Digital quantum simulation of fermion-boson systems in a two-dimensional quantum computing system

Abstract

One aspect of the present disclosure relates to a method for configuring a quantum computing system, the quantum computing system including a plurality of qubits arranged on a two-dimensional (2D) lattice and configured to perform a plurality of quantum computation operations. The method of the present disclosure includes receiving a selection of a first plurality of qubits and a second plurality of qubits of the plurality of qubits, the first plurality of qubits and the second plurality of qubits representing two different degrees of freedom associated with respective components of a physical system mapped onto the first and second plurality of qubits. Adjacent qubits of the first aspect are configured to transmit quantum information between each other. Some chains of the first plurality of qubits and some ladders of the second plurality of qubits are configured to perform a plurality of quantum computation operations.

Description

[Technical Field]

[0001] FIELD OF THE INVENTION The present specification relates to methods for configuring a quantum computing system. Related aspects relate to quantum computing systems and remote computing systems. [Background technology]

[0002] There is growing interest in implementing quantum computing on various physical systems to solve various real-world problems, such as those dealing with chemistry, biology, solid-state physics, and cryptography (see, for example, E. Grumbling and M. Horowitz, “Quantum Computing: Progress and Prospects,” Washington, DC: The National Academies Press, 2019; https: / / doi.org / 10.17226 / 25196). The goal is to speed up computations compared to classical computers and / or to solve classes of problems that cannot be solved even on supercomputers that perform classical computations based on classical algorithms. In the field of physics simulation on quantum computers, there are many applications, for example, in solid-state physics and chemistry, that require the combined quantum simulation of fermionic and non-interacting buses on a quantum computer. These can be physical problems in which the non-interacting buses take the form of boson modes or non-interacting fermionic modes.

[0003] Some known prior art relates to the implementation of both fermionic degrees of freedom and bosonic modes and their possible interactions on quantum computer qubits. However, the number of quantum computing operations required for quantum simulations of physics problems involving interacting fermions and bosons executed on some prior art quantum computer architectures is so large that the entire quantum simulation takes time, and as a result, the qubits may lose coherence in time intervals shorter than the time interval required to complete the quantum computing task. [Prior art documents] [Non-patent literature]

[0004] [Non-Patent Document 1] E. Grumbling and M. Horowitz, “Quantum Computing: Progress and Prospects” Washington, DC: The National Academies Press, 2019; https: / / doi.org / 10.17226 / 25196 Summary of the Invention [Problem to be solved by the invention]

[0005] Therefore, there is a need to develop new efficient techniques to reduce the number of quantum computational operations required to perform quantum simulations of fermion-boson interacting systems on quantum computers. [Means for solving the problem]

[0006] A first aspect of the present disclosure relates to a method for configuring a quantum computing system, the quantum computing system including a plurality of qubits arranged on a two-dimensional (2D) lattice and configured to perform a plurality of quantum computation operations. The method of the present disclosure includes receiving a selection of a first plurality of qubits of the plurality of qubits, the first plurality of qubits including several chains of qubits. In the method of the first aspect, each qubit of the several chains of the first plurality of qubits represents a first degree of freedom associated with a respective component of a physical system mapped onto the several chains of the first plurality of qubits. Furthermore, each qubit of the first plurality of qubits is configured to transmit quantum information of the qubit to another qubit of the first plurality of qubits adjacent to the qubit, and the other qubit of the first plurality of qubits is configured to receive the quantum information of the qubit of the first plurality of qubits. The method further includes receiving a selection of a second plurality of qubits of the plurality of qubits, the second plurality of qubits including several ladders of qubits, each qubit ladder representing a second degree of freedom associated with a respective component of the physical system mapped onto the several ladders of the second plurality of qubits. The second degree of freedom of the first aspect is different from the first degree of freedom. In a first aspect of the present disclosure, each qubit of the second plurality of qubits is configured to transmit its quantum information to another qubit of the second plurality of qubits adjacent to the qubit, and the other qubit of the second plurality of qubits is configured to receive the quantum information of the qubit of the second plurality of qubits. In the method of the first aspect, one or more qubits of several chains of the first plurality of qubits are adjacent to one or more qubits of each of several ladders of the second plurality of qubits and are configured to transmit quantum information to and / or receive quantum information from one or more qubits of each of several ladders of the second plurality of qubits.The several chains of the first plurality of qubits and the several ladders of the second plurality of qubits of the method are configured to perform several quantum computing operations.

[0007] A second aspect provides a quantum computing system configured according to any of the steps of the technique according to the first aspect.

[0008] A third aspect provides a quantum computing system configured to perform a plurality of quantum computation operations and adapted to perform any of the steps of the technique according to the first aspect.

[0009] A fourth general aspect of the present disclosure relates to a remote computing system including a quantum computing system and configured to perform a quantum computing task, the quantum computing task including a plurality of quantum computing operations according to the first aspect. The plurality of quantum computing operations of the fourth aspect can be performed according to any one of the method steps of the first aspect. The remote computing system of the fourth aspect is further configured to transmit results of the computing task to the computer-implemented system. [Effects of the Invention]

[0010] The techniques of the first to fourth aspects can have advantageous technical effects.

[0011] First, the techniques of this disclosure involve performing digital quantum simulations of fermionic systems (e.g., fermion-boson interacting systems) interacting with a non-interacting bus on a quantum computing system including a hardware architecture (e.g., one or more chips) with qubits arranged in a 2D lattice, with a reduced number of quantum computational operations required compared to some prior art techniques. For example, some prior art techniques using quantum computers with two-dimensional connectivity require 100 quantum computation operations per interaction between a fermionic degree of freedom and a boson mode.

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[0012] Second, the use of the qubit configuration of the present technology can reduce the number of quantum gates required to accomplish a quantum computing task, thereby reducing the overall decoherence or inaccuracy occurring within a quantum computing system. Each quantum gate can be a potential source of decoherence and / or inaccuracy related to the fact that the physical realization of the quantum gate may not match the user-specified logical gate operation (this problem is referred to as gate infidelity). Therefore, the gains of the present technology in reducing the total number of quantum computing operations on qubits in a 2D lattice and the resulting computation time become even more significant compared to some prior art techniques when the two above-mentioned factors are considered together.

[0013] Third, the techniques disclosed herein enable the parallel execution of some quantum computing operations that cannot be performed in parallel using some prior art techniques, which can provide additional speedup in performing quantum computing operations, which can lead to the preservation of coherence between qubits throughout the execution time of a quantum computing task.

[0014] In this specification, several terms are used as follows:

[0015] The term "qubit" (or quantum bit) can refer to a quantum mechanical system having (at least) two quantum states or any superposition of these quantum states, also referred to as a two-level system for short. A two-level system is the basic unit that carries quantum information, within which quantum information can be encoded and from which it can be extracted. For example, the spin of an electron in a magnetic field, with two energy levels and corresponding spin-up and spin-down states, is a physical realization of a qubit. Another physical realization deals with the polarization of a single photon, whose two orthogonal polarizations can be considered as the two qubit states. In some cases, the two quantum states can be associated with two different energy levels, e.g., two selected energy levels within the anharmonic energy spectrum (or, in other words, an anharmonic ladder of energy levels) of a physical system that serves as a physical realization of a qubit (this can be the case, for example, of a superconducting qubit). In other cases, the two quantum states can be associated with two degenerate energy levels (i.e., they share the same energy value), which can be the case in photonic quantum computers. The quantum state of a single qubit can be described by a wave function, which can be represented as a vector in two-dimensional complex space, and changes in that quantum state (e.g., due to the time evolution of the qubit state and / or as a result of applying quantum gate operations) can be visualized on the Bloch sphere (see, e.g., MA Nielsen and I.L. Chuang, "Quantum Computation and Quantum Information": 10th Anniversary Edition, Cambridge University Press, 2010). Quantum computing can involve quantum computational operations on multiple qubits (see also below), thereby manipulating and modifying their multi-qubit quantum states.In some cases, each qubit in a plurality of qubits can be treated independently of one another, in which case the multi-qubit quantum state can be written as a separable quantum state, i.e., can be represented as a tensor product of each single-qubit state (which can ultimately be represented as a corresponding superposition of the quantum states of the individual qubits). In other cases, when at least two qubits from a plurality of qubits cannot be treated independently of one another (or, in other words, cannot be described separately from one another), the multi-qubit quantum state represents an entangled state that cannot be represented in terms of tensor products of the individual qubit states (see also below, where both situations are explained in more detail).

[0016] There are several physical realizations of systems that can be used as qubits (i.e., as two-level systems) in the context of quantum computing. The qubits of the present disclosure are not limited to a particular physical realization. An example of such a physical realization is a quantum computer based on cavity quantum electrodynamics (cQED), in which the qubits are provided by the internal states of trapped atoms coupled to a high-finesse cavity. One example of quantum computing using circuit quantum electrodynamics is superconducting quantum computing based on superconducting qubits coupled to a microwave cavity (referred to as a quantum bus) and radiating in the microwave range, whose quantum states are manipulated by electromagnetic pulses to control the magnetic flux, charge, or phase difference across nanofabricated Josephson junctions (see, e.g., https: / / doi.org / 10.1038 / nature07128). Another example relates to a solid-state nuclear magnetic resonance (NMR) Kane quantum computer with qubits realized as nuclear spin states of donor atoms (e.g., phosphorus donor atoms) embedded in respective host lattices (e.g., pure silicon lattices). In some other examples, the physical implementation of a quantum computer may be based on neutral atoms in an optical lattice, where qubits are implemented by the internal states of neutral atoms (e.g., Rydberg atoms) trapped in the optical lattice (e.g., interacting with each other via the Rydberg interaction), see, e.g., https: / / doi.org / 10.1088 / 0953-4075 / 49 / 20 / 202001. In yet other examples, the quantum computer may be a quantum dot computer, where the qubits are given by the respective spin states of trapped electrons.

[0017] The term "quantum computing operation" and the related term "quantum computation" can refer to an operation on a qubit that can change its quantum state. A quantum computing operation on one or more qubits can be performed by quantum gates that manipulate the quantum state of the qubits, or in other words, using the quantum information carried by the qubits. As further disclosed below, a single qubit can form a single-qubit quantum state (e.g., a ground state, an excited state, or a superposition of both). In some cases, multiple qubits can form a multi-qubit quantum state, which may be a tensor product state or an entangled state (see the discussion below for more details). In some cases, a quantum computing operation can be represented by a sequence of quantum gates acting on each qubit. Quantum gates can be represented by a unitary operator U (e.g., represented by a respective unitary matrix) that guarantees norm conservation of the qubit's wave function in the absence of dissipation, such that the product of this operator with its Hermitian conjugate is the identity operator UU. †=I, where † denotes Hermitian conjugate and I is the identity operator. A quantum gate is therefore configured to perform a unitary transformation on a qubit, or in other words, the unitary operator representing the quantum gate performs a unitary transformation on the quantum state of the qubit (see also below). The Hadamard gate H, the phase gate S, the π / 8 gate, as well as the Pauli X-gate, Pauli Y-gate, and Pauli Z-gate are examples of single-qubit gates whose action on qubits can be visualized on the Bloch sphere mentioned above (see, for example, the book by MANielsen and I.L. Chuang mentioned above. Any quantum computation on one or more qubits can be generated by a finite set of qubit gates that are said to be universal for quantum computation. In this case, any unitary operation that represents this quantum computation on a qubit can be decomposed into a set of operations performed by a quantum circuit that includes gates from this finite set. Any unitary operation (e.g., a unitary operation performed on any multi-qubit logic gate) can be composed of a two-qubit controlled-NOT (CNOT) gate and a corresponding number of single-qubit gates, i.e., single-qubit rotations, with some free parameters that characterize the unitary operation under consideration. For example, any unitary operation can be approximated (to a given accuracy) using Hadamard, phase, CNOT, and π / 8 gates, also referred to as universal quantum gates (see, for example, the book by MANielsen and I.L. Chuang mentioned above). (See I. L. Chuang, "Quantum Computation and Quantum Information": 10th Anniversary Edition, Cambridge University Press, 2010.) For example, in the context of superconducting qubits, a single-qubit gate can be realized by a rotation between two energy levels of a single superconducting qubit induced by a microwave pulse sent down a transmission line coupled to the qubit, the frequency of which is resonant with the energy separation between the levels.Furthermore, two-qubit gates can be realized by coupling two superconducting qubits, for example, via a microwave cavity or intermediate electrical coupling circuit (see, e.g., https: / / doi.org / 10.1038 / nature02851). In the context of neutral atom quantum computing, two-qubit gates can be realized using controllable Rydberg interactions between neutral atoms that are strong enough to perform two-gate operations; see, e.g., https: / / doi.org / 10.1103 / PhysRevX.10.021054.

[0018] The term "unitary transformation" as used herein as applied to a qubit should be interpreted broadly in this disclosure and may refer to a unitary transformation of the quantum state of a qubit caused by a unitary operation acting on the qubit, which may be defined by a unitary operator acting on the qubit's quantum state. For example, a quantum computing operation, such as applying one or more different quantum gates (represented by respective unitary operators) to a qubit, can result in a unitary transformation of the qubit's quantum state. In other words, one or more unitary transformations of the quantum state of one or more qubits associated with respective gates applied to one or more qubits can perform the gates in a corresponding subspace of the qubit's Hilbert space, while a unity transformation (or, in other words, a unity operation) is applied to the remainder of the Hilbert space. (Hilbert space in this context can be understood as a complex vector space spanned by vectors representing the quantum states of qubits with a defined dot product between them. In some cases, the quantum states of qubits can evolve unitarily according to the qubit's Hamiltonian (e.g., the Jaynes-Cummings Hamiltonian in cQED), which is a Hermitian operator that determines their interaction with an external control field (e.g., a magnetic field), as well as the qubit's coupling with a host lattice or cavity (e.g., a quantum bus in the context of superconducting quantum computing) and their possible interactions (e.g., dipole-dipole interactions in the context of Rydberg atoms, etc.). This unitary time evolution of the qubit's quantum state (represented by the unitary time evolution operator) that occurs during quantum computation is also referred to herein as a “unitary transformation.” As can be seen from the above discussion, a single-qubit rotation is a particular case of a unitary transformation.

[0019] In the present disclosure, “quantum computing operations” performed on qubits of a quantum computing system can be performed to simulate the time evolution of a physical system under consideration governed by a respective quantum Hamiltonian H. The physical system of interest can be “encoded into the qubits” of the quantum computing system, such that the unitary time evolution of the components of the physical system (see the next paragraph for details) is transformed from the components to the qubits, allowing the quantum computing system to simulate the unitary time evolution of the physical system under consideration. In other words, the quantum states of the qubits of the quantum computing system can evolve unitarily according to the Hamiltonian of the physical system H encoded in the qubits, in which case the time evolution of the physical system can be implemented, for example, as a sequence of quantum gates (i.e., gates available in a particular architecture of the quantum computer and / or qubit topology) acting on the qubits. Thus, the “quantum computing operations” of the present disclosure performed on the qubits of a quantum computing system may be referred to as a “digital quantum simulation” of the physical system of interest on the quantum computing system. For example, fermionic orbitals (more precisely, electron distributions in atomic or molecular orbitals) and / or bosonic mode occupations can be represented by a respective qubit (i.e., a respective quantum state of that qubit). In this disclosure, "encoding onto a qubit" can be used together with "mapping onto a qubit," which has the same meaning.

[0020] As used herein, a "physical system" can be any quantum physical system that includes corresponding components (e.g., one or any combination selected from the following non-exhaustive list: atoms, ions, molecules, quantum dots, photons, holes, phonons, Cooper pairs, excitons, polaritons, magnons, polarons) that evolve according to respective quantum Hamiltonians H and can interact with each other. For example, components of one species can interact with components of the same species (e.g., electron-electron interactions) and / or another species (e.g., electron-phonon interactions). In this regard, the term "degrees of freedom" can be used in some cases of the present disclosure to indicate whether the spin quantum number of each component has an integer value (e.g., 0, 1, 2, or any other integer value) or a half-odd value (e.g., 1 / 2, 3 / 2, 5 / 2, or any other half-odd value), resulting in the corresponding quantum statistics of these components. For example, one or more components of a physical system may be fermionic particles (e.g., electrons or other particles) with spins having half-odd integer values, or fermionic quasiparticles (e.g., holes, polarons, or other quasiparticles) that obey Fermi-Dirac statistics, and "fermionic degrees of freedom" are associated with such components. One or more other components of the physical system may be bosons, e.g., single particles (e.g., photons), composite particles (e.g., some real or artificial atoms), or quasiparticles that obey Bose-Einstein statistics (e.g., Cooper pairs, collective excitations, e.g., phonons, excitons, magnons, etc.), in which case they may be referred to as bosonic modes. In other examples of the present disclosure, bosonic modes can represent non-interacting fermionic modes, e.g., non-interacting fermionic modes of the Dynamical Mean Field Theory (DMFT) bath. An example of a fermionic system in contact with a non-interacting fermionic bath deals with single-electron nanoelectronic circuits that provide electrons (fermionic degrees of freedom) and their conductor contacts that act as a non-interacting fermionic bath (in some cases, such a non-fermionic bath may be equivalent to a bosonic mode for the purposes of this specification). See, e.g., https: / / doi.org / 10.1109 / 5.752518.

