Method for finding an optimal quantum state minimizing the energy of a Hamiltonian operator with a quantum processor by using a VQE method, determining a quantum state of a chemical compound, and determining physical quantum properties of materials

Abstract

A method for finding an optimal quantum state minimizing the energy of a Hamiltonian operator with a quantum processor and a classical processor comprising a quantum circuit for producing trial quantum states for the Hamiltonian operator and parametric quantum gates with associated parameters, by using a VQE method, the method comprising: providing the Hamiltonian operator in an orbital basis and iteratively, until a predefined stopping criterion is satisfied: (i) applying the VQE method to find optimized values for the parameters that yield an intermediate optimal quantum state which minimizes the energy of the Hamiltonian operator, (ii) computing a one particle reduced density matrix (1-RDM) based on the intermediate optimal quantum state, (iii) determining an updated orbital basis in which the 1-RDM is diagonal, and an associated transformation matrix, and (iv) modifying the Hamiltonian operator with the transformation matrix; and then returning, as the optimal quantum state the intermediate optimal quantum state that minimizes the most the energy.

Claims

1. A method for finding an optimal quantum state minimizing an energy associated with a Hamiltonian operator with a quantum processor and a classical processor by using a Variational Quantum Eigensolver (VQE) method, wherein the Hamiltonian operator represents an energy of a molecule, the quantum processor comprising a predetermined quantum circuit for producing trial quantum states for the Hamiltonian operator, said predetermined quantum circuit comprising at least parametric quantum gates associated with one or more parameters to be optimized, the method comprising: providing the Hamiltonian operator in an orbital basis and iteratively, until a predefined stopping criterion is satisfied: performing the VQE method to find optimized values for at least some of the one or more parameters associated with the parametric quantum gates of the predetermined quantum circuit that yield an intermediate optimal quantum state which minimizes the energy associated with the Hamiltonian operator; computing a one particle reduced density matrix based on the intermediate optimal quantum state; diagonalizing the one particle reduced density matrix to obtain a transformation matrix, and determining an updated orbital basis in which the one particle reduced density matrix is diagonal, based on the transformation matrix; and modifying the Hamiltonian operator by using the transformation matrix, to express the Hamiltonian operator in the updated orbital basis; wherein when the predefined stopping criterion is satisfied, the method further comprises returning, as the optimal quantum state minimizing the energy associated with the Hamiltonian operator, the intermediate optimal quantum state which minimizes the most the energy associated with the Hamiltonian operator.

2. The method according to claim 1, wherein performing the VQE method is an iterative scheme in which the quantum processor is used in conjunction with the classical processor, the quantum processor preparing a trial quantum state for the Hamiltonian operator and performing measurements representative of the energy associated with the Hamiltonian operator for said trial quantum state, and the classical processor updating values of the parameters of the parametric quantum gates of the predetermined quantum circuit based on the measurements performed by the quantum processor, the iterative scheme being executed until a second predefined stopping criterion is satisfied, the VQE method returning the optimized values of the parameters.

3. The method according to claim 1, wherein the predetermined quantum circuit is a product quantum circuit comprising only one-qubit quantum gates in a form of rotations, or a quantum circuit comprising fSim quantum gates or a Low-Depth Circuit Ansatz, LDCA, quantum circuit.

4. The method according to claim 1, wherein the method is applied on a Hubbard model for which the Hamiltonian operator is provided.

5. The method according to claim 4, wherein the Hamiltonian operator is a second-quantized Hamiltonian.

6. The method according to claim 1, wherein a physical quantum state is encoded into a qubit state by means of a Jordan-Wigner transformation from which the Hamiltonian operator is decomposed accordingly in terms of qubit observables.

7. The method according to claim 6, wherein the optimal quantum state corresponding to an eigenvector associated a lowest eigenvalue.

8. The method according to claim 7, wherein the molecule is a H2, LiH and/or H2O molecule.

9. The method according to claim 7, wherein a number of qubits in the predetermined quantum circuit corresponds to a number of spin-orbitals used to describe the molecule.

