Computer-implemented method for determining a control sequence for executing a series of qubit interactions to simulate a fermion hamiltonian, computer program product, quantum circuit and method for determining system characteristics

Abstract

1. A computer-implemented method for determining control sequences for performing qubit interactions on a plurality of qubits on a quantum device having a square qubit layout to simulate a fermionic Hamiltonian, the method including: receiving input parameters of the fermionic Hamiltonian to be simulated; and projecting the associated fermionic lattice onto the qubit layout such that qubits that are assigned fermionic modes are referred to as physical qubits P and qubits that are not assigned any fermionic modes are referred to as ancilla qubits A, wherein the arrangement of the physical qubits P and ancilla qubits A in each single horizontal line of the qubit layout is the same. The method further includes associating each physical qubit P with at least one edge operator E and one vertex operator V, mapping each fermionic operator to a qubit operator based on the edge operator E and the vertex operator V, and determining a control sequence for the qubit operators that includes at least the qubit operator determined by the mapping.

Description

[Technical Field]

[0001] The present invention relates generally to quantum devices, and more particularly to fermion Hamiltonians and enabling their simulation on quantum devices. [Background technology]

[0002] A quantum computer or quantum device is a machine that uses the properties of quantum physics to store data and perform calculations. Compared to classical computers, which encode information in the form of bits, e.g., 0s or 1s, quantum computers use quantum bits, called qubits, which can be in two states simultaneously in a coherent superposition. A qubit can refer to a fundamental unit of quantum information or a quantum device, such as a two-level quantum mechanical system, used to store a unit of quantum information. Thus, a quantum computer generally comprises an array of qubits and hardware for manipulating these qubits.

[0003] There are three fundamental quantum computing methods: the analog quantum model, the universal quantum gate model, also known as the digital quantum computing model or quantum circuit model, and quantum annealing. In the quantum gate model, operations on or interactions between qubits are called gates, and a sequence of one or more gates arranged to be applied to qubits constitutes a control sequence for a quantum device, which may be called a quantum circuit, corresponding to instructions for manipulating units of quantum information to perform a desired computation. Quantum gates may also be referred to as unitary operators, represented by unitary matrices. Quantum gates that operate on multiple qubits may also be referred to as interactions between multiple qubits. Implementing gates that operate on multiple qubits on a quantum device corresponds to performing qubit interactions on multiple qubits.

[0004] Hamiltonian equations can be used to study the properties of quantum many-body systems, thereby determining, for example, the electronic structure of molecules. Because the complexity of classical simulations of quantum many-body systems typically grows exponentially with the dimensionality of the system, quantum computers can provide a powerful tool for simulating many-body problems through the implementation of the associated Hamiltonian by simulating system interactions through qubit interactions.

[0005] Quantum computers are well suited to simulating quantum many-body systems, but current quantum computers are limited by the number of available qubits and errors in the form of noise, malfunctions, and loss of quantum coherence. The accuracy of quantum computations can degrade rapidly as the number of gate operations and circuit depth increase.

[0006] The study of physical fermionic quantum systems, which can be characterized by a fermionic Hamiltonian, is important in many technological fields, including superconductivity, battery design, chemical reaction optimization, fertilizers, and the research of novel drugs. When implementing a Hamiltonian, its operations must be mapped to unitary matrices that can be implemented on a quantum computer. Due to the incompatibility of the commutation relations between fermionic operations and Pauli operators, which are unitary matrices considered in terms of qubit interactions, fermionic quantum systems cannot be easily simulated using quantum computers.

[0007] To be able to efficiently simulate fermion Hamiltonians on quantum devices, fermion-qubit mappings may be utilized, which show how the modes of a fermion system are represented in terms of qubits and quantum gates. Many types of fermion-qubit mappings exist, but they may not be optimal with respect to the particular type of hardware and / or the particular type of Hamiltonian considered. Summary of the Invention

[0008] It is an object of the present invention to alleviate at least some of the problems of the prior art. According to one aspect of the present invention, there is provided a method for determining a control sequence for performing a series of qubit interactions on a plurality of qubits on a quantum device to simulate a fermion Hamiltonian H that can be expressed as a sum of one or more tensor products of Pauli matrices, the method comprising: the quantum device includes a plurality of qubits arranged in a two-dimensional square lattice qubit layout, each qubit arranged to interact with up to four neighboring qubits; The method comprises: receiving parameters of a fermion Hamiltonian, the parameters comprising at least the number of fermion lattice sites L, the number of fermion modes M in the fermion lattice, and receiving a fermionic mode M, including a fermionic operator corresponding to the interaction between the fermionic modes M; projecting a fermionic lattice onto a qubit layout of the quantum device such that all fermionic modes are assigned to qubits of the quantum device, said qubits being referred to as physical qubits P, the projection between the fermionic modes and the physical qubits being one to one, and a plurality of further qubits of the quantum device being referred to as ancilla qubits A, the ancilla qubits A not being assigned to any fermionic modes; the physical qubits P and the ancilla qubits A are arranged on single horizontal lines of a two-dimensional square lattice, each single horizontal line including at least one string P' and at least one ancilla qubit A, each string P' including one or more physical qubits P, and the arrangement of the physical qubits P and the ancilla qubits A in each single horizontal line of the qubit layout is the same; and associating each physical qubit P with at least one edge operator E and one vertex operator V; mapping each fermion operator to a qubit operator based on an edge operator E and a vertex operator V; and determining a control sequence of qubit interactions that includes at least the qubit operators determined by the mapping.

[0009] The present invention can provide fermion-qubit mappings that are well-suited or optimal for use in solving problems involving a range of considered fermionic systems and / or Hamiltonians. Optimal or well-suited means that the methods of the present invention can be used to provide mappings for a variety of use cases without having to adjust the mapping for each case separately, which would require more (human) effort. Optimal or well-suited can additionally or alternatively mean that the methods of the present invention can provide high-quality fermion-qubit mappings when considering one or more aspects related to performance when performing calculations or simulations on a quantum device based on the determined mappings.

[0010] The present invention can provide fermion-to-qubit mappings that are well suited for use in conjunction with square lattice qubit layouts of quantum devices. Well suited to a particular qubit layout may refer to the possibility of utilizing the number of available qubits to a higher or more efficient extent than is the case with corresponding prior art.

[0011] Mapping according to embodiments of the present invention can provide control sequences corresponding to quantum circuits with shallower depths than quantum circuits obtained with prior art mappings for the same problem or simulation (properties determined for a particular fermionic system given a selected fermionic Hamiltonian). The required computation time can then be reduced, thus, for example, reducing errors and / or saving energy. In some embodiments, circuit depth can be utilized as a primary optimization criterion to obtain a maximally reduced (shortest) circuit depth.

[0012] In some embodiments, the present invention can provide fermion-qubit mappings that result in constant circuit depth. The constant circuit depth can result from having edge and vertex operators that do not scale with system size in relation to the number of fermionic lattice sites in the fermionic Hamiltonian. This can be true for at least two-dimensional fermionic lattices.

[0013] In some embodiments, the circuit depth can scale linearly with the considered number of fermionic modes per fermionic lattice site. Furthermore, the considered qubit operators can include up to 4 Pauli weights, independent of the size of the considered fermionic system. In contrast, for example, in Jordan-Wigner mappings known from the prior art, at least some of the considered qubit operators have a size N 2 Scaling with N for the fermion square lattice.

[0014] By having each row of the qubit layout identical (in terms of the arrangement of physical qubits and ancilla qubits), it is possible to obtain an edge operator that connects two physical qubits via a vertical stack of ancilla qubits. This arrangement can reduce the depth of the quantum circuit obtained by a determined control sequence.

[0015] Each qubit operator can include a product of vertex operators V, each of which operates on at least one of the physical qubits P assigned a fermionic mode of the fermionic operator.

[0016] Each qubit operator can include a product of at least one of the vertex operators V and one of the edge operators E, where each of the vertex and edge operators in the product operates on at least one of the physical qubits P assigned a fermionic mode of the fermionic operator.

[0017] Associating each physical qubit P with at least one edge operator E and one vertex operator V can include determining the edge operators E and vertex operators V such that the edge and vertex operators acting on each pair of the same physical qubits anticommutate, and the edge and vertex operators acting on all other operator pairs commute.

[0018] Associating each physical qubit P with at least one edge operator E and one vertex operator V can, in some embodiments, be performed by: Each physical qubit P is assigned to a vertex operator V p V p is a first type of Pauli operator selected from types X, Y, and Z of Pauli operators and acting on physical qubit p; For any pair of physical qubits p and q, they are either directly adjacent in the horizontal dimension with no physical qubits or ancilla qubits in between, or separated by one or two ancilla qubits, and there is a horizontal edge operator E associated with said qubits. pq H To define E pq H is the product of the number of Pauli operators, at least two Pauli operators of a second or third type, respectively, selected from Pauli operator types X, Y, and Z, acting on qubits p and q, respectively; and a product of the number of Pauli operators, each of which includes an additional Pauli operator of a first type, acting on each of said ancilla qubits, if any, between physical qubits p and q along the horizontal dimension; When two horizontal edge operators act on the same qubit q, the first of the two horizontal edge operators, E pq H1 acts on the qubit q using a second kind of Pauli operator, the second horizontal edge operator E pq H2acts on the qubit q using a third kind of Pauli operator, and vice versa, For any pair of physical qubits p and q, which are immediately adjacent in the vertical dimension without any physical qubits or ancilla qubits in between, such pair of qubits is adjacent to a pair of ancilla qubits a and b, such ancilla qubits a and b are immediately adjacent in the vertical dimension and are located adjacent to qubits p and q, respectively, and a vertical edge operator E associated with such qubits p, q, a, and b is pq V To define E pq V is a product of four Pauli operators, each of the second or third kind acting on one of the qubits p, q, a, b, such that each of the four Pauli operators acts on a different qubit; The Pauli operators acting on the ancilla qubits a and b are of different types, the Pauli operator acting on physical qubit p is of the same kind as the Pauli operator acting on physical qubit q and forming part of the horizontal edge operator acting on at least physical qubit p and ancilla qubit a; similarly, the Pauli operator acting on physical qubit q is of the same kind as the Pauli operator acting on physical qubit p and forming part of the horizontal edge operator acting on at least physical qubit q and ancilla qubit b; The vertical edge operator is defined as follows: if ancilla qubits a and b are located on a first side of physical qubits p and q along the horizontal dimension, respectively, then the first vertical edge operator E pq V1 or if ancilla qubits a and b are located on the second side of physical qubits p and q along the horizontal dimension, a second vertical edge operator E pq V2 and defining, referred to as, When two vertical edge operators act on the same ancilla qubit, if one of the two vertical edge operators acts on said ancilla qubit with a second type of Pauli operator, then the other of the two vertical edge operators acts on said ancilla qubit with a third type of Pauli operator, and vice versa.

[0019] Preferably, in a row of qubits there are at most two consecutive ancilla qubits A. Although more may be utilized, at most two consecutive ancilla qubits may be more efficient.

[0020] Receiving parameters of the simulated fermion Hamiltonian may further include obtaining a number of fermion modes within each fermion lattice site, a number of fermion lattice sites within a first dimension L1, a number of fermion lattice sites within a second dimension L2, and a number of fermion lattice sites within a third dimension L3, where the total number of fermion lattice sites is L=L1L2L3, and the fermion lattice sites may be identified by indices i, j, k indicating the fermion lattice site position along the respective dimensions, where i=[1, L1], j=[1, L2], k=[1, L3], and the number of fermion modes within each fermion lattice site identified by indices i, j, k indicates that the total number of fermion modes in the fermion lattice is M=Σ i=1 L1 Σ j=1 L2 Σ k=1 L3 M ijk As M ijk and each fermion mode in the fermion Hamiltonian can be identified by four indices i, j, k, l, where l = [1, M ijk ]. Projecting the fermionic lattice onto the square lattice qubit layout of the quantum device then maps each fermionic mode, identified by four indices i, j, k, and l, of the fermionic lattice onto a physical qubit P of the quantum device. ijkl, where the index of each physical qubit identifies the fermion mode to which the physical qubit is assigned.