[0021] An example of a physical system containing fermionic degrees of freedom interacting with bosonic modes is a physical system containing electrons and phonons interacting with each other, the evolution of which can be described by a Frohlich or Holstein Hamiltonian (see, for example, G.D. Mahan, “Many Particle Physics”, Springer, New York, 2000; https: / / 10.1103 / PhysRevLett.109.200501). A non-exhaustive list of further examples includes physical systems involving fermions in contact with a bosonic bath in the context of tunneling problems used to derive boson-mediated interactions of fermionic modes as commonly found in so-called P(E) theory (see, e.g., https: / / arxiv.org / pdf / cond-mat / 0508728.pdf), the interaction of electron systems with electromagnetic radiation fields (see, e.g., https: / / arxiv.org / pdf / 1804.07142.pdf or https: / / arxiv.org / pdf / 1501.00803.pdf), high-energy physics (see, e.g., http: / / arxiv.org / abs / 1404.2868), ultracold fermion-boson mixtures (see, e.g., https: / / arxiv.org / pdf / 1212.3535.pdf), and various other physical systems arising in the context of solid-state physics and quantum chemistry.

[0022] As used herein, “transmitting quantum information” (or “exchanging quantum information,” as may be used in a similar context) between qubits within the same or different types of qubits should be interpreted broadly. In some cases, transmitting / exchanging quantum information between two adjacent qubits can involve applying a unitary transformation to the qubits, e.g., using one or more two-qubit gates, or in some examples, one or more single- or multi-qubit gates in addition to the two-qubit gate. Note that “transmitting quantum information” between two adjacent qubits involving a unitary transformation can include direct (physical) interaction between the qubits and / or interaction of the qubits via, for example, a host lattice / quantum bus (in the sense described above). In some examples herein, “quantum information” can be transmitted / exchanged between distant qubits via respective unitary transformations applied between adjacent qubits located between the distant qubits. In the present techniques, “quantum information” between adjacent qubits can be transmitted / exchanged, which is a step that can be part of a quantum computation operation performed on a quantum computing system. In some cases, "quantum information" may be transmitted / exchanged, e.g., one or more times during the course of a quantum computation operation, e.g., to simulate the unitary time evolution of a physical system (or, in other words, its quantum Hamiltonian) on a quantum computing system within a (predetermined) time step (see also below). Also, in the above sense, if "quantum information" can be communicated from one qubit to another adjacent qubit, the qubit to which "quantum information" is communicated may be said to be configured to receive "quantum information."

[0023] In the present disclosure, one type of "degrees of freedom" ("first degrees of freedom") described above, e.g., "fermionic degrees of freedom," may be encoded (or, in other words, mapped) by some qubits representing the "first degrees of freedom," while another number of qubits may represent another type of "degrees of freedom" ("second degrees of freedom"), e.g., "bosonic modes." Such a division of qubits into different species encoding different degrees of freedom may be interpreted, e.g., such that a "quantum computational operation" performed on a qubit representing a "fermionic degree of freedom" is associated (at least in part) with the "fermionic degree of freedom" (as described by a respective term in the system's quantum Hamiltonian, H), while a "quantum computational operation" performed on another qubit representing a "bosonic mode," e.g., is associated (at least in part) with the "bosonic mode." In some cases, adjacent qubits of different species (e.g., where one qubit represents a "fermionic degree of freedom" and another adjacent qubit represents a "bosonic mode") may involve "quantum computation operations" associated with both degrees of freedom.

[0024] As used herein, the encoding of "degrees of freedom" may be nonlocal in nature, i.e., some qubits (e.g., two or more qubits from this number, or all of them) representing a "degree of freedom" of the type under consideration may contain "at least partial information" about a single degree of freedom of this type (e.g., the quantum state of some qubits may depend on that single degree of freedom). For example, the Jordan-Wigner transformation (see, e.g., https: / / doi.org / 10.1103 / PhysRevLett.120.110501) used to encode "fermionic degrees of freedom" on qubits is nonlocal, so that the fermionic parity of a single orbit can be encoded in several qubits (e.g., all qubits of this type representing "fermionic degrees of freedom"). In some cases, the encoding of "bosonic modes" may also involve nonlocality, in which case, for example, all qubits of a ladder (e.g., n qubits) together encode the "bosonic mode." For example, in binary encoding, the Fock state |n> b (n represents any integer or 0, and the subscript b represents a bosonic mode) the number of occupancies of the bosonic mode is 2 n It can be represented as a binary number with occupied qubit states. Another example related to unary encoding is |0> b From |n> b The number of occupancies of bosonic modes up to |k> can be expressed by n qubits, and the number of occupancies b For , one qubit nk is in state |1〉 and all other qubits are in state |0〉. Binary, unary, as well as other possible encodings are described in https: / / doi.org / 10.1038 / s41534-020-0278-0. In the following, the term "qubit ladder" (e.g., used to encode "boson modes") will also be used below and is a synonym for the term "ladder of qubits".

[0025] In the present disclosure, for example, in the course of performing a “quantum computation operation” on qubits representing “first degrees of freedom” (e.g., implemented as a sequence of respective quantum gates), an entangled multi-qubit state can be established (see discussion above), even though the initial quantum states of these qubits are selected as separable quantum states. In this case, each qubit involved in forming the multi-qubit state may contain “at least partial information” with respect to some first degrees of freedom (e.g., two or more first degrees of freedom). Similar considerations can be applied to “second degrees of freedom.” In some cases, when neighboring qubits of different species exchange quantum information, entangled states may be formed on qubits of different species, such as those representing both “first degrees of freedom” and “second degrees of freedom.” In this case, each qubit involved in forming this multi-qubit state can be said to contain “at least partial information” with respect to both types of degrees of freedom (e.g., one or more fermionic degrees of freedom and one or more bosonic modes).

[0026] The term "adjacency" (or the attribute "adjacent") with respect to qubits (e.g., arranged on a 2D lattice) should be interpreted broadly in this disclosure, such that two qubits may be classified as adjacent if a universal quantum gate operating on the two qubits can be realized without requiring one or more separate quantum gates between any of the two qubits and a third qubit. For example, if a quantum computer provides all unitary operations associated with a universal gate ensemble operating on the two qubits without involving the third qubit, or if they are physically coupled in hardware, or if a hardware-specific two-qubit gate operating on the two qubits can be realized without requiring separate quantum gates between each individual qubit of the two qubits and the third qubit. In some examples, qubits may be classified as adjacent if, for example, they are nearest neighbors among qubits of the same or different species (e.g., one species may represent a qubit representing a fermionic degree of freedom, while the other may be a ladder of qubits, e.g., a bosonic mode), or if the distance between the qubits under consideration is equal to or less than a predetermined characteristic distance (see also later discussion). Spatial separation of qubits may be a determining factor, for example, when qubits interact directly with each other via dipole-dipole interactions, as in the case of dipole-dipole interactions in optically trapped Rydberg atoms; see, e.g., https: / / doi.org / 10.1088 / 0953-4075 / 49 / 20 / 202001. In other examples, spatial distance between qubits may not be a relevant factor, or at least may not be the only relevant factor, in determining qubit adjacency. For example, semiconductor qubits can be coupled to each other via a quantum bus to which they are coupled, such that the qubit coupling can be tuned by flux control of the qubits, and their spatial separation may not be a determining factor.For example, when the qubit coupling exceeds a certain critical value, the qubits can be considered adjacent qubits, such that a two-qubit gate can be realized based on these two qubits (without a third qubit), or the two qubits can form an entangled state. In this case, their spatial separation is not a determining factor. In some other examples, the inter-qubit coupling of superconducting qubits can be adjusted by connecting them to intermediate electrical coupling circuits; see, e.g., https: / / doi.org / 10.1038 / s41586-019-1666-5.

[0027] The term "chain of qubits," as used herein, should be broadly interpreted in this disclosure to refer to a one-dimensional (1D) spatial arrangement of qubits of the same species on the plane of a 2D lattice (e.g., along a 1D curve or line). In some cases, a species of qubits may include one or more connected chains of qubits extending in one or more directions. Additionally or alternatively, a species of qubits may include one or more disconnected chains of qubits extending in one or more directions, for example, interrupted by one or more chains of another species of qubits. [Brief explanation of the drawings]

[0028] [Figure 1a] FIG. 1 is a flow diagram illustrating a method for configuring a quantum computing system according to a first aspect. [Figure 1b] FIG. 4 is a flow diagram illustrating further possible method steps according to the first aspect. [Figure 1c] FIG. 4 is a flow diagram illustrating further possible method steps according to the first aspect. [Figure 1d] FIG. 4 is a flow diagram illustrating further possible method steps according to the first aspect. [Figure 2a]Two possible quantum computing systems 1000 of the present technology are illustrated schematically, each having qubits arranged on a 2D lattice. In FIG. 2a, a plurality of connected qubit chains representing a first degree of freedom (e.g., fermionic degrees of freedom) includes three vertically aligned qubit chains 15a and four horizontally aligned qubit chains 15b. The plurality of connected qubit chains are arranged in a vertically extending serpentine pattern. Each of the four horizontally aligned qubit chains 15b is adjacent to a respective ladder from several qubit ladders 16 representing a second degree of freedom (e.g., bosonic modes) (see also FIG. 3a for details). On the left side of the illustration in FIG. 2a, a quantum exchange register 17 is shown, which includes a chain that can be used to exchange quantum information between qubit ladders from different sets of qubit ladders 16, for example, by using a SWAP operation (see also FIG. 4). [Figure 2b] Two possible quantum computing systems 1000 of the present technology with qubits arranged on a 2D lattice are shown schematically in Figure 2b. Figure 2b shows one pair of horizontally aligned qubit chains 15b connected by a vertically extending chain 15a, with two horizontally aligned qubit chains 15b representing a first degree of freedom. Each of the four horizontally aligned qubit chains 15b is adjacent to a respective ladder from several qubit ladders 16 representing a second degree of freedom (see also Figure 3b for details). Two vertical auxiliary qubit chains 15c can be used to exchange quantum information between subsequent disconnected horizontally aligned qubit chains 15b (e.g., by using a SWAP operation). On the left side of the diagram in Figure 2b, a quantum exchange register 17 is shown, which includes chains that can be used to exchange quantum information between qubit ladders from different sets of qubit ladders 16, e.g., by using a SWAP operation (see also Figure 4). [Figure 3a]We present two possible topologies that specify the arrangement of qubits shown in Fig. 2a: a 2D lattice containing multiple square cells with four qubits at the vertices of the square cells. [Figure 3b] Two possible topologies are presented for the qubit arrangement shown in Figure 2b: a 2D lattice containing multiple square cells with four qubits at the vertices of the square cells. For simplicity, only the upper portions of the corresponding Figures 2a and 2b are shown in Figures 3a and 3b (the lower qubit ladder 16 and lower horizontal qubit chain 15b of Figures 2a and 2b are omitted in Figures 3a and 3b). For simplicity, the quantum exchange register 17 is omitted in these figures. The qubit arrangements of Figures 3a and 3b may also be distinct topologies independent of the topologies shown in Figures 2a and 2b. Each ladder 16, representing a respective second degree of freedom (e.g., boson mode), contains two qubits, 2a, 2b; 2c, and 2d, enclosed by dashed ellipses. In total, three sets of qubit ladders are displayed in both figures, with each set containing four ladders. The solid lines in Figures 3a and 3b indicate the possible exchange of quantum information between adjacent qubits representing the same type of degree of freedom, i.e., a pair of qubits q1, q2 representing a first degree of freedom (e.g., fermionic degree of freedom) and a pair of qubits qb1,1;qb1,2 belonging to the respective qubit ladders representing a second degree of freedom, and additionally in Figure 3b, a pair of qubits qh1;qh2 belonging to the respective auxiliary qubit chains 15c. The dashed lines between qubits in Figures 3a and 3b indicate the possible exchange of quantum information between adjacent qubits representing different types of degrees of freedom. [Figure 4]2a and 2b show schematic diagrams illustrating possible topologies for designating qubit arrangements with quantum exchange register 17. In this example, quantum exchange register 17 is a chain of six qubits qR1 through qR6 that can be used to exchange quantum information between qubit ladders belonging to two different sets of ladders, e.g., between a ladder having qubits qb5,1 and qb5,2 (a ladder in a lower set of qubit ladders) and a ladder having qubits qb4,1 and qb4,2 (a ladder in an upper set of qubit ladders). The horizontal dashed lines indicate possible quantum information exchanges between adjacent qubits in different adjacent qubit ladders, as well as the exchange of quantum information (e.g., by using a SWAP operation) between qubits qR2 through qR5 of quantum exchange register 17 and adjacent qubits qb1,1, qb1,2, qb8,2, and qb8,1, respectively, of two quantum ladders belonging to two different sets of ladders. Other notations are similar to those defined in connection with Figures 2a and 3a. [Figure 5a] We present an example of implementing quantum computation operations on a quantum computing system with four qubits q1-q4 that initially encode fermionic degrees of freedom and two qubits of ladder 16 with qubits qb2,1;qb2,2 that initially encode bosonic modes, i.e., (1,↑) → q1, (1,↓) → q2, (2,↓) → q3, (2,↑) → q4, and |Mode1>b → qb2,1 qb2,2. [Figure 5b]We show an example of implementing quantum computation operations on a quantum computing system with four qubits, q1-q4, that initially encode fermionic degrees of freedom and two qubits in a ladder 16 with qubits qb2,1;qb2,2 that initially encode bosonic modes, i.e., (1,↑) → q1, (1,↓) → q2, (2,↓) → q3, (2,↑) → q4, and |Mode1>b → qb2,1 qb2,2. Here, (i,σ) for i = 1, 2 labels an electron with spin σ = ↑,↓ in orbital i, and |Mode1>b denotes the quantum state of the bosonic mode. Each row in Figure 5b depicts the ordering of fermionic orbitals, indicated by circles, encoded on qubits q1-q4. Back and forth arrows indicate fermionic swap operations (FSWAPs) between adjacent orbitals in each row. The top row shows the initial ordering of fermionic orbitals as defined above. In each subsequent row, starting from the top row and proceeding to the bottom row, the resulting fermion orbitals of the new order are displayed (see below for further details). Available quantum computation operations are performed on qubits q1-q4, qb2,1, and qb2,2 each time an FSWAP operation is performed within the respective row (not shown in Figure 5b). [Figure 6a] 5a and 5b, which includes a qubit q2 representing a first degree of freedom (e.g., a fermionic degree of freedom) and two qubits qb2,1 and qb2,2 of a qubit ladder 16 encoding a second degree of freedom (e.g., a bosonic mode). In this figure, H denotes a Hadamard gate, reference numerals 42 and 46 denote a sequence of controlled NOT (CNOT) operations, and Rz(2γ) is a rotational transformation about rotation axis z by angle 2γ applied to qubit qb2,1 of FIG. 6a. [Figure 6b]5a and 5b, which includes a qubit q2 representing a first degree of freedom (e.g., a fermionic degree of freedom) and two qubits qb2,1 and qb2,2 of a qubit ladder 16 encoding a second degree of freedom (e.g., a bosonic mode). In this figure, H denotes a Hadamard gate, reference numerals 42, 43, 45, and 46 denote a sequence of controlled NOT (CNOT) operations, and Rz(γ) is a rotation transformation about rotation axis z by angle γ applied to qubit qb2,2 of FIG. 6b. [Figure 6c] An example of a quantum circuit for performing possible quantum computation operations 40-48 on the quantum computing system of Figures 5a and 5b is shown, including a qubit q2 representing a first degree of freedom (e.g., a fermionic degree of freedom) and two qubits qb2,1 and qb2,2 of a qubit ladder 16 encoding a second degree of freedom (e.g., a bosonic mode). Qubits q2 and qb2,1 are adjacent, and qubits qb2,1 and qb2,2 are also adjacent (see Figure 5a). In this figure, H denotes a Hadamard gate, reference numerals 42, 43, 45, and 46 denote a sequence of controlled NOT (CNOT) operations, and Rz(γ) is a rotation transformation about rotation axis z by angle γ, applied to qubit qb2,2 in Figure 6c. RX(π / 2) represents the basis rotation transformation, and RX(-π / 2) represents the inverse basis rotation transformation. DETAILED DESCRIPTION OF THE INVENTION

[0029] Before describing some possible implementations, some general aspects related to configuring a quantum computing system are first described. An overview of a first general aspect of the present disclosure related to a method for configuring a quantum computing system is provided with reference to the flowcharts shown in FIGS. 1a-1d. Additional aspects of methods related to performing quantum computing operations on a quantum computing system are also presented with reference to these figures. Next, exemplary topologies having qubits arranged on a quantum computing system according to techniques of the present disclosure are described with reference to FIGS. 2a, 2b, 3a, 3b, and 4. Next, examples of implementing quantum computing operations on a quantum computing system are described with reference to FIGS. 5a and 5b. Finally, three example quantum circuits for performing possible quantum computing operations on a quantum computing system are presented in FIGS. 6a-6c.