10. The method according to claim 1, wherein the predefined stopping criterion is a maximum number of iterations and/or a minimum change of a variance between the minimums of the energy associated with the Hamiltonian operator obtained after two consecutive iterations.

11. The method according to claim 1, wherein the parametric quantum gates comprise rotation quantum gates, and wherein the parameters associated with said rotation quantum gates comprise values of angles.

12. A method for determining a quantum state of a chemical compound, comprising: a method for finding an optimal quantum state minimizing the energy associated with a Hamiltonian operator with a quantum processor and a classical processor according to claim 1, wherein an expectation value of the Hamiltonian operator over a given quantum state corresponds to the energy of said quantum state.

13. A method for determining physical properties of materials comprising: a method for finding an optimal quantum state by minimizing the energy associated with a Hamiltonian operator with a quantum processor and a classical processor according to claim 1.

Description

BRIEF DESCRIPTION OF DRAWINGS

[0031] Other features, details and advantages will be shown in the following detailed description and on the figures, on which:

[0032] FIG. 1 is a flow diagram of the method for finding a minimum of a cost function with a noisy quantum processor.

[0033] FIG. 2 is a flow chart illustrating some of the main steps of the method of FIG. 1.

[0034] FIG. 3A illustrates an example of a quantum circuit comprising a product of rotations.

[0035] FIG. 3B illustrates an example of an fSim quantum circuit comprising fSim quantum gates

[0036] FIG. 3C illustrates an example of a LDCA quantum circuit.

[0037] FIG. 4A illustrates the performance of the method applied on the Hubbard model at half-filling on two sites, with μ=U/2 and U=0

[0038] FIG. 4B illustrates the performance of the method applied on the Hubbard model at half-filling on two sites, with μ=U/2 and U=1.

[0039] FIG. 5A illustrates the performance of the method applied on the Hubbard model at half-filling on four sites, with μ=U/2 and U=0.

[0040] FIG. 5B illustrates the performance of the method applied on the Hubbard model at half-filling on four sites, with μ=U/2 and U=1.

[0041] FIG. 6A illustrates the performance of the method applied on the Hubbard model at half-filling on four sites, with μ=U/2 and U=0, on a noisy processor with a depolarizing noise of magnitude 0.001 for the 1-qubit quantum gates and 0.01 for the 2-qubits quantum gates

[0042] FIG. 6B illustrates the performance of the method applied on the Hubbard model at half-filling on four sites, with μ=U/2 and U=1, on a noisy processor with a depolarizing noise of magnitude 0.001 for the 1-qubit quantum gates and 0.01 for the 2-qubits quantum gates.

[0043] FIG. 7A illustrates the performance of the method applied on the Hubbard model at half-filling on four sites, with μ=U/2 and U=0, on a noisy processor a depolarizing noise of magnitude 0.005 for the 1-qubit quantum gates and 0.05 for the 2-qubits quantum gates

[0044] FIG. 7B illustrates the performance of the method applied on the Hubbard model at half-filling on four sites, with μ=U/2 and U=1, on a noisy processor a depolarizing noise of magnitude 0.001 for the 1-qubit quantum gate and 0.01 for the 2-qubits quantum gates.

DETAILED DESCRIPTION

[0045] It is now referred to FIGS. 1 and 2 respectively illustrating a flow diagram and a flow chart of the method for finding an optimal quantum state minimizing the energy of a Hamiltonian operator.

[0046] At step S1, a predetermined quantum circuit U.sub.θ, or simply “quantum circuit U.sub.θ” in the following specification, for providing trial quantum states |ψ.sub.θ is provided. In the field of quantum mechanics, the quantum circuit U.sub.θ can be referred to as an ansatz quantum circuit. The quantum circuit U.sub.θ comprises at least parametric quantum gates associated with parameters θ to be optimized. The trial quantum states can be expressed as |ψ.sub.θ=U.sub.θ|0.sup..Math.M where |0.sup..Math.M is the initial quantum state of the M-qubit quantum processor.

