Method of encoding a fermionic state of a fermionic system in a qubit state of qubits of a quantum processing device, computer system and computer program

Abstract

The present invention relates to a method of encoding a fermionic state of a fermionic system described by a plurality of N fermionic mode operators in a qubit state of a plurality of n qubits of a quantum processing device according to a target Fermion-to-qubit mapping, wherein said method comprises: deriving a plurality of tree-based Fermion-to-qubit mappings for the encoding, wherein each mapping of the plurality is derived on the basis of a pair of a tree and an instruction for deriving the mapping on the basis of said tree, said tree comprising a plurality of nodes and a plurality of links, wherein a link is either an edge connecting two nodes or a leg connected only to one node, wherein said instruction comprises a first instruction of assigning the fermionic mode operators, the qubits, and a plurality of Pauli operators, each Pauli operator being associated with a Hilbert space of a qubit of the quantum processing device, to the nodes and links of said tree to thereby obtain a labeled tree, 'and said instruction further comprises a second instruction for deriving the mapping on the basis of said labeled tree; estimating, for each of the Fermion-to-qubit mappings, a computational cost of encoding the fermionic state in the qubit state of the qubits of the quantum processing device according to said mapping; identifying the target fermion-to-qubit mapping as the mapping among the Fermion-to-qubit mappings of the plurality for which the computational cost fulfills an optimization criterion.

Description

[0001] Method of encoding a fermionic state of a fermionic system in a qubit state of qubits of a quantum processing device, computer system and computer program

[0002] The present invention is related to a method of encoding a fermionic state of a fermionic system described by a plurality of N fermionic mode operators in a qubit state of qubits of a quantum processing device according to a target fermion-to-qubit mapping, to a computer program and to a computing system.

[0003] Quantum computing has made significant strides in the past decade. However, achieving fault-tolerant quantum computing remains a challenging goal. Current quantum devices have limitations such as a small number of qubits, restricted qubit connectivity, and error-prone gates, making it difficult to execute deep circuits required for paradigmatic quantum algorithms [1]. Nevertheless, recent experiments have demonstrated the potential of today’s quantum devices, and have shown success in solving complex problems [2-5]. This potential offers valuable computational resources, particularly when combined with classical computing [6—9], especially in mitigating the detrimental effects of noise [10—14].

[0004] Among the diverse applications of quantum computing, simulating many-body Fermionic quantum systems with quantum devices presents an intriguing prospect, especially in computational chemistry [15-17]. This extends to fields like material science

[0018] and drug discovery [19—21], among others. Various approaches exist for addressing these quantum chemical problems which have a computational complexity that is prohibitive for classical computers, on quantum computers [22-24], Many approaches utilize the physical qubit state of the quantum device to represent the desired many-body Fermionic system. Properties of the system are then inferred through measurements of the qubit state [25, 26]. One such approach is the Variational Quantum Eigensolver (VQE)

[0027] , which approximates the qubit representation of a target Fermionic state, such as the ground state of a Fermionic Hamiltonian. The algorithm begins by deriving a qubit Hamiltonian from the desired Fermionic Hamiltonian using a Fermion-to-qubit mapping. Next, a parameterized quantum circuit, known as an ansatz, is designed. Finally, the circuit parameters are optimized using a classical optimizer, to minimize the energy of the current quantum state for the qubit Hamiltonian.

[0005] For near-term quantum devices, circuit noise robustness is crucial. VQE offers the potential to discover such circuits, characterized by a reduced presence of noisy entangling two-qubit CNOT gates compared to far-term approaches like qubitization

[0028] . The reduction of these gates is important as they take longer and have lower fidelities compared to single- qubit gates, contributing to computation errors

[0029] . One promising variant of VQE is the Adaptive Derivative-Assembled Pseudo-Troter (ADAPT) VQE algorithm. It starts with a reference state, like the Hartree-Fock state, and sequentially adds elements from a predefined candidate gate set, known as a pool, to optimize for the target state

[0030] . The choice of the operator pool significantly affects the convergence and circuit cost in qubit space. Often, pools originating from Fermionic systems are chosen to produce such circuits [30-32], The Fermionic pool

[0030] consisting of single- and double-excitation operations present in the Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz

[0033] , However, mapping Fermionic operators to qubits can result in highly non-local operations incurred from mapping indistinguishable Fermions to distinguishable qubits. To address this challenge, the Qubit-Excitation-Based (QEB) pool was introduced, which modifies elements of the Fermionic pool to disregard Fermionic anti-symmetry. This enables efficient implementation with a fixed number of CNOT gates for full connectivity, making it a leading method for state preparation [31 , 34, 35], Another approach, the qubit-pool, reduces CNOT gate requirements further by splitting QEB pool elements into individual 4-local Pauli strings

[0032] . The unitaries in the non-Fermionic pools do not have a straightforward representation in Fermionic space. Although a representation does exist, we refer to them as non-Fermionic pools. Additionally, some entangler-circuit approaches aim to minimize gate count by avoiding Fermionic operations altogether

[0036] .

[0006] With most approaches to solving Fermionic problems on quantum computers, a Fermion- to-qubit mapping is selected. The mapping encodes a many-mode Fermionic Hamiltonian and target state, as a multi-qubit Hamiltonian of Pauli operators and qubit state, |VQ). The choice of mapping is not unique, and different mappings result in different qubit states with varying challenges in preparation on the quantum device. Moreover, the interest lies not only in simulating \Wf), but also in determining physical properties via the expectation value, IO / IVJ-), of certain Fermionic observable operators Of. The chosen Fermlon-to-qubit mapping maps Of to its qubit counterpart, Oq, and the evaluation involves measurements on a physical quantum device, incurring measurement costs depending on the chosen mapping [37, 38].

[0007] A significant obstacle in implementing these Fermionic operations is the connectivity of the quantum device which is used to simulate the state. Limited connectivity devices, such as those based on superconducting qubits, can incur large circuit overheads when compared to full connectivity due to the necessity of SWAP gates needed to transpile the circuit to the device and due to the non-local nature of the mapped Fermionic operations. To address this issue,

[0039] introduces a versatile class of mappings and presents the Bonsai algorithm. This algorithm tailors the Fermion-to- qubit mapping to the device, reducing the SWAP gate overhead by aligning the mapping’s tree structure with the qubit connectivity graph. Subsequent research has built on this approach by employing the framework to encode double excitations within two-qubit subspaces to simplify the entanglement structure and lower the computational cost of VQEs and tensor-network simulations for chemical systems

[0040] .

[0008] The significance of Fermion-to-qubit mappings has thus fueled extensive research toward designing mappings beyond the traditional Jordan-Wigner (JW) transformation

[0041] . Many efforts are directed towards lowering Pauli weight, that is the number of qubits that the encoded Fermionic operators act on, from the O(N) scaling of JW to more favourable 0(log(7V)) scaling of Bravyi-Kitaev

[0042] where N is the number of modes simulated. Certain mappings have succeeded in reducing the number of qubits from the N -qubits required to simulate N -modes usually

[0043] . A substantial body of work has concentrated on reducing both these Pauli and qubit requirements in lattice models [44-49], Others have optimized measurement costs by introducing mappings with provably optimal Pauli weight

[0050] . Recently, the connection between ternary trees and mappings has been explored

[0051] .

[0009] Additional work involves the study of custom encodings to reduce circuit overhead in the context of VQE, as highlighted in

[0052] , In

[0053] , a general scheme that employs a brute force search over the space of encodings mapping from Majorana monomials to Pauli operators is explored. These mappings are also optimized for limited qubit connectivity settings, with resulting encodings providing fairly general optimality guarantees on solutions. However, due to the high computational cost of the brute force method, only small systems are feasible with a focus on symmetric lattice models. In

[0044] , the enumeration scheme between Fermionic modes and qubit operators representing said modes is explored to minimize various simulation costs with the JW encoding.

[0010] Due to these problems in the prior art, it Is therefore an object of the present invention to provide a method of encoding a fermionic state in a qubit state of qubits which allows for an efficient implementation on a quantum processing device.

[0011] According to a first aspect of the present invention, there is provided a method of encoding a fermionic state of a fermionic system described by a plurality of N fermionic mode operators in a qubit state of a plurality of n qubits of a quantum processing device according to a target Fermion-to-qubit mapping, wherein said method comprises: deriving a plurality of tree-based Fermion-to-qubit mappings for the encoding, wherein each mapping of the plurality is derived on the basis of a pair of a tree and an instruction for deriving the mapping on the basis of said tree, said tree comprising a plurality of nodes and a plurality of links, wherein a link is either an edge connecting two nodes or a leg connected only to one node, wherein said instruction comprises a first instruction of assigning the fermionic mode operators, the qubits, and a plurality of Pauli operators, each Pauli operator being associated with a Hilbert space of a qubit of the quantum processing device, to the nodes and links of said tree to thereby obtain a labeled tree, and said instruction further comprises a second instruction for deriving the mapping on the basis of said labeled tree; estimating, for each of the Fermion-to-qubit mappings, a computational cost of encoding the fermionic state in the qubit state of the qubits of the quantum processing device according to said mapping; identifying the target Fermion-to-qubit mapping as the mapping among the Fermion-to- qubit mappings of the plurality for which the computational cost fulfills an optimization criterion.

[0012] The present invention was originally filed in colour, and any reference to colour in the present copy relates to the colours as originally filed.

[0013] The target Fermion-to-qubit mapping allows to encode the Fermionic state in the qubit state of the qubits of the quantum processing device in a highly efficient manner as the associated computational cost fulfills the optimization criterion. In this way, computational problems, for example in the field of computational quantum chemistry, material science and drug discovery, that may otherwise not be efficiently solvable by the quantum processing device may now be solvable.

[0014] The quantum processing device comprises a plurality of n qubits and means to apply quantum gates to their state. Each qubit is associated with a Hilbert space C2with basis states described by state vectors |0) and |1). A pure state of the n qubits of the quantum processing device may thus be represented by a state vector in the Hilbert space (C2)®wwith complex coefficients qXo..J;w-1fulfilling

[0015] An N-mode Fermionic system in second quantization may be described in terms of N creation operators anc' N annihilation operators that satisfy the canonical anticommutation relations:

[0016] Mathematically, the / V-mode Fermionic system is equivalent to the / V-dimensional Fock space ?(CN), a 2N-dimensional Hilbert space spanned by the so-called Fock basis. The operators defined above allow us to define the basis as follows. First, the Fermionic vacuum \vacf) is defined to be the unique vector such that aj\vacf) = 0 for all j = 0, . . . . N - 1. The remaining basis elements can be constructed by considering all possible combinations of occupation numbers nj e {0, 1}: some fermionic operator / .

[0017] Creation and annihilation operators are not the only operators that may define the Fermionic space. It is also common to define an equivalent set of so-called Majorana operators MS1as

[0018] Such operators obey many useful properties, such as being unitary and self-adjoint. Additionally, one can show that they obey the anticommutation relation {mt. mj} = 250-l.

[0019] Any pure state of N Fermionic modes may thus be represented as wherein the complex coefficients cno^nN-1fulfill Xnanw-1=o|cna...njv_J2= 1 and are such that they reflect the fermionic anti-commutation relations (e.g. cn„n n„ - -cn n n„„ ). The Fermionic Fock space and the Hilbert space of N qubits, (C2)®w, are both 2W- dimensional Hilbert spaces. Thus, one may define a unitary mapping between them, called Fermion-to- qubit mapping. The Fermion-to-qubit mapping maps Fermionic states and operators to their representation in the Hilbert space and the operator space of qubits. A popular Fermion-to-qubit mapping is the Jordan-Wigner Transformation which maps Fock basis states to computational basis states. In the basis of Majorana operators, the Jordan-Wigner transformation is of the following form: for j=0,...,N-1. Here, Xk, Yk, and Zk are the Pauli -X operator, the Pauli -Y operator, and the Pauli -Z operator acting on the qubit qk.

[0020] Mapping Fermionic operators to qubits may result in highly non-local operators incurred from mapping indistinguishable fermions to distinguishable qubits. These non-local operators may make the implementation of the encoding on the quantum processing device computationally costly. Within this application, the computational cost may be indicative, for example, of the computation time, the number of entangling gates, etc. The computational cost is indicative of the computational cost to prepare the Fermionic state of the N modes as a qubit state of qubits of a quantum computer, or to measure the expectation value of a certain Fermionic operator Ofvia its representation in qubit space, <VQ|OQl^), wherein 0qis an encoding of the Fermionic operator Ofand is an encoding of the Fermionic wave function |4y) according to the Fermion-to-qubit mapping. The Fermionic state is not limited to a pure state, though, and the method also comprises the encoding of mixed fermionic states.

[0021] The computational cost may depend, for example, on the chosen Fermion-to-qubit mapping, the Fermionic state and / or Fermionic operator to be encoded and properties of the quantum processing device, in particular its connectivity which specifies between which pairs of qubits two-qubit entangling gate may be implemented directly, without the use of SWAP gates.

[0022] The computational costs may be measured by a cost function, wherein a value of the cost function is indicative of the computational cost. In one example, a low value of the cost function may be indicative of a mapping that is efficiently implementable on the quantum processing device. I.e., a low value of the cost function is desirable.

[0023] The method according to the present invention comprises deriving a plurality of treebased Fermion-to-qubit mappings for the encoding of the Fermionic state of the Fermionic system described by the plurality of N fermionic modes. In particular, N > 2. The quantum processing device comprises n > 2 qubits. In particular, the number n of qubits is at least the number N of fermionic modes, n > N in one example. In one example, n = N. I.e., the number of qubits is equal to the number of Fermionic modes. Tree-based Fermion-to-qubit mappings are a special class of Fermion-to-qubit mappings (see for example A. Vlasov, “Clifford algebras, spin groups and qubit trees”, arXiv:1904.00912v6, A. Miller et.al, “Bonsai Algorithm: Grow Your Own Fermion-to-qubit Mappings”, PRX Quantum 4, 030314 (2023), Z. Jiang et.al “Optimal Fermion-to-qubit mapping via ternary trees with applications to reduced quantum state learning”, arXiv:1910.10746v2). Terminology used throughout the present specification is also in line with standard terminology used in the cited documents.

[0024] > ;ed Fermion-to-qubit mapping is a mapping derived on the basis of a pair of a tree and an instruction for deriving the mapping on the basis of the tree. A tree is a special type of a graph. A graph G = (V, E) is an ordered pair, wherein V represents a set of nodes (also called vertices) and a set E of edges which are connections between two nodes (or simply, and edge is a pair of nodes). A tree is a graph in which any two vertices are connected by exactly one path, wherein a path is a sequence of edges which joins a sequence of distinct nodes. Within this application, a tree may also comprise at least one leg, which is a “half-edge” connected with only one node. Within this application, a link is either an edge or a leg.

[0025] The instruction comprises a first instruction of assigning the N Fermionic mode operators, the n qubits and a plurality of Pauli operators, each Pauli operator being associated with a Hilbert space of a qubit of the quantum processing device, to the nodes and links of the tree. Possible embodiments for such an assignment will be provided below. In particular, the Fermionic mode operators and the qubits may be assigned with the nodes and the Pauli operators may be assigned with the links in one example. Then, the number of nodes is in particular at least the maximum of the number of qubits and the number of Fermionic modes. In particular, when the node v is associated with the qubit qu, the Pauli operators Xu, Yu, and Zudefined on the Hilbert space of the qubit qumay be associated with the links, in particular the outgoing links, of the node v. In one example, the plurality of Pauli operators may comprise the plurality of Pauli -X, -Y and -Z operators defined on the single-qubit Hilbert spaces of the qubits of the quantum processing device. The assignment results in a labeled tree, wherein the nodes and links are associated with the Fermionic mode operators, the qubits and the Pauli operators. The instruction further comprises a second instruction for deriving the mapping on the basis of said labeled tree. Possible examples for said second instruction will be provided below.

