METHOD FOR SIMULATING THE BEHAVIOR OF A PHYSICAL SYSTEM

Abstract

A method for simulating the behavior of a physical system using a quantum computer comprising a plurality of qubits, wherein the qubits are first evaluated with respect to their system properties and categorized into high-performance qubits and low-performance qubits, wherein the physical system is described in a Hilbert space spanned by orbitals, and the spin-like system is assigned to the high-performance qubits during a mapping process, the behavior of the physical system is simulated using the qubits of the quantum computer, and physical observables of the physical system are determined by measurements of the qubits of the quantum computer, characterized in that, in an identification step, spin-like linear combinations of orbitals in the physical system are identified, and these are spin-like orbitals in the spin-like system.which, in addition to other remaining orbitals, span the Hilbert space, wherein the physical system thus described is transformed such that, after the transformation, changes in the number of electrons and / or the electron density on the spin-like orbitals are reduced or completely eliminated, and the transformed physical system is then described on a Hilbert space spanned only by the orbitals identified as spin-like.

Description

[0001] The present invention relates to a method for simulating the behavior of a physical system using a quantum computer comprising a plurality of qubits, wherein the qubits are first evaluated with respect to their system properties and categorized into high-performance qubits and low-performance qubits, wherein the physical system is described in a Hilbert space spanned by orbitals, and the spin-like system is assigned to the high-performance qubits during a mapping process, the behavior of the physical system is simulated using the qubits of the quantum computer, and physical observables of the physical system are determined by measurements of the qubits of the quantum computer.

[0002] Such a method is already known from DE 10 2019 109 816 A1. This method uses the high-performance qubits of a quantum computer to model a system with system elements consisting of two states, where the system is formed from a conventionally describable bath and a quantum mechanically describable cluster.

[0003] Auf die weitere Literatur in diesem Zusammenhang, insbesondere die Dokumente BROWN, Katherine L.; MUNRO, William J.; KENDON, Vivien M. Using quantum computers for quantum simulation. Entropy, 2010, 12. Jg., Nr. 11, S. 2268-2307. DOI: https: / / doi.org / 10.3390 / e12112268, STADLER, Pascal, et al. Demonstration of system-bath physics on a gate-based quantum computer. arXiv preprint arXiv:2404.18828v1, 29.04.2024. DOI: https: / / doi.org / 10.48550 / arXiv.2404.18828, SHIRAZI, Reza G., et al. Understanding Radicals via Orbital Parities. arXiv preprint arXiv:2404.18787, 29.04.2024. DOI: https: / / doi.org / 10.48550 / arXiv.2404.18787, SCHOENAUER, Benedikt M. [et al.]: A method to derive material-specific spin-bath model descriptions of materialsdisplaying prevalent spin physics (for simulation on NISQ devices). 30.04.2024, 17 S. https: / / static1.squarespace.com / static / 5f34fa00fe2696063683642a / t / 663347a384911934b32331c1 / 1714636709312 / HQS+Spin+Mapper_White+Paper_HQS-Quantum-Simulations.The PDF is referenced for further information.

[0004] When simulating quantum systems, the problem lies in identifying or characterizing the relevant degrees of freedom. For correlated quantum systems, spin physics is usually relevant, even if this is not apparent in the initial description.

[0005] Previous methods identify spin degrees of freedom via natural spin orbitals. Natural orbitals are eigenstates of the (reduced) single-particle density matrix of the physical system. However, these exhibit strong fluctuations in electron density, which makes spin determination imprecise. Natural orbitals are unsuitable for characterizing spin degrees of freedom because they do not account for population fluctuations.

[0006] It is known from the prior art to apply the Schrieffer-Wolff transformation method to a Hamiltonian operator.

[0007] Furthermore, the implementation of a Schrieffer-Wolff transformation is known from the prior art, which executes a generator using a previously known approach. However, since no closed-form solution exists for such an approach even for the one-dimensional Hubbard model, this method is also not helpful.

[0008] Another method involves a perturbative Schrieffer-Wolff transform calculation. Here, the generator is determined through a perturbative calculation. While this method works, it requires evaluating the Hamiltonian operator for all arbitrary states of Hilbert space. While the evaluation is sufficiently simple for a standard lattice model, it can become quite complex for molecular systems or other radicals, thus increasing the computational effort.

[0009] Another method performs a Schrieffer-Wolff transformation via variational calculus, starting with a general approach whose free parameters are optimized to minimize the difference between [S, H0] and V. However, for larger molecules or radicals, this method is extremely complex and computationally expensive.