[0021] The step of projecting the fermion lattice onto the qubit layout of the quantum device includes projecting physical qubits P, each assigned a fermion mode identified by indices i, j, k, and l in the qubit layout. ijkl may also include placing Each horizontal single line of the qubit layout includes L strings P′, each string being a physical qubit P where the string is assigned a fermionic mode associated with a fermionic lattice site with position indices i and j. ijkl To show that P' contains the number ij and the string P' ij The number of physical qubits in Σ k=1 L3 M ijk is equal to each physical qubit with a lower index i is positioned in the horizontal dimension before any physical qubit with a higher index i; the number of horizontal single lines in the qubit layout involved in the projection is equal to L, and each physical qubit with a lower index j is positioned in front of any physical qubit with a higher index j in the vertical dimension; The physical qubits in each string P' are associated with varying indices k and l, which place their respective orderings according to fermionic interactions.

[0022] The physical qubits of any string P' in the horizontal dimension may be arranged such that the order of physical qubits with indexes k and l is reversed in successive strings P'.

[0023] If one or more of the fermionic lattice sites includes two or more modes associated with that lattice site, the control sequence may further include one or more fSWAP operators for permuting the fermionic modes assigned to the physical qubits by the projection, to implement interactions between the fermionic modes included in the Hamiltonian, to which no qubit operators have been applied prior to said permutation.

[0024] When a fermion Hamiltonian describes a system with more than one spin type within a fermion lattice site, there is a mode M per each fermion lattice site. ijk may be further assigned to a number o of orbitals, each containing a number s of spins, and M ijk = o*s, where physical qubits assigned fermionic modes having a spin of one orbital are each separated by one physical qubit, and where if the l indices of the physical qubits assigned the orbitals are odd, then the fermionic modes associated with the physical qubit with index l may be arranged such that increasing index l is associated with increasing spin types, whereas if the l indices of the physical qubits representing the orbitals are even, then the fermionic modes associated with the physical qubit with index l may be arranged such that increasing index l is associated with decreasing spin types s.

[0025] The association of physical qubits with edge and vertex operators may take into account native gates of the quantum device in which the obtained control sequence is implemented. In one embodiment, the physical qubits in a row of qubits in the horizontal dimension may be grouped into pairs of physical qubits alternately labeled even or odd, where ancilla qubits are not taken into account, and adjacent even and odd pairs share a physical qubit, and associating each physical qubit with at least one edge operator may For each even pair of physical qubits p and q and, if present, one or more ancilla qubits therebetween, an even horizontal edge operator E associated with the qubits pq H1 to define If p and q are not separated by an ancilla qubit but are directly adjacent, then E pq H1 is the product of two Pauli operators of the second kind acting on the physical qubits p and q, respectively, and optionally E pq H1 =X p X q and If p and q are separated by one ancilla qubit a, then E pq H1 is a product of a second type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, and a second type Pauli operator acting on the physical qubit p, and optionally E pq H1 =X p Z a X q and If p and q are separated by two ancilla qubits a and b, then E pq H1 is a product of a second type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, a first type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit p, and optionally E pq H1 =X p Z a Z b X q and defining For each pair of odd physical qubits p and q, and, if present, one or more ancilla qubits between them, an odd horizontal edge operator E associated with the qubits pq H2 to define If p and q are not separated by an ancilla qubit but are directly adjacent, then E pq H2 is the product of two Pauli operators of a third kind acting on the physical qubits p and q, respectively, and optionally E pq H2 =Y p Y q and If p and q are separated by one ancilla qubit a, then E pq H2 is a product of a third type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, and a third type Pauli operator acting on the physical qubit p, and optionally E pq H2 =Y p Z a Y q and If p and q are separated by two ancilla qubits a and b, then E pq H2 is a product of a third type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, a first type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit p, and optionally E pq H2 =Y p Z a Z b Y q and defining For each pair of physical qubits p and q, there is a vertical edge operator E pq V to define If ancilla qubits a and b are directly adjacent to p and q in the horizontal dimension in a first direction, and p and q have no other ancilla qubits directly horizontally adjacent to them, and a and b are associated with an even horizontal edge operator, then E pq Vis a product of a second type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit q, and optionally E pq V =X p X a Y b X q and If ancilla qubits a and b are directly adjacent to p and q in the horizontal dimension in a first direction, and p and q have no other ancilla qubits directly horizontally adjacent to them, and a and b are associated with odd horizontal edge operators, then E pq V is a product of a third type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit q, and optionally E pq V =Y p X a Y b Y q and If ancilla qubits a and b are directly adjacent to p and q in the horizontal dimension in a second direction, and p and q have no other ancilla qubits directly horizontally adjacent to them, and a and b are associated with an even horizontal edge operator, then E pq V is a product of a second type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit a, a second type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit q, and optionally E pq V =X p Y a X b X q and If ancilla qubits a and b are directly adjacent to p and q in the horizontal dimension in a second direction, and p and q have no other direct ancilla qubits horizontally adjacent to them, and a and b are associated with odd horizontal edge operators, then E pq V is a product of a third type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit a, a second type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit q, and optionally E pq V =Y p Y a X b Y q and If p and q are immediately adjacent to ancilla qubits a and b in the horizontal dimension in a first direction, and ancilla qubits c and d are immediately adjacent in a second direction, then For the first direction, E pq V is a product of a second type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit q, and optionally E pq V =X p X a Y b X q or E pq V is a product of a third type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit q, and optionally E pq V =Y p X a Y b Y q and For the second direction, E pq Vis a product of a second type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit c, a second type Pauli operator acting on the ancilla qubit d, and a second type Pauli operator acting on the physical qubit q, and optionally E pq V =X p Y c X d X q or E pq V is a product of a third type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit c, a second type Pauli operator acting on the ancilla qubit d, and a third type Pauli operator acting on the physical qubit q, and optionally E pq V =Y p Y c X d Y q and defining that: The Pauli operators of the first kind, second kind, and third kind are all different from each other, at least when two or more of them are considered in an edge operator. The choice of which Pauli operator is the first, second, and third may be different. pq V =X p Y c X d X q If, as given, E pq V =Y p Y c X d Y q Alternatively, E pq V =Y p X c Y d Y q Furthermore, for example, E in the given example above may be pq H1 =X p Z a Z b X q is Epq H1 =Y p Z a Z b Y q It can also be done as follows.

[0026] The qubits may be grouped into a pattern that repeats in the horizontal dimension, the pattern being selected from the group consisting of P', P'P'A, P'P'AA, and P'AA. The selection of the pattern may be based on criteria related to the Pauli weight of the operator, the circuit depth of the associated quantum circuit, and / or the type of associated gate. The selected or optimal pattern may be different, for example, for different types of Hamiltonians.

[0027] A pattern may be selected by determining the maximum Pauli weight of the edge operator associated with at least two patterns and selecting the pattern associated with the lowest maximum Pauli weight.

[0028] A pattern may be selected by determining the circuit depths of control sequences separately associated with at least two patterns and selecting the pattern associated with the lowest circuit depth.

[0029] A pattern may be selected by determining the number and / or type of gates included in a control sequence separately associated with at least two patterns, and selecting the pattern associated with the lowest number of gates and / or the pattern associated with the selected type of gates, and / or the pattern associated with the selected qubit-mode ratio.

[0030] In one aspect of the invention there may be provided a computer program product as claimed in independent claim 17. In yet another aspect of the invention there may be provided a method for determining at least one property of a quantum circuit as claimed in claim 18 and a system as claimed in claim 19.

[0031] The expression "number" as used herein can refer to any positive integer starting from 1.

[0032] The novel features which are believed to be characteristic of the invention are set forth with particularity in the appended claims. However, the invention itself, both as to its structure and its method of operation, together with additional objects and advantages thereof, will best be understood from the following description of specific illustrative embodiments when read in connection with the accompanying drawings.

[0033] The invention will now be described in more detail with reference to exemplary embodiments according to the accompanying drawings. [Brief explanation of the drawings]

[0034] [Figure 1] 10A-10C illustrate examples of arrangements of qubits in patterns and rows in the horizontal and vertical dimensions according to embodiments of the present invention. [Figure 2] Some examples of vertex and edge operators are given below. [Figure 3A] 1 illustrates edge and vertex operators that may be utilized in connection with one exemplary embodiment of the present invention. [Figure 3B] 1 illustrates edge and vertex operators that may be utilized in connection with one exemplary embodiment of the present invention. [Figure 3C] 1 illustrates edge and vertex operators that may be utilized in connection with one exemplary embodiment of the present invention. [Figure 3D] 1 illustrates edge and vertex operators that may be utilized in connection with one exemplary embodiment of the present invention. [Figure 3E] 1 illustrates edge and vertex operators that may be utilized in connection with one exemplary embodiment of the present invention. [Figure 4A] 10 illustrates edge and vertex operators that may be utilized in connection with one further exemplary embodiment of the present invention. [Figure 4B] 10 illustrates edge and vertex operators that may be utilized in connection with one further exemplary embodiment of the present invention. [Figure 4C] 10 illustrates edge and vertex operators that may be utilized in connection with one further exemplary embodiment of the present invention. [Figure 4D] 10 illustrates edge and vertex operators that may be utilized in connection with one further exemplary embodiment of the present invention. [Figure 4E] 10 illustrates edge and vertex operators that may be utilized in connection with one further exemplary embodiment of the present invention. [Figure 5] Figure 1 shows a diagram of the edge and vertex operators associated with an example multimode fermion lattice site and the associated fSWAP network. [Figure 6] Some examples of fermion lattice geometries are shown. [Figure 7] An example of a fermion lattice and the connectivity and interaction terms considered are shown. [Figure 8] 1 shows the qubit hardware layout. [Figure 9] 1 illustrates an exemplary hopping operator. [Figure 10] 1 shows an fSWAP network with modes and orbitals for an exemplary fermionic system. DETAILED DESCRIPTION OF THE INVENTION

[0035] Electronic structure Hamiltonian H es is expressed as follows in the second quantization format: JPEG2026505191000002.jpg19164 where p, q, r, and s can represent different fermion modes, and c i and c i + are the annihilation and creation operators that respectively create and annihilate fermions in mode i, and h pq , h pqrsrepresent the one- and two-electron integrals, respectively, which can be thought of as known constants. The quadratic terms in the first summation of the Hamiltonian (including the two fermion creation / annihilation operators) may be called the hopping operators, while the quartic terms may be called the interaction operators.

[0036] When characterizing a physical system to be solved or simulated, a Hamiltonian equation can be considered in which selected terms, connectivity, and / or spin are considered, where M is the total number of fermionic modes considered, O(M 4 The above electronic structure Hamiltonian H es This leads to a Hamiltonian that is sparser, or less dense (meaning that there are fewer fermionic operators involved) than O(M 2 ) is a Hamiltonian with degree JPEG2026505191000003.jpg19164 where t and U can be considered as constants. The Hamiltonian to be considered can vary with respect to the total number of fermion operators (denseness / sparseness). Typically, the Hamiltonian considered relevant for quantum chemistry problems is, for example, the Fermi-Hubbard Hamiltonian H FHM may be denser than

[0037] Fermions are particles with half-integer spin that obey the Pauli exclusion principle, so only one fermion can occupy a particular quantum state at a given time. Therefore, quantum states must be antisymmetric under exchange, and the annihilation and creation operators must be JPEG2026505191000004.jpg67133 It must be anticommutative, like this:

[0038] Furthermore, the Pauli spin operator σ i, and its tensor product are regularly used as quantum gates in quantum computers to perform qubit operations, may be used to simulate Hamiltonians on quantum devices, are anticommutative, and {σ p ,σ q}=2δ pq I, (6) In the formula, σ i ∈[X, Y, Z] and JPEG2026505191000005.jpg25121 JPEG2026505191000006.jpg79129 is.