[0026] Once the plurality of tree-based Fermion-to-qubit mappings for the encoding is derived, the computational cost of encoding the Fermionic state in the qubit state of the qubits of the quantum processing device according to the respective mapping is estimated. Ways how to perform this estimation will be provided below. Then, the target Fermion-to-qubit mapping is identified among the Fermion-to-qubit mappings of the plurality for which the computational cost fulfils an optimization criterion. In one example, the optimization criterion may be fulfilled when the computational cost is minimized for the respective Fermion-to-qubit mapping of the plurality.

[0027] In one example, at least some of the steps of the method according to the present invention may be performed by a classical computer. For example, the deriving of the plurality of Fermion-to-qubit mappings and / or the estimating of the computational cost and / or the identifying of the target Fermion-to-qubit mapping may be completely or partially performed by the classical computer.

[0028] In one example, the method further comprises a step of encoding the fermionic state in the qubit state l1!^) of the qubits of the quantum processing device according to the target Fermion-to-qubit mapping. In a further example, the method further comprises selecting a Fermionic operator Of, encoding the Fermionic operator in a qubit operator Oqaccording to the target Fermion-to-qubit mapping and deriving an estimator value of the expectation value of the Fermionic operator Of in the Fermionic state (Vj), by measuring the qubit operator in the encoded qubit state to thereby obtain

[0029] According to an expedient embodiment of the present invention, the method may further comprise generating a plurality of pairs of trees and instructions for the deriving of the plurality of mappings, said pairs being outputs of an iteration in which each iteration step accepts an input pair of an input tree and an input instruction as an input and outputs an updated pair of an updated tree and an updated instruction which is used as the input for the next iteration step, wherein the updated pair is generated on the basis of the input pair by an application of a transformation to said input pair, said transformation comprising a treechanging transformation applied to the input tree transforming said input tree into the updated tree and / or an instruction transformation applied to the input instruction transforming said input instruction into the updated instruction such that the estimated computational cost of the Fermion-to-qubit mapping derived from the updated pair fulfills an acceptance criterion, and terminating the iteration when a termination criterion is fulfilled. The method may also include selecting an initial pair of an initial tree and an initial instruction as the initial input of the iteration. The initial pair may be randomly generated in one example. In another example, the input pair may be provided by the user of the method based on well-thought considerations. The iteration may comprise deriving the Fermion-to-qubit mapping on the basis of the updated pairs in one example. The iteration may be completely or partially performed by a classical computer in one example. In this way, the plurality of tree-based Fermion-to-qubit mappings may be efficiently generated.

[0030] The updated pairs of trees and instructions which are outputs of the iteration are thus used for the deriving of the plurality of tree-based Fermion-to-qubit mappings. The iteration allows to efficiently explore the space of Fermionic-to-qubit mappings that are generated by the pairs of trees and instructions obtained as the updated trees and updated instructions of the iteration and allows to efficiently determine a target mapping with an optimized computational cost In one example, the acceptance criterion may be fulfilled when the computational cost of the mapping derived from the updated tree and updated instruction is smaller (or bigger) than the computational cost of the mapping derived from the input tree and the input instruction of the iteration step. However, the invention is not limited to this. The iteration is terminated when a termination criterion is fulfilled. In one example, the termination criterion may be a maximum number of iterations steps. In another example, the termination criterion may be fulfilled when the estimated computational cost is below a threshold value or fulfills a minimization criterion. In yet another example, the termination criterion may be fulfilled, when the change of the estimated computational cost between two subsequent iteration steps is below a threshold value.

[0031] The updated pair of the iteration is generated on the basis of the pair of the previous iteration step by an application of a transformation to said pair. Said transformation comprises a tree-changing transformation and / or an instruction transformation. The tree-changing transformation is applied to the tree and maps the tree of the previous iteration to the updated tree. The updated tree is such that it allows to derive a Fermion-to-qubit mapping for the N Fermionic modes. Possible examples of the tree-changing transformation are presented below. The instruction transformation is applied to the first instruction of assigning the Fermionic mode operators, the qubits and the Pauli operators to the nodes and links of the tree and / or to the second instruction for deriving the mapping on the basis of the labeled tree. Therefore, the instruction of the previous iteration is mapped to an updated transformation associated with the updated tree. The updated instruction is such that it allows to derive a Fermon-to-qubit mapping for the N Fermionic modes on the basis of the updated tree. Possible examples of the instruction transformation are presented below. In one embodiment, the treechanging transformation and the instruction transformation may be changed independently from each other by application of the tree-changing and the instruction transformation. In one expedient example, only the tree-changing transformation may be applied to the input tree in at least one, more than one, or all iteration steps. Then, the updated instruction is identical to the input instruction for the respective iteration steps. In another embodiment, only the instruction transformation may be applied to the input instruction in at least one, more than one, or all iteration steps. Then, the updated tree is identical to the input tree for the respective iteration steps. In a further embodiment, the tree-changing transformation may be applied to the input tree and the instruction transformation may also be applied to the input instruction in at least one, more than one, or all iteration steps. In further expedient examples, it may vary from iteration step to iteration step whether the tree-changing transformation, the instruction transformation or both are applied. In one example, a predetermined set of allowed treechanging transformations and a predetermined set of allowed instruction transformations may be selected by the user of the method, and the tree-changing transformation and the instruction transformation may be selected from the respective sets in the iteration steps. In one example, the selection may be a random selection according to a probability distribution.

[0032] By the application of the iteration, the set of all possible tree-based Fermion-to-qubit mappings or a subset thereof may be reachable, depending on the allowed transformations and the input pair.

[0033] In one embodiment of the present invention, obtaining the updated pair may comprise an application of a probabilistic optimization procedure, and in particular simulated annealing. Simulated annealing is a probabilistic optimization technique inspired by the annealing process in metallurgy. Starting from a solution it interactively explores potential solutions by introducing random changes. I.e. starting from the input pair as the solution, a random pair of a candidate tree and a candidate instruction is derived by an application of a random tree-changing transformation and a random instruction transformation randomly selected from a predetermined set of allowed tree-changing transformations and allowed instruction transformation, respectively. The computational cost Cinis estimated for the input pair, and the computational cost Ccandis estimated for the candidate pair. If the computational cost for the input pair, the candidate pair is identified as the updated pair. If, however, the computational cost for the candidate pair is larger than the computational cost for the input pair, the candidate pair is only accepted as the updated pair with a certain probability, in particular with probability wherein is a real parameter, also called “temperature”, at the iteration step frwhich may change with k. If the candidate pair is not accepted, a new candidate pair is created by the application of another randomly selected tree-changing transformation and another randomly selected instruction transformation. This probabilistic acceptance of worse solutions may allow to escape local minima. In a further embodiment of the present invention, the tree may be a directed rooted tree comprising n nodes, one of the nodes being a root node, each node comprises three outgoing links, and the tree comprises 2N legs, wherein said first instruction comprises, for each node, bijectively assigning one qubit with said node and bijectively assigning each of the three outgoing links of said node with Pauli operators defined on the Hilbert space of the one qubit assigned with said node, and said second instruction comprises constructing for each of 2N paths from the root node to the legs a Pauli string, wherein the Pauli operators of the Pauli string are the ones assigned with the links along said path, and a pairing instruction for expressing each of the N fermionic mode operators as a linear combination of two Pauli strings.

[0034] A directed tree has nodes with both, incoming and outgoing edges. For each node, the degree of a connectivity of the node refers to the number of incoming and outgoing edges. There is also an inward degree which corresponds to the number of incoming edges and an outward degree which corresponds to the number of outgoing edges. A directed rooted tree is one in which a single node is designated as the root node with an inward degree of 0. In such a tree, there are terminal nodes without any outgoing edges which are denoted as leaves. A leaf has an outward degree of 0. Every node, including the leaves, may be associated with at least one leg as introduced above. The leg is an outgoing “half-edge”. The tree according to the present embodiment comprises for each node three outgoing links, that is three outgoing edges or legs. In one example, the tree may be a directed rooted ternary tree which is a tree where the maximum outward degree of connectivity of all vertices is three. Then, for the tree of the embodiment, each leaf, that is each terminal node with no outgoing edges has three legs, each node with a single outgoing edge has two legs, each node with two outgoing edges has one leg, and each node with three outgoing edges has zero legs. As each node is bijectively assigned with one qubit, and there are three Pauli operators, namely Pte {XbYbZi} defined on the Hilbert space associated with each qubit q, (Xfis the Pauli-X, Ytis the Pauli-Y and Zi is the Pauli-Z operator acting on the qubit qi), the three Pauli operators are distributed among the three outgoing links of the node associated with said qubit.

[0035] According to the above embodiment, the tree comprises 2N legs (and has exactly 2N+1 legs for the case of the ternary tree). Therefore, the tree is such that there are 2N paths from the root node to the legs. I.e. , to each of 2N legs there is associated one path from the root node to the leg. The associated path comprises said leg and all the edges on the path from the root node to the leg. The second instruction comprises, for each path Pj of the 2N paths, constructing a Pauli string Sj =®(}uEpjPu, wherein Pudenotes the Pauli operators assigned with the edges and the leg along the path Pj. Identity operators acting on the qubits which are not along the path are omited in the following. According to the pairing instruction of the second instruction, each of the N Fermionic mode operators is expressed as a linear combination of two Pauli strings, i.e., at= Spl&+ Sp2ra, wherein p1 and p2 are two functions that map the index i of the fermionic mode operator atto first and second indices, p1 (!) and p2(i) of the 2N Pauli strings. The pairing instruction is such that the Fermionic anticommutation relations are fulfilled for the Fermionic mode operators.

[0036] When the plurality of pairs of trees and instructions is generated by the iteration according to one of the embodiments mentioned above, the tree-changing transformation and the instruction transformation may be such that all generated pairs of trees and instructions are a directed rooted tree and the associated instruction as explained above.

[0037] In one expedient embodiment, the quantum processing device may comprise n = N qubits, the tree may be a directed rooted ternary tree with N nodes, said instruction further comprises bijectively assigning each of the N nodes with one of the N fermionic modes, said pairing instruction further comprises assigning a pairing scheme with each node, said pairing scheme comprises a selection instruction for selecting, among the 2N constructed Pauli strings, two Pauli strings which act non-tri vially and differently on the qubit associated with said node, and a combination instruction specifying the linear combination of the two selected Pauli strings in the fermionic mode operator associated with said node.

[0038] Here, each node v may be associated with a triple (j , u, b), wherein j is the index of the Fermionic mode aj bijectively assigned with said node, u is the qubit index of the qubit qubijectively assigned with said node, and b specifies the combination instruction. All the constructed Pauli strings anticommute with each other, as they exhibit non-trivial differences at a single qubit, that is, there is exactly one qubit on which they both act non-trivially, but with a different Pauli operator. On all other qubits, they act either identical or only one of them acts non-trivially. Preferably, the 2N constructed Pauli strings are such that they do not contain the Pauli string that consists only of Pauli Z-operators.

[0039] When the tree is a ternary tree, each node v has exactly three links according to the above embodiment, and the three Pauli operators Xu, Yuand Zuassociated with the qubit quwhich is assigned with the node v are distributed among the three links. Then, the pairing scheme may be as follows:

[0040] The selection instruction may comprise the following instruction for every node v in the ternary tree: Starting from the node v assigned with the qubit qu, follow its outgoing link labelled with Xu. If the link is not a leg, keep traveling downward, away from the root, always along the links labeled with a Z, until a leg is reached. Denote this leg by s^. Start from the same node v again, and follow its outgoing link labelled with Yu. Again, if the link is not a leg, keep traveling downward, away from the root, always along the links labeled with a Z, until a leg is reached. Denote this leg by SyU\ The Pauli string which is defined by the path from the root to the leg is denoted with S (Uy. The Pauli string which is defined by the path from the root to the leg sx s X£u)is denoted with S Sy <«). The Pauli strings S Sx(«),S Sy («) may be represented as S SX(u) = are sets of qubits that S S*w and S (u) act non-trivially on below the qubit quassigned with the node v in the tree (i.e. , on the sy path to the leg, away from qubit qutowards the leg sxu\ respectively s^), and Guis a common Pauli string of the two Pauli strings S (U) and S comprising the Pauli operators associated sxsy with the edges on the path from the node assigned with the qubit quto the root node. The combination instruction associated with said node may then be as follows: For the fermionic mode operator associated with the node v, assign the two Pauli strings S <u),Swwith the Majorana operators m2j = aj + aj, m2j+1= either according to m2 / S wand

[0041] J sxm2j+i -S’ Sy («) (in the following also denoted as “combination instruction b=’+’ ”), or according to m2j — S (u)and m2j+1>-* S (U) (in the following also denoted as “combination instruction b- ”). The fermionic mode operator a}- is then obtained from the linear combination of the Majorana operators. Changing the combination instruction from b- +’ ” to b=’-‘ ” may also be understood as braiding of the Majorana operators.

[0042] In one example, the pair of the tree and the instruction may be such that the derived treebased Fermion-to-qubit mapping is a Product-Preserving Ternary Tree (PPTT) based Fermion- to-qubit mapping. These mappings possess the property of product preservation, transforming basis product states in the Fermionic space into qubit computational product states. This ensures the separability in qubit space for states such as the Hartree-Fock states. Another important feature is that the Fermionic vacuum is transformed to the all zero qubit state. This is particularly significant, as these states are the initial starting points for numerous quantum algorithms, including algorithms for quantum chemistry. The tree-based mappings m2t S («> sx and m2,-+1•-» s (U) , or m2 / « — 5 («) defined above are within the class of

[0043] PPTT mappings. Thus, they are particularly useful for various applications, including applications in quantum chemistry. When the plurality of pairs of trees and instructions is generated by the iteration as explained above, the transformation applied to the input tree and the input instruction may be chosen to always map the input pair to an updated pair such that the Fermion-to-qubit mapping derived from said updated pair is PPTT. For example, the treechanging transformation is such that the updated tree is always a directed rooted ternary tree in one example. In one example, the initial tree is also a directed rooted ternary tree.

[0044] In a further embodiment of the present invention, a connectivity of the qubits of the quantum processing device may be specified by a connectivity graph of the qubits, and the graph defined by the nodes and edges of the tree for deriving of the Fermion-to-qubit mapping may be a subgraph of said connectivity graph. Two qubits of a quantum processing device have connectivity when a qubit gate may be implemented between them directly, that is, without the use of SWAP gates. The graph defined by the nodes and edges of the tree for the deriving of the mapping may be understood as the tree for the deriving of the mapping without the legs. The connectivity of the qubits induces a connectivity graph, wherein the connectivity graph nodes correspond to the qubits of the quantum processing device and there is a connectivity graph edge between two nodes when the corresponding qubits are connected. This embodiment is particularly useful, as the Fermion-to-qubit mapping derived from such a tree may be implemented efficiently on the quantum processing device by use of the connectivity of the qubits. In one expedient example of this embodiment, the tree-changing transformation is connectivity preserving, i.e. , when the tree-changing transformation is applied to a tree which is, when its links are omitted, the subgraph of the connectivity graph, the updated tree is, when its links are omitted, also a subgraph of the connectivity graph. Then, the iteration allows to efficiently explore the space of Fermion-to-qubit mappings which are efficiently implementable on the quantum processor.

[0045] As mentioned above, there are embodiments wherein the method comprises generating a plurality of pairs of trees and instructions by application of the iteration. The iteration may comprise an application of a tree-changing transformation to the tree of the previous iteration or the initial tree. In one embodiment, the tree-changing transformation may comprise changing the number of nodes of the tree, and / or moving a leaf which is a node without an outgoing edge to an open end of a selected leg thereby turning said selected leg into a new edge and turning the edge originally connected with the leaf into a new leg, and / or for the case of a rooted tree changing a designation of the root node in the tree. These tree-changing transformations may be efficiently implemented by a classical computer.

