Quantum computing system

Abstract

A quantum computing system configured to determine a Hamiltonian and a corresponding quantum circuit for a physical system (e.g. an element, molecule, crystal structure, or other material). The quantu

Description

This disclosure relates to quantum computing systems, and in particular systems and methods associated therewith for determining energy levels of physical systems. Background Determining the energy states of a physical system such as a particular element, molecule, crystal structure or other material has many applications in modern technology. For example, understanding the energy states and electronic structure of various materials enables the discovery of new materials and the interpretation of experimental data that would otherwise be impossible. It is important that energy level and electronic structure estimations must be produced with a very high degree of accuracy in order for the results to have technical application. It is well known that understanding such properties of a material, and in particular identifying the energy levels of a material, involves solving the Schrodinger equation to find the energy eigenvalues. For complex materials (materials comprising more than one element, complex crystal structures or other such materials), a key step to solving the Schrodinger equation is to identify the Hamiltonian operator, H, associated with the material. For complex systems that do not give rise to analytic solutions to the Schrodinger equation, identifying the Hamiltonian is a difficult task and solving Schrodinger's equation involves computational simulation, taking up vast amounts of computing resources. One such technique for deriving a Hamiltonian for a complex system is the projector augmented wave (PAW) method (P. E. BIbchl, Projector augmented-wave method, Phys. Rev. B 50, 17953 (1994)). The PAW method is capable of producing results with a high degree of accuracy which can be determined relatively efficiently on a classical computer. However, one drawback is that the PAW method produces a Hamiltonian with non-orthogonal eigenstates, leading to additional complexity in the computation simulation. In addition, identifying a Hamiltonian using the PAW method may give rise to other complications that may require additional computing resources to address. For example, the PAW method generally leads to an N-body interaction term unless an inverse PAW transformation is implemented. This complicates the use of orthonormal basis sets such as planes waves, while the use of non-orthogonal basis functions complicates the description of many-body techniques. The present disclosure seeks to address these and other disadvantages encountered in the prior art by providing an improved technique for deriving a Hamiltonian and estimating the energy levels of a physical system. Summary According to a first aspect, there is provided a quantum computing system comprising a quantum compilation system and a quantum processing unit comprising a control system coupled to one or more registers of quantum devices. The quantum compilation system is configured to: receive information indicative of one or more properties of a physical system; determine, based on the received information, a factorised Hamiltonian for the physical system, wherein determining the factorised Hamiltonian comprises performing a projector augmented-wave method using a plane wave basis; generate a quantum circuit based on the factorised Hamiltonian; and transmit the quantum circuit to the quantum processing unit. The quantum processing unit is configured to receive the quantum circuit and prepare the one or more registers of quantum devices according to the quantum circuit. Optionally, each register of quantum devices comprises one or more qubits. Optionally, the one or more properties of the physical system comprise one or more of: a cell structure of the physical system; the chemical elements comprised within the physical system; and the geometry of the physical system, including the position of atoms within the physical system. Optionally, the projector augmented-wave method is a unitary projector augmented-wave method that determines a factorised Hamiltonian with orthonormal eigenstates. Optionally, the quantum computation system is configured to generate the quantum circuit by decomposing the factorised Hamiltonian into a Linear combination of unitary matrices (LCU). Optionally, the quantum computation system is configured to generate the quantum circuit by block encoding the LCU decomposition of the factorised Hamiltonian into a unitary operator, wherein the unitary operator represents the quantum circuit, and wherein the quantum compilation system is configured to transmit the unitary operator to the quantum processing unit. Optionally, the quantum processing unit is configured to prepare the plurality of registers of quantum devices by constructing an arrangement of quantum gates according to the received unitary operator, and operate on the registers of quantum devices using the arrangement of quantum gates to prepare the one or more registers in a predetermined state. The predetermined state may correspond to the factorised Hamiltonian. Optionally, the quantum processing unit is configured to prepare the plurality of registers of quantum devices by performing data loading on a first set of the one or more registers, wherein the data loading comprises operating on the first set of registers using a first sub-arrangement of quantum gates to encode the first set of registers with information corresponding to the factorised Hamiltonian. The first sub-arrangement of quantum gates may corresponds to a classical lookup table comprising the information for the factorised Hamiltonian. Optionally, the factorised Hamiltonian comprises a plurality of terms each comprising a sign function, and wherein the quantum processing unit is configured to perform data loading on a dedicated register of the first set of registers to encode one or more qubits of the dedicated register with information corresponding to the value of the sign function for each of the plurality of terms. The quantum circuit may comprises a controlled-Z gate configured to act on the dedicated register after data loading on the dedicated register has been performed. The controlled-Z gate may be conditioned on the success of the state preparation of one or more other ones of the one or more registers. Optionally, the quantum processing unit is further configured to perform a quantum computing algorithm with the prepared register of quantum devices. The quantum computing algorithm may be a quantum phase estimation algorithm. Optionally, the quantum processing unit is configured to estimate one or more energy levels of the physical system using the quantum phase estimation algorithm. According to a second aspect, there is provided a method for preparing a register of quantum devices using a quantum computing system according to the first aspect. The method comprises receiving information indicating one or more properties of a physical system; determining, based on the received information, a factorised Hamiltonian for the physical system, wherein determining the factorised Hamiltonian comprises performing a projector augmented-wave method using a plane wave basis; generating a quantum circuit based on the encoded representation of the factorised Hamiltonian; and preparing one or more registers of quantum devices according to the quantum circuit. Optionally, each register of quantum devices comprises one or more qubits. Optionally, the one or more properties of the physical system comprise one or more of: a cell structure of the physical system; the chemical elements comprised within the physical system; and the geometry of the physical system, including the position of atoms within the physical system. Optionally, the projector augmented-wave method is a unitary projector augmented-wave method that determines a factorised Hamiltonian with orthonormal eigenstates. Optionally, the method comprises generating the quantum circuit by decomposing the factorised Hamiltonian into a Linear combination of unitary matrices (LCU). Optionally, the method further comprises generating the quantum circuit by block encoding the LCU decomposition of the factorised Hamiltonian into a unitary operator, wherein the unitary operator represents the quantum circuit, and the method comprises transmitting the unitary operator to the quantum processing unit. Optionally, preparing the plurality of registers of quantum devices comprises constructing an arrangement of quantum gates according to the received unitary operator, and operating on the registers of quantum devices using the arrangement of quantum gates to prepare the one or more registers in a predetermined state. The predetermined state may correspond to the factorised Hamiltonian. Optionally, preparing the plurality of registers of quantum devices comprises performing data loading on a first set of the one or more registers, wherein the data loading comprises operating on the first set of registers using a first sub-arrangement of quantum gates to encode the first set of registers with information corresponding to the factorised Hamiltonian. The first sub-arrangement of quantum gates may correspond to a classical lookup table comprising the information for the factorised Hamiltonian. Optionally, the factorised Hamiltonian comprises a plurality of terms each comprising a sign function, and the method comprises performing data