[0010] Against this background, the method of the present invention solves this problem without using a specific solution approach, while simultaneously avoiding the introduction of variations. Furthermore, the method is intended to require less computational effort when describing the behavior of a physical system, to provide more accurate solutions, and to enable the simulation of the system on the high-performance qubits of a quantum computer.

[0011] This problem is solved by a method according to claim 1. Advantageous embodiments of the method can be found in the dependent claims.

[0012] A method is proposed for describing the behavior of a physical system using a quantum computer comprising a plurality of qubits, whereby the qubits are first evaluated with regard to their system properties and categorized into high-performance qubits and low-performance qubits.

[0013] The method is characterized according to the invention in that a model of the physical system is described by identifying spin-like linear combinations of orbitals in the physical system in an identification step, by describing the physical system in a Hilbert space spanned by these spin-like orbitals and the remaining orbitals, wherein the physical system thus described is transformed such that, after the transformation, changes in the number of electrons and / or the electron density on the spin-like orbitals are reduced or completely eliminated, and the transformed physical system is then described on a Hilbert space spanned only by the orbitals identified as spin-like, wherein the spin-like system is assigned to the high-performance qubits during a mapping process.The behavior of the physical system is simulated using the qubits of the quantum computer, and physical observables of the physical system are determined by measurements of the qubits of the quantum computer.

[0014] Specifically, such a method can be characterized by representing the remaining non-spin-like orbitals as a bath coupled to the spin-like system, with the orbitals of the bath being simulated by low-performance qubits of the quantum computer.

[0015] The transformation can also be modified so that the spin-like system is mapped onto the hardware of the quantum computer by rotating coupling terms between spin-like orbitals to correspond to two-qubit gates available on the quantum computer.

[0016] Furthermore, it may be provided that during the identification step, linear combinations of orbitals are considered spin-like if the fluctuation of the number of electrons on the orbital is negligible compared to the average occupation, where the average occupation of an orbital is n, with n=1 for spin-like orbitals, and a limit of the fluctuation dn is defined as 0<=ε<=n, where the fluctuation dn is considered small if ε>=dn / n.

[0017] Preferably, during the identification step, a value can be determined which quantifies the spin similarity of the basis orbitals by means of a systematic procedure that runs on the quantum computer or a conventional computer, whereby the orbitals are optimized to minimize the value.

[0018] In a more specific development, the procedure can provide that during the identification step a value is determined which measures the spin similarity of the basis orbitals using an observable that is the local parity of the orbitals.

[0019] Furthermore, during the identification step, an iterative optimization of the parity can be performed, whereby a sequence of unitary rotations of the fermionic operators for the orbitals is carried out, which is also performed in the same way on Hermitian conjugates of the operations, until an orbital basis is reached in which the smallest local parities have assumed their extreme values.

[0020] In a specific implementation, it may be provided that orbitals with a local parity 〈P i 〉0 ≅ -1 are excluded from the parity optimization.

[0021] Furthermore, it may be provided that after the optimization step the complete Hamiltonian operator, i.e. the Hamiltonian operator which describes the complete physical system in a suitable basis, is block diagonalized within the framework of a generalized Schrieffer-Wolff transformation.

[0022] In a preferred embodiment, the bath degrees of freedom can also be preconditioned to accelerate the Schrieffer-Wolff transformations by splitting the full Hamiltonian into a spin Hamiltonian, a bath Hamiltonian and a spin bath Hamiltonian, wherein the bath Hamiltonian relates exclusively to the bath and is used within the full Hamiltonian in a diagonalized form or in an approximate form such as Hartree-Fock decoupling or mean-field approximation.

[0023] The preferred solution is to reformulate the Schrieffer-Wolff transformations in a vector space as a system of linear equations.

[0024] Furthermore, it may be provided that, via block specifiers, observables are used in the partitioning of the complete Hamiltonian operator which extend beyond the Standard Model to describe spin, whereby operators acting on spin-like orbitals are split into components acting on different symmetry sectors, in particular on sectors with different electron numbers.

[0025] Preferably, the physical system to be described here can be a radical or a molecule containing radicals.

[0026] Alternatively, it can be assumed that the physical system to be described is a solid-state system or molecule containing atoms with open d- and f-shells.

[0027] Finally, another alternative may provide that the physical system to be described is a solid-state system or a molecule that is described by spin physics and / or in which transitions between states with different spin degrees of freedom are relevant.