[0046] In the following, moving the leaf may also be denoted as “leaf move”. Changing the designation of the root node may also be denoted as “root change”. In particular, the node selected as the new root node may be such that it has outgoing degree of at most two (i.e., it has at most two outgoing edges). Then, the path from the old root node to the new root node is identified, and the tree is updated so that child and parent designations of the nodes are swapped along the path so that the updated tree is obtained.

[0047] In yet another embodiment, said instruction transformation may comprise selecting first and second nodes, the first node being assigned with a first qubit and the second node being assigned with a second qubit according to the input instruction, and interchanging the assignment of the qubits such that the updated instruction comprises assigning the first node with the second qubit and assigning the second node with the first qubit. Le., when the first and second nodes Vi and V2 are selected, and the first qubit u is assigned with the node Vi and the second qubit u is assigned with the node V2 according to the input instruction, the updated instruction obtained after the application of the instruction transformation assigns the first node vi with the second qubit u and the second node V2 with the first qubit u.

[0048] In a further expedient embodiment, the instruction transformation may comprise selecting third and fourth nodes, the third node being assigned with a third Fermionic mode and the fourth node being assigned with a fourth Fermionic mode, and interchanging the assignment of the Fermionic modes such that the updated instruction comprises assigning the third node with the fourth Fermionic mode and assigning the fourth node with the third Fermionic mode. The third and fourth nodes may be identical to or different from the first and second nodes. This instruction transformation may also be denoted as “mode association swap” in the following.

[0049] I.e., when the third and fourth nodes V3 and v4are selected, and the input instruction assigns the third Fermionic mode aj with the third node V3 and assigns the fourth Fermionic mode ak with the fourth node v4, the updated instruction assigns the fourth Fermionic mode ak with the third node V3 and assigns the third Fermionic mode a, with the fourth node v4. I.e., when the third node V3 is associated with a triple (J3, U3, ba), and the fourth node v4is associated with a triple (j4, u4, b4), after the application of the above instruction transformation the third node v3is associated with a triple (j4, U3, bs), and the fourth node v4is associated with a triple (j3, u4, b4)

[0050] In a further embodiment, the instruction transformation may comprise selecting a fifth node and changing the combination instruction assigned with said fifth node such that the updated instruction assigns the changed combination instruction with said fifth node. The fifth node may be identical to one of the first, second, third, and fourth nodes, or it may be different from them. For the example above, the combination instructions b - “+” may be changed to b = and vice versa. This instruction transformation may also be denoted as “Majorana braiding change” in the following.

[0051] In a further expedient embodiment, said instruction transformation may comprise selecting a sixth node and interchanging the Pauli operators assigned with the outgoing links of said sixth node so that the updated instruction assigns the Pauli operators to the outgoing links in the interchanged way compared to the input instruction. The sixth node may be identical to one of the first to fifth nodes, or it may be different from them. In particular, the selected node is not a leaf in an expedient embodiment. This instruction transformation may also be denoted as “Pauli shuffle” in the following. For example, the node v may be selected. The node v has three outgoing links, denoted as h, k, h. Assume, as an example, that the qubit u is assigned with the node v, and that Xuis assigned with the linkh , Yuis assigned with the link l2, and Zuis assigned with the link b according to the input instruction. Then, the updated instruction obtained after the application of the instruction transformation may assign Yuwith I1, Xuwith l2, and Zuwith l3in one example, or it may assign Zuwith h , Xuwith I2, and Yuwith I3 in another example or similar.

[0052] The leave move, the root change the mode association swap, the Majorana braiding change and the Pauli shuffle may be efficiently implemented by a classical computer.

[0053] As will be shown below at least for the heavy hexagonal and 2D grid hardware graphs the leaf move, the root change, the Pauli shuffle, the mode association swap and the Majorana braiding change allow to convert any given PPTT Fermion-to-qubit mapping into another one using only connectivity-preserving leaf moves and the four other transformations. Thus, these transformations provide sufficient control to derive target Fermion-to-qubit mappings which are implementable with low computational cost on the quantum processing device.

[0054] In a further embodiment, estimating the computational costs may comprise estimating a number of quantum gates native to the quantum processing device and required for encoding the Fermionic state and the qubit state of the qubits of the quantum processing device. In one example, the quantum gates native to the quantum processing device may comprise the CNOT gate, and the computational costs may be estimated on the basis of the number of the CNOT gates required for the encoding. The number of CNOT gates is a crucial metric for analyzing the feasibility of applying quantum circuits on quantum hardware. CNOT gates typically take more time and introduce errors that are approximately 10 times higher than those of single qubit gates. In another expedient embodiment, the computational costs may be estimated on the basis of a Pauli weight of Pauli strings present in a qubit representation of the Fermionic state according to the Fermion-to-qubit mapping. The Pauli weight of the Pauli string is the number of (non-identity) Pauli operators in the Pauli string. This embodiment may be particularly useful in a case where the Fermionic state is expressed as a product of r-unitaries applied to a reference state, where uk(0) = exp (0krk) are unitaries parameterized by a parameter 0kand generated by Fermionic generators rk. This Fermionic circuit is subsequently mapped to the qubit space by an application of a Fermion-to-qubit mapping

[0055] M K) = nkefl^ |¥re / ) , (21 ) where Tkand |¥re / ) are the qubit generators and reference state obtained by the application of the Fermion-to-qubit mapping. Such states naturally arise in VQE calculations. Ways how to estimate the Pauli weight for fully-connected architectures and limited connectivity architectures are presented below.

[0056] In another expedient embodiment, the method may further comprise deriving a quantum circuit for encoding the fermionic state in the qubit state of the qubits of the quantum processing device according to the Fermion-to-qubit mapping, and estimating the computational cost for the encoding is on the basis of said quantum circuit. The quantum circuit comprises a plurality of quantum gates. In one embodiment, deriving of said quantum circuit may comprise use of a transpiler, like those in Qiskit or TKET. The computational cost may be estimated on the basis of the number of CNOT gates or other entangling gates in the circuit, or on the basis of a circuit depth of the quantum circuit in certain examples.

[0057] In a further embodiment of the present invention, the method may comprise deriving a target quantum circuit configured for encoding the fermionic state in the qubit state of the quantum processing device according to the target Fermion-to-qubit mapping and encoding the Fermionic state in the qubit state of the qubits by executing the target quantum circuit by the quantum processing device. In this way, the fermionic state may be efficiently encoded in the qubit state of the qubits. The method may further comprise encoding a fermionic operator of an observable of interest, like the energy, in an encoded qubit operator by use of the target fermion-to-qubit mapping, and inferring the expectation value of the fermionic observable for the fermionic state by measuring the encoded qubit operator.

[0058] In a further expedient embodiment, the method may further comprise approximating an eigenstate of the fermionic system by application of a quantum-classical variational algorithm, wherein each iteration step of the algorithm comprises defining a fermionic ansatz state by applying an operator selected from a predetermined pool of fermionic operators according to instructions obtained in a preceding iteration step of the algorithm to a fermionic reference state, deriving the target Fermion-to-qubit mapping for the fermionic ansatz state according to anyone of the above, encoding the fermionic ansatz state in the qubits of the quantum processing device according to the target Fermion-to-qubit mapping and estimating a value of an energy of the fermionic ansatz state by measuring a qubit operator which is an encoding of a fermionic operator associated with the energy according to the target Fermion-to-qubit mapping to thereby obtain instructions for selecting the fermionic operator for the next iteration step on the basis of the estimated value of the energy.

[0059] In one example, the variational algorithm may be the Variational Quantum Eigensolver or a similar algorithm. In one example, the quantum-classical variational algorithm may be the ADAPT-VQE algorithm. The pool of fermionic operators is a pre-defined set of operators. In one example, the energy may be minimized according to the variational principle. The choice of a pool significantly impacts both, the convergence of the ADAPT algorithm and the gates in the quantum circuit representation of the ansatz. In one embodiment, the fermionic pool comprises single- and double-excitation operators that preserve the spin and particle number of the resulting state. Another example is the Quantum Excitation Base (QEB)-pool which is derived by mapping the fermionic spin pool using the Jordan-Wigner transformation and removing the trailing Z strings. Explicit representations of these pools are presented below. However, the pre-determined pool of fermionic operators in this embodiment is not limited to the given examples within this application.

[0060] According to a second aspect for the present invention, there is provided a computer program comprising instructions, which, when the computer program is executed by a classical computer, causes the classical computer to carry out the steps: receiving a number N of fermionic mode operators, data representative of a fermionic state described by said N fermionic mode operators and a number n of qubits; deriving a plurality of tree-based Fermion-to-qubit mappings for the encoding of said fermionic state in a qubit state of the n qubits, wherein each mapping of the plurality is derived on the basis of a pair of a tree and an instruction for deriving the mapping on the basis of said tree, said tree comprising a pluraiity of nodes and a plurality of links, wherein a link is either an edge connecting two nodes or a leg connected only to one node, wherein said instruction comprises a first instruction of assigning the fermionic mode operators, the qubits, and a plurality of Pauli operators, each Pauli operator being associated with a Hilbert space of a qubit, to the nodes and links of said tree to thereby obtain a labeled tree, and said instruction further comprises a second instruction for deriving the mapping on the basis of said labeled tree; estimating, for each of the Fermion-to-qubit mappings, a computational cost of encoding the fermionic state in the qubit state of the qubits of the quantum processing device according to said mapping; identifying the target fermion-to-qubit mapping as the mapping among the fermion-to- qubit mappings of the plurality for which the computational cost fulfills an optimization criterion.

[0061] The number N of fermionic modes, the number n of qubits, and the data representative of the fermionic state may be received by the computer by a user input.

[0062] According to a third aspect of the present invention, there is provided a computing system comprising a quantum processing device and a classical computer, wherein the classical computer is configured to execute the computer program according to the second aspect of the present invention, and to derive a target quantum circuit configured for encoding the fermionic state in the qubit state of the quantum processing device according to the target fermion-to-qubit mapping, and said quantum processing device is configured to receive the target quantum circuit as an input and to encode the fermionic state in the qubit state of the qubits by executing the target quantum circuit.

[0063] According to a fourth embodiment of the present invention, there is provided a computing system comprising a quantum processing device and a classical computer, said computing system being configured to approximate an eigenstate of the fermionic system by application of a quantum-classical variational algorithm, wherein the computing system is operative to implement an iteration of the algorithm, wherein each iteration comprises defining, by the classical computer, a fermionic ansatz state by applying an operator selected from a predetermined pool of fermionic operators according to instructions obtained in a preceding iteration of the algorithm to a fermionic reference state, deriving, by the classical computer, the target fermion-to-qubit mapping for the fermionic ansatz state according to anyone of the above, providing the target fermion-to-qubit mapping to the quantum processing device, wherein said quantum processing device is configured to receive the target fermion-to-qubit mapping as an input, to encode the fermionic ansatz state in its qubits according to the target fermion-to-qubit mapping, to measure a qubit operator which is an encoding of a fermionic operator associated with an energy of the fermionic system according to the target fermion-to- qubit mapping to thereby obtain a measurement outcome, to provide said measurement outcome to the classical computer, and wherein the classical computer is further configured to estimate a value of the energy on the basis of the measurement outcome to thereby obtain instructions for selecting the fermionic operator for the next iteration on the basis of the estimated value of the energy.

[0064] In the following, the invention will be described in greater detail by way of example with reference to the drawings:

[0065] Fig. 1 is an infographic of the treespilation algorithm.

[0066] Fig. 2 Example of a mapping derived from a ternary tree. The enumeration of the vertices (v) corresponds to the qubits while the black lines (e) represent edges connecting two qubits and the red lines (L) represent legs which we label with a Majorana string Su. \Ne represent the Pauli labelling on the links (both edges and legs) based on their position: the leftmost, central, and rightmost links below node-u are labelled with Xu, Yuand zurespectively. This labelling is explicit along the rightmost path (0-3-9). Each leg in the tree is associated with a Pauli string by following the path from the root node (node-0, in this case) to the leg. The strings are generated as follows: each time a link with label P stemming downward from a qubit- u is crossed, the Pauli operator P acting on qubit-u is added to the string. The resulting string acts trivially on all qubits not visited along the path, while differs by only a single Pauli from any other string. To define the mapping, we remove the rightmost all-Z operator (S22= Z0Z3Z9in this case), and the remaining strings are paired into qubit modes according to the pairing algorithm outlined in

[0039] and in doing so we associate the Pauli strings Suwith Majorana operators m,. Pairings (p) are represented by the green lines in the figure. This process ensures the separability of the Fermionic product states in qubit space and maps the Fermionic vacuum to the zero qubit state. For example, the leg S18corresponds to the string S18= Z0-Z3X9, while S19= Z0Z3Y9. These strings are paired to represent the ith-mode, (X9+ iY9)Z0Z3. For clarity purposes, we omit the mode index and braiding flag from the node label.

[0067] Fig. 3 The basic tree transformations allow us to transition between different tree-based mappings, (a) depicts the deformation process of moving external leaves to free legs (L). This transformation may modify the underlying edges(e) of the ternary tree, as demonstrated by the removal of edge (2,6) and the addition of edge (1 ,6). Furthermore, the labelling of the edges (e) can be adjusted, as shown by the change from a Z label to a Y label for edge (3,7). (b) illustrates the ability to change the root node (old root node: vo; new root node: v7).

[0068] Fig. 4 Illustration of Connectivity Preserving (CP) and Non-Connectivity Preserving (NCP) updates for a 9-mode PPTT mapping on a 37-qubit heavy-hexagon processor (H; left part of the figure). The qubits (q) are labelled with the Arabic numerals 0-8. The underlying mapping tree for CP, (a), is made of edges (e) that are connected on the device targeted. For NCP, (b), the edges (e) of the tree do not exist on the device.

[0069] Fig. 5 Visualization of results tabulated in I.

[0070] Fig. 6 Here the energy convergence of the ADAPT-VQE simulations with respect to number of CNOT gates is displayed for the groundstate simulations across six molecules, considering full, sycamore, and eagle connectivity. For each molecule, six curves are presented: one for QEB- and qubit-pools, and one each before and after treespilation with the Fermionic- and Majoranic pools. Before treespilation, the ansatz is encoded using the Jordan-Wigner transformation. The dashed lines in the treespilation curves represent points where the optimized mapping from the previous point serves as the starting point for the algorithm, and the resulting CNOT count is plotted. Due to the stochastic nature of the qiskit transpiler used, the lines do not exhibit monotonic convergence.

[0071] Fig. 7 An example of the ternary tree (green nodes, (g)) which cannot produce any other ternary tree. The only node to which any leaf could be attached already has degree 4 in the tree, and degree 5 is not allowed.

[0072] Fig. 8 A visualization of the proof presented in in Theorem B.3. We start with the F2Q (Fermion-to-Qubit) mapping which is fully on W (top left), and we wish to obtain the F2Q mapping as on top right. This can be done in three steps as depicted below, where the green nodes (g) are nodes that are already in the correction location in the tree, and the orange ones (o) are yet-to-be-moved nodes.

[0073] Fig. 9 Figure (a) illustrates the topology of the IBM Washington device, and (b) is the Google Sycamore. CNOT gates between qubits (depicted as circles with Arabic numerals) are allowed only along the edges (solid lines between the qubits) of the graphs in these devices. Fig. 10 Convergence against ADAPT-VQE iterations to within 10'3Hartree energy error of the exact ground-state energy. The number of iterations is equivalent to the number of variational parameters in the ansatz. For LiH only two lines are discernable, as the qubit and Majoranic pools, and the Fermionic and QEB pools converge identically.