loading on a dedicated register of the first set of registers to encode one or more qubits of the dedicated register with information corresponding to the value of the sign function for each of the plurality of terms. The quantum circuit may comprise a controlled-Z gate configured to act on the dedicated register after data loading on the dedicated register has been performed. The controlled-Z gate may be conditioned on the success of the state preparation of one or more other ones of the one or more registers. Optionally, the method further comprises performing a quantum phase estimation algorithm using the prepared one or more registers. Optionally, the method further comprises determining one or more energy levels of the physical system using the quantum phase estimation algorithm. According to a third aspect, there is provided a computer-readable medium storing instructions that, when executed by one or more quantum computing systems, cause the quantum computing systems to carry out the method of the second aspect. Figures Specific embodiments are now described, by way of example only, with reference to the drawings, in which: Figure 1 depicts a quantum circuit according to an embodiment of the present disclosure; Figure 2 depicts a flowchart of a method according to the present disclosure; Figure 3 is an example embodiment of a quantum computer system; and Figure 4 is an example embodiment of a computer program product. Detailed Description In overview, and without limitation, the application discloses systems and methods for preparing one or more registers of quantum devices, such as qubits, according to a quantum circuit. The quantum circuit is generated according to a Hamiltonian for a physical system, where the Hamiltonian is derived from information relating to known properties of the physical system, such as a cell structure, chemical make-up, atomic geometry, or other suitable known properties of the physical system. In more detail, the Hamiltonian for the physical is derived using the known physical properties using a technique that is based on the projector augmented-wave method. In particular, as described in more detail below, a novel unitary version of the projector augmented-wave technique that uses plane wave basis is used to generate a factorised Hamiltonian for the physical system. The factorised Hamiltonian is mapped to qubit representation by decomposing the Hamiltonian into a linear combination of unitary matrices, which is then block encoded into a unitary operator to generate the quantum circuit used to prepare the one or more registers of quantum devices. The prepared registers of quantum devices may subsequently be used in quantum computation algorithms, such as a quantum phase estimation algorithm. For example, a quantum phase estimation algorithm may be run by a quantum computer, and using the prepared registers of quantum devices, to determine one or more energy levels of the physical system. The novel technique for deriving the Hamiltonian, described herein as a unitary projector augmented-wave method (UPAW method), differs from the known projector augmented-wave method (PAW) in that it produces a Hamiltonian for a physical system that has orthogonal eigenstates. Unlike conventional PAW techniques, which are incompatible with quantum processing due the lack of orthogonality of the eigenstates of the resultant Hamiltonian, the UPAW method-produced Hamiltonian disclosed herein is compatible with quantum processing due to the orthonormality of the eigenstates. As such, the Hamiltonian derived using the novel UPAW method can be efficiently block encoded into a unitary operator, thereby producing an efficient quantum circuit (in the sense that the resultant quantum circuit has a limited qubit cost and toffoli complexity) for the state preparation. Being able to encoded the factorised Hamiltonian derived using the UPAW method into a unitary operator means that the efficiency gains of quantum processing (relative to classical computation) in certain applications can be leveraged for solving Schrodinger's equation for the Hamiltonian to accurately derive energy level estimates for of the physical system. In other words, while many current methods, including the PAW method, rely on classical algorithms to identify energy eigenvalues and eigenstates for a physical system, the disclosed UPAW-method and corresponding block encoding of the Hamiltonian make it possible to identify eigenvalues and eigenstates using more efficient and established quantum computing algorithms that are familiar to the skilled person, such as quantum phase estimation. In the following section, the known techniques of Quantum Phase Estimation and LCU decomposition are briefly described for context of the present disclosure. The PAW method is an all-electron approach which is exact in theory but relies on several approximations in practice. The classical PAW approach carries advantages in terms of accuracy and precision of results, however it brings additional complications in its implementation, one of which is that the electronic integrals are not generally analytically computable in the PAW formulism and plane-wave basis set. Another complication is that the PAW method generally leads to non-orthogonal orbitals. This leads to certain levels of inefficiency when using the PAW method with classical algorithms for finding energy levels of a physical system. These issues are addressed by the novel unitary PAW method discussed below and the formulation of a many-body PAW transformation for the generic chemistry Hamiltonian in the first and second quantization. As mentioned above, an example quantum algorithm used for finding energy levels of a physical system is qubitization-based quantum phase estimation (QPE). Quantum phase estimation is a known efficient and accurate technique for finding the energy levels of a physical system using a quantum processor. In order to apply QPE for solving Schrodinger's equation to find the energy levels, the Hamiltonian for the physical system must be implemented in a quantum circuit. An efficient means for implementing a Hamiltonian in a quantum circuit is to map the Hamiltonian to a linear combination of unitary matrices, or linear combination of unitaries (LCU). As is well known by the skilled person, a Hamiltonian can be decomposed into a LCU using the following equation: Equation I.1 where aq are real numbers, and Wt are unitary matrices. The unitary operator used in qubitized QPE, i.e. the resultant quantum circuit that implements the Hamiltonian in a quantum processing unit, is the block-encoding of the Hamiltonian whereby H is embedded into a larger unitary matrix V: / H / A junki \junk2 junk3 Equation I. 2 The subnormalization factor A is often closely related to the one-norm ex 3 = I of the Hamiltonian's LCU on which the block encoding is based. The asymptotic complexity of qubitized QPE is then determined by T, the amount of information needed to specify the LCU decomposition (Equation 1.1), and A. Using a QROAM (also known as select-swap network) for data loading, the Toffoli complexity is VTA / eQPE and the qubit number complexity is O(Vr), where eQPE is the error budget in the QPE. The properties of the LCU determine the efficiency of the quantum algorithm, and thus developing an LCU decomposition which lowers the overall quantum computational cost (in terms of qubits and quantum gates required) is preferable. In the following section, the UPAW method using plane wave basis is described for constructing a factorised Hamiltonian for a physical system in the form of a linear combination of unitaries (LCU). The corresponding block encoding of the LCU is provided in Fig. 1 as described in more detail below. Within the Born-Oppenheimer approximation, the motion of electrons and nuclei are decoupled and the electrons are described using the clamped-nuclei approach, i.e., the electrons move in the field generated by nuclei with charges {Za} fixed at the positions {Pa}. The resulting Hamiltonian, which is well known to the skilled person, can be written as Equation II. 1 j>m where N is the number of electrons, NA is the number of atomic centers (nuclei). Apart from the Laplacian V?, which is a differential operator appearing in the kinetic energy term for electron j, the potential terms are multiplication operators V that multiply the functions they act on by a potential function of the form i. e.,V = ^({^ / )) / (^) / where {ry} are the spatial coordinates of the electrons. The constant term C consists of the repulsion nuclear term, the one-electron operator hj acts on the jth electron and contains the kinetic energy contribution of that electron and its potential energy at position Tj in the field of all the nuclei, and finally, the two-electron terms merely consists of the repulsion between electrons. Using Schrodinger's equation, goal is to estimate the ground state energy E (or any other energy eigenvalue) of the Hamiltonian: .....xN) = .....xN), Equation II. 3 where x = (r, o) is a combined coordinate variable consisting of the spatial (r) and spin (o) degrees of freedom. In order to solve this equation, the wavefunction 'P is expressed in terms of orbitals, such as plane waves. Because the wavefunction demonstrates rapid oscillations near the nuclei, one would need a large number of plane waves. In order to reduce the number of plane wave coefficients, the projector augmented-wave (PAW) method is generalized as described below. A key feature of the PAW approach is that it establishes