[0047] In an embodiment, the parametric quantum gates of the quantum circuit U.sub.θ comprise, for instance, quantum rotation gates, the parameters θ to be optimized comprising, for instance, angles of the quantum rotation gates.

[0048] In an embodiment, the method is computed on a Noisy Intermediate Scale Quantum (NISQ) processor, which is a quantum processing unit (QPU), comprising only a few tens of qubits and operating with relatively high error rates. Due to their limited capacity, NISQ processors are generally used in conjunction with classical processors (CPUs), for instance to minimize a cost function. The predetermined quantum circuit should comply with the constraints of the NISQ processor. In particular, the quantum circuit U.sub.θ that is used to prepare the trial quantum state |ψ.sub.θ=U.sub.θ|0.sup..Math.M (with M the number of qubits) should fulfill the connectivity and gateset constraints of the quantum processor, and more importantly be compatible with its decoherence properties. On the other hand, the quantum circuit U.sub.θ needs to contain enough quantum gates so that the quantum state |ψ.sub.θ it prepares can approximate with sufficient accuracy the expected quantum ground state, also called the optimal quantum state, to be found.

[0049] In an embodiment, the quantum circuit U.sub.θ may be a “product” quantum circuit, comprising only one-qubit gates in the form of rotations as illustrated on FIG. 3A. In another embodiment, the quantum circuit U.sub.θ may be an “fSim” quantum circuit, illustrated on FIG. 3B, and comprising fSim gates developed by the Google© company. In yet another embodiment, the quantum circuit U.sub.θ may be the so-called “Low-Depth Circuit Ansatz”, LDCA, circuit.

[0050] The product quantum circuit only produces product quantum states, also known as Slater determinants, and is adapted only to non-interacting Hamiltonian operators expressed in their diagonal basis.

[0051] The fSim quantum gates of the fSim quantum circuit conserve the number of excitations of a given quantum state.

[0052] The LDCA circuit was introduced in the following piece of literature Low-depth circuit ansatz for preparing correlated fermionic states on a quantum computer, Pierre-Luc Dallaire-Demers, Jonathan Romero, Libor Veis, Sukin Sim, Alan Aspuru-Guzik, Jan. 4, 2018.

[0053] It should be noted that other types of quantum circuits may be used for preparing the trial quantum states, and the choice of specific type of quantum circuit corresponds to a specific embodiment of the present disclosure.

[0054] In an embodiment, the type of quantum gates, and the order of the qubits of the quantum circuit U.sub.θ may be fixed, such that a qubit expressed in a scheme is always associated to the same qubit in the hardware, while the quantum circuit U.sub.θ comprises at least parametric quantum gates which are tuned by the parameters θ to be optimized.

[0055] In another embodiment, the quantum circuit U.sub.θ may comprise parametric quantum gates, as well as non-parametric quantum gates.

[0056] In yet another embodiment, only some of the parametric quantum gates may be tuned by the parameters θ to be optimized, while some other parametric quantum gates are associated with parameters which may not change during the implementation of the method.

[0057] At step S2, the Hamiltonian operator is provided in an orbital basis. In other words, the Hamiltonian operator is expressed in said orbital basis. The orbital basis in which the Hamiltonian operator is initially expressed is referred to as original orbital basis in the sequel.

[0058] In an embodiment, the Hamiltonian operator H is a second-quantized Hamiltonian written as:

[00001]$H={.Math.}_{\mathrm{pq}}{h}_{\mathrm{pq}}{c}_p^{†}{c}_q+\frac{1}{2}{.Math.}_{\mathrm{pqrs}}{h}_{\mathrm{pqrs}}{c}_p^{†}{c}_q^{†}{c}_r{c}_s$

with p, q, r, sϵM.sup.4, M being the number of spin-orbitals.

[0059] In this embodiment, the Hamiltonian operator may represent the energy of an electronic system, such as a cloud of electrons in the field of an atomic potential in a molecule, with c.sub.p.sup.† and c.sub.q the creation and annihilation operators in the original orbital basis ϕ.sub.p(r), h.sub.pq the overlap of atomic orbitals of a molecule and h.sub.pqrs the collision of two orbitals.