[0074] I. SHORT SUMMARY

[0075] Quantum computers hold great promise for efficiently simulating Fermionic systems, benefiting fields like quantum chemistry and materials science. To achieve this, algorithms typically begin by choosing a Fermion-to-qubit mapping to encode the Fermionic problem in the qubits of a quantum computer. In this application, we introduce ’’treespilation,” a technique for efficiently mapping Fermionic systems using a large family of favourable tree-based mappings previously introduced in the art. We use this technique to minimise the number of CNOT gates required to simulate chemical groundstates found numerically using the ADAPT-VQE algorithm. We observe significant reductions, up to 74%, in CNOT counts on full connectivity. For limited qubit connectivity- type devices such as IBM Eagle and Google Sycamore, we observe similar reductions in CNOT counts. In fact, in many instances, the reductions achieved on these limited connectivity devices even surpass the initial full connectivity CNOT count. Additionally, we find our method improves the CNOT and parameter efficiency of QEB- and qubit- ADAPT-VQE protocols, which are, to our knowledge, the most CNOT-efficient VQE protocols for molecular state preparation.

[0076] II. INTRODUCTION

[0077] Quantum computing has made significant strides in the past decade. However, achieving fault-tolerant quantum computing remains a challenging goal. Current quantum devices have limitations such as a small number of qubits, restricted qubit connectivity, and error-prone gates, making it difficult to execute deep circuits required for paradigmatic quantum algorithms [1]. Nevertheless, recent experiments have demonstrated the potential of today’s quantum devices, and have shown success in solving complex problems [2-5]. This potential offers valuable computational resources, particularly when combined with classical computing [6-9], especially in mitigating the detrimental effects of noise [10-14],

[0078] Among the diverse applications of quantum computing, simulating many-body Fermionic quantum systems with quantum devices presents an intriguing prospect, especially in computational chemistry [15-17]. This potential transformation extends to fields like material science

[0018] and drug discovery [19-21], among others. Various approaches exist for addressing these quantum chemical problems on quantum computers [22-24], with many utilizing the physical qubit state of the quantum device to represent the desired many-body Fermionic system. Properties of the system are then inferred through measurements of the qubit state [25, 26]. One such approach is the Variational Quantum Eigensolver (VQE)

[0027] , which approximates the qubit representation of a target Fermionic state, such as the ground state of a Fermionic Hamiltonian. The algorithm begins by deriving a qubit Hamiltonian from the desired Fermionic Hamiltonian using a Fermion-to-qubit (F2Q) mapping. Next, a parameterized quantum circuit, known as an ansatz, is designed. Finally, the circuit parameters are optimized using a classical optimizer, to minimize the energy of the current quantum state for the qubit Hamiltonian.

[0079] For near-term quantum devices, circuit noise robustness is crucial. VQE offers the potential to discover such circuits, characterized by a reduced presence of noisy entangling two-qubit CNOT gates compared to far-term approaches like qubitisation

[0028] . The reduction of these gates is important as they take longer and have lower fidelities compared to single-qubit gates, contributing to computation errors

[0029] . One promising variant of VQE is the Adaptive Derivative- Assembled Pseudo-Trotter (ADAPT) VQE algorithm. It starts with a reference state, like the Hartree-Fock state, and sequentially adds elements from a predefined candidate gate set, known as a pool, to optimize for the target state

[0030] . The choice of the operator pool significantly affects the convergence and circuit cost in qubit space. Often, pools originating from Fermionic systems are chosen to produce such circuits [30-32], The Fermionic pool

[0030] consisting of single- and double-excitation operations present in the Unitary Coupled Cluster Singles and Doubles (UCCSD) ansatz

[0033] . However, mapping Fermionic operators to qubits can result in highly non-local operations incurred from mapping indistinguishable Fermions to distinguishable qubits. To address this challenge, the Qubit-Excitation-Based (QEB) pool was introduced, which modifies elements of the Fermionic pool to disregard Fermionic antisymmetry. This enables efficient implementation with a fixed number of CNOT gates for full connectivity, making it a leading method for state preparation [31, 34, 35], Another approach, the qubit-pool, reduces CNOT gate requirements further by splitting QEB pool elements into individual 4- local Pauli strings

[0032] , The unitaries in the non-Fermionic pools do not have a straightforward representation in Fermionic space. Although a representation does exist, we refer to them as non-Fermionic pools. Additionally, some entangler-circuit approaches aim to minimize gate count by avoiding Fermionic operations altogether

[0036] .

[0080] With most approaches to solving Fermionic problems on quantum computers, a Fermion-to-qubit mapping is selected. The mapping encodes a many-mode Fermionic Hamiltonian and target state, | Kfr ), as a multi-qubit Hamiltonian of Pauli operators and qubit state, [’IQ). The choice of mapping is not unique, and different mappings result in different, qubit states with varying challenges in preparation of IfiQ) on the quantum device. Moreover, the interest lies not only in simulating |^f) but also in determining physical properties, ('Pf | Of |$f ), of certain Fermionic observable operators Of. The chosen Fermion-to-qubit mapping maps Of to its qubit counterpart, Oq, and the evaluation of ('Pql OqI'fq) involves measurements on a physical quantum device, incurring measurement costs depending on the chosen mapping [37, 38] .

[0081] A significant obstacle in implementing these Fermionic operations is the connectivity of the quantum device we use to simulate the state. Limited connectivity devices, such as those based on superconducting qubits, can incur large circuit overheads when compared to full connectivity due to the necessity of SWAP gates needed to transpile the circuit to the device and due to the non-local nature of the mapped Fermionic operations. To address this issue,

[0039] introduces a versatile class of mappings and presents the Bonsai algorithm. This algorithm tailors the Fermion-to- qubit mapping to the device, reducing SWAP gate overhead by aligning the mapping’s tree structure with the qubit connectivity graph. Subsequent research has built on this approach by employing the framework to encode double excitations within two-qubit subspaces to simplify the entanglement structure and lower the computational cost of VQEs and tensor- network simulations for chemical systems

[0040] .

[0082] The significance of Fermion-to-qubit mappings has thus fueled extensive research toward designing mappings beyond the traditional Jordan- Wigner (JW) transformation

[0041] , Many efforts are directed towards lowering Pauli weight, that is the number of qubits that the encoded Fermionic operators act on, from the O(N) scaling of JW to more favourable O(IogN) scaling of Bravyi-Kitaev

[0042] where N is the number of modes simulated. Certain mappings have succeeded in reducing the number of qubits from the W-qubits required to simulate IV-modes usually

[0043] . A substantial body of work has concentrated on reducing both these Pauli and qubit requirements in lattice models

[0083] Others have optimized measurement costs by introducing mappings with provably optimal Pauli weight

[0050] . Recently, the connection between ternary trees and mappings has been explored

[0051] .

[0084] Additional work involves the study of custom encodings to reduce circuit overhead in the context, of VQE, as highlighted in

[0052] . In

[0053] , a general scheme that employs a brute force search over the space of encodings mapping from Majorana monomials to Pauli operators is explored. These mappings are also optimized for limited qubit connectivity settings, with resulting encodings providing fairly general optimality guarantees on solutions. However, due to the high computational cost of the brute force method, only small systems are feasible with a focus on symmetric lattice models. In

[0044] , the enumeration scheme between Fermionic inodes and qubit operators representing said modes is explored to minimize various simulation costs with the JW encoding.

[0085] When simulating a Fermionic state we are free to choose the mapping of the Fermionic-based operations comprising said state. In this application, we introduce “treespilation” , a technique that leverages this understanding and extends the Bonsai algorithm

[0039] . The algorithm optimises a mapping of to prepare |$?) with a low CNOT count. To illustrate this approach, we optimize the encodings for numerically produced ADAPT- VQE ansatz, using Fermionic and introduced Majoranic pools on setups with full and limited connectivity. Comparing our approach to the non-Fermionic QEB and qubit pools, we observe that across the molecules considered, our method significantly outperforms these state-of-the-art approaches on setups with limited connectivity. When considering setups with full connectivity, on average, our approach shows improvement over using QEB and qubit pools, challenging the benefits of employing these widely used non-Fermionic operations in state preparation. Of the methods considered, we find the Majoranic pool combined with treespilation to be by far the most CNOT-efficient pool for state preparation on limited connectivity hardware. Specifically, we observe that the limited connectivity overhead is eliminated, and in certain cases, the CNOT count is reduced compared to the full connectivity ansatz in the JW encoding. Figure 1 illustrates the method. y

[0086] An W-mode Fermionic system in second quantization can be described in terms of N creation operators {«! QTQ1and annihilation operators {aQ^o1that satisfy the canonical anticommutation relations:

[0087] (1) (2)

[0088] Mathematically, the Af-mode Fermionic system is equivalent to the IV-dimensional Fock space a 2N- diiiieiisioiial Hilbert space spanned by the so-called Fock basis. The operators defined above allow us to define the basis as follows. First, the Fermionic vacuum |vacf) is defined to be the unique vector such that a,j |vacf) = 0 for all j = 0, . . . , N — 1. The remaining basis elements can be constructed by considering all possible combinations of occupation numbers n3€ {0, 1}: where f”» = n3f + (1 nQl for some fermionic operator / .

[0089] Creation and annihilation operators are not the only operators that, can define the Fermionic space. It is also common to define an equivalent set of so-called Majorana operatorsas(4) (5)

[0090] Such operators obey many useful properties, such as being unitary and self-adjoint. Additionally, one can show that they obey the anticommutatioii relation {rnt, mj} = 24Q1.

[0091] The above ways of defining Fermionic systems allow us to provide two equivalent forms of an IV-mode second- quantized Fermionic Hamiltonian: j j for coefficients h.ij, hijki-, Qj and c, The equivalence between these forms comes directly from the linear dependency presented in Eq. (4).

[0092] An important operation we consider in this application is the Majorana braiding transformation fop,

[0054] , This unitary swaps the roles of the fc’th and j’tli Majorana modes (up to a sign), leaving other Majoranas unchanged. From consideration of the Fermionic parity

[0055] , the unitary can be expressed as the Clifford operator: with the Majoranas transforming as:

[0093] (8)

[0094] Considering bT2j,2j+i:we seeit exchanges the role of Majoranas within Fermionic mode-y, that is rn>2j — > “Wj+i and TOaj+i — > rri2j- with creation and annihilation operators transforming as,

[0095] (9)

[0096] Under this particular transformation, Fock basis states are mapped to the same state with a phase shift, i.e., |non4 . . . nff j) — > ini \norii . . .njy-i). Importantly, the vacuum state is invariant for this transformation. For arbitrary exchanges, this is not the case, as we see when we consider swapping mi and m? that partially constitute modes 0 and 1. In this case, the vacuum state |00) is transformed into a nontrivial linear combination of the original Fock states

[0097] These features, wherein the Fock basis states are mapped to basis states and vacuum state preservation, hold significance for subsequent sections in which our objective is to perform Fermion-to-qubit transformations that maintain these features. Specifically, we aim to map Fock product states to qubit states while encoding the Fermionic vacuum state as the all-zero qubit state.

[0098] B. Fermion-to-Qnbit mappings

[0099] The Fermionic Fock space, jp (CM) , and the Hilbert space of N qubits are both 2JV-dimensional Hilbert spaces; thus, we can unitarily map between them. A natural unitary mapping is to map Fock basis states computational basis states of the qubits such that the occupation number of the J ’th Fermionic mode matches with the state of the y’th qubit

[0041] : i=0

[0100] This mapping is known as the Jordan- Wigner ( JW) transformation, and under it, the basis of Majoranas is mapped to qubit space as: k=0 k=O for j = 0, 1, . . . N — 1. Here and in the rest of the application, we denote Pj with P G {X, Y, Z\ for an operator that, acts as the Pauli operator P on the J ’th qubit, and as the identity on all others.

[0101] The JW mapping is part of the class of the so-called Majorana string Fermion-to-qubit mappings, as it associates a Majorana operator with a single Pauli string Si while preserving the commutation relations:

[0102] (13) for i, j 6 {(), . . . , 2N — 1}. This identification is implicitly used in other canonical mappings [41, 42, 50, 56], and we refer to the associated Pauli operators Sj as Majorana strings. To complete the mapping, these Majorana strings are paired into qubit mode operators and A,|. Then, the qubit vacuum state |vac)qis found by solving Aj |vac) for j e 1 . . . N. Note that many mappings exist outside this class [42, 57, 58]; however, this class proves particularly useful for quantum chemistry.

[0103]

[0104] FIG. 2. Example of a mapping derived from a ternary tree. The enumeration of the vertices corresponds to the qubits while the black lines represent edges connecting two qubits and the red lines represent legs which we label with a Majorana string Su. We represent the Pauli labelling on the links (both edges and legs) based on their position: the leftmost, central, and rightmost links below node-u are labelled with Xu, Yuand Zurespectively. This labelling is explicit along the rightmost path (0-3-9). Each leg in the tree is associated with a Pauli string by following the path from the root node (node-0, in this case) to the leg. The strings are generated as follows: each time a link with label P stemming downward from a qubit-u is crossed, the Pauli operator P acting on qubit-u is added to the string. The resulting string acts trivially on all qubits not visited along the path, while differs by only a single Pauli from any other string. To define the mapping, we remove the rightmost all-Z operator (S22 = Z0Z3Z9 in this case), and the remaining strings are paired into qubit modes according to the pairing algorithm outlined in

[0039] and in doing so we associate the Pauli strings Suwith Majorana operators mi. Pairings are represented by the green lines in the figure. This process ensures the separability of the Fermionic product states in qubit space and maps the Fermionic vacuum to the zero qubit state. For example, the leg Sig corresponds to the string Sis = Z0Z3X9, while S19 = Z0Z3Y9. These strings are paired to represent the ith-mode, a, =— > |(Xe + iYgjZoZs. For clarity purposes, we omit the mode index and braiding flag from the node label.

[0105] C. Ternary tree based Fermion-to-qubit mappings

[0106] We now present a useful class of Maj or ana-string mappings introduced in

[0039] , the product-preserving ternarytree (PPTT) based Fermion-to-qubit mappings. These mappings possess the crucial property of product preservation, transforming Fock basis product states into qubit, computational basis product states, i.e., |non,i • • ■ -> for Xi E {0, 1}. This ensures the separability in qubit space for states such as the Hartree- Fock. Another notable feature is that the Fermionic vacuum is transformed to the all-zero qubit state, i.e. |vac)f— > |0 . . . 0). This is particularly significant, as these states are the initial starting points for numerous quantum chemical algorithms, including UCCSD and ADAPT- VQE, and can thus be prepared without entangling gates. Additionally, the authors establish a connection between the tree structure of the mapping and the encoding of Fermionic mode occupancy information in qubits. The methodology encompasses well-known mappings such as the Jordan- Wigner

[0041] , Bravyi-Kitaev

[0042] , Ternary Tree

[0050] , and Parity

[0043] encodings. Furthermore, they introduce the Bonsai algorithm, which leverages the flexibility of this methodology to design mappings that reduce SWAP gate requirements by aligning the generating ternary tree of the Ferniioii-to-qubit mapping with the limited connectivity of the quantum device.

[0107] A PPTT mapping is uniquely defined by a labelled ordered Ternary Tree (TT). The ordered TT is a directed graph and a tree with N nodes such that there is a unique node with indegree 0, and all the nodes point to at most 3 other nodes. These nodes are usually labelled as left, middle, and right child, but for convenience, we will label them with X, Y, and Z, respectively.

