Topological qubits in quantum spin liquids

Abstract

In various embodiments, a device is provided that includes a two-dimensional array of particles, each of which comprises: [Equation 1] TIFF2023552091000460.tif8150; each particle has a first state and an excited state; each particle belonging to at least three unit cells of the ruby ​​lattice has a blockade radius sufficient to block, when in an excited state, each of at least six nearest neighboring particles in the ruby ​​lattice from transitioning from its first state to its excited state, and the array has at least one exterior edge configured to be in a first boundary condition.

Description

Technical Field

[0001] Cross - reference to related applications This application claims the benefit of U.S. Provisional Application No. 63 / 116,321, filed Nov. 20, 2020, and U.S. Provisional Application No. 63 / 166,165, filed Mar. 25, 2021, each of which is incorporated herein by reference in its entirety.

[0002] Statement regarding federally - sponsored research or development This invention was made with government support under Grants No. 1734011 and 2012023 awarded by the National Science Foundation; W911NF2010082 awarded by the U.S. Army Research Laboratory; and DE - SC0021013 awarded by the U.S. Department of Energy. The government has certain rights in this invention.

Background Art

[0003] Background The key to fault - tolerant quantum computing is quantum codes that protect quantum information from decoherence and errors due to the environment. By far the most studied error - correcting code is the so - called surface code. However, the actual implementation of the surface code lags behind the theory.

[0004] Aspects of the present disclosure relate to the generation of quantum spin liquids and the execution of qubits and qubit operations therein.

Summary of the Invention

[0005] Brief Summary In a first exemplary aspect, the invention is a device. In a first aspect, the device includes a two - dimensional array of particles, and each particle

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[0006] In a second exemplary embodiment, the present invention is a system. The system includes a confinement system for aligning particles in a two-dimensional array and an excitation source for exciting at least some of the particles from a first state to an excited state. The confinement system includes a laser source aligned to generate a plurality of confinement regions; the atomic cloud source may be arranged so as to at least partially overlap the plurality of confinement regions. In the first aspect of the second exemplary embodiment, in the two-dimensional array, each particle is located at the vertices of a ruby ​​lattice; each particle has a first state and an excited state; each particle belonging to at least three unit cells of the ruby ​​lattice has a blockade radius sufficient to block each of at least six nearest neighboring particles in the ruby ​​lattice from its transition from its first state to its excited state when it is in an excited state, and the array has at least one outer edge configured to be in a first boundary condition.

[0007] In a third exemplary embodiment, the present invention is

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[0008] In a fourth exemplary embodiment, the present invention is

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[0009] In a fifth exemplary embodiment, the present invention is

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[0010] In a sixth exemplary embodiment, the present invention is

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[0011] In a seventh exemplary embodiment, the present invention is

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[0012] In an eighth exemplary embodiment, the present invention relates to a method for manipulating a topological qubit. The method includes the step of preparing a topological qubit according to a method defined in any of the fourth exemplary embodiment or aspects thereof. In the first aspect of the eighth exemplary embodiment, the first boundary condition is an e-boundary condition, and the method includes: generating first and second e-anyons in an array; removing the first e-anyon from the array via a first outer edge; and removing the second e-anyon from the array via a third outer edge.

[0013] In a ninth exemplary embodiment, the present invention relates to a method for manipulating a topological qubit. The method includes the step of preparing a topological qubit according to a method defined in any of the fifth exemplary embodiment or aspects thereof. In the first aspect of the ninth exemplary embodiment, the method further includes the steps of generating first and second e-anyons in an array; pinning the first e-anyon; and defining the boundary of at least one internal edge and moving the second e-anyon along a circular path having an endpoint at the location of the first e-anyon, thereby destroying the first and second e-anyons.

[0014] In a tenth exemplary embodiment, the present invention is carried out according to a method defined by the fourth exemplary embodiment or any aspect thereof.

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[0015] In an eleventh exemplary embodiment, the present invention is a method for manipulating first and second topological qubits, comprising the step of encoding the first and second topological qubits according to a method defined in a tenth exemplary embodiment. In the first aspect of the eleventh exemplary embodiment, the method further comprises the step of moving the first internal edge along a closed continuous path that defines the boundary of the second internal edge.

[0016] In a twelfth exemplary embodiment, the present invention is a computer program comprising a computer-readable storage medium having program instructions thereby implemented, the program instructions being executable by the processor in a manner defined by any of the fourth to eleventh exemplary embodiments or aspects thereof. [Brief explanation of the drawing]

[0017] A brief explanation of some of the figures in the drawing. The foregoing is evident from the following more specific description of exemplary embodiments of the invention, as illustrated in the attached drawings, where similar reference numerals refer to the same parts through different figures. The drawings are not necessarily drawn at a fixed rate, and instead emphasis is placed on illustrating embodiments of the invention. [Figure 1] Figure 1 is a schematic flowchart illustrating a process for preparing a quantum spin liquid and measuring the state of a topological qubit according to an aspect of this disclosure. [Figure 2] Figure 2 is a schematic diagram of an apparatus for parallel classification of trapped particles according to an aspect of the present disclosure. [Figure 3] Figures 3A-3C illustrate exemplary ruby ​​lattices according to embodiments of this disclosure. [Figure 4] Figure 4 shows an ideal unit cell of a ruby ​​lattice according to an embodiment of the present disclosure. [Figure 5] Figures 5A and 5B show an approximate Kagome grid according to an embodiment of the present disclosure. [Figure 6]Figure 6 illustrates an exemplary state preparation protocol for a quantum spin liquid according to an aspect of this disclosure. [Figure 7] Figure 7 illustrates an exemplary m-type string according to an embodiment of the present disclosure. [Figure 8] Figure 8 illustrates an exemplary e-type string according to an embodiment of the present disclosure. [Figure 9] Figure 9 illustrates an invalid e-type string according to an aspect of this disclosure. [Figure 10] Figures 10A-C illustrate exemplary closed m-type strings and corresponding open strings according to embodiments of the present disclosure. [Figure 11] Figures 11A-C illustrate exemplary closed m-type strings and corresponding open strings according to embodiments of the present disclosure. [Figure 12] Figure 12 shows an example of atoms at a lattice boundary according to an aspect of the present disclosure. <n>This is the plot. [Figure 13] Figure 13 shows a plot of the boundary against the boundary P and Q string correlation according to an aspect of the present disclosure. [Figure 14] Figure 14 is a schematic diagram of an exemplary grid illustrating an e-anyon according to an aspect of the present disclosure. [Figure 15] Figures 15A-E show exemplary dimer covers of a grid according to embodiments of this disclosure. [Figure 16] Figure 16 is a schematic diagram of an exemplary grid having one external edge and three internal edges according to an aspect of the present disclosure. [Figure 17] Figure 17 is a schematic diagram of an exemplary grid having four external edges according to an aspect of the present disclosure. [Figure 18] Figure 18 illustrates an exemplary conditioning and quenching protocol according to an aspect of this disclosure. [Figure 19] Figure 19 is a schematic diagram of the measurement of a topological qubit corresponding to an internal edge according to an embodiment of the present disclosure. [Figure 20] Figure 20 is a schematic diagram of a grid illustrating the application of rotation to a topological qubit according to an embodiment of the present disclosure. [Figure 21] Figures 21A and 21B are schematic diagrams of a grid illustrating the application of a CNOT gate to a pair of topological qubits according to an embodiment of the present disclosure. [Figure 22] Figure 22 shows a computer computing node according to an aspect of this disclosure. [Figure 23] Figures 23A-C illustrate the relationship between the Rydeberg blockade model and the dimer model according to aspects of this disclosure. [Figure 24] Figure 24 is a phase diagram of the Rydebell blockade model for the connection of kagome grids according to an aspect of this disclosure. [Figure 25] Figures 25A and 25B illustrate the detection of a phase transition via the filling fraction according to an embodiment of this disclosure. [Figure 26] Figures 26A-C illustrate the topological entanglement entropy according to the embodiments of this disclosure. [Figure 27] Figures 27A-27C illustrate topological string operators according to embodiments of this disclosure. [Figure 28] Figure 28 illustrates a phase diagnosis for a topological string operator according to an aspect of this disclosure. [Figure 29] Figures 29A and 29B illustrate the ground state and module transformation according to the embodiments of this disclosure. [Figure 30] Figure 30 illustrates the topological ground state degeneracy according to the embodiments of this disclosure. [Figure 31] Figures 31A-31C illustrate ruby ​​lattices according to embodiments of this disclosure. [Figure 32] Figure 32 illustrates a spin liquid on a ruby ​​lattice according to an embodiment of the present disclosure. [Figure 33] Figure 33 illustrates a module transformation on a ruby ​​lattice according to an embodiment of the present disclosure. [Figure 34] Figures 34A and 34B illustrate the measurement of off-diagonal string operators using a quench protocol according to an aspect of this disclosure. [Figure 35] Figure 35 illustrates the trapping potential for an e-anyon according to an embodiment of this disclosure. [Figure 36] Figures 36A to 36D are boundary phase diagrams of a blockade model according to an embodiment of this disclosure. [Figure 37] Figures 37A and 37B illustrate topological degeneracy in plane geometry according to the embodiments of this disclosure. [Figure 38] Figure 38 illustrates the reading of a topological ground state according to an embodiment of this disclosure. [Figure 39] Figure 39 is a connectivity graph for van der Waals interactions on a ruby ​​lattice according to an embodiment of the present disclosure. [Figure 40] Figures 40A and 40B illustrate the FM string order parameter in a blockade model according to an embodiment of this disclosure. [Figure 41] Figures 41A and 41B illustrate the FM string order parameters according to the embodiments of this disclosure. [Figure 42] Figures 42A-E illustrate dimer models in a Rydberg atomic array according to an embodiment of the present disclosure. [Figure 43] Figures 43A-43C illustrate dimer phase detection via a diagonal string operator according to an embodiment of the present disclosure. [Figure 44] Figures 44A-44F illustrate the exploration of coherence between dimer states via off-diagonal string operators according to aspects of this disclosure. [Figure 45] Figures 45A-I illustrate string order parameters and quasiparticle excitations according to embodiments of this disclosure. [Figure 46] Figures 46A-46C illustrate the topological properties of an array having holes according to an embodiment of this disclosure. [Figure 47] Figure 47 illustrates the preparation of a quasi-adiabatic state according to an embodiment of this disclosure. [Figure 48] Figure 48 illustrates a double Z- and X-loop according to an embodiment of the present disclosure. [Figure 49] Figures 49A to 49D illustrate the average Rydberg density for each region according to the embodiments of this disclosure. [Figure 50] Figures 50A-C illustrate snapshots of the dimer phase according to the embodiments of this disclosure. [Figure 51] Figure 51 illustrates the density correlation between individual Rydberg excitations according to an embodiment of this disclosure. [Figure 52] Figures 52A and 52B illustrate the phase dependence of the quench according to the embodiments of this disclosure. [Figure 53] Figure 53 illustrates the improved state-adjusted Z-loop parity according to the embodiments of this disclosure. [Figure 54] Figures 54A and 54B illustrate the correlation between parity loops according to the embodiments of this disclosure. [Figure 55] Figure 55 illustrates a magnetic anyon according to an embodiment of the present disclosure. [Figure 56] Figures 56A-D illustrate the scaling of Z and X parity with loop sizes according to embodiments of this disclosure. [Figure 57] Figure 57 illustrates the distinctions between topological sectors according to the embodiments of this disclosure. [Figure 58] Figures 58A to 58F illustrate boundary-to-boundary string operators according to embodiments of this disclosure. [Figure 59] Figures 59A-F are ground state phase diagrams of the linked Kagome model for two truncation distances according to the embodiments of this disclosure. [Figure 60] Figure 60 is a ground state phase diagram of the linked Kagome model according to an embodiment of this disclosure. [Figure 61] Figures 61A and 61B illustrate the dynamic state preparation in the PXP model according to the embodiments of this disclosure. [Figure 62] Figures 62A and 62B illustrate the dynamic state adjustment in a van der Waals model according to an embodiment of this disclosure. [Figure 63] Figures 63A-63F show a comparison between experimental results and numerical simulations of dynamic state preparation according to the embodiments of this disclosure. [Modes for carrying out the invention]

[0018] Detailed explanation Referring to Figure 1, a flowchart is provided outlining the process for preparing a quantum spin liquid and measuring the state of a topological qubit according to an aspect of this disclosure. Each step of this process is described in more detail in the following sections.

[0019] In step 101, Rb-87 atoms are provided in a magneto-optical trap. In step 102, these atoms are loaded into a two-dimensional optical tweezers array, which can be generated, for example, using a spatial light modulator (SLM). In step 103, the atoms are rearranged into a desired lattice configuration, for example, using a two-dimensional acoustic-optical deflector (AOD). In step 104, a quasi-adiabatic preparation of the spin liquid state is performed.

[0020] In step 111, fluorescence imaging is used to read out all atoms in the ground-Rydberg basis in order to measure the state of the qubits in the Z-basis. In step 112, the parity of Rydberg excitations (dimers) on the Z-string is measured.

[0021] In step 121, a quenched time evolution is performed to measure the state of the X-referenced qubit and achieve the reference rotation. In step 122, fluorescence imaging is used to read out all atoms in the ground Rydberg reference. In step 123, a parity measurement of Rydberg excitations (dimers) on the double Z string is performed.

[0022] An exemplary device for preparing quantum spin liquids and measuring the states of topological qubits includes a two-dimensional array of optical tweezers configured to provide confinement for atoms. Rearranging atoms to form a desired defect-free array with arbitrary geometric structure can be provided using the two-dimensional AOD described below. A laser is provided to excite the atoms from their electronic ground state to Rydberg states (highly excited electronic states), and the atoms interact with each other via strong van der Waals interactions. Readout of the atomic states is provided via fluorescence imaging. This allows detection of atoms in the ground state, while atoms in the Rydberg state are detected as losses (due to the anti-trapping effect of the optical tweezers).

[0023] Formation of particle arrays using optical tweezers Optical trapping of neutral atoms is a powerful technique for isolating atoms in a vacuum. Atoms are polarized, and the oscillating electric field of a light beam induces an oscillating electron dipole moment within the atom. The relevant energy shift within the atom from the induced dipole, averaged over the optical oscillation period, is called the AC Stark shift. Based on the AC Stark shift induced by light detuned (i.e., offset in wavelength) from the atomic resonance transition, atoms are attracted to light below the resonance frequency and are therefore trapped at a local maximum intensity (for detuned red, i.e., longer wavelength trapping light). The AC Stark shift is proportional to the intensity of the light. Thus, the shape of the intensity field is the shape of the relevant atomic trap. Optical tweezers utilize this principle by focusing a laser to a constriction on the micron scale, where individual atoms are trapped at the focal point. Two-dimensional (2D) arrays of optical tweezers can be generated, for example, by irradiating a spatial light modulator (SLM) that imprints computer-generated holograms onto the wavefront of the laser field. A 2D array of optical tweezers overlaps with a cloud of laser-cooled atoms within a magneto-optical trap (MOT). The precisely focused optical tweezers operate in a "collision blockade" configuration where a single atom is loaded from the MOT, and pairs of atoms are ejected for optically assisted collisions, ensuring that at most one atom is loaded onto the tweezers, although the loading is expected, so there is approximately a 50-60% probability that a single atom will be loaded into the trap.

[0024] To prepare a deterministic atomic array, a real-time feedback procedure identifies randomly loaded atoms and rearranges them into a pre-programmed geometric structure. Atomic rearrangement requires moving atoms in tweezers, which can be smoothly carried out to minimize heating by deflecting the laser beam by an adjustable angle controlled by the frequency of the acoustic waveform applied to the AOD crystal, for example, using an acoustic-optical deflector (AOD). Dynamic tuning of the acoustic frequency translates to smooth movement of the optical tweezers. Multi-frequency acoustic waves generate an array of laser deflections, which, after being focused through a microscope objective lens, form an array of optical tweezers with adjustable position and amplitude, both controlled by the acoustic waveform. Atoms are rearranged using a further set of dynamically moving tweezers placed over the top of the SLM tweezers array.

[0025] Exemplary hardware Optical tweezers arrays constitute a powerful and flexible method for constructing large-scale systems composed of individual particles. Each optical tweezers traps a single particle, including individual neutral atoms and molecules, for applications in quantum technology, but not limited to. Loading individual particles into such tweezers arrays is a statistical process, where each tweezers in the system is filled with a single particle with a finite probability p<1, e.g., p about 0.5, in the case of many neutral atom tweezers. To compensate for this random loading, real-time feedback can be obtained by measuring which tweezers are loaded and then classifying the loaded particles into programmable geometric structures. This can be done by moving one particle at a time or in parallel.

[0026] Parallel classification can be achieved by using two acoustic-optical deflectors (AODs) to generate multiple tweezers capable of picking up particles from an existing particle trap structure, moving them simultaneously, and releasing them elsewhere. This may involve moving particles around within a single trap structure (e.g., a tweezers array) or transporting and classifying particles from one trap system to another (e.g., between one tweezers array and another type of optical / magnetic trap). This classification is flexible and allows for the programmed placement of each particle. Each movable trap is formed by an AOD, and its position is dynamically controlled by the frequency components of the radio frequency (RF) driven field for the AOD. Since the RF drive of the AOD can be controlled in real time and may include any combination of frequency components, it is possible to generate any grid of traps (e.g., lines of arbitrarily placed traps) by changing the number, scale, and distribution of frequency components in the RF driven field of the AOD, move rows or columns of the grid, and add or remove rows and columns of the grid.

[0027] In an exemplary embodiment, an optical tweezers array is generated using liquid crystals on a silicon space light modulator (SLM) that can programmatically generate a flexible arrangement of tweezers. These tweezers are fixed in space for a given experimental sequence and loaded with individual atoms statistically, so that each tweezers is loaded with a probability of p approximately 0.5. Fluorescence images of the loaded atoms are captured to identify in real time which tweezers are loaded and which are empty.

[0028] After detecting which tweezers are loaded, the movable tweezers, which overlap with the optical tweezers array, can dynamically rearrange atoms from their starting positions to fill the target arrangement of the trap with a nearly uniform packing. The movable tweezers are generated using pairs of intersecting AODs. These AODs can be used to move one atom at a time to fill the target arrangement or to generate a single movable trap that moves many atoms in parallel.