[0028] The method for modeling and simulating the physical system provides that, in an identification step, spin-like linear combinations of orbitals in the physical system are first identified, and the physical system characterized by these linear combinations is described in a complete Hamiltonian operator. In an optimization step, the complete Hamiltonian operator is simplified to a static Hamiltonian by removing the charge degrees of freedom, which connects the remaining spin degrees of freedom as a cluster on the one hand with the bath of the system on the other. Preferably, during the mapping process, the spin degrees of freedom of high-performance qubits and the bath are assigned to the remaining, especially low-performance, qubits.

[0029] In concrete terms, during the identification step, linear combinations of orbitals can be considered spin-like if they are occupied by exactly one electron and the fluctuation of the electron number is low.

[0030] Preferably, during the identification step, the degree of spin-like character of the basis orbitals can be determined by means of a systematic procedure which can be run on the quantum computer or a binary auxiliary system.

[0031] During the identification step, the degree of spin-like character of the basis orbitals can be measured using an observable corresponding to the local parity.

[0032] In concrete terms, an iterative optimization of the parity can be carried out during the identification step, whereby a sequence of unitary rotations of the fermionic operators in the orbitals can be performed, which can also be carried out in the same way on Hermitian conjugates of the operations, until an orbital basis can be reached in which the smallest local parities have assumed their extreme values.

[0033] To identify spin-like orbitals, their occupation n i analyzed. Is 〈n i 〉 exactly 1 and the fluctuation 〈δn i 〉 ≈ 0. If this is the case, then the charge degree of freedom of the electrons in these orbitals is negligible. The electron spin then remains the only relevant degree of freedom. Here, an optimization of the local parities P can be used. ia systematic approach can be taken to determine the special linear combinations of basis orbitals that satisfy both criteria for a spin-like character.

[0034] Local parity is defined as Pi=(−1)ni↑+ni↓≡(1−2ci↑†ci↑)(1−2ci↓†ci↓)=1−2(ci↑†ci↑+ci↓†ci↓)+4ci↑†ci↓†ci↓ci↑, and its expected value can be represented as 〈P i 〉 = 1 - 2 (ρ1) ii + 4(ρ2) iii , where (ρ1)qp=∑σ〈cqσ†cpσ〉, the reduced single-particle density matrix and (ρ2)qprs=〈cq↑†cp↓†cr↓cs↑〉, The reduced two-particle density matrix. If an orbital in the ground state |Ψ0〉 is strictly singly occupied, the local parity takes the value 〈Ψ0|P. i |Ψ0〉 = -1. Non-vanishing fluctuations δn i ≠ 0 of the local occupation lead to 〈Ψ0|P i |Ψ0〉 > 1. An orbital i with parity 〈P iThe system 〉0 = -1 + ε with ε → 0 fulfills both criteria to be considered spin-like in the ground state and probably also in low-lying excited states. For spin-like systems with multiple flavors or fermions in a higher spin representation, the parity naturally extends to Pi=(−1)∑niα where α runs across all flavors / spin representations.

[0035] An iterative optimization procedure for determining the specific orbital basis in which parity is optimized is presented. To optimize parity, the orbitals can be optimized pairwise as follows. Alternatively, the Jacobian transformations can be replaced by a Householder transformation. Thus, to optimize parity, a sequence of unitary rotations of the fermionic operators in the orbital basis is used, given by ciσ=cos ϕciσ+sin ϕcjσ,cjσ=−sin ϕciσ+cos ϕcjσ the same rotation is also performed on the Hermitian conjugates of the operators. As a result of the rotation between orbitals i and j, the local parity of the updated orbital, expressed in terms of the original fermionic operators, where the rotation angle ϕ is yet to be determined, is given by: 〈P i The function 〉 after the rotation is therefore an analytical, periodic function of the rotation angle ϕ. The extreme points of the function 〈P are determined. i 〉(ϕ) in the range ϕ ∈ [0,2π) determined by d〈Pi〉dϕ=0 is solved, whereby the solution is chosen for which d2〈Pi〉d2ϕ2|ϕn≠0 This applies. The first and second derivatives of the function 〈P i 〉(ϕ) are known analytically. If a solution ϕ m exists for the d2〈Pi〉d2ϕ2|ϕm>0 applies and for the simultaneous Pi(ϕm)≤Pi(ϕl)∀ϕl∈ ϕn This applies, resulting in a pairwise rotation of the orbitals i and j by the angle ϕ. m performed. Otherwise, the rotation is discarded. By repeatedly performing the described procedure for each pair of orbital indices, an orbital basis is obtained in which the smallest local parities have reached their extreme values. Each orbital i of the final basis resulting from the parity optimization and for which 〈P i 〉0 = -1 + ε holds, where ε can be an arbitrarily small value, can be called a spin degree of freedom or spin-like orbital.