[0108] Given such a labelled ordered Ternary Tree (TT), one can uniquely construct a PPTT Fermion-to-qubit mapping as follows. First, following the procedure presented in

[0039] , one can generate a basis of pairs of anticommuting Pauli strings from the tree that are later connected to particular Majoranas to complete the mapping. The procedure starts by adding “legs” to a ternary tree, which are labels associated with each vertex so that the total number of outward edges plus legs is equal to three for each node. Then, Pauli- A”, Pauli- T) or Pauli-H labels are assigned to each of the legs of the tree, so that each node has three edges or legs stemming from it with each of these labels. For such a tree with JV-nodes, there are (22V + l)~paths from the root to the legs. Pauli operators are associated with each of these paths by taking the tensor product of each Pauli label along the path where the Pauli acts on the qubit-u. All the Paulis generated this way pairwise anticommute, as they exhibit nontrivial differences at a single qubit, that is, there is one qubit at which they simultaneously have different a Pauli assigned such that neither is identity. This procedure generates (2N + l)-anticommuting Pauli strings that are to be associated with 22V-Majorana operators. This is achieved by removing the Pauli string that consists of only Z-Paulis, leaving us with 22V-strings. Then we apply the pairing algorithm outlined in

[0039] to form 2V~Pauli operator pairs called qubit modes. In doing so, we connect each string with a Majorana. Each qubit mode is associated with the qubit in the TT at which the corresponding Majorana string pair exhibits a nontrivial difference.

[0109] A bijection is then established between the 2V-qubit modes and the 2V-Fermionic modes of our system. Any association ensures the PP property, as proven in

[0039] . Moreover, as demonstrated in Sec. Ill A, we can interchange the roles of our Majoranas within mode pairs while maintaining this property. It is important to note that the Fermionic labelling of the Majoranas and modes involve Fermionic operations that do not alter the Pauli structure of the basis of strings generated by this construction.

[0110] Thus, we redefine a PPTT mapping by a set of IV-nodes, each labelled with a triple (j, u, b) e [N] x [Q] x {+, — }, where j is the index for the Fermionic mode, u is the qubit index of the Q-qubit machine, and b specifies the Majorana order in the mode pair. Note that each value of j and u can appear only once in the tree. For each node with a label (j, u, b), mode-j is assigned to the assigned pair of Pauli strings that exhibit nontrivial difference at qubit-w.

[0111] It can be shown that the Majorana operators assigned to a particular node for b =‘+’ correspond to

[0112] When we exchange the Majoranas, i.e. b =£W we get where Z^ and Zy **U^ are sets of qubits that SSr «r and S Sy « act non-trivially on below qubit u in the tree, and Guis a common Pauli string. This equation can be graphically understood as Gubeing the common path of and Sgpo from the root to qubit-u and sets Z^ and ZyU^ are qubits along the Z-paths bifurcating from the X- and V-legs of qubit-u. Note that Z^ A ZyU^ = 0 for any x, y.

[0113] The mapped creation and annihilation mode operators for b =‘+’ are thus, )

[0114] When the roles in the pair are switched, i.e. b =‘ — a complex phase term emerges as in Eq. (9).

[0115] The PP property is a crucial attribute of the mappings, and the pairing algorithm outlined in

[0039] is proven to ensure the PP property. To understand the significance of this property, consider the vacuum state in a PPTT mapping Fpp, i.e., |vac) —> |0). In Fpp, we associate Majoranas with Pauli strings, i.e. (mo, mi, . . . , m2W-i) (So, Si, . . . , S2N-I) where this notation means we associate m0O So, mo <-> Si and so on. Now envision a mapping FNPP that differs from Fpp by the assignment of Majoranas to Pauli operators. This deviation is captured by permutation p from the initial order in Fpp, i.e. (mo, mo, .. . , maw-i) (Sp(p) , Sp^ , . . . , 5p(2W-i))- The vacuum state in the new mapping can be related to the initial state by |0) = T |0) , for some unitary T. To identify this operator, we can break down the permutation to pairwise transpositions, p = (to, ti)(t2, tg) . . . (£&-— 1, ft). Each transposition can be interpreted as a pairwise exchange of the Majorana strings within qubit modes, where the Fermionic representation is given by Eq. (7). Thus, we can express the vacuum state in FNPP as,

[0116] Therefore, to prepare the vacuum trial state in a non-PP TT mapping, one must apply several of these pairwise braiding unitaries to the original PPTT vacuum state. In practice, there may be prohibitively many transpositions, and the product SiSj can be highly nonlocal, resulting in a large gate cost to merely implement the vacuum state. Likewise, these exchanges must be implemented to prepare the Hartree-Fock state defined as the Fermionic product state, Ilfego °'l lvacK where O is the set of occupied spin-orbitals. In an example where we seek to reduce the number of CNOT gates, we strictly consider the space of PPTT mappings.

[0117] D. State preparation

[0118] Quantum chemistry calculations using VQE-aiisat.z-based approaches involve the preparation of a parameterised quantum circuit on the qubits of a quantum device, followed by appropriate measurements to determine the desired chemical properties. On a quantum computer, we are usually restricted to applying unitary operations to construct this ansatz. In quantum chemistry, these militaries can be derived from chemical principles such as the single- and double-excitation operators in Unitary Coupled- Cluster techniques. These operations in particular are important as they can construct an ansatz to approximate an electronic wavefunction to arbitrary accuracy [59, 60] . We can express such a Fermionic state, denoted |Q / )>88 aproduct of r- unitaries applied to a reference state, \ipref), as: where «t(fi) = exp(0Tk) are unitaries parameterised by generated by the Fermionic generators rk. This Fermionic circuit is subsequently mapped to qubit space using a Ferniioii-to-qubit mapping, k where Tkand |Wref) are the qubit generators and reference state. The parameters Q, are then optimized on the quantum computer with a classical algorithm.

[0119] The ADAPT-VQE approach

[0030] has emerged as a promising avenue for quantum-based chemical state preparation. This technique involves iteratively applying parameterized qubit unitaries from a predefined set of operators, referred to as a “pool”, to a reference state while minimizing the energy following the variational principle. The choice of the pool significantly impacts both the convergence of the ADAPT algorithm and the gates in the quantum circuit representation of the ansatz.

[0120] A common choice is the Fermionic pool

[0030] , comprising single- and double-excitation operations that preserve the spin and particle number of the resulting state. The single and double excitation operators generate the pool:

[0121] For j, k G {0, . . . , iV — 1}. We can express these excitation elements in terms of a linear combination of products of underlying Majorana Fermions using Eq. (4). We define the Majoranic pool by taking each element of this linear combination and adding it to the pool separately for each element in the Fermionic pool: for u, v, r, s E {0, . . . , 2N . 1} .

[0122] Both of the aforementioned pools possess a proper Fermionic representation, thus ansatz constructed from them have a representation in Fermionic space given by Eq. (20). However, accounting for Fermionic anticommutation when mapping to qubits generally leads to highly nonlocal operators in qubit space. The Quantum-excitation-based (QEB) pool

[0031] aims to rectify this nonlocality by not implementing the exact, commutation relations of the operators. It is derived by mapping the Fermionic-spin pool using the JW transformation and removing the trailing Z-strings, yielding qubit operators:

[0123] The elimination of the parity-checking Z-strings is motivated by the existence of efficient circuit representations that can be achieved using a constant number of CNOT gates, resulting in at most 4- local gates. Due to the removal of the parity-checking strings, the Fermionic interpretation for this pool is unclear, thus it has been coined “pseudo- Fermionic”. Similar to how we did with the Majoranic pool, the QEB pool can be broken down further into the so-called qubit-pool

[0032] , where in this case the unitaries generated by single strings of Pauli operators of weight 2 or 4. The QEB and qubit, -pool are defined only for the Jordan-Wigner mapping, unlike the Fermionic-based pools.

[0124] IV. TREESPILATION: OPTIMIZING THE FERMION TO QUBIT MAPPING

[0125] A Fermion-to-qubit mapping is chosen to encode Fermionic operations and the Hamiltonian in preparing ansatz states. The choice of mapping is not unique, and it can significantly impact the efficiency of the resulting circuit. In the case of the Fermionic and Majoranic pools, we are entirely free to choose a mapping. In a typical workflow, however, a single mapping is selected, and the resulting circuit is optimized with a transpiler.

[0126] We can in general, exploit this freedom of mapping to optimise a given state. To do so, one would need the following elements:

[0127] 1. A target Fermionic state, denoted |Qy): The quantum state must be in a form that allows one to effectively find the corresponding qubit representation. Quantum states in Eq. (20) satisfy this requirement.

[0128] 2. A cost function C: For a given Fermionic state |Qy) and mapping F, it computes the quality of the qubit state C(|Qf) , F). In an example that focuses on minimizing the number of CNOTs for a transpiled circuit representing the state F(|Qy)), a natural choice is to use this number as the cost function. However, as we will show later, other cost functions can be used.

[0129] 3. A set of rules that allow us to transform a given F2Q mapping F into another F': This is essential for optimization algorithms. Various methods, such as simulated annealing, may require rules for generating new mapping candidates based on optimization history.

[0130] 4. An optimization algorithm: Equipped with the cost function defined above, it searches for high-quality Fermion-to-qubit mappings. Since the space of PPTT mappings is discrete and large, a natural choice is to use metaheuristic algorithms.

[0131] In this section, we develop an approach to systematically explore various Fermion-to-qubit mappings to find efficient circuit representations of jQ). Specifically, we will start by proposing multiple cost functions considered in this application. Then we will introduce the simulated annealing optimization algorithm considered in this application and present transformations allowing us to explore the space of PPTT mappings. Finally, we will present the treespilation algorithm.

[0132] A. Cost Functions

[0133] We now discuss the cost functions used to optimize the qubit representations, |ffg), of |Qy). In present-day quantum devices, the number of CNOT gates is a crucial metric for analysing the feasibility of applying quantum circuits on quantum hardware. These gates typically take more time and introduce errors that are approximately ten times higher than those of single-qubit gates

[0029] . Consequently, the cost functions employed in this context will prioritize minimizing the CNOT count. To effectively optimize, an appropriate cost function must be accurate and fast. a. Transpiler cost The ideal cost function involves processing a qubit state using a transpiler, like those in Qiskit

[0061] or TKET

[0062] , to calculate metrics such as depth or CNOT count. Although this approach offers high accuracy and optimization specific to the transpilation scheme, it might be too slow in practice. b. Pauli string cost Alternatively, one can consider a simpler cost based on the Pauli representation of the generators Tt that make up as defined in Eq. (21). On a fully connected architecture, implementing a Pauli string represented as P requires 2(k ~~~ 1) CNOT gates, where k is the Pauli weight, i.e. the number of non-identity terms in the string. However, on a limited connectivity architecture, implementing the same string necessitates 2(2n — k — 1) CNOT gates, where n is the number of nodes in the Steiner tree spanning qubits acted on by Paulis in the string

[0063] (i.e., a minimal subtree of the hardware architecture containing the qubits acted on by the Paulis in this string). For our purposes, this cost estimates the number of CNOTs by iterating over all the generators T;, that appear in the qubit state |d / g) as defined in Eq. (21), and then estimating the sum of the number of CNOTs required for all the Paulis that appear in the following generator. The problem of finding an optimal Steiner tree is NP-hard in general

[0064] , However, provided the underlying mapping tree is a subtree of the hardware architecture, i.e. it is a mapping defined by the Bonsai algorithm in

[0039] , the strings resulting from single and double excitation operations can be implemented optimally in polynomial time, as we prove in Appendix A. This cost function provides a robust measure for Majoranic pools but overlooks CNOT cancellations between adjacent gates. In the Fermionic pool, where substantial CNOT cancellations occur within generators, this simplistic approach less accurately tracks CNOT costs. Improvement may be achieved by considering the collective structure of Pauli strings within these generators.

[0134] B. Updating a tree-based mapping

[0135] When defining a PPTT mapping, we can adjust four degrees of freedom:

[0136] 1. Structure of the ternary tree.

[0137] 2. Choice of the root node.

[0138] 3. Association of Fermionic modes to qubit operators.

[0139] 4. Ordering of Majoranas in the Fermionic mode pairs.

[0140] The most basic transformation updating point 1 involves moving a leaf to a leg, effectively converting that leg into an edge of the ternary tree. In this way, we can change both the Pauli-labelling and the edge structure of the tree. The second degree of freedom in point 2 is the ability to switch the designation of the root node. Coupled with 1, we can deform between this class of mappings as we can from one ordered ternary tree to any other.

[0141] Item 3 relates to our freedom to choose a bijection between the Fermionic modes and the pairs of qubit Majorana strings representing them. Furthermore, 4 exploits the freedom to switch or braid the Majoranas within the pairs defined in Eq. (4). Unlike the tree transformations, these updates do not change the Paulis in the Majorana strings that express the Fermionic system, they only change how said strings are associated with Fermionic mode operators.

[0142] All these transformations result in mappings that fall within the vacuum preserving and PP categories. Notably, the braiding in 4 expands the space of possible PPTT mappings beyond the original presentation in

[0039] , where braiding was not considered. Figure 3 visually illustrates some of these degrees of freedom.

[0143] C. Optimisation

[0144] Using a cost function and transformation rules, we now employ an optimization procedure to find a mapping that minimises the cost of implementing |VT)- In this application, we adopt simulated annealing as the chosen scheme.

[0145] Simulated annealing is a probabilistic optimization technique inspired by the annealing process in metallurgy. It begins with a solution and iteratively explores potential solutions by introducing random changes. Once a new candidate x' is created from x, the algorithm evaluates the objective values E(x) and JE(X') and decides whether a new candidate should be accepted based on the difference between these objective values. If candidate x' has a smaller objective value, then it is always accepted. At the same time, if the objective value is larger, it is only accepted with a decreasing probability depending on the “temperature” t that changes during the optimization. The probability of accepting a worse solution at iteration k is exp{(E(x) — E(x')) where is the temperature at step k. This probabilistic acceptance of worse solutions allows the algorithm to escape local optima. This method applies to a wide range of complex optimization problems, especially those with non-convex or high-dimensional solution spaces where finding optimal solutions is challenging. Simulated annealing requires defining a way of applying “local” changes to a solution, and the aforementioned transformation rules offer exactly this. We note that more advanced optimization

[0146]

[0147] FIG. 3. The basic tree transformations allow us to transition between different tree-based mappings, (a) depicts the deformation process of moving external leaves to free legs. This transformation may modify the underlying edges of the ternary tree, as demonstrated by the removal of edge (2, 6) and the addition of edge (1, 6). Furthermore, the labelling of the edges can be adjusted, as shown by the change from a Z label to a Y label for edge (3, 7). (b) illustrates the ability to change the root node. algorithms like Tabu search may be used. Bringing this together we introduce treespilation, for optimisation of a Fermionic states’ mapping.

[0148] We now describe the methodologies used to produce the results presented in this application. On full-connectivity devices (FC), we initiate the annealing process with a randomly generated mapping tree. For limited-connectivity

[0149] (LC), we start with a Bonsai transformation of the device

[0039] so the tree under lying the mapping is connected on the device, and from it, we can determine the mapping from virtual to physical qubits.

[0150] For optimization purposes, we use the following adapted mapping tree transformations:

[0151] 1. Leaf move'. For FC, choose a random terminal node-v with three legs and attach it to the free leg of another node-w that is neither v nor its parent. For LC constraints, two choices are available: If starting from a Bonsai transformation, i.e., the mapping tree is connected on the device, the new qubit that, node-v assumes must be physically connected on the device to the qubit that parent node-w represents. We refer to this restriction as

[0152] Connectivity Preserving (CP), as the tree remains a connected tree on the underlying connectivity. Alternatively, one can perform Non-Connectivity Preserving (NCP) optimization, where this constraint is not applied, and the process proceeds as if it were FC. The CP and NCP optimization strategies are demonstrated in Figure 4

[0153] 2. root change'. A. node v different than the root with out-degree at most 2 is chosen as a new root. The path from root to v is identified, and the tree is updated so child and parent designations are swapped along the path

[0154] 3. Pauli shuffle: A random node with an out-degree of at least 1 is chosen and the Pauli operators associated with the links are changed

[0155] 4. mode association swap: Two nodes with labels (icu. b) and (i'puf b1) are chosen at random and their labels are changed to (i , u. b) and (?', «, , & ) respectively.