[0039] A mapping between the Hilbert space of a fermionic system and a set of qubits in a quantum device can enable the representation of the fermionic system on the device. The fermion-qubit mapping (also referred to herein as "mapping") can map fermionic operators to a string of Pauli operators, which can be implemented as a quantum circuit that induces qubit interactions on the quantum device to simulate the fermionic interactions of the Hamiltonian while maintaining fermionic parity (meaning that the qubit interaction representation should include an equal number of interactions corresponding to creation and annihilation operators, respectively, as a fermionic Hamiltonian has). Fermionic interactions here can refer to interactions that include both interactions and hopping operators.

[0040] The fermion Hamiltonian can then be rewritten or expressed in the following form: JPEG2026505191000007.jpg20164 where the product is a tensor product and a j is the coupling constant, the index f runs over all F-operators included in the fermion Hamiltonian, the index n runs over the total number of qubits in the device N, and JPEG2026505191000008.jpg1562 The Pauli weight associated with the Hamiltonian may be determined as the maximum Pauli weight of any summand.

[0041] Further properties of the Hamiltonian that characterize the system to which it relates include, for example, the fermion lattice dimension and the number of lattice sites (L = L x L y L z , where L x is the number of first-dimensional lattice sites, L y is the number of second-dimensional lattice sites, L z is the number of lattice sites in the third dimension), the total number of fermion modes (M) and the number of modes per lattice site (M / L).

[0042] Hamiltonians may vary widely in terms of the degree of connectivity, i.e., the interactions between fermions that are considered. Connectivity can include nearest-neighbor interactions only, nearest-neighbor and next-nearest-neighbor interactions, or nearest-neighbor, next-nearest-neighbor, and higher-neighbor interactions.

[0043] A considered site (single qubit) on the qubit layout may be assigned to represent a mode of the considered Hamiltonian. The fermion-qubit mapping generates a qubit connectivity graph containing commutation relations corresponding to those of the fermions (E pq =-E qp ) and vertices (V p ) operators first. After defining such edge and vertex operators, any fermion operator can be transformed into a product of the edge and vertex operators and then expressed in terms of operators acting on qubits. The edge and vertex operators may be defined as follows: JPEG2026505191000009.jpg19164 JPEG2026505191000010.jpg20164 It may be further shown that the edge and vertex operators obey an anti-commutation relation.

[0044] {E pq , V p}=0 and (14) {E pq , E qr}=0, (15) Furthermore, the operators obey the following commutation relation ([A, B]=AB-BA) and anticommutation relation ([A, B]=AB+BA).

[0045] For p≠q≠r≠s, [E pq , E rs ]=0, (16) [V p , V q ]=0, and (17)[E pq , V r ]=0(18).

[0046] These relationships can be summed up by stating that any two distinct vertex or edge operators are mutually anticommutative if they act on a common fermionic mode and would otherwise commute. Finding a suitable fermion-qubit mapping may then involve determining a string of Pauli operators corresponding to the vertex and edge operators so as to satisfy the above relationships. The Pauli weight associated with the mapping may be determined as the maximum Pauli weight associated with either the edge or vertex operators. The Pauli weight can refer to the number of different qubits associated with the utilized operator.

[0047] A further condition to be considered is the product of edge operators on the closed path {p1, p2, ...}, i.e., E 12 , E 23 , and E 31 The edge operators that connect qubits to form closed paths such as must be equal to identity.

[0048] JPEG2026505191000011.jpg20164 With respect to the above condition, the initial state may be chosen to lie within the +1 eigenspace of the product of edge operators over all closed paths. As long as this condition on the initial state is met, there is no need to consider Equation 19 when constructing the mapping.

[0049] Suitable edge and vertex operators may be determined, for example, by determining several different combinations (such as all possible combinations) of tensor products of Pauli operators and selecting a set of operators that satisfy selected criteria, such as at least the anticommutation and commutation criteria.

[0050] After determining a set of appropriate edge and vertex operators, these can be used to map each interaction between fermionic modes to qubit operators, finding expressions corresponding to the fermionic operators of the problem Hamiltonian under consideration.

[0051] To determine the set of qubit interactions corresponding to the interactions between fermionic modes, we consider the relationship between the edge and vertex operators, the fermionic Majorana operators, JPEG2026505191000012.jpg1421 , as well as fermion annihilation and creation operators. Such a relation can be JPEG2026505191000013.jpg93134 is given by

[0052] By using these edge, vertex, Majorana, and commutation properties of the annihilation and creation operators, different fermion interaction expressions required for each use case may be determined. Some examples may be given as follows: JPEG2026505191000014.jpg62164

[0053] A consideration that can be used in determining the mapping is that not all possible edges, i.e., all possible connectivity between fermion modes, need to be directly represented in the connectivity graph: the qubit sites corresponding to any edge can be connected by paths involving intermediate edge operators to form a "composite" edge operator. E pr =E pq E qr (28) The connection may be made as follows.

[0054] Furthermore, it is also possible to dynamically modify the connectivity graph itself by using a fermion swap (fSWAP) operation, which exchanges the positions of two fermion modes connected by an edge. Such an fSWAP operator is JPEG2026505191000015.jpg24164 It is defined as follows:

[0055] Via the fSWAP operator, modes of the Hamiltonian that interact with each other may be brought closer together, thus reducing the Pauli weight of the corresponding Hamiltonian operator. However, it can be taken into account that this comes at the cost of having to implement the fSWAP operation itself, which may increase the computational effort.

[0056] There are various considerations or criteria that may be taken into account in selecting edge and vertex operators, and thus in determining the fermion-qubit mapping. Adhering to any one or more criteria may result in weaknesses with respect to other aspects. Criteria may relate to performance aspects related to the computational resources required to perform a simulation / computation on a quantum device using the determined mapping and the control sequence determined therefrom (e.g., circuit depth, number of two-qubit gates, qubit-mode ratio, i.e., number of qubits required to encode one fermion mode, and / or Pauli weights), limitations imposed by hardware aspects (e.g., number of qubits utilized and / or connectivity of qubit layouts and / or types of one-qubit and / or two-qubit gates allowed, depending on gates specific to the hardware in question), error-related aspects or error-correction requirements related to known or estimated errors with respect to the computation performed via the quantum device, and / or generality aspects related to the types of Hamiltonians to which the mapping may be applied. Thus, it may not be possible to determine a fermion-qubit mapping that is optimal in performance with respect to any kind of Hamiltonian, any kind of fermion system, and any type of hardware.

[0057] Many known mappings, such as the Jordan-Wigner transformation, are not local or of constant depth, so the Pauli weights of the associated edge and / or vertex operators do not remain constant as the size of the fermionic system increases. This can be a drawback because the computational burden increases as the system size increases. In the Jordan-Wigner mapping, a fermionic mode is mapped to a single qubit, and all fermionic vertex operators are mapped to single-qubit operators. All fermionic modes are arranged on a linear chain, and nearest neighbors (NN) on this connectivity graph are connected by fermionic edge operators, which translate to two-qubit operators. In this mapping, for interacting non-NN fermionic modes, the Pauli weights of the qubit operators scale with the distance on the chain and, in the worst (exhaustive) case, with the system size.

[0058] Prior art mappings that utilize the fSWAP operation in conjunction with Jordan-Wigner mapping as an fSWAP network have allowed for improved scalability with system size, but in order to combine all two-mode pairs in the mapping, these methods require circuit depths that are not optimal for many use cases.

[0059] Known alternatives to the Jordan-Wigner mapping include the use of ancilla qubits to "fix" fermionic exchange relations and allow for fermionic connectivity graphs higher than one dimension. Ancilla qubits may be qubits that are not associated with fermionic modes (i.e., not associated with vertex operators). Here, the operator weights are constant over the size of the fermionic system, but the problem with such known mappings is that they are tailored to a specific fermionic lattice and cannot be easily generalized to Hamiltonians with dimensions greater than two and / or high degrees of connectivity, such as greater than eight. These known mappings can be tailored in some cases and may be suitable for simulating condensed matter physics systems, such as the Fermi-Hubbard model.

[0060] However, the prior art does not offer a strategy that can be used to obtain optimal or feasible mappings that encompass a range of different types of fermionic systems and associated Hamiltonians. The present invention can provide such mappings.

[0061] The present invention aims to provide a fermion-to-qubit mapping that is well adapted for use on hardware that includes a square lattice qubit layout, where one qubit can be connected to up to four other qubits. Prior art mappings have not been specifically designed or optimized for square qubit layouts. However, square qubit layouts are technically feasible and are layouts that can realistically be offered by current hardware providers.

[0062] A further consideration taken into account by the present invention is to provide a mapping with low-weight composite edge operators for far-neighbor (NN-on-NN) edges that allows for the simulation of a wide class of two-dimensional fermion lattices, for example from condensed matter physics.

[0063] Furthermore, the mappings described herein can provide lower Pauli weights than prior art mappings and / or mappings in which the Pauli weights are independent of the fermion lattice dimensions and / or the number of modes per fermion lattice site.

[0064] The optimality of the mapping for a selected use case, taking into account a particular type of Hamiltonian and / or fermionic system and / or a particular type of hardware, or providing a mapping that takes into account such criteria, has not previously been considered in the prior art.

[0065] More specifically, the mappings considered herein may be determined with respect to the following criteria:

[0066] A qubit can preferably interact (via two-qubit gates) with up to four neighboring nodes on the square qubit lattice connectivity graph, All vertex and edge operators should have Pauli weights that are constant in the total number of fermion lattice sites and the number of modes per lattice site M / L, and the Pauli weights are preferably as low as possible, or at least lower than those considered in some prior art mappings; The number of fSWAPs used is preferably two-dimensional (L x and L y etc.), preferably scaling linearly with the number of fermionic lattice sites and fermionic modes per third dimension; and / or The mapping should preferably be universal to at least a selected degree, so that it can produce (nearly) optimal results when applied to at least a selected amount of different fermionic systems with respect to the number of fermionic lattice sites, number of fermionic modes, and / or number of fermionic operator terms (sparsity / density) considered.

[0067] When the mapping includes edge and vertex operators selected to provide low or optimized Pauli weights, the resulting quantum circuit may also be provided with a low or optimized circuit depth, as circuit depth may be strongly related to the Pauli weights of the operators. The depth of a quantum circuit may refer to the number of time steps required for its completion. Therefore, because quantum computations can experience increased errors as the time required for computation increases, reducing circuit depth may result in reduced errors in addition to reduced computation time. In the context of devices in the noise-intermediate-scale quantum (NISQ) era, reducing circuit depth may be important due to low fidelity. The circuit depth provided by the present invention may be advantageous, where circuit depth is measured in terms of the number of unique two-qubit gates per Trotter layer.

[0068] If the number of fSWAPS is constant in two dimensions and can be scaled linearly with the number of fermion modes per site and in the third dimension, then the number of utilized fSWAPS is O(L z M / L). This can also ensure that the circuit depth does not increase with increasing system size more than in the prior art.

[0069] If fermion-qubit mapping were available for a selection of different types of fermionic systems, this could reduce the otherwise significant overhead in human labor that results from investigating the mapping of different fermion models on a case-by-case basis.

[0070] Many prior art mappings have been determined with a view to optimizing low qubit-mode ratios, but these may result in unfavorable circuit depths. Mappings determined through the present invention may be determined to optimize or consider lower or lowest possible circuit depth as an optimization criterion more important than qubit-mode ratio, which may be considered a secondary optimization criterion.

[0071] The present invention may be performed by one first computing device or processor, or by multiple computing devices. The determination of the mapping and / or the determination of the associated control sequence may be performed by a first computing device comprising a classical computer or processor. The first computing device may receive one or more inputs provided, for example, by a user of the first computing device or by another computing program that determines the input(s) based on additional information provided by the user; the additional information may be, for example, properties describing the quantum device in which the particular fermionic system is implemented (such as the number of qubits utilized and / or information about native gates) and / or desired properties of the fermionic system or associated Hamiltonian. Any inputs may additionally or alternatively be obtained, for example, via a database.