[0029] Referring to Figure 2, a schematic diagram of an apparatus 200 for parallel classification of trapped particles according to an aspect of the present disclosure is provided. As shown in Figure 2, using a beam generated by a light source 202 (e.g., a coherent light source; in some exemplary embodiments - a monochromatic light source), an SLM 204 forms an array of trap beams (i.e., a tweezers array), which is imaged onto a trap surface 208 within a vacuum chamber 210 by a train of optical components including elements 206a, 206c, 206d and a high numerical aperture (NA) objective lens 206e, in the exemplary embodiment shown in Figure 2. Other suitable trains of optical components may be used as readily understood by those skilled in the art. Using a beam generated by a light source 212 (e.g., a coherent light source; in some exemplary embodiments - a monochromatic light source), a pair of AODs 214 and 216 having non-parallel directions (e.g., orthogonal directions) of sound wave transmission generate dynamically movable classification beams. Using a series of optical components such as those shown in Figure 2 (elements 217, 206b, 206c, 206d, and 206e), the classification beam overlaps with the trap beam. It is understood that the same result can be achieved using other series of optical components. For example, sources 202 and 212 could be a single source, and the trap beam and classification beam could be generated by a beam splitter.

[0030] The dynamic movement of the steering beam is achieved using two non-parallel AODs 214, 216 arranged in series. In the exemplary embodiment shown in Figure 2, one AOD defines the direction of the “row” (“horizontal” - 'X' AOD) and the other AOD defines the direction of the “column” (“vertical” - 'Y' AOD). Each AOD is driven by an arbitrary RF waveform originating from an arbitrary waveform generator 220, which is generated in real time by a computer 222 that processes a feedback routine after analyzing an image of the positions where atoms are loaded. When each AOD is driven by a single frequency component, a single steering beam ("AOD trap") is generated in the same plane 208 as the SLM trap array. The frequency of the X AOD drive determines the horizontal position of the AOD trap, and the frequency of the Y AOD drive determines the vertical position; in this way, a single AOD trap can be advanced to overlap with any SLM trap.

[0031] In Figure 2, laser 202 irradiates the SLM 204 with a beam of light. The SLM 204 can be controlled by computer 222 to generate a beam pattern ("trap beam" or "tweezers array"). The beam pattern is focused by lens 206a, passes through mirror 206b, and is parallelized by lens 206c on mirror 206d. The reflected light passes through objective lens 206e to focus the optical tweezers array in the vacuum chamber 210 on the trap surface 208. The laser light from the optical tweezers array continues to pass through objective lens 224a, through dichroism mirror 224b, and is detected by a charge-coupled device (CCD) camera 224c.

[0032] The vacuum chamber 210 may be illuminated by an additional light source (not shown). Fluorescence from atoms trapped on the trap surface also passes through the objective lens 224a but is reflected by the dichroic mirror 224b to the electron-amplified CCD (EMCCD) camera 224d.

[0033] In this example, laser 212 directs a beam of light to AODs 214, 216. AODs 214, 216 are driven by an arbitrary wave generator (AWG) 220, which is then controlled by a computer 222. The intersecting AODs 214, 216 emit one or more beams as described above, which are directed to the focusing lens 217. The beams then enter the same set of optical components 206b...206e as described above with respect to the optical tweezers array and are focused onto the trapping surface 208.

[0034] It is understood that an optical tweezers array suitable for the use described herein can be fabricated using an alternative set of optical components.

[0035] Excitation of atoms in an optical tweezers array to the Rydberg state With separate optical tweezers on a micrometer scale, the atoms in their ground electronic states have negligible van der Waals interactions. Fortunately, neutral atoms provide a remarkable way to switch on the strong interaction through coherent excitations of the atom to the Rydberg state.

[0036] The properties of atomic states are dramatically proportional (scale) to the principal quantum number. The Rydberg state is a highly excited electronic state of an atom, where one of the atom's electrons has a high principal quantum number n in the range of 30-100. In a classical photograph of an atom, this situation corresponds to a single (negatively charged) electron orbiting far away from a (positively charged) ion core on an atomic length scale, thus forming an oscillating electrical dipole. Two atoms excited to the same Rydberg state can exhibit very strong dipole interactions over distances of tens of microns. Interaction energy

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[0037] Consider an ideal two-level atom having a ground state |g> and a Rydberg state |r>. These two states are laser-coupled with a coupling intensity set by an angular Ribi frequency Ω, also called a Rabiflop, which is the inverse function of the duration of the Rabi cycle. This is the periodic absorption and stimulated emission of quanta of energy by the two-level atom in the presence of an oscillatory driving field. The Rabi frequency is proportional to the coupling intensity between light and atomic transitions and the amplitude of the electric field of light. For two such atoms, also referred to herein as Rydberg atoms, the van der Waals interaction energy

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[0038] Several methods for optical excitation from an atomic ground state to a target Rydberg state are available. Direct laser excitation using a single-photon transition is the simplest. The wavelength for such a transition in a Rydberg atom is typically in the ultraviolet range. For example, 87 The single-photon wavelength for Rb is 297 nm. Ultraviolet lasers present significant experimental difficulties, for example, due to material decomposition and the lack of availability of optical fibers and low-loss optical instruments. Alternatively, two-photon laser excitation can be used to couple the atomic ground state to a target Rydberg state via an intermediate electronic excitation state by irradiating an atom from opposite sides with two counterpropagating laser beams.

[0039] In line with the above description, the term “blockade” is used herein to refer to the phenomenon in which a laser-stimulated transition of an atom in an interacting pair of atoms from a first state (e.g., ground state) to an excited state cannot be achieved (blocked) due to a mismatch between the laser frequency and the shifted energy level of the excited state, where the energy level shift is electrically or magnetically induced. For example, a blockade can be achieved by a dipole-dipole interaction between two adjacent atoms, where one atom is excited to a Rydberg state.

[0040] Detuning from resonances with excited states The coherent evolution of two atoms under laser excitation from the ground state |g> to the Rydberg state |r> is given by the Hamiltonian

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[0041] Furthermore, in a two-photon laser excitation scheme, two excitation lasers are typically used, one having a frequency in the blue region of the optical spectrum, e.g., 420 nm, and the other having a frequency in the red or infrared region, e.g., 1013 nm, to create an intermediate state.

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[0042] Array Geometric Structures Figures 3A–C illustrate exemplary ruby ​​lattices according to the present disclosure. Referring to Figure 3A, the kagome lattice 302 is shown by gray lines overlapping the ruby ​​lattice 301. When a particle 303 is located at a vertex of the ruby ​​lattice 301, it is understood that the particle is located at the corresponding edge of the kagome lattice 302.

[0043] A ruby ​​lattice has a free parameter ρ that corresponds to the aspect ratio of the quadrilateral portion of the lattice. Figure 3A shows that

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[0044] In addition to the lattice, Figure 3C shows seven interaction distances from particle 304. As shown, each continuity radius includes further particles in the lattice as follows: radius 311-2 particles; radius 312-4 particles; radius 313-6 particles; radius 314-10 particles; radius 315-11 particles; radius 316-13 particles; radius 317-17 particles. As will be fully explained below, the selection of the number of particles within a given interaction radius provides the necessary conditions for realizing a quantum spin liquid (QSL).

[0045] Figure 4 shows an ideal unit cell of a ruby ​​lattice. Any unit cell of the lattice can be deformed to such an extent that the deformation does not exclude the quantum spin liquid properties of the lattice as described herein. For example, the left and right triangles A and A' may deviate from being equilateral, and the quadrilateral B may deviate from being rectangular.

[0046] As used herein, the term “edge” means a one-dimensional manifold of vertices in a ruby ​​lattice occupied by particles. An edge may be a division having two endpoints (endopoints) or a closed loop. An edge may be an outer edge or an inner edge. An “outer edge” is a subset of lattice vertices occupied by particles that can be approached from an infinite distance without crossing any unit cell having vertices occupied by at least one particle. An “inner edge” is a subset of lattice vertices occupied by particles that define the boundary of a continuous region of the lattice, and vertices of an inner edge also do not belong to an outer edge.

[0047] As further described below, internal edges can define the boundaries of regions containing lattice vertices not occupied by particles. Alternatively, internal edges can define the boundaries of regions containing lattice vertices occupied by particles, but these particles do not interact with most of the particles in the lattice via van der Waals interactions because they differ in properties or states from particles elsewhere in the lattice. For example, a bounded particle can be driven to a ground state.

[0048] With respect to the unit cell in Figure 4, it is understood that various path divisions can be defined for a unit cell or a grid containing multiple unit cells. Firstly, a division can extend between two vertices within a triangular portion of the unit cell. Such a division corresponds to one edge of the triangular portion (A or A') of the unit cell in the figure above. Secondly, a division can extend between two vertices within a quadrilateral portion of the unit cell. Such a division corresponds to either an edge of the quadrilateral portion (B) of the unit cell or a diagonal connection between non-adjacent vertices of the quadrilateral portion. Thirdly, a division can extend between vertices in different unit cells within a ruby ​​grid. For example, if a division extends between two vertices in different unit cells of the ruby ​​grid without intersecting any unit cell of the ruby ​​grid, this refers to a division that extends through a hexagonal gap, as seen when multiple unit cells are assembled within a grid.

[0049] For a quadrilateral portion that is a rectangle with a width of one, its length is equal to the aspect ratio of the quadrilateral portion, and is therefore understood to be equal to ρ, as shown in Figure 4. In the unit cell, a is the length of the leg of the equilateral triangular portion, which generally corresponds to the smallest distance between atoms in the array. Thus, the edges of the quadrilateral portion have lengths a and ρa.

[0050] Realization of quantum spin liquids For a specific choice of ρ for a ruby ​​lattice, three further parameters are selected to fabricate a quantum spin liquid: Ω sets the Rabi frequency term (causing transitions between ground and excited states); δ sets the detuning (can be analogized to a chemical potential that prefers excited states for positive δ and ground states for negative δ); and the blockade radius R. b (This sets a distance within which the likelihood of encountering two excited Rydberg atoms is low due to repulsive dipole interactions.) Ω sets the overall scale of the Hamiltonian (which does not directly affect physics), and therefore Ω can be set to 1 without loss of generality (this can be interpreted as choosing our energy unit such that Ω=1, and can always be done). Thus,

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[0051] A quantum spin liquid is realized on a ruby ​​lattice in the parameter space domain as follows: R b The detuning parameter δ(or

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[0052] This covering can be visualized in relation to the Kagome grid. As mentioned above,

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[0053] The dimer visualization provides a framework for visualizing anions. In particular, if a particular vertex of the kagome lattice (corresponding to an edge within the ruby lattice between particles) has zero or more than one dimer contacting it, this indicates the presence of an e - anion (discussed further below). A vertex without an adjacent dimer can be termed a monomer, and a vertex with two adjacent dimers can be termed a double dimer.

[0054] In addition to selecting the geometric structure and frustration as described above, sufficient quantum fluctuations are required to prevent the state from freezing or settling into a classical pattern. Instead of forming a quantum superposition of many different dimer coverings, such fluctuations are quantified by the Rabi oscillation term. Whether the Rabi oscillations are strong enough to give rise to a QSL can be determined numerically as described in the following examples. In an exemplary case, ρ = 3, δ / Ω = 5.3 and R b = 3.8a gives

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[0055] Therefore, the adjustable parameters can be controlled as follows: The aspect ratio of the rectangle in the ruby ​​lattice (ρ) can be controlled by moving the particles in the optical tweezers array to new positions according to a selected value of ρ. The Rabi oscillation, parameterized by Ω, is controlled by changing the laser intensity. The detuning term δ is controlled by changing the laser frequency (and how much of the laser is non-resonant between the ground state and excited state energy separation).

[0056] Blockade radius R for a given atomic species and selected Rydberg state b Once Omega is known, it becomes known. In particular, R b It is understood in the art that this depends on the sixth root of Ω. b Although it is not an independent parameter, this is because R is more important than Ω. b It is understood that this is convenient for explaining the conditions regarding QSL.

[0057] As mentioned above,

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[0058] however,

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[0059] This relationship is related to the range of ρ and R b This gives the corresponding range. As mentioned above, once the atomic type is fixed, R b The range of is equal to the range of Ω. What needs to be determined is the range of detuning δ. As mentioned above, δ adjusts the likelihood that an atom is in the ground state or an excited state. In particular, when δ → -∞, all atoms are in the ground state, and when δ → +∞, all atoms are excited. Thus, δ can be used to adjust this likelihood. A desired value of δ is chosen such that the probability of finding an excited state atom is about 0.25. As mentioned above, this probability corresponds to a dimer covering where each vertex of the kagome lattice has one adjacent dimer. This probability can be measured experimentally by taking a snapshot of the atomic array at a given δ and determining the ratio of ground state atoms to excited state atoms. In this method, δ can be determined by iteratively measuring the system and increasing or decreasing δ until the desired density is achieved.

[0060] It is understood that the generation of anyons and boundary conditions involves deviations from this type of precise coverage. Therefore, it is understood that the observed state of the QSL may deviate from the ideal 0.25 probability of excitation.

[0061] For any given set of parameters within the above range,

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[0062] Referring to Figure 6, an exemplary state preparation protocol for a quantum spin liquid is provided. As shown, the detuning and amplitude of the optical transition to the Rydberg state are dynamically adjusted to achieve a quasi-adiabatic preparation of the spin liquid state. The quasi-adiabatic preparation is achieved by amplitude (Ω, line 6001)

number

number

number

number

number

[0063] This quasi-adiabatic preparation of the spin liquid phase is

number

[0064] Read the state Various methods can be used to read out the state of an atomic array. Generally, these methods use fluorescence imaging to read out all atoms. From such snapshots, the ground state and excited (Rydberg) states of the atoms can be identified.

[0065] Fluorescence imaging detects atoms with |g>, while atoms with |r> are detected as losses. As mentioned above, Rydberg atoms are not trapped by optical tweezers (anti-trapped), so they are lost as soon as we turn the tweezers on in reverse. In some embodiments, a strong microwave (MW) pulse is applied to ionize the Rydberg atoms and increase the probability of loss.

[0066] String parity measurement To experimentally confirm quantum spin liquids and read out the states of topological qubits, string correlation properties are measured based on snapshots of atomic arrays. In particular, parity is measured for “strings” or “paths” (used interchangeably) that satisfy specific constraints, as described below.

[0067] A string is a collection of atoms that make up a one-dimensional subset of the vertices of a two-dimensional ruby ​​lattice. For any such subset, parity can be measured. The parity of a single vertex in the lattice is defined as +1 if the vertex has atoms in the ground state and -1 if the vertex has atoms in the excited state. The parity of a string or path is the product of the parities of the component vertices.

[0068] Depending on the specific cross-section of the ruby ​​lattice, the strings can be m-shaped or e-shaped.

[0069] Referring to Figure 7, an exemplary m-type string is illustrated. The m-type string is assembled piecewise from piece to piece, each of which either extends along the edges of the triangular portions of the ruby ​​lattice or across the codes of the hexagonal cells of the ruby ​​lattice. In this context, a code refers to a line extending between two vertices of a hexagon.

[0070] In the example in Figure 7, sections 601, 603, 605, 607, and 609 extend along the edges of the triangular parts of the grid. Sections 602, 604, 606, and 608 extend across the codes of the hexagonal cells.

[0071] The parity of an m-type string extending between different edges with m-type boundary conditions corresponds to the logical state of the topological qubit, which is further described below.

[0072] Referring to Figure 8, an exemplary e-type string is illustrated. An e-type string is piecewise assembled from partitions, each extending along either an edge of a triangular portion of the ruby ​​lattice or an edge or diagonal of a rectangular portion of the ruby ​​lattice. Furthermore, an e-type string must not contain more than one edge of any given triangular portion of the lattice, nor more than one edge or diagonal of any given rectangular portion of the lattice (i.e., each triangular or rectangular portion must give the string one or fewer partitions).

[0073] In the example in Figure 8, sections 701, 703, 705, 707, and 709 extend along the edges of the triangular portions of the grid. Sections 702 and 706 extend along the edges of the rectangular portions of the grid. Sections 704 and 708 extend along the diagonals of the rectangular portions of the grid.

[0074] Referring to Figure 9, an invalid e-type string is illustrated. The rest of the string satisfies the criteria for an e-type string, but the rectangular portion 801 violates one of the e-type string criteria because it contains two divisions.

[0075] To measure the parity of a string, the state of each vertex on the string is determined, for example, by fluorescence imaging. Each vertex is assigned a value of +1 for parity or the ground state and -1 for the excited state. These values ​​are multiplied together to produce a parity of +1 or -1 for the string.

[0076] In an experimental context, a single string measured in a single snapshot will always yield a value of +1 or -1. However, if these are expected values, simulations will yield fractional values ​​toward +1 or -1, reflecting the quantum superposition of states. Similarly, multiple measurements of the same string can be performed and averaged to achieve a value that reflects the expected distribution of QSL.

[0077] Confirmation of quantum spin liquid The existence of QSLs can be experimentally confirmed by measuring the parity of specific m-type and e-type strings. In the following discussion, both closed-loop and open strings are considered.

[0078] Referring to Figure 10A, an exemplary closed m-type string 901 is illustrated. Both of these strings meet the criteria for an m-type string and have no endpoints. As described above, the parity of 12 atoms in string 901 can be measured. For ease of reference, the value obtained is P 12 This can be expressed as follows. Referring to Figures 10B and 10C, open strings 902 and 903 are shown that together form a closed loop 901. The parity of open string 902 (containing 6 atoms) is:

number

number

[0079] If the topological state of atoms within a lattice is QSL, then the ratio of open to closed strings must be very small. More precisely,

number

number

[0080] The same analysis can be applied using e-type strings, and similarly, the ratio described above decreases exponentially with respect to the string length for QSLs. This adherence to this property for both m-type and e-type strings provides a unique fingerprint for QSLs. For a given set of configuration parameters, the presence of QSLs can be determined at once. Therefore, devices with given configuration parameters do not need to be retested for the presence of QSLs during operation.

[0081] Realization of boundary conditions For the purpose of topological quantum computer calculations

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[0082] As numerically shown in the following examples, the termination of a lattice without modification of other parameters produces an m-condensation boundary. That is, for a given lattice and the δ and Ω that produce the QSL, the termination of the lattice without modification of δ produces an m-condensation boundary.

[0083] By changing the detuning δ at the edge, a phase transition is induced along the edge from the m-condensation boundary condition to the second boundary condition, e-condensation. Referring to Figure 12, the ratio between detuning at the lattice boundary and detuning in the rest of the lattice.

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number

[0084] therefore, <n>Measuring the correlation length can already be used to distinguish between two boundary conditions. Furthermore, the correlation length along the boundary diverges in boundary phase transitions in the case of adjustments from m-boundary conditions to e-boundary conditions and vice versa. Therefore, measuring the correlation length along the boundary is another method for locating boundary transitions.