a linear transformation T between a set of Hartree-Fock or Kohn-Sham orbitals {ip(x)} and a set of smooth pseudo orbitals {ip(x)} that have no cusps around the nuclei, ip(x) = 7\p(x). Equation II. 4 Crucially, the orbitals and the transformation operator itself are then constructed from atomic contributions. To achieve this, the system is divided into atom-centered augmentation spheres within which ip(x) is expanded in terms of localized (atom-like) partial waves (r), where j identifies one such function belonging to the atom labelled by a. Similarly, the pseudo orbitals ip(x) are also expanded using smooth pseudo partial waves denoted as ^(r). The functions and c|)“(r) are identical outside the augmentation sphere around atom a, and they are related by the transformation T in the same way as the orbitals ip(x) and ip(x). As a result, the atomic-centre expansion of orbitals can be given as: Na na ¢0) = $(*) + zz(4>“(r) —       J d3r'p“(r')t|j(x'). a=l;=1 Equation II. 5 Here, na is the total number of projector functions, [pf (r')}, localized on an atom a, which can be constructed from any set of linearly independent functions and which must be biorthogonal to pseudo partial waves. To simplify the notation, the combined index k = (a,j) is used, as well as the functions: Xk(r) = 4>k(r) — 4>k(r). Equation li. 6 Both xk(r) and p^(r) are localized at the nuclei, a, and vanish outside the augmentation ball. The PAW operator, T acting on the pseudo orbitals is then defined as f = l + ^Tk, k Equation li. 7 where 7^ is an integral operator defined by its action on the trial function / (r) as ^f(r) = Xk(T) J d3rp^(r')f(r'). Equation li. 8 Since the spin orbitals ip(x) are products of a spatial part of the form / (r) and a pure spin part depending on the spin variable o, it should be noted that when 7^ acts on ip(x) it leaves the spin part entirely unaffected. This can be expressed in the bracket notation as: T=-I + ^\xk) fel k Equation li. 9 where Is is identity operator acting on spin degrees of freedom. Next, the PAW transformation is generalized to a many-body wavefunction, 'P. In order to do this, a procedure in which the PAW transformation is applied recursively coordinate by coordinate is followed: 'Pj_1(x1,x2, ...,xN) = 'P / (x1,x2, ...,xN) + J^Xkj(rj} J drj ...,Xj,...,x^, kj j = 1 ...N, Equation li. 10 Here, % is the original many-body wavefunction and *PW is a smooth many-body wavefunction which does not have cusps around the nuclei, %(x1,x2,...,xw) := ^Cx^Xz,..., xN\ -, XN) := -, XN). Equation it. 22 Using this recursive relation, the many-body PAW transformation is obtained as follows: Tmb = 7-(1) 0 f(2) 0 • ■ ■ 0 f(W) = 1 + >2 521», (ii)) (MOI + ii ki + 52 —+ 52 1^1(^)---^(-^))(^1(^^ •j 1 <%2 <^'2 • '"CAOv Equation H. 13 Where |xki(1) ... / / ^QV)}, \pki(1) — P^tN')) are the Slater determinants, and the PAW-transformed Schrodinger equation for the pseudo many-body wavefunction can be written as follows: H\$) =ES\V), Equation li. 14 With H = Equation it. 15 S = TmBTmb- Equation it. 16 and E is the same energy as in the all-electron Schrodinger Equation II. 3). While the wavefunction becomes smooth around nuclei, the PAW transformation brings additional complications as mentioned above. First, the eigenstates of the Hamiltonian are not orthonormal anymore, that is: Equation it. 17 As a result, one has to solve a generalized eigenvalue problem. Secondly, the PAW transformation generally leads to an N-body interaction term which must be addressed using the inverse PAW transformation, This complicates the use of orthonormal basis sets such as plane waves, while the use of non-orthogonal basis functions complicates the mathematical description of many-body techniques. In order to overcome these shortcomings in the Many-Body PAW method, the unitary PAW method (UPAW) which eliminates the complications of a conventional PAW method is described below. In order to overcome at least some of the complications described above, modifications to the above approaches are introduced at the level of constructing pseudo partial waves. In particular, the modifications involve making T a unitary operator, = T-1. This means that the one-electron overlap operator: O = Pt =1 + ^0, Equation II. 18 Where « Equation II. 19 Equation il. 20 becomes the identity. This requires pseudo partial waves c|)“ to be selected in such a way that O^j is zero: Equation II. 21 One way to enforce this orthogonality condition is to make both the partial and pseudo partial waves orthogonal but such an approach would create additional oscillations of the wavefunction which would in turn increase the plane-wave cutoff. The procedure for constructing pseudo partial waves without making them orthogonal is described below in the section titled "Appendix A". For the unitary PAW transform, the pseudo many-body Schrodinger equation is W = EW, Equation II. 22 Where i ' ' 1 7 &=c+52 nppt + 2 S t)p(j)^T(z)tij) = c+52 p + - 52 i Li .i ~ ij Equation II. 23 For practical applications, such as identifying the energy levels of a physical system, the matrix elements of the Hamiltonian are evaluated in either first or second quantization: N oc i oo ~mihwvH)»+ 252 52    52^^7(&\)ddT}(M).r i PSI O’ i^j P>tpT,s v\r Equation it. 24 And / 7 C i £ hpq^p^qa + qprs ^pcr ^rr , p-.q^ Equation it. 25 respectively. In the expressions above, orthonormal smooth basis sets [pa) are used with a spin-restricted approach. The one- and two-body matrix elements calculated from PAW-transformed one- and two-body operators are: h„ = (p^kriq), Equation it. 26 Kqprs = (pj 0 T^gT 0 T |s), Equation it. 27 where h. and g are conventional one- and two-body operators which in the |r) basis can be written as: 71%) = V» - r') -       77 - r9 2 i1 “ Equation li. 28 (rir2|.g|r3r4) = £(ri - r3)6(r2 - r4)-j--------r Equation it. 29 The operators, \pa), (qa\, dpa, aqa in Equation II. 24 and Equation II. 25, act directly on the pseudo space of many-body wavefunctions that require a significantly smaller number of basis functions, especially if plane wave basis sets are used. The frozen-core approximation is used in the following description and mathematical formulation, where only valence and sometimes semicore electrons are explicitly considered. While the energetic contributions from the core electrons change the constant and the one-body terms, they are absent in the two-body term. It is also convenient to rewrite the Hamiltonian in terms of the generators of the unitary group U(n), i.e., spin-summed excitation operators (excitons) 7^ 22 0=0,1 Equation / / / . 1 This yields a spin-free formulation in which all dependence on the spin is included in the summation in Epg. Equation HI. 2 where summations run over band indices p, q, r, s and the total number of bands is Nb. hpq, Krprq are the matrix elements which are calculated using some orbitals (single-electron wavefunctions), Crucially, the ground state energy will depend on the number of orbitals, Nb, and this number should be large enough to ensure sufficient convergence up to an acceptable error (relative to the Nb -> energy). In computational quantum chemistry, the standard choice of orbitals is either canonical Hartree-Fock or Kohn-Sham orbitals, which are obtained by solving the one-electron Schrodinger equation ^p(r) = ^p(r), p = 1... Equation III. 3 Since it is very difficult to solve these equations exactly without any numerical approximation, additional basis functions, x5(r), are introduced to expand orbitals: w(r) ^22^^)^- 9 Equation III. 4 Equation III. 3 can then be rewritten in an algebraic form that is easier to solve using computational algorithms. From Equations III.2 and II 1.4, it can be seen that there are now two parameters Ng and Nb. Ng should be large enough so as to give a good approximation to ipp(r) and Nb should be large enough so as to estimate the ground state energy of the Hamiltonian in Equation III. 2 to a good accuracy. In some examples, Nb = Ng especially when one employs Gaussian basis functions. However, in other examples, for example is natural orbitals are used instead of canonical orbitals with plane wave basis set, Nb « Ng. As a default, MP2-natural orbitals are used throughout the present disclosure unless stated otherwise (see Appendix C). The notation is also changed from Ng to Npw to be more explicit, since a plane wave basis set is used. In order to implement the Hamiltonian in Equation III. 2 in the quantum circuit with low cost, it is necessary to factorize the Hamiltonian so as to reduce the amount of information needed for the block encoding and to have the smallest possible subnormalization factor. In order to do this, firstly, the two-body term in Hamiltonian in Equation III. 2 can be expanded into a soft (pseudo) contribution and atomic-centered PAW corrections: <w.-(^ia.)