[0030] 1a-1d disclose and propose a method for configuring a quantum computing system 1000 according to a first general aspect of the present disclosure. The quantum computing system of the present disclosure includes a plurality of qubits arranged on a two-dimensional (2D) lattice (see, e.g., FIGS. 2a, 2b, 3a, 3b, and 4, which schematically illustrate various embodiments of 2D lattices, and further discussion). The plurality of qubits of the first aspect are configured to perform a plurality of quantum computation operations (see above), which may include, e.g., performing a unitary transformation on each qubit implemented as a sequence of quantum gates acting on those qubits. Method steps of corresponding independent claims are summarized within boxes delineated with solid lines in FIGS. 1a-1d, while method steps of dependent claims are shown within boxes delineated with dashed lines.

[0031] The present techniques for configuring a quantum computing system include receiving a selection 100 of a first plurality of qubits 15a-15c; 1a-1d; 3a-3c of a plurality of qubits, where the first plurality of qubits includes several chains of qubits. Further, each qubit 1a-1d of the several chains of qubits 15a-15b of the first plurality of qubits represents a first degree of freedom associated with a respective component of a physical system that maps onto the several chains of the first plurality of qubits. In some examples, consistent with the discussion above, one or more components of the physical system described by the quantum Hamiltonian H may be fermionic particles (e.g., fermionic orbitals), in which case the first degree of freedom may be referred to as a fermionic degree of freedom. In the example 2D lattice shown in Figures 2a and 3a, several chains of qubits representing a first degree of freedom (e.g., fermionic degree of freedom) consist of four horizontal chains 15b and three vertical chains 15a (depicted as solid bars 15a, 15b in Figure 2a) arranged in a serpentine pattern (in total, Figure 3a shows 21 qubits q1 through q21 representing the first degree of freedom). In the embodiment of Figures 2b and 3b, several chains of qubits representing the first degree of freedom (depicted as black bars 15a, 15b in Figure 2b) consist of two disconnected horizontal chains 15b and two other horizontal chains 15b connected by vertical chains 15a (a total of 12 qubits q1 through q12 representing the first degree of freedom are shown in Figure 3b). In some examples, some chains of the first plurality of qubits may include 1 or more, 2 or more, 3 or more, 4 or more, 5 or more, 7 or more, 10 or more, or 50 or more qubit chains. A chain of qubits (e.g., one or more qubit chains) of a number chain of the first plurality of qubits can include 2 or more, 5 or more, 10 or more, 20 or more, or 50 or more qubits.In some cases, some qubits in some chains of the first plurality of qubits may be equivalent to several fermionic degrees of freedom mapped to (or, in other words, encoded by) this numbered chain of qubits. In some examples, consistent with the above discussion, a qubit (e.g., each qubit) representing a first degree of freedom may contain “at least partial information” about more than one first degree of freedom due to the nonlocal nature of the mapping of the first degree of freedom onto qubits in some qubit chains, and due to multi-qubit entanglement that may arise during the course of quantum computation (see also discussion below).

[0032] In this technique, each qubit (q1) of the first plurality of qubits is configured to transmit its quantum information to another qubit (q2) of the first plurality of qubits that is adjacent to the qubit. Furthermore, another qubit of the first plurality of qubits is configured to receive the quantum information of the qubit of the first plurality of qubits. Possible neighboring qubits of several chains of the first plurality of qubits are designated by solid vertical and horizontal lines in FIGS. 3a and 3b. As can be seen from FIGS. 3a and 3b, qubit q1 of the upper chain of the first plurality of qubits is nearest to qubit q2 of this chain, and qubit q2 can be classified as a neighboring qubit to qubit q1 according to the above definition. For the same reason, qubit q3 in FIGS. 3a and 3b can be considered adjacent to qubit q2, and so on. Thus, qubit q1 can be configured to transmit quantum information carried thereby to qubit q2 (e.g., using a fermionic SWAP operation, see further discussion), and vice versa, and qubit q2 can be configured to transmit the quantum information it carries to qubit q1. Similarly, qubit q2 can be configured to transmit quantum information carried thereby to qubit q3, and vice versa, and qubit q3 can be configured to transmit the quantum information it carries to qubit q2. Furthermore, qubit q2 can be configured to receive quantum information carried by qubit q1, and vice versa, and qubit q1 can be configured to receive quantum information carried by qubit q2. In one example, qubit q3 can be configured to receive quantum information carried by qubit q2, and vice versa, and qubit q2 can be configured to receive quantum information carried by qubit q3.Similar considerations can be applied to other qubits in some chains of the first plurality of qubits shown in Figure 3a (qubits q4 through q21) and Figure 3b (qubits q4 through q12).

[0033] A next step of the technique involves receiving a selection 200 of a second plurality of qubits 16; 16a-16e; 2a-2d of the plurality of qubits, the second plurality of qubits including several ladders of qubits, where each ladder of qubits represents a second degree of freedom associated with a respective component of the physical system that is mapped onto several ladders of the second plurality of qubits. (As noted above, "qubit ladder" is used interchangeably with "ladder of qubits," which has the same meaning. The second degree of freedom of the first aspect is different from the first degree of freedom. In some examples, consistent with the discussion above, one or more components of the physical system described by the quantum Hamiltonian H may be bosonic modes, in which case the second degree of freedom may be referred to as a bosonic mode. In the 2D lattice embodiment shown in Figures 3a and 3b, each qubit ladder consists of two qubits, each enclosed by a dashed oval. Twelve qubit ladders are shown in the embodiment of Figures 3a and 3b (e.g., the qubit ladder in the top left corner is composed of qubits qb 1,1 and qb 1,2 and the qubit ladder in the bottom right is the qubit qb 12,1 and qb 12,2 As can be seen in these figures, the qubits in each ladder extend vertically, while the collection of qubit ladders extends horizontally. In the embodiment of Figures 3a and 3b, three collections of ladders are shown, each consisting of four ladders, e.g., the top collection of ladders has qubits qb 1,1 , qb 1,2 , qb 2,1 , qb 2,2 , qb 3,1 , qb 3,2 , qb 4,1 , qb 4,2(The first subscript enumerates the ladder, and the second subscript corresponds to the qubit number within the ladder.) In Figures 2a and 2b, the number of ladders of the second plurality of qubits is displayed as four solid bars 16 extending horizontally. Each horizontal bar 16 in these figures is thicker than the bars representing the chains of the first plurality of qubits 15a, 15b to schematically depict that a vertically extending qubit ladder can include more than one qubit. In some cases, a qubit (e.g., each qubit) of a ladder (e.g., each ladder) representing a second degree of freedom can contain "at least partial information" about one or more of the second degrees of freedom due to the nonlocal nature of the mapping of the second degree of freedom onto the qubits of the qubit ladder and the entanglement of multiple qubits that can arise during the course of quantum computation (see discussion above and below).

[0034] In some examples, the number of qubit ladders representing the second degree of freedom may include one or more, two or more, five or more, ten or more, fifty or more, or one hundred or more qubit ladders. The qubit ladders of the number chain (e.g., one or more qubit ladders) can include two or more, five or more, ten or more, twenty or more, or fifty or more qubits. In some cases, the number of ladders of the second plurality of qubits can be made equal to the number of boson modes mapped to (or alternatively, encoded by) this number of qubit ladders. In some embodiments, similar to those shown in FIGS. 2b and 3b, the number of qubits in some chains 15a; 15b representing the first degree of freedom can be made equal to the number of qubit ladders representing the second degree of freedom. In the embodiment of FIG. 3b, there are, for example, twelve qubit ladders 16 that can represent twelve boson modes, and twelve qubits q1~q12 in three chains of the first plurality of qubits that can represent, for example, twelve fermionic degrees of freedom. In other embodiments, as in FIGS. 2a and 3a, the number of qubits in some chains 15a; 15b representing the first degree of freedom can be made larger than the number of qubit ladders representing the second degree of freedom. Similar to the embodiment of FIG. 3b, the embodiment of FIG. 3a has twelve qubit ladders 16, which can represent, for example, twelve boson modes. On the other hand, in the 2D lattice of FIG. 3a, there are twenty-one qubits q1~q21 in six chains of the first plurality of qubits, which can represent, for example, twenty-one fermionic degrees of freedom. In still other embodiments, the number of qubits in some chains representing the first degree of freedom can be made smaller than the number of qubit ladders representing the second degree of freedom (not shown). In some cases, the total number M of qubits in some chains of the first plurality of qubits can be made less than the total number N of qubits in some ladders of the second plurality of qubits, i.e., M < N. For example, the corresponding ratio M / N may be a value of 9 / 10 or less, 1 / 2 or less, 1 / 4 or less, 1 / 10 or less.

[0035] In the disclosed technique, each qubit 2a; 2c of the second plurality of qubits is configured to transmit its quantum information to another qubit 2b; 2d of the second plurality of qubits that is adjacent to the qubit. Furthermore, another qubit of the second plurality of qubits is configured to receive the quantum information of the qubit of the second plurality of qubits. Possible neighboring qubits of the second plurality of qubits are shown in Figures 3a and 3b with solid lines. In some cases, only qubits within the same ladder, such as those shown in these figures, may be considered adjacent to each other. In the embodiment of Figures 3a and 3b, the qubit qb of the top-left qubit ladder is 1,1 is the same ladder qubit qb 1,2 is nearest neighbor to qubit qb 1,2 is the qubit qb according to the above definition. 1,1 Therefore, the qubit qb 1,1 is the quantum information carried by the qubit qb 1,2 (e.g., using a SWAP operation, see further discussion) to send qubit qb 1,2 is the quantum information it carries, qubit qb 1,1 Furthermore, the qubit qb 1,2 is a qubit qb 1,1 can be configured to receive quantum information carried by the qubit qb 1,1 is a qubit qb 1,2 qubits of different ladders may be adjacent to one another, as in the embodiment shown in Figure 4 (this possible adjacency between qubits of adjacent qubit ladders is indicated by dashed horizontal lines in Figure 4).

[0036] In a first embodiment, one or more qubits 1a; 1c of some chains of the first plurality of qubits are adjacent to one or more qubits 2a; 2c of each of some ladders of the second plurality of qubits. Furthermore, one or more qubits of some chains of the first plurality of qubits can be configured to transmit quantum information to one or more qubits of each of some ladders of the second plurality of qubits. Additionally or alternatively, one or more qubits of some chains of the first plurality of qubits can be configured to receive quantum information from one or more qubits of each of some ladders of the second plurality of qubits. In the example of Figures 3a and 3b, possible adjacent qubits of some chains of the first plurality of qubits and some ladders of the second plurality of qubits are designated by vertical dashed lines. For example, qubit q2 of the top chain of the first plurality of qubits is adjacent to qubit qb of the top ladder of qubits. 2,1 is nearest neighbor to qubit qb 2,1 can be classified as a neighboring qubit with respect to qubit q. Therefore, qubit q transmits the quantum information carried by it to qubit qb. 2,1 (e.g., by one or more of the quantum circuits disclosed in connection with Figures 5a-5c, see the description below). Additionally or alternatively, qubit q2 can be configured to transmit to qubit qb 2,1 In some cases, one or more qubits 2a; 2c of each of several ladders of the second plurality of qubits adjacent to one or more qubits 1a; 1c of several chains of the first plurality of qubits are configured to receive quantum information from and / or transmit quantum information to one or more qubits 1a; 1c of several chains of the first plurality of qubits. 2,1 Going back to our example, the qubit qb in the upper ladder of qubits 2,1can be configured to transmit the quantum information carried thereby to qubit q2. Additionally or alternatively, qubit qb 2,1 is the quantum information carried by the qubit qb 2,1 Similar considerations apply to the other 11 ladder qubits presented in Figures 3a and 3b (e.g., q1 and qb) adjacent to each qubit in the chain of the first plurality of qubits. 1,1 , q3 and qb 3,1 , as well as other pairs of qubits).

[0037] In the techniques of the present disclosure, and further in accordance with the above definition, some chains of the first plurality of qubits and some ladders of the second plurality of qubits are configured to perform a plurality of quantum computing operations (see examples and embodiments disclosed further below for details). In other words, some or all qubits from some chains of the first plurality of qubits and from some ladders of the second plurality of qubits may be involved in performing a quantum computing operation.

[0038] In the above description related to the embodiments of Figures 3a and 3b, nearest-neighbor qubits within qubits of the same or different species are referred to as neighboring qubits, which is one possible example of when a pair of qubits can be classified as adjacent (the fact that a pair of qubits is adjacent is indicated in these figures by the respective solid or dashed lines). In other cases, other criteria can be used to determine whether a pair of qubits (within the same or different species of qubits) is adjacent. As mentioned above, the crucial factor for classifying two qubits as adjacent is the possibility of realizing a universal quantum gate that operates on these two qubits without the need to include a third qubit for this purpose. For example, one qubit from a plurality of qubits on a 2D lattice can be considered adjacent to another qubit from the plurality if the distance from the one qubit to the other qubit is equal to or less than a predetermined characteristic distance, and as a result, the universal gate described above can be realized based only on these two qubits. In some examples, the predetermined characteristic distance may be proportional to the average distance between qubits in a plurality of qubits arranged on a 2D lattice (e.g., between the first and second qubits introduced above). In some cases, the proportionality factor between these quantities may be selected to be equal to or less than 0.5, 0.9, 1.2, or 1.5. In some cases, the predetermined characteristic distance may be equal to the average distance between qubits in the plurality of qubits. These two alternatives may be preferred, for example, when the 2D lattice includes square cells. In yet other examples, when the average distance between qubits in the first plurality of qubits is different from the average distance between qubits in the second plurality of qubits, several predetermined characteristic distances can be used to identify neighboring qubits. For example, the predetermined characteristic distance may be proportional to or equal to the respective average distances between qubits in the same or different plurality of qubits. In some cases, the value of the proportionality factor may be selected in the same way as for a single predetermined characteristic distance.

[0039] In other cases, according to the above definition, spatial distance between qubits may not be the only relevant factor for classifying a pair of qubits as adjacent. For example, inter-qubit coupling can be increased (e.g., via flux control or additional electrical circuitry in the context of superconducting qubits), such that two qubits can be considered adjacent to one another in some instances despite their spatial separation. In other words, the indication of adjacency between each pair of qubits by respective solid or dashed lines in Figures 3a and 3b serves to indicate the fact that the pair of qubits under consideration is adjacent without specifying which of the above-listed factors is crucial for said adjacency (e.g., the distance between qubits q1 and q2 in the upper chain of the first plurality of qubits may differ from the distance between qubits q2 and q3 in the same chain, but both pairs may still be classified as adjacent pairs because other aforementioned factors are involved).

[0040] In the present technology, one qubit ladder 16 (e.g., each qubit ladder) of several ladders of the second plurality of qubits can include several qubits, and the several qubits in the ladder extend in a first direction (e.g., the vertical direction shown in Figures 3a, 3b, 4, and 5a). In a first aspect of the present specification, each chain of several chains of the first plurality of qubits can extend in a first direction 15a (e.g., the vertical direction shown in Figures 3a and 3b) or a second direction 15b different from the first direction (e.g., the horizontal direction shown in Figures 3a and 3b). It should be noted that the number of ladders, the several qubits in the qubit ladder (e.g., each qubit ladder), and the extension of several chains of the first plurality of qubits should be broadly interpreted in this disclosure. In some examples, the shape of the qubit chain, qubit ladder, or qubits in the qubit ladder can form a planar one-dimensional (1D) curve of a 2D lattice extending in the respective direction. In some examples, this consideration may apply to one or more (eg, all) of the qubit chains, qubit ladders, and qubits within the qubit ladders.

[0041] In some cases, some ladders of the second plurality of qubits may include sets of qubit ladders 16a-16d extending in the second direction. In some cases, the sets of qubit ladders may be aligned linearly in the second direction. This is illustrated in FIG. 5a, which shows a set of four ladders (each consisting of two qubits) aligned linearly in the horizontal direction. In other examples, some ladders of the second plurality of qubits may include two or more disconnected sets of qubit ladders extending in the second direction. In some cases, each set of qubit ladders of the two or more disconnected sets may be aligned linearly in the second direction. For example, in the embodiment of FIGS. 3a and 3b, three disconnected sets of qubit ladders (each consisting of two qubits) are aligned linearly in the horizontal direction. In some cases, two sets of qubit ladders may be classified as disconnected from each other if there are no adjacent pairs of qubits (in the sense defined above) belonging to the two qubit ladders from different sets of qubit ladders. For example, the two superset qubit ladders depicted in Figure 3a are composed of qubit pairs (qb 1,2 ,qb 8,2 ), (qb 2,2 ,qb 7,2 ), (qb 3,2 ,qb 6,2 ), and (qb 4,2 ,qb 5,2 ) are adjacent qubits, so it can be classified as a disconnected-ensemble qubit ladder. In other cases, some ladders of the second plurality of qubits can include two or more connected-ensemble qubit ladders extending in the second direction (not shown). Note that, for example, two qubit ladders of different ensembles can be classified as connected to each other when at least a single pair of adjacent qubits belonging to the two qubit ladders can be found that are from different ensembles.