[0085]

number

number

[0086] The method described above, which achieves the desired boundary conditions by adjusting δ, is the only possible way to do so. Other options include, for example, introducing more atoms closer to the edge, whose repulsive force acts as effectively as the chemical potential over atoms in the system. In this example,

number

[0087] In addition to identifying boundary conditions as described above, string correlations can also be used to identify boundary conditions. As shown in Figure 13, the string correlations of P and Q between boundary-to-boundary (as defined in the examples) are:

number

[0088] The generation of e-anyons, immobilization, and movement. As mentioned above, e-anyons correspond to defects in dimer covering when considering a kagome lattice. In particular, if there are no dimers in contact with it, or if there are two dimers in contact with it, then we can say that there is an e-anyon at the vertex of the kagome lattice.

[0089]

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[0090] Figure 14 illustrates an example of an e-anyon on a Kagome lattice. The e-anyon is located at the vertices of the Kagome lattice in regions 1301 and 1302. In each example, the four atoms adjacent to the vertex (hollow circles) are in the ground state. Therefore, in the dimer model, it can be said that there are no dimers in contact with the vertex.

[0091] In one method of generating e-anyons, detuning applied to four atoms adjacent to the vertices of a kagome lattice is adjusted to maintain these atoms in the ground state during state preparation. Referring again to Figure 14, a laser beam with a large negative delta applied to region 1301 biases the atoms to their ground state.

[0092] The exemplary system illustrated in Figure 14 is cylindrical and periodic in the y-direction. Therefore, the dashed line represents a closed loop around the circumference. The numbers associated with each string give the result of a parity measurement of each string. String parity is positive to the left and right and negative between two e-anyons. This provides an example of how parity string measurements can be used to measure the presence of e-anyons. By locally varying the detuning as a function of space and time, these e-anyons can be moved around, as further illustrated below.

[0093] Instead of changing δ on four atoms as described above (immobilizing e-anyons at the vertices), we can change δ on a single atom. This generates two e-anyons on two adjacent vertices of the kagome lattice to this atom. However, these anyons are not immobilized and tend to move away unexpectedly due to heat or quantum fluctuations.

[0094] This is illustrated in Figures 15A and 15B. In particular, Figure 15A shows a dimer coating lacking e-anyons. In Figure 15B, the atom at 1401 is driven to the ground state by adjusting the detuning δ applied to it by the laser source. This generates the pair of anyons 1402, 1403 at adjacent vertices of the kagome lattice. As shown, the detuning at 1401 does not prevent the dimer from occupying an edge adjacent to anyons 1402, 1403, e.g., edge 1404. Thus, anyons 1402, 1403 are not constrained to move across the lattice.

[0095] To generate a pair of e-anyons whose movement is controlled, a pair of immobile anyons are simultaneously generated using the method described with respect to Figure 14, and then moved across the grid. This is illustrated in Figures 15C-E.

[0096] As shown in Figure 15C, by changing the detuning on the atom at 1411...1417, anyons 1418 and 1419 are generated and immobilized. These e-anyons are then gradually moved to the surroundings and separated. In Figure 15D, anyon 1419 is moved from its original position by returning the atom's detuning to its bulk value at 1416...1417 and driving the atom to the ground state at 1421...1424. Similarly, in Figure 15D, anyon 1418 is moved from its original position by returning the detuning to its bulk value at 1411...1414 and driving the atom to the ground state at 1425...1428.

[0097] Generally, anyons can be immobilized by adjusting the detuning applied to four adjacent atoms. In relation to the previous diagram, since the dimer is prevented from reaching its vertex, it is understood that this configuration prevents the e-anyon from moving unless further detuning is performed. In contrast, anyons produced by adjusting the detuning on a single atom can move more freely across the lattice.

[0098] In the exemplary protocol, the detuning of atoms in the direction of travel is gradually decreased, and the detuning of atoms in the opposite direction is gradually increased. In this way, the gradual fade between immobilized positions is achieved to reposition a given anyon between adjacent vertices. This can be achieved by gradually moving a local detuning beam across the lattice. The timescale of these transitions should be slow compared to the energy cost of generating one of the anyons.

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[0099] e-anyons can also be generated by removing Rydberg atoms from the lattice rather than driving them to the ground state. However, it is understood that selective detuning offers various engineering advantages over the physical removal of atoms.

[0100] Initialization of one or more topological qubits In the previous section,

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[0101] There are two categories of methods for generating qubits using the tools described above. The first option is to generate a ruby ​​lattice with m-boundary conditions, and then generate several internal edges that also have m-boundary conditions. The internal edges may contain vertices that are either occupied or unoccupied by particles in the ruby ​​lattice. In the case of occupied vertices, the particles bounded by the internal edges are driven to the ground state by adjusting the laser detuning in the same manner as described above. With external edges, the internal edges are in m-boundary conditions without further adjustment of detuning at the edges. The region inside this type of internal edge may be referred to as a "hole".

[0102] Referring to Figure 16, a schematic diagram of an exemplary grid having one external edge and three internal edges is provided. Grid 1501 has an external edge 1502 with m-condensation boundary conditions. Grid 1501 also includes internal edges 1503, 1504, and 1505. The ground state degeneracy of such a system is 2 N It is proportional to the number of internal edges, where N is the number of internal edges. Therefore, in a configuration that uses internal edges, each internal edge corresponds to one topological qubit.

[0103] The second option is to generate a system without internal edges. This requires generating both m-condensation and e-condensation boundary conditions.

[0104] Referring to Figure 17, an exemplary grid with four external edges is illustrated. In this example, grid 1600 forms a square slab with four edges 1601...1604. Detuning on the boundary connections of the edges is selected such that edges 1603 and 1604 have m-boundary conditions and edges 1601 and 1602 have e-condensed boundary conditions. Such a system has a 2x ground state degeneracy, which can encode a single logical qubit. More generally, if one has several finite regions with 2N alternations of boundary conditions (alternating between e-boundary conditions and m-boundary conditions), then this region can host N-1 topological qubits (Figure 17 corresponds to N=2).

[0105] Reading topological qubits corresponding to external edges in the Z-reference. Referring again to Figure 17, the state of the topological qubits in the case of the outer edges can be read by determining the parity of a particular string. In particular, the dashed line 1605 corresponds to an m-type string extending between edges 1603 and 1604, which has an m-boundary condition. As mentioned above, the parity of this string can be determined by multiplying together the parity of each vertex of the ruby ​​lattice that the string crosses (which is +1 for the ground state and -1 for the excited states). In the context of the Kagome lattice, this can be equivalently described as counting whether the string crosses an odd or even number of dimers. An odd intersection results in a parity of -1, and an even intersection results in a parity of +1.

[0106] A quantum state is a superposition of distinct dimer coatings. To measure this number, the quantum state can be sampled multiple times. For each sample, a classical snapshot is obtained in which the parity can be computed by computer. The results can then be averaged across multiple snapshots. The resulting number constitutes a readout of a Z-referenced logical qubit. This string correlation can be called a P-string (short for dimer parity).

[0107] Since the system does not form a complete dimer covering, it is desirable to standardize the parity measurement described above. This can be done by simultaneously measuring two strings that each extend between edges 1603 and 1604. As shown in Equation 3, the parity of the first string is divided by the square root of the product of the parities of the two strings.

Number

[0108] In Equation 3, the gray boxes are an abbreviated representation for the finite system shown in FIG. 17. The logical 0 and 1 states in the z - basis are

Number

[0109] As described above, the number of topological qubits is proportional to the number of alternations between the e - boundary condition and the m - boundary condition. The above example contains four alternations and thus one topological qubit. A system with six alternations, for example a lattice with approximately hexagonal outer edges, encodes two topological qubits. In such a case, the above process can be applied to each pair of m - condensation boundaries to measure two qubits.

[0110] Reading of topological qubits corresponding to outer edges in the X - basis Bloch sphere (

Number

[0111] In particular, the system is exposed to a high-density laser (corresponding to a large value of Ω) for a fixed time. For a given large selection of Ω (constrained only by the available laser power), the pulse is:

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number

[0112] As mentioned above, the parity averaged over many different snapshots of the state yields an x ​​component. Again, as mentioned above, it is desirable to standardize the value by the square root of the product of the following two strings relative to each other. This is again shown in Equation 4.

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[0113] Referring to Figure 18, an exemplary conditioning and quenching protocol is provided. The left portion of this graph corresponds to the conditioning in Figure 6, and the protocol for reference rotation follows without delay. As depicted, after conditioning, the detuning Δ is rapidly changed to 0, and the blockade radius is reduced so that only atoms within the same triangle of the ruby ​​lattice are blocked (i.e., R b (This is reduced to cover only the two nearest neighbors of a given atom.)

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number

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[0114] Next, under these conditions, the characteristic time

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[0115] As mentioned above, e-type strings are measured with or without standardization. Alternative methods for understanding these measurements are discussed in the examples related to Figures 34, 44C, and 48.

[0116] Reading topological qubits corresponding to internal edges The above example focuses on the case where the outer edges are used to correspond to the topological qubits illustrated in Figure 17. Referring now to Figure 19, the measurements of the topological qubits corresponding to the inner edges are illustrated. In this example, grid 1900 includes the inner edge 1901. Measurements of the topological qubits on the z-reference ( [Number] ) corresponds to measuring the parity of the m-type string 1902 that extends from the external edge to the internal edge 1901. As described above, multiple measurement values can be obtained for multiple snapshots and can be averaged to reach a final value. The measured value of the topological qubit with respect to the x-axis ( [Number] ) corresponds to measuring the parity of the e-type string 1903 that extends around the internal edge 1901 within the closed loop. As described above, multiple measurement values can be obtained for multiple snapshots and can be averaged to reach a final value.

[0117] Single qubit operation Referring again to the exemplary system shown in FIG. 17, a 90° rotation around the x-axis of the Bloch sphere can be achieved by generating and moving a pair of e-ions. Such a rotation generally [Number] results in a change in the sign of. To achieve this rotation, a pair of e-ions is generated in the system and then pulled apart as described above with respect to FIG. 15. As described above, by changing the detuning δ at individual vertices of the ruby lattice, the ions are pulled across the lattice. Referring again to the exemplary system of FIG. 17, to achieve this rotation, one ion is pulled to edge 1601 and one is pulled to edge 1602. Upon achievement at the edge, the ions are annihilated and the system is returned to its original state, but a phase is imparted.

[0118] As described above regarding qubit measurement, this protocol is generalized to cases where further alternations in boundary conditions allow for the realization of further topological qubits. In such cases, the movement of anyons to each pair of edges in the m-boundary conditions corresponds to one rotation of the qubit.

[0119] Referring again to the exemplary system in Figure 19, a single qubit operation on a topological qubit corresponding to an internal edge can be performed by using a local addressing laser beam.

[0120] In particular, rotation around the x-axis (and for that purpose)

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[0121] Referring to Figure 20, rotation around the z-axis (and for that purpose)

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[0122] Two qubit gates Referring to Figures 21A-B, a CNOT gate is illustrated that includes two topological qubits corresponding to internal edges. The grid 1700 includes two internal edges 1701 and 1702. Internal edge 1701 has an e-condensation boundary condition, and internal edge 1702 has an m-condensation boundary condition. The grid 1700 also includes two external edges. The first external edge 1703 extends to one side of the grid 1700 in a hexagonal shape and has an e-condensation boundary condition, and the second external edge extends to the remaining side of the grid 1700 and has an m-condensation boundary condition. As discussed in the embodiment, this topology provides a ground state degeneracy of a 4-dimer model, where further degeneracy is possible by adding more internal or external edges.

[0123] As described above, m boundaries can be prepared by removing atoms from the array or by biasing atoms to the ground state through the application of local detuning. Also as described above, e boundaries can be prepared by shifting the detuning applied to edge atoms to half the detuning of atoms in the majority of the lattice.

[0124] Similar to the procedures discussed with respect to anyon movement, internal edges can expand, contract, or move within the majority of the lattice by altering local detuning at or with respect to atoms proximal to the edge. In particular, edges can be induced to shift within the majority by decreasing detuning in the direction of movement while increasing detuning in the opposite direction. Movement can also be achieved by multiple expansion and contraction steps in which detuning is first decreased in the direction of movement to expand the edge, and then detuning is increased in the opposite direction to contract the edge.

[0125] This movement is reflected in Figures 21A and 21B. Comparing these two figures, it can be seen that the internal edges 1702 of the 10 atoms are moved from one position to another within the lattice 1700 by changing the detuning of the direction of movement, thereby biasing the atoms toward the ground state.

[0126] Referring again to Figures 21A-B, the internal edge 1701 with the e-condensation boundary condition is held stationary within the grid 1701. Simultaneously, the second internal edge 1702 with the m-condensation boundary condition is moved along the closed path 1704, defining the boundary of internal edge 1701. This applies CNOT to the qubits corresponding to internal edges 1701 and 1702.

[0127] In some embodiments, the internal edge 1702 is moved along the closed path 1704, provided that the same atoms as in the initial condition are present at the edge until it reaches its original position. In some embodiments, the internal edge 1702 is moved along the closed path 1704, provided that at least one atom present at the edge is present in the initial condition until it overlaps with its original position. It is understood that an equivalent effect can be achieved by moving the internal edge 1702 along the closed path by an amount sufficient to contribute the quantum phase to the QSL, e.g., 50%, 60%, 70%, 80%, or 90% of the path to its original position.

[0128] The above example describes the movement of an edge with an m-condensed boundary condition around an edge with an e-condensed boundary condition, but the same effect can be achieved by moving an edge with an e-condensed boundary condition around an edge with an m-condensed boundary condition. Also, the above example describes two internal edges and two external edges, but two qubits can also be realized using three internal edges and one external edge. In this case, of the four edges (three internal and one external), two must have m-boundaries and two must have e-boundaries. Therefore, the two internal edges must have the same boundary condition, and one internal edge must have the same boundary condition as the external edge.

[0129] Referring here to Figure 22, an schematic example of a computer compute node is shown. Computer compute node 10 is merely one example of a suitable computer compute node and is not intended to imply any limitation on the scope of use or functionality of the embodiments described herein. Nevertheless, computer compute node 10 may implement and / or perform any of the functions described herein.

[0130] On the computer computing node 10, there is a computer system / server 12 that is operable with many other general or specialized computer computing system environments or configurations. Examples of well-known computer computing systems, environments and / or configurations that may be suitable for use with the computer system / server 12 include, but are not limited to, personal computer systems, server computer systems, thin clients, thick clients, handheld or laptop devices, multiprocessor systems, microprocessor systems, set-top boxes, programmable home electronic devices, network PCs, minicomputer systems, mainframe computer systems and distributed cloud computing environments that include any of the systems or devices mentioned above.

[0131] The computer system / server 12 may be described in the general context of computer system executable instructions, such as program modules, that are executed by the computer system. Generally, a program module may include routines, programs, objects, components, logic, data structures, etc., that perform a specific task or execute a specific abstract data type. The computer system / server 12 may be implemented in a distributed cloud computing environment where tasks are performed by remote processing devices connected via a communication network. In a distributed cloud computing environment, program modules may reside in both local and remote computer system storage media, including memory storage devices.

[0132] As shown in Figure 22, the computer system / server 12 in the computer computing node 10 is represented in the form of a general-purpose computer computing device. Components of the computer system / server 12 may include, but are not limited to, one or more processors or processing units 16, system memory 28, and a bus 18 that connects various system components such as the system memory 28 to the processor 16.

[0133] Bus 18 represents one or more of several types of bus structures, including a memory bus or memory controller, a peripheral bus, an accelerated graphics port, and a processor or local bus using any of the various bus architectures. By example, but not limited to, such architectures include the Industry Standard Architecture (ISA) bus, the Microchannel Architecture (MCA) bus, the Enhanced ISA (EISA) bus, the Video Electronics Standards Association (VESA) local bus, the Peripheral Component Interconnect (PCI) bus, the Peripheral Component Interconnect Express (PCIe), and the Advanced Microcontroller Bus Architecture (AMBA).

[0134] The computer system / server 12 typically includes various computer system-readable media. Such media can be any available media accessible by the computer system / server 12, and include both volatile and non-volatile media, and removable and non-removable media.

[0135] The system memory 28 may include computer system-readable media in the form of volatile memory, such as random access memory (RAM) 30 and / or cache memory 32. The computer system / server 12 may further include other removable / non-removable, volatile / non-volatile computer system storage media. For illustrative purposes only, the storage system 34 may be provided for reading from and writing to a non-removable, non-volatile magnetic medium (not shown, typically referred to as “hard drive”). Not shown, a magnetic disk drive for reading from and writing to a removable, non-volatile magnetic disk (e.g., “floppy disk”), and an optical disk drive for reading from or writing to a removable, non-volatile optical disk, e.g., CD-ROM, DVD-ROM or other optical medium may be provided. In such examples, each may be coupled to the bus 18 by one or more data medium interfaces. As further shown and described below, the memory 28 may include at least one program product having a set (e.g., at least one) of program modules configured to perform the functions of the embodiments of this disclosure.

[0136] A program / utility 40 having a set (at least one) of program modules 42, as well as an operating system, one or more application programs, other program modules, and program data (without being limited to being an example), may be stored in memory 28. Each of the operating system, one or more application programs, other program modules, and program data, or any combination thereof, may include the execution of a network environment. The program modules 42 generally perform functions and / or methodologies of the embodiments described herein.

[0137] The computer system / server 12 may also communicate with one or more external devices 14, such as a keyboard, position indicator, or display 24; one or more devices that enable a user to exchange information with the computer system / server 12; and / or any devices (e.g., network cards, modems, etc.) that enable the computer system / server 12 to communicate with one or more other computer computing devices. Such communication may occur via the input / output (I / O) interface 22. Furthermore, the computer system / server 12 may communicate with one or more networks, such as a local area network (LAN), a general wide area network (WAN), and / or a public network (e.g., the Internet), via the network adapter 20. As shown, the network adapter 20 communicates with other components of the computer system / server 12 via the bus 18. It should be understood that other hardware and / or software components may be used with the computer system / server 12, although not shown. Examples include, but are not limited to, microcode, device drivers, redundant processing units, external disk drive arrays, RAID systems, tape drive and data archival storage systems, etc.

[0138] This disclosure may be embodied as a system, method, and / or computer program. The computer program may include one or more computer-readable storage media having computer-readable program instructions on the computer-readable storage media for causing a processor to carry out aspects of this disclosure.

[0139] Computer-readable storage media can be specific devices capable of holding and storing instructions for use by instruction-executing devices. Computer-readable storage media can be, for example, but are not limited to, electronic storage devices, magnetic storage devices, optical storage devices, electromagnetic storage devices, semiconductor storage devices, or any suitable combination of the foregoing. A non-exclusive list of more specific examples of computer-readable storage media is as follows: portable computer diskettes, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), static random access memory (SRAM), portable compact disk read-only memory (CD-ROM), digital versatile disks (DVDs), memory sticks, floppy disks, mechanically encoded devices such as punch cards or raised structures in grooves with instructions recorded on the grooves, and any suitable combination of the foregoing. As used herein, a computer-readable storage medium is not construed as a transient signal itself, such as radio waves or other freely propagating electromagnetic waves, electromagnetic waves propagating through waveguides or other transmitting media (e.g., light pulses through fiber optic cables), or electrical signals transmitted through wires.