+E E a ifia 4314 Equation Hi. 5 Respectively, the notation for Coulomb matrix elements is introduced here as: Equation Hi. 6 other notations are described in more detail below in Appendix C. In order to factorize the soft term, the plane wave expansion of the Coulomb kernel is used as follows: - = 22 eiGr”(G) I ! G Equation Hi. 7 In this expansion, Wigner-Seitz regularization is used for the divergent G = 0 component. For the PAW term, eigendecompositions of the tensor f f and the atomic orbital-pair density matrices, ^pq,1^21 are used. The resulting Hamiltonian can then be written as follows (see Appendix C for detailed derivation and notations): ' / '[ \ ApW / 2 / \ 2 H = H^+ E E ^(G)+ PQ \ r j G>0 J = \ p / A’a / A’s \2 E £ a i] <z2 \ P / Equation Hi. 8 with fpj(G) being the eigenvalues of the soft reciprocal orbital-pair density matrix r|pqj(G). When the Hamiltonian in Equation III. 8 is mapped onto qubits, an additional shift appears in the one-body term r Equation Hi. 9 And the subnormalization factor in the block encoding becomes Equation III. 10 where ep is the pth eigenvalue of h', and = / / A* = G>0 y ’ ’ Equation III. 11 This factorization is of double factorization form apart from the fact that there is an additional sign in the PAW contribution, sign(e“j2), which appeared to due the fact that f j is not positive definite. Equation III.8 provided above therefore gives the factorised Hamiltonian for a physical system that is derived using the Unitary PAW method described above and using plane wave basis. As described above, the disclosed methods and systems involve mapping this factorised Hamiltonian to a qubit representation by decomposing the Hamiltonian into a linear combination of unitary matrices, which is then block encoded into a unitary operator to generate the quantum circuit used to prepare the one or more registers of quantum devices. The following section describes the block encoding of the factorised Hamiltonian given in equation III.8. With reference to Fig. 1, a quantum circuit 100 for the block encoding of the factorised UPAW Hamiltonian is shown. As would be understood by the skilled person, the block encoding shown in Fig. 1 is a quantum circuit comprising a specific arrangement of quantum gates configured to operate on a plurality of quantum registers 110A- 110M. Each quantum register is represented by the horizontal lines as shown in Fig. 1 and in accordance with standard quantum circuit notation as would be understood by the skilled person. Each quantum register 110A - 110M may comprise one or more qubits. In general, the number of qubits and the number of quantum gates may be ultimately dependent on the physical system being represented by the factorised Hamiltonian, since the known properties of the physical system will dictate the complexity of the Hamiltonian. Each register in the quantum circuit is described in more detail below. In more detail, the quantum circuit 100 depicted in Fig. 1 is a schematic representation of the unitary operator V as given by equation 1.2 above, where the Hamiltonian H in equation 1.2 has been decomposed into a LCU of the form given in equation 1.1. In other words, the quantum circuit 100 depicted in Fig. 1 is a quantum circuit that is generated using block encoding based on the factorised Hamiltonian that is determined using the disclosed UPAW method (equation III.8). As would be appreciated by the skilled person, the output of the quantum circuit 100 is the plurality of quantum registers 110A - 110M prepared in predetermined states by the specific arrangement of quantum gates of the quantum circuit. In other words, the output of the circuit is the action of the unitary operator V on the plurality of registers. As described above, this output may be used in subsequent quantum computational algorithms, such as quantum phase estimation, to estimate energy levels of the physical system represented by the initial factorised Hamiltonian from which the unitary operator V represented in the circuit 100 is derived. Referring now to the plurality of registers shown in Fig. 1, the quantum circuit 100 is configured to operate on a plurality of registers 110A - 110M. Each register 110A - 110M is operated on by one or more quantum gates or quantum sub-circuits as described in more detail below. Some of the gates / sub-circuits may be controlled by the states of other quantum registers, and some of the gates / sub-circuits may be operated on more than one register at the same time. Register HOB shown in Fig. 1 upon which the circuit 100 is configured to operate is labelled by the index I, which represents the one body term (for I = 0) and the various factorised two-body terms (for I >0). In other words, the one body terms and factorised two body terms in the Hamiltonian given in equation III.8 are indexed by I = 0,1 ... L, where I = 0 represents the one body term (taken from equation III.8 above): e -4 e^W and all other values I = 1 ... L represent the various factorised two-body terms indexed by G,j, a, in equation III.8 above. The register HOB labelled I is therefore a multiqubit register which is prepared in an equal superposition over L + 1 basis states by the quantum sub-circuit 120 labelled "prep". In other words, the sub-circuit 120A labelled "prep" prepares the equal superposition of the L + 1 basis states on the I register. The register 110A labelled succ I is a register associated with register HOB that flags the success of the preparation of the equal superposition over the L + 1 states for register HOB by the "prep" sub-circuit 120A. The quantum circuit 100 further comprises a sub-circuit 120B, labelled Ini and data^ which is configured to operate on registers HOC - 110G labelled I #= 0, E®, "offset", "rot" and sign |0) respectively. As would be appreciated by the skilled person, sub-circuit 120B is used for data loading on the I #= 0, E®, "offset", "rot" and sign |0) registers using QROM or QROAM data loading techniques as would be known by the skilled person. In more detail, datat represents a classical lookup table storing information about the factorised Hamiltonian for the physical system which has been converted into a quantum subcircuit. The lookup table maps indices enumerating the terms (indexed by I) of the Hamiltonian with corresponding coefficients in the Hamiltonian. For any given value I which goes into the Int part of the subcircuit 120B (based on the state preparation on the I register by sub-circuit 120A), the datat part of the subcircuit 120B loads the corresponding value(s) from the datat lookup table. As would be appreciated by the skilled person, the datat sub-circuit encodes classical data from the lookup table into a plurality of C-NOT quantum gates. In particular, the particular values of the data in the lookup table are encoded into the presence or absence of C-NOT gates in the datat sub-circuit. The data loading takes place on registers HOC - 110G, that is, the subcircuit 120B performs data loading on the I #= 0, E®, "offset", "rot" and sign \6) registers according to the information stored in the datai lookup table for the particular Hamiltonian in question. Referring to each of these registers HOC - 110G in more detail, the register HOC labelled I #= 0 is a temporary register used to keep the result of an inequality test checking that I #= 0 to ensure that inner block encoding that is performed twice (to represent the square in the factorised two-body terms) has no effect for the I = 0 state (which represents the non-squared one body term shown above). In other words, the squared elements of the factorised Hamiltonian given in equation III.8 applies only to the two body terms (I #= 0), which is represented in the circuit by performing the inner block encoding twice as can be seen in Fig. 1. As an example, the I #= 0 register may be prepared in the 11) state by the datai sub-circuit when any positive I value goes into the Ini subcircuit. For the one body term, when I = 0, the I #= 0 register may be prepared in the |0) state by the datat sub-circuit. The registers 110D, 110E and 110F labelled E®, "offset", and "rot" respectively are some of the outputs of the data loading (e.g. using a QROM, QRAM, or QROAM as described above) on the I register based on the classical Hamiltonian information stored in the datat lookup table. As shown by the subcircuit 120C in Fig. 1, these registers are used for the controlled state preparation on the p register 1101. The p register and it's state preparation is discussed in more detail below. The register 110G labelled sign \6) is also an output of the data loading based on the I register and using the datat lookup table. In more detail, the specific implementation of the unitary version of the projector augmented-wave method as described above introduces a "sign" function element in the unitary projector augmented-wave (UPAW) part of the two-body term of the factorised Hamiltonian given in equation III.8. The PAW term (extracted from equation III.8) is given as follows: Here it can be seen that in this part of the factorised Hamiltonian, termed the UPAW contribution to the Hamiltonian, each term has a coefficient given by sign(e“i These signs cannot be absorbed into a square bracket (since the sign may be negative) and are therefore treated separately by loading the information for sign(e“i into a dedicated ancilla qubit in the register sign |0). In other words, the lookup table implemented in the quantum circuit datat contains information regarding the value of sign(e“i iz) for each term in the UPAW contribution to the Hamiltonian. This information is loaded into one or more qubits in the sign \6) register during the data loading (subcircuit 120B) indexed by I. As an example, the value of sign(e“i iz) for any given term may be encoded in a single ancilla qubit in the register HOG such that |0) indicates sign(e“i f ) = +1 and |1) indicates s'gn«i2) = -1- In general, the lookup table for data loading for the sign |0) uses the actual value of sign(e“i taken from the Hamiltonian for each term in the UPAW contribution. As a non-limiting example, the lookup table may take the form of {Z: 0} = {1: 0,2:1,3: 0,4: 0,5:1,...