[0072] The mapping and / or control sequence (corresponding to the quantum circuit) may be provided as output by the first computing device. The output may be used to control the quantum device. One embodiment of the present invention may be considered a compiler for providing a quantum circuit.

[0073] The present invention may be usable with at least one quantum device comprising at least one quantum processor. The quantum processor may comprise a plurality of qubits or other quantum elements arranged in a square lattice layout. Each quantum element may be connectable / coupleable with up to four other adjacent quantum elements on the square lattice. The quantum processor may also include a plurality of qubits arranged in a plurality of square lattice qubit layouts.

[0074] In one embodiment, the invention can include a method for determining a fermion-qubit mapping used to simulate a fermion Hamiltonian on a quantum device. The method can further include determining a control sequence for performing a series of qubit interactions on a plurality of qubits on the quantum device based on the mapping, the plurality of qubits on the quantum device arranged in a two-dimensional square lattice qubit layout having horizontal and vertical dimensions, with each qubit arranged to interact with up to four neighboring qubits.

[0075] The method may include receiving or obtaining parameters of a fermionic Hamiltonian to be considered. The obtained parameters may include at least the number of fermionic lattice sites L, the number of fermionic modes M in the fermionic lattice, and a fermionic operator corresponding to interactions between the fermionic modes, i.e., the connectivity of the Hamiltonian to be considered, including information about which interaction terms (e.g., NN-only or nonlocal interactions) should be considered.

[0076] The parameters of the fermion Hamiltonian may be obtained from a user, for example as user input. Additionally or alternatively, at least some parameters of the fermion Hamiltonian may be received, for example via a database.

[0077] The method may then include projecting the fermionic lattice onto a qubit layout of the quantum device such that every fermionic mode is assigned to a distinct qubit of the quantum device. A qubit associated with a fermionic mode may be referred to as a physical qubit P, where the mapping between the fermionic mode and the physical qubit is one-to-one. In the method, not all of the qubits of the quantum device involved in the projection are physical qubits P associated with fermionic modes. Multiple additional qubits involved in the projection are referred to as ancilla qubits A, which are not assigned any fermionic modes.

[0078] It may be noted that "physical" and "ancilla" qubits may be physically equivalent: the same hardware qubit used as a physical qubit P may also be used as an ancilla qubit A.

[0079] The physical qubits P and ancilla qubits A may be arranged in rows in the horizontal dimension of a two-dimensional square lattice qubit layout, with each row including one or more strings P' and one or more ancilla qubits A. The strings P' and ancilla qubits A may be arranged within each row such that there are any number, and more preferably up to two consecutive strings P' and up to two consecutive ancilla qubits. Each string of physical qubits P' may include at least one or more physical qubits P.

[0080] The strings P' and ancilla qubits A may be arranged to follow one or more selected patterns. A pattern may refer to a selected number of consecutive strings P' and ancilla qubits A. A pattern may include any number of strings P' and any number of ancilla qubits A. In advantageous embodiments, a pattern may include one or two strings P' and one or two ancilla qubits A. A maximum of two consecutive ancilla qubits A in a row may yield optimal results. In some embodiments, three or more consecutive strings P' in a row may be utilized. Some use cases for this may be using particular fermion models with rare vertical edge operators and / or arbitrarily reducing the qubit-to-mode ratio.

[0081] The resulting row of qubits in the horizontal dimension obtained by the selected arrangement of physical qubits P and ancilla qubits A is repeated along the vertical dimension to obtain the two-dimensional lattice of physical qubits and ancilla qubits utilized in the mapping. Thus, the arrangement of physical qubits and ancilla qubits in each row of the qubit layout is the same, and each column of the qubit layout used in the mapping contains only either physical qubits P or ancilla qubits A. This can be particularly useful because the qubit operation may involve fewer physical qubits P, and therefore the columns of ancilla qubits can be utilized to perform qubit interactions more efficiently than some other mappings.

[0082] After determining the arrangement of the physical qubits P and ancilla qubits A in the same row, each physical qubit is associated with at least one edge operator E and one vertex operator V. The edge and vertex operators are then utilized to map each interaction between fermionic modes to a qubit operator, after which a control sequence of the qubit interactions may be determined, the control sequence including at least the qubit operator determined by the mapping.

[0083] The patterns may be arranged sequentially in a row of qubits to provide a successive set of qubits, and the patterns may be repeated in a row of qubits. For example, a row of qubits may include one selected pattern that is repeated, such as P'A, as further illustrated in the example below. A row of qubits may also include sets of qubits with multiple different patterns; for example, a row of qubits may include a first set having a first pattern, such as P'A, a second set having a second pattern, such as P'P'A, and a third set having a third pattern. Any combination of different types of patterns in successive sets may be considered.

[0084] 1 shows an example of an arrangement of string P' and an ancilla qubit A into rows of qubits or horizontal single lines of qubits that are repeated to obtain a square lattice qubit layout. The total number of qubits shown, the number of patterns, the number of times the pattern is repeated in a horizontal row, and the number of rows in the vertical dimension are merely exemplary. The quantum device used may also include more qubits than the qubits utilized in the projection or mapping.

[0085] In an exemplary embodiment, the pattern may be selected from the group consisting of P'A, P'P'A, P'AA, and P'P'AA. FIG. 1A shows an arrangement of physical qubits and ancilla qubits in pattern 102, including an arrangement of set P' and ancilla qubit A as P'A. FIG. 1B includes an arrangement in which pattern 102 is P'P'A, while FIGS. 1C and 1D show arrangements with patterns P'AA and P'P'AA, respectively. Set P' is shown as a square, and ancilla qubits are shown as circles. The qubit-to-mode ratios of possible patterns, when considering only modes per fermion site, i.e., P'=P, are P'A2.0, P'P'A1.5, P'AA3.0, and P'P'AA2.0.

[0086] A row of qubits can be obtained by selecting a pattern and repeating the pattern in the horizontal dimension to obtain a desired / determined total number of strings P' and ancilla qubits A for that row. The pattern may be repeated a selected number of times, or the pattern may be repeated such that the pattern is only partially provided at the beginning and / or end of a row or horizontal single line of qubits. For example, a row of qubits having pattern P'P'A may include strings P' and ancilla qubits A arranged as P'AP'P'AP'P'AP', i.e., the pattern is repeated twice in its entirety, with the sequence "cut" as P'A at the beginning of the row and P' at the end of the row.

[0087] The number of consecutive physical qubits P that the string P' corresponds to or includes may be obtained based on the fermionic system and / or Hamiltonian considered. For example, if the fermionic system includes two fermionic modes per fermionic lattice site, then the pattern of P'AA corresponds to PPAA when implemented in a row of qubits in a qubit layout.

[0088] The horizontal rows of qubits can correspond to a sequence of qubits that can be obtained by further consideration of the fermionic system. The obtained or known information may indicate the number of fermionic lattice sites in two or three dimensions, including the number of fermionic lattice sites in a first dimension L1, the number of fermionic lattice sites in a second dimension L2, and optionally the number of fermionic lattice sites in a third dimension L3, where the total number of fermionic lattice sites is L = L1L2 when two dimensions are considered, or L = L1L2L3 when three dimensions are considered.

[0089] In one embodiment, each fermionic mode in the fermionic Hamiltonian can be identified by an index i, j, k that indicates the location of the fermionic lattice site along the respective dimension, where i = [1, L1], j = [1, L2], k = [1, L3], and the total number of fermionic modes in the fermionic lattice is M = M = Σi=1 L1 Σ j=1 L2 Σ k=1 L3 M ijk Let M be the number of fermionic modes at each of the fermionic lattice sites identified by indices i, j, and k, so that ijk where each fermion mode in the fermion Hamiltonian can be identified by four indices i, j, k, l, where l = [1, M ijk ]. Projecting the fermion lattice onto the square lattice qubit layout of the quantum device then maps each fermion mode, identified by the four indices i, j, k, and l of the fermion lattice, onto the physical qubit P of the quantum device. ijkl , where the index of each physical qubit identifies the fermion mode to which the physical qubit is assigned.

[0090] A physical qubit P is assigned a fermionic mode, each identified by indices i, j, k, and l. ijkl may be arranged in a qubit layout such that each row of the qubit layout includes L strings P′, each string being a physical qubit P where the string is assigned a fermionic mode associated with a fermionic lattice site having position indices i and j. ijkl Show that P' contains the number ij and the string P' ij The number of physical qubits in Σ k=1 L3 M ijk is equal to each physical qubit with a lower index i is positioned in the horizontal dimension before any physical qubit with a higher index i; the number of rows in the qubit layout involved in the projection is equal to L, and each physical qubit with a lower index j is positioned in front of any physical qubit with a higher index j in the vertical dimension; The physical qubits in each string P' are associated with varying indices k and l, which place their respective orderings according to fermionic interactions. Associating each physical qubit P with at least one edge operator E and one vertex operator V comprises: Each physical qubit is assigned to the vertex operator V p V p is a first type of Pauli operator selected from types X, Y, and Z of Pauli operators and acting on physical qubit p; For any pair of physical qubits p and q, they are either directly adjacent in the horizontal dimension with no physical qubits or ancilla qubits in between, or separated by one or two ancilla qubits, and there is a horizontal edge operator E associated with said qubits. pq H To define E pq H is the tensor product of the number of Pauli operators, at least two Pauli operators of a second or third type, respectively, selected from Pauli operator types X, Y, and Z, acting on qubits p and q, respectively; and if there are any ancilla qubits between physical qubits p and q along the horizontal dimension, defining a tensor product of a number of Pauli operators including: When two horizontal edge operators act on the same qubit q, the first of the two horizontal edge operators, E pq H1 acts on the qubit q using a second kind of Pauli operator, the second horizontal edge operator E pq H2 acts on the qubit q using a third kind of Pauli operator, and vice versa.

[0091] To relate is For any pair of physical qubits p and q, which are directly adjacent in the vertical dimension with no physical qubits or ancilla qubits in between, such qubit pair is adjacent to a pair of ancilla qubits a and b, such ancilla qubits a and b are directly adjacent in the vertical dimension and are located adjacent to qubits p and q, respectively, and a vertical edge operator E associated with such qubits p, q, a, and b is pq V and E pq V is a product of four Pauli operators, each of the second or third kind, acting on one of the qubits p, q, a, b, such that each acts on a different qubit, The Pauli operators acting on the ancilla qubits a and b are of different types, the Pauli operator acting on the physical qubit p is of the same kind as the Pauli operator acting on the physical qubit q and forming part of the horizontal edge operator acting on at least the physical qubit p and the ancilla qubit a; similarly, the Pauli operator acting on the physical qubit q is of the same kind as the Pauli operator acting on the physical qubit p and forming part of the horizontal edge operator acting on at least the physical qubit q and the ancilla qubit b; If ancilla qubits a and b are located along the horizontal dimension on a first side, e.g., to the right, of physical qubits p and q, respectively, then the vertical edge operator is a first vertical edge operator E pq V1 and if the ancilla qubits a and b are located along the horizontal dimension on a second side, e.g., to the left, of the physical qubits p and q, then a second vertical edge operator E pq V2 It is called.

[0092] Furthermore, when two vertical edge operators act on the same ancilla qubit, if one of the two vertical edge operators acts on said ancilla qubit using a second type of Pauli operator, then the other of the two vertical edge operators can act on said ancilla qubit using a third type of Pauli operator, and vice versa.

[0093] In most use cases, for any first and any second vertical edge operators acting on ancilla qubits a1, b1 and a2, b2, respectively, if ancilla qubits a1 and a2 are located on the same side along the vertical dimension of ancilla qubits b1 and b2, respectively, the Pauli operator of the first vertical edge operator acting on ancilla qubit a1 is of a different type than the Pauli operator of the second vertical edge operator acting on ancilla qubit a2. A first qubit located on a first side of a second qubit can refer to a qubit adjacent to the second qubit in a row or horizontal line of qubits, and a third qubit on a second side of the second qubit is adjacent to the second qubit in the row of qubits but is located on the opposite side from the first qubit.