[0036] The orbitals with a local parity 〈P i〉0 ≅ +1 can be excluded from parity optimization. If the reduced density matrices ρ1 and ρ2 were calculated in a previous ab initio calculation, typically the majority of the basis orbitals are either completely occupied or empty. The corresponding local parities of such basis orbitals are 〈P i 〉0 ≅ +1. It is possible to significantly reduce the runtime and memory consumption of the procedure if orbitals with initial local parity 〈P are used. i 〉0 ≅ +1 are excluded from parity optimization because they are in linear combinations of orbitals with local parity 〈P iThe difference between 0 and -1 does not occur significantly. Instead of pairwise optimization, a so-called Householder optimization can also be performed. Furthermore, it is possible to cluster the systems to avoid unwanted mixing. If, for example, one assumes localized orbitals, the optimization could be restricted to those orbitals that are specific to a particular atom, thus preserving the local character.

[0037] If the set of spin-like orbitals is a proper subset of the set of all system orbitals, the block diagonal Hamiltonian is further subdivided into H0=H0Spins+H0Bad+H0Spin−Bath, where H0spins=∑ij∈spins∑σσ'tijσσ'ciσ†cjσ'+∑ij∈spinsVijklci↑†cj↓†ck↓cl↑, This refers exclusively to the spin-like orbitals. The block-diagonal Hamiltonian, which acts exclusively on the remaining fermionic orbitals, is: H0Bad=∑qp∈Bad∑σσ'tqpσσ'cqσ†cpσ' +∑qprs∈BadVqprs(〈cq↑†cs↑〉cp↓†cr↓+〈cp↓†cr↓〉cq↑†cs↑−〈cq↑†cr↓〉cp↓†cs↑ −〈cp↑†cs↑〉cq↑†cr↓) The Hartree-Fock approximation can be used for the bath-specific interaction terms. If the reduced density matrix ρ1 is available from a previous calculation, the expected values ​​are taken; otherwise, they can optionally be considered negligible.

[0038] The Hamiltonian operator, which couples via interaction terms between both spin-like orbitals and bad orbitals, is given as H0Spin−Bad=∑i∈SpinsViqpici↑†cq↓†cp↓ci↑+Viqipci↑†cq↓†ci↓cp↑ +Vqiipcq↑†ci↓†ci↓cp↑+Vqipicq↑†ci↓†cp↓ci↑, and preserves the number of particles in spin-like orbitals.

[0039] In a subsequent step, the bad-exclusive part of the Hamiltonian operator can be diagonalized according to H0Bad=U DBadU†, to eliminate all nonlocal terms for the bath, since these increase the required size of the equations for a subsequent Schrieffer-Wolff transform, thus saving considerable computing power. A basic rotation U is used. † applied to all fermionic operators acting on bad orbitals c˜qσ=∑q'∈BadUqq'†cq'σ. In this updated base, the respective block-offdiagonal elements of the Hamiltonian operator become H0→H0Spins+DBad+H˜0Spin−BadV→V˜, where H̃0 and Ṽ are transformed fermionic operators. The resulting Hamiltonian operator H = H0 + V is then used for the Schrieffer-Wolff transformation procedure, which will be introduced later. The extension of the procedure to anomalous pairing terms such as ĉ pσ ĉ qσThis can also be done directly. During the optimization step, the complete Hamiltonian operator can be block diagonalized within the framework of a generalized Schrieffer-Wolff transformation.

[0040] In concrete terms, the bath degrees of freedom can be preconditioned to accelerate the Schrieffer-Wolff transformations in a procedure as described above by splitting the full Hamiltonian into a spin Hamiltonian, a bath Hamiltonian and a spin bath Hamiltonian, where the bath Hamiltonian can refer exclusively to the bath and can be used in a diagonalized form within the full Hamiltonian.

[0041] The observables of the complete Hamiltonian operator can be partitioned via block specifiers, which divide the standard operators into their contributions in the different symmetry sectors.