} where I is the index and 9 represents the sign. As described below, when 0 = 0 the sign is positive, and when 0 = 1 the sign is negative. For all other terms (outside of the UPAW contribution), the lookup table stores the value +1 (since all other terms are positive). Referring still to register HOG, the quantum circuit 100 comprises a controlled-Z gate 160 configured to operate on the sign |0) register to implement the actual sign values loaded into the sign |0) qubit(s). In more detail, the controlled-Z gate is capable of implementing a sign in front of any given state. For a given state |R), a sign qubit |0) can be added and the controlled-Z gate will act as follows: Z\9,R) = (—1)^9,R) It follows that for 0 = 0, the Z-gate gives Z|0,R) = |0, R) whilst for 0 = 1, the Z-gate gives Z\1,R) = —11, R). In other words, the Z operator on |0) produces a + or - sign in front of the state. Referring again to gate 160 shown in Fig. 1, the controlled-Z gate 160 is controlled by succ I, whether the state preparation of the I register by subcircuit 120A was successful. This is to ensure that the gate 160 only acts on factorized terms of the Hamiltonian. In other words, the controlled-Z gate is only applied if the I register was successfully prepared, in which case succ I = 1. Otherwise, if the I register was not successfully prepared, succ I =0 and the gate 160 is not applied to the sign |0) register. As mentioned above, Fig. 1 shows quantum register 1101, denoted by the p register. The index p indexes the unitary matrices in the linear combination of unitaries representation of the factorised Hamiltonian. In more detail, the p register 1101 is used to prepare a superposition of all states needed to implement the linear combination of unitaries (LCU) representing the Hamiltonian in qubit representation. In more detail, for the two-body terms, the p register enumerates all unitaries in the range [0, LE — 1] where E is output in register HOD and is used to control the state preparation of the p register 1101. Similar to register 110A labelled succ I, the register 110H labelled succ p is used to flag the success of the preparation of the superposition over the LE states for register 1101. The state preparation of the p register is carried out using subcircuit 120C labelled prepi, which is controlled (as shown by the In portion of the subcircuit 120C) by the data loading on the E®, "offset", and "rot" registers. In other words, the state preparation on the p register 1 lOi is based on the output of the datat operation on the E®, "offset", and "rot" registers. The quantum circuit 100 further comprises a subcircuit 120D labelled Inp and datap which is configured to operate on register 110J labelled "Rotations". Similar to the subcircuit 120B described above, the subcircuit 120D is used for data loading on the "Rotations" register 110J using QROM or QROAM data loading techniques. In particular, datap represents a classical lookup table storing rotation information for each term (indexed by p) in the LCU representation of the factorised Hamiltonian. In other words the look up table for datap maps indices p enumerating the terms in the LCU with corresponding rotations. For any given value p which goes into the Inp part of the subcircuit 120D (based on the state preparation of the p register by subcircuit 120C) the datap part of the subcircuit 120D loads the corresponding rotations from the datap lookup table. In other words, the subcircuit 120D performs data loading on the "Rotations" register according to the rotations information stored in the datap lookup table for the particular Hamiltonian in question. Finally, the register 110K labelled |0) is used to select the spin, and registers 110L and 110M labelled \ipi) and \ipx) respectively represent the spin-down and spin-up components of the target system. In other words, as would be understood by the skilled person, registers 110L and 110M are the system qubits which are prepared according to the factorised Hamiltonian and which, when prepared according to the factorised Hamiltonian, may be used in subsequent quantum computing algorithms, such as quantum phase estimation, to estimate energy levels of the physical system in question. As shown in Fig. 1, the quantum circuit 100 comprises quantum gates that act on the system registers \ip^) and ) in order to prepare the system registers. These quantum gates acting on the system registers involve: a controlled swap 130 between the system registers 110L and 110M, with the spin qubit register |0) 110K as control. A controlled rotation 140 controlled by the state of the "Rotations" register 110J, which is prepared as described above. A controlled-Z gate 150, which is controlled on the succ I and succ p registers 110A and 110H respectively (i.e. controlled on the success of the state preparation of the I and p registers. Following these operations, the reverse operations take place so as to disentangle all ancilla qubits (those in registers 110A - 110K) from the system qubits (registers 110L and 110M) so that the ancilla qubits do not affect the system states when performing measurements on the system states (e.g. as part of the subsequent quantum computation algorithms, such as quantum phase estimation). As described above, the inner block encoding steps (indicated by subcircuits 120C, 120D, 130,140, 150) and their reverses are repeated for all the two body terms (I #= 0) to account for the squares as given in equation III.8. The reverse of the processes of 120B and 120A (i.e. the inverse of the data loading and state preparation on the I register) then take place to complete the uncomputation to disentangle all ancilla qubits from the system qubits before any measurements can be performed on the system qubits as part of a further algorithm. Figure 2 illustrates a flowchart of a method 200 according to the present disclosure. The method comprises a first step 202 of receiving, by a classical computer (such as a quantum compilation device 350 as described below), information indicative of known properties of a physical system. The system may be an atom, molecule, crystal structure or some other physical material. The known properties may include one or more of: cell structure, atomic geometry (e.g. relative positives of atoms in a molecule), information indicated of the chemical elements making up the physical system, or any other suitable information that can be used to derive a Hamiltonian for the physical system. At step 204, the quantum compilation system determines, using a classical computing processor and using a projector augmented-wave method as described above, a factorised Hamiltonian for the physical system, based on the received known properties. The projector augmented wave method may be the unitary projector augmented-wave method described above, which produces a Hamiltonian of the form given in equation III.8. At step 206, the quantum complication system generates, based on the factorised Hamiltonian, a quantum circuit corresponding to the factorised Hamiltonian. As described above, generating the quantum circuit may involve mapping the factorised Hamiltonian to a qubit representation by decomposing the factorised Hamiltonian into a linear combination of unitaries (LCU) (according to equation 1.1). The quantum circuit, or unitary operator corresponding to the quantum circuit, is generated by block encoding the LCU representation of the Hamiltonian into a unitary operator. The unitary operator or quantum circuit corresponding to the unitary operator is then transmitted to a quantum processing unit 310, whereby at step 208, one or more quantum registers 302 of the quantum processing unit are prepared in a state according to he received quantum circuit. Although not shown in Fig. 4, the method may further comprises performing a quantum computation algorithm using the prepared quantum registers to determine one or more energy levels of the physical system. Figure 3 illustrates a block diagram of one implementation of a computing system 300, e.g. a quantum computing system, within which a set of instructions for causing the computing system to perform any one or more of the methodologies of the present disclosure may be executed. The computing system may comprise a single computing device as illustrated, although the term "computing device" shall also be taken to include any collection of machines (e.g., computers) that individually or jointly execute a set (or multiple sets) of instructions to perform any one or more of the methodologies discussed herein. The computing system 300 comprises a quantum processing unit 310 and a classical computing system 350, which may otherwise be referred to as a quantum compilation system. The quantum processing unit 310 is in communication with the quantum compilation system 350. The quantum compilation system 350 is arranged to perform classical computing operations and instruct the quantum computing system to prepare quantum states, and to perform measurements on those quantum states, according to instructions stored in memory. As described above with reference to Fig. 2, the quantum compilation system is configured to determine a factorised Hamiltonian for a physical system and generate a quantum circuit in the form of a unitary operator, which is transmitted to the quantum processing unit 310 for preparing a plurality of quantum registers of the quantum processing unit according to the generated quantum circuit. The quantum processing unit 310 comprises a control system 306 coupled to one or more registers 302 of quantum devices. In general, the quantum processing unit 310 uses registers of qubits 302, although one skilled