[0156] 5. Majorana braiding change: A node with label (i, u, b) is chosen at random, and the braiding b is changed to the opposite one, i.e., ‘+’ is changed to ’ and vice versa

[0157] While it is feasible to propose an alternative set of transformations, we demonstrate in Appendix B that for heavy- hexagonal and 2D grid hardware graphs, it is possible to convert any given PPTT F2Q mapping into another using C ti i P i

[0158] FIG. 4. Illustration of Connectivity Preserving (CP) and Non-Connectivity Preserving (NCP) updates for a 9-mode PPTT mapping on a 37-qubit heavy- hexagon processor. The underlying mapping tree for CP, (a), is made of edges that are connected on the device targeted. For NCP, (b), the edges of the tree do not exist on the device only connectivity-preserving leaf moves and the four other transformations. This proof establishes that these transformations provide sufficient control to derive a high-quality mapping. Additionally, one might anticipate a significant increase in possibilities with the growth of qubit numbers in the device. However, as detailed in Appendix C, the number of PPTT F2Q mappings on bounded-degree hardware graphs is roughly equal to the total number of PPTT F2Q mappings. Furthermore, the dependency on the number of qubits can effectively become negligible.

[0159] We iteratively update the mapping with simulated annealing, choosing to propose one of the random updates above with equal likelihood. For the Fermionic pool, we note the CNOT count of the generator is invariant to pairwise braiding so we do not use them here. As mentioned, we consider the CP and NCP search settings of the algorithm. Moreover, we also search in the restricted space of mappings generated by fixing the underlying mapping as JW and optimizing the assignment of qubit modes to Fermionic modes. We name this setting Mode Shuffling (MS).

[0160] To benchmark our approach, we compare it with Fermionic and Majoranic pools mapped using the JW encoding with gate compilation and transpilation being pool and mapping dependant. For the Fermionic pool, we utilize a circuit representation of excitation generators from

[0065] , known for its CNOT efficiency in the JW encoding to compile the ansatz initially. For MS with this pool, we benefit from this representation as we use the JW encoding. After optimizing the mapping beyond simple mode association permutation, we employ the TKET compilation pass from

[0066] for effective gate compilation.

[0161] For the Majoranic pool on FC and LC, we initially compile generators using the standard CNOT staircase approach. After treespilation on FC, we apply the same scheme. On LC with CP optimization, we use a Steiner-tree compilation

[0063] with an optimal Steiner-tree generation (details in Appendix A). The assignment from logical to physical qubits, necessary for the latter approach, is provided by the underlying mapping tree. For NCP, we compile strings using the standard CNOT staircase approach since the underlying tree is not connected on the device.

[0162] QEB and qubit pools are analyzed using the representations in

[0065] and the staircase approach, respectively. In all cases, after compiling pool elements in the ansatz, we use the Qiskit transpiler at optimization level 3 for circuit optimization. For all cases on LC, bar CP, the transpiler is used to find an assignment between virtual and physical qubits. Refer to Table V in the appendix for a summary of compilation and transpilation passes used.

[0163] We assess our technique with both the Pauli and Transpiler cost functions for the Fermionic and Majoranic pools. We use the same passes as described previously for the transpiler cost function. To mitigate the stochastic nature of the Qiskit transpiler and the simulated annealing optimization scheme, was run five times and the minimal cost was selected. With each call of the transpiler cost, we run the transpiler a single time.

[0164]

[0165] From analysing the number of generators in the ADAPT simulations, in each case, we see the Fermionic and Majoranic pools converge to 10““3precision in fewer or the same number of parameters compared to their non- Fermionic counterparts, the QEB and qubit pools respectively. This suggests that the Fermionic ground state can be more easily expressed using true Fermionic operations. Thus, we identify a potential tradeoff: using Fermionic operators allows for reaching a given precision in fewer iterations while using non-Fermionic counterparts may require more iterations but potentially fewer CNOTs.

[0166] TABLE I. CNOT counts for groundstate preparation circuits of various molecules obtained using the ADAPT-VQE algorithm with different choices of pools transpiled on both full and limited connectivity quantum devices. The number of qubits used is indicated in brackets beside the molecule label. Counts in bold correspond to the minimum of the rows. The initial (In.) columns display counts of transpiled results using the JW encoding, while the final (Fin.) column shows the results after treespilation and transpilation. Treespilation is only applied to ansatz with clear Fermionic representations, and as such, it is not used with qubit or QEB pools. For detailed information regarding molecular geometries, ADAPT-VQE convergence, device topologies, and transpilers used refer to the Appendix section D.

[0167] Ill Tab and IV, we present the Full Connectivity (FC) and Limited Connectivity (LC) results of our treespilation algorithm. For FC, we considered unconstrained and Mode Shuffling (MS) search settings with both Transpiler Cost (TC) and Pauli Cost (PC) functions. On LC, we explore MS, and Connectivity Preserving (CP) and Non-Connectivity Preserving (NCP) settings with both costs. Overall, we observe a significant reduction in the CNOT cost of implementing these states with treespilation.

[0168] Specifically, for the Majoranic pool on FC, the CNOT count reduced by an average of 49%, and with the Mode Shuffling (MS) search space, the reduction was 33%, with little difference between the Transpiler Cost (TC) and Pauli Cost (PC) functions. For LC, we see reductions of 51% and 49% for the Connectivity Preserving (CP) and Non-Connectivity Preserving (NCP) search spaces, respectively. This indicates that the CP space, corresponding to Bonsai mappings with pairwise mode braidings, contains high-quality solutions. We also found that the PC cost effectively represents the CNOT cost and can replace the more time-consuming TC. MS on LC performed worse, and a significant disparity between cost functions was observed, with TC and PC achieving reductions of 33% and 28%, respectively, likely due to cancellations not accounted for with the JW mapping.

[0169] For the Fermionic pool on FC, we observe an improvement of 25% and 19% for TC and PC, respectively. With MS, TC outperformed PC with a 27% and 25% reduction, respectively. On LC, we saw improvements of 28% and 25% for TC and PC, with no discernible difference between the CP and NCP search spaces. For MS, we see a 33% and 21% reduction in TC and PC, respectively. Now, the PC cost does not faithfully represent the resulting CNOT cost. With this pool, we find MS generally performs best. We partially attribute this success to the ability to use efficient circuit compilation passes with these mappings. The PC cost did not faithfully represent the resulting CNOT cost for the Fermionic pool on LC, likely due to the compilation scheme of the Fermionic Pauli generators not sufficiently accounting for LC constraints and CNOT cancellations.

[0170] In Figure 5 a, nd Table I, we present the best-performing setting and cost function of treespilation, showcasing the resulting CNOT counts. We achieved an overall average improvement of 28% and 49% for the Fermionic and Majoranic pools compared to the initial JW encoded ansatz on FC. The largest reduction was for the Majoranic pool with LiH. reaching three-quarters. For the Sycamore device, we report a respective average improvement of 34% and 52%, and for the Eagle, 37% and 52%.

[0171] For reference, we include counts of the QEB and qubit pool ansatz. On FC, the best-treespiled result exhibits an average improvement of 23% compared to the best performance of the QEB and qubit pools. For LC, the improvement is 44% and 51% for Sycamore and Eagle connectivities, respectively. We observe that CNOT counts on the Sycamore device are lower than those of the Eagle. This is pronounced in the case of the Fermionic and QEB pools, where the higher average degree of connectivity of each qubit makes transpiling these operators easier, as they demand a high degree of connectivity to implement efficiently.

[0172] We found that treespilation shifted the aforementioned tradeoff in favour of the Fermionic pools, resulting in improved performance compared to their non-Fermionic counterparts. One might attribute this to the notion that the Fermionic and Majoranic pools generally have fewer elements in their ansatz. However, this is not entirely the case as, for example, with LiH where the number of ansatz elements was identical, the Majoranic pool outperforms the qubit pool.

[0173] We analyzed the performances across the entire energy convergence against CNOTs of the ansatz in Figure 6. We opted for unrestricted, and CP search spaces with PC for the Majoranic pool on both FC and LC, as we noted satisfactory performance with these selections. Selecting CP ensures that the resulting mapping for quadratic- and quartic-Majorana-products, represented by the pool elements in qubit space, can be optimally implemented in polynomial time, as demonstrated in Appendix A. MS with TC performed best for the Fermionic pool, so we highlight this setting and cost. The treespiled Majoranic pool performed exceptionally well on LC, often outperforming the initial JW-encoded FC result. On the other hand, the qubit pool’s performance was underwhelming partially due to more generators in the ansatz. Another factor is the increased necessity for costly SWAP gates to interconnect the sparse generators on LC. This is in contrast to the treespiled Majoranic pool, where we use the mappings tree structure to reduce this sparsity, replacing it with Pauli operators on qubits where SWAPs would have been necessary. For example, with the CP setting the mapping tree is connected on the device with nodes representing physical qubits, causing the resulting Pauli string generators to act on ‘mostly connected’ qubits. Moreover, we cannot perform the efficient Steiner-tree pass, as we do not know the optimal layout of this pool ab-initio.

[0174] Overall, we find that using proper Fermionic pools can significantly reduce the number of CNOTs required for implementing ansatz compared to non-Fermionic pools. Treespilation provides an effective approach to achieve these reductions, particularly for the Majoranic pool on LC. However, more efficient compilation methods are needed for the Fermionic pool on LC.

[0175] VII, CONCLUSION

[0176] In this application, we have presented a Fermion-to-qubit mapping scheme to reduce the number of CNOT gates required to implement Fermionic states through quantum circuits. To achieve this, we defined a space of “good” mappings characterized by product-preserving ternary tree-based mappings combined with Majorana braidings within Fermionic mode pairs. Additionally, we introduced fundamental tree-mapping transformations allowing for the deformation of any mapping within this class, along with a procedure for optimization within this space. Furthermore, we establish cost functions that quantify a mapping’s CNOT cost within this space. Using these tools, we introduce the “treespilation” algorithm that tailors a mapping to the Fermionic representation of the ansatz while considering the potentially limited-connectivity architecture of the quantum device. Essentially, our method can be seen as a meta-compilation approach that augments the results from a given transpilation scheme by optimizing the qubit representation of a Fermionic state.

[0177] To illustrate our approach, we applied it to ansatz generated through statevector ADAPT- VQE simulations, representing ground-state approximations of molecules with different Fermionic-based operator pools. In summary, our method yielded encodings of Fermionic states that significantly reduced the number of CNOTs required to represent the qubit state on both full and limited connectivity quantum computers. For instance, in the case of LiH on full connectivity, we observed a remarkable 74% reduction in CNOTs compared to the initial Jordan-Wigner encoded ansatz. Additionally, when comparing our method to some similar, CNOT-efficient, noii-Ferniionic-based pools, we found that, on average, our approach significantly outperforms them.

[0178] In summary, our scheme presents a promising approach to reducing the CNOT requirements for implementing Fermionic ansatz on full and limited connectivity quantum computers. Using it, we show that we can essentially (a) Full connectivity (b) Sycamore (b) Eagle

[0179] No. CNOTs

[0180] FIG. 6. CNOT convergence of ADAPT-VQE groundstate simulations across six molecules, considering full, sycamore, and eagle connectivity. For each molecule, six curves are presented: one for QEB- and qubit-pools, and one each before and after treespilation with the Fermionic- and Majoranic pools. Before treespilation, the ansatz is encoded using the Jordan- Wigner transformation. The dashed lines in the treespilation curves represent points where the optimized mapping from the previous point serves as the starting point for the algorithm, and the resulting CNOT count is plotted. Due to the stochastic nature of the qiskit transpiler used, the lines do not exhibit monotonic convergence. eliminate much of the circuit burden usually incurred by the mapping of many-body Fermionic states to qubits. Furthermore, we anticipate that the tools introduced in this application may be utilized to optimize various aspects of Fermionic simulations, including measurement cost, state fidelity, circuit depth and so on. It’s important to note that while we use ADAPT-VQE as an example, our methodology is broadly applicable in optimizing the representation of Fermionic unitaries, such as chemical Hamiltonians. Further work may explore more advanced optimization schemes than simulated annealing. Here, we explore mappings where the number of modes is equal to the number of qubits simulated, however, it would be interesting to study the effect of introducing ancillary and removing qubits through symmetries.

[0181] >

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[0246] Appendix A: Polynomial-time algorithm for Steiner-based implementation of Pauli exponentiation

[0247] Using the results from

[0063] , one can find an efficient implementation of arbitrary exponentiation exp(— itF) for any real t and any Pauli string P on limited connectivity defined through a connected graph G = (V, E). First, we apply one-qubit gates to transform the operation exp(— itF) into exp(— itPz), where Pz is a Pauli string defined over {I, Z}. The transformed Pauli, Pz, acts non-trivially on the same set of qubits Vp C V as P. We proceed then exactly as described in

[0063] . A Steiner tree T = (VT, ET) of Vp over G is found, which is the minimal in the number of edges subgraph of G such that Vy D Vp. This generates the minimal set of qubits that need to interact to implement exp(— itPz). Then, the following operations are applied iteratively:

[0248] 1. we choose a particular leaf v of the tree and a vertex w connected to it,

[0249] 2. we apply CNOT with control on v and target on w; if w $ Vp, in addition, we apply CNOT with control on w and target on v,

[0250] 3. we remove v from Vp, {v, w} from ET, and we repeat the above steps.

[0251] The procedure ends when we are left with just one node v', on which we apply the rotation exp(— Finally, we uncompute all the steps done before the single-qubit rotation. The procedure requires 2(2Vp — Up — 1) CNOTs.

[0252] Thus, the difficulty of finding a CNOT efficient circuit is reduced to the problem of finding the Steiner tree, which is known to be NP-hard

[0064] . However, if the ternary tree of PPTT-based F2Q is a subgraph of the connectivity graph G, one can show that the problem of finding Steiner trees for any product of a Pauli string resulting from the product of 2 or 4 Majoranas can be done efficiently. These cases are important as the resulting strings compose the Fermionic single- and double-excitation operations described in the main text and also form the Pauli strings in the second quantised Hamiltonian. The rest of the section is dedicated to formally proving this fact. We start by introducing a variant of the Steiner tree problem. We adopt the notation G[W] for the induced subgraph of G generated by the vertex subset V .

[0253] Definition V C V be a at least one 1. G|W] has at most two connected components, or

[0254] 2. G[lz / ] is disconnected and there is exactly one vert, ex v <E ¥ \ ¥' s.t. the G[¥' U {»}] is a connected component

[0255] Let us start by showing that it is easy to solve the PPTT F2Q STEINER TREE PROBLEM exactly.

[0256] Theorem A.l. PPTT F2Q STEINER TREE PROBLEM is in P.

[0257] Proof. Suppose that G[F'] is connected. Then it is enough to find any spanning tree of the graph, which can be done in polynomial time.

[0258] Suppose that G[F'] consists of two connected components, defined over vertex sets V{ and Vf. Let (v, ly, . . . , Vk, w) be the shortest path over («,w) € V{ x Vfi Such a path can be found by applying e.g. Floyd- Warshall algorithm that runs time, and exhaustively searches for the smallest distance which takes O(|Fj'| • IFJ’I) = O(|F|2). Note that Vi G V \ V', as otherwise shortest path could be found that connects vertices from Vf and Vf. Note that adding edges from this path to G[F'] makes a new connected graph, for which we can find a spanning tree efficiently. By the definition, such a graph is the Steiner tree of V . On the other hand, one cannot hope for finding a smaller Steiner tree as it would contradict the chosen path (v, vi , . . . to be the shortest path.