[0042] Preferably, Schrieffer-Wolff transformations in a vector space can be reformulated as a system of linear equations. In principle, any Hamiltonian operator can be completely diagonalized by a certain unitary transformation U. Finding such a unitary transformation U often requires a complete diagonalization of the Hamiltonian operator, which is equivalent to solving its characteristic polynomial. A Schrieffer-Wolff transformation aims to find an approximate transformation, or the generator of a transformation, that does not completely diagonalize the Hamiltonian operator but yields a block-diagonal Hamiltonian. The blocks consist of all orthonormal states in Hilbert space that have the same properties with respect to a given choice of conditions, for example, the local particle number n. iThe set of terms in the Hamiltonian operator that are initially block-diagonal are denoted H0. The remaining terms connect separate blocks, making them block-off-diagonal, and are denoted V. The complete Hamiltonian operator H = H0 + V is subjected to a unitary similarity transformation, given by H˜=H†HU=e−s(H0+V)es=e−sH0es+e−sVes, where S is an arbitrary anti-Hermitian operator, which is hereafter referred to as the generator of the Schrieffer-Wolff transform. The main problem is to find the generator S such that the transformed Hamiltonian operator H becomes completely block diagonal. To obtain an equation for S, the Campbell-Baker-Hausdorff formula can be used as H˜=esHe−s=∑m=0∞1m![S,H]m=H+[S,H]+12[S,[S,H]]+⋯, where a suitable norm for the generator S can be found, where this expression for ||S|| << ||H0||, where ||.|| is a suitable norm, can be approximated by H˜=H0+V+[S,H0]+[S,V]+O(S2). where it was assumed that the commutators consist of pairs of two block-diagonal operators and pairs of two block-off-diagonal operators are generally block-diagonal, while a commutator of a block-diagonal operator with a block-off-diagonal operator is generally block-off-diagonal, [S,H0]=−V↔[H0,S]=V, which allows the generator S to be determined. If the generator meets the above criteria, [S,[S,H0]]=[S,−V]=−[S,V]∈O(S1) can be used to express the transformed Hamiltonian operator as H˜=e−s(H0+V)es≅H0+12[S,V]+O(S2), where the block-diagonal corrections of H0 are given by [S,V] / 2, which includes all perturbing corrections resulting from two consecutive block-off-diagonal excitations. In the case [S,V], unwanted off-diagonal terms would be generated; these can be added to the basis in the same way as the generator S is determined.

[0043] A block-offdiagonal part of a Hamiltonian operator V comprises a sum of block-offdiagonal terms. These, in turn, comprise products of individually block-offdiagonal operators X. A procedure for individually partitioning block-offdiagonal operators X into separate components is presented below, whereby each component exclusively connects two separate blocks of the Hilbert space H, often accompanied by separate quantum numbers of a system symmetry.

[0044] A given block-offdiagonal operator X: H → H must satisfy the condition [H0,X]=εZ≠0 Satisfy, where ε can be any scalar, and Z:H → H can be any operator. Let A be a diagonal operator in the initial basis. A can be identical to the symmetry operator that distinguishes the blocks of the Hilbert space, but this is not necessary. One can use the spectrum of A to determine the operator X. X=∑qXq=∑qβq∏i≠q(A−ai)X to extend, where the different x q the target subspace, which has the eigenvalue a q A is connected to other subspaces of Hilbert space. Furthermore, if the operator A satisfies the condition... [X,A]=0, Thus the two subspaces, initial and final, are defined via X q connected and by the eigenvalue a q determined. This is possible for the fermionic creation and annihilation operators. ciσ† and c iσ The coefficients β q are solutions of the equation

[0045] β q = ∏ i≠q (a q - a i ) = 1. The symmetry-specific, block-offdiagonal operators X q then fulfill [Xq,Xq†,]∝δqq', where Xq† the Hermitian conjugation of the operator X q represents.

[0046] In the next step, the Hamiltonian operator of the system will be H=H0+V decomposed into its block diagonals and its block off-diagonal components.

[0047] The block-offdiagonal component V comprises every block-offdiagonal term V with V=∑{v}αv[(∏jOj∏iXi)+(∏jOj∏iXi)†]=∑{v}αv[(∏i,jOj∑q(i)Xq(i)i)+(∏i,jOj∑q(i)Xq(i)i)†], where Π i X i Sequences of individually block-offdiagonal operators are denoted, ∏ j O j In contrast, it refers to sequences of individually block-diagonal operators and Xq(i)i the symmetry-specific components of the operator X i .