[0042] In some cases of the first aspect, similar to those shown in the embodiments of Figures 3a and 3b, the qubit ladders within a set of qubit ladders (e.g., each individual set of qubit ladders) can be disconnected from one another. In this case, it is not possible to find adjacent qubits that reside in different qubit ladders; that is, pairs of qubits within each individual ladder can be classified as adjacent qubits only. For example, if four pairs of adjacent qubits (qb 1,1 , qb 1,2 ), (qb 2,1 ,qb 2,2 ), (qb 3,1 , qb 3,2 ), and (qb 4,1 , qb 4,2 ) form four truncated ladders of the superset of ladders shown in FIG. 3a. In other cases of the first aspect, as shown in the embodiment of FIG. 4, one or more pairs of adjacent qubit ladders within a set of qubit ladders (e.g., each individual set of qubit ladders) can be connected to each other. In this case, it is possible to find at least pairs of adjacent qubits belonging to pairs of adjacent qubit ladders within the same set of qubit ladders. In the embodiment of FIG. 4, (qb 1,1 ,qb 2,1 ), (qb 2,1 ,qb 3,1 ), (qb 3,1 ,qb 4,1 ), (qb 1,2 ,qb 2,2 ), (qb 2,2 ,qb 3,2 ), and (qb 3,2 ,qb 4,2 ) are the six adjacent pairs of qubits that belong to each pair of adjacent qubit ladders.

[0043] In some cases of the first aspect, the first plurality of qubit chains (representing the first degree of freedom) can include multiple connected qubit chains, with each chain of the multiple connected qubit chains extending in one of the first and second directions. Furthermore, one or more chains from the multiple connected chains of the first plurality of qubit chains extending in the second direction (e.g., horizontally) can be connected to a corresponding chain from the multiple connected chains of qubits extending in the first direction (e.g., vertically). In the present technology, each chain of the multiple connected chains of the first plurality of qubits can be aligned in a respective line in one of the first and second directions. In the embodiment of FIG. 2a, four chains of the first plurality of qubits 15b are aligned in a line in the horizontal direction and connected by three chains of the first plurality of qubits 15a aligned in a line in the vertical direction. In Figure 3a, which identifies the top of Figure 2a, three chains 15b of a first plurality of qubits are aligned horizontally and connected to two qubit chains 15a of a first plurality of qubits aligned vertically. In a first embodiment, the multiple connected chains may be arranged in a serpentine extending in a first direction that is at least partially adjacent to some ladders of a second plurality of qubits (e.g., such a serpentine of qubit chains extending vertically and adjacent to qubit ladders 16 in their horizontal portions is shown in Figures 2a and 3a). In some cases, multiple chains from the multiple connected chains of the first multiple of qubits aligned in the second direction may be interrupted by two corresponding sets of qubit ladders from two or more disconnected sets of qubit ladders of the second multiple of qubits, for each of the first or second chains (e.g., see Figure 2a, where the lowest qubit chain 15b of the serpentine is interrupted by two sets of qubit ladders 16, and the two subsequent horizontal qubit chains 15b of the serpentine are interrupted by two subsequent sets of qubit ladders 16).In one particular case, one or more chains of the plurality of connected chains aligned in a respective line in the second direction (e.g., horizontally) may be adjacent to a respective one or more ladders from several ladders of the second plurality of qubits (e.g., the top qubit chain having horizontally aligned qubits q1, q2, q3, and q4 in the embodiment of FIG. 3a may be adjacent to a respective one or more ladders from several ladders of qubits (qb). i,2 ,qb i,2 ), where index i enumerates the ladders, going from 1 to 4).

[0044] In alternative embodiments of the first aspect, some chains of the first plurality of qubits (representing the first degree of freedom) may include several disconnected qubit chains aligned in a second direction (e.g., horizontally) and / or one or more pairs of chains aligned in a second direction. Furthermore, two chains in a pair of chains may be connected to each other by respective chains extending in a first direction (e.g., vertically). In the present technology, two chains in a pair of chains may be connected to each other by respective chains extending in the first direction. In some examples, a pair of chains may be disconnected from one or more subsequent chains of some chains of the first plurality of qubits. In the embodiment of FIG. 2b, four chains 15b of the first plurality of qubits are aligned in a horizontal direction. Two of them form a pair of qubit chains 15b connected by a chain 15a of the first plurality of qubits aligned in a vertical direction. This pair of chains is disconnected from the other two qubit chains 15b (the upper and lower qubit chains 15b in Figure 2b). In Figure 3b, which identifies the top of Figure 2b, the three chains 15b of the first plurality of qubits are aligned horizontally. Two of them form a pair of qubit chains 15b connected by a vertically aligned chain 15a of the first plurality of qubits. This pair of chains is disconnected from the third (top) qubit chain 15b of the first plurality of qubits shown in Figure 3b. In some cases, each chain of some disconnected qubit chains or each chain of one or more pairs of chains can be at least partially adjacent to one or more respective ladders from some ladders of the second plurality of qubits. For example, each of the four qubit chains of the first plurality of qubits aligned horizontally is adjacent to the qubit ladder of Figure 2b. In the embodiment of Figure 3b, each of the three qubit chains of the first plurality of qubits aligned in a horizontal line is adjacent to four ladders of each of the second plurality of qubits.

[0045] In this alternative embodiment of the first aspect, the first plurality of qubits may further include one or more auxiliary chains of qubits 15c; 3a-3c configured to exchange quantum information between subsequent disconnected chains of the first plurality of qubits (e.g., using a series of SWAP operations, see below). Note that these auxiliary chains of qubits do not represent first or second degrees of freedom. The embodiment of Figure 3b illustrates two auxiliary chains of qubits, including qubits qh1-qh6 in an upper auxiliary chain and qubits qh7-qh9 in a lower auxiliary chain. (A qubit in an auxiliary chain may be referred to as an auxiliary qubit.) In some cases, each chain of one or more auxiliary chains 15c of the first plurality of qubits may include a first qubit 3a;qh1 and a second qubit 3b;qh6, where the first qubit 3a;qh1 is adjacent to a qubit q4 of the respective chain 15b from some disconnected chain or one or more paired chains (this adjacency is indicated in FIG. 3b by a dashed line between qubits qh1 and q4, and another dashed line between qubits qh7 and q12). Further, the second qubit 3b;qh6 is adjacent to a qubit q5 of another chain from some disconnected chain or one or more paired chains subsequent to the respective chain, where the respective chain and subsequent chain are disconnected from each other (this adjacency is indicated in FIG. 3b by a dashed line between qubits qh6 and q5). In this embodiment of the first aspect, a first qubit 3a;qh1 adjacent to a qubit in a respective chain may be configured to receive quantum information from and / or transmit quantum information to a qubit q4 in a respective chain from some of the disconnected chains or one or more pairs of chains. Additionally, a second qubit 3b;qh6 adjacent to a qubit q5 in another chain that follows the respective chain may be configured to receive quantum information from and / or transmit quantum information to a qubit in another chain from some of the disconnected chains or one or more pairs of chains.In some cases, then, qubit q4 of each chain from some disconnected chain or one or more paired chains adjacent to the first qubit 3a;qh1 may be configured to transmit quantum information to and / or receive quantum information from the first qubit 3a;qh1. Additionally, qubit q5 of the other chain following each chain may be configured to transmit quantum information to and / or receive quantum information from the second qubit 3b;qh6.

[0046] In the embodiment shown in Figures 2b and 3b, the auxiliary chains of qubits 15c; 3a-3c extend in a first direction (i.e., in this example, they are aligned vertically). Additionally or alternatively, one or more chains from one or more auxiliary chains of qubits 15c; 3a-3c can extend in a second direction (e.g., they can be aligned horizontally). This may be the case, for example, when one or more qubits from qubit chain 15b representing the first degree of freedom shown in Figures 2b and 3b are replaced by corresponding auxiliary qubits. Returning to the embodiment shown in Figure 3b, if qubit q4 in the upper horizontal chain is replaced by auxiliary qubit qh10 and / or qubit q5 in the other horizontal chain is replaced by auxiliary qubit qh11 (not shown), the auxiliary chains of qubits 15c; 3a-3c can extend in both the first and second directions. In this situation, the number of qubit ladders representing the second degree of freedom (e.g., bosonic modes) can be greater than the number of qubits in some chains 15a; 15b representing the first degree of freedom (e.g., fermionic degrees of freedom).

[0047] The method of the first aspect may further include receiving a selection of a third plurality of qubits 17; 4a-4b from the plurality of qubits. In some cases, some qubits (qR2-qR5) from the third plurality of qubits may be adjacent to two or more ladders 16e; 16a from some ladders of the second plurality of qubits and may be configured to receive quantum information from and / or transmit quantum information to two or more ladders from some ladders of the second plurality of qubits. Two or more ladders 16e; 16a from some ladders of the second plurality of qubits adjacent to some qubits (qR2-qR5) from the third plurality of qubits may be configured to transmit quantum information to and / or receive quantum information from some qubits (qR2-qR5) from the third plurality of qubits. The third plurality of qubits herein may include one or more chains of qubits extending in a first direction (e.g., vertically). This situation is illustrated in the embodiment of Figures 2a and 2b, where the third plurality of qubits, also referred to as quantum exchange register 17, is represented by a vertical solid bar. Figure 4 shows in more detail a schematic representation of the qubit arrangement with quantum exchange register 17 shown in Figures 2a and 2b. Four of the six qubits in quantum register 17 of Figure 4, namely qubits qR2, qR3, qR4, and qR5, are each connected to qubits qb of the two qubit ladders. 1,1 , qb 1,2 , qb 8,2 , and qb 8,1 and configured according to the definition above. In one non-limiting example of Figure 4, the third plurality of qubits is shown as a single chain extending in a vertical direction. In other examples, the third plurality of qubits can include two or more qubit chains, three or more qubit chains, five or more qubit chains, or ten or more qubit chains. In some cases, some or all of the qubit chains can extend in the first direction (not shown).

[0048] In the techniques of this disclosure, qubit ladders 16 (e.g., each qubit ladder) of some ladders of the second plurality of qubits may include a single qubit adjacent to a qubit in a respective chain 15b of the first plurality of qubits, the single qubit being spaced a first predetermined distance from the qubit in the respective chain. For example, the qubit qb of the four ladders of the top set of ladders shown in FIG. 3a 1,1 , qb 2,1 , qb 3,1 , and qb 4,1 are adjacent to qubits q1, q2, q3, and q4, respectively, and are separated therefrom, in a non-limiting example, by the same first predetermined distance. Thus, universal quantum gates operating on any two qubits of each of these four qubit pairs from two different qubit species can be realized without the need to include a third qubit, as described in further detail above. In some cases, when spatial separation is the determining factor for classifying a pair of qubits as adjacent, the first predetermined distance is less than a threshold. The threshold can be defined as the threshold separation distance between qubits above which the two qubits are no longer adjacent.

[0049] In some possible topologies of the present disclosure, spatial qubit placements and relative distances between qubits of the same and / or different types may be defined similarly to those shown in the embodiments of Figures 2a and 3a. In some cases, subsequent pairs of chains from a first plurality of connected chains of qubits may be separated by a second predetermined distance when the chain pair is uninterrupted. For example, the second predetermined distance shown in Figure 3a may be the distance between any one of qubit pairs (q13, q16), (q12, q17), and (q11, q18), which may be the same in some embodiments. Alternatively, when a subsequent pair of chains is interrupted, the subsequent pair of chains may be separated by a third predetermined distance. For example, the third predetermined distance may be the distance between any one of qubit pairs (q1, q14), (q3, q12), or (q5, q10), which may be the same in some embodiments. In a first embodiment, the qubit ladders (e.g., each qubit ladder) of some ladders of the second plurality of qubits can have a predetermined length. In some cases, pairs of subsequent sets of ladders can be separated by a fourth predetermined distance if the sets of pairs are uninterrupted, e.g., the fourth predetermined distance shown in FIG. 3a is 1 / 4 the length of the qubit pairs (qb 1,2 qb 8,2 ), (qb 2,2 ,qb 7,2 ), and (qb 4,2 ,qb 5,2 ), which may be the same in some embodiments. Alternatively, when a subsequent pair of sets is interrupted, the subsequent pair of sets may be separated by a fifth predetermined distance. For example, the fifth predetermined distance may be the distance between the pair of qubits (qb 8,1 qb 9,1 ), (qb 7,1 qb 10,1 ) and (qb 5,1 ,qb 12,1), which may be the same in some embodiments. In some particular examples, the first, second, and fourth predetermined distances may be equal. Additionally, the predetermined length of the qubit ladder may be equal to the product of the first predetermined distance and the number of qubits in the plurality of qubits in the qubit ladder and a value obtained by subtracting 1. In some particular cases, the third predetermined distance may be equal to twice the sum of the predetermined length and the first predetermined distance plus the fourth predetermined distance. Furthermore, in one example, the fifth predetermined distance may be equal to the sum of the second predetermined distance and twice the first predetermined distance.

[0050] In one example of the present technology, the 2D lattice can be a rectangular lattice. In another example, the 2D lattice can be a square lattice. In yet another example, the 2D lattice can have any other 2D shape (e.g., a polymorph, a square, a pentagon, a hexagon, a parallelogram, a circle, or a triangle). In some examples, multiple qubits from a plurality of qubits arranged in the 2D lattice can be equally spaced. For example, all qubits from one or more of the first, second, and third pluralities of qubits can be equally spaced. In some cases, all qubits in the 2D lattice can be equally spaced. Furthermore, in some cases, the 2D lattice (e.g., a rectangular or square lattice) can include multiple square cells having four qubits at the vertices of the square cell, where each qubit from the four qubits is from one of the first, second, and third pluralities of qubits. In the qubit topologies of Figures 3a, 3b, and 4, there are square cells with various combinations of qubits belonging to one or more of: (i) qubit chains of the first plurality of qubits, (ii) qubit ladders of the second plurality of qubits, and (iii) qubit chains of the third plurality of qubits (i.e., qubits from a quantum exchange register). For example, one square cell can contain only qubits of the same species (i), (ii), or (iii), while the other can contain two or all qubits of the listed species. In an alternative topology (not shown), a 2D lattice can include multiple square cells rotated 45° about an axis perpendicular to the plane of the page in Figure 3a (for more details, see, e.g., Figure 1a at https: / / doi.org / 10.1038 / s41586-019-1666-5). Additionally or alternatively, the 2D lattice may include a plurality of quadrilateral (e.g., rectangular) cells having four qubits at the vertices of the quadrilateral (e.g., rectangular) cell, each qubit from the four qubits being a qubit from one or more of a first, second, and third plurality of qubits.Additionally or alternatively, the 2D lattice may include a plurality of triangular cells having three qubits at the vertices of the triangular cells, each qubit from the three qubits being from one or more of a first, second, and third plurality of qubits.

[0051] In the present technology, the first plurality of qubits, the second plurality of qubits, and the third plurality of qubits can be selected during the design phase of the quantum computing system. Additionally or alternatively, the first plurality of qubits, the second plurality of qubits, and the third plurality of qubits can be selected, for example, automatically (e.g., by a program / algorithm depending on the computational task to be performed). In other examples, the first plurality of qubits, the second plurality of qubits, and the third plurality of qubits can be selected by a user (e.g., via an appropriate user interface). In the technology of the present disclosure, steps 100, 200, and 250 of receiving the selection of the first one or more qubits and the second and third plurality of qubits are not particularly limited, and in some cases, the qubits may be redistributed among these three plurality of qubits (e.g., by a user or automatically as described above). For example, for one quantum computing task (see below for further details), some qubits from a plurality of qubits of a quantum computing system may be selected as members of a first plurality of qubits, and for another quantum computing task, one or more qubits from the plurality of qubits (e.g., all of the qubits) may be selected to belong to a second and / or third plurality of qubits (e.g., to perform the quantum computing task more efficiently). In other examples, similar considerations apply to some qubits from the second and / or third plurality of qubits.

[0052] A next step of the method, after performing the receiving selection (related to configuring the quantum computing system), may include performing 300 a plurality of quantum computing operations on the quantum computing system. In the present technique, performing 300 a plurality of quantum computing operations on the quantum computing system may include performing 400 a plurality of quantum computing operations on several chains 15a-15b of the first plurality of qubits. Possible non-limiting arrangements of several chains of the first plurality of qubits representing a first degree of freedom are further described above in connection with Figures 2a, 2b, 3a, 3b, and 4, see, for example, the 21 qubits q1-q21 in the six chains 15a, 15b of the first plurality of qubits in Figure 3a and the 12 qubits q1-q12 in the four chains 15a, 15b of Figure 3b. In a first aspect, performing a plurality of quantum computation operations on several chains of the first plurality of qubits may include exchanging quantum information between two qubits (q1, q2) from one or more pairs of adjacent qubits (q1, q2; q2, q3) of several chains 15a-15b of the first plurality of qubits (e.g., by applying respective unitary transformations to the qubits, further implemented as one or more quantum gates according to the definitions above). For example, qubits q1 and q2 from the top chain of the first plurality of qubits depicted in FIGS. 3a and 3b, which may be adjacent according to any one of the criteria described above, may exchange quantum information with each other (e.g., using the fermion SWAP operation described below). In some cases, any other one or more pairs of adjacent qubits among several chains of the first plurality of qubits, e.g., q2 and q3, q3 and q4, q10 and q11, as well as other pairs of adjacent qubits among several chains shown in these figures, may exchange quantum information between each other in a manner similar to that of qubits q1 and q2.