[0060] In this embodiment, the original orbital basis in which the Hamiltonian operator is expressed initially is the local orbital basis, i.e. the orbital basis that is centered on the atoms.

[0061] A generic cost function may be expressed as the expectation value over a quantum state of some quantum observable.

[0062] In this embodiment, the cost function to be minimized is the energy associated with the Hamiltonian operator, which is defined as:

E.sub.θ.sup.(k)=ψ(θ)|H.sup.(k)|ψ(θ)

[0063] Mathematically, E.sub.θ.sup.(k) may then be expressed as

[00002]${.Math.}_{\mathrm{pq}}{h}_{\mathrm{pq}}{P}_{\mathrm{pq}}+\frac{1}{2}{.Math.}_{\mathrm{pqrs}}{h}_{\mathrm{pqrs}}{P}_{\mathrm{pqrs}},$

with P.sub.pq=ψ(θ)|c.sub.p.sup.†c.sub.q|ψ(θ), P.sub.pqrs=ψ(θ)|c.sub.p.sup.†c.sub.q.sup.†c.sub.rc.sub.s|ψ(θ) and where P.sub.pq and P.sub.pqrs can be obtained from measurements with the quantum processor.

[0064] However, the Hamiltonian operator can be expressed differently, depending on the physical or chemical entity to be studied, without modifying the steps of the method for finding an optimal quantum state minimizing the energy associated with a Hamiltonian operator.

[0065] In particular, the Hamiltonian operator may be expressed with a form more general than the one used in this embodiment, which may be specific to chemistry and physics problems. A generic Hamiltonian would contain terms of higher order in the creation and annihilation operators c.sub.p.sup.† and c.sub.q. Moreover, according to the system to be described with the cost function, the parameters h.sub.pq and h.sub.pqrs of the Hamiltonian may take different expressions.

[0066] Finally, step S2 may also comprise the initialization of the parameters h.sub.pq and h.sub.pqrs such that h.sub.pq.sup.(k=0)=h.sub.pq and h.sub.pqrs.sup.(k=0)=h.sub.pqrs, with k being the current iteration, for the Hamiltonian operator H.sup.(k=0) expressed in the original orbital basis (with c.sub.p.sup.†(k=0)=c.sub.p.sup.† and c.sub.q.sup.(k=0)=c.sub.q).

[0067] The order of steps S1 and S2 is purely illustrative.

[0068] Steps S3 to S5 are then performed iteratively until a first predefined stopping criterion is met. The first predefined stopping criterion may be satisfied when a maximum number of iterations is performed and/or when the minimum energy associated with the Hamiltonian operator calculated during successive iterations no longer varies.

[0069] During steps S3 to S5, the VQE method is applied to find optimized values θ*.sup.(k) of the parameters θ that yield an intermediate optimal quantum state which minimizes the energy E.sub.θ.sup.(k)=ψ(θ)|H.sup.(k)|ψ(θ) associated with the Hamiltonian operator H.sup.(k) in the current orbital basis, with |ψ(θ)=U(θ)|0.

[0070] More specifically, the quantum processor (QPU on FIG. 1, step S3 on FIG. 2) prepares a trial state for the Hamiltonian operator and measures the values P.sub.pq and P.sub.pqrs representative of the energy associated with the Hamiltonian operator, as defined above. Then the quantum processor transmits these values to a classical processor (CPU on FIG. 1, step S4 on FIG. 2) that determines updated values for the parameters θ for preparing a trial quantum state that may further reduce the energy of the quantum state that is prepared, based on the measurements performed by the quantum processor.

[0071] As visible on FIGS. 1 and 2, the VQE method is an iterative scheme which ends when a second predefined stopping criterion is satisfied.

[0072] In an embodiment, the second predefined stopping criterion may be satisfied when a maximum number of iterations is performed and/or when the optimized values θ*.sup.(k) of the parameters θ updated during successive iterations no longer varies.