[0140] The computer-readable program instructions described herein may be downloaded from a computer-readable storage medium to a computer computing / processing device, or to an external computer or external storage device via a network, such as the Internet, a local area network, a wide area network, and / or a wireless network. The network may include copper transmission cables, optical transmission fibers, wireless transmissions, routers, firewalls, switches, gateway computers, and / or edge servers. A network adapter card or network interface in each computer computing / processing device receives computer-readable program instructions from the network and transfers the computer-readable program instructions for storage in the computer-readable storage medium within the computer computing / processing device.

[0141] Computer-readable program instructions for performing the operations of the Disclosure may be assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state setting data, or source code or object code written in any combination of one or more programming languages, such as object-oriented programming languages ​​like Smalltalk and C++, and traditional procedural programming languages ​​such as the C programming language or similar languages. Computer-readable program instructions may run entirely on a user's computer, partially on a user's computer as a standalone software package, partially on a user's computer and partially on a remote computer, or entirely on a remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer through any type of network, including a local area network (LAN) or wide area network (WAN), or such connection may be made to an external computer (for example, via the Internet using an Internet service provider). In some embodiments, electronic circuits including, for example, a programmable logic circuit, a field-programmable gate array (FPGA), or a programmable logic array (PLA) may execute computer-readable program instructions using state information of computer-readable program instructions for personalizing the electronic circuit in order to carry out aspects of the present disclosure.

[0142] Aspects of this disclosure are described herein with reference to flowcharts and / or block diagrams of methods, apparatus (systems) and computer programs as described herein. It is understood that each block in a flowchart and / or block diagram, as well as any combination of blocks within a flowchart and / or block diagram, can be executed by computer-readable program instructions.

[0143] These computer-readable program instructions may be provided to the processor of a general-purpose computer, a special-purpose computer, or other programmable data processing device for manufacturing machines, and the instructions executed via the processor of the computer or other programmable data processing device create means for performing functions / actions identified in a flowchart and / or block diagram block(s). These computer-readable program instructions may also be stored in a computer-readable storage medium that can direct a computer, programmable data processing device, and / or other device to a function in a particular manner, and a computer-readable storage medium having instructions stored in it includes a manufactured article containing instructions for performing a function / action aspect identified in a flowchart and / or block diagram block(s).

[0144] Computer-readable program instructions may also be loaded onto a computer, other programmable device, or other device to create a computer execution process by having a series of operations performed on the computer, other programmable device, or other device, and such instructions performed on the computer, other programmable device, or other device perform functions / actions identified in one or more blocks of a flowchart and / or block diagram.

[0145] The flowcharts and block diagrams in the figures illustrate the architecture, functionality, and possible execution operations of systems, methods, and computer programs according to various aspects of the present disclosure. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of instructions containing one or more executable instructions for performing a particular logical function. In some alternative executions, the functions described in a block may occur out of the order shown in the figure. For example, two blocks shown consecutively may actually be executed substantially simultaneously, or they may be executed in reverse order depending on the functions they contain at times. It should also be noted that each block in a block diagram and / or flowchart representation, and combinations of blocks in a block diagram and / or flowchart representation, may be executed by a system based on special-purpose hardware that performs a particular function or action or a combination of special-purpose hardware and computer instructions.

[0146] Exemplary embodiments of the present invention are described below.

[0147] Therefore, in a first exemplary embodiment, the present invention is a device. In the first aspect, the device includes a two-dimensional array of particles, each particle being

number

[0148] In the third aspect of the first exemplary embodiment, the blockade is a dipole blockade, for example, a Rydberg blockade. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first and second aspects.

[0149] In the fourth aspect of the first exemplary embodiment, the particle is an atom, the first state is the ground state, and the blockade is a Rydberg blockade. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to third aspects.

[0150] In the fifth aspect of the first exemplary embodiment, the array includes at least a first and a third outer edge, each having a first boundary condition, and at least a second and a fourth outer edge, each having a second boundary condition different from the first boundary condition. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to fourth aspects.

[0151] In the sixth aspect of the first exemplary embodiment, the array has a plurality of external edges, each external edge being under either the first or second boundary condition, and each external edge being under a different boundary condition than any adjacent external edge. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to fifth aspects.

[0152] In the seventh phase of the first exemplary embodiment, an external edge configured to be under the first boundary condition is e-condensed, and an external edge configured to be under the second boundary condition is m-condensed. Other phases and exemplary features of the first exemplary embodiment are as they are described with respect to the first to sixth phases.

[0153] In the eighth aspect of the first exemplary embodiment, the array includes at least one internal edge. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to seventh aspects.

[0154] In the ninth aspect of the first exemplary embodiment, each vertex enclosed by at least one internal edge is not occupied by a particle. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to eighth aspects.

[0155] In the tenth aspect of the first exemplary embodiment, at least one internal edge has the same boundary conditions as at least one external edge. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to ninth aspects.

[0156] In the eleventh aspect of the first exemplary embodiment, at least one internal edge encloses at least four vertices. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to tenth aspects.

[0157] In the twelfth aspect of the first exemplary embodiment, at least one internal edge surrounds a vertex occupied by a particle. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to eleventh aspects.

[0158] In the thirteenth aspect of the first exemplary embodiment, at least one internal edge is subject to a first boundary condition distinct from at least one external edge. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to twelfth aspects.

[0159] In the 14th aspect of the first exemplary embodiment, the array has a plurality of internal edges, each internal edge enclosing a plurality of corresponding vertices, each not occupied by a particle. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to 13th aspects.

[0160] In the 15th aspect of the first exemplary embodiment, the array has a plurality of internal edges, each internal edge enclosing a plurality of corresponding vertices, at least one of which is occupied by a particle. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to 14th aspects.

[0161] In the 16th aspect of the first exemplary embodiment, the internal edges surrounding the vertices occupied by the particles have different boundary conditions from at least one external edge. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to 15th aspects.

[0162] In the 17th phase of the first exemplary embodiment, the edges configured to be under different boundary conditions are selected from e-condensation or m-condensation edges. Other phases and exemplary features of the first exemplary embodiment are as they are described with respect to the first to 16th phases.

[0163] In the 18th aspect of the first exemplary embodiment, the two-dimensional array includes at least 96 particles, for example, at least 200 particles. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to 17th aspects.

[0164] In a second exemplary embodiment, the present invention is a system. The system includes a confinement system for arranging particles in a two-dimensional array and an excitation source for exciting at least some of the particles from a first state to an excited state. The confinement system includes a laser source arranged to generate a plurality of confinement regions; and a source of atomic clouds, the atomic clouds may be arranged to at least partially overlap the plurality of confinement regions. In the first aspect of the second exemplary embodiment, in the two-dimensional array, each particle is located at the vertices of a ruby ​​lattice; each particle has a first state and an excited state; each particle belonging to at least three unit cells of the ruby ​​lattice, when in an excited state, has a blockade radius sufficient to shield each of at least six nearest neighboring particles in the ruby ​​lattice from its transition from its first state to its excited state; and the array has at least one outer edge configured to be in a first boundary condition.

[0165] In the second aspect of the second exemplary embodiment, the particle is an atom, and the excitation source is configured to excite at least some of the atoms to a Rydberg state. Other aspects and exemplary features of the second exemplary embodiment are as they are described with respect to the first aspect.

[0166] In a third aspect of the second exemplary embodiment, the two-dimensional array includes at least 96 particles, for example, at least 200 particles. Other aspects and exemplary features of the second exemplary embodiment are as they are described with respect to the first and second aspects.

[0167] In a third exemplary embodiment, the present invention is

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[0168] In the second aspect of the third exemplary embodiment, the particle is an atom and the excited state is a Rydberg state. Other aspects and exemplary features of the third exemplary embodiment are as they are described with respect to the first and second aspects.

[0169] In a fourth exemplary embodiment, the present invention is

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[0170] In the second aspect of the fourth exemplary embodiment, the array has a plurality of external edges, and the method further includes the step of imposing either a first boundary condition or a second boundary condition on each external edge, such that each external edge has a different boundary condition from any adjacent external edge. Other aspects and exemplary features of the fourth exemplary embodiment are as they are described with respect to the first aspect.

[0171] In the third aspect of the fourth exemplary embodiment, an external edge configured to be under the first boundary condition is e-condensed, and an external edge configured to be under the second boundary condition is m-condensed. Other aspects and exemplary features of the fourth exemplary embodiment are as they are described with respect to the first and second aspects.

[0172] In a fifth exemplary embodiment, the present invention is

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[0173] In the second aspect of the fifth exemplary embodiment, each vertex enclosed by at least one internal edge is not occupied by a particle. Other aspects and exemplary features of the fifth exemplary embodiment are as they are described with respect to the first and second aspects.

[0174] In the third aspect of the fifth exemplary embodiment, at least one internal edge encloses at least four vertices. Other aspects and exemplary features of the fifth exemplary embodiment are as they are described with respect to the first and second aspects.

[0175] In the fourth aspect of the fifth exemplary embodiment, at least one internal edge surrounds a vertex occupied by a particle. Other aspects and exemplary features of the first exemplary embodiment are as they are described with respect to the first to third aspects.

[0176] In a fifth aspect of the fifth exemplary embodiment, the method further includes the step of imposing boundary conditions on at least one internal edge that are different from the boundary conditions of at least one external edge. Other aspects and exemplary features of the fifth exemplary embodiment are as they are described with respect to the first to fourth aspects.

[0177] In a sixth aspect of the fifth exemplary embodiment, the method further includes the step of imposing the same boundary conditions on at least one internal edge as the boundary conditions on at least one external edge. Other aspects and exemplary features of the fifth exemplary embodiment are as they are described with respect to the first to fifth aspects.

[0178] In the seventh aspect of the fifth exemplary embodiment, the array has a plurality of internal edges, each internal edge enclosing a plurality of corresponding vertices, and at least one of the enclosed vertices is occupied by a particle. Other aspects and exemplary features of the fifth exemplary embodiment are as they are described with respect to the first to sixth aspects.

[0179] In the eighth aspect of the fifth exemplary embodiment, the method further includes the step of imposing boundary conditions on internal edges surrounding vertices occupied by particles different from the boundary conditions of at least one external edge. Other aspects and exemplary features of the fifth exemplary embodiment are as they are described with respect to the first to seventh aspects.

[0180] In the ninth phase of the fifth exemplary embodiment, the edges configured to be under different boundary conditions are selected from e-condensation or m-condensation edges. Other phases and exemplary features of the fifth exemplary embodiment are as they are described with respect to the first to eighth phases.

[0181] In a sixth exemplary embodiment, the present invention is

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[0182] In the second phase of the sixth exemplary embodiment, a reference rotation is applied to the first topological qubit before determining the first path. Other phases and exemplary features of the sixth exemplary embodiment are as they are described with respect to the first phase.

[0183] In a third aspect of the sixth exemplary embodiment, the method further includes the steps of: determining a second path through an array from a first outer edge of a plurality of outer edges to a second outer edge of a plurality of outer edges via a second plurality of vertices of a ruby ​​lattice having a second plurality of second particles thereon; assigning a second value to the second path based on the state of each of the second plurality of particles; and determining the state of a first topological qubit based on the first and second values. Other aspects and exemplary features of the sixth exemplary embodiment are as they are described with respect to the first and second aspects.

[0184] In the fourth aspect of the sixth exemplary embodiment, the method further includes the steps of: determining a third path through an array from a first outer edge of a plurality of outer edges to a third outer edge of a plurality of outer edges having a first boundary condition, via a third plurality of vertices of a ruby ​​lattice having a third plurality of third particles therein; assigning a third value to the third path based on the state of each of the third plurality of particles; and determining the state of a second topological qubit based on the third value. Other aspects and exemplary features of the sixth exemplary embodiment are as they are described with respect to the first to third aspects.

[0185] In a seventh exemplary embodiment, the present invention is

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[0186] In a second aspect of the seventh exemplary embodiment, the method further includes the steps of: determining a second path through the array from at least one internal edge to at least one external edge via a second plurality of vertices of a ruby ​​lattice having a second plurality of particles therein; assigning a second value to the first path based on the state of each of the second plurality of particles; and determining the state of a first topological qubit based on the first and second values. Other aspects and exemplary features of the seventh exemplary embodiment are as they are described with respect to the first aspect.

[0187] In a third aspect of the seventh exemplary embodiment, the array has at least a second internal edge, and the method further includes: determining a third path through the array from the second internal edge to at least one external edge via a third plurality of vertices of a ruby ​​lattice having a third plurality of particles thereon; assigning a third value to the third path based on the state of each of the third plurality of particles; and determining the state of a second topological qubit based on the third value. Other aspects and exemplary features of the seventh exemplary embodiment are as they are described with respect to the first and second aspects.

[0188] In a further exemplary embodiment of either the sixth exemplary embodiment or an aspect of the seventh exemplary embodiment, the step of determining one of the paths comprises: piecewise assembling paths from a plurality of divisions, each division extending between two vertices in a ruby ​​lattice, where each division either extends between two vertices in a triangular portion of a unit cell of the ruby ​​lattice, or extends between two vertices in different unit cells of the ruby ​​lattice without intersecting any unit cell of the ruby ​​lattice. Other aspects and exemplary features of the sixth and seventh exemplary embodiments are as they are described with respect to any of those aspects.

[0189] In any further exemplary embodiment of the sixth exemplary embodiment and any aspect of the seventh exemplary embodiment, determining any one of the paths involves: piecewise assembling the paths from a plurality of divisions, each division extending between two vertices in a ruby ​​lattice, where each division extends between two vertices in a quadrilateral portion of the unit cell of the ruby ​​lattice. Other aspects and exemplary features of the sixth and seventh exemplary embodiments are as they are described with respect to any of those aspects.

[0190] In an eighth exemplary embodiment, the present invention relates to a method for manipulating a topological qubit. The method includes the step of preparing a topological qubit according to a method defined in any of the fourth exemplary embodiment or aspects thereof. In the first aspect of the eighth exemplary embodiment, the first boundary condition is an e-boundary condition, and the method includes: generating first and second e-anyons in an array; removing the first e-anyon from the array via a first outer edge; and removing the second e-anyon from the array via a third outer edge.

[0191] In a ninth exemplary embodiment, the present invention is a method for manipulating a topological qubit. The method includes the step of preparing a topological qubit according to a method defined in any of the fifth exemplary embodiment or aspects thereof. In the first aspect of the ninth exemplary embodiment, the method further includes the steps of generating first and second e-anyons in an array; immobilizing the first e-anyon; and defining the boundary of at least one internal edge and moving the second e-anyon along a circular path having an endpoint at the location of the first e-anyon, thereby destroying the first and second e-anyons.

[0192] In the tenth exemplary embodiment, the present invention is carried out in accordance with the method specified in the fourth exemplary embodiment or any aspect thereof.

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[0193] In an eleventh exemplary embodiment, the present invention is a method for manipulating first and second topological qubits, comprising the step of encoding the first and second topological qubits in accordance with a method defined in a tenth exemplary embodiment. In a first aspect of the eleventh exemplary embodiment, the method further comprises the step of moving the first internal edge along a closed continuous path that defines the boundary of the second internal edge.

[0194] In a second aspect of either the 10th or 11th exemplary embodiment, either (i) the first boundary condition is m-condensation and the second boundary condition is e-condensation, or (ii) the first boundary condition is e-condensation and the second boundary condition is m-condensation. Other aspects and exemplary features of the 10th and 11th exemplary embodiments are as they are described with respect to the first aspect of each of these embodiments.

[0195] In a twelfth exemplary embodiment, the present invention is a computer program comprising a computer-readable storage medium having program instructions implemented by the computer-readable storage medium, wherein the program instructions are executable by a processor in such a way that the processor performs any one of the third to eleventh exemplary embodiments or aspects thereof.

[0196] In one exemplary embodiment, the present invention is a device comprising a two-dimensional array of particles, each particle positioned at the vertices of a ruby ​​lattice, and each particle belonging to at least three unit cells of the ruby ​​lattice having a ground state and an excited state, the excited state having a blockade radius sufficient to block at least six nearest adjacent particles in the ruby ​​lattice.

[0197] In another exemplary embodiment, the present invention is a system for aligning particles in a two-dimensional array, wherein each particle is located at the vertices of a ruby ​​lattice, and each particle belonging to at least three unit cells of the ruby ​​lattice has a ground state and an excited state, the excited state having a blockade radius sufficient to block at least six nearest neighboring particles in the ruby ​​lattice; the confinement system includes a laser source aligned to generate a plurality of confinement regions; an atomic cloud source, where the atomic cloud may be positioned to at least partially overlap the plurality of confinement regions; and an excitation source for developing at least some of the plurality of particles from a ground state to an excited state.

[0198] In another exemplary embodiment, the present invention relates to a method for reading the state of a topological qubit, the method comprising the steps of receiving a representation of the state of each particle of a two-dimensional array of particles, where each particle is located at the vertices of a ruby ​​lattice, and each particle belonging to at least three unit cells of the ruby ​​lattice is in a ground state or an excited state, the excited state having a blockade radius sufficient to block at least six nearest adjacent particles in the ruby ​​lattice, the ruby ​​lattice having a plurality of edges, each edge having either a first boundary condition or a second boundary condition, each edge having a boundary condition different from any adjacent edge; therein a first plurality of particles in the ruby ​​lattice The process includes: determining a first path through a ruby ​​lattice from a first edge of a plurality of edges having a first boundary condition to a second edge of a plurality of edges having a first boundary condition via the vertices of the first plurality of particles; assigning a first value to the first path based on the state of each of the first plurality of particles; determining a second path through a ruby ​​lattice from a third edge of a plurality of edges having a second boundary condition to a fourth edge of a plurality of edges having a second boundary condition via a second plurality of vertices of the ruby ​​lattice having a second plurality of particles thereon; assigning a second value to the second path based on the state of each of the second plurality of particles; and determining the state of a topological qubit based on the first and second values.

