricted to any specific characteristics, features, or aspects of the technology with which that terminology is associated. In general, the terms used in the following claims should not be construed to limit the invention to the specific implementations disclosed in the specification, unless the above Detailed Description section explicitly defines such terms. Accordingly, the actual scope of the invention encompasses not only the disclosed implementations, but also all equivalent ways of practicing or implementing the invention under the claims.

Claims

ClaimsWhat is claimed is:

1. A quantum computer for encoding electronic spectra comprising:A unit for generating eigenfunctions of quantum angular momenta to transform the Coulomb operator of electronic spectra into an isospectral diagonal formAnd a unit for decomposing said isospectral diagonal form into atomic orbital angular momentum blocks.

2. The quantum computer of claim 1 for executing quantum computing simulations of molecules and condensed matters.

3. The quantum computer of claim 1 wherein there is reduced entanglement between the quantum angular momenta at a same center.

4. A method for encoding electronic spectra comprising: generating eigenfunctions of quantum angular momenta to transform the Coulomb operator of electronic spectra into an isospectral diagonal form decomposing said isospectral diagonal form into atomic orbital angular momentum blocks.

5. The method of claim 4 for executing quantum computing simulations of molecules and condensed matters.

6. The method of claim 4 wherein there is reduced entanglement between the quantum angular momenta at a same center.