[0053] The techniques of the present disclosure may include performing 500 a plurality of quantum computing operations on a number of ladders 16; 16a-16e of the second plurality of qubits. Possible non-limiting arrangements of the number of ladders 16; 16a-16e of the second plurality of qubits are further described above in connection with Figures 2a, 2b, 3a, 3b, and 4, see, for example, the three horizontally extending sets of qubit ladders in Figures 3a and 3b, each set of qubit ladders having qubits qb i,1 , qb i,2 , where each ladder extends vertically, and the subscript i ranges from 1 to 12. In a first aspect, performing a plurality of quantum computation operations on some ladders of the second plurality of qubits includes performing a plurality of quantum computation operations on two qubits (qb) from one or more pairs of adjacent qubits of each ladder 16; 16a-16e from some ladders of the second plurality of qubits. 2,1 ,qb 2,2 ) of the top left qubit ladder of the second plurality of qubits shown in FIGS. 3a and 3b, which may be adjacent according to any one of the criteria described above. 1,1 and qb 1,2 can exchange quantum information with one another (e.g., using one or more controlled-NOT (CNOT) operations and / or rotational transformations similar to those disclosed further below in connection with the embodiments of Figures 6a-6c). In some cases, adjacent qubits in any other one or more of the several ladders of the second plurality of qubits, i.e., qb shown in these figures, can exchange quantum information with one another. i,1 , qb i,2 (i is 2 to 12) is the qubit qb in the upper left ladder 1,1 and qb 1,2 can exchange quantum information between them in a similar way.

[0054] The technique may further include exchanging 600 quantum information between two qubits from one or more pairs of adjacent qubits 1 a, 2 a; 1 c, 2 c, where one qubit 1 a; 1 c of the pair is from some chain of a first plurality of qubits and another qubit 2 a; 2 c of the pair is from a respective ladder of some ladder of a second plurality of qubits adjacent to the qubit from some chain. Returning to the embodiment depicted in Figures 3a and 3b, a qubit q1 from a top chain of the first plurality of qubits and a qubit qb of an upper left qubit ladder of the second plurality of qubits depicted in Figures 3a and 3b may be exchanged. 1,1 can exchange quantum information with each other (e.g., using one or more controlled NOT (CNOT) operations and / or rotational transformations, as further disclosed below in connection with the embodiments of Figures 6a-6c). In some cases, one or more other pairs of adjacent qubits belonging to different species of qubits as defined above (i.e., one qubit is from a chain of the first plurality of qubits and another qubit belongs to a qubit ladder), e.g., q and q 2,1 , q3 and qb 3,1 , q4 and qb 4,1 , and other pairs of adjacent qubits shown in these figures are represented by the qubits q and q 1,1 can exchange quantum information between them in a similar way.

[0055] In some cases herein, the number of times quantum information should be exchanged between pairs of qubits belonging to the same or different species of qubits during performing 300 a plurality of quantum computing operations on a quantum computing system may depend on one or more of: i) the particular qubit arrangement of the quantum computing system on the 2D lattice; ii) the particular physical system that is mapped and simulated on the quantum computing system; iii) the particular encoding used to encode the physical system into a plurality of qubits of the quantum computing system; and iv) the decomposition of the quantum computing operations performed on the physical system under consideration into corresponding quantum computing operations performed by native hardware gates.

[0056] According to the first aspect, performing 400 a plurality of quantum computing operations on a chain of several qubits and performing 500 a plurality of quantum computing operations on several ladders may further include performing 410 a plurality of quantum computing operations on one or more qubits 1 a; 1 c of several chains of the first plurality of qubits adjacent to one or more qubits 2 a; 2 c of each of several ladders of the second plurality of qubits. In the embodiment of FIG. 3 a, twelve qubits q1-q4 and q11-q18 of several chains of the first plurality of qubits are adjacent to qubit qb of several ladders of the second plurality of qubits. i,1 ~qb i,1 , where i is 1 through 4 and 5 through 12, respectively (these qubit pair neighbors are shown with dashed lines in accordance with the above discussion). In another embodiment shown in FIG. 3b, all 12 qubits q1 through q12 in some chains of the first plurality of qubits are adjacent to qubit qb in some ladders of the second plurality of qubits. i,1 ~qb i,1, where i ranges from 1 to 12, respectively. In some cases, performing 400, 500 the plurality of quantum computing operations can include performing 510 some of the plurality of quantum computing operations on the respective one or more qubits 2a; 2c of some ladders of the second plurality of qubits. For example, such quantum computing operations can be performed on 12 ladder qubits qb adjacent to corresponding qubits of some chains of the first plurality of qubits shown in these figures. i,1 ~qb i,1 (i is 1 to 12).

[0057] In the present technique, the step of performing 400, 500 the plurality of quantum computing operations may further include performing 520 some quantum computing operations of the plurality of quantum computing operations on some qubits 2b; 2d of some ladders of the second plurality of qubits for which no neighboring qubits from some chains of the first plurality of qubits are available. In the embodiment of Figures 3a and 3b, qubits qb of some ladders of the second plurality of qubits are available. i,2 (i is 1 to 12) does not have a neighboring qubit from the first plurality of qubits. In some cases, a qubit in each qubit ladder for which no neighboring qubit from some chain of the first plurality of qubits is available can exchange quantum information with a corresponding qubit in the same qubit ladder that is neighboring the qubit in some chain. Returning to the example of Figures 3a and 3b, each qubit qb i,2 is a qubit qb i,1(i is 1 to 12) are adjacent to each other, so that they can exchange quantum information between one another. Additionally, if a qubit ladder (e.g., one or more or each qubit ladder) includes more than two qubits (a situation not shown in the figures), adjacent qubits of the qubit ladder can exchange quantum information between one another in accordance with the above description (i.e., if, for example, none of these adjacent qubits of the qubit ladder has an adjacent qubit from some chain of the first plurality of qubits).

[0058] In some examples of the present disclosure, performing 400 a plurality of quantum computing operations on a number of chains of qubits may further include performing 420 a number of quantum computing operations on a number of qubits in a number of chains of a first plurality of qubits for which a neighboring qubit from a number of ladders of a second plurality of qubits is not available. For example, in the embodiment of FIG. 3a, none of qubits q5-q10 and q19-q21 in a number of chains of a first plurality of qubits has a neighboring qubit belonging to one of the qubit ladders shown in this figure. In other cases, such as those disclosed in connection with the embodiment of FIG. 3b, each qubit q1-q12 in a number of chains of a first plurality of qubits is neighbored by a respective qubit from a qubit ladder of a number of qubit ladders. In other words, in such embodiments, performing 420 a number of quantum computing operations defined above is irrelevant.

[0059] In some cases, the first aspect may further include receiving quantum information from a qubit q4 of a respective chain 15b of several disconnected chains or one or more paired chains by a first qubit 3a;qh1 of one or more auxiliary chains 15c of the first plurality of qubits. This method step may function for those embodiments of the first aspect when the first plurality of qubits includes one or more auxiliary chains 15c;3a-3c of qubits, as further explained above in connection with the discussion related to configuring a quantum computing system and FIG. 3b. The method may then further include transmitting quantum information from a first qubit 3a;qh1 of one or more auxiliary chains 15c of the first plurality of qubits to a second qubit 3b;qh6 by iteratively applying several subsequent SWAP operations between adjacent qubits of chains qh2-qh5 of one or more auxiliary chains 15c of the first plurality of qubits located therebetween (see FIG. 3b). Finally, in connection with embodiments in which auxiliary chains are available, the techniques can further include transmitting quantum information from a second qubit 3b;qh6 of one or more auxiliary chains 15c of the first plurality of qubits to a qubit q5 of another chain from some subsequent disconnected chain or one or more paired chains for the respective chain. Note that qubits q4 and qh1 are adjacent qubits, as discussed further above. Similarly for adjacent qubits qh6 and q5.

[0060] In some examples of the present technology, the above-described SWAP operations can be performed by respective quantum circuits (not shown) for swapping two qubits. In some examples, each quantum circuit can include three CNOT quantum gates known to those skilled in the art (see, for example, MA Nielsen and ILChuang, "Quantum Computation and Quantum Information": 10th Anniversary Edition, Cambridge University Press, 2010). Additionally or alternatively, one or more SWAP operations can include decompositions of the respective SWAP operations into corresponding quantum computation operations performed by native hardware gates.

[0061] In the techniques of the present disclosure, some ladders of the second plurality of qubits may include a first set of qubit ladders 16a-16d, including a first ladder 16a and one or more qubit ladders 16b-16d. In a non-limiting example, for purposes of further discussion, the sub-set of qubit ladders in the embodiment of FIG. 4 may be referred to as the “first set of qubit ladders,” the left ladder 16a from this set may be referred to as the “first ladder,” and the remaining three qubit ladders 16b-16d may be referred to as “one or more qubit ladders.” In some cases, the method of the first aspect may include transmitting quantum information from ladders 16b-16d of one or more qubit ladders to first ladder 16a by applying a SWAP operation between the ladder 16b of the one or more qubit ladders and the first ladder, if the ladder 16b is adjacent to the first ladder 16a. As a result, the ladder 16b (adjacent to the first ladder) and the first ladder can exchange quantum information between each other. As explained further above, a pair of adjacent qubit ladders in the same set of qubit ladders can be classified as an adjacent qubit ladder pair if at least one pair of adjacent qubits belonging to the pair of qubit ladders can be found. In some cases, a SWAP operation between two adjacent ladders can include several SWAP operations applied to individual qubits of these ladders. For example, the number of SWAP operations can increase as the number of qubits in the ladder under consideration increases and as the number of adjacent pairs of qubits in these two ladders decreases. For example, the qubit ladder 16b in FIG. 4, which is located next to the left ladder 16a in the lower set of qubit ladders, can exchange quantum information between two pairs of adjacent qubits, i.e., (qb 8,1 ,qb 7,1 ) and (qb 8,2 ,qb 7,2), it can be considered as a qubit ladder adjacent to the left ladder of qubit 16b. In this case, a SWAP operation between two adjacent ladders 16a, 16b (each of these ladders consists of two qubits) involves swapping two pairs of adjacent qubits (qb 8,1 , qb 7,1 ) and (qb 8,2 , qb 7,2 ) can include applying two SWAP operations between

[0062] Alternatively, if the ladder 16d is not adjacent to the first ladder 16a, the method of the first aspect may include transmitting quantum information from ladders 16b-16d of one or more qubit ladders to the first ladder 16a by applying several subsequent SWAP operations between the ladder 16d and adjacent ladders 16b-16c of the one or more qubit ladders that are disposed between the ladder 16d and the first ladder 16a of the one or more qubit ladders. Returning to the embodiment of FIG. 4, ladder 16c is not adjacent to ladder 16a (i.e., the "first ladder" according to the terminology used above). Thus, quantum information from ladder 16c that is not adjacent to ladder 16a may be transmitted, for example, through two pairs of adjacent qubits (qb 7,1 ,qb 6,1 ) and (qb 6,2 ,qb 7,2) can be first transmitted to its adjacent ladder 16b by applying two SWAP operations between ladder 16c and the first ladder 16a. As a result, these two adjacent ladders 16c and 16b can exchange quantum information between each other, and as a result, the quantum information of ladder 16c is physically located on ladder 16b. In the next step, a SWAP operation can be applied between ladder 16b and the first ladder 16a to exchange the quantum information of ladders 16b and 16a, according to the above description. Due to the final SWAP operation between ladders 16b and 16a, the quantum information of ladder 16c, which is physically located on qubit ladder 16b as a result of the first SWAP operation between ladders 16c and 16b, will finally be located on the first ladder 16a. Following this strategy, the quantum information of qubit ladder 16d, the furthest one in the subset of qubit ladders, can be transmitted to the first ladder 16a via three subsequent SWAP operations between qubit ladders 16c, 16b, and 16a.

[0063] In some cases, swapping quantum information along a set of qubit ladders representing a second degree of freedom (e.g., bosonic modes) in parallel with other quantum computing operations, e.g., operations involving using several chains of a first plurality of qubits representing a first degree of freedom (e.g., fermionic degrees of freedom), can potentially speed up quantum computations and / or reduce the overall depth (i.e., path length, representing the number of gates that must be executed along the path) of the quantum circuits involved in these computations. For example, when it is necessary to perform a quantum computing operation that represents an interaction between a first degree of freedom and a second degree of freedom of a physical system under consideration, for example, quantum information located on one of the ladders from a set of qubit ladders can be swapped along the set of qubit ladders toward the corresponding qubit representing the first degree of freedom.

[0064] In embodiments of the present technology, when multiple qubits 17;4a-4b of the multiple qubits (i.e., the quantum exchange register 17 introduced above in the context of the method steps related to configuring a quantum computing system) are present, the method of the first aspect may further include transmitting quantum information from a first ladder 16a of two or more ladders from several ladders of the second multiple of qubits to first several qubits qR4;qR5 of several qubits of a third multiple of qubits adjacent to the first ladder 16a. In some cases, a SWAP operation may be applied between the first ladder 16a and the first number of qubits qR4;qR5, which may include several SWAP operations applied to individual qubits of the first ladder and the first number of qubits in a manner similar to that described above in the context of swapping quantum information between adjacent qubit ladders in the same set of qubit ladders. For example, in the embodiment of Figure 4, ladder 16a can be referred to as the "first ladder" adjacent to two qubits qR4;qR5 of the third plurality of qubits (i.e., the two qubits qR4;qR5 of quantum exchange register 17). In this case, a SWAP operation between first ladder 16a (consisting of two qubits) and the two qubits qR4;qR5 is performed between two pairs of adjacent qubits (qb 8,1 ,qR5) and (qb 8,2 , qR4).

[0065] In a next step, the method of the first aspect including an embodiment having the quantum exchange register 17 may include transmitting quantum information from the first number of qubits qR4;qR5 to a second number of qubits qR2;qR3 of a number of qubits of a third plurality of qubits adjacent to a second ladder 16e of two or more ladders from a number of ladders of the second plurality of qubits by iteratively applying a number of subsequent SWAP operations between the first number of qubits qR4;qR5 and the second number of qubits qR2;qR3, where the iterative application of a number of subsequent SWAP operations may include iteratively applying a number of subsequent SWAP operations between adjacent qubits of the third plurality of qubits located between the first number of qubits qR4;qR5 and the second number of qubits qR2;qR3 when adjacent qubits between the qubits are available. Returning to the example of FIG. 4 , the quantum information of the first ladder 16a, which is physically located on qubits qR4;qR5 of the quantum exchange register 17 (as a result of the transmitting step defined above), can be transmitted to a second number of qubits, i.e., qubits qR2;qR3, adjacent to ladder 16e. Also, ladder 16e can be read as a “second ladder.” In the non-limiting embodiment shown in FIG. 4 , there are no qubits available between the first number of qubits qR4;qR5 and the second number of qubits qR2;qR4. In a next step, the method of the first aspect can include transmitting quantum information from second some qubits qR2;qR3 of some qubits of the third plurality of qubits to a second ladder 16e of two or more ladders adjacent to the second some qubits. For this purpose, a SWAP operation can be applied between the second number of qubits qR2;qR3 and the second ladder 16e, which may include several SWAP operations applied to the second number of qubits and to individual qubits of the second ladder (see above). In the example of Figure 4, the SWAP operation between the second ladder 16e (consisting of two qubits) and the two qubits qR2;qR3 is performed by swapping two pairs of adjacent qubits (qb 1,1,qR2) and (qb 1,2 , qR3) of the first qubit ladder 16a. 8,1 , qb 8,2 ) of the second qubit ladder 16e. 1,1 , qb 1,2 ) can be sent.

[0066] In one non-limiting example, the qubits (qb 8,1 , qb 8,2 ) and the qubit of the second qubit ladder 16e (qb 1,1 , qb 1,2 ) can be exchanged via the quantum exchange register 17 by applying the following sequence of SWAP operations starting from the initial encoding of quantum information on these qubits: 1,1 →qb 1,1 ,qb 1,2 →qb 1,2 ,qb 8,2 →qb 8,2 ,qb 8,1 →qb 8,1 : 1)qR2→qb 1,1 ,qR3→qb 1,2 ,qR4→qb 8,2 ,qR5→qb 8,1 ; 2) qR2 → qb 1,1 ,qR3→qb 8,2 ,qR4→qb 1,2 ,qR5→qb 8,1 ; 3) qR2 → qb 8,2 ,qR3→qb 1,1 ,qR4→qb 8,1 ,qR5→qb 1,2 ; 4) qR2 → qb 8,2 ,qR3→qb 8,1 ,qR4→qb 1,1 ,qR5→qb 1,2 ; 5) quarterback 1,1 →qb8,2 ,qb 1,2 →qb 8,1 ,qb 8,2 →qb 1,1 ,qb 8,1 →qb 1,2 .