[0199] Methods for preparing a spin liquid state are provided in various embodiments. A coherent light beam is directed to a two-dimensional array of particles. Each particle is positioned at the vertices of a ruby ​​lattice. Each particle belonging to at least three unit cells of the ruby ​​lattice has a ground state and an excited state, the excited state having a blockade radius sufficient to block at least six nearest neighboring particles in the ruby ​​lattice. Detuning of the coherent light beam is performed using a frequency sweep, changing it from a negative to a positive value. In some embodiments, the frequency sweep is cubic. In some embodiments, Rydberg coupling is activated before the detuning change. In some embodiments, the activation is performed for a period of at least equal to the reciprocal of the Rabi frequency of each particle. [Examples]

[0200] Example A: Prediction of toric code topological order from Rydberg blockade When encountered in exemplary trick code

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[0201] preface Nearly 50 years ago, Anderson proposed that quantum fluctuations could give rise to a liquid of resonating valence bonds, which has inspired a tremendous amount of theoretical effort to this day. Further work has linked this idea to a more precise concept of a gapped quantum spin liquid, an exotic state that can potentially be realized in frustrated magnets. At the same time, it has been suggested that such a gapped quantum liquid contains topological order, the simplest example being in two spatial dimensions

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[0202] Phases of matter possessing topological order exhibit several remarkable properties. Firstly, although the gauge groups and other details differ, they suggest the emergence of gauge fields similar to those that explain fundamental forces. Therefore,

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[0203] Ultimately, the crucial property of topological order, a long-range characteristic of this entanglement, was pointed out. On the one hand, this implies that a topologically ordered state of matter, unlike any other conventional ground state realized to date, realizes a novel form of entangled quantum matter as a whole. On the other hand, this observation also has profound relevance in fields such as quantum error correction and fault-tolerant quantum computing.

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[0204] For these considerations,

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[0205] Referring to Figure 23, the relationship to the Rydebell blockade model and the dimer model is illustrated. (Figure 23A) Hardcore bosons on the kagome lattice connections (forming the ruby ​​lattice) are strongly repelled and dispose of (punish) the double occupation in the disk.

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[0206] Recently, a novel approach has emerged for investigating quantum many-body physics. This approach is based on an array of neutral atoms trapped in an optical tweezers array. Tunerable atomic interactions can be engineered in such systems using a Rydberg blockade mechanism and mediated by laser excitation of atoms to Rydberg states. Significant progress has been made in realizing two-dimensional quantum lattice models from atomic arrays, and rich phase diagrams of symmetry breaking orders have been predicted and observed. Simultaneously, the unique characteristics of Rydberg atomic interactions make them an attractive platform for realizing nascent lattice gauge theories and quantum dimer models. We note that symmetry-protected topological phases have been realized in one-dimensional Rydberg chains; this requires no symmetry and is distinct from the intrinsic topological order considered in this work, which features nascent anyons.

[0207] Here, we

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[0208] Our approach to realizing topological spin liquids is based on the Rydberg blockade: when a neutral atom is excited to a Rydberg state with a high principal quantum number, resonance excitations of nearby atoms are suppressed by strong atom-atom interactions. The possibility of exciting atoms to Rydberg states is explained by a two-level system. The minimum effective Hamiltonian for a Rydberg array is the so-called PXP model.

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[0209] Because it is unique, we first focus on a PXP model on a so-called ruby ​​lattice with a blockade radius containing six nearby regions, an equivalent Kagome lattice linkage (see Figure 23A). By adjusting δ, we find a phase transition from an ordinary phase of the material to another featureless phase. We find that the latter can be analyzed using various probes such as topological entanglement entropy, ground state degeneracy, and modular transformation.

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[0210] These results can be understood by noting that for the aforementioned lattices and Rydebelly blockade radii, the Hamiltonian is equivalent to the dimer-monomer model on a Kagome lattice. The dimer model on lattices that are not divided into two (e.g., triangular and Kagome lattices) is

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[0211] Furthermore, we show that the above findings do not fine-tune for the PXP model. More precisely, we show realistic inter-Rydberg atoms on a specific example of a ruby ​​lattice.

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[0212] In experimentally relevant models

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[0213] These string operators also serve as very useful probes for detecting spin liquids experimentally. The ability to measure nonlocal observables is a truly remarkable advantage of certain cold atom platforms. In more general solid-state systems, we must rely on local probes that are suitable for identifying local order parameters but cannot directly detect topological order. In contrast, the Rydberg platform allows us to take snapshots of quantum states with single-site resolution, opening up the possibility of extracting nonlocal correlation functions. We describe in detail how this feature can be extended to diagnose topological order. While diagonal string operators can be readily measured, we further show how string operators for e-anyons that innately (a priori) contain off-diagonal operations that are difficult to measure in the laboratory can be transformed into diagonal string operators by time evolution using a Hamiltonian with a quenched blockade radius. Thus, we show that both string operators become measurable on a diagonal basis.

[0214] Ultimately, we discuss how generating and manipulating quantum information stored in a topologically degenerate ground state can solidify the path for the potential exploration of topological quantum memory. Two crucial pieces of the puzzle we identify are the ability to trap e-anyons and the ability to generate distinct topological boundary conditions, both of which are achieved directly by locally varying laser detuning. As we will explain, these two components already provide access to topologically degenerate qubits on a surface that can be initialized and read out.

[0215] The remainder of this embodiment includes a discussion of the Rydeberg blockade model, for example, comparing it to the general dimer model and distinguishing it from the dimer model. (Minor phases,)

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[0216] Ludberg Rockade 'PXP' Model We have a two-dimensional version of the Kagome lattice and the Fendley-Sengupta-Sachdev model:

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[0217] We set Ω > 0. i to -b i By substituting this, the sign of Ω can be preserved, which is n i Note that this remains unchanged. Only where the sign of Ω matter is important in this document is within the definition of the topological string operator. For Rydberg atoms,

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[0218] Here, lattice space a is the shortest distance between two atoms. As shown in Figure 23A, using this interaction range, a given site is connected to six other sites, which is the distance.

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[0219] Relationship with the Dimer Model and Differences from It Regarding the dimer state on the Kagome lattice,

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[0220] The constraint of a dimer model having exactly one dimer per vertex can be interpreted as Gauss's law. More precisely, the presence or absence of a dimer is,

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[0221] In contrast, the Rydebell-Brockade model (Equation 5) is a dimer-monomer model. That is, Gauss's law in lattice gauge theory is expressed here.

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[0222] Therefore, although it does not realize a strict dimer model, it has clear advantages in realizing a dimer-monomer model, and it is also advantageous that it is an approximation of the dimer model (i.e., it has a low monomer density). Firstly, as mentioned above, the dimer model on the kagome lattice cannot realize the minor phases of the material, so this is a good place to look for spin liquids. Secondly,

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[0223] Referring to Figure 24, a phase diagram of the Rydebell blockade model on the connection of the Kagome grid is provided.

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[0224] phase diagram Here, we use the density matrix renormalization group (DMRG) to test the phase diagram of the model of Equation 5 with blockades in Equation 6. We explicitly do this by working in a reduced Hilbert space.

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[0225]

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[0226] As a first indication that this intermediate phase is still within the approximate dimer model, we have the filling fraction shown by the red curve in Figure 25A. <n>We will consider this.

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[0227] moreover, <n>The derivative of diverges at the transition between the trivial phase and the spin liquid, indicating a continuous transition. In fact, the theoretical prediction is that this belongs to the 2+1D Ising general class (where the trivial phase corresponds to the 'ordered' side), but our available system size is not large enough to accurately extract the scaling dimension. Figure 25A shows that there is no singularity between the spin liquid and the VBS phase. However, it turns out to be a first-order transition that is very difficult to diagnose this way (due to the small energy scale associated with the VBS phase). This is between different parts <n>This can be explained quite easily by considering the changes in : Figure 25B shows that this jumps discontinuously.

[0228] Topological entanglement entropy One characteristic property of the topological phases of matter can be found in the scaling of entanglement entropy. The gapped phases of matter satisfy the area law: for a region with a boundary L, we can determine that

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[0229] Topological entanglement entropy can be efficiently extracted from cylindrical geometric structures. We take the midpoint δ / Ω = 1.7 of the estimated spin liquid in Figure 24 and numerically obtain the entanglement entropy when bipartitioning an infinitely long cylinder in two halves. By doing this for different circumferences, as shown in Figure 26C, we

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[0230] Importantly, it has been previously observed that false values ​​of γ can be obtained for certain cuts in a particular lattice model, i.e., one can be tricked into thinking that trivial phases are actually topologically ordered. In all such reported cases, false values ​​can be detected by comparing the results for different cuts. For this purpose, we extracted γ for two different geometric structures: XC (where a finite periodic direction bisects the triangles of the kagome lattice) and YC (where the circumference runs parallel to one of the axes of the kagome lattice) for the explanation of this naming convention, see below. Both linear fits are:

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[0231] Referring to Figure 25, it is illustrated that the phase transition can be detected via the packing density. This data is obtained for an infinitely long cylinder having an XC-8 geometric structure. (Figure 25A) The packing density exhibits unique behavior during the transition from a minor phase to a spin liquid, after which the system...

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[0232] To confirm that the above is not a finely tuned feature of a specific point in the phase diagram, we,

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[0233] Referring to Figure 26, topological entanglement entropy is illustrated. We have the Area Law

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[0234] Referring to Figure 27, the topological string operators are illustrated. (a) Two different string operators are defined by their action on a single triangle. We refer to them as the diagonal and off-diagonal string operators P and Q, respectively. (b) An example of the action of string operators on a classical dimer state. (c) The definition of the Fredenhagen-Marcu order parameter is shown for the diagonal string.

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[0235] String operators and anyon condensation The advantage of measuring topological entanglement entropy is that it is well defined for any model, even in the absence of microscopic identification of operators corresponding to the nascent gauge theory. However, in our Rydebell-Brockade model, a more microscopic understanding of the spin liquid is available. Here, we have, as with the trick code model, this

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[0236]

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[0237] In the dimer criterion, binary string

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[0238] this

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[0239] Spin liquids are defined by the unconfined nature of these excitations. Nearby phases correspond to condensing either e or m, which, for mutual statistics, confine either m or e, respectively. Historically, e-condensates have been called Higgs phases, and m-condensates have been called confined phases (because charged e excitations are confined). In odd gauge theory, the non-zero background gauge charge at each lattice site implies that the latter actually exhibits spontaneous symmetry breaking (i.e., a valence bond solid). The reason for this is that m-anyons,

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[0240] These condensates can be diagnosed by open P or Q strings that achieve long-range order. It is important to standardize these string operators in order to properly define what this means. In fact, these strings generally decay to 0, as the ground state has de facto e and m variations. For this purpose, the standardized string operators shown in Figure 27C are introduced, which are referred to as FM string order parameters. These two string order parameters are very useful tools for diagnosing different phases of lattice gauge theory: confinement in pure gauge theory can be investigated by the area law, but in the presence of dynamic matter (as we have in our model), the loop operators are typically proportional to the perimeter law.

[0241] Referring to Figure 28, the phase diagnosis with respect to the topological string operator is illustrated. Upper panel: The Fredenhagen-Marcu (FM) string order parameters indicate that the minor phase is an e-condensate (=Higgs phase) and the VBS phase is an m-condensate (=confined phase). These string order parameters decay to 0 in a spin liquid, and that

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[0242] The remaining technical matters to discuss are the phase factors in the definition of the off-diagonal string Q in Figure 27A.

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[0243] Here we can evaluate the open string and loop operators in the Rydebell-Blockade model. The results are shown in Figure 28. As expected, we find that Q has long-range order in the trivial phase corresponding to the e-condensate, and P has long-range order in the VBS phase corresponding to the m-condensate. P also achieves long-range order deep in the trivial phase: note that this is possible because the definitions of e and m anyons (and the distinction between them) are strictly significant only in the unconfined phase. In the intermediate spin liquid, both FM order parameters decay to 0, which is consistent with the requirement of the lattice gauge theory that it is an unconfined phase. Figure 28 shows the FM string order for only a specific length of string (as shown on both sides of the panel), but a more careful scaling analysis below confirms that in the spin liquid, these strings decay exponentially to 0 with respect to the length of the string. We emphasize that this is a very non-trivial property that is extremely difficult to explain without the presence of topological order. Accordingly, in this region, the loop operator evaluated around the circumference is not suppressed and has a considerable value (despite decreasing with the circumference). In fact, the sign of this non-zero number labels topologically distinct ground states, as we will discuss next.

[0244] In fact, contributions can arise only from endpoints that naively affect only a finite region due to their finite correlation length, for the sake of string standardization. Generally, in the absence of further symmetry properties, the expectation value of an operator with a non-zero finite support is expected. This is an emerging one-form symmetry of a topologically ordered topology that constrains it to be zero (up to an exponentially small correction connecting two endpoints).

[0245] Referring to Figure 29, the ground state and module transformation are illustrated. From the ground state on an infinitely long cylinder, we can obtain the smallest entangled ground state on a torus geometric structure. For smaller geometric structures, we can,

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[0246] Topological ground state degeneracy and modular matrices Another distinguishing feature of a topological spin liquid is its topological ground state degeneracy over manifolds that are themselves topologically non-trivial. For an abelian topological order on an infinitely long cylinder, theoretically, each anyon has a corresponding ground state. Conceptually, these distinct states can be associated by nucleating anyon pairs, starting with one of the ground states, and separating them infinitely along the infinite directions of the cylinder. Naturally, instead, they could be chosen to be wound around a finite direction, generating different criteria in this four-dimensional space. However, these states will never be minimally entangled on the cylinder, for which DMRG is optimal. Thus, in the case of the present invention, we predict four distinct topological ground states corresponding to the 1, e, m, and f lines that pass along an infinite axis. For mutual statistics, these distinct ground states can be diagnosed by measuring the P and Q loops around the circumference.

[0247] Numerically, if we iterate through DMRG with different random initializations, we can iterate around the circumference.

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[0248] Referring to Figure 30, the topological ground state degeneracy is illustrated. In the topological phase, we obtain the 1 and e ground states from a DMRG with random initial states. The m and f states are obtained by starting from a fixed-point that resonates the dimer state, followed by the application of imaginary time evolution. The energies shown are

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[0249] About smaller tori

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[0250] Another way to confirm that these two ground states correspond to 1 and e anyon is by constructing a fixed-point wave function in which we find a large overlap. More precisely, we find around the circumference

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[0251] Referring to Figure 31, the ruby ​​lattice is illustrated. (a) The atoms on the linkage of the kagome lattice are rectangular in aspect ratio.

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[0252] Referring to Figure 32, a spin liquid on a ruby ​​lattice (ρ=3) is

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[0253] Further features beyond topological order include the implementation of symmetry, i.e., symmetry enrichment of topological order. This can be inferred from the relationship to the Kagome lattice dimer model, even in the absence of spin rotational symmetry (since monomers do not possess spin). We consider the boson mean field

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[0254] Predictions regarding implementation and detection We have found that the Rydeberg blockade model has parameters within a certain range.

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[0255]

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[0256] For specificity, we consider a ruby ​​lattice with ρ=3 shown in Figure 31C. By the empirical rule of Equation 8, we can see that

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[0257] To numerically simulate models with long-range interactions, we,

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[0258] The above applies to models with van der Waals interactions.

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[0259] Referring to Figure 33,

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[0260] In short, we also note that our numerical results are for cylindrical geometric structures, but experimental realizations naturally have open boundary conditions. The main difference is the absence of topologically non-trivial loops (i.e., all loops are contractile), and consequently, the ground state is unique. Nevertheless, the topological ground state degeneracy can be restored by either considering boundary conditions that penetrate the system or mixed boundary conditions. Both mechanisms are described in detail below, where we also consider numerical results for strip geometric structures.

[0261] Measuring off-diagonal strings by converting them to diagonal strings. We

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[0262] Referring to Figure 34, the measurement of off-diagonal string operators by the quench protocol is illustrated. (a) The vertical direction is periodic with respect to the cylinder. The off-diagonal string Q (blue dashed line) is the time of the nearest adjacent Rydebell Blockade Hamiltonian.

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[0263] To perform this rotation, we consider the Rydberg Hamiltonian at 0 detuning, which has a complex phase factor in Rabi oscillations. This is achieved by combining the original Hamiltonian with the appropriate time evolution in which detuning is dominant, i.e.

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[0264] The essential idea is that the lydeberg blockade is localized on each triangle of the ruby ​​lattice, that is

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[0265] Here, since the blockade acts only within the triangles of the ruby ​​lattice, the time evolution using the Hamiltonian described above is equivalent to the on-site unitary transformation (amount to). Therefore, it is sufficient to consider a single triangle, and by describing the P and Q operators defined in Figure 27A as 4x4 matrices acting on the Hilbert space of a single triangle:

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[0266] Therefore, Q can be effectively measured along the string by first evolving it over time using H' and then measuring the P string on the resulting state.

[0267] If the aspect ratio ρ of the ruby ​​lattice is not too close to 1 (unity), then the R between the first two radii... b By quenching this, the adjacent blockade Hamiltonian at the nearest position can be approximated, i.e.

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[0268] Referring to Figure 35, the trapping potential for an e-anyon is illustrated. The ground state for a lattice (here on a cylinder) with four parts around the vertices removed traps an e-anyon. This can be read from the expectation value of the parity loop (orange dashed line) around the circumference: if two adjacent loops have opposite signs, the charge is trapped.

[0269] Referring to Figure 36, the boundary phase diagram of the blockade model is illustrated. We consider an infinitely long strip of the XC-8 geometric structure: the bulk is a spin liquid with δ / Ω = 1.7, but we adjust δ on the outermost boundary coupling. (a) The correlation length diverges at two boundary phase transitions; in the intermediate shaded region, entanglement increases. (b) Small and large

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[0270] Moving towards fault-tolerant quantum memory. Part of the reason why the topologically ordered phases of matter are so interesting is that they can potentially act as a means of generating fault-tolerant quantum memory based on degenerate topological ground states.

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[0271] e-Enion's trap If braids containing anyions are desired, they need to be able to be localized to specific regions. Since the e-anyons in this model correspond to monomers, a natural way to trap them is by ensuring that no dimers contact them at specific vertices. This can be done either by simply removing atoms on these bonds or by reducing the detuning δ. We numerically confirm that this works: Figure 35 shows the result of removing two such vertices on XC-8 for a blockade model with δ / Ω = 1.7. Since the parity loop measures the charge enclosed within a given loop, the non-zero charge localized on these defects can be inferred by comparing the symbols of the parity loop along the cylinder. In fact, we even understand that two e-anyons are connected by a gauge string where the parity loop is negative.

[0272] Note that actual removal of atoms is not necessary: ​​detunement

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[0273] boundary phase diagram

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[0274] Simply terminating the lattice and preserving all Hamiltonian terms that fit the remaining geometric structure tends to stabilize the m-boundary. Indeed, boundary dimers experience less repulsion, so it is preferable that they align in a classical pattern with some variation to give long-range order to the diagonal string operator P. To stabilize the e-boundary condition, we need to increase such boundary variation. One way to do this is by varying the detuning δ along the boundary site to find the suspended sweet spot where the dimer is suspended between two classical (empty or filled) arrangements.

[0275] We numerically determined the obtained boundary phase diagram for a blockade model on an infinitely long strip geometric structure, where we selected the bulk deep in the spin liquid with δ / Ω = 1.7. The results are shown in Figure 36. Consistent with the above prediction, we found that before changing the boundary detuning, i.e.