[0094] The selection of the types of Pauli operators to be used (i.e., which Pauli operators are selected as the first type, second type, and third type) may depend on the quantum device considered by the mapping. Consideration may be given to which types of gates are native to the quantum device, and the types of Pauli operators assigned to vertex and edge operators may be adapted to tailor the mapping according to the native gates of the qubit hardware utilized.

[0095] The type of Pauli operator to be used may, in one embodiment, be selected separately for each physical qubit P. For example, for the first physical qubit, the first type of Pauli operator is σ Z and for the second physical qubit, the Pauli operator of the first kind is σ YIn such a case, for example, the vertex operators may be different for different physical qubits P.

[0096] However, in connection with some advantageous mappings, the vertex operators may be equivalent for all physical qubits P. An example of a choice of Pauli operator type for the vertex operators is V p =Z p and Z p is the Pauli operator σ acting on the qubit p Z The vertex and edge operators corresponding to this choice are shown in FIG. 2. FIG. 2A shows the vertex operators, FIG. 2B shows possible horizontal edge operators, and FIG. 2C shows possible vertical edge operators that may be used in connection with most qubits and qubit layouts. FIG. 2D shows a vertical edge operator that may be used in rare cases where two related physical qubits P can be connected without an ancilla qubit A. This situation may arise at the boundary of finite systems, for example, in connection with honeycomb lattice geometries, where an ancilla qubit may not need to be used between at least some of the possible pairs of physical qubits at the edges of the qubit lattice.

[0097] The different possibilities of the edge operators depend on the placement of ancilla qubits in the vicinity of the considered physical qubit. In Figure 2, the physical qubit (or physical qubit string P') is shown as a square and the ancilla qubit is shown as a circle, but the notations X / Y and X / Y relating to physical qubits within the same operator are used when both qubits are in σ X operator or σ Y (In the following diagrams, the notation X / Y and Y / X relating to physical qubits within the same operator indicates that one qubit is acted upon by an operator corresponding to σ X or σ Y , and the other qubit is acted upon by an operator σ that is different from the operator acting on the other qubit. X or σ Y These operator choices indicate that the native gate is JPEG2026505191000016.jpg1433 and JPEG2026505191000017.jpg1452 Such an operator choice is also advantageous for quantum devices whose native gates correspond to fSIM gates (covering both previous and further gates), JPEG2026505191000018.jpg14164 is.

[0098] Figure 2B shows that the Pauli ring corresponding to the horizontal edge operator is σ z σ on the two end qubits combined with up to two intermediate ancilla qubits acted upon by the operator X or σ Y We show that it can be formed by either of the operators σ X and σ Y may be varied between adjacent horizontal edges, ensuring that they anti-commutate with each other. Thus, depending on the choice of pattern used, the Pauli weights of horizontal edges can be chosen between 2 and 4.

[0099] As can be seen in Figure 2C, all vertical edge operators as illustrated in relation to the qubit layout typically have a square shape containing two physical qubits P and two ancilla qubits A. The physical qubit P may be either to the left or to the right of the ancilla qubit A, and both Pauli operators acting on the physical qubit P in the vertical edge operator, as well as the horizontal edge operators acting on the ancilla qubit A and the physical qubit P, have a square shape σ X or σ Y , and the choice of which horizontal edge operator to anticommutate with must be the same as the horizontal edge operator used in the same mapping.

[0100] In the present invention, a placement of ancilla qubits A and physical qubits P in a row of qubits or patterns may be first selected, and then edge and / or vertex operators may be selected based on the placement of the qubits or patterns. Alternatively, edge and / or vertex operators may be determined for a number of different placements or patterns, and placements of physical qubits P and ancilla qubits A may then be selected based on criteria that may optionally depend at least in part on properties of the operators.

[0101] In one embodiment, the physical qubits P of a row of qubits in the horizontal dimension may be grouped into pairs of physical qubits labeled alternatingly as even or odd, where ancilla qubits A are not considered, and such odd or even neighboring pairs of labeling qubit pairs share a physical qubit P, and associating each physical qubit P with at least one edge operator comprises: For each even pair of physical qubits p and q, and one or more ancilla qubits between them, if present, an even horizontal edge operator E associated with the qubits pq H1 to define If p and q are directly adjacent and not separated by an ancilla qubit a, then E pq H1 is the product of two Pauli operators of the second kind acting on the physical qubits p and q, respectively, and optionally E pq H1 =X p X q and X is the Pauli operator σ acting on the physical qubit p or q. X and If p and q are separated by one ancilla qubit a, then E pq H1 is a product of a second type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, and a second type Pauli operator acting on the physical qubit p, and optionally E pq H1 =Xp Z a X q and X is the Pauli operator σ acting on the physical qubit p or q. X and Z is the Pauli operator σ acting on the ancilla qubit a. Z and If p and q are separated by two ancilla qubits a and b, then E pq H1 is a product of a second type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, a first type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit p, and optionally E pq H1 =X p Z a Z b X q and defining For each odd pair of physical qubits p and q and, if present, one or more ancillary qubits in between, an odd horizontal edge operator E associated with said qubits pq H2 to define If p and q are not separated by an ancilla qubit but are directly adjacent, then E pq H2 is the product of two Pauli operators of a third kind acting on the physical qubits p and q, respectively, and optionally E pq H2 =Y p Y q and Y is the Pauli operator σ acting on the physical qubit p or q. Y and If p and q are separated by one ancilla qubit a, then E pq H2 is a product of a third type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, and a third type Pauli operator acting on the physical qubit p, and optionally E pq H2 =Y p Z a Y qand If p and q are separated by two ancilla qubits a and b, then E pq H2 is a product of a third type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, a first type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit p, and optionally E pq H2 =Y p Z a Z b Y q and defining For each pair of physical qubits p and q, a vertical edge operator E pq V to define If ancilla qubits a and b are directly adjacent to p and q in the horizontal dimension in a first direction, and p and q have no other ancilla qubits directly horizontally adjacent to them, and a and b are associated with an even horizontal edge operator, then E pq V is a product of a second type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit q, and optionally E pq V =X p X a Y b X q and If ancilla qubits a and b are directly adjacent to p and q in the horizontal dimension in a first direction, and p and q have no other ancilla qubits as direct horizontal neighbors, and a and b are associated with odd horizontal edge operators, then E pq Vis a product of a third type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit q, and optionally E pq V =Y p X a Y b Y q and If ancilla qubits a and b are directly adjacent to p and q in the horizontal dimension in a second direction, and p and q have no other ancilla qubits as direct horizontal neighbors, and a and b are associated with an even horizontal edge operator, then E pq V is a product of a second type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit a, a second type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit q, and optionally E pq V =X p Y a X b X q and If ancilla qubits a and b are directly adjacent to p and q in the horizontal dimension in the second direction, and p and q have no other direct ancilla qubits in the horizontal dimension, and a and b are associated with odd horizontal edge operators, then E pq V is a product of a third type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit a, a second type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit q, and optionally E pq V =Y p Y a X b Y q and If p and q are immediately adjacent to ancilla qubits a and b in the horizontal dimension in a first direction, and ancilla qubits c and d are immediately adjacent in a second direction, then For the first direction, E pq V is a product of a second type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit q, and optionally E pq V =X p X a Y b X q or E pq V is a product of a third type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit q, and optionally E pq V =Y p X a Y b Y q and For the second direction, E pq V is a product of a second type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit c, a second type Pauli operator acting on the ancilla qubit d, and a second type Pauli operator acting on the physical qubit q, and optionally E pq V =X p Y c X d X q or E pq V is a product of a third type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit c, a second type Pauli operator acting on the ancilla qubit d, and a third type Pauli operator acting on the physical qubit q, and optionally E pq V =Y p Y c X d Y qand defining that:

[0102] In the above-described embodiments, preferably, X corresponds to a second type of Pauli operator acting on either a physical qubit or an ancilla, e.g., X when acting on a physical qubit p p and when acting on an ancilla qubit c, it is denoted as X c Preferably, Y corresponds to a third type of Pauli operator acting on either a physical qubit or an ancilla, e.g., Y when acting on an ancilla qubit c. c and when acting on a physical qubit p, Y p Preferably, Z corresponds to a first type of Pauli operator acting on either a physical qubit or an ancilla, e.g., Z when acting on an ancilla qubit c. c and when acting on the physical qubit p, Z p It is shown as follows.

[0103] A pattern may be selected based on determining the maximum Pauli weight of the edge operators associated with at least two patterns and selecting the pattern associated with the lowest or selected maximum Pauli weight.

[0104] The pattern may be selected based on determining the circuit depth of control sequences separately associated with at least two patterns and selecting the pattern associated with the lowest or selected circuit depth.

[0105] The patterns may be selected by determining the number and / or types of gates included in control sequences separately associated with at least two patterns, and selecting the pattern associated with the lowest number of gates or selected gates and / or the pattern associated with the selected type of gates that includes at least a selected amount of native gates. The qubit-to-mode ratio may also be considered when selecting the patterns.

[0106] The operators according to Figures 2A, 2B, 2C and 2D can determine a pool of operators based on which selected operators can be used to determine the mapping and select the pattern.

[0107] When selecting a pattern or row of qubits based on an operator from a pool of operators and considering different criteria for selection, when selecting horizontal edges of weights −2 and −3 as opposed to horizontal edges of weight −4, a trade-off may be required between lowering the qubit-to-mode ratio (edges of weights −2 and −3) and higher parallelism in implementing the corresponding vertical edge operator (horizontal edge of weight −4).

[0108] 3A-3E show an example of vertex and edge operators for a qubit layout corresponding to the selection of a P'A pattern that is repeated in the horizontal dimension to obtain rows of qubits. This example may be particularly well suited for simulations involving a spinless Fermi-Hubbard model, in terms of the pattern selection for the qubit layout and / or the selection of edge and vertex operators. In this example, one fermion lattice site contains only one mode. In this example, the fermion lattice contains five lattice sites in the first dimension and five lattice sites in the second dimension (L1=5, L2=5, L3=1). The patterns, selected operators, and properties of the fermion system are merely exemplary.

[0109] In the example of Figure 3, a fermionic Hamiltonian may be mapped to a qubit lattice containing at least 10 qubits in the horizontal dimension and at least 5 qubits in the vertical dimension. In Figure 3, squares represent physical qubits P, and circles represent ancillary qubits A. Figures 3A and 3C show horizontal edge operators. Figures 3B and 3D show vertical edge operators, and Figure 3E shows a vertex operator. Each diagram represents a group of operators that may be implemented in parallel because they act on different / independent qubits. Operators are depicted by labeling the associated qubit with a letter corresponding to the type of associated Pauli operator and showing the associated qubits and connections with bold lines. Selected regions 302, 304, 306, 308, and 310 can help visualize one of the depicted operators, with 302 and 306 referring to horizontal edge operators, 304 and 308 referring to vertical edge operators, and 310 referring to a vertex operator.

[0110] With respect to the fermion Hamiltonian to be considered, it can be shown that (quadratic) fermion hopping operators are of the same Pauli weight as the edge operators connecting the physical qubits associated with the modes for which they are defined (this implies a Pauli weight of 2, 3 or 4). For (quadratic) interaction operators acting on only two modes, as in the Fermi-Hubbard model, the Pauli weight is twice that of a single vertex operator, implying a Pauli weight of 2.

[0111] If there are two or more fermionic modes per fermionic lattice site, the physical qubits corresponding to these modes may be positioned adjacent to each other in a horizontal row of qubits. Thus, a string P' may contain at least the number of physical qubits P corresponding to the number of modes contained in the associated fermionic lattice site. A string P' may be associated with only one fermionic lattice site in the first dimension of the fermionic lattice. Physical qubits P associated with the same fermionic lattice site may be connected by two horizontal weighted edges. This may be permissible because only two vertical edge operators may be required per fermionic lattice site. Different modes in the same fermionic lattice site may utilize an fSWAP network to group the modes together, thus eliminating the need for further vertical connections via edge operators.

[0112] The different modes of a fermionic lattice site may refer to different modes associated with different orbitals or different spin types considered in association with a single orbital. A fermionic system may include any number of orbitals, and each orbital may include any number of spin types.