[0073] At step S6, the VQE method scheme returns the optimized values θ*.sup.(k) of the parameters which yield and intermediate optimal quantum state |ψ(θ*.sup.(k)) minimizing the energy associated with the Hamiltonian operator in the current orbital basis, and transmits it to perform the diagonalization step illustrated on FIG. 1 and FIG. 2.

[0074] At step S7, a one particle reduced density matrix D.sub.ij.sup.(k), also called 1-RDM, associated to the intermediate optimal quantum state |ψ(θ*.sup.(k))is measured on the quantum processor (QPU on FIG. 1) such that D.sub.ij.sup.(k)(θ*.sup.(k))=ψ(θ*.sup.(k))|c.sub.i.sup.†(k)c.sub.j.sup.(k)|ψ(θ*.sup.(k)), with θ*.sup.(k) the optimized values of the parameters θ yielding the intermediate optimal quantum state |ψ(θ*.sup.(k)) minimizing the energy associated with the Hamiltonian operator expressed in the current orbital basis, and c.sub.i.sup.†(k) and c.sub.j.sup.(k) the creation and annihilation operators in the current orbital basis (i.e. the 1-RDM matrix is measured in the current orbital basis), defined as:

c.sub.i.sup.†(k)=Σ.sub.p[V.sub.pi.sup.(k−1)]c.sub.p†.sup.(k−1)

c.sub.j.sup.(k)=Σ.sub.p[V.sub.pj.sup.(k−1)]*c.sub.p.sup.(k−1)

[0075] At step S8, the classical processor (CPU on FIG. 1) diagonalizes the 1-RDM to obtain a transformation matrix V.sup.(k). An updated orbital basis is obtained which corresponds to the orbital basis in which the 1-RDM matrix is diagonal.

[0076] At step S9, the parameters h.sub.pq.sup.(k) and h.sub.pqrs.sup.(k) of the Hamiltonian operator are modified, for instance by the CPU, to h.sub.pq.sup.(k+1) and h.sub.pqrs.sup.(k+1) using the transformation matrix V.sup.(k) such that:

[00003]$\begin{array}{c}{h}_{\mathrm{pq}}^{\left(k+1\right)}=\underset{p^{\prime }{q}^{\prime }}{.Math.}{V}_{p^{\prime }p}^{\left(k\right)}{h_{p^{\prime }{q}^{\prime }}^{\left(k\right)}\left[V^{\left(k\right)†}\right]}_{{\mathrm{qq}}^{\prime }}\\ h_{\mathrm{pqrs}}^{\left(k+1\right)}=\underset{p^{\prime }{q}^{\prime }{r}^{\prime }{s}^{\prime }}{.Math.}{V}_{p^{\prime }p}^{\left(k\right)}{V}_{q^{\prime }q}^{\left(k\right)}{{h_{p^{\prime }{q}^{\prime }{r}^{\prime }{s}^{\prime }}^{\left(k\right)}\left[V^{\left(k\right)†}\right]}_{{\mathrm{rr}}^{\prime }}\left[V^{\left(k\right)†}\right]}_{{\mathrm{ss}}^{\prime }}\end{array}$

[0077] The method is then iteratively processed to step S3 with the new parameters h.sub.pq.sup.(k+1) and h.sub.pqrs.sup.(k+1) which are used to compute the Hamiltonian operator H.sup.(k+1) expressed in the updated orbital basis (wherein the updated orbital basis at the end of iteration k is referred to as current orbital basis at the beginning of iteration k+1), used for the subsequent VQE method iteration, until the first predefined stopping criterion is satisfied (step S11).

[0078] When the first predefined stopping criterion (step S11) is satisfied, the minimum value of the energy associated with the Hamiltonian operator is considered to have been found and the optimal quantum state, which corresponds to one of the intermediate quantum states obtained, is returned.

[0079] In an embodiment, the optimal quantum state corresponds to the last intermediate optimal quantum state determined.

[0080] In another embodiment, the optimal quantum state is the intermediate quantum state which minimizes the most the energy associated with the Hamiltonian, among all the intermediate quantum states obtained during the implementation of the method.