[0048] In the next step, new vector spaces will be developed. V0h and V0a defined in which each unique pair of Hermitian or anti-Hermitian, symmetry-specific operator sequences in V corresponds to an orthonormal basis vector e^v={(∏jOj∏ixq(i)i)+(∏jOj∏ixq(i)i)†∈V0h(∏jOj∏ixq(i)i)−(∏jOj∏ixq(i)i)†∈V0a corresponds to where V h the Hermitic vector space and V a denoted as the antihermitic vector space. For a linear mapping process, the mapping must first be defined. L:V0h→V1aV0a→V1h be defined, where the effect of the linear transformation on a vector ê V is given by Le^v=[H0,e^V] corresponds. Here, (Le^V)∈V1a if e^V∈V0h and (Le^V)∈V1h if e^V∈V0a with dim(V1) ≥ dim(V0), since V1 contains additional unique operator sequences that are defined by [H0, ê V ] can be generated. In vector spaces V, the equation for the generator S for the Schrieffer-Wolff transform can be transferred, which LS=V↔[H0,S]=V must fulfill, whereby S∈V0A and V∈V1h This applies. Unlike originally intended, the search for the generator S of the Schrieffer-Wolff transform is solved here by a linear system of equations, instead of a nonlinear system. In general, rank(L) ≤ dim(V0) ≤ dim(V1). Consequently, there is no unique solution S for the set of linear equations. It is evident that the terms in V are approximate transitions between different eigenstates of H0, which belong to separate blocks of the Hilbert space. This is reflected in [H0,e^Vh]=(ΔE0,V∏lOl)e^Va, where ΔE 0,Vdenotes the eigenvalue difference between the two eigenstates of H0 and Π l O l represents an arbitrary sequence of individually block-diagonal operators. The solution to the equation above is obtained. ‖S‖≈∑{V}(aVΔE0,V)2 approximate, which necessitates a significant energy gap ΔE 0,V between the separated subspaces of the Hilbert space, which are connected by V, in order to derive from the Taylor series expansion from H to O(S 2 ) to obtain a good approximation. It can be assumed that terms with |aV2 / ΔE0,V|≥1 The resonant terms of V should be preserved in the transformed Hamiltonian. To identify the resonant terms of V, a singular value decomposition of the linear map is performed, resulting in L=U∑W†=U(∑>+∑<)W†=Lgapped+Lresonant, where Σ > the singular values σi>1NV∑{V}aV2 include, where Nv describes the number of terms V.

[0049] A Moore-Penrose pseudoinverse of L can be defined that acts exclusively on the non-resonant sectors of V, with Lgapped+=(W∑>+U†), where ∑>+ the pseudoinverse of Σ > denoted by. Since there is no unique solution for the generator S, the best approximate solution is given by S=(Lgapped+V)∈v0a, so that the resulting transformed Hamiltonian H has the following representation: H˜=H0+Vresonant+[S,Vresonant]+12[S,Vgapped] if V gapped = [H0, S] and V resonant = (V - V gapped ) applies.

[0050] The foregoing invention will be explained in more detail below with reference to an exemplary embodiment.

[0051] They show Fig. 1 Schematic representation of the method for simulating the spin states of a physical system, as well as Fig. 2 an energy scheme of the 3d orbitals of the chromium(III) ion according to crystal field theory, as well as Fig. 3 a relative energy spectrum of the CrBr6 molecule, which for three different Hamiltonians and the block-diagonal part of the original Hamiltonian represents the energy difference of the eigenstate to the ground state for different excited states.

[0052] The quality of the approximate transformed Hamiltonian H̃ and the corresponding H̃ block-diagona The following discussion focuses on the description of the low-energy dynamics of a given system, using molecular chromium bromide (CrBr6) as an example. 3+ ).

[0053] In Fig. Figure 1 shows a schematic diagram of the method according to the invention. In a first identification step, spin-like linear combinations of orbitals are recognized as spin degrees of freedom. For this purpose, the local parity is calculated as 〈P i 〉0 ≅ -1 is optimized, meaning that the fluctuation of the electron density approaches zero, in order to identify spin-like orbitals. For this purpose, both the expectation value of the parity itself 〈P i The first and second derivatives of the parity with respect to the rotation angle are also analyzed. This step can also be used for systems that are not spin-symmetric. Here, the individual spin channels are optimized separately. In a next step, the Hamiltonian is optimized according to... Fig. The Hamiltonian operator is divided into individual terms to separately describe the spin, the bath, and the spin-bath interaction. A single-particle Hamiltonian is diagonalized by a basis rotation, eliminating non-local bath terms to significantly minimize computational effort. A Schrieffer-Wolff transform is then applied to Bock-diagonalize the Hamiltonian using a system of linear equations, where the blocks are characterized by different electron densities in the spin-like orbitals. By removing the charge degrees of freedom, the complete Hamiltonian is simplified to a single Hamiltonian that links the remaining spin degrees of freedom as a cluster to the system's bath. During quantum simulation, the spin degrees of freedom are assigned to high-performing qubits, and the bath to the remaining, particularly low-performing, qubits.