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[0276] Topological degeneracy on a surface With the knowledge of the boundary phase diagram described above, constructing a rectangular geometric structure with topological ground state degeneracy is now straightforward. A schematic diagram is shown in Figure 37A: a square slab with four alternate e- and m-condensates. One way to understand this twofold degeneracy is as follows: one can imagine extracting a single e-anyon from the top boundary (which is ultimately an e-condensate), dragging it through the unconfined bulk, and placing it at the bottom boundary. Similarly, the same can be done for the m-anyon from left to right. Due to the mutual statistics of e and m, these two processes are anti-exchange, implying degeneracy.

[0277] Referring to Figure 37, topological degeneracy in planar geometric structures is illustrated. (a) Alternating e- and m-condensate boundaries implies a doubling degeneracy. One way to understand this lies in the Majorana 0-mode (red dot) term at the point where the boundary conditions change; for the comprehensive nascent fermion parity which must be 1 (unity), only these four Majorana modes result in a doubling degeneracy. If we label the states using a P-string connecting the left and right boundaries, pulling an e-anyon from one e-condensate boundary to another effectively retains the state of this two-level system. (b) Ring-shaped geometric structures with m-condensate boundaries also have a doubling degeneracy. Moving an e-anyon around a hole retains the state. Since e-anyons can only be produced in pairs, there is another e-anyon (not shown) that we do not move.

[0278] Here we address how to physically label this two-level system, or equivalently, how to read out a given state. If the spin liquid is in a fixed-point limit similar to the trick code, then the topological string operators P and Q are exactly symmetric to the model. That is, locally

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[0279] It is worth noting that, unlike the numerator of Equation 11, the denominator does not depend on the logical state of the system, and therefore only needs to be determined once for any particular architecture. To understand this, degeneracy could be interpreted as the result of moving m- or e-anyons between corresponding condensed boundaries, which can be exchanged with pairs of topological string operators.

[0280] Referring to Figure 38, the topological ground state readout is illustrated. We consider a blockade model for δ / Ω = 1.7 for a finite sample with an open boundary, as shown. Furthermore, we perform laser detuning on the vertex and bottom boundaries.

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[0281] To demonstrate that this procedure is significant and well-defined, we consider a simulated example shown in Figure 38. The top and bottom boundaries are synchronized to be e-condensed, using the boundary phase diagram in Figure 36.

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[0282] Therefore, we have a way to label and read out our topological quantum states. Here we consider the question of initialization. We,

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[0283] In the sequence described above, we also show a parity string whose value does not change throughout this process so as to achieve |0>. To initialize to |1>, here we can use the fact that we know how to immobilize the e-anyon. Thus we can dynamically change the detuning, as shown in Figure 37A, to pull the e-anyon out of the top e-condensation boundary and move it to the bottom e-condensation boundary. This is logical.

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[0284] The above steps can be repeated for the ring-shaped alternative architecture shown in Figure 37B. In particular, in this case, the logical states are retained by weaving e-anyons around m-condensation holes. More generally, multiple e- and m-condensation holes can be generated in a given sample. Weaving these (by dynamically changing the parameters of the Hamiltonian) potentially provides another handle to the topological processing of quantum information.

[0285] Outlook We use a lyde-belli blockade on a ruby ​​lattice,

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[0286] Specifically, the theoretical predictions outlined above can be investigated using a programmable quantum simulator based on an array of neutral atoms. In particular, the required atomic alignment can be achieved using the atomic sorting techniques shown, while the relevant effective blockade range can be readily implemented using laser excitation to Rydberg states with large principal quantum numbers of 60 < n < 100. In the design of an appropriate atomic array, care should be taken that the atomic separation and the careful selection of Rydberg states are made to avoid molecular resonances that can modify the blockade constraints. The spin liquid phase can be generated via an adiabatic sweep of the laser detuning that starts from a disordered phase to a desired value of positive detuning, as previously shown for one - and two - dimensional systems. For typical parameters corresponding to effective Rabi frequencies within the range of a few MHz, such an adiabatic sweep can be potentially implemented with minimal decoherence in systems exceeding 200 atoms. We note that topologically ordered states are separated from trivial product states by a preferred single continuous transition for preparation. Several tools can be developed to identify and probe the transition to a spin liquid state lacking local order parameters. The transition point can be identified by measuring the filling fraction (see Fig. 25), while more detailed investigations can be made by measuring the predicted values of parity operators (Fig. 28) associated with various loops. Notably, both the P and Q operators can be efficiently measured either by directly analyzing single - shot images or, after this analysis, by performing qubit rotations in the dimer basis associated with individual triangles. The latter can be achieved by using resonant atomic driving with appropriately chosen parameters. Furthermore, the topological entanglement entropy can potentially be obtained by measuring the second Renyi entropy for different regions. Together with the control over boundaries and the investigation of samples with non - trivial topology, these methods constitute a unique opportunity for a detailed investigation of spin liquid states with a precision and sophistication not accessible by any other existing approach.

[0287] Furthermore, this work opens up several very interesting avenues that can be explored within the framework presented here. These range from investigating the non-equilibrium dynamical properties of spin liquid states in response to rapid changes in various Hamiltonian parameters to the experimental realization and detection of anyons using simple statistics. In particular, anyon blades can be investigated using time-varying local potentials. Moreover, approaches to improve the stability of TQL and the realization of more unusual spin liquid states can potentially be realized by further engineering modifications of the interaction potential, for example, using long-lasting hyperfine atomic states. In particular, approaches involving optical lattices and Rydebell dressings could be investigated to realize a wider range of spin liquid states. Finally, we note that the blockade model is essentially an Ising model on a ruby ​​lattice. Such a model could be implemented in other ways, e.g., arrays of superconducting qubits, magnets with strongly anisotropic exchange, or perhaps even in recently developed two-dimensional materials. Potentially, these systems could be used to realize topologically protected qubits, with the aim of developing novel, robust approaches to manipulating quantum information.

[0288] numerical details In this work, we consider two types of cylinders in the Kagome lattice, denoted as XC or YC. Considering the Kagome lattice shown in Figure 24, the XC cylinder has its infinite orientation along the x-axis, while for the YC cylinder, this is along the y-axis (and in both cases, 'C' simply stands for 'cylinder'). Consequently, we understand that the finite periodic orientation of the YC cylinder runs along the bonds of the Kagome lattice.

[0289] We performed DMRG simulations using the Open Access Tensor Network Python (TeNPy) package developed by Johannes Hauschild and Frank Pollmann, version 0.7.2. While DMRG is a method for one-dimensional systems, it can be used for cylindrical geometric structures by meandering through the system (i.e., giving all parts a one-dimensional label). The damage paid for this is that couplings used to be close in a two-dimensional geometric structure typically become a wider range of couplings in an effective one-dimensional label. To obtain the ground state, we start with a low bond dimension, e.g., χ=100 or χ=200, and iterate through DMRG for continuously large values ​​of χ until we see that the physical observable no longer changes. For most plots in this work, χ=1000 is sufficient, but in certain cases, we reached χ=2000. As a further soundness check that the bond dimensions were chosen to be large enough to accurately encode ground state physics, density at sites that are equivalent on the cylinder but not equivalent in effective one-dimensional labeling. <n>Considering this is very useful: if χ is too low, their predictive values ​​typically do not match; the densities on such a site match only if the ground state converges exactly to the ground state on a two-dimensional cylinder. Therefore, this is a very powerful indicator of convergence.

[0290] For a system on an infinitely long cylinder, we used a translation-invariant ansatz consisting of a specific number of rings. If this number is chosen to be so small that it cannot fit a particular VBS pattern, this problem reveals that the DMRG cannot converge to a stable state (and often this leads to a large norm error due to the tendency to form cat states). In such cases, the number of independent rings was increased until the state converged. This is how we found the VBS phase in Figure 24. Often, this VBS phase can fit into a local minimum: for example, we also found a ground state in Figure 24 where the two pin gears have opposite directions. For this reason, we found that starting with various distinct initial states, for the VBS patterns shown in Figure 24, an overall minimum occurs. If the phase was trivial or a spin liquid phase, we found that the single-ring hypothesis is sufficient to obtain a converged state, except for the YC-6 geometric structure with the Lieb-Schultz-Mattis anomaly (where entanglement entropy appears in Figure 26) (although we confirmed that the results do not change when the number of rings is increased), and here the two-ring hypothesis was necessary even for the trivial and spin liquid phases. We also found that the correlation length ξ is obtained by a standard MPS procedure: the absolute value of the second largest eigenvalue is obtained when the transition matrix is ​​diagonalized and standardized so that its largest eigenvalue is 1 (unity).

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[0291] Referring to Figure 39, a coupling graph for van der Waals interactions on a ruby ​​lattice is provided. The black dots indicate ruby ​​lattices with ρ=3 (see also Figure 31C). Each line is included in the numerical values ​​for the phase diagram in Figure 32.

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[0292] We considered a model on a ruby ​​lattice with long-range van der Waals interactions, specifically with ρ=3 and blockade radius R. b For a ruby ​​lattice with =3.8a, the data in Figures 32 and 33 are for r≦9a.

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[0293] Scaling of the Fredenhagen-Marcu order parameter In Figure 28, we show the FM string order parameter for n=2.

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[0294] Referring to Figure 40, the FM string order parameters in the blockade model are illustrated. (a) The grid shown is an XC-12 cylinder (periodic along the vertical direction, infinite along the horizontal direction). The blue line represents the FM order parameters.

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[0295]

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[0296] Referring to Figure 41,

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[0297] Topological ground states on a torus The ground state on a torus geometric structure can first be approximated by using DMRG to obtain a ground state on an infinitely long cylinder, and then (by identifying appropriate actual indicators) the obtained matrix product state wavefunction on the torus can be simply evaluated. This can further be used to construct, more precisely, the least entangled state (MES) on the torus. The topologically distinct ground state observed by DMRG on an infinitely long cylinder is naturally an MES, and if the finite-size effect is small, this gives the MES on the torus.

[0298] However, this does not have to be true if the finite-size effect is strong enough to induce actual anyon loops that wrap around the torus. For specificity, we represent the direction along the circumference of the cylinder as 'vertical' and the infinite direction along the cylindrical axis as 'horizontal'. If we place this wavefunction on a torus (i.e., the horizontal direction is finite and periodic), the actual anyon fluctuations can wrap around the horizontal direction and connect separate topological sectors. This means that the resulting state is no longer an MES.

[0299] To make this more precise, it is useful to characterize the MES as a state that is an eigenstate of a topological line operator along the vertical direction.

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[0300] In the blockade model, we,

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[0301] In conclusion, we:

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[0302] For these MES, in Figure 33 we plot the overlap after π / 3 rotation.

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[0303] The duality between topological string operators Here we prove equation 10. To do this, we first mark the four reference states in a single triangle as follows:

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[0304] Next, the P and Q string operators (defined in Figure 27A) are:

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[0305] Since the Hamiltonian defined in Equation 9 does not connect separate triangles, it is sufficient to prove the requirement for a single triangle. Then, Equation 9 is,

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[0306] Therefore, the time evolution operator is,

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[0307] Next,

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[0308] Example B: Investigation of topological spin liquids on a programmable quantum simulator Quantum spin liquids, unusual phases of matter possessing topological order, have been a major focus of exploration in physical science over the past few decades. Such phases are characterized by long-range quantum entanglement, which could potentially be leveraged to enable robust computer calculations. We use a 219-atom programmable quantum simulator to investigate quantum spin liquid states. In our approach, arrays of atoms are arranged on a kagome lattice, and evolution under a lydebelle blockade generates frustrated quantum states lacking local order. The initiation of a model trickcode-type quantum spin liquid phase is detected by evaluating a topological string operator that provides a direct signature of topological order and quantum correlation. Its properties are further revealed by using an atomic array with non-trivial topology, representing the first step toward topological encoding. Our observations enable a controlled experimental exploration of topological quantum matter and protected quantum information processing.

[0309] Spurred on by the unrealistic theoretical work done over the past 50 years, extensive research is currently underway to identify the sign of quantum spin liquid (QSL) in novel materials. Furthermore, influenced by intriguing predictions from quantum information theory, techniques for engineering such systems for topological protection of quantum information are being actively explored. Systems with failures caused by lattice geometry or long-range interactions offer promising avenues for studying QSL. In particular, such systems can be used to implement a class of so-called dimer models, which are the most promising candidates for accepting quantum spin liquid states. However, realizing and investigating such states is difficult because they are often surrounded by other competing phases. Moreover, in contrast to topological systems involving time-reversal symmetry breaking, such as in the fractional quantum Hall effect, these states cannot be readily investigated via, for example, quantized conductance or edge states. Instead, access to nonlocal observables, such as topological string operators, is essential to diagnose the spin liquid phase. While some indications of the QSL phase in correlated materials have been reported previously, these unusual states of materials have so far avoided direct experimental detection.

[0310] Programmable quantum simulators are well suited to the controlled exploration of these strongly correlated quantum phases. In particular, recent work has shown that various phases of the quantum dimer model can be efficiently executed using Rydberg atom arrays and that trick-code type dimer spin liquid states can potentially be generated in certain failed lattices. We note that trick-code states were dynamically generated in small systems using quantum circuits. However, some important properties, such as topological robustness, are difficult to realize in such systems. Spin liquids have also been investigated using quantum annealers, but the lack of coherence in these systems has ruled out observations of quantum properties.

[0311] Dimer Model in Rydberg Atomic Arrays. A key idea in our approach is based on the correspondence between Rydberg atoms arranged on a Kagome lattice (or equivalently a portion of a ruby ​​lattice) linkage, as shown in Figure 42A, and a dimer model on the Kagome lattice. Rydberg excitations can be seen as “dimer couplings” linking two adjacent vertices of the lattice (Figure 42B). Due to the Rydberg blockade, strong and precisely tuned interactions constrain the excitation density such that each vertex is in maximum contact with one dimer. At 1 / 4 packing, each vertex is in contact with exactly one dimer, resulting in a complete dimer coverage of the lattice. Smaller packing densities result in a finite density of vertices without a proximal dimer, called monomers. A quantum spin liquid can emerge within this dimer-monomer model near 1 / 4 packing and can be seen as a coherent superposition of exponentially many degenerate dimer coverages and small mixtures of monomers (Figure 42C). This corresponds to a resonance valence bond (RVB) state, which has long been predicted in any experimental system but has remained unnoticed until now.

[0312] Referring to Figure 42, a dimer model in a Rydberg atom array is illustrated. Figure 42A shows fluorescence images of 219 atoms arranged on a Kagome lattice linkage. Atoms initially in the ground state |g> evolve according to many-body dynamics U(t). The final state of the atoms is determined by the fluorescence image of the ground state atoms. Rydberg atoms are marked with red dimers on the Kagome lattice links. In Figure 42B, we set the blockade radius R by selecting Ω = 2π × 1.4 MHz and a = 3.9 μm. b Since we adjust / a=2.4, the nearest neighbor of all 6 atoms in |r> is the blockade radius R. b It is located within. Then, the state that coincides with the Rydberg blockade at maximum packing can be seen as a dimer coating of the kagome lattice, where each vertex is in contact with exactly one dimer. In Figure 42C, the quantum spin liquid state corresponds to a coherent superposition of exponentially many dimer coatings. In Figure 42D, the detuning Δ(t) and Rabi frequency Ω(t) are used for quasi-adiabatic state preparation. In Figure 42E, Rydberg excitations are present in most of the system, except for the three outer layers (top). <n>The average density. (Below) The probability of empty vertices, single dimers, or double dimers (weakly obstructing blockades) within a bulk (monomer).

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[0313] Referring to Figure 43, the detection of the dimer phase via the diagonal string operator is shown. In Figure 43A, the Z-string operator measures the parity of dimers along the string. In Figure 43B, since a complete dimer coverage always has exactly one dimer in contact with each vertex of the array, around a single vertex... <z>= -1, and for larger loops

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[0314] To experimentally generate and test such a state, we used 219 individual optical tweezers, as shown in Figure 42A, which were placed on a linked Kagome grid. 87 A two-dimensional array of Rb atoms is used. The atoms are initialized in their electronic ground state |g> and coupled to a Rydberg state |r> via a two-photon optical transition with a Rabi frequency Ω. The atoms in the Rydberg state |r> are subjected to a strong van der Waals potential.

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[0315] To explore the many-body phase in this system, we utilize a quasi-adiabatic evolution by slowly turning on the Rydberg coupling Ω and then using a third-frequency sweep over approximately 2 μs to change the detuning Δ from negative to positive (Figure 42D). We stop the third-frequency sweep at different endpoints and first induce the Rydberg excitation. <n>We measure the density of the Rydberg atoms. Away from the array boundary (which produces an edge effect that permeates the bulk across just two layers), we observe that the average density of Rydberg atoms is uniform across the array (see Figure 49). Focusing on the bulk density, we see that

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[0316] Measurement of Topological String Operators. Defining the properties of a phase with topological order means that it cannot be examined locally. Therefore, to investigate the possible existence of QSL states, it is essential to measure topological string operators, similar to those used in trick code models. In this model, there are two such string operators: the first features a valid dimer description, and the second investigates quantum coherence between dimer states. We first measure the parity of Rydberg atoms along strings S perpendicular to the couplings of the Kagome lattice.

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[0317] About the adult loop <z>To measure (Figure 43C), we evaluate the string observability directly from single-shot images and average it over many experimental replicates and over all loops of the same shape in the bulk of the grid.

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[0318] Referring to Figure 44, it is illustrated that we can examine the coherence between dimer states via the off-diagonal string operator. Figure 44A shows the definition of the X string operator on a single triangle of the Kagome lattice. Figure 44B shows on an arbitrary closed loop, <x>We show that the X operator maps any dimer coating to another reliable dimer coating, as does the coherence between pairs of dimer configurations. In Figure 44C, the X operator is measured by evolving the initial state under a Hamiltonian (Equation 22) with Δ=0 and a reduced blockade radius, enclosing only the atoms within each individual triangle, and performing a reference rotation that maps X to Z. Figure 44D shows that in the experiment, after state preparation, we laser detuning

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[0319] Next, we investigate the quantum coherence properties of the prepared state. For this purpose, we consider an off-diagonal X operator acting on strings along the couplings of the Kagome lattice. This is defined in Figure 44A by its action on a single triangle. Applying X to an arbitrary closed string maps dimer coatings to other reliable dimer coatings (see Figure 44B for a loop around a single hexagon, e.g.). Thus, a finite expectation value for X implies that the state contains a coherent superposition of one or more pairs of dimer states coupled by a specific loop that is pre-required for the quantum spin liquid. Measurement of X can be performed by performing a collective reference rotation illustrated in Figure 44C. This rotation has Δ=0 and a reduced blockade radius so that only atoms within the same triangle are subject to the Rydebelly blockade constraint.