[0259] Finally, if G[V'] is disconnected up to one vertex v, it, is clear that the spanning tree of G\V U {v}] is a minimal Steiner tree for V . >

[0260] In addition, we can show that adding edges to the graph G does not make the problem harder.

[0261] Theorem A.2. Suppose we’re given a PPTT F2Q STEINER TREE PROBLEM defined, over graph G = [V, E) and set of terminals V' C V. Then, adding any new edge, to E makes it still an instance of PPTT F2Q STEINER TREE PROBLEM.

[0262] Proof. Let G' be the new graph formed by adding a new edge to G. It. is enough to show that G' tV'] satisfies the same properties as G[F?] required by the definition of PPTT F2Q STEINER TREE PROBLEM.

[0263] If G[F] has at most two connected components, then the same can be said for G'[F']. Furthermore, if there is a unique vertex v s.t. G[VfU {«}] is connected, adding an edge will either make G'[V'] a connected subgraph which makes the problem of the first type as introduced in the definition of PPTT F2Q STEINER TREE PROBLEM, or Gfi¥fU {v}] is connected. >

[0264] The practical implication of the theorem above is that demonstrating the polynomial complexity of finding a Steiner tree on the ternary tree implies its polynomial complexity on the hardware connectivity graph. It, is essential to note that this doesn’t guarantee the Steiner tree will be the same, as increasing the number of edges in the hardware connectivity graph may allow finding a smaller Steiner tree.

[0265] We conclude this section by demonstrating that, for any Pauli string representing the product of 2 or 4 Majoranas, solving the Steiner tree problem can be accomplished in polynomial time. The proof will be presented by establishing that the Steiner problem will conform to the form defined in PPTT F2Q STEINER TREE PROBLEM. We begin with the product of 2 Majoranas, which can be shown to consistently form a connected path on the ternary tree in qubit space.

[0266] Theorem A.3. Let the hardware connectivity graph G be a connected graph. Let there, be. an arbitrary PPTT F2Q defined, through a ternary tree which is a subgraph of G. Then for any product of two strings generated by the PPTT mapping, the qubits on which the products act non-trivially induce a connected component on G.

[0267] Proof. Let 5j and S2be two Majorana strings (i.e., Pauli strings representing Majorana operators) generated by the PPTT mapping as described in the theorem statement. Each is constructed by taking a path from the root to a leg (tq = root, . . . , Vk, I), where Vk is the last node to which the leg I is assigned. This path defines each Majorana string. For each node Vi with the label (j, u, c), we take the tq+i-th Pauli label P G {X, Y, Z} and apply it to qubit j. The product over xq, . . . ,Vk (for the last one, we take the leg’s Pauli operator) defines the Pauli string.

[0268] Now, suppose that ay is different for both Majorana strings, indicating that the strings diverge at the root 'ip. This implies that there is no shared ternary tree node for the strings except for the root. In addition, the Pauli operator acting on the root is different. Since for any two non-identity Paulis Pj and P-2, we have P1P2 fi I, the product of these two Majorana strings will be a path of non-identity Paulis acting on qubits from the leg of one string, through the root, to the leg of another string. Such a Pauli operator acts on qubits forming a connected path on the mapping’s underlying tree, and hence a connected path on G.

[0269] Note that if i,y is the same for both strings, then there must be a node vm(possibly a leaf) in the path at which the strings diverge. In this case, the product of those strings does not include the root, as we act with the same Pauli on it, and on all the nodes up to vmexclusively. However, in this case, the strings form a connected path from one leg to another through vmto the leg of another string. In this scenario, we can also observe that the qubits form a connected (possibly one-node) path.

[0270] >

[0271] Note that in the proof, we utilized the fact that if two Majorana strings diverge at any point, they form a connected component. This is because the two strings are identical up to the point of divergence and cancel out when multiplied together. This fact will be frequently used in the following theorem for the product of 4 Majorana strings. We adopt the notation A ex B if A = cB for some complex number c.

[0272] Theorem A.4. Let the hardware connectivity graph G = (V, E) be a connected graph. Assume there is an arbitrary PPTT mapping defined on a ternary tree that is a subgraph of G. Let S be the product of any 4 Majorana strings. Then, finding the Steiner tree over G for qubits on which S acts non-trivially can be done in polynomial time.

[0273] Proof. The proof will proceed by showing that finding a Steiner tree on the ternary tree is a PPTT F2Q STEINER TREE PROBLEM.

[0274] We will prove the statement by considering several scenarios in which Majorana strings will diverge at different nodes and in various combinations of directions.

[0275] 1. Majorana strings leave the root in all X, Y, Z directions: In this case, the root is a terminal as PPP'P" oc P for any pairwise different non-identity P, P', P". With this example, all the nodes that are on the paths of Majorana strings leaving into P' and P" directions form a connected path that passes the root and are terminals. On the other hand, for Majorana strings leaving into P direction, the case reduces to what was observed for the product of two Majorana strings analyzed in Theorem A.3. This means that all the nodes up to a diverging node (excluding the diverging node and the root) will not be terminals, and all the other nodes from these two Majorana strings will be terminals. Eventually, we will obtain a single connected component if the Majorana strings diverge at the P-children of the root, or they will form independent connected components if they diverge later, reducing the problem to PPTT F2Q STEINER TREE PROBLEM.

[0276] 2. Majorana strings leave the root in pairs in two directions: In this case, the root is not included as PPP'P' = I for any Paulis P, Pf. Two pairs of Majorana strings will always form two disconnected components, following the reasoning in Theorem A.3 for each pair independently, again reducing the problem to PPTT F2Q STEINER TREE PROBLEM.

[0277] 3. Three Majorana strings leave the root in one direction, and the 4th one leaves in another direction: In this case, the root is always a terminal, as PPPP' I for any P, P' different than identity. Similarly, all the nodes on the path before the next diverging point for the three Majorana strings will be terminals until they diverge. At the diverging qubit, one of the two possibilities can occur:

[0278] (a) All Majorana strings leave in different directions: The diverging node is not a terminal as XY Z oc I, however, all other nodes are terminals, forming in total 4 connected components including the one already created with the root. In this case, adding the diverging node will make the induced graph connected, which reduces the problem to PPTT F2Q STEINER TREE PROBLEM.

[0279] (b) Two Majorana strings leave in one direction, and the 3rd Majorana string leaves in another direction: In this case, the diverging node is a terminal as PPP' = P' for any non-identity P, P' , and so are the nodes from the 3rd Majorana string. The two Majorana strings going in the same direction might form the second connected component if they do not diverge at the children of the diverging node. Since eventually we have at most two connected components, the problem reduces to PPTT F2Q STEINER TREE PROBLEM.

[0280] 4. All Majorana strings leave in the same direction: In this case, neither the root nor the nodes on the path before the first diverging node are terminals as P4= I for any Pauli P. Then, the analysis reduces to one of the cases above except instead of the root, the diverging node is a starting point.

[0281] As shown in the analysis above, in all cases, the problem reduces to PPTT F2Q STEINER TREE PROBLEM. By Theorem A.2, the result generalizes to a hardware connectivity graph G after adding missing nodes from G that are not in the ternary tree, allowing finding the Steiner tree in polynomial time thanks to Theorem A.l. >

[0282] The two last theorems above can be concluded with the following lemma.

[0283] Theorem A.5. Let S be a Pauli string, representing a. product of 2 or 4 Majorana strings defined over k qubits, coming from the PPTT F2Q mapping with the ternary tree as a subgraph of the hardware connectivity graph. One can find an optimal Steiner tree with n > k nodes in the hardware connectivity graph in polynomial time, which, in turn, allows the implementation of exp(— ?tS) for any real t with only 2(2n — k — 1) CNOTs.

[0284]

[0285] FIG. 7. An example of the ternary tree (green nodes) which cannot produce any other ternary tree. The only node to which any leaf could be attached already has degree 4 in the tree, and degree 5 is not allowed

[0286] Proof. The fact that the Steiner tree can be found in polynomial time comes directly from Theorems A.3 and A.4. The number of CNOTs comes from the implementation proposed in

[0063] , >

[0287] Appendix B: Reachability of PPTT F2Q mappings

[0288] In this section, we analyze whether the PPTT F2Q transformations, as presented in Sec. V, allow transforming any hardware; connectivity-preserving PPTT F2Q to any other hardware connectivity-preserving PPTT F2Q. Starting from now, we will assume that we are given an undirected graph G = (VI E) representing the quantum hardware connectivity, and the ordered ternary tree (OTT) used for any considered PPTT F2Q is a subgraph of G without explicitly stating it.

[0289] For general graphs, it can be shown that transforming one F2Q mapping into another is not always possible with the transformations from Sec. V. Consider a full OTT. For sufficiently many nodes (at least 16), there is a node v that has both a parent and all three children, none of which are leaves. Now, suppose that the graph G is the underlying ternary tree, and to v we attach a long path tree only. The aforementioned F2Q mapping, as well as, for example, the JW mapping defined on this path graph, are liardware-connectivity preserving PPTT F2Q mappings. However, transforming the former to the latter would require creating a tree with one node connected to 5 nodes at once, as shown in Fig. 7. Since our PPTT F2Q mappings require the tree to be a ternary ordered tree, such a move is not allowed. Note that if we relax this restriction that the OTT must be a subgraph of G, then of course we can reach any PPTT mapping.

[0290] On the other hand, it is possible to provide sufficient conditions that allow showing that any PPTT mapping can be generated from any other on a sufficiently large heavy-hexagonal and two-dimensional grid graph. We dedicate the rest of the section to proving this fact. We start with the following lemma, which reduces the complexity of changing between F2Q mappings, as long as the rules allow changing any subtrees with at most degree 3. For this, we introduce the term “underlying tree.” Let H be the OTT used for a particular PPTT F2Q. The underlying tree of H is a simple graph (without a root pointed) such that all the arcs in the OTT are replaced with edges. Note that this is a proper subgraph of G and a tree. The underlying tree of an F2Q mapping is the underlying tree of the OTT used in the mapping.

[0291] Lemma B.l. Let G = (V, E) be an undirected connected graph, and let there be two PPTT F2Q mappings with underlying trees TI, T2. Assuming that moving leaves allows transforming I\ into T2, one can transform Fi into F2with the steps introduced in Sec. V.

[0292] Proof. First, let us note that for any fixed OTT, swapping modes associated with any two nodes allows the production of any mode association. Similarly, changing braiding can also be done independently of the other changes to the PPTT F2Q mappings. Moreover, children for any node can be reassigned with a new Pauli without changing the underlying tree. Finally, since one can change any root to any other node that can be a root, the only issue with changing one F2Q to another is with changing the underlying tree. >

[0293] Note that the lemma almost allows us to focus solely on the underlying trees. However, at this moment, it remains unclear if we can move leaves freely in the underlying tree, as we may accidentally connect the 4-th node to the root of the tree. Fortunately, by changing the root, we don’t have to worry, as one can attach a node ‘to the root’. Lemma B.2. Let Ti be an underlying tree of OTT, and let T2be another underlying tree of OTT that differs from by reattaching the leaf. Then it is possible with the. steps introduced in Sec. V to reach PPTT F2Q with underlying tree T2out of PPTT F2Q with underlying tree 1\ .

[0294] Proof. Note that the only problem that arises might be if we would assign the moved leaf to a root and make it a degree-4 node. However, such a node can’t be a root for T2, so before moving the leaf we simply have to move the root to any node which is not the moved leaf. Fortunately, one can show that for any tree, one can find at least two nodes with degree 1, as otherwise we contradict the degree sum formula:

[0295] (Bl)

[0296] In light of Lemma B.l which allows us to change mode association, braiding and root position arbitrarily, we have proved the statement of the lemma. >

[0297] The lemmas above allow us to think about the reachability of any PPTT F2Q mapping from any other PPTT F2Q only in terms of underlying graphs, without even being concerned about the root position. Before showing sufficient conditions for reaching any PPTT F2Q mapping, let’s introduce a definition that will turn out to be useful for demonstrating how to construct an arbitrary underlying tree on particular hardware connectivity graphs.

[0298] Definition B.l. Led. G = (F, E) be arbitrary undirected connected graphs and let V C V. Let W C V s.t. G[W] and G[V \ W] are connected, and W n V 0. We call sequence of sets (fo , . . . , Vjf) a ping-pong partition of the V over (G, IV) if simultaneously

[0299] 1. is partition of W,

[0300] 2. Vi C W for odd i and If c V \ W for even i,

[0301] 3. Vf is connected for any 1 < k' < k.

[0302] Note that 3. implies that G[V'] is connected. A particular algorithm for generating one goes as follows: first, we choose an arbitrary vertex w G F'n W and find the maximum number of nodes in the induced graph inside G[TF]. The vertex set of this induced graph is our fo. Then we find the maximum set of nodes F2A V \ W such that G[Vi L) V2] is connected. We repeat the process, constructing consecutive Fi, . . . , Vf until they form a partition of W.

[0303] Equipped with such definitions, we are ready to prove the main theorem of this section.

[0304] Theorem B.3. Let G = (F, E) be a simple, connected graph and suppose we are given a n-mode PPTT F2Q mapping defined over G. Then one can create any other PPTT F2Q mapping provided the sufficient conditions on G hold:

[0305] 1. the maximum, degree of G is f,

[0306] 2. there exists W C V s.t. G[W] and G[V \ W] are connected and | W\, \V \ W\ > n.

[0307] Proof. Let there be two PPTT mappings with underlying trees TI, T2= (VT2, ET2). In light of Lemma B.l and Lemma B.l!. it is enough to show that by moving leaves such that, all intermediate graphs are subtrees, we can transform tree Tj into T2. Note that using the fact that the maximum degree of nodes in G is 4, we can never produce a tree with a maximum degree of 5 or more. Thus, we can always find a valid PPTT mapping for each of them.

[0308] Without loss of generality, we will assume that T) has nodes in V\W only, and T2has nodes in both F\ W and W. This is because if we can transform Ti into T2, we can also transform T2to Tj . Additionally, the same conclusions by symmetry can be done by swapping V \ W and W. Finally, such transformations can be chained to eventually allow the transformation of any tree to any other tree.

[0309] Furthermore, we will assume that 7) has a node that has a neighbour in W n VT2, a so-called element of the border of G[V \ W]. Otherwise, we could transform the tree so that it will satisfy this assumption as follows. First, we look for a path over nodes from V \ W that connects a particular node in Ti with any node v from the border such that intermediate nodes are not in Tj. Finally, we iteratively move leaves from the consecutively generated trees and add them along the path, up to v inclusive. This way we can transform '1\ appropriately.

[0310] Now let (Fi, . . . Vif) be a ping-pong partition of Fy2over (G, W'). Given I\, we create a tree by moving leaves from Ti to V2. Note that since the border element v is a neighbouring element of W fl Ty2, each leaf moved from Ti can be already assigned to some element from V2, connecting them according to T2. This way, after moving |Fi| nodes, we have a subtree with all the nodes from V) and edges as in T2[V)]. Finally, we move all the remaining nodes from I) and assign them arbitrarily to W so that they will form a connected tree T^ with all nodes in W and = T2[FI]. Note that since G[W] is a connected subgraph with n nodes, one can always find free nodes in W f Vi.

[0311] The steps above are repeated for Vk> with k1= 2. . . . , k with the following updates:

[0312]

[0313] FIG. 8. A visualization of the proof presented in in Theorem B.3. We start with the F2Q mapping which is fully on W (top left), and we wish to obtain the F2Q mapping as on top right. This can be done in three steps as depicted below, where the green nodes are nodes that are already in the correction location in the tree, and the orange ones are yet-to-be-moved nodes.

[0314] 1. We are not moving nodes that are already in Vi-

[0315] 2. The nodes from Vk' should be connected to along the T2structure.