[0054] In Fig. Figure 2 shows the energy scheme of the 3d orbitals of the chromium(III) ion according to crystal field theory. This is due to the splitting of the 3d orbitals in the crystal field at t 2g and e g Orbitals and the simple occupation of the t 2gChromium(III) represents the most stable oxidation state of the transition metal. It has three unpaired electrons, each with a spin of ½. The advantageous application of the present method can be illustrated using the molecule at hand, as it is a molecule for which spin modeling cannot be solved using prior art methods, or only with considerable computational effort. The present method enables spin modeling via a portion of the complete Hamiltonian operator that describes the bath. This portion does not require a specific solution approach and avoids the introduction of variations that would subsequently need to be approximated. This significantly reduces the computational effort while simultaneously yielding a more accurate result.The computational operations of the present method can also be performed on the high-performance qubits of a quantum computer.

[0055] In Fig. Figure 3 is a schematic diagram of the low-energy spectrum of the individual Hamiltonians, using the CrBr6 molecule as an example. The defined active space encompasses the 3d and 4d orbitals of the chromium atom. The diagram shows the energy difference between the eigenstate and the ground state for different excited states. Plotted are the relative spectra of the original Hamiltonian, the Schrieffer-Wolff transformed Hamiltonian, the effective Hamiltonian with effective spin-bad coupling, and the zeroth-order approximation of the Hamiltonian.

[0056] The original Hamiltonian is represented by a solid line and initially remains unchanged below an energy difference of 0.1, before beginning to rise slightly from an index value of n=3 and remaining flat again from n=4. A steep rise is observed between n=6 and n=7, where the energy difference exceeds 0.1, and then remains unchanged again until n=9.

[0057] The Schrieffer-Wolff transformed Hamiltonian closely follows the curve of the original Hamiltonian, represented in the diagram by a short dashed line. Unlike the original Hamiltonian, the Schrieffer-Wolff transformed curve rises earlier and more gently before running along the same plateau as the curve of the original Hamiltonian between n=5 and n=6. The subsequent rise is less steep than that of the original Hamiltonian, but this rise transitions into a gentle slope from n=7 onwards, at which point the curve of the original Hamiltonian is intersected by the Schrieffer-Wolff transformed curve at n=8.

[0058] The zeroth-order approximation is represented by a dotted line and begins at a higher energy difference below 0.2. It can be seen that the curve is subject to slight fluctuations at smaller values ​​before a plateau sets in from n=4, which extends to n=7, after which the curve rises steeply and stops at the eigenstate index n=9 below an energy difference of 0.3.

[0059] The term of the Hamiltonian operator, which describes only the spin-bad coupling, was plotted with a normal dashed line and exhibits a plateau with the lowest energy difference from n=0 to n=6, followed by a rise to above the curves of the original Hamiltonian operator and the Schrieffer-Wolff transformed Hamiltonian at n=7, followed by another plateau. This illustrates that the energy difference to the ground state is described significantly better by the effective spin-bad Hamiltonian than by the simple model description using a zeroth-order approximation. The effective spin-bad Hamiltonian with spin-bad coupling was determined using the present method and can be used in a further step to simulate spin-like orbitals on a quantum computer.

[0060] Furthermore, generators can be constructed to adapt the Hamiltonian of a quantum system to the available two-qubit gates of quantum hardware by suppressing or transforming away terms that are not or only with great difficulty realizable.

[0061] Thus, using the generator S=v(S^0+S^0−−S^1+S^0−) the S^0+S^1−+hc The coupling is transformed into a SöSi coupling, including its effective local magnetic fields. v=arctan(JPB0z−B1z) / 2 This is even achieved exactly for a two-spin model.

[0062] The coupling, and thus the two-qubit gates, between qubits differs depending on the technology used. While ion traps often implement XX or YY coupling, superconducting systems frequently exhibit ZZ interactions. The transformation described above allows the spin system to be represented as closely as possible to the hardware-native implementation on the target quantum hardware. This eliminates many operations on the quantum computer that would otherwise lead to longer computation times or increased errors.