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[0320] Experimentally, the reference rotation is Δ of the laser detuning. q Quench to =0, increase laser intensity by approximately 200 times, and set blockade radius R b The state adjustment is performed by reducing / a to 1.53 (Figure 44D). We adjust the state with Δ / Ω=4 and calibrate τ by evolving under the quench Hamiltonian over the time of variability. We measure the parity of the Z string, which is doubled relative to the target X loop, and observe a sharp recovery of the parity signal at τ approximately 30 ns (Figure 44E). With the quench time τ fixed, we examine the detuning Δ for different values ​​at the end of the third sweep. <x>We measured (Figure 44F) and observed a finite X parity signal for loops extending over large portions of the array. In light of experimental imperfections, we emphasize that the observation of finite parity for string observables of up to 28 atoms in the experiment, with lengths of μs, is rather remarkable. These observations clearly indicate the presence of long-range coherence in the prepared state.

[0321] Referring to Figure 45, the string order parameters and quasiparticle excitations are illustrated. Figure 45A shows the open string operator acting on the dimer state |D>.

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[0322] Investigation of spin liquid properties. Testing of closed string operators shows that we prepare approximate dimer phases with quantum coherence between dimer coatings. These closed loops exhibit topological order, but it is important to distinguish between topological effects and trivial ordering by comparing their properties with those of open strings, the former being sensitive to the topology of the loops. This comparison is shown in Figures 45D,E, which show several distinct regions. For small Δ, we show that the loop parity of both Z and X can be factored into the product of the parity on the half-loop open string, in particular for finite... <z>However, we find that this is a minor result of low-density Rydberg excitation. In contrast, the dimer phase (

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[0323] Figures 45F and 45G show the measured values ​​of these order parameters.

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[0324] at the same time,

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[0325] The measured values ​​of the closed-loop operator in Figures 43 and 44 are: | <z> |,| <x>The fact that |<1 and the decrease in signal amplitude with loop size indicate that this arises from a finite density of quasiparticle excitations. Specifically, defects in dimer coatings such as monomers and double dimers can be interpreted as electron (e) anyons in the language of lattice gauge theory. The presence of defects inside a closed loop changes the sign of Z, so the parity on the loop is,

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[0326] A second type of quasiparticle excitation that can arise in this model is the so-called magnetic (m) anyon. Similar to the e-anyons present at the endpoints of the open X-string (Figure 45A), the m-anyons are generated by the open Z-string, and they correspond to phase errors during dimer coating (Figure 55). These excitations cannot be directly identified from individual snapshots, but they are detected by measurements of the closed X-loop operator. The prominent circumferential scaling observed in Figure 45I indicates that the m-anyons arise only in pairs with short correlation lengths. These observations highlight the potential of using topological string operators to detect and investigate quasiparticle excitations in this system.

[0327] Referring to Figure 46, the topological properties of an array with holes are illustrated. Figure 46A shows a lattice with a non-trivial topology obtained by removing three atoms at the center to create a small hole. The dimer state can be divided into two distinct topological sectors 0 and 1. The Z strings connecting the holes to the boundary always have well-defined expected values ​​within each sector and opposite signs between the two sectors, although the correlation between the two such strings Z1 and Z2 is identical for both sectors. Figure 46B shows the operator Z as defined in the inset. L and X L This shows the expected value measured for, which is in the QSL region (shaded region) that we |+> state ( <X L A superposition state of two topological sectors having a finite overlap with > >0) <Z L This shows that we prepare (>=0). Figure 46C shows the finite expectation value for the correlation between pairs of hole-to-boundary Z-strings (inset) and is consistent with Figure 46A.

[0328] We move toward a topological qubit. To further investigate the topological properties of the spin liquid state, we generate an atomic array with a small hole by removing three atoms on a central triangle, which generates a valid internal boundary (Figure 46). This gives rise to two distinct topological sectors for the dimer coating, where states belonging to different sectors can only be converted to each other via a large X-loop surrounding the hole, constituting a highly nonlocal process (involving at least 16-atom resonance) (Figure 57). We define the logical state |0 as a superposition of all dimer coatings from topological sectors 0 and 1, respectively. L >and|1 L > defines these. These are opposite for two sectors but have sufficiently defined eigenvalues ​​±1 for all dimer states in the same sector, so any Z connecting the hole and the outer boundary L Logical operators proportional to string operators

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[0329] After the same quasi-adiabatic preparation as in Figure 42D, we placed Z on the string defined in the inset in Figure 46B. L and X L We measure the initiation of the QSL phase and related

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[0330] Discussion and Outlook. Note that it is not possible to classically simulate quantum dynamics for a sufficient experimental system, and we compare our results with several theoretical approaches. We first note that our observations do not qualitatively agree with the ground state phase diagram obtained from density matrix renormalization group (DMRG) simulations on an infinitely long cylinder. For the largest accessible system size, including van der Waals interactions only up to an intermediate distance (about 4a), we consider the ground state

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[0331] To gain further insight, we perform time-dependent DMRG calculations simulating the same state preparation protocol as in our experiment on an infinitely long cylinder with a circumference the length of seven atoms. The results of these simulations qualitatively agree well with our experimental observations (see Figure 63). Specifically, as with the results in Figure 45, we perform region

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[0332] Our experiments offer unprecedented insights into elusive topological quantum materials, opening up several new directions where these tests can be extended to improving the robustness of QSLs by using modified lattice geometry structures and boundaries, and optimizing state preparation to minimize quasiparticle excitations; understanding and mitigating environmental effects associated with phase shifts and spontaneous emission, for example; and optimizing string operator measurements using potentially associated quasi-local transformations with the aid of quantum algorithms. Simultaneously, hardware-efficient techniques for robust operation and braiding of topological qubits can be investigated. Furthermore, methods for any-on trapping and annealing can be investigated with a final application toward fault-tolerant quantum information processing. Improved programmability and control allow for the effective implementation of a broader range of topological quantum materials and lattice gauge theories, opening the door to detailed exploration of them under controlled experimental conditions and providing novel pathways for the design of quantum materials that can complement precisely solvable models and classical numerical methods.

[0333] Experimental system Our experiment uses the second-generation atomic array configuration described in Ebadi, et al., Quantum Phases of Matter on a 256-Atom Programmable Quantum Simulator (https: / / arxiv.org / abs / 2012.12281). In our experiment, atoms are in the ground state 5S. 1 / 2 From intermediate state 6P 3 / 2 Up to 420nm laser and intermediate state to Rydberg state 70S 1 / 2 The Rydberg state is excited using a two-photon excitation scheme consisting of a 1013nm laser up to [a certain point].

[0334] In this operation, we used a laser with an intermediate 6P. 3 / 2 The state is adjusted to have a detuning of δ = 2π × -450 MHz, where the 420 nm laser is detuned red from the intermediate state. The 1013 nm laser is always applied at maximum optical power (approximately 3 W total for atoms) and the single-photon Rabi frequency Ω 1013 This produces 2π × 50MHz. The 420nm laser output varies depending on the protocol. During the quasi-adiabatic preparation of the dimer phase, we apply 420nm light at low power, which reduces the two-photon Rabi frequency and thus the target blockade radius R. b / a increases up to 2.4. This low power setting consists of a total of approximately 0.5 mW for an atom and the single-photon Rabi frequency Ω 420 = 2π × 25MHz. Therefore, during the quasi-adiabatic preparation, we

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[0335] Referring to Figure 47, a quasi-adiabatic state preparation is illustrated. Ω(t) and Δ(t) are used for state preparation. To examine the phase diagram at different Δs, we stop the tertiary sweep at different endpoints and turn off Ω accordingly.

[0336] To measure the X operator, after dimer phase preparation, we apply a short quench with significantly higher blue power. This high-power setting corresponds to a maximum power of approximately 100 mW for the atom and the single-photon Rabi frequency Ω 420 This corresponds to 2π × 360 MHz. The corresponding two-photon Rabi frequency is Ω = 2π × 20 MHz, and R b / a = 1.53. In this configuration, the 420nm laser introduces a substantially larger optical shift for a 2π × 36MHz Rydberg transition. To avoid a systematic offset in effective detuning from resonance, we calibrate the resonance conditions separately for both low and high power. The 420nm laser amplitude is controlled using a double-path AOM with a rise time of approximately 10 ns. In the ideal model for quenching, the optimal quench time is given for high-power Rabi frequencies.

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[0337] Throughout this process, Z and X parity measurements are averaged across the system over identical loops, including reflection and rotational symmetry. However, loops touching the edges of the system are excluded to avoid boundary effects. Error bars are used.

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[0338] Reference rotation for X and Z parity loops A reduced blockade radius is applied to the reference rotation used to measure the X parity loop, which, in the ideal limit, eliminates interactions between different triangles while maintaining rigid blockade constraints on Rydberg excitations within a single triangle. Thus, the rotation can be understood by its action on individual well-occluded triangles.

[0339] The Hilbert space for each triangle is four-dimensional, allowing either a Rydberg excitation of 0 or a Rydberg excitation on any of the three connected states.

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[0340] The reference rotation is shown in Figure 44C, which relates the X and Z parities under this evolution by Quench-Hamiltonian equation 23 and can be proven directly by computer calculation. Here we offer an alternative derivation. First, we take two edges on the triangle defined in Figures 43, 44 (

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[0341] The X and Z parity operators can be diagonalized relative to each other by changing to an appropriate symmetrized criterion: [Table 1]

[0342] In this standard, Quench-Hamiltonian equation 23 is:

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[0343] This Hamiltonian generates a ring permutation between the base states |0>, |1>, and |2> while keeping |3> invariant. The permutation |0>→|1>→|2>→|0> is, for each initial state,

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[0344] We further see that this relationship is also maintained for parity operators defined on other faces of the triangle, for example.

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[0345] Refer to Figure 48, which illustrates double Z and X loops. An example of a double Z loop (dashed line) relative to a closed X loop (solid wave line).

[0346] Supplemental experimental data Mean Rydberg density and boundary effects

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[0347] Referring to Figure 49, the average Rydberg density for each site is illustrated. For both the complete array (Figure 49A) and the array with holes (Figure 49C), we show the average Rydberg excitation density for each site in the dimer phase with Δ / Ω=4. <n>We measure this. (Figure 49B, D) Next, we plot the average density corresponding to each layer as a cross-section from the edge into the bulk, which is within the outer 2-3 layers, the bulk is

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[0348] Lack of spatial order in the spin liquid phase The lack of spatial order within the spin liquid phase is a key feature that separates this phase from any possible nearby solid phase. At its simplest level, spatial order can be assessed by observing individual projection measurements of atomic states within the set. We present three examples of snapshots taken in Figure 50, where the measured states of individual atoms are represented as small circles on a packed or unpacked kagome lattice linkage, each representing either a Rydberg state or a ground state, respectively. In mapping to the monomer-dimer model, we can alternatively consider the vertices of the kagome lattice in relation to how many adjacent Rydberg excitations (dimers) are present. In practice, a vertex can have zero attached dimers (so-called monomers), a single attached dimer (corresponding to an ideal dimer covering), or many attached dimers (interfering with the long-range blockade constraint). In Figure 50, we further color each vertex according to the number of such attached dimers. The extensive abundance of vertices linked to a single dimer (snapshots in Figures 42E and 50) indicates the occupation of the dimer phase.

[0349] Refer to Figure 50, which provides a snapshot of the dimer phase.

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[0350] Referring to Figure 51, the density correlations between individual Rydberg excitations are illustrated. We have the Rydberg density-density correlation between the central atom and all the other atoms in the system.

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[0351] Furthermore, spatial correlations can be used to search for spatial order in solids (Figure 51). We measured Rydberg density-density correlations on atomic arrays and found non-vanishing correlations for atoms within a single triangle or between adjacent triangles, while correlations vanish over longer distances. This observation confirms the lack of spatial order in the dimer phase we prepared.

[0352] Quench phase dependence The quench, which induces a reference rotation for measuring X parity, is used to detune the laser after the preparation of the dimer phase. q To quickly switch to =0 and simultaneously change the phase of the laser field

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[0353] Referring to Figure 52, the phase dependence of the quench is illustrated. (Figure 52A) After preparing the dimer phase with Δ / Ω=4, we perform a quench with a variable quench phase for a pre-calibrated time τ and measure the resulting X-loop parity around a single hexagon. (Figure 52B) For the fixed quench phase φ=π / 2 or φ=0, we measure the X-parity after a pre-calibrated quench time as a function of the final detuning of the third sweep. The data for φ=π / 2 is reproduced from Figure 44F.

[0354] Phase change is a process that lasts for a period of time φ.

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[0355] Z parity measurement with improved conditioning All data presented in the above discussion are taken for two-photon Rydberg excitations with intermediate detuning δ = 2π × -450 MHz. This choice allows for our maximum dynamic range of Rabi frequencies, which is important because we can perform state preparation at a low Ω and then apply a quench at a larger Ω to the reduced blockade radius. Larger intermediate detuning requires performing state preparation at an even lower initial Rabi frequency, where we observe worse results. However, small intermediate detuning introduces stronger decoherence due to increased spontaneous emission from the intermediate state. To supplement these results, we further perform state preparation and measure Z parity at increased intermediate detuning δ = 2π × 1 GHz. To further optimize this state preparation, we use a larger Rabi frequency Ω = 2π × 1.7 MHz and a smaller lattice spacing a = 3.7 μm, which should improve adiabaticity during preparation. In this configuration, we actually observe a larger Z-loop parity (Figure 53), but we are unable to measure the corresponding X-loop parity. This highlights that the large dynamical range required to measure the X operator is one of the main technical challenges of this experimental work. At the same time, this suggests that the quality of state preparation can be improved by working with this increased intermediate detuning and higher Rabi frequency using more available laser power for Rydberg excitations (and using smaller lattice spacing to achieve the same blockade radius).

[0356] Referring to Figure 53, the Z-loop parity with improved state conditioning is illustrated. We measure Z on the closed loop using a larger Rabi frequency during state conditioning, with a larger intermediate state detuning for two-photon Rydberg excitations to reduce the spontaneous emission rate. We observe a larger parity than that shown in the equivalent Figure 43.

[0357] Correlation between parity loops We use the string operator in this work to evaluate long-range topological order. However, the large loop under test can be decomposed into a product of smaller loops around a subdomain: for example, an X-loop can be decomposed into a product of enclosed hexagons. To demonstrate that the parity measured on the large loop actually exhibits long-range order, we extract correlations between separate parity loops containing larger loops, rather than those that emerge individually from the ordering of each hexagon.

[0358] We first examine the parity loops surrounding adjacent hexagons in the Kagome lattice. The smallest X-parity loop is exactly equal to the product of the parities around the two enclosed hexagons. The connected correlation field of the parities around these two interior hexagons is:

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[0359] Similarly, the Z-loop enclosing two hexagons can be decomposed into the product of the Z-parities around the two hexagons, and further multiplied by the parity around the central interior vertex (which should always be -1 in dimer covering). We have two analogous hexagonal connected correlations with respect to Z.

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[0360] The higher-order connected correlations between the three adjacent hexagons forming the triangle further emphasize non-local correlations in this system. We obtain a connected three-point correlation body by subtracting contributions from the underlying two-point correlation.

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[0361] As shown in Figure 54, we observe linked correlations between two non-zero hexagons and three hexagons within the dimer phase region, indicating that the parity measured on the double and triple hexagon loops does not emerge independently of the determined parity around their respective internal sub-regions, but instead emerges due to non-trivial correlations over longer scales.

[0362] Referring to Figure 54, the correlations between parity loops are illustrated. We measure the two-point and three-point connected correlations between parities around adjacent hexagons. (Figure 54A) Z-parity correlations between loops surrounding pairs and triplets of adjacent hexagons. (Figure 54B) X-parity correlations between pairs and triplets of adjacent hexagons.

[0363] Quasiparticle excitation Within the dimer-monomer model, two types of quasiparticle excitations are generated by the application of open X and Z strings: these are electron (e) and magnetic (m) anyons, respectively. The open X strings generate monomers (or double dimers) at their endpoints, and thus the e-anyons are identified as defects in the dimer coating.

[0364] Referring to Figure 55, a magnetic anyon is illustrated.

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[0365] On the other hand, the open Z-string gives relative phase between various dimer configurations, corresponding to the m-anyon. To understand the m-anyon, we first note that all dimer coverings in the QSL superposition are related to one another by the application of a appropriately selected closed X-loop (first row of Figure 55). The open Z-string then applied to the QSL state results in different dimer coverings that acquire a ± phase factor depending on the number of dimers crossed by the string. Whenever two dimer configurations are related by a closed X-loop surrounding one of the Z-string endpoints, they acquire opposite signs (Figure 55). Then, after the application of the open Z-string, around the endpoint of the open X-string (defect) <z>As with how it is reversed, <x>This is inverted for any closed loop around one endpoint of the Z string. Since the open Z string terminates within a hexagon of the Kagome lattice, we associate the resulting magnetic (m) anyions as they lie on these hexagons, and thus the X parity around the hexagons detects the presence of m-anyon excitations.

[0366] In Figure 56, we report the Z and X loop parity, which are redesigned to the Area Law and circumferential Law for different values ​​of Δ within the relevant range of detuning. We observe that the excellent circumferential Law scaling for X reported in Figure 45I extends over the entire range of Δ. For Z, instead, we find that the initial approximate Area Law scaling converges to the circumferential Law for larger loops.

[0367] Referring to Figure 56, the proportion of Z and X parity with respect to loop size is illustrated. We redesign the parity for different loop sizes in (Figure 56A).

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[0368] By comparing the scaling behavior observed in experiments with the scaling predicted from theory, we can elucidate the scaling behavior observed in experiments. We first note that the general equilibrium conjecture for both string operators is circumferential scaling. This can be seen as a result of the mutual statistics of e- and m-anyons: since there are de facto fluctuations in both anyons, these induce correlations for other types of anyons, resulting in the circumferential law. This general conjecture for the circumferential law is well known in the (lattice) gauge theory community and can be associated with the phenomenon of string breaking. Experimentally, we observe the circumferential law for X-loops and the (approximate) area law for Z-loops (with substantial deviations for larger loop sizes). This can be understood by noting that we input a QSL-like state from a trivial phase, which can be interpreted as a condensation of e-anyons (i.e., both closed and open X-strings give a non-zero correlation): thus the circumferential law for closed X-strings already exists in the trivial phase and naturally remains in the QSL-like state (the correlation for open X-strings disappears). In contrast, Z-correlations are absent in the trivial phase closest to QSL: they arise only at quantum critical points, and since we sweep this at a finite rate, Z-loop correlations arise only for characteristic length scales, which implies the Elia law. Numerically, we actually confirm that Z-loop correlations are significantly increased as we increase the preparation time, which is consistent with our observation in Figure 53. We note that this imperfect generation of Z-loop correlations can be interpreted equivalently as generating a density of e-anyon excitations. Dynamically inducing the initiation of QSL and possible metastable states is a valuable phenomenon that deserves further detailed examination.