[0067] In some cases, the qubit qb 1,1 and QB 1,2 can be additionally performed as the sixth operation in the above example, so that after the swap, the quantum information originally located on the first (second) qubit of the first ladder 16a is located on the first (second) qubit of the second ladder 16e, and vice versa, the quantum information originally located on the first (second) qubit of the second ladder 16e is located on the first (second) qubit of the first ladder 16a, i.e., 6)qb 1,1 →qb 8,1 ,qb 1,2 →qb 8,2 ,qb 8,2 →qb 1,1 ,qb 8,1 →qb 1,2 In the above notation, a physical qubit is shown to the left of each arrow, and the quantum information of the qubit located at this physical qubit is shown to the right of each arrow. For example, in step 1) above, qR2 → qb 1,1 is a qubit qb 1,1 This means that the quantum information is physically located on the qubit qR2.

[0068] In some cases, exchanging quantum information between different qubit ladders representing a second degree of freedom (e.g., boson modes) with quantum exchange register 17 in parallel with other quantum computation operations can potentially speed up quantum computations and / or reduce the overall depth of the quantum circuits involved in these computations. For example, quantum information of a particular qubit ladder can be efficiently transported through a 2D lattice of qubits of the present disclosure toward a corresponding qubit representing, for example, a first degree of freedom to perform a required quantum computation operation.

[0069] In the techniques of the present disclosure, the quantum information of each qubit of several chains of a first plurality of qubits (e.g., the qubit chains further described above in connection with Figures 2a, 2b, 3a, and 3b) carried by that qubit can include at least partial information regarding one or more first degrees of freedom (e.g., regarding fermionic degrees of freedom), or regarding one or more first degrees of freedom and one or more second degrees of freedom (e.g., regarding bosonic modes). In a first aspect, the partial information carried by a qubit of several chains of the first plurality of qubits can correspond to the quantum state of the qubit. In other words, the partial information encoded in a qubit can be represented by the quantum state possessed by the qubit. In some cases, the quantum information of each qubit of a ladder from several ladders of the second plurality of qubits (e.g., the qubit ladders discussed further above in connection with FIGS. 2a, 2b, 3a, and 3b) carried by the qubit may include at least partial information regarding one or more second degrees of freedom, or regarding one or more second degrees of freedom and one or more first degrees of freedom. Herein, the partial information of each qubit of a ladder from several ladders may correspond to the quantum state of the qubit of the ladder. As further explained above, when quantum computing operations are performed, a multi-qubit entangled state may be formed on qubits of the same and / or different types, and thus the qubits involved in forming this multi-qubit state may include at least partial information regarding different degrees of freedom of the same and / or different types (e.g., one or more fermionic degrees of freedom and / or one or more bosonic modes).

[0070] As used herein, multiple quantum computation operations can be performed to simulate the time evolution of a physical system on quantum computing system 1000, where the time evolution of the physical system can be defined by the unitary time evolution of the quantum Hamiltonian of the physical system. In some cases, the unitary time evolution can be expressed by a unitary time evolution operator. For example, the unitary time evolution of a physical system at time t in physical units can be expressed by a unitary time evolution operator where the reduced Planck constant is

number

number

[0071] In this technique, the unitary time evolution of the quantum Hamiltonian of a physical system within a given time interval can be decomposed into several time steps. For example, a given time interval T can be written as T = n τ, where τ represents the time step and n is the number of steps (e.g., n is 10 or greater, e.g., 10). 2 That's it, 10 3 That's it, 10 4 That's it, 10 5(This may be an integer equal to or greater than τ. In some cases, multiple quantum computing operations, as further introduced above, may be performed to simulate the unitary time evolution of a quantum Hamiltonian of a physical system on the quantum computing system 1000 within each time step. To this end, the unitary time evolution of a quantum Hamiltonian within a time step (e.g., within each time step) may be decomposed into the unitary time evolution of a first sub-Hamiltonian, the unitary time evolution of a second sub-Hamiltonian, and the unitary time evolution of a third sub-Hamiltonian. For example, the unitary time evolution of a physical system within a time step τ given by the unitary time evolution operator defined above:

number

[0072] In the technique of the present disclosure, multiple quantum computing operations can be performed on several chains of the first plurality of qubits to simulate the unitary time evolution of the first sub-Hamiltonian and the third sub-Hamiltonian (e.g., the sub-Hamiltonians h1 and h3 introduced above). In other words, several chains of the first plurality of qubits representing the first degree of freedom can be involved in the quantum computing operations of those portions of the Hamiltonian of the physical system including the first degree of freedom. Next, multiple quantum computing operations can be performed on several ladders of the second plurality of qubits to simulate the unitary time evolution of the second sub-Hamiltonian and the third sub-Hamiltonian (e.g., the sub-Hamiltonians h2 and h3 introduced above). Similarly, several ladders of the second plurality of qubits representing the first degree of freedom can be involved in the quantum computing operations of those portions of the Hamiltonian of the physical system including the first degree of freedom.

[0073] As used herein, the first degree of freedom may be a fermionic degree of freedom. In this case, the first sub-Hamiltonian (e.g., h1) may include an operator associated with the fermionic degree of freedom, where the operator associated with the fermionic degree of freedom may be a fermion creation operator, a fermion annihilation operator, or a product thereof (e.g., such a product may include one or more, two or more, three or more fermion creation operators and / or fermion annihilation operators if the product is a Hermitian operator). In some examples, the first sub-Hamiltonian may include an interaction term describing the interaction between two or more of the fermionic degrees of freedom. Additionally or alternatively, the first sub-Hamiltonian may include a hopping term describing the hopping between two or more of the fermionic degrees of freedom. In some cases, the unitary time evolution of the first sub-Hamiltonian within a time step τ can be decomposed into the unitary time evolutions of the interaction term and the hopping term within this time step if both terms, the interaction term and the hopping term, are present in the first sub-Hamiltonian.

[0074] In the first embodiment, the second degree of freedom may be a bosonic mode. In this case, the second sub-Hamiltonian (e.g., h2) may include an operator associated with the bosonic mode, which may be a bosonic creation operator, an annihilation operator, or a product thereof (e.g., such a product may include one or more, two or more, or three or more bosonic creation operators and / or annihilation operators, provided that the product is a Hermitian operator). Furthermore, the third sub-Hamiltonian (e.g., h3) may include an operator that is a product of an operator associated with a fermionic degree of freedom and a bosonic mode (the Hermiticity condition should also be satisfied for these operator products). The third sub-Hamiltonian of the first embodiment may describe the interaction between the bosonic mode and each fermionic degree of freedom.

[0075] A non-limiting example of a coupled fermion-boson system that can be simulated using the techniques of this disclosure is a physical system governed by the following Hamiltonian:

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number

number

[0076] Another non-limiting example of a coupled fermion-boson system that can be simulated using the techniques of the present disclosure is a physical system governed by the following Hamiltonian (all notations are kept the same as those used in the previous example):

number

[0077] For a physical system described by Hamiltonian (1a), the number of bosonic modes is smaller than the number of fermionic degrees of freedom. In this case, the physical system can be mapped onto a quantum computing system arranged on a 2D lattice with the topology shown in Figure 2a (as described above in connection with Hamiltonian (1)). For example, for N = 12 bosonic modes, there are 2N = 24 fermionic degrees of freedom. In this particular case, the physical system can be mapped onto a quantum computing system arranged on a 2D lattice similar to that shown in Figure 3a (where the 12 ladders 16 represent the bosonic modes and the 21 qubits q1 through q21 represent the fermionic degrees of freedom). The only difference to be implemented in the embodiment of Figure 3a is the insertion of three additional qubits, namely qubits 22, 23, and 24, into the chain of the first plurality of qubits. In one example, the so-called jellium model, which describes electrons in a solid using a simplified on-site Coulomb repulsion coupled to phonons with only one polarization (see, for example, "Many-Body Quantum Theory in Condensed Matter Physics. An Introduction", Bruus K. and Flensberg K., Oxford University Press 2007), can be given by Hamiltonian (1): (In some cases, the only possible deviation from Hamiltonian (1) may be related to the influence of boundary conditions.)

[0078] A third non-limiting example of a coupled fermion-boson system that can be simulated by the present technique is a physical system described by the following Hamiltonian (all notations are kept the same as those used in the previous examples):

number

[0079] For a physical system governed by the above Hamiltonian, the number of bosonic modes is equal to the number of fermionic degrees of freedom. In this case, the physical system can be mapped onto a quantum computing system arranged on a 2D lattice with the topology shown in Figure 2b. For example, for 2N = 12 bosonic modes, there are 2N = 12 fermionic degrees of freedom. In this particular case, the physical system can be mapped onto a quantum computing system arranged on a 2D lattice as depicted in Figure 3b (where 12 ladders 16 represent bosonic modes and 12 qubits q1 to q12 represent fermionic degrees of freedom). In another example of 2N = 4 bosonic modes and 2N = 4 fermionic degrees of freedom, the physical system can be mapped onto a quantum computing system arranged on a 2D lattice as depicted in Figure 5a (where 4 ladders 16 represent bosonic modes and 4 qubits q1 to q4 represent fermionic degrees of freedom).

[0080] A fourth non-limiting example of a coupled fermion-boson system that can be simulated by the present technique is a physical system described by the following Hamiltonian (all notations are kept the same as those used in the previous examples):

number

[0081] For a physical system governed by Hamiltonian (2a), the number of bosonic modes is greater than the number of fermionic degrees of freedom. For example, there are 3N = 12 bosonic modes and 2N = 8 fermionic degrees of freedom. In this particular case, the physical system can be mapped onto a quantum computing system arranged on a 2D lattice with a topology similar to that depicted in Figure 3b (12 ladders 16 represent the bosonic modes and 8 qubits 15b represent the fermionic degrees of freedom). The only difference to be implemented in the embodiment of Figure 3b is the replacement of four qubits from the first chain of qubits with four ancillary qubits (e.g., the four qubits q4, q5, q11, and q12 can be replaced with four ancillary qubits). In one example, the jellium model described above in connection with Hamiltonian (1a), which treats electrons in a solid with simplified on-site Coulomb repulsion, can yield Hamiltonian (2a) if several polarizations are considered in the model (e.g., optical and acoustic modes are considered). (In some cases, when considering coupling to phonons with three polarizations, the only possible deviation from Hamiltonian (2a) may be related to the influence of boundary conditions).

[0082] In some examples of the present technology, the hopping coefficient V may be different for all hopping terms of the Hamiltonian H (e.g., for one or more of the Hamiltonians given by equations (1), (1a), (2), and (2a) above). In other examples, two or more (e.g., all) hopping coefficients V may be the same. In some cases, all interaction coefficients U may be different, or two or more (e.g., all) interaction coefficients U may be the same. In some cases, all coupling strengths g i,σ may be different, or two or more (e.g., all) of the interaction coefficients g i,σ may be the same. In some cases, all boson frequencies ω i may be different, or two or more (e.g., all) boson frequencies ω i may be the same.

[0083] In the techniques of the present disclosure, the step 300 of performing the plurality of quantum computation operations may further include initially ordering a plurality of first degrees of freedom (e.g., fermionic degrees of freedom) onto several chains of the first plurality of qubits, each of the plurality of first degrees of freedom being mapped onto a respective qubit from the several chains of the first plurality of qubits. For example, the fermionic creation operator and the fermionic annihilation operator introduced above may be used.

number

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[0084] In other words, the quantum information associated with these four fermionic degrees of freedom is physically located on qubits q1 through q4 in the order defined above, which can be represented, for example, by the quantum states of these qubits (see, e.g., https: / / doi.org / 10.1103 / PhysRevLett.120.110501 for further details). The ordering of the four fermionic degrees of freedom in this non-limiting example is shown schematically in the upper dashed rectangle 20a of Figure 5b.

[0085] In a first aspect, the step 300 of performing the plurality of quantum computing operations can further include initially mapping one or more of the second degrees of freedom (e.g., bosonic modes) to respective one or more ladders from several ladders of the second plurality of qubits. For example, in the embodiment of FIG. 5a, the first bosonic mode, |Mode1>, b qubit (qb 2,1 ,qb 2,2 ) can be mapped to the qubits of ladder 16 (this ladder is enclosed in a dashed oval in this figure). |Mode1> b →qb 2,1 qb 2,2 (4).

[0086] In this non-limiting embodiment, when only a single bosonic mode is present, the other three ladders do not participate in the quantum computation. For example, no quantum computation operations are performed on the qubits of these remaining ladders. In the binary encoding defined above, the first bosonic mode, |Mode1>, b The four Fock states for the two qubits (qb 2,1 ,qb 2,2 ), which can be given, for example, in bracket notation as known in the art: |00>=|0> b ,|01>=|1> b ,|10>=|2> b ,|11>=|3> b (see also the following discussion). Here, the ket vector, e.g., |01>, is the vector of the qubit qb 2,1 is in an excited state, and another qubit qb 2,1 In the case of unary encoding, the first bosonic mode, |Mode1>, corresponds to the case where b The two Fock states of the ladder 16 shown in Figure 5a are only 2,1 ,qb 2,2 ) In one example, this unary encoding can beb , |10>=|1> b Note that the exact implementation of a digital quantum simulation (at the level of individual quantum operations) may depend on the chosen encoding of the boson modes onto the qubits of the qubit ladder.

[0087] The disclosed method may further include initializing one or more qubits from some chains of the first plurality of qubits and one or more qubits from some ladders of the second plurality of qubits to an initial quantum state. In a first aspect, one or more qubits from some chains of the first plurality of qubits and one or more qubits from some ladders of the second plurality of qubits may be configured to be initialized to an initial quantum state. According to the above description, the initial quantum state of the qubits may represent a quantum state associated with a first degree of freedom and a second degree of freedom. In some cases, the quantum state is a product quantum state of a first quantum state associated with the first degree of freedom and a second quantum state associated with the second degree of freedom, where the first quantum state is represented by one or more qubits (e.g., each qubit) from some chains of the first plurality of qubits and the second quantum state is represented by one or more qubits (e.g., each qubit) from some ladders of the second plurality of qubits.

[0088] In some cases, the initial quantum state of the qubit can be a multi-qubit quantum state, which is a tensor product state (also referred to as a separable quantum state) including the tensor product of each single-qubit quantum state of one or more qubits from some chains of the first plurality of qubits and one or more qubits from some ladders of the second plurality of qubits. In one example, the quantum state of the qubit can be, for example, the zero-numbered quantum state corresponding to the ground state of the qubit (i.e., the state with the lowest energy, or in other words, the unexcited state). This state can be represented, for example, by the ket vector |0> in bracket notation known to those skilled in the art. In another example, the quantum state of the qubit can be, for example, a first quantum state corresponding to the first excited state of the qubit (i.e., the excited state closest in energy to the ground state). This state can be represented by the ket vector |1>. In yet another example, the quantum state of the qubit of the first one or more qubits can be a linear superposition of the zeroth and first quantum states. In this case, the quantum state of the qubit can be written as |ψ>=α|0>+β|1>, for example, where α and β are some non-zero amplitudes. Returning to the tensor product state above, in some cases

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[0089] In a first aspect, in accordance with the above explanation, the initial tensor product state can be expressed as a product state of the quantum state of the fermionic degrees of freedom and the quantum state of the bosonic modes. In one particular case, the fermionic states can correspond to the ground states of the completely non-interacting fermionic part of the Hamiltonian of the physical system, and the bosonic modes can correspond to the non-excited states |0> b In other cases, the fermionic states can correspond to the ground states of the fully interacting fermionic part of the Hamiltonian of the physical system, and the bosonic modes correspond to those de-excited states |0> b In some cases, the ground states of the fully interacting fermionic part of the Hamiltonian can be generated using corresponding quantum algorithms.

[0090] In the disclosed technique where the first degree of freedom is a fermionic degree of freedom and the second degree of freedom is a bosonic mode, the step 300 of performing a plurality of quantum computing operations may further include exchanging quantum information 700 between each qubit from some odd-numbered qubits (q1; q3) of some chains of the first plurality of qubits and a respective adjacent even-numbered qubit (q2; q4) located on a predetermined side of the odd-numbered qubit, if there is an even-numbered qubit located on a predetermined side of the odd-numbered qubit. Here, each adjacent even-numbered qubit may be a qubit from some even-numbered qubits of some chains of the first plurality of qubits. For example, the second qubit q2, which is an even-numbered qubit representing the first encoded fermionic degree of freedom (1,↓), is located to the right of the first qubit q1, which is an odd-numbered qubit representing the first encoded fermionic degree of freedom (1,↑). These qubits can exchange their quantum information, as shown schematically by the arrows in the upper dashed rectangle 20a of Figure 5b. Similarly, the fourth qubit, q4, which is an even-numbered qubit representing the first encoded fermionic degree of freedom (2,↑), is located to the right of the first qubit, q3, which is an odd-numbered qubit representing the first encoded fermionic degree of freedom (2,↓). These qubits can also exchange their quantum information, as shown schematically by the arrows in the upper dashed rectangle 20a of Figure 5b. In one example, both exchange steps can be performed within the first step of a digital quantum simulation (see also further discussion).