[0316] Constructing a new T(~k\ With such rules, for each k', we have T^k[IJi=1K]=^MUiLr K], which eventually will produce T2for k1= k. The visualization of the process described in the last two paragraphs can be found in Fig. 8.

[0317] □

[0318] Note it is rather easy to relax the conditions of the theorem. First, we don’t need a partition into W and W \ V both having n or more nodes as long as we can fit all the remaining nodes in the process. With similar arguments, we don’t necessarily require G[W] and G[V \ W] to be connected as already depicted in Fig. 8. Finally, the condition on maximum degree could also be relaxed as long as we can guarantee that the intermediate trees will always have maximum degree 4.

[0319] However, this theorem is sufficient to show that reaching any PPTT F2Q mapping is possible on sufficiently large heavy-hexagonal and 2D grid graphs. For both classes, the maximum degree of nodes are 3 and 4 respectively, and since they are two-dimensional structures, it is easy to split them vertically or horizontally into two connected halves, see Fig. 9 for the arrangement of qubits.

[0320] Appendix C: Estimates on the number of PPTT F2Q mappings

[0321] In this section, we estimate the number of possible PPTT F2Q mappings for a particular hardware connectivity graph G. Two scenarios will be considered: in the first, we will assume G is a complete graph; in the second, we will assume the maximum degree is bounded by 3. a. A complete graph G Let n be the number of modes, and Q be the number of qubits. First, note that mode associations and braiding can be chosen arbitrarily, giving us 2"n! possibilities. Now, let’s count the number of ternary trees available. Starting with a root, we can attach a new node to any of its 3 possible children. Then we can attach a new leaf in 5 possible ways, and so on. In total, the number of ordered ternary trees is at most n"=i(2'i 1) — (2n — 1)1! = (2n)! / (2nn!). Additionally, we can assign any of the separately chosen physical qubits and assign them to nodes in any of n! ways. Note that with the procedure above, some ternary trees can be constructed in more than one way. This gives us an upper bound on the number of PPTT F2Q mappings as where, at the end, we used Stirling’s formula Iog(n!) = n logn — nloge + O(logn). Note that for indistinguishable qubits, we can just choose Q = n, which simplifies the formula to

[0322] 2^(2 log ra-log(e))+O(log2(n)) b. A bounded- degree graph G Here, we assume the maximum degree of the graph is d > 2. All the steps are as before, except for how many trees we can find. In the i-th step, instead of having 2i — 1 possibilities of assigning a node and an arbitrary physical qubit to be attached, we have to consider only those qubits which are neighbouring.

[0323] Therefore, first, we choose a root among one of Q qubits. Then, for one of the 3 legs, one of d neighbours of the root is chosen. Since all the used nodes in the trees can have at most d — 1 neighbours, we can upper bound the number of possibilities a physical qubit can be attached as the i-th node for i = 3, . . . , n as 3(d — l)(i — 1), where 3 comes from upper bounding the number of free legs for each node, d — 1 from the number of nodes that can be attached to each node in the tree, and i — 1 from the total number of nodes in the current tree. Therefore, we can construct at most

[0324] Thus, the total number of PPTT F2Q mappings is where we again used Stirling’s formula. Note that the dependency on the number of qubits Q is essentially lost, suggesting that for bounded-degree graphs like the heavy hexagonal with d = 3 or 2D grid with d = 4, the number of F2Q mappings is not significantly larger (if larger at all) than for the complete graphs with indistinguishable qubits. Note that here, all the qubits are assumed to be indistinguishable, so apart from overestimating the number of trees in Eq. (C’3), we did not account for possible isomorphism between ternary trees. Furthermore, for heavy- hexagonal trees, many of the nodes have a degree of 2, which further decreases the number of different PPTT F2Q mappings.

[0325] Appendix D: Simulation data and Results

[0326] FIG. 9. Figure (a) illustrates the topology of the IBM Washington device, and (6) is the Google Sycamore. CNOT gates between qubits are allowed only along the edges of the graphs in these devices.

[0327]

[0328] FIG. 10. Convergence against ADAPT-VQE iterations to within 103Hartree energy error of the exact ground-state energy. The number of iterations is equivalent to the number of variational parameters in the ansatz.

[0329] TABLE II. Full connectivity results for various configurations of the treespilation algorithm. Mode Shuffling (MS) refers to a scenario where the mapping tree is fixed as JW, and the assignment of Fermionic modes to qubit modes is optimized. This allows us to leverage the efficient circuit representation of the excitation operators presented in

[0031] . Pauli Cost (CP) and Transpiler Cost (TC) represent the possible cost functions employed. The last two columns labelled PC and TC display the results when full treespilation is applied with the labelled cost function. The lowest CNOT counts are displayed in bold.

[0330] TABLE III. Sycamore connectivity results for various configurations of the treespilation algorithm. Connectivity and Nonconnectivity Preserving (CP and NCP) settings, where the algorithm searches inside, and outside the space of connected subtrees of the device, respectively, are displayed.

[0331] TABLE IV. Eagle connectivity results for various configurations of the treespilation algorithm TABLE V. Transpilation passes used for different pools. The Efficient Circuits pass (ECP) employs circuit representations of the Fermionic pool in the JW encoding and the QEB pools as introduced in

[0065] . When possible, this pass is used as we found it more efficient than the TKET pass that utilizes advanced compilation techniques from

[0066] . This includes the use of ECP for the ansatz produced by MS. The Qiskit pass uses the qiskit transpiler with optimisation level 3 and is employed in all cases to allow CNOTs cancellation and map the circuit to the device on LC for QEB and qubit pools. The Staircase pass involves the standard approach of compiling exponentiated Pauli strings into a “staircase” of CNOTs. The Treespilation pass encompasses novel mapping optimization techniques described in this application. The Steiner pass is exclusively used to compile strings resulting from the LC Treespilation strategy. Treespilation is not applicable for QEB and qubit pools due to the operators not having an exact Fermionic representation.

[0332]

Claims

PATENT CLAIMS1 . Method of encoding a fermionic state of a fermionic system described by a plurality of N fermionic mode operators in a qubit state of a plurality of n qubits of a quantum processing device according to a target Fermion-to-qubit mapping, characterized in that said method comprises: deriving a plurality of tree-based Fermion-to-qubit mappings for the encoding, wherein each mapping of the plurality is derived on the basis of a pair of a tree and an instruction for deriving the mapping on the basis of said tree, said tree comprising a plurality of nodes and a plurality of links, wherein a link is either an edge connecting two nodes or a leg connected only to one node, wherein said instruction comprises a first instruction of assigning the fermionic mode operators, the qubits, and a plurality of Pauli operators, each Pauli operator being associated with a Hilbert space of a qubit of the quantum processing device, to the nodes and links of said tree to thereby obtain a labeled tree, and said instruction further comprises a second instruction for deriving the mapping on the basis of said labeled tree; estimating, for each of the Fermion-to-qubit mappings, a computational cost of encoding the fermionic state in the qubit state of the qubits of the quantum processing device according to said mapping; identifying the target fermion-to-qubit mapping as the mapping among the Fermion-to- qubit mappings of the plurality for which the computational cost fulfills an optimization criterion.

2. Method according to claim 1 , wherein the said method further comprises generating a plurality of pairs of trees and instructions for the deriving of the plurality of mappings, said pairs being outputs of an iteration in which each iteration step accepts an input pair of an input tree and an input instruction as an input and outputs an updated pair of an updated tree and an updated instruction which is used as the input for the next iteration step, wherein the updated pair is generated on the basis of the input pair by an application of a transformation to said input pair, said transformation comprising a tree-changing transformation applied to the input tree transforming said input tree into the updated tree and / or an instruction transformation applied to the input instruction transforming said input instruction into the updated instruction such that the estimated computational cost of the Fermion-to-qubit mapping derived from the updated pair fulfills an acceptance criterion, and terminating the iteration when a termination criterion is fulfilled.

3. Method according to claim 2, wherein obtaining the updated pair comprises an application of a probabilistic optimization procedure, and in particular simulated annealing.

4. Method according to anyone of the preceding claims, wherein the tree is a directed rooted tree comprising n nodes, one of the nodes being a root node, each node comprises three outgoing links, and the tree comprises 2N legs, wherein said first instruction comprises, for each node, bijectively assigning one qubit with said node and bijectively assigning each of the three outgoing links of said node with Pauli operators defined on the Hilbert space of the one qubit assigned with said node, and said second instruction comprises constructing for each of 2N paths from the root node to the legs a Pauli string, wherein the Pauli operators of the Pauli string are the ones assigned with the links along said path, and a pairing instruction for expressing each of the N fermionic mode operators as a linear combination of two Pauli strings.

5. Method according to claim 4, wherein the quantum processing device comprises n = N qubits, the tree is a directed rooted ternary tree with N nodes, said first instruction further comprises bijectively assigning each of the N nodes with one of the N fermionic modes, said pairing instruction further comprises assigning a pairing scheme with each node, said pairing scheme comprises a selection instruction for selecting, among the 2N constructed Pauli strings, two Pauli strings which act no n-tri vially and differently on the qubit associated with said node, and a combination instruction specifying the linear combination of the two selected Pauli strings in the fermionic mode operator associated with said node.

6. Method according to anyone of the preceding claims, wherein a connectivity of the qubits of the quantum processing device is specified by a connectivity graph of the qubits, and the graph defined by the nodes and edges of the tree for deriving of the Fermion-to-qubit mapping is a subgraph of said connectivity graph.

7. Method according to anyone of claims 2-6, wherein the tree-changing transformation comprises changing the number of nodes of the tree, and / or moving a leaf which is a node without outgoing edges, to an open end of a selected leg thereby turning said selected leg into a new edge and turning the edge originally connected with the leaf into a new leg, and / or for the case of a rooted tree changing a designation of the root node in the tree.

8. Method according to anyone of claims 4-7, wherein said instruction transformation comprises selecting first and second nodes, the first node being assigned with a first qubit and the second node being assigned with a second qubit according to the input instruction, and interchanging the assignment of the qubits such that the updated instruction comprises assigning the first node with the second qubit and assigning the second node with the first qubit.

9. Method according to anyone of claims 5 to 8, wherein said instruction transformation comprises selecting third and fourth nodes, the third node being assigned with a third Fermionic mode and an associated third pairing scheme and the fourth node being assigned with a fourth Fermionic mode and an associated fourth pairing scheme, and interchanging the assignment of the Fermionic modes and their pairing schemes such that the updated instruction comprises assigning the third node with the fourth Fermionic mode and the fourth pairing scheme and assigning the fourth node with the third Fermionic mode and the third pairing scheme.

10. Method according to anyone of claims 5-9, wherein said instruction transformation comprises selecting a fifth node and changing the combination instruction assigned with said fifth node such that the updated instruction assigns the changed combination instruction with said fifth node.

11. Method according to anyone of claims 4-10, wherein said instruction transformation comprises selecting a sixth node and interchanging the Pauli operators assigned with the outgoing links of said sixth node so that the updated instruction assigns the Pauli operators to the outgoing links in the interchanged way compared to the input instruction.

12. Method according to anyone of the preceding claims, wherein estimating the computational cost comprises estimating a number of quantum gates native to the quantum processing device and required for encoding the fermionic state in the qubit state of the qubits of the quantum processing device.

13. Method according to anyone of the preceding claims, wherein the computational cost is estimated on the basis of a Pauli weight of Pauli strings present in a qubit representation of the fermionic state according to the Fermion-to-qubit mapping.

14. Method according to anyone of the preceding claims, wherein the method further comprises deriving a quantum circuit for encoding the fermionic state in the qubit state of the qubits of the quantum processing device according to the Fermion-to-qubit mapping, and estimating the computational cost for the encoding is on the basis of said quantum circuit.

15. Method according to claim 14, wherein deriving of said quantum circuit comprises use of a transpiler.

16. Method according to anyone of the preceding claims, wherein the method comprises deriving a target quantum circuit configured for encoding the fermionic state in the qubit stateof the quantum processing device according to the target Fermion-to-qubit mapping and encoding the fermionic state in the qubit state of the qubits by executing the target quantum circuit by the quantum processing device.

17. Method according to anyone of the preceding claims, wherein said method further comprises approximating an eigenstate of the fermionic system by application of a quantum- classical variational algorithm, wherein each iteration step of the algorithm comprises defining a fermionic ansatz state by applying an operator selected from a predetermined pool of fermionic operators according to instructions obtained in a preceding iteration step of the algorithm to a fermionic reference state, deriving the target Fermion-to-qubit mapping for the fermionic ansatz state according to anyone of the preceding claims, encoding the fermionic ansatz state in the qubits of the quantum processing device according to the target Fermion- to-qubit mapping and estimating a value of an energy of the fermionic ansatz state by measuring a qubit operator which is an encoding of a fermionic operator associated with the energy according to the target Fermion-to-qubit mapping to thereby obtain instructions for selecting the fermionic operator for the next iteration step on the basis of the estimated value of the energy.

18. Computer program comprising instructions which, when the computer program is executed by a classical computer, cause the classical computer to carry out the following steps: receiving a number N of fermionic mode operators, data representative of a fermionic state described by said N fermionic mode operators and a number n of qubits; deriving a plurality of tree-based Fermion-to-qubit mappings for the encoding of said fermionic state in a qubit state of the n qubits, wherein each mapping of the plurality is derived on the basis of a pair of a tree and an instruction for deriving the mapping on the basis of said tree, said tree comprising a plurality of nodes and a plurality of links, wherein a link is either an edge connecting two nodes or a leg connected only to one node, wherein said instruction comprises a first instruction of assigning the fermionic mode operators, the qubits, and a plurality of Pauli operators, each Pauli operator being associated with a Hilbert space of a qubit, to the nodes and links of said tree to thereby obtain a labeled tree, and said instruction further comprises a second instruction for deriving the mapping on the basis of said labeled tree; estimating, for each of the Fermion-to-qubit mappings, a computational cost of encoding the fermionic state in the qubit state of the qubits of the quantum processing device according to said mapping;identifying the target fermion-to-qubit mapping as the mapping among the fermion-to- qubit mappings of the plurality for which the computational cost fulfills an optimization criterion.

19. Computing system comprising a quantum processing device and a classical computer, wherein the classical computer is configured to execute the computer program according to claim 18, and to derive a target quantum circuit configured for encoding the fermionic state in the qubit state of the quantum processing device according to the target fermion-to-qubit mapping and said quantum processing device is configured to receive the target quantum circuit as an input and to encode the fermionic state in the qubit state of the qubits by executing the target quantum circuit.

20. Computing system comprising a quantum processing device and a classical computer, said computing system being configured to approximate an eigenstate of the fermionic system by application of a quantum-classical variational algorithm, wherein the computing system is operative to implement an iteration of the algorithm, wherein each iteration comprises defining, by the classical computer, a fermionic ansatz state by applying an operator selected from a predetermined pool of fermionic operators according to instructions obtained in a preceding iteration of the algorithm to a fermionic reference state, deriving, by the classical computer, the target fermion-to-qubit mapping for the fermionic ansatz state according to anyone of claims 1-15, providing the target Fermion-to-qubit mapping to the quantum processing device, wherein said quantum processing device is configured to receive the target fermion-to-qubit mapping as an input, to encode the fermionic ansatz state in its qubits according to the target Fermion-to-qubit mapping, to measure a qubit operator which is an encoding of a fermionic operator associated with an energy of the fermionic system according to the target Fermion- to-qubit mapping to thereby obtain a measurement outcome, to provide said measurement outcome to the classical computer, and wherein the classical computer is further configured to estimate a value of the energy on the basis of the measurement outcome to thereby obtain instructions for selecting the fermionic operator for the next iteration on the basis of the estimated value of the energy.

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