[0081] Eventually, by performing the method iteratively, the updated orbital basis, in which the Hamiltonian operator is expressed, will come closer and closer to the natural orbital basis relative to the Hamiltonian's quantum ground state, allowing to reach this quantum ground state while limiting the number of gates in the quantum state preparation circuit. The main property of the natural orbital basis in quantum chemistry is that it is the orbital basis in which the description of a quantum state is the simplest, i.e. the basis that necessitates the least number of Slater determinants to represent the quantum state.

[0082] By representing the Hamiltonian operator in an orbital basis that is closer and closer to its natural orbital basis, at each iteration, it results in a higher and higher expressivity of the quantum circuit U.sub.θ. In other words, the transformation of the minimization problem to the natural orbital basis endows the method with a guarantee that the problem is placed in the most economical basis, due to the property that the natural orbital basis associated with the Hamiltonian's ground state is the basis in which it can be expressed in a minimal fashion.

[0083] As a consequence, the quantum circuit U.sub.θ may not perform well in the initial orbital basis but it will perform better and better as the method iteratively converges to the natural orbital basis. Moreover, since the quantum circuit U.sub.θ may be shallow, the minimization is robust to noise. Thus, the method allows using shallow quantum circuits, such as a product quantum circuit, or an fSim quantum circuit or a LDCA quantum circuit, as illustrated on FIGS. 3A to 3C.

[0084] Mathematically, given a Hamiltonian H and a quantum state |Ψ, the natural orbital basis NO is defined as the basis that diagonalizes the one-particle reduced density matrix D.sub.pq 1-RDM D.sub.pq≡Ψ|c.sub.p.sup.†c.sub.q|Ψ, with c.sub.p.sup.† and c.sub.q the creation and annihilation operators in the original orbital basis ϕ.sub.p(r).

[0085] More specifically, if

D.sub.pq=V.sub.pαn.sub.αV.sub.αq.sup.†, the natural orbitals are defined as {tilde over (c)}.sub.α.sup.†≡Σ.sub.pV.sub.pαc.sub.p.sup.†.

[0086] The main property of the NO basis is that it is the basis where the quantum state |Ψ can be represented as a linear combination of the least number of Slater determinants, or, in quantum computing terms, of computational basis quantum states.

[0087] These quantum states are defined, in terms of creation operators, as

[00004]$\begin{array}{c}{.Math.{\tilde{n}}_1,{\tilde{n}}_2,.Math.,{\tilde{n}}_M.Math.=\underset{\alpha =1}{\overset{M}{.Math.}}{\left({\tilde{c}}_{\alpha }^{†}\right)}^{{\tilde{n}}_{\alpha }}.Math.0.Math.}^{.Math.M}\\ {.Math.n_1,n_2,.Math.,n_M.Math.=\underset{p=1}{\overset{M}{.Math.}}{\left(c_p^{†}\right)}^{n_p}.Math.0.Math.}^{.Math.M}\end{array}$

[0088] The quantum state |Ψ can be represented either in the original basis, {|n.sub.1, n.sub.2, . . . , n.sub.M}:

[00005]$.Math.\Psi .Math.=\underset{n_1,n_2,.Math.,n_M}{.Math.}{a}_{n_1,n_2,.Math.,n_M}.Math.n_1,n_2,.Math.,n_M.Math.$

[0089] Or in the natural orbital basis

[00006]$.Math.\Psi .Math.=\underset{{\tilde{n}}_1,{\tilde{n}}_2,.Math.,{\tilde{n}}_M}{.Math.}{\tilde{a}}_{{\tilde{n}}_1,{\tilde{n}}_2,.Math.{\tilde{n}}_M}.Math.{\tilde{n}}_1,{\tilde{n}}_2,.Math.,{\tilde{n}}_M.Math.$

[0090] The above-mentioned property means that the number of nonzero coefficients in this expansion is minimal for the natural orbital basis:

[00007]$\begin{array}{c}\#\left\{{\tilde{a}}_{{\tilde{n}}_1,{\tilde{n}}_2,.Math.,{\tilde{n}}_M},.Math.{\tilde{a}}_{{\tilde{n}}_1,{\tilde{n}}_2,.Math.,{\tilde{n}}_M}.Math.>0\right\}\\ \le \#\left\{a_{n_1,n_2,.Math.,n_M},.Math.a_{n_1,n_2,.Math.,n_M}.Math.>0\right\}\end{array}$

[0091] As a consequence, the quantum circuit to prepare the quantum state |Ψ is simpler in the natural orbital basis than in the original orbital basis.

[0092] FIGS. 4A to 8B illustrate the performance of the method applied on the Hubbard model at half-filling, with μ=U/2 on an increasing number of sites, the number of sites being 2, 3 and 4. The method has been applied on the three quantum circuits illustrated on FIGS. 3A to 3C.

[0093] The Hubbard model is a model used in the field of condensed-matter physics to describe phase transitions in so-called correlated materials, for instance the transition between conducting and insulating systems, such as metals and Mott insulators.

[0094] In these specific and non-limiting examples, the method is used to study a chemical compound such as a molecule. The molecule may be a H2, LiH or H20 molecules.

[0095] The method further comprises providing an encoding scheme (step S2 of FIG. 2). More precisely, a mapping from physical states defined with fermionic variables to qubit states defined with spin variables is provided.

[0096] In an exemplary embodiment, this mapping is achieved through the Jordan-Wigner transformation.

[0097] Once the transformation is chosen, the Hamiltonian operator is decomposed accordingly in terms of qubit observables. The quantum circuit U.sub.θ is also provided.

[0098] In an embodiment, the number of qubits in the quantum circuit U.sub.θ is equal to the number of orbitals of the molecule.

[0099] In an embodiment, the number of quantum gates in the quantum circuit U.sub.θ is also proportional to the number of qubits. For example, if a fSim quantum circuit is used, the number of gates scales as O(Ml), with M the number of qubits in the circuit and 1 the number of fSim layers (l=1 in the non-limiting example provided). If a product quantum circuit is used, the number of quantum gates is M, M being the number of qubits in the circuit. If a LDCA quantum circuit is used, the number of gates scales as O(Ml), with M the number of qubits in the circuit and 1 the number of layers of the LDCA routine employed (l=1 in the example provided).

[0100] FIGS. 4A and 4B show the simulated the performance of the method when computed on a noiseless quantum processor.

[0101] FIG. 4A illustrates the performance of the method for each of the three quantum circuits, where the line (A) represents the performance for the product circuit, the line (B) represents the fSim circuit and the line (C) represents the LDCA circuit on a Hubbard model for two sites at half-filling where U=0, and where the expected energy value, represented by the dotted line, is approximately equal to −2.0 in this non limiting example.

[0102] FIG. 4B illustrates the performance of the method for each of the three quantum circuits, where the line (A) represents the performance for the product circuit, the line (B) represents the fSim circuit and the line (C) represents the LDCA circuit on a Hubbard model for two sites at half-filling where U=1, and where the expected energy value, represented by the dotted line, is approximately equal to −2.5 in this non limiting example.

[0103] It can be seen from FIGS. 4A and 4B that for U=0 (FIG. 4A), the product circuit is able to recover the exact energy after a few steps. For U=1 (FIG. 4B), the product circuit does not reach the exact expected energy value because the expected optimal state is entangled due to interactions. However, the energy value reached by the product circuit gets lower as the orbital basis is rotated in a basis closer to the natural orbital basis.

[0104] The LDCA circuit reaches the expected energy value for U=0 and U=1 and at each step of the method. This is because the LDCA circuit is very expressive and is numerically able to reach the expected energy value for the size of this circuit.

[0105] The fSim circuit is intermediate with respect to the product circuit and LDCA circuit in terms of expressivity. After a few steps, the obtained energy value is closer to the expected energy value as the orbital basis is rotated closer to the natural orbital basis.

[0106] FIGS. 5A and 5B show the performance for the computing of the method on a Hubbard model at half-filling for 4 sites.

[0107] FIG. 5A illustrates the performance of the method for each of the three quantum circuits, where the line (A) represents the performance for the product circuit, the line (B) represents the fSim circuit and the line (C) represents the LDCA circuit on a Hubbard model for four sites at half-filling where U=0, and where the expected energy value, represented by the dotted line, is approximately equal to −4.0 in this non limiting example.

[0108] FIG. 5B illustrates the performance of the method for each of the three quantum circuits, where the line (A) represents the performance for the product circuit, the line (B) represents the fSim circuit and the line (C) represents the LDCA circuit on a Hubbard model for four sites at half-filling where U=1, and where the expected energy value, represented by the dotted line, is approximately equal to −4.5 in this non limiting example.

[0109] The illustrated performances show that the observations made with regards to FIGS. 4A and 4B can be applied to FIGS. 5A and 5B.

[0110] FIGS. 6A and 6B show simulated performance for the computing of the method on a noisy processor.

[0111] Since noise is going to penalize long circuits, going to the orbital basis that requires the shortest circuits will be advantageous.

[0112] FIG. 6A illustrates the performance of the method for each of the three quantum circuits when computed on a noisy quantum processor, where the line (A) represents the performance for the product circuit, the line (B) represents the fSim circuit and the line (C) represents the LDCA circuit on a Hubbard model for four sites at half-filling where U=0, where the depolarizing noise is equal to 0.001 for the 1-qubit quantum gates and 0.01 for the 2-qubits quantum gates and where the expected energy, represented by the dotted line, value is approximately equal to −4.0 in this non limiting example.

[0113] FIG. 6B illustrates the performance of the method for each of the three quantum circuits when computed on a noisy quantum processor, where the line (A) represents the performance for the product circuit, the line (B) represents the fSim circuit and the line (C) represents the LDCA circuit on a Hubbard model for four sites at half-filling where U=1, where the depolarizing noise is of magnitude 0.001 for the 1-qubit quantum gates and 0.01 for the 2-qubits quantum gates and where the expected energy value, represented by the dotted line, is approximately equal to −5.25 in this non limiting example.

[0114] It can be seen from FIGS. 6A and 6B that the convergence of the obtained energy to the expected energy value is much faster for the quantum circuits with a small number of parameters such as the product and the fSim quantum circuits.

[0115] More precisely, both product and fSim quantum circuits retain their high accuracy at U=0 (FIG. 6A), whereas at U=1, noise slightly degrades their performances. However, due to the large depth of the LDCA circuit, the energy obtained by optimizing the latter in presence of noise is far off the expected energy value.

[0116] FIGS. 7A and 7B show the performance for the computing of the method on noisier quantum processors.

[0117] FIG. 7A illustrates the performance of the method for each of the three quantum circuits when computed on a noisy quantum processor, where the line (A) represents the performance for the product circuit, the line (B) represents the fSim circuit and the line (C) represents the LDCA circuit on a Hubbard model for four sites at half-filling where U=0, where the depolarizing noise is of magnitude0.005 for the 1-qubit quantum gates and 0.05 for the 2-qubits quantum gates and where the expected energy value, represented by the dotted line, is approximately equal to −4.0 in this non limiting example.

[0118] FIG. 7B illustrates the performance of the method for each of the three quantum circuits when computed on a noisy quantum processor, where the line (A) represents the performance for the product circuit, the line (B) represents the fSim circuit and the line (C) represents the LDCA circuit on a Hubbard model for four sites at half-filling where U=1, where the depolarizing noise is of magnitude 0.005 for the 1-qubit quantum gates and 0.05 for the 2-qubits quantum gates and where the expected energy value, represented by the dotted line, is approximately equal to −5.25 in this non limiting example.

[0119] The observations made with reference to FIGS. 6A and 6B can be applied on the performance illustrated on FIGS. 7A and 7B. More precisely, the accuracy of the obtained energy, compared to the expected energy value, for the product and fSim quantum circuits is maintained, while the energy reached by the LDCA circuit is far off the expected energy value, due to the large depth of the LDCA circuit.

[0120] In an embodiment (not shown), the method can be applied for a Hubbard model at half-filling for three sites.