[0369] To further clarify this, we note that since the ground state has so-called 'virtual' fluctuations when it is not an idealized fixed-point state, the monomers (and double dimers) observed in the experimental snapshot do not need to directly correspond to physical excitations. These can be interpreted as correlated e-anyons. In contrast, the ideal

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[0370] Further data on arrays with non-trivial topologies The distinction between two distinct topological sectors can be better understood by looking at the transition graphs between pairs of dimer states. These are constructed by superimposing two dimer coverings and removing the overlapping dimer (Figure 57). If the remaining dimers form an odd number of closed loops around the hole, the dimer state belongs to the opposite topological sector, which represents the set of nonlocal moves required to transform one into the other.

[0371] Referring to Figure 57, the distinctions between topological sectors are illustrated. To determine whether the three dimer coverings |D1>, |D2>, and |D3> belong to the same or opposite topological sector, we use a transition graph.

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[0372] To demonstrate that the removal of three atoms at the center of the array generates an actual internal boundary, we measure the Z and X operators on a string with endpoints on both the internal and external boundaries (Figure 58). The relevant range of detuning (

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[0373] Referring to Figure 58, boundary-to-boundary string operators are illustrated. We measure the Z (Figure 58A) and X (Figure 58B) operators on open strings connecting two points on the outer (Figure 58C, D) or inner (Figure 58E, F) boundary of the array. Observing the same features for both, we confirm that a small central hole actually generates a valid inner boundary.

[0374] Numerical Test Below, we report a numerical examination of a Rydberg atomic array. We first discuss the 0-temperature equilibrium phase diagram established using the density matrix renormalization group (DMRG). Next, we directly simulate a quasi-adiabatic sweep using both exact diagonalization and dynamic DMRG calculations. To minimize boundary effects due to the limits of the numerically accessible system size, these calculations are performed on a torus (exact diagonalization) or on an infinite cylinder (DMRG).

[0375] Ground state phase diagram For the initial approximation, the Hamiltonian is the effective 'PXP' model.

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[0376] Here, P is the blockade radius R of site i. b This is the projection operator for |g> for all parts within the blockade radius. This model approximates the Rydberi-Miltonian by treating all pairwise interaction energies as either infinite if within the blockade radius or 0 if beyond. b Regarding =2.4a, this corresponds to blocking the first three interaction distances. This 'blockade model' is,

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[0377] To include sufficient van der Waals interactions, we use R b Cutting distance having = 2.4a

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[0378] Referring to Figure 59, ground state phase diagrams of the connected Kagome model for two section distances are provided. All data are taken with respect to the blockade radius R on the XC-8 cylinder. b This is for a van der Waals model with =2.4a. (Figure 59A~C) Section distance

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[0379] Referring to Figure 60, the ground state phase diagram of the connected Kagome model is provided.

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[0380] Regarding the intermediate section distance, we find spin liquids in the ground state phase diagram. In particular,

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[0381] However, we found that spin liquids are destabilized when longer-range interactions are involved:

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[0382] Numerical simulation of dynamic state adjustment Referring to Figure 61, the dynamic state preparation in the PXP model is illustrated. (Figure 61A) The lowest instantaneous eigenstates of the Hamiltonian of Equation 31 for 36 atoms on a torus. The colors indicate the set of states generated in real-time quench dynamics with the Hamiltonian parameters given in the inset (data is total sweep time).

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[0383] The detuning gradient Δ(t) used to generate various states is stimulated by the adiabatic principle. For sufficiently slow gradients, the system follows an instantaneous ground state adiabatically. Indeed, finite coherence time limits the maximum evolution time and requires a faster sweep than adiabatically. This is expected to induce a non-adiabatic process, especially near the critical point, where the finite-size gap is minimal.

[0384] Referring to Figure 62, the dynamic state adjustment in the van der Waals model is illustrated. The results are in R b This is for the XC-4 cylinder for =2.4a. As shown, the two rows correspond to two different section distances. For each panel, we show both ground state results (blue dashed lines, obtained by DMRG) and dynamic state preparation using the protocol in Figure 47 (red solid lines, obtained by time-dependent DMRG; the bright solid lines are for sweeps at half speed). For shorter section distances, the ground state accepts a spin liquid (blue shaded region). The diagonal loops around the hexagon are

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[0385] To gain an understanding of the quantum many-body states generated in such a quasi-adiabatic sweep, we consider the wave function.

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[0386] To further confirm this figure, we use an approach based on the matrix multiplication operator to show the van der Waals (1 / r 6 We also performed dynamic DMRG calculations for state preparation in realistic models with interactions. We consider an infinitely long XC-4 cylinder. For the XC-8 results reported above, a small section distance

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[0387] The results in Figure 62 suggest several notable points. First, as far as dynamic state preparation is concerned, the results for the two cross-section distances are very similar: state preparation appears to be less sensitive to longer-distance interactions that break the intermediate spin liquid in the ground state. Second, in both cases, the characteristics of the time-evolved state are such that in the spin liquid region

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[0388] Referring to Figure 63, a comparison between experimental results and numerical simulations of dynamic state preparation is provided. The experimental data (Figures 63A, C, E) are reproduced from Figures 42, 45, and (Figures 63B, D, F) we perform the simulation on an infinitely long cylinder (XC-4) with a circumference of 7 atomic lengths.

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[0389] All patents, published applications, and references cited herein are incorporated by reference in their entirety.

[0390] The descriptions of various aspects of this disclosure are provided for illustrative purposes only and are not intended to be exhaustive or limiting to the aspects disclosed. Many modifications and variations will be apparent to those skilled in the art without departing from the scope and spirit of the aspects described. The terms used herein have been chosen to best describe the principles of the aspects, their practical applications or technological advancements to the art available on the market, or to enable those else skilled in the art to understand the aspects disclosed herein.< / z> < / x> < / z> < / x> < / z> < / n> < / z> < / z> < / x> < / z> < / x> < / z> < / x> < / x> < / x> < / z> < / z> < / z> < / z> < / z> < / n> < / z> < / z> < / n> < / n> < / n> < / n> < / q> < / n> < / n> < / n> < / n> < / n> < / n> < / n> < / n> < / n> < / n> < / n> < / n> < / n>

Claims

1. A device including a two-dimensional array of particles, Each particle (303) corresponds to the aspect ratio of the quadrilateral portion of the ruby ​​lattice. [Math 1] Located at the vertices of a ruby ​​lattice (301) with a larger parameter ρ; Each particle has a first state and an excited state; Each particle belonging to the three unit cells of the ruby ​​lattice, when in an excited state, has a blockade radius (313) sufficient to block each of at least six nearest neighboring particles in the ruby ​​lattice from its transition from its first state to its excited state. The array is a device having at least one external edge configured to be in a first boundary condition.

2. The device according to claim 1, wherein the particles are atoms, ions, or molecules.

3. The device according to claim 1, wherein the blockade is a dipole blockade.

4. The device according to claim 1, wherein the blockade is a lydeberg blockade.

5. The device according to claim 1, wherein the particles are atoms, the first state is the ground state, and the blockade is a Rydebell blockade.

6. The device according to claim 1, wherein the array includes at least a first external edge (1601) and a third external edge (1602), each of which is subject to a first boundary condition, and at least a second external edge (1603) and a fourth external edge (1604), each of which is subject to a second boundary condition different from the first boundary condition.

7. The device according to claim 1, wherein the array has a plurality of external edges, each external edge is in either a first boundary condition or a second boundary condition, and each external edge is in a boundary condition different from any adjacent external edge.

8. The device according to claim 1, wherein the array includes at least one internal edge (1503).

9. The device according to claim 8, wherein each vertex enclosed by at least one internal edge is not occupied by a particle.

10. The device according to claim 9, wherein at least one internal edge has the same boundary conditions as at least one external edge.

11. The device according to claim 8, wherein at least one internal edge is on a first boundary condition different from at least one external edge.

12. The device according to claim 8, wherein the array has a plurality of internal edges, each internal edge enclosing a plurality of corresponding vertices, each not occupied by a particle.

13. The device according to claim 8, wherein the array has a plurality of internal edges, each internal edge enclosing a plurality of corresponding vertices, and at least one enclosed vertex is occupied by a particle.

14. The device according to claim 13, wherein the internal edges surrounding the vertices occupied by the particles have different boundary conditions from at least one external edge.

15. The device according to claim 14, wherein the edges configured to be under different boundary conditions are selected from e-condensation or m-condensation edges.

16. - A confinement system for aligning particles within a two-dimensional array, here: Each particle (303) is placed at a vertex of the ruby ​​lattice (301); Each particle has a first state and an excited state; Each particle belonging to the three unit cells of the ruby ​​lattice, when in an excited state, has a blockade radius (313) sufficient to block each of at least six nearest neighboring particles in the ruby ​​lattice from its transition from its first state to its excited state, and the array has at least one outer edge configured to be in a first boundary condition; -The confinement system is: Laser sources (202) aligned to generate multiple confinement regions; The source of the atomic cloud, where the atomic cloud may be arranged to at least partially overlap with multiple confinement regions, Including; and - An excitation source to excite at least some of the particles from a first state to an excited state. A system that includes this.

17. The system according to claim 16, wherein the particles are atoms, and the excitation source is configured to excite at least some of the atoms to a Rydberg state. [Request Item 18] [Number 2] Quantum spin liquid ( [Math 3] A method for creating QSLs: - The process of aligning a two-dimensional array of particles, here: Each particle (303) corresponds to the aspect ratio of the quadrilateral portion of the ruby ​​lattice. [Math 4] Located at the vertices of a ruby ​​lattice (301) with a larger parameter ρ; Each particle has a first state and an excited state; The array has at least one external edge; - A step of exciting approximately 25% of the particles into an excited state so that each particle in the excited state belonging to the three unit cells of the ruby ​​lattice has a blockade radius (313) sufficient to block at least six nearest neighboring particles in the ruby ​​lattice; and -Optionally, a step of imposing a first boundary condition on at least one external edge. Methods that include...

19. The method according to claim 18, wherein the particle is an atom and the excited state is a Rydberg state. [Request Item 20] [Number 5] Quantum spin liquid ( [Math 6] A method for encoding topological qubits within a QSL: -According to the method described in claim 18 [Number 7] The process of preparing a QSL, wherein the array includes at least a first outer edge (1601), a second outer edge (1603), a third outer edge (1602), and a fourth outer edge (1604); - A process of imposing a first boundary condition on the first and third outer edges, and imposing a second boundary condition on the second and fourth outer edges. Methods that include...

21. The method according to claim 20, wherein the array has a plurality of external edges, and the method further includes the step of imposing either a first boundary condition or a second boundary condition on each external edge, such that each external edge has a boundary condition different from any adjacent external edge.

22. The method according to claim 20 or 21, wherein an external edge configured to meet the first boundary condition is e-condensed, and an external edge configured to meet the second boundary condition is m-condensed. [Request Item 23] [Number 8] Quantum spin liquid ( [Number 9] A method for encoding topological qubits in QSL, wherein the method is: According to the method described in claim 18, [Number 10] A method comprising the step of preparing QSLs, wherein the array includes at least one internal edge (1503).

24. The method according to claim 23, wherein each vertex enclosed by at least one internal edge is not occupied by a particle.

25. The method according to claim 24, wherein at least one internal edge encloses at least four vertices.

26. The method according to claim 23, wherein at least one internal edge surrounds a vertex occupied by a particle.

27. The method according to claim 26, further comprising the step of imposing boundary conditions on at least one internal edge that are different from the boundary conditions of at least one external edge.

28. The method according to claim 26, further comprising the step of imposing boundary conditions on at least one internal edge that are the same as those on at least one external edge.

29. The method according to claim 23, wherein the array has a plurality of internal edges, each internal edge enclosing a plurality of corresponding vertices, and at least one enclosed vertex is occupied by a particle.

30. The method according to claim 29, further comprising the step of imposing boundary conditions on internal edges surrounding vertices occupied by particles that are different from boundary conditions on at least one external edge.

31. The method according to claim 29, wherein the edges configured to be under different boundary conditions are selected from e-condensation or m-condensation edges. [Request Item 32] [Number 11] Quantum spin liquid ( [Math 12] A method for reading the state of a topological qubit encoded in a QSL, wherein the method is: A process of receiving a display of the state of each particle in a two-dimensional array of particles, Here, each particle (303) is placed at a vertex of the ruby ​​lattice (301); Each particle has a first state and an excited state; Each particle belonging to the three unit cells of the ruby ​​lattice, when in an excited state, has a blockade radius (313) sufficient to block each of at least six nearest neighboring particles in the ruby ​​lattice from its transition from its first state to its excited state. The array has multiple external edges, each external edge being under either a first or second boundary condition, and each external edge being under a different boundary condition than any adjacent external edge; The process involves determining a first path through an array from a first outer edge (1601) of a plurality of outer edges having a first boundary condition to a second outer edge (1602) of a plurality of outer edges having a first boundary condition, via a first plurality of vertices of a ruby ​​lattice having a first plurality of particles; A step of assigning a first value to a first path based on the state of each of the first set of particles; A process to determine the state of a first topological qubit based on a first value. Methods that include...

33. The method according to claim 32, wherein a reference rotation is applied to the first topological qubit before determining the first path.

34. The process of determining a second path through an array from a first outer edge (1601) of a plurality of outer edges to a second outer edge (1602) of a plurality of outer edges, via a second plurality of vertices of a ruby ​​lattice having a second plurality of particles; A step of assigning a second value to a second path based on the state of each of the second set of particles; and A step in which the state of the first topological qubit is determined based on the first and second values. The method according to claim 32, further comprising:

35. The process of determining a third path through an array from a first outer edge of a plurality of outer edges to a third outer edge of a plurality of outer edges having a first boundary condition, via a third plurality of vertices of a ruby ​​lattice having a third plurality of particles; A step of assigning a third value to a third path based on the state of each of the third set of particles; and A step to determine the state of the second topological qubit based on the third value. The method according to claim 34, further comprising: [Request Item 36] [Number 13] Quantum spin liquid ( [Number 14] A method for reading the state of a topological qubit encoded in QSL, wherein the method is: The process of receiving a display of the state of each particle in a two-dimensional array of particles, here Each particle (303) is placed at a vertex of the ruby ​​lattice (301); Each particle has a first state and an excited state; Each particle belonging to the three unit cells of the ruby ​​lattice, when in an excited state, has a blockade radius (313) sufficient to block each of at least six nearest neighboring particles in the ruby ​​lattice from its transition from its first state to its excited state. The array includes at least one external edge (1502) and at least one internal edge (1503); The process involves determining a first path through an array from at least one internal edge (1503) to at least one external edge (1502) via a first set of vertices of a ruby ​​lattice having a first set of particles; A step of assigning a first value to a first path based on the state of each of the first set of particles; A process to determine the state of a first topological qubit based on a first value. Methods that include...

37. The process involves determining a second path through an array from at least one internal edge (1503) to at least one external edge (1502) via a second set of vertices of a ruby ​​lattice having a second set of particles; A step of assigning a second value to a first path based on the state of each of the second set of particles; A step in which the state of the first topological qubit is determined based on the first and second values. The method according to claim 36, further comprising:

38. The array has at least a second internal edge (1504), and the method: The process involves determining a third path through an array from a second internal edge (1504) to at least one external edge (1502) via a third set of vertices of a ruby ​​lattice having a third set of particles; A step of assigning a third value to a third path based on the state of each of the third set of particles; A step to determine the state of the second topological qubit based on the third value. The method according to claim 36, further comprising:

39. The process of determining one of the routes is: A method comprising piecewise assembling paths from multiple partitions, wherein each partition extends between two vertices in a ruby ​​lattice, and each partition is: It stretches between two vertices of the triangular portion of the unit cell of the ruby ​​lattice, or The method according to any one of claims 32 to 38, wherein the extension is either between two vertices in different unit cells of the ruby ​​lattice without intersecting any unit cell of the ruby ​​lattice.

40. The process of determining one of the routes is: A method comprising piecewise assembling paths from multiple partitions, wherein each partition extends between two vertices in a ruby ​​lattice, and each partition is: The method according to any one of claims 32 to 38, wherein the extension is between two vertices of the quadrilateral portion of the unit cell of a ruby ​​lattice.

41. A step of preparing a topological qubit according to the method of claim 20, wherein the first boundary condition is an e-boundary condition; A step of generating first and second e-anyons in an array; A step of removing a first e-anyon from the array via a first outer edge and a step of removing a second e-anyon from the array via a third outer edge. A method for manipulating topological qubits, including [specific example].

42. A step of preparing a topological qubit according to the method described in claim 23, A step of generating first and second e-anyons in an array; A process to immobilize the first e-anyon; and A process of moving a second e-anyon along a circular path that surrounds the boundary of at least one internal edge and has an endpoint at the location of the first e-anyon, thereby destroying the first and second e-anyons. A method for manipulating topological qubits, including [specific example]. [Request Item 43] [Number 15] Quantum spin liquid ( [Number 16] A method for encoding first and second topological qubits in a QSL, wherein the method is: According to the method described in claim 18 [Number 17] This includes the process of preparing QSL cards. The array includes a first internal edge and a second internal edge, the first internal edge having a first boundary condition, the second internal edge having a second boundary condition different from the first boundary condition, the first topological qubit corresponding to the first internal edge, and the second topological qubit corresponding to the second internal edge. The array includes a first outer edge, and the first outer edge has a first boundary condition. A method wherein the array includes further edges, the further edges having a second boundary condition, and being either internal or external edges.

44. A step of encoding first and second topological qubits according to the method described in claim 43; A process of moving the first internal edge along a closed continuous path that defines the boundary of the second internal edge. A method for manipulating the first and second topological qubits, including the following.

45. The method according to claim 43 or 44, wherein (i) the first boundary condition is m-condensation and the second boundary condition is e-condensation, or (ii) the first boundary condition is e-condensation and the second boundary condition is m-condensation.

46. A computer program comprising a computer-readable storage medium having program instructions implemented by the computer program, wherein the program instructions are executable by a processor and perform the method described in any one of claims 18 to 45.

47. The device according to claim 6 or 7, wherein an external edge configured to meet a first boundary condition is e-condensed, and an external edge configured to meet a second boundary condition is m-condensed.

48. The device according to claim 8, wherein at least one internal edge surrounds at least four vertices.

49. The device according to claim 8, wherein at least one internal edge surrounds a vertex occupied by a particle.

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