[0113] If the orbitals associated with a fermion lattice site contain multiple different spin types, then these corresponding modes can be advantageously assigned to different physical qubits P such that they are separated by one physical qubit P. This ensures that these modes are no more than one physical qubit P apart when using an fSWPAP network.

[0114] Figure 4 shows an example of edge and vertex operators that may be utilized when the pattern P'AA is selected; if the pattern is repeated, it is truncated so that each row actually begins and ends with one ancilla, and the fermionic system contains two modes per lattice site. Thus, in this case, the string P' corresponds to two physical qubits, namely, PP. The squares represent the physical qubits P, and the circles represent the ancilla qubits A. In this example, each lattice site is considered to contain one orbital, each with two spin types, here "up" and "down." Filled (black) squares correspond to one spin-type mode, e.g., up, and empty (unfilled) squares correspond to the other spin-type mode, e.g., down.

[0115] Figures 4A and 4C show horizontal edge operators, Figures 4B and 4D show vertical edge operators, and Figure 4E shows a vertex operator. The groups of operators in each separate figure act on different qubits and may therefore be implemented in parallel. The operators are depicted by labeling the associated qubit with a letter corresponding to the type of Pauli operator associated with it and showing the associated qubit and connection with a bold line. Selected regions 402 and 406 show horizontal edge operators, 404 and 408 show vertical edge operators, and 410 shows a vertex operator, each shown for both spin-up and spin-down modes. Thus, each edge operator acts on or includes physical qubits assigned corresponding fermionic modes on different fermionic lattice sites.

[0116] 3 and 4 show examples of edge and vertex operators that may be selected relative to a selected pattern or row of qubits. Other types of edge and vertex operators are possible. While the example operators may yield optimal results, e.g., taking into account certain considerations or criteria regarding the resulting quantum circuit, other choices of edge and vertex operators may also, e.g., give the operators lower Pauli weights than prior art mappings.

[0117] It should also be noted that if the fermionic lattice is three-dimensional, each lattice site in the third dimension (L3) may be treated in the projection in the same manner as an additional mode or orbital of the fermionic lattice site in the first dimension. Thus, a string P' may include a number of physical qubits P corresponding to the number of modes contained in its associated lattice site in the first dimension, and a number of physical qubits P corresponding to the number of modes contained in each fermionic lattice site in the third dimension. However, in this case, the resulting circuit depth may not be a constant depth or may scale linearly with the size of the fermionic system, although it should be noted that the circuit depth scales linearly with the third dimension and the number of fermionic modes per lattice site.

[0118] When more than one fermion mode per fermion lattice site is considered, the different modes may be associated with different orbitals and / or different spin types of the same orbital. However, in these cases, string P' may include multiple consecutive physical qubits P forming a chain of modes. A mode associated with a physical qubit at the end of one chain or string P' may readily engage in interaction with one or more modes of further fermion lattice sites associated with physical qubits of further strings P'; such interactions are enabled via edge and vertex operators that can connect qubits associated with modes.

[0119] Thus, in some cases, there may be a chain of fermionic modes associated with the qubits in a row, with one or more modes initially located at intermediate sites in the string being internal modes associated with the internal physical qubits of string P. Such internal modes may be advantageously rearranged to be involved in one or more interactions such that they may be located at the end positions of the chain of fermionic modes.

[0120] The rearrangement of fermionic modes can be performed by utilizing fSWAP operators such that one or more internal modes are shifted in a first direction or a second direction (left or right) along a row of qubits. A network of fSWAP operators including M / L-2 parallel fSWAP layers can swap fermionic modes in a string P' of M / L modes by alternating two layers of fSWAP between all adjacent qubit pairs, which can be alternately labeled as either even (e.g., physical qubits numbered 2n, 2n+1) or odd (e.g., physical qubits numbered 2n+1, 2n+2).

[0121] The assignment of modes to an array of rows of qubits may include ordering the modes such that an fSWAP network can be efficiently used to place related modes adjacent to each other on a connectivity graph corresponding to the qubit layout projection. Mode assignment as disclosed herein may lead to efficient simulation of fermionic Hamiltonians, as multiple operations may be performed in parallel. Advantageously, there may be no physical qubits assigned fermionic modes that are not involved in qubit interactions in a layer of a control sequence or quantum circuit to which a qubit interaction corresponding to the fermionic interaction applies.

[0122] 5 shows an example of a mode associated with one fermion lattice site and how the fSWAP network may be selected to provide an optimal fSWAP network for this particular example. The fSWAP network may comprise multiple fSWAP operators, where the network comprises multiple layers, each layer comprising one or more fSWAP operators, with the operators in one layer operating on separate / different qubits.

[0123] 5A-5C each show a physical qubit P having a mode assigned to a fermion lattice site. The physical qubits in FIG. 5 may represent, for example, a portion of a qubit layout or row of qubits, and may be included in a larger qubit layout that constitutes a string P' of qubit layouts. In the example of FIG. 5, the fermion lattice site under consideration includes eight modes.

[0124] 5A and 5B show horizontal edge operators (of weight 2), with one operator highlighted by selections 502 and 504, and FIG. 5C shows vertex operators, with 506 highlighting one of the vertex operators.

[0125] Figure 5D illustrates an optimal fSWAP network that can be used in conjunction with the modes illustrated in Figures 5A-5C. The network in Figure 5D can ensure that, during one complete control sequence involving at least the fSWAP network and qubit operators corresponding to the desired Hamiltonian and fermionic system under consideration, each mode of the fermionic lattice site is associated at some point with both ends of the illustrated chain or string, and all mode pairs can be brought together, meaning that they are associated at some point with directly adjacent physical qubits. The first fSWAP layer l1 can exchange the fermionic modes assigned to the physical qubits in each pair of physical qubits labeled with a first label (here, "even"), and the second fSWAP layer l2 can exchange the fermionic modes assigned to the physical qubits in each pair of physical qubits labeled with a second label (here, "odd").

[0126] In the mapping of the present invention, the number of required fSWAP operations may be independent of the dimension of the considered fermionic lattice, at least considering two-dimensional fermionic lattices.

[0127] If the fermionic lattice sites of a fermionic system contain different numbers of modes per fermionic lattice site, one or more "extra" physical qubits P can be utilized as "dummy" modes in at least one string P' (corresponding to one or more strings associated with a fermionic lattice site having a fewer number of modes than another lattice site) to enable efficient utilization of the fSWAP network.

[0128] A composite edge operator can be defined that corresponds to an interaction between any two physical qubits that do not have a direct edge operator defined between them, as long as there are two or more other edge operators defined that act on and correspond to the interaction between the physical qubits, ultimately defining a chain of physical qubits, a chain containing physical qubits, whose endpoints are physical qubits acted on by the composite edge operator, and adjacent physical qubits in the chain are acted on by mutual edge operators.

[0129] The possibility of utilizing composite edge operators allows the simulation of fermionic systems or Hamiltonians with higher than square connectivity (NN) to determine the qubit interactions considered. For example, a version of the Fermi-Hubbard model including next-nearest-neighbor (NNN) hopping terms is thought to represent a minimal description of high-temperature cuprate superconductors. The fourth-order interaction term (U) also makes it extremely difficult to solve with classical methods, especially when U is large.

[0130] The Fermi-Hubbard model on a fermion square lattice geometry is considered the canonical version of the model. However, the Fermi-Hubbard model has also been extensively studied for other lattice geometries, particularly triangular, honeycomb, and kagome lattices, and actual material realizations of these exist. Prior art methods for fermion-qubit mapping often do not consider these other non-square lattice geometries. For example, several studies of fermion-qubit mapping using honeycomb and kagome lattices have been performed, but these determine separate mappings for each fermion lattice geometry. Furthermore, these mappings are often not designed to fit the square lattice of four-connected qubit layouts and are optimized for low qubit-to-mode ratios instead of circuit depth.

[0131] Other models can be considered, such as the Hubbard-Kanamori model, which includes a wider range of fourth-order terms than the Fermi-Hubbard model: interband density between different spins (U1), interband density between same spins (U2), and pair hopping and spin exchange interactions (J).

[0132] The present invention allows for the determination of edge operators (either as direct edge operators determined directly in the connectivity graph, or as composite edge operators that are determined combinations of direct edge operators) that enable all possible NN interactions between the relevant fermionic modes and qubit interactions between all pairs of physical qubits P that correspond to NNN interactions, so that alternative fermionic lattice geometries can also be projected onto the square lattice qubit layout.

[0133] Figures 6A-6I illustrate alternative fermionic lattice geometries (two-dimensional) and associated connectivity between sites that can be efficiently embedded into a square lattice qubit layout using the projection, fermionic site-to-physical qubit association, and mapping of the present invention. Figure 6A illustrates a square lattice with NNN connectivity, Figure 6B illustrates a square lattice with NNN connectivity, Figure 6C illustrates a Shastry-Sutherland lattice, Figure 6D illustrates a checkerboard lattice, Figure 6F illustrates a triangular lattice, Figure 6G illustrates a honeycomb lattice, Figure 6H illustrates a kagome lattice, and Figure 6I illustrates a tetrakis lattice. The methods of the present invention can maximize the use of available qubits or available qubits to associate with fermionic lattice sites present in a square lattice qubit layout. Within at least a selected subset or selected square lattice portion of the square lattice layout of a quantum device, all (physical) qubits can be utilized in a projected lattice geometry.

[0134] The provided circuit depths associated with at least some of the lattice geometries of Figure 6 may be optimal for the present invention. Although the provided circuit depths associated with the lattice geometries of Figure 6 may not be optimal for all lattice geometries, the present invention may still provide advantages because the circuit depth may be at least close to or comparable to that obtained with prior art solutions while providing a high level of abstraction, i.e., the mapping is suitable for use with many types of fermion Hamiltonians, and the additional effort of providing a mapping associated with each lattice geometry individually may be avoided.

[0135] The present invention may enable determining edge operators that connect two physical qubits via a vertical stack of ancilla qubits, where multiple vertical edges may be configured to reach additional neighbors (not connected by the directly determined edge operator), and the cost may increase by one ancilla qubit per unit of distance on the fermionic graph.

[0136] In the present invention, min(D x , D y )=1, and D x , y =max(d x , y ) and d x , y where is the horizontal / vertical component of the Manhattan distance between pairs of physical qubits acted on by a composite edge operator that corresponds to an existing edge operator on the fermionic connectivity graph. The Pauli weight of the determined composite edge operator is d x +d y +d A -1, where d A is the number of vertical ancillary qubit layers between physical qubits.

[0137] 7-9 show an example of applying the method of the present invention to the simulation of a spinless Fermi-Hubbard model (one mode per fermion lattice site) with NN and NNN hopping / interactions. The example considered is a 3x3 fermion lattice (L1=3, L2=3, L3=1) projected onto a qubit layout with three horizontal rows, one row having a PAPAPA (since only one mode per lattice site is considered, a PA pattern where P' equals P is used) arrangement of physical qubits and ancilla qubits.

[0138] Figure 7 shows, in 7A, a fermionic lattice and the connectivity considered. Figures 7B-7I specify (highlighted with bold lines) the different hopping terms on the fermionic lattice to be considered: the first horizontal interaction in Figure 7B, the second horizontal interaction in Figure 7C, the first vertical interaction in Figure 7D, the second vertical interaction in Figure 7E, the first diagonal interaction in Figure 7F, the second diagonal interaction in Figure 7G, the third diagonal interaction in Figure 7H, and the fourth diagonal interaction in Figure 7I. In Figure 7, circles represent different fermionic lattice sites. In separate subfigures, operators that can be applied in parallel are shown on the same lattice.

[0139] Figure 8 shows a qubit layout that can be used to simulate the 3x3 fermion lattice and fermion Hamiltonian, including interactions, shown in Figure 7. In Figure 8, squares represent physical qubits and circles represent ancillary qubits, and Figure 8 also shows the qubit numbering that may be used.