[0063] The above describes a method for describing the behavior of a physical system which requires less computational effort, provides more accurate solutions, and is also executable on the high-performance qubits of a quantum computer.

Claims

[1] Method for simulating the behavior of a physical system using a quantum computer comprising a plurality of qubits, wherein the qubits are first evaluated with respect to their system properties and categorized into high-performance qubits and low-performance qubits, wherein the physical system is described in a Hilbert space spanned by orbitals and the spin-like system is assigned to the high-performance qubits during a mapping process, the behavior of the physical system is simulated using the qubits of the quantum computer and physical observables of the physical system are determined by measurements of the qubits of the quantum computer, characterized by, that in an identification step spin-like linear combinations of orbitals are identified in the physical system, and that these are spin-like orbitals in the spin-like system which, along with other remaining orbitals, span the Hilbert space, wherein the physical system thus described is transformed such that, after the transformation, changes in the number of electrons and / or the electron density on the spin-like orbitals are reduced or completely eliminated, and the transformed physical system is then described on a Hilbert space spanned only by the orbitals identified as spin-like. [2] Method according to claim 1, characterized by , that the remaining non-spin-like orbitals are represented as a bath coupled to the spin-like system, with the orbitals of the bath being simulated by low-performance qubits of the quantum computer. [3] Method according to claim 1 or 2, characterized by , that the transformation is modified so that the spin-like system is mapped onto the hardware of the quantum computer by rotating coupling terms between spin-like orbitals to correspond to the two-qubit gates available on the quantum computer. [4] Method according to any one of the preceding claims, characterized by , that during the identification step, linear combinations of orbitals are considered spin-like if the fluctuation of the number of electrons on the orbital is negligible compared to the average occupation, where the average occupation of an orbital is n, with n=1 for spin-like orbitals, and a limit ε of the fluctuation dni2=〈(ni−1)2〉 is defined as 0<=ε=<=n, where the fluctuation dn is considered small if ε>=dn / n>=-ε. [5] Method according to any one of the preceding claims, characterized by, that during the identification step a value is determined which quantifies the spin similarity of the basis orbitals by means of a systematic procedure which runs on the quantum computer or a conventional computer, whereby the orbitals are optimized so that the value becomes extremal. [6] Method according to any one of the preceding claims, characterized by , that during the identification step a value is determined which measures the spin similarity of the basis orbitals using an observable which is the local parity of the orbitals. [7] Method according to any one of the preceding claims, characterized by, that during the identification step an iterative optimization of the parity is performed, whereby a sequence of unitary rotations of the fermionic operators for the orbitals is carried out, which is also performed in the same way on Hermitian conjugates of the operations, until an orbital basis is reached in which the smallest local parities have assumed their extreme values. [8] Method according to claim 7, characterized by , that orbitals with a local parity 〈P i 〉0 ≅ -1 are excluded from the parity optimization. [9] Method according to any one of the preceding claims, characterized by , that after the optimization step the complete Hamiltonian operator is block diagonalized within the framework of a generalized Schrieffer-Wolff transformation. [10] Method according to claim 7, characterized by, that the bad degrees of freedom are preconditioned to speed up the Schrieffer-Wolff transformations by splitting the full Hamiltonian operator into a spin Hamiltonian, a bad Hamiltonian and a spin bad Hamiltonian, where the bad Hamiltonian refers exclusively to the bad and is used in a diagonalized form within the full Hamiltonian. [11] Method according to any one of claims 7 to 10, characterized by , that the Schrieffer-Wolff transformations in a vector space are reformulated as a system of linear equations. [12] Method according to claim 11, characterized by, that those operators which span the vector space and couple different symmetry sectors, in particular sectors with different numbers of electrons in the spin-like orbitals, are split into components that act on different symmetry sectors, in particular on sectors with different numbers of electrons in the spin-like orbitals. [13] Method according to any one of the preceding claims, characterized by that the physical system to be described is a radical or a molecule containing radicals. [14] Method according to any one of claims 1 to 12, characterized by , that the physical system to be described is a solid-state system or molecule containing atoms with open d- and f-shells. [15] Method according to any one of claims 1 to 12, characterized bythat the physical system to be described is a solid-state system or a molecule that is described by spin physics and / or in which transitions between states with different spin degrees of freedom are relevant.

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