in the art will appreciate that the present disclosure is also applicable to quantum computing systems that use other quantum devices, such as qutrits and qudits. Accordingly, it should be understood that any reference herein to qubits is applicable to any type of quantum devices that can be used to encode quantum information. The registers of quantum devices are controlled by a control system 306 having one or more classical processors. The control system 306 transmits control signals (e.g. RF pulses) to the registers of qubits 302 for performing operations on the qubits (including measurement operations) and receives measurement information from the qubits. The measurement information will generally be analogue data signals, although the analogue signals may alternatively be converted to digital signals before being transmitted to the control system 306 in some implementations (e.g. the registers of qubits 302 may be provided with one or more analogue to digital converters). The control system 306 may receive high-level instructions from the quantum complication system 350, such as instructions for preparing a predetermined state on the registers of quantum devices using a particular arrangement of quantum gates (e.g. the quantum circuit generated by the quantum compilation system). The control system 306 can then convert these high-level instructions (such as logic gates) into low-level qubit instructions (e.g. microwave pulses etc.), which may be in analogue format. As would be appreciated by one skilled in the art, the qubits may be physically implemented using, for example, photons, trapped ions, electrons, one or more nuclei, superconductor circuits and / or quantum dots. In other words, a qubit may be physically implemented in a variety of means, including the polarization state of a single photon; the spatial optical path of a single photon; two differing energy states of an atom or an ion; the spin orientation of a particle or plurality of particles such as a nucleus. The quantum processing unit 310 also comprises means for storing the qubits and maintaining the qubits in a suitable environment to allow quantum computation, for example means for supercooling the qubits. The qubits may be operated upon by one or more quantum circuits, formed by a suitable arrangement of quantum gates. A quantum gate acts on some number of qubits and can be thought of as the quantum analogue of a basic low-level instruction in a classical circuit such as a NOT or AND gate. Typically, quantum circuits are decomposed into a sequence of single and two-bit gates taken from a universal gate set along with state preparation and the measurement or read-out of the qubits. The results of the measurements are classical data that are then processed by a classical computer (such as quantum compilation system 350). Many quantum computers based on superconducting circuits and trappedions have already demonstrated all of the capabilities at a small scale that are required for a large quantum computing device. Possible implementations and methods of manipulation of the qubits in the quantum computer are now described. These implementations are by way of example only, and the skilled person will be aware of other methods of implementing a quantum computer. Birefringent wave plates may be used to manipulate the polarization state of a single photon, for example, to cause a linear polarization or horizontal polarization of the photon, signifying two distinct states of the photon. The qubits may also be implemented using a beam splitter. For example, the presence or absence of a photon along particular optical path can be implemented using a beam splitter that splits a beam of photons into two separate paths. The presence of the photon in either path represents two distinct states of the photon. Alternatively or additionally, two separate electronic energy states for an atom or ion can represent two separate distinct states for a qubit. For example, transition energies between these levels may correspond to the energy of electromagnetic radiation of a certain frequency and so the separate energy states of the atom or ion may be addressed using a source of radiation such as a laser or microwave emitter. Alternatively or additionally, the two distinct spin states (spin "up" and spin "down") of a particle or a plurality of particles, for example a nucleus, can represent the two distinct states of a qubit. Manipulations of nuclear spin may be implemented using a magnetic field using methods known to the person skilled in the art. Alternatively or additionally, superconducting electronic circuits may be used to create qubits. These systems are supercooled to below 100K and use Josephson junctions, a non-linear inductor that allows the creation of anharmonic oscillators. Anharmonic oscillators do not have evenly spaced energy levels (unlike harmonic oscillators) and therefore two of the states can be separately controlled, and used to store a qubit. The qubits are connected with microwave cavities and single and two-qubit gates can be performed using microwave signals. The quantum computing device 310 also comprises measurement means 304. The measurement means 304 may comprise measurement hardware and / or a measurement device. The measurement means comprises hardware configured to take a measurement from a state prepared by the control means 306 using the qubit registers 302. The example classical computing device 350 includes a processor 352, a main memory 354 (e.g., read-only memory (ROM), flash memory, dynamic random access memory (DRAM) such as synchronous DRAM (SDRAM) or Rambus DRAM (RDRAM), etc.), a static memory 356 (e.g., flash memory, static random access memory (SRAM), etc.), and a secondary memory (e.g., a data storage device), which communicate with each other via a bus. Processing device 352 represents one or more general-purpose processors such as a microprocessor, central processing unit, or the like. More particularly, the processing device 352 may be a complex instruction set computing (CISC) microprocessor, reduced instruction set computing (RISC) microprocessor, very long instruction word (VLIW) microprocessor, processor implementing other instruction sets, or processors implementing a combination of instruction sets. Processing device 352 may also be one or more special-purpose processing devices such as an application specific integrated circuit (ASIC), a field programmable gate array (FPGA), a digital signal processor (DSP), network processor, or the like. Processing device 352 is configured to execute the processing logic for performing the operations and steps discussed herein. The data storage device may include one or more machine-readable storage media (or more specifically one or more non-transitory computer-readable storage media) on which is stored one or more sets of instructions embodying any one or more of the methodologies or functions described herein. The instructions may also reside, completely or at least partially, within the main memory 354 and / or within the processing device 352 during execution thereof by the computer system, the main memory 354 and the processing device 352 also constituting computer-readable storage media. In general, the classical computer 350 instructs the control means 306 of the quantum processing unit 310 to prepare a particular state in the register of qubits 302. This may be done by transmitting, from the classical computer 350 (i.e. quantum compilation system) a quantum circuit (e.g. in the form of a unitary operator) to the quantum processing unit for the control means to implement using an arrangement of quantum gates. The control means 306 manipulates the qubits in the register 302 based on the instructions. Once the qubits have been manipulated such that the desired state has been constructed, i.e. once the registers of qubits have been prepared according to the quantum circuit, the measurement means 304 may be configured takes a measurement from the state, e.g. as part of a quantum computation algorithm such as quantum phase estimation. The quantum processing unit 310 then communicates the measurement result to the classical computer. The various methods described herein may be implemented by a computer program stored on a computer-readable medium. Figure 4 depicts a computer-readable medium 400 according to the present disclosure storing a computer program. The computer program may include computer code (e.g. instructions) 410 arranged to instruct a computer to perform the functions of one or more of the various methods described above. The computer program and / or the code 410 for performing such methods may be provided to a computing system 300 for execution. The computer readable media may be transitory or non-transitory. The one or more computer readable media 400 could be, for example, an electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, or a propagation medium for data transmission, for example for downloading the code over the Internet. Alternatively, the one or more computer readable media could take the form of one or more physical computer readable media such as semiconductor or solid state memory, magnetic tape, a removable computer diskette, a random access memory (RAM), a read-only memory (ROM), a rigid magnetic disc, and an optical disk, such as a CD-ROM, CD-R / W or DVD. The instructions 410 may also reside, completely or at least partially, within the memory 354 and / or within the controller circuitry 306 during execution thereof by the computing