Lateral state-dependent forces for entanglement of trapped ions

Abstract

The present disclosure provides embodiments for entangling two or more trapped ions. To this end, a common motional mode of two or more trapped ions is used by conditionally exciting and / or de-exciting the common motional mode depending on the internal states of the two or more trapped ions. The common motional mode is conditionally excited / de-excited by inducing, on each of the two or more trapped ions, a respective perpendicular state-dependent force (SDF) that is modulated depending on the frequency of the motional mode. More specifically, each perpendicular SDF is induced by a laser beam and acts perpendicular to the propagation direction of the laser beam inducing the perpendicular SDF. The SDF may be modulated by (i) modulating the intensity and / or amplitude of the electromagnetic field of a first laser beam, (ii) changing the position and / or direction of the first laser beam relative to the first trapped ion, and / or (iii) including light of a different frequency in the laser beam.

Description

[Technical Field]

[0001] Embodiments of the present invention relate to the field of entanglement of trapped ions. Background

[0002] In many technical applications (e.g., quantum computing, quantum simulation, atomic and molecular experiments, spectroscopy, atomic clocks, and measurements), laser beams are used to trap and manipulate individual ions. For example, in the field of quantum computing, trapped ions are used to represent qubits, and quantum computations are performed by irradiating the ions with an appropriate laser beam. A key aspect of such applications, and especially a key aspect of performing quantum computations using trapped ions, is the generation of entangled states from two or more ions that are initially in an unentangled state.

[0003] Ion traps, also known as ion traps, typically contain multiple electrodes that generate an electromagnetic field to confine ions to a small area within a vacuum chamber. However, the components of such trap setups can restrict optical access, making it difficult to address specific ions (only) in an ion / qubit register, and / or address ions from a particular direction, especially along the axis of a linear ion sequence.

[0004] Current approaches to entangle m (two or more) ions / qubits within a register of n trapped ions typically require simultaneous high numerical aperture (NA) optical access from at least two different directions to reach a focal point small enough to illuminate only each of the m ions. For example, current implementations of light-shift (LS) gates are based on standing waves generated by two laser beams with specific properties from different directions. This can make technical implementation difficult, reduce the fidelity of the entanglement operation, increase setup costs, and / or limit the available motion modes. [Prior art documents] [Non-patent literature]

[0005] [1] D. Leibfried et al.: “Experimental demonstration of a robust, high-fidelity geometric two ion-qubit phase gate”, Nature 422, 412 to 415, (2003); [2] PJ Lee etal.: “Phase control of trapped ion quantum gates”, J. Opt. B: QuantumSemiclass. Opt. 7 (2005) S371-S383, arXiv:quant-ph / 0505203; [3] L. Aolita et al.: “High-fidelity ion-trap quantum computing with hyperfine clock states”, Phys. Rev. A 76, 040303(R), October19, 2007, arXiv:0707.3916. [4] B. Sawyer et al.: “A Wavelength-Insensitive, Multispecies Entangling Gate for Group-2Atomic Ions”, Phys. Rev. A 103, 022427 (2021), arXiv:2010.04526. Overview

[0006] It would be desirable to provide a simple and versatile technology for entanglement of trapped ions.

[0007] In some embodiments, this is achieved by exposing each of the trapped ions to a laser beam having a transverse gradient to induce a transverse state-dependent force (SDF) in the trapped ions, which is modulated for ion entanglement.

[0008] The present invention is defined by the independent claims. Some of the advantageous embodiments are the subject of the dependent claims.

[0009] In some embodiments of the present invention, a method is provided for entanglement of first and second trapped ions. This method utilizes the motion modes of the first and second trapped ions and includes the steps of inducing a first SDF (Sensitive Dispersion Factor) on the first trapped ion using a first laser beam and inducing a second SDF on the second trapped ion using a second laser beam. The steps of inducing the first and second SDFs are performed simultaneously, with the first SDF acting perpendicular to the propagation direction of the first laser beam and the second SDF acting perpendicular to the propagation direction of the second laser beam. In the steps of inducing the first and second SDFs, the motion modes are excited according to the internal states of the first and second trapped ions by modulating the first and second laser beams according to the frequency of the motion modes, thereby modulating the first and SDFs, respectively.

[0010] Details of one or more embodiments are described in the accompanying drawings and the following description. Other features, purposes, and advantages will become apparent from this description, drawings, and claims. [Brief explanation of the drawing]

[0011] Embodiments of the present invention will be described in further detail below with reference to the attached figures and drawings. [Figure 1] Figure 1 is a schematic diagram showing different energy levels resulting from different excitation modes of motion. [Figure 2] Figure 2 is a flowchart showing exemplary steps for entanglement of trapped ions. [Figure 3] Figure 3 is a block diagram showing an exemplary setup for entanglement of trapped ions. [Figure 4] Figure 4 is a block diagram illustrating the exemplary functional structure of the controller shown in Figure 3. [Figure 5] Figure 5 is a schematic diagram illustrating the addressing of two ions using a single objective lens. [Figure 6] Figure 6 is an illustrative diagram of an example optical path of a laser beam. [Figure 7] Figure 7 shows the radial intensity profile of a Gaussian laser beam and the positioning of addressed ions at the location of the maximum intensity gradient. [Figure 8] Figure 8a) shows a laser beam composed of two beam components having the same but displaced beam shape. Figure 8b) shows a laser beam composed of two beam components having different beam shapes. Figure 8c) shows a laser beam composed of two beam components having different beam shapes and orthogonal linear polarization. [Figure 9]Figure 9a) shows the radial dependence of (i) the laser beam component and (ii) the amplitude of the electric field relative to the laser beam in Figure 8b. Figure 9b) shows the radial dependence of (i) the laser beam component and laser beam intensity in Figure 8b and (ii) the gradient of the laser beam intensity. [Figure 10] Figure 10 shows a first example of energy levels and laser frequencies for performing SDF on ions. [Figure 11] Figure 11 shows a second example of energy levels and laser frequencies for performing SDF on ions. [Figure 12] Figure 12 shows the conditional excitation of the axial center-of-mass mode of an ion row using SDF, Figures 12a) to 12d) are xy cross-sectional views (z=0) showing the SDF of two laser beams and their effects on the |00>, |01>, |10>, and |11> states of two addressed qubits, respectively, and Figure 12e) shows the geometric situation and the coordinate system used in a three-dimensional view. [Figure 13] Figure 13 shows exemplary beam geometries (in particular, the position of the beam axis relative to the addressed ion) for entanglement of two ions using axial motion modes and Gaussian beams, with Figures 13a)–c) showing different two-dimensional cross-sectional views and Figure 13d) showing the geometric situation and the coordinate system used in a three-dimensional view. [Figure 14] Figure 14 shows exemplary beam geometries (in particular, the position of the beam axis relative