[0091] Herein, exchanging quantum information 700 between adjacent qubits q1, q2; q3, q4 from several chains of the first plurality of qubits can be performed by applying a fermionic swap operation (FSWAP) 21 between these adjacent qubits, where the fermionic swap operation preserves the fermionic anticommutation relation between the operators associated with the fermionic degrees of freedom encoded in the qubits. For example, the FSWAP operation can be defined as a unitary transformation that swaps two fermionic degrees of freedom. In one example, the FSWAP transformation is:

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[0092] In a next step, the method of the first aspect may include executing 710 several available quantum computation operations from the plurality of quantum computation operations on one or more qubits 1a-1d from several chains of the first plurality of qubits. In some cases, several available quantum computation operations on qubits from several chains of the first plurality of qubits may be executed to simulate the unitary time evolution of the first sub-Hamiltonian within a time step (e.g., within each time step). In one example, the unitary time evolution of the first sub-Hamiltonian within time step τ can be written as exp(-iτh1), consistent with the above description. In some cases, the "availability" of quantum computation operations may depend on one or more of: i) the structure of the Hamiltonian under consideration; ii) the mapping (e.g., based on the Jordan-Wigner transformation) used for fermionic degrees of freedom; and iii) the steps of the digital quantum simulation. In other words, some quantum computing operations may be available in a first step of the digital quantum simulation, while some other quantum computing operations may be available in a second step of the digital quantum simulation, etc. Note that after performing multiple quantum computing operations on the 2D qubit lattice of the present disclosure, the unitary time evolution of all terms in the quantum Hamiltonian of the physical system within a time step (e.g., within time step τ) can be simulated as described above.

[0093] If a physical system is governed by Hamiltonian (2) and the system is mapped onto the 2D lattice shown in Figure 5a, the first step of the digital quantum simulation can describe the unitary time evolution of the interaction terms associated with the fermionic degrees of freedom (1,↑), (1,↓), (2,↓), and (2,↑), while the unitary time evolution of the hopping terms is not available at this stage of the digital quantum simulation (see also the initial ordering of the four fermionic degrees of freedom given by Equation (3)). For example, the unitary time evolution of the hopping terms can be simulated in the second step of the digital quantum simulation (see further discussion below). Specifically, the following unitary time evolution of the interaction terms of the first sub-Hamiltonian within time step τ can be simulated in the first step of the digital quantum simulation: exp(-iUn 1,↑ n 1,↓ τ)exp(-iUn 2,↑ n 2,↓ τ).

[0094] In some cases of the present disclosure, for convenience, a quantum operation describing the simulation of the first sub-Hamiltonian within time step τ can be combined with the FSWAP operation. This quantum operation can be referred to as a fermion simulation (FSIM) operation. In the example of Hamiltonian (2), the FSIM operation acting on each qubit can be written as:

number

[0095] The disclosed technique involves two adjacent qubits (q1, q2) among several chains of a first plurality of qubits and a qubit (qb 2,1 ), where each qubit ladder is from several ladders of a second plurality of qubits, and a corresponding boson mode is encoded by the respective qubit ladder. In some cases, several available quantum computing operations including two adjacent qubits from several chains of the first plurality of qubits and each qubit ladder having a qubit adjacent to one of the adjacent qubits from several chains of the first plurality of qubits can be executed to simulate a unitary time evolution of the third sub-Hamiltonian within a time step (e.g., within each time step). In one example, the unitary time evolution of the first sub-Hamiltonian within time step τ can be written as exp(-iτh3), consistent with the above description.

[0096] Returning to the embodiment of Figure 5a, the first bosonic mode, |Mode1> b is a qubit (qb 2,1 ,qb 2,2 ) can be mapped to the qubit of ladder 16 with qubit qb 2,1 is adjacent to qubit q2 in the chain of the first plurality of qubits (as described above, qubit qb 2,1 (The adjacency between qb and q2 is shown in this figure by a dashed line.) If a physical system is governed by Hamiltonian (2) and the system is mapped onto the 2D lattice shown in Figure 5a, the first step in the digital quantum simulation is to map the qubit qb of the qubit ladder 16. 2,1 and qb 2,2We can further describe the unitary time evolution of the fermion-boson interaction associated with the first boson mode, encoded in q and the fermion degrees of freedom (1,↑) and (2,↑). Note that the fermion degrees of freedom (1,↑) and (2,↑) are encoded in qubits q and q, as shown schematically in dashed rectangle 20b in FIG. 5b. Because two FSWAP operations have already been performed, see upper dashed rectangle 20a and previous discussion. Specifically, the following unitary time evolution of the fermion-boson interaction term of the third sub-Hamiltonian within time step τ can be simulated in the first step of the digital quantum simulation:

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[0097] In the disclosed technique, the step 300 of performing a plurality of quantum computing operations may further include exchanging 730 quantum information between each qubit from some even-numbered qubits (q2) of some chains of the first plurality of qubits and a respective adjacent odd-numbered qubit (q3) located on a predetermined side of the even-numbered qubit, if there is an odd-numbered qubit located on a predetermined side of the even-numbered qubit. Here, each adjacent odd-numbered qubit is a qubit from some odd-numbered qubits of some chains of the first plurality of qubits. For example, the third qubit q3, which is an odd-numbered qubit representing fermionic degrees of freedom (2,↑) after performing the FSWAP operation during the first step of the digital quantum simulation, is located to the right of the second qubit q2. Qubit q2 is an even-numbered qubit and represents fermionic degrees of freedom (1,↑) after performing the FSWAP operation during the first step of the digital quantum simulation. Qubits q2 and q3 can exchange their quantum information, as shown schematically by arrow 22 within dashed rectangle 20b in Figure 5b. In one example, this exchange step can be performed within the second step of the digital quantum simulation.

[0098] In a next step, the method of the first embodiment can include performing 740 some available quantum computation operations from the plurality of quantum computation operations on one or more qubits 1a-1d from some chains of the first plurality of qubits. This method step can be performed similarly to method step 710 described above. If a physical system is governed by Hamiltonian (2) and the system is mapped onto the 2D lattice shown in FIG. 5a, the second step of the digital quantum simulation can describe the unitary time evolution of the hopping terms associated with the fermionic degrees of freedom (2,↑) and (1,↑), while the unitary time evolution of other terms in the first sub-Hamiltonian is not available at this stage of the digital quantum simulation. Specifically, the following unitary time evolution of the hopping terms of the first sub-Hamiltonian within time step τ can be simulated in the second step of the digital quantum simulation:

number

[0099] Thus, the second step of the digital quantum simulation in the above example can be written using the FSIM operation as FSIM(τV,0)q2 q3. As a result of this operation, the fermionic degrees of freedom are arranged as shown in dashed rectangle 20c (third from the top) in Figure 5b.

[0100] The disclosed technique involves two adjacent qubits (q1, q2) among several chains of a first plurality of qubits, and a qubit (qb 2,1), and executing 750 some available quantum computing operations from a plurality of quantum computing operations including a respective qubit ladder 16 having a qubit number of 1 and a qubit number of 2. This method step may be performed similarly to method step 720 further disclosed above. Returning to Hamiltonian (2) mapped onto the 2D lattice depicted in FIG. 5 a, the quantum computing operations describing the unitary time evolution of the fermion-boson interaction involving the qubits defined in method step 750 are not available in the second step of the digital quantum simulation. In this case, the digital quantum simulation may continue in a third step, as described below.

[0101] In a next step, the method of the first aspect may include step 760 of iteratively repeating exchanging information and executing some available quantum computing operations until all available quantum computing operations from the plurality of quantum computing operations have been executed. If a physical system is governed by Hamiltonian (2) and the system is mapped onto the 2D lattice shown in Figure 5a, further possible steps of the digital quantum simulation may be as follows.

[0102] 3) FSIM(0,0)q1 q2 and FSIM(0,0)q4 q3: These are the two FSWAP operations (both FSIM operations have zero arguments) required to introduce the fermionic degrees of freedom (1,↓) and (2,↓) and simulate the hopping terms involving these fermionic degrees of freedom. As a result of this operation, the fermionic degrees of freedom are arranged as shown in dashed rectangle 20d (fourth from the top) in Figure 5b.

[0103] Next, the fermionic degrees of freedom (1,↓) and (2,↓) (physically located on qubits q2 and q3) and the qubit qb of qubit ladder 16 are 2,1 and qb 2,2This can be done in a manner similar to that described above in connection with method step 720.

[0104] 4) FSIM(τV,0)q2 q3. This step exchanges the fermionic degrees of freedom (1,↓) and (2,↓) and simulates the hopping terms that involve them. As a result of this operation, the fermionic degrees of freedom are arranged as shown in the rectangle 20e below the dashed line in Figure 5b.

[0105] The techniques of the present disclosure may further include performing 770 some quantum computing operations of the plurality of quantum computing operations on each qubit ladder 16 (e.g., on one or more ladders from some ladders of the second plurality of qubits). In some cases, some quantum computing operations on each qubit ladder may be performed to simulate a unitary time evolution of the second sub-Hamiltonian within a time step (e.g., within each time step). In one example, the unitary time evolution of the second sub-Hamiltonian within time step τ can be written as exp(-iτh2), consistent with the above description. The qubit qb of qubit ladder 16 2,1 and qb 2,2 In the particular case of the first boson mode encoded in, such a unitary time evolution within a time step τ is exp(-iωb † bτ), where ω represents the frequency of the first bosonic mode. Returning to the Hamiltonian (2) mapped onto the 2D lattice depicted in Figure 5a, in some cases, the quantum computational operation describing the unitary time evolution of the second sub-Hamiltonian within time step τ in this particular example is performed as the final, fifth step 5) of the digital quantum simulation (the configuration of the fermionic degrees of freedom after performing the fourth step 4), shown in the dashed rectangle 20e in Figure 5b).

[0106] In the present disclosure, the unitary time evolution of a physical system determined by Hamiltonian H is said to be simulated within a single time step (e.g., within time step τ) when all of the above available quantum computing operations and the number of quantum computing operations from the plurality of quantum computing operations on each qubit ladder 16 are completed. As further described above, this procedure can be repeated to propagate the unitary time evolution over a predetermined time interval (e.g., over a time interval T=n−τ), where n is an integer as further described above.

[0107] As used herein, each of the boson creation and annihilation operators can be decomposed into a Pauli operator and / or a superposition of products of Pauli operators, where each Pauli operator can act on the quantum state of one of the qubits of the respective qubit ladder (e.g., on each qubit of the respective ladder) that encodes a corresponding boson mode. Here, the Pauli operator is one of the Pauli X, Y, or Z operators. The qubit qb of the qubit ladder 16 shown in FIG. 5a is 2,1 and qb 2,2 In the particular case of the first bosonic mode encoded in , the bosonic number operator is, in the binary encoding, b † b=3 / 2-2Z2-Z1, where Z1 is the qubit qb 2,1 represents the Pauli Z operator acting on the qubit qb of ladder 16 in Fig. 5a. 2,2 is the Pauli Z operator acting on . Therefore, the unitary time evolution within a time step τ can be written in this particular example as

number

[0108] In the method of the first aspect, the above decomposition may be implemented as a sequence of single-qubit rotations known to those skilled in the art.

[0109] In the technique of the present disclosure, two adjacent qubits (q1, q2) among several chains of a first plurality of qubits and a qubit (qb 2,1 Executing 720; 750 several available quantum computation operations from the plurality of quantum computation operations, including a respective qubit ladder 16 having a first plurality of qubits (q1, q2), can include using several quantum two-qubit gates and / or single-qubit gates arranged in a corresponding order, acting on two adjacent qubits (q1, q2) of several chains of the first plurality of qubits. In some cases of the present technology, this step allows the unitary time evolution of a fermion-boson interaction term involving two adjacent qubits and a bosonic mode of several chains of the first plurality of qubits (encoding two respective fermionic orders) to be expressed as a fermion-boson interaction term involving only one of these two qubits and a bosonic mode. Returning to the embodiment of FIG. 5a, the two qubits q2, q3 of the chains of the first plurality of qubits are adjacent qubits. One of these qubits, namely qubit q2, is also connected to qubit qb of qubit ladder 16. 2,1 If a physical system is governed by Hamiltonian (2) and the system is mapped onto the 2D lattice shown in FIG. 5a, then the fermion-boson interaction term (3) described above with the fermionic degrees of freedom (1,↑) and (2,↑) encoded in qubits q2 and q3 in the first step of the digital quantum simulation can be rewritten, in one non-limiting example, as follows:

number

number

[0110] In the method of the first aspect, the above decomposition may be implemented as a sequence of single-qubit rotations and controlled-NOT (CNOT) operations, as illustrated in Figures 6a-6c and described in detail below.

[0111] The next step in the method is to: i) load each qubit (qb 2,1 ), one (q2) of the adjacent qubits (q1, q2) of the several chains of the first plurality of qubits that is adjacent to the adjacent qubit (q2) of the several chains of the first plurality of qubits, and ii) a qubit (qb 2,1 ) and iii) one or more qubits (qb 2,2 ) and performing several quantum computation operations, including: (i) and (iii). For illustrative purposes, the implementation of these three steps i), ii), and iii) can be performed using the unitary transformation exp(-igτZ) shown in Figures 6a, 6b, and 6c, respectively. f X1), exp(-igτZ f X1X2 / 2), and exp(-igτZ f This is elucidated using three non-limiting examples of quantum circuits for simulating the quantum state (Y1Y2 / 2).

[0112] In the present technology, performing the above-mentioned several quantum computing operations, including i), ii), and iii), includes performing the quantum computation operations on a qubit qb of a respective qubit ladder adjacent to one of the adjacent qubits (q2) of the several chains of the first plurality of qubits. 2,1 The next step of the method may involve applying a basis rotation transformation 40 to one or more qubits (qb) of each qubit ladder that are not adjacent to one of the adjacent qubits (q2) of the several chains of the first plurality of qubits. 2,2 ), where the basis rotation transformation corresponds to rotating a given axis on the rotation axis of the respective qubit (e.g., on the z-axis). In the embodiment of Figure 6b, one Hadamard gate H, 40 is applied to the qubit qb of Figure 5a adjacent to qubit q2. 2,1 Another Hadamard gate H, 41 can be applied to the qubit ladder qb 2,1 qubits qb that are not adjacent to 2,2 In one example, the Hadamard gate can convert between the z-basis and the x-basis. In the embodiment of Figure 6c, R X (π / 2) represents the basis rotation transformation, which is the rotation operator for converting between the z- and y-bases. The method of the first aspect then calculates the qubits (qb 2,1 ) and one of the adjacent qubits (q2) of some chains of the first plurality of qubits adjacent to the qubit ladder (qb 2,1 ) (this CNOT operation is indicated by reference numeral 42 in Figures 6a-6c).

[0113] The method of the first embodiment may further comprise, where the one or more qubits include at least two qubits of a respective qubit ladder (see reference numeral 43 in FIGS. 6b and 6c), a qubit (qb 2,1 ), and then add one or more qubits (qb 2,2 ) to the two subsequent qubits (qb 2,1 ;qb 2,2 ) to the last qubit qb of the one or more qubits in each qubit ladder to which the number of CNOT operations has been applied. 2,2 A rotation transformation around the rotation axis to (R z (γ))44. In some cases, the rotation transformation about the rotation axis applied to the last qubit of the one or more qubits of the respective qubit ladder can be defined by a predetermined rotation angle about the rotation axis. In some examples, the predetermined rotation angle can be proportional to the product of the time step and the coupling strength between the fermionic degrees of freedom encoded by two adjacent qubits of some chains of the first plurality of qubits and the bosonic mode encoded by the respective qubit ladder. In the particular example of the unitary transformation simulated in Figures 6a, 6b, and 6c, the predetermined rotation angle about the rotation axis can be given by γ = gτ.

[0114] The next step in the technique is to determine the last qubit (qb) of the one or more qubits of each qubit ladder, when the one or more qubits include at least two qubits of the respective qubit ladder. 2,2 ), and then select the qubits (qb2,1 ) so that the two subsequent qubits (qb 2,2 ;qb 2,1 ) in reverse order (see reference numerals 45 in Figures 6b and 6c). The next step of the method is to iteratively apply several successive CNOT operations 45 between the qubits (qb 2,1 ) and one of the adjacent qubits (q2) of some chains of the first plurality of qubits adjacent to the qubit ladder (qb 2,1 ) (see reference numeral 46 in Figures 6a-6c). In a next step of the method, an inverse basis rotation transformation 47 is applied to one or more qubits (qb) of the respective qubit ladders that are not adjacent to one of the adjacent qubits (q2) of the several chains of the first plurality of qubits. 2,2 ) (see reference numeral 47 in Figures 6b and 6c). The next step of the technique is to apply an inverse basis rotation transformation 48 to the qubits (qb) of each qubit ladder adjacent to one of the adjacent qubits (q) of the several chains of the first plurality of qubits. 2,1 ) (see reference numeral 48 in FIGS. 6a-6c). The inverse basis rotation transformation of the first aspect may correspond to rotating the rotation axis of each qubit back to a predetermined axis. In some examples, the inverse basis rotation transformation may be defined as a transformation defined by operators (or their respective matrix representations) that are Hermitian conjugates to the operators defining the basis rotation transformation. In some embodiments of the present technology, the rotation axis may be the z-axis.

[0115] In some examples of the present technology, one or more CNOT operations used above can be performed by respective quantum CNOT gates. Additionally or alternatively, one or more CNOT operations may include respective decompositions of the one or more CNOT operations into corresponding quantum operations performed by native hardware gates, where native hardware gates are gates available in a particular architecture of a quantum computer and / or qubit topology.