system 300, the memory 354 and the controller circuitry 306 also constituting computer-readable storage media. In an implementation, the modules, components and other features described herein can be implemented as discrete components or integrated in the functionality of hardware components such as ASICS, FPGAs, DSPs or similar devices. In addition, the modules and components can be implemented as firmware or functional circuitry within hardware devices. Further, the modules and components can be implemented in any combination of hardware devices and software components, or only in software (e.g., code stored or otherwise embodied in a machine-readable medium or in a transmission medium). Unless specifically stated otherwise, as apparent from the following discussion, it is appreciated that throughout the description, discussions utilizing terms such as "receiving", "determining", "comparing ", "enabling", "maintaining," "identifying," or the like, refer to the actions and processes of a computer system, or similar electronic computing device, that manipulates and transforms data represented as physical (electronic) quantities within the computer system's registers and memories into other data similarly represented as physical quantities within the computer system memories or registers or other such information storage, transmission or display devices. It is to be understood that the above description is intended to be illustrative, and not restrictive. Many other implementations will be apparent to those of skill in the art upon reading and understanding the above description. Although the present disclosure has been described with reference to specific example implementations, it will be recognized that the disclosure is not limited to the implementations described, but can be practiced with modification and alteration within the spirit and scope of the appended claims. Accordingly, the specification and drawings are to be regarded in an illustrative sense rather than a restrictive sense. The scope of the disclosure should, therefore, be determined with reference to the appended claims, along with the full scope of equivalents to which such claims are entitled. Supplementary information The following appendices are referenced above in the description of the unitary projector augmented-wave method and are intended to be supplementary but not essential information for understanding the various aspects of the present disclosure. Appendix A: Unitary PAW Setups The partial and pseudo partial waves are atomic like orbitals, ^(r) = rl Rnip(r)Yim(r), where n is the principle quantum number, I is the angular momentum, m is magnetic quantum number and p is an additional index enumerating the number of partial waves per angular momentum. While partial waves are solutions of the atomic Schrodinger equation, the pseudo partial waves c|)“ must satisfy the following properties: 1. Radial part of c|)“(r), Rnip(r) is a smooth function for r G Ba. 2. = Rnipir) for r G Ba. 3. At the boundary, we require 3lRnip(ra) = 3lRnip(ra), i G {0,1 ...,P — 1}. 4. O^j = 0 for any i, j, a. The polynomial approximation to the radial part of pseudo partial waves inside the augmentation spheres is used: R^r) = H Equation A. 1 where M coefficients are allocated to satisfy the orthonormality constraints (property 4). In order to find coefficients, cp, non-linear optimization is carried out until the constraints (1-4) are satisfied. In order to improve the smoothest of pseudo partial waves, for some elements the optimization procedure also attempts to remove high Fourier components of pseudo partial waves. Appendix B: Compression of virtual space with approximate MP2-natural orbitals In order to reduce the size of the virtual space, approximate MP2 natural orbitals are used. By definition, these orbitals diagonalize the approximate-MP2 density matrix (¾ + €e ~ 2e,) (ea + ec ~ 2c" tvirt. tgocc. Equation B. 1 This matrix can be calculated using 0(NpwN0CC N?iri) operations and 0(NpwN0CC Nviri ) memory requirements in the plane-wave basis set. Appendix C: Matrix elements in PAW formalism and plane wave basis set The constant term is Equation C. 1 where prime over the sum indicates that self-interaction energy is not included, and where introduced the following notation is introduced Equation C. 2 and Za = Za8(r - P^. The one-body matrix elements consist of the valence electron kinetic and external potential contributions, the interaction of valence electrons with core electrons, and the PAW correction: Equation C. 3 Where M?) =    +        ^(r) = Equation C. 4 L 1'1*2 Equation C. 5 where Dpq i t is the atomic orbital-pair density matrix: 2?” ■ ■ == ' (pa Ij / a,v> pq,il*2 \ * pU *i* Ui? Irq / Equation C. 6 The atomic compensation charges £aZ“(r) are introduced to ensure that the Coulomb potential created by atomic-centered densities are zero outside the augmentation spheres, which allows for separation of the original Hamiltonian into soft and atomic parts only. The same approach will be used below for two-body term. The atomic constants due to kinetic energy and external potential contribution, H“i , Hartree energy of valence and core electrons, V^, and the exchange energy between valence and core electrons, , are ’ lll2' = J- lv2 -iO - Equation C. 7 Equation C. 8 7-1 Equation C. 9 and ga(r) is a Gaussian function localized at the atom a, with angular and magnetic numbers L = (I, m). The two-body term is « *112*3*4 Equation C. 10 The matrix element is expanded into the sum of soft contribution and atomic PAW correction. The most computationally intensive part of such a Hamiltonian is the soft two-body contribution since it is a 4-rank tensor. A plane-wave basis set is used in order to derive an LCU decomposition of the two-body term. 1. Factorization of soft two-body part using plane waves The soft part of two body term can be expanded as: Kpqra — (ppg|P™) — Equation C. 11 Where / \ Ppg (G Ppq{~~%-) / \ / \ / \ = —--> w(r) = w(”r) = Wr) Equation C. 12 f v Wr) ” Pw(”r) f \ \ / \ = —-------, W = = Vgp,i (r)- Equation C. 13 It is convenient to introduce the orbital-pair densities, T]pqj, because their plane wave coefficients are Hermitian matrices with reflection and anti-reflection symmetry, respectively: C^j(G) = <WG), (G), Equation C. 14 unlike the plane-wave coefficients of pair-density matrix, ppq(r), which satisfy the following: Equation C. 15 Then, the plane-wave expansion of the Coulomb potential is used: Equation C. 16 To derive sm„ = 2 x £ V'(GK'Z^ j = l,2 G>0 Equation C. 17 with v'(0) = v(0) / 2, and v'(G) = v(G) otherwise. The zero G component is not omitted and instead Wigner-Seitz regularization is used. The soft two-body term can be rewritten then as follows: / \ 2 = 52 22 v'w (52 cp^g^} j G>0 \ pq / Equation C. 18 Cpq j^G} can be further diagonalized using the fact that for a given G it is a Hermitian matrix: r Equation C. 19 then the soft two-body can be rewritten using free-fermionic unitaries, Uj(G) as follows: Equation C. 20 2. Factorization of PAW two-body part In order to factorize the PAW two-body term, the fact that the atomic Coulomb coefficients f f are symmetric with respect to swapping of a pair of indices (j1; j2) and (j3, j4) is used. Unlike the electron repulsion integrals, f f are not positive definite and one cannot use Cholesky decomposition which is usually used in the single- and double-factorization methods. Instead one could use singular-value decomposition but in order to reduce the number of angles that need to be loaded from QROAM, regular eigendecomposition is used and later the sign of each term in the LCU is loaded. By introducing the eigendecomposition for ij <f2: -¾ <i4, k<l Equation C. 21 The PAW two-body term can be rewritten as follows: A-Y % Equation C. 22 Since the matrix V na. , . (Da - + isSU Equation C. 23 is Hermitian with respect to p, q, we can make use of additional eigendecompostion pq^ilh Equation C. 24 and introducing the free fermionic unitaries, f / “[2, leads to Equation C. 25 Appendix F: Block encoding of a square matrix A main feature of the double factorised Hamiltonian is that it contains terms that are a square X2 of another expression, see Equation III. 8. Starting from a block encoding of A / a. (with subnormalisation a), in principle the square can be implemented by multiplying block-encoded matrices, yielding a block encoding —A . Instead, when the double factorisation algorithm was conceived, the square was implemented by applying the second Chebyshev polynomial: T2(x) = 2x2 - 1 Equation F. 1 2 9 to A / ol. This results in a block encoding of — A — 1, whose constant shift can be computed classically. Compared to multiplication, the subnormalisation of the Chebyshev polynomial is better by a factor of 2, at the same query complexity. This factor of two has been taken into account when computing subnormalisation in Equation III. 10. Chebyshev polynomials can be implemented via qubitisation. Here we explicitly demonstrate T2(x). Let Equation F. 2 be a Hermitian block encoding of the Hermitian X / a, and the reflection around the coding subspace. Due to unitarity we have (X / oc)2 + BB^ = 1, such that A / a B\ (A / a B \ ((AM2 - BB^ A _ ((A / af + (.4 / a)2 A (^A / a)2 - 1 ■ W CJ L / F cj I ' ■ 7 \ ■ J k Equation F. 3 where calculation of the junk blocks of the final block encoding have been omitted. The reflection 5? must implement a —1 phase outside of the coding subspace, otherwise —T2(x) is implemented. To this end, a CZ has been added to the implementation of the reflection shown in Fig. 1.