[0140] 9A-9H show hopping operators (i.e., operators including one edge operator multiplied by one vertex operator) that may be determined to be utilized in the fermion-qubit mapping and associated determination of qubit operators that include the qubit layout of FIG. 8 and correspond to the fermion lattice and fermion interactions of FIG. 7.

[0141] The hopping operator shown by Figure 9A corresponds to the first horizontal interaction of Figure 7B, Figure 9B corresponds to the second horizontal interaction of Figure 7C, Figure 9C corresponds to the first vertical interaction of Figure 7D, Figure 9D corresponds to the second vertical interaction of Figure 7E, Figure 9E corresponds to the first diagonal interaction of Figure 7F, Figure 9F corresponds to the second diagonal interaction of Figure 7G, Figure 9G corresponds to the third diagonal interaction of Figure 7H, and Figure 9H corresponds to the fourth diagonal interaction of Figure 7I.

[0142] The horizontal or vertical hopping operator is V i E ij may correspond to E ij are the corresponding edge operators in the horizontal or vertical direction, and the diagonal hopping operators may be obtained as composite operators, and V i E jk E kj where one of the edge operators is a corresponding horizontal edge operator and the other edge operator is a corresponding vertical edge operator.

[0143] Table 1 below shows in column 2 (labeled "PA") the Pauli weights of the different edge operators determined for the example described above with reference to Figures 7-9. Columns 2-5 show the Pauli weights of the edge operators determined in a similar manner for qubit layouts using different patterns. Column 6 shows the Pauli weights of the edge operators associated with the Derby-Klassen (DK) fermion-qubit mapping known in the prior art. The column labels specify the pattern of strings P and ancilla qubits A utilized in the horizontal row of qubits in the qubit layout (each row is identical). The discussion is limited to the case where one mode per lattice site is considered, i.e., P' = P. In prior art mappings, the Pauli weights of the corresponding operators are usually higher. Even if similar or even lower Pauli weights corresponding to some of the determined operators could be achieved with prior art methods, these methods failed to allow for the definition of at least some of the diagonal edge operators without violating the mode independence criterion satisfied by the mapping of the present invention, i.e., the Pauli weights of the diagonal edge operators are independent of the number of modes per fermion lattice site (at least considering square lattice qubit layouts). Even if such a criterion were not considered, the Pauli weights of the corresponding edge operators using prior art methods would not be lower than the Pauli weights achievable with the present invention, at least considering all the determined edge operators. [Table 1] Table 1: Pauli operator weights for edge operators in one embodiment of the present invention

[0144] The qubit layouts considered in the examples of Figures 7-9, which utilize a P'A pattern that is at least partially repeated in each row of the qubit layout, can be advantageous due to the edge operators corresponding to each interaction acting on a different, mutually exclusive set of qubits. This can be readily seen from the example of each of the separate diagrams in Figure 9. Thus, the qubit operations illustrated by each of the separate diagrams can be performed in parallel, thereby reducing the circuit depth of the associated control sequences.

[0145] Furthermore, as can be seen from Table 1 and FIG. 9, the maximum Pauli weight of the considered edge operators in column 2 is 4. This is lower than the Pauli weights of edge operators obtained with other qubit layouts, even when considering the prior art. Therefore, the selected pattern of P′A can provide additional advantages, since the circuit depth can be reduced compared to other cases. It can also be shown that for one fermion mode per lattice site, where P′=P, the edge and vertex operators are arranged in a different topology, similar to the Verstraete-Cirac (VC) mapping known in the prior art, but with different connectivity. This is because VC considers a square grid of qubits, with each qubit attached to an ancillary qubit, which requires qubits with at least five connections, which is not the case for the four-connected square grid of qubits usable in the present invention.

[0146] Furthermore, a mapping that utilizes a P'P'A pattern may also be beneficial because it may be the only mapping (considering prior art mappings) that has a qubit-to-mode ratio of 1.5 while utilizing all qubits in at least a selected portion of the square lattice qubit layout and limiting all horizontal and vertical edge operators to a maximum Pauli weight of 4.

[0147] When a fermion Hamiltonian describes a system with more than one spin type within a fermion lattice site, there is a mode M per each fermion lattice site.ijk may be further assigned to a number o of orbitals, each containing a number s of spins, and M ijk = o*s, where the physical qubits assigned fermionic modes having a spin of one orbit are each separated by one physical qubit, and where if the l indices of the physical qubits assigned to the orbitals are odd, then the fermionic modes associated with the physical qubit with index l are arranged such that increasing index l is associated with increasing spin type, whereas if the l indices of the physical qubits representing the orbitals are even, then the fermionic modes associated with the physical qubit with index l are arranged such that increasing index l is associated with decreasing spin type s.

[0148] The assignment of fermionic modes to a physical qubit, considering one fermionic lattice site containing multiple orbitals and two spin types, is demonstrated in the example of Figure 10. In Figure 10, each row shows a chain of modes, with circles corresponding to the first spin type ("up") and squares corresponding to the second spin type ("down"). (Note that Figure 10 illustrates fermionic modes; the circles and squares in this figure are used to represent types of modes, not physical qubits and ancilla qubits.) Different orbitals are indicated by filling shapes with different patterns, and Figure 10 shows an exemplary case of six orbitals. The numbered rows indicate how the mode placement is altered when using an fSWAP network of the type described in connection with Figure 5D. Thus, the example of Figure 10 shows 12 steps of the fSWAP network being used. It should also be noted that FIG. 10 shows only some of the steps performed in an actual simulation involving an fSWAP network, and that the network includes additional steps to ensure that all modes are shifted across the lattice arrangement so that each mode resides at some point on both the left and right edges of the lattice arrangement.

[0149] As seen in Figure 10, the spin modes of each orbital (which in this case can have two spin modes, one for each spin mode) can be arranged so that they are two-spaced (meaning that there is one spin mode of an adjacent orbital located between spin modes of the same orbital). Once the fermion modes are arranged in this way, when an fSWAP network is utilized, the modes can be moved through the qubit configuration so that all pairwise combinations of the same spin type—here, "up" and "down" modes—are adjacent to each other at some point within the fSWAP network. While the complete fSWAP network is periodic, and only some interactions are required, if not all of the interactions resulting from the entire fSWAP network are required, it is possible to limit fSWAP to a subset of the fSWAP network that allows the desired interactions, thereby saving some steps and avoiding implementing the entire network when not necessary.

[0150] For modes of the same orbit (such as the pair in the example) moving in a first direction when the fSWAP network is applied, the spin types may be further arranged such that the spin of the first type is positioned in a second direction of the spin of the second type. Here, for pairs moving in a rightward direction, the spin "up" type is positioned to the left of the "down" spin type, and for pairs moving in a leftward direction, the spin "up" type is positioned to the right of the spin "down" type. However, the spin types may also be arranged in reverse. The solid arrows in FIG. 10 indicate pairs of modes moving in a leftward direction, and the dashed arrows indicate pairs of modes moving in a rightward direction.

[0151] By projecting fermionic modes onto physical qubits and ancilla qubits as described herein (repeat the arrangement of physical qubits and ancilla qubits in each row of utilized qubits of the quantum device), preferably taking into account the native gates of the device, the assignment of edge and vertex operators can then lead to the determination of qubit operators corresponding to the fermionic operators required by the fermionic Hamiltonian to be considered.

[0152] Note also that in the presently disclosed mapping, the fourth-order spin-exchange and pair-hopping terms always act on two pairs of spin modes within two orbitals, so the fourth-order terms necessary to simulate the Hubbard-Kanamori model can also be considered. These terms can be made adjacent in the fSWAP network by ensuring that the spin pairs of the orbitals always move in the same direction. Considering the example of four orbitals, this can be ensured by arranging the modes in the pattern 1↑, 2↓, 1↓, 2↑, 3↑, 4↓, 3↓, 4↑. It can then be seen that any two spin modes of the orbitals are at most one qubit apart at each step of the fSWAP network.

[0153] The control sequence thus obtained, including the qubit operator, is a string of Pauli operators that can be used to implement the evolution operator of the fermionic Hamiltonian on a quantum device as a quantum circuit. As is well known to those skilled in the art, an exponential function including the fermionic Hamiltonian may be Trotterized, and an exponential function in terms may be implemented. An appropriate fSWAP network can be selected to achieve a desired or optimal (lowest) circuit depth. In our mapping, the fSWAP network can be used to obtain control sequences corresponding to shallower quantum circuits than control sequences determined by prior art methods (prior art mappings, specifically) for the same fermionic system.

[0154] As known to those skilled in the art, the determined control sequence may be implemented on a quantum device to determine at least one property of a fermionic system, such as the ground state energy. Determining the property may also be referred to as simulating the fermionic system. A method for determining at least one property of a fermionic system may include determining, obtaining, or receiving a fermionic Hamiltonian, where at least one property of the system is characterized by the Hamiltonian; determining a control sequence according to a method described herein; implementing the determined control sequence on a quantum device; and applying one or more measurement gates to determine the property of the system.

Claims

1. 1. A computer-implemented method for determining a control sequence for performing a series of qubit interactions on a plurality of qubits on a quantum device to simulate a fermion Hamiltonian H, expressible as a sum of one or more tensor products of Pauli matrices, comprising: the plurality of qubits on the quantum device are arranged in a two-dimensional square lattice qubit layout, with each qubit arranged to interact with up to four neighboring qubits; receiving input parameters of a fermion Hamiltonian to be simulated, the input parameters comprising at least: the number of fermion lattice sites L, the number of fermion modes M in the fermion lattice, and receiving a signal including a fermionic operator corresponding to an interaction between the fermionic modes M; projecting the fermionic lattice onto the qubit layout of the quantum device such that all fermionic modes are assigned to qubits of the quantum device, the qubits being referred to as physical qubits P, the projection between the fermionic modes and the physical qubits P being one to one, and a plurality of further qubits of the quantum device being referred to as ancilla qubits A, the ancilla qubits A not being assigned to any fermionic modes; the physical qubits P and the ancilla qubits A are arranged on single horizontal lines of the two-dimensional square lattice, each single horizontal line including at least one string P′ and at least one ancilla qubit A, and each string P′ including one or more physical qubits P; a projection in which the arrangement of physical qubits P and ancilla qubits A in each single horizontal line of the qubit layout is the same; associating each physical qubit P with at least one edge operator E and one vertex operator V; mapping each fermion operator to a qubit operator based on the edge operator E and the vertex operator V; determining a control sequence for qubit interactions that includes at least the qubit operator determined by the mapping.

2. 2. The method of claim 1 , wherein each qubit operator comprises a product of vertex operators V, each of the vertex operators V operating on at least one of the physical qubits P assigned to the fermionic mode of the fermionic operator.

3. 2. The method of claim 1 , wherein each qubit operator comprises a product of at least one of the vertex operators V and one of the edge operators E, and wherein each of the vertex operators V and the edge operators E in the product operates on at least one of the physical qubits P assigned to the fermionic mode of the fermionic operator.

4. 4. The method of claim 1, wherein associating each physical qubit P with at least one edge operator E and one vertex operator V comprises determining the edge operators E and vertex operators V such that the edge operators and vertex operators acting on each pair of the same physical qubit P anti-commutate.