[0116] In the disclosed technique, step 770 of performing some of the quantum computing operations on each qubit ladder 16 includes: i) performing some of the quantum computing operations on a qubit (qb 2,1 ii) one or more qubits (qb 2,2 ) and performing several quantum computation operations, including: (i) performing a predetermined rotation angle γ about the axis of rotation γ; (ii) performing a predetermined rotation angle γ about the axis of rotation γ; (iii) performing a predetermined rotation angle γ about the axis of rotation γ; (iv) performing a predetermined rotation angle γ about the axis of rotation γ; (v) performing a predetermined rotation angle γ about the axis of rotation γ; (vi ...

[0117] The techniques of the present disclosure may further include executing a quantum computing task, where the quantum computing task includes a plurality of quantum computing operations performed on several chains of the first plurality of qubits and several ladders of the second plurality of qubits. In addition, the quantum computing task may further include a plurality of quantum computing operations performed on one or more auxiliary chains of qubits of the first plurality of qubits. In some cases, the quantum computing task may further include a plurality of quantum computing operations performed on a third plurality of qubits. The techniques of the present disclosure may perform one or more of the plurality of quantum computing operations to simulate the unitary time evolution of a quantum Hamiltonian of a physical system on a quantum computing system within a time step (e.g., each time step). In a first aspect, the plurality of quantum computing operations may include a respective number of one or more quantum computing operations performed to simulate the unitary time evolution of a quantum Hamiltonian of a physical system within a predetermined time interval.

[0118] In some cases, the quantum computing task may include one or more problems in fields such as simulation of quantum systems, computational chemistry, computational biology, solid-state physics, quantum annealing, quantum machine learning, search problems, cryptography, etc. In some examples, one or more quantum computing operations may be performed in parallel on different multiples of qubits. For example, during a first time interval, a first number of computational operations may be performed sequentially on a first number of qubits from several chains of a first plurality of qubits and in parallel with a second number of computational operations on a second number of qubits from those several chains. Alternatively or additionally, a third number of computational operations may be performed sequentially on a first number of qubits from one qubit ladder of the second plurality of qubits and in parallel with a fourth number of computational operations performed on another qubit ladder of the second plurality of qubits during the same or a different time interval.

[0119] A second aspect provides a quantum computing system 1000 configured according to any of the steps of the technique according to the first aspect.

[0120] A third aspect provides a quantum computing system adapted to perform a plurality of quantum computation operations and to perform any of the steps of the technique according to the first aspect.

[0121] In some examples, the quantum computing system of the third aspect is configured according to the quantum computing system of the second aspect. The present disclosure also relates to a computer program adapted to perform any of the steps of the technique according to the first aspect. The present disclosure also relates to a computer-readable medium (e.g., a machine-readable storage medium such as an optical storage medium or a read-only memory, e.g., a flash memory) and a signal that stores or encodes the computer program of the present disclosure.

[0122] A quantum computing system of the second and / or third aspects may include at least one processor (e.g., a quantum processor), at least one memory (which may include a program that, when executed, performs the method steps according to the first aspect or the computer program of the present disclosure), and at least one interface for input and output. In some examples, the quantum computing system may include a hardware architecture including, for example, one or any combination of one or more chips, a quantum data plane, a control plane, a measurement plane, a control processor plane, a host processor, etc. The hardware architecture of the quantum computing system of the second and / or third aspects may be based on qubits coupled to a high-finesse cavity (e.g., superconducting qubits coupled to a microwave cavity), as further described above. In other examples, the quantum computing system may include a hardware architecture based on qubits realized as nuclear spin states of donor atoms embedded in respective host lattices. In yet other examples, the quantum computing system may include a hardware architecture based on neutral atoms in an optical lattice. In some examples, the quantum computing system may be a standalone computing device. In other examples, the quantum computing system may be integrated into a computing device or system that serves purposes other than performing the steps of the techniques of the present disclosure. In yet another example, the quantum computing system may be a distributed system that communicates over a network (e.g., the Internet).

[0123] A fourth general aspect of the present disclosure relates to a remote computing system configured to perform a quantum computing task, the quantum computing task including a plurality of quantum computing operations according to the first aspect. The plurality of quantum computing operations (performed by the remote computing system) of the fourth aspect can be performed according to any one of the method steps of the first aspect. In some examples, the remote computing system can be configured to receive a query regarding the quantum computing task from a computer-implemented system (e.g., external to the remote computing system). The remote computing system of the fourth aspect is further configured to transmit results of the computational task (e.g., based on controlled quantum computing operations performed thereon) to the computer-implemented system (e.g., to the computer-implemented system from which the query was sent). In some examples, the hardware architecture of the quantum computing system of the fourth aspect can include one or more building blocks (or blocks of elements) of the hardware architecture of the quantum computing system from the second and / or third aspects disclosed above. In some cases, the hardware architecture of the quantum computing system of the fourth aspect can be the same as the hardware architecture of the quantum computing system from the second and / or third aspects disclosed above. [Explanation of symbols]

[0124] 1a-1d, 2a-2d, 3a-3c, 4a-4b, 15a-15c, 16, 16a-16e, 17 qubits q1~q21 qubits qh1~qh9 qubits qR1~qR6 qubits qb 1,1 , qb 1,2 , qb 2,1 , qb 2,2 , qb 3,1 , qb 3,2 , qb 4,1 , qb 4,2 , qb5,1 , qb 5,2 , qb 6,1 , qb 6,2 , qb 7,1 , qb 7,2 , qb 8,1 , qb 8,2 , qb 9,1 , qb 10,1 , qb 12,1 , qb 12,2 , qb i,1 , qb i,2 Qubit 15a15b~15c Cubit Chain 15c16, 16a~16e Cubit Ladder 17 Quantum Switch Register 20a rectangle 40, 41 Base rotation transformation 42, 43, 45, 46 CNOT operations 44 Rotation Transform 47, 48 Inverse basis rotation transformation 100 selections 1000 Quantum Computing Systems

Claims

1. 1. A method for configuring a quantum computing system (1000), the quantum computing system including a plurality of qubits arranged on a two-dimensional (2D) lattice and configured to perform a plurality of quantum computation operations, the method comprising: receiving a selection (100) of a first plurality of qubits (15a-15c; 1a-1d; 3a-3c) of said plurality of qubits, said first plurality of qubits comprising several chains of qubits; each qubit (1a-1d) of the several chains (15a-15b) of the first plurality of qubits represents a first degree of freedom associated with a respective component of a physical system mapped onto the several chains of the first plurality of qubits; each qubit (q1) of the first plurality of qubits is configured to transmit quantum information of the qubit to another qubit (q2) of the first plurality of qubits that is adjacent to the qubit; the other qubit of the first plurality of qubits is configured to receive the quantum information of the qubit of the first plurality of qubits. Steps and receiving a selection (200) of a second plurality of qubits (16; 16a-16e; 2a-2d) of said plurality of qubits, said second plurality of qubits including some ladder of qubits; each ladder of qubits representing a second degree of freedom associated with a respective component of the physical system mapped onto the ones of the ladders of the second plurality of qubits; the second degree of freedom is different from the first degree of freedom; each qubit (2a; 2c) of the second plurality of qubits is configured to transmit the quantum information of the qubit to another qubit (2b; 2d) of the second plurality of qubits that is adjacent to the qubit; Steps and Including, the other qubit of the second plurality of qubits is configured to receive the quantum information of the qubit of the second plurality of qubits; one or more qubits (1 a; 1 c) of the several chains of the first plurality of qubits are adjacent to one or more qubits (2 a; 2 c) of each of the several ladders of the second plurality of qubits and are configured to transmit the quantum information to and / or receive the quantum information from the respective one or more qubits of the several ladders of the second plurality of qubits; the number of chains of the first plurality of qubits and the number of ladders of the second plurality of qubits are configured to perform a number of quantum computing operations. method.

2. After performing the step of receiving the selection of claim 1, the method includes performing (300) a plurality of quantum computing operations on the quantum computing system, the plurality of quantum computing operations comprising: performing (400) a plurality of quantum computing operations on the several chains (15a-15b) of the first plurality of qubits, performing the plurality of quantum computing operations on the several chains of the first plurality of qubits includes exchanging the quantum information between two qubits (q1, q2) from one or more pairs of adjacent qubits (q1, q2; q2, q3) of the several chains of the first plurality of qubits (15a-15b); Step (400); performing (500) a plurality of quantum computing operations on said several ladders (16; 16a-16e) of said second plurality of qubits, The step of performing the plurality of quantum computing operations on the number of ladders of the second plurality of qubits includes performing quantum computing operations on two qubits (qb ２，１ , qb ２，２ ) exchanging said quantum information between Step (500); a step (600) of exchanging the quantum information between two qubits from one or more pairs of adjacent qubits (1a, 2a; 1c, 2c), one qubit (1a; 1c) of the pair being from the number of chains of the first plurality of qubits and another qubit (2a; 2c) of the pair being from a respective ladder of the number of ladders of the second plurality of qubits adjacent to the qubit from the number of chains; The method of claim 1 , comprising:

3. The step of performing the plurality of quantum computing operations on the number of chains of qubits (400) and the step of performing the plurality of quantum computing operations on the number of ladders (500) include: performing (410) some of the quantum computing operations on one or more qubits (1 a; 1 c) of the some chains of the first plurality of qubits adjacent to the respective one or more qubits (2 a; 2 c) of the some ladders of the second plurality of qubits; performing (510) some quantum computing operations of the plurality of quantum computing operations on the respective one or more qubits (2a; 2c) of the some ladders of the second plurality of qubits; performing (520) some quantum computing operations of the plurality of quantum computing operations on some qubits (2b; 2d) of some ladders of the second plurality of qubits for which no adjacent qubits from some of the chains of the first plurality of qubits are available; The method of claim 2 further comprising:

4. The step of performing (400) the plurality of quantum computing operations on the number of chains of qubits comprises: performing (420) some quantum computing operations of the plurality of quantum computing operations on some qubits of some chains of the first plurality of qubits for which no adjacent qubits from some ladders of the second plurality of qubits are available; The method of claim 3 further comprising:

5. a qubit ladder (16) of the number of ladders of the second plurality of qubits includes a plurality of qubits; the plurality of qubits in the ladder extend in a first direction; each chain of the several chains of the first plurality of qubits extends in the first direction (15a) or in a second direction (15b) different from the first direction; 5. The method according to any one of claims 1 to 4.

6. The method further includes receiving (250) a selection of a third plurality of qubits (17; 4a-4b) of the plurality of qubits; some qubits of the third plurality of qubits (qR2-qR5) are adjacent to two or more ladders (16e; 16a) of the number of ladders of the second plurality of qubits and are configured to receive the quantum information from and / or transmit the quantum information to the two or more ladders of the number of ladders of the second plurality of qubits; the two or more ladders (16e; 16a) from the number of ladders of the second plurality of qubits adjacent to the number of qubits (qR2-qR5) of the third plurality of qubits are configured to transmit the quantum information to the number of qubits (qR2-qR5) of the third plurality of qubits and / or receive the quantum information from the number of qubits (qR2-qR5) of the third plurality of qubits; Optionally, the third plurality of qubits comprises one or more chains of qubits extending in the first direction.

6. The method according to any one of claims 1 to 5.

7. the quantum information of each qubit of the several chains of the first plurality of qubits carried by the qubit comprises at least partial information with respect to one or more first degrees of freedom, or with respect to the one or more first degrees of freedom and one or more second degrees of freedom; the partial information carried by the qubits of the some chains of the first plurality of qubits corresponds to a quantum state of that qubit; the quantum information of each qubit of the ladder from the number of ladders of the second plurality of qubits carried by the qubit includes at least partial information with respect to one or more second degrees of freedom, or with respect to the one or more second degrees of freedom and the one or more first degrees of freedom; the partial information of each qubit of the ladder from the number of ladders corresponds to a quantum state of the qubit of the ladder.

7. The method according to any one of claims 1 to 6.

8. The step of performing the plurality of quantum computing operations (300) comprises: initially ordering the first plurality of degrees of freedom onto the several chains of the first plurality of qubits; each of the plurality of first degrees of freedom is mapped onto a respective qubit from the several chains of the first plurality of qubits; Steps and initially mapping one or more of the second degrees of freedom onto respective one or more ladders from the number of ladders of the second plurality of qubits; 8. The method of claim 1, comprising:

9. The method further includes initializing one or more qubits from the some of the chains of the first plurality of qubits and one or more qubits from the some of the ladders of the second plurality of qubits to an initial quantum state; the one or more qubits from the some chains of the first plurality of qubits and the one or more qubits from the some ladders of the second plurality of qubits are configured to be initialized to the initial quantum state; the initial quantum state of the qubit represents a quantum state associated with the first degree of freedom and the second degree of freedom; Optionally, the quantum state is a product quantum state of a first quantum state associated with the first degree of freedom and a second quantum state associated with the second degree of freedom; the first quantum state is represented by the one or more qubits from the several chains of the first plurality of qubits, and the second quantum state is represented by the one or more qubits from the several ladders of the second plurality of qubits.

9. The method according to any one of claims 1 to 8.

10. the first degree of freedom is a fermionic degree of freedom and the second degree of freedom is a bosonic mode; The step of performing the plurality of quantum computing operations (300) comprises: exchanging (700) the quantum information between each qubit from some odd-numbered qubits (q1; q3) of the some chains of the first plurality of qubits and a respective adjacent even-numbered qubit (q2; q4) located on a predetermined side relative to the odd-numbered qubit, if there is an even-numbered qubit located on the predetermined side relative to the odd-numbered qubit, the respective adjacent even-numbered qubits are qubits from some even-numbered qubits of the some chains of the first plurality of qubits. Step (700); performing (710) some available quantum computing operations from the plurality of quantum computing operations on one or more qubits (1a-1d) from the some chains of the first plurality of qubits; Two adjacent qubits (q1; q2) of the several chains of the first plurality of qubits and a qubit (qb ２，１ and executing (720) some available quantum computing operations from the plurality of quantum computing operations, each of the quantum computing operations comprising a respective qubit ladder (16) having: the respective qubit ladder is a qubit ladder from the number of ladders of the second plurality of qubits; a corresponding boson mode is encoded by the respective qubit ladder; Step (720); exchanging (730) the quantum information between each qubit from the several even-numbered qubits (q2) of the several chains of the first plurality of qubits and a respective adjacent odd-numbered qubit (q3) located on the predetermined side relative to the even-numbered qubit, if there is an odd-numbered qubit located on the predetermined side relative to the even-numbered qubit; the respective adjacent odd-numbered qubits are qubits from some odd-numbered qubits of the some chains of the first plurality of qubits. Step (730); performing (740) some available quantum computing operations from the plurality of quantum computing operations on the one or more qubits (1a-1d) from the some chains of the first plurality of qubits; The two adjacent qubits (q1, q2) of the several chains of the first plurality of qubits and the qubit (qb) adjacent to the one (q2) of the adjacent qubits (q1, q2) of the several chains of the first plurality of qubits. ２，１ providing (750) a number of available quantum computing operations from the plurality of quantum computing operations, the number of available quantum computing operations including the respective qubit ladders (16) having a Iteratively repeating the steps of exchanging information and executing the available quantum computing operations (760) until all available quantum computing operations from the plurality of quantum computing operations have been executed; performing (770) some of the quantum computing operations on each of the qubit ladders (16); 10. The method of claim 1, further comprising:

11. The two adjacent qubits (q1, q2) of the several chains of the first plurality of qubits and the qubit (qb) adjacent to the one (q2) of the adjacent qubits (q1, q2) of the several chains of the first plurality of qubits. ２，１ and executing (720; 750) the number of available quantum computing operations from the plurality of quantum computing operations including: using a number of quantum two-qubit gates and / or single-qubit gates arranged in a corresponding order to act on the two adjacent qubits (q1, q2) of the several chains of the first plurality of qubits; i) the qubits (qb ２，１ ii) the one (q2) of the adjacent qubits (q1, q2) of the several chains of the first plurality of qubits that is adjacent to the one (q2) of the adjacent qubits of the several chains of the first plurality of qubits; and ii) the qubit (qb ２，１ ) and iii) one or more qubits (qb ２，２ ) and performing a number of quantum computing operations, including: The method of claim 10, comprising:

12. 12. The method of claim 1, further comprising performing a quantum computing task, the quantum computing task comprising the plurality of quantum computing operations performed on the number of chains of the first plurality of qubits and the number of ladders of the second plurality of qubits.

13. 13. A quantum computing system (1000) configured according to the method steps of any one of claims 1 to 12.

14. A quantum computing system (1000) configured to perform said plurality of quantum computing operations and adapted to perform the method steps of any one of claims 2 to 12.

15. A remote computing system including a quantum computing system (1000), the remote computing system comprising: Executing a quantum computing task, the quantum computing task comprising a plurality of quantum computing operations according to the method of claim 12; The plurality of quantum computing operations are performed according to the method steps of any one of claims 2 to 12. To carry out transmitting results of said computational tasks to a computer-implemented system; a remote computing system adapted to perform the steps of:

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