Claims

1. A quantum computing system comprising:a quantum compilation system; anda quantum processing unit comprising a control system coupled to one or more registers of quantum devices;wherein the quantum compilation system is configured to:receive information indicative of one or more properties of a physical system; determine, based on the received information, a factorised Hamiltonian for the physical system, wherein determining the factorised Hamiltonian comprises performing a projector augmented-wave method using a plane wave basis;generate a quantum circuit based on the factorised Hamiltonian; and transmit the quantum circuit to the quantum processing unit;where the quantum processing unit is configured to:receive the quantum circuit; andprepare the one or more registers of quantum devices according to the quantum circuit.

2. The quantum computing system of claim 1, wherein each register of quantum devices comprises one or more qubits.

3. The quantum computing system of claim 1 or claim 2, wherein the one or more properties of the physical system comprise one or more of:a cell structure of the physical system;the chemical elements comprised within the physical system; andthe geometry of the physical system, including the position of atoms within the physical system.

4. The quantum computing system of any preceding claim, wherein the projector augmented-wave method is a unitary projector augmented-wave method that determines a factorised Hamiltonian with orthonormal eigenstates.

5. The quantum computing system of any preceding claim, wherein the quantum computation system is configured to generate the quantum circuit by decomposing the factorised Hamiltonian into a Linear combination of unitary matrices (LCU).

6. The quantum computing system of claim 5, wherein the quantum computation system is configured to generate the quantum circuit by block encoding the LCU decomposition of the factorised Hamiltonian into a unitary operator, wherein the unitary operator represents the quantum circuit, and wherein the quantum compilation system is configured to transmit the unitary operator to the quantum processing unit.

7. The quantum computing system of claim 6, wherein the quantum processing unit is configured to prepare the plurality of registers of quantum devices by constructing an arrangement of quantum gates according to the received unitary operator, and operate onthe registers of quantum devices using the arrangement of quantum gates to prepare the one or more registers in a predetermined state.

8. The quantum computing system of claim 7, wherein the predetermined state corresponds to the factorised Hamiltonian.

9. The quantum computing system of any preceding claim, the quantum processing unit is configured to prepare the plurality of registers of quantum devices by performing data loading on a first set of the one or more registers, wherein the data loading comprises operating on the first set of registers using a first sub-arrangement of quantum gates to encode the first set of registers with information corresponding to the factorised Hamiltonian.

10. The quantum computing system of claim 9, wherein the first sub-arrangement of quantum gates correspond to a classical lookup table comprising the information for the factorised Hamiltonian.

11. The quantum computing system of claim 9 or 10, wherein the factorised Hamiltonian comprises a plurality of terms each comprising a sign function, and wherein the quantum processing unit is configured to perform data loading on a dedicated register of the first set of registers to encode one or more qubits of the dedicated register with information corresponding to the value of the sign function for each of the plurality of terms.

12. The quantum computing system of claim 11, wherein the quantum circuit comprises a controlled-Z gate configured to act on the dedicated register after data loading on the dedicated register has been performed.

13. The quantum computing system of claim 12, wherein the controlled-Z gate is conditioned on the success of the state preparation of one or more other ones of the one or more registers.

14. The quantum computing system of any preceding claim, wherein the quantum processing unit is further configured to perform a quantum computing algorithm with the prepared register of quantum devices15. The quantum computing system of claim 14, wherein the quantum computing algorithm is a quantum phase estimation algorithm.

16. The quantum computing system of claim 15, wherein the quantum processing unit is configured to estimate one or more energy levels of the physical system using the quantum phase estimation algorithm.

17. A method for preparing a register of quantum devices using a quantum computing system of any preceding claim, the method comprising:receiving information indicating one or more properties of a physical system;determining, based on the received information, a factorised Hamiltonian for the physical system, wherein determining the factorised Hamiltonian comprises performing a projector augmented-wave method using a plane wave basis;generating a quantum circuit based on the encoded representation of the factorised5 Hamiltonian; andpreparing one or more registers of quantum devices according to the quantum circuit.

18. The method of claim 17, further comprising performing a quantum phase estimation algorithm using the prepared one or more registers.1019. The method of claim 18, further comprising determining one or more energy levels of the physical system using the quantum phase estimation algorithm.

20. A computer-readable medium storing instructions that, when executed by one or more15 classical or quantum processors, cause the one or more processors to execute the methodof any of claims 17 to 19.

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