5. Associating each physical qubit P with at least one edge operator E and one vertex operator V comprises: Each physical qubit P is assigned to a vertex operator V ｐ and V ｐ is a first type of Pauli operator selected from types X, Y, and Z of Pauli operators and acting on the physical qubit p; For any pair of physical qubits p and q, they are either directly adjacent in the horizontal dimension with no physical qubits or ancilla qubits in between, or separated by one or two ancilla qubits, and a horizontal edge operator E associated with the qubits ｐｑ Ｈ Defining E ｐｑ Ｈ is the product of the number of Pauli operators, at least two Pauli operators of a second or third type, respectively, selected from Pauli operator types X, Y, and Z, acting on qubits p and q, respectively; and a product of a number of Pauli operators, each of which includes an additional Pauli operator of a first type, if present, acting on each of said ancilla qubits, if any, between said physical qubits p and q along the horizontal dimension; When two horizontal edge operators act on the same qubit q, the first horizontal edge operator E ｐｑ Ｈ１ acts on the qubit q using a second type of Pauli operator, the second of the two horizontal edge operators E ｐｑ Ｈ２ acts on the qubit q using a Pauli operator of the third kind, and vice versa; For any pair of physical qubits p and q, the physical qubits p and q are immediately adjacent in the vertical dimension, the pair of qubits is adjacent to a pair of ancilla qubits a and b, the ancilla qubits a and b are immediately adjacent in the vertical dimension, the ancilla qubits a and b are located adjacent to the qubits p and q, respectively, and a vertical edge operator E associated with the qubits p, q, a, and b is ｐｑ Ｖ Defining E ｐｑ Ｖ is a product of four Pauli operators, each of a second or third type acting on one of the qubits p, q, a, b, such that each of the four Pauli operators acts on a different qubit; the Pauli operators acting on the ancilla qubits a and b are of different types, the Pauli operator acting on the physical qubit p is of the same type as the Pauli operator acting on the physical qubit q and forming part of the horizontal edge operator acting on at least the physical qubit p and the ancilla qubit a; similarly, the Pauli operator acting on the physical qubit q is of the same type as the Pauli operator acting on the physical qubit p and forming part of the horizontal edge operator acting on at least the physical qubit q and the ancilla qubit b; a first vertical edge operator E if the ancilla qubits a and b are located on a first side of the physical qubits p and q, respectively, along the horizontal dimension; ｐｑ Ｖ１ or, if the ancilla qubits a and b are located on a second side of the physical qubits p and q along the horizontal dimension, a second vertical edge operator E ｐｑ Ｖ２ and defining, 5. The method of claim 1, wherein, when two vertical edge operators act on the same ancilla qubit, if one of the two vertical edge operators acts on the ancilla qubit with a Pauli operator of a second type, then the other of the two vertical edge operators acts on the ancilla qubit with a Pauli operator of a third type, and vice versa.

6. 6. The method of any one of claims 1 to 5, wherein a single horizontal line of qubits comprises at most two consecutive ancilla qubits A.

7. The step of receiving parameters of the fermion Hamiltonian to be simulated includes receiving parameters of the fermion Hamiltonian, the number of fermion modes in each fermion lattice site, the first dimension L １ the number of fermion lattice sites in the second dimension L ２ the number of fermion lattice sites in the ３ and obtaining the number of fermion lattice sites in the total number of fermion lattice sites L = L １ L ２ L ３ where the fermion lattice sites are identified by indices i, j, k that indicate the fermion lattice site position along the respective dimensions, where i = [1, L １ ], j = [1, L ２ ], k=[1, L ３ ], and the number of fermionic modes in each fermionic lattice site identified by indexes i, j, k is such that the total number of fermionic modes in the fermionic lattice is M=Σ ｉ＝１ Ｌ１ Σ ｊ＝１ Ｌ２ Σ ｋ＝１ Ｌ３ M ｉｊｋ As M ｉｊｋ and each fermion mode of the fermion Hamiltonian is identified by four indices i, j, k, l, where l = [1, M ｉｊｋ ], and the step of projecting the fermion lattice onto the qubit layout of the square lattice of the quantum device includes projecting each fermion mode identified by four indices i, j, k, and l of the fermion lattice onto a physical qubit P of the quantum device. ｉｊｋｌ 7. The method of claim 1, comprising assigning an index for each physical qubit to the fermion mode to which the physical qubit is assigned.

8. 10. The quantum device of claim 1, wherein the step of projecting the fermion lattice onto the qubit layout of the quantum device includes projecting the fermion lattice onto the qubit layout of the physical qubits P, each of which is assigned a fermion mode identified by the indices i, j, k, l in the qubit layout. ｉｊｋｌ so that: Each horizontal single line of the qubit layout is L １ strings P′, each string being assigned a physical qubit P to which the string is assigned a fermionic mode associated with a fermionic lattice site having position indices i and j. ｉｊｋｌ To show that P' contains the number ｉｊ and the string P' ｉｊ The number of physical qubits in Σ ｋ＝１ Ｌ３ M ｉｊｋ is equal to each physical qubit with a lower index i is positioned in the horizontal dimension before any physical qubit with a higher index i; The number of horizontal single lines in the qubit layout involved in the projection is L ２ where each physical qubit with a lower index j is positioned in the vertical dimension before any physical qubit with a higher index j; 8. The method of claim 7, wherein the physical qubits in each string P' are associated with varying indices k and l to arrange their respective orderings according to fermionic interactions.

9. 9. The method of claim 8, wherein the method comprises arranging the physical qubits in any string P′ in the horizontal dimension such that the order of physical qubits with indexes k and l is reversed in successive strings P′.

10. 10. The method of claim 1, wherein if one or more of the fermion lattice sites include two or more modes associated with the fermion lattice site, the control sequence further includes one or more fSWAP operators for reordering the fermion modes assigned to the physical qubits by the projection to implement interactions between fermion modes included in the fermion Hamiltonian, for which no qubit operators have been applied prior to the reordering.

11. If the fermion Hamiltonian describes a system with more than one spin type within a fermion lattice site, then the fermion modes M per each fermion lattice site are ｉｊｋ is further assigned to a number o of orbitals, each of which contains a number s of spins, and M ｉｊｋ 11. The method of claim 1, wherein σ = o*s and the physical qubits assigned to fermionic modes having the spin of one orbit are each separated by one physical qubit.

12. the physical qubits P in a single horizontal line of qubits are grouped into pairs of physical qubits, the pairs being alternately labeled as odd or even, and such odd or even neighboring pairs labeling qubit pairs share a physical qubit P; and associating each physical qubit P with at least one edge operator comprises: For each even pair of physical qubits p and q, an even horizontal edge operator E associated with said qubits ｐｑ Ｈ１ to define If p and q are not separated by an ancilla qubit but are directly adjacent, then E ｐｑ Ｈ１ is the product of two Pauli operators of a second kind acting on the physical qubits p and q, respectively, and optionally E ｐｑ Ｈ１ =X ｐ X ｑ and If p and q are separated by one ancilla qubit a, then E ｐｑ Ｈ１ is a product of a Pauli operator of a second type acting on the physical qubit p, a Pauli operator of a first type acting on the ancilla qubit a, and a Pauli operator of a second type acting on the physical qubit p, and optionally E ｐｑ Ｈ１ =X ｐ Z ａ X ｑ and If p and q are separated by two ancilla qubits a and b, then E ｐｑ Ｈ１ is a product of a second type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, a first type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit p, and optionally E ｐｑ Ｈ１ =X ｐ Z ａ Z ｂ X ｑ and defining For each odd pair of physical qubits p and q, an odd horizontal edge operator E associated with said qubits ｐｑ Ｈ２ to define If p and q are not separated by an ancilla qubit but are directly adjacent, then E ｐｑ Ｈ２ is the product of two Pauli operators of a third kind acting on the physical qubits p and q, respectively, and optionally E ｐｑ Ｈ２ = Y ｐ Y ｑ and If p and q are separated by one ancilla qubit a, then E ｐｑ Ｈ２ is a product of a third type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, and a third type Pauli operator acting on the physical qubit p, and optionally E ｐｑ Ｈ２ = Y ｐ Z ａ Y ｑ That is, If p and q are separated by two ancilla qubits a and b, then E ｐｑ Ｈ２ is a product of a third type Pauli operator acting on the physical qubit p, a first type Pauli operator acting on the ancilla qubit a, a first type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit p, and optionally E ｐｑ Ｈ２ = Y ｐ Z ａ Z ｂ Y ｑ and defining For each pair of physical qubits p and q, a vertical edge operator E associated with the qubits that are directly adjacent in the vertical dimension without any physical qubits or ancilla qubits between them is ｐｑ Ｖ to define If ancilla qubits a and b are directly adjacent to p and q in the horizontal dimension in a first direction, and p and q have no other ancilla qubits that are direct horizontal neighbors, and a and b are associated with an even horizontal edge operator, then E ｐｑ Ｖ is a product of a second type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit q, and optionally E ｐｑ Ｖ =X ｐ X ａ Y ｂ X ｑ and Ancillary qubits a and b are aligned in a first direction L １ If a and b are directly adjacent to p and q in the horizontal dimension of ｐｑ Ｖ is a product of a third type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit q, and optionally E ｐｑ Ｖ = Y ｐ X ａ Y ｂ Y ｑ and The ancilla qubits a and b are aligned in the second direction L ２ , and p and q have no other ancilla qubits that are direct horizontal neighbors, and a and b are associated with an even horizontal edge operator, then E ｐｑ Ｖ is a product of a second type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit a, a second type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit q, and optionally E ｐｑ Ｖ =X ｐ Y ａ X ｂ X ｑ and The ancilla qubits a and b are aligned in the second direction L ２ qubits, and p and q have no other direct horizontal neighbors, and a and b are associated with odd horizontal edge operators, then E ｐｑ Ｖ is a product of a third type Pauli operator acting on the physical qubit p, a third Pauli operator acting on the ancilla qubit a, a second type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit q, and optionally E ｐｑ Ｖ = Y ｐ Y ａ X ｂ Y ｑ and p and q are in the first direction L １ and ancilla qubits c and d are directly adjacent to ancilla qubits a and b in the horizontal dimension of the second direction L. ２ If directly adjacent to First direction L １ About E ｐｑ Ｖ is a product of a second type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a second type Pauli operator acting on the physical qubit q, and optionally E ｐｑ Ｖ =X ｐ X ａ Y ｂ X ｑ or E ｐｑ Ｖ is a product of a third type Pauli operator acting on the physical qubit p, a second type Pauli operator acting on the ancilla qubit a, a third type Pauli operator acting on the ancilla qubit b, and a third type Pauli operator acting on the physical qubit q, and optionally E ｐｑ Ｖ = Y ｐ X ａ Y ｂ Y ｑ and Second direction L ２ About E ｐｑ Ｖ is a product of a second type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit c, a second type Pauli operator acting on the ancilla qubit d, and a second type Pauli operator acting on the physical qubit q, and optionally E ｐｑ Ｖ =X ｐ Y ｃ X ｄ X ｑ or E ｐｑ Ｖ is a product of a third type Pauli operator acting on the physical qubit p, a third type Pauli operator acting on the ancilla qubit c, a second type Pauli operator acting on the ancilla qubit d, and a third type Pauli operator acting on the physical qubit q, and optionally E ｐｑ Ｖ = Y ｐ Y ｃ X ｄ Y ｑ and defining that:

13. 13. The method of any one of claims 1 to 12, wherein the qubits are arranged in a pattern, the pattern being repeated continuously within a single horizontal line of qubits, the pattern being selected from the group consisting of P'A, P'P'A, P'P'AA, and P'AA.

14. The method of claim 13 , wherein the pattern is selected by determining a maximum Pauli weight of an edge operator associated with at least two patterns and selecting the pattern associated with the lowest maximum Pauli weight.

15. 14. The method of claim 13, wherein the pattern is selected by determining circuit depths of control sequences separately associated with at least two patterns and selecting the pattern associated with the lowest circuit depth.

16. 14. The method of claim 13, wherein the pattern is selected by determining the number and / or type of gates included in control sequences separately associated with at least two patterns, and selecting the pattern associated with the lowest number of gates and / or the pattern associated with the selected type of gates, and / or the pattern associated with the selected qubit-mode ratio.

17. A computer program product comprising program code means adapted to carry out the method of any one of claims 1 to 16 when said program product is executed on a computer.

18. A quantum circuit comprising a sequence of qubit interactions determined according to the method of any one of claims 1 to 16, executable on a quantum device for simulating a fermion Hamiltonian.

19. 1. A method for determining at least one property of a fermionic system, comprising: receiving a fermion Hamiltonian, wherein at least one property of the fermion system is characterized by the fermion Hamiltonian; determining a control sequence according to any one of claims 1 to 16; implementing the determined control sequence on a quantum device; and applying one or more measurement gates to determine the at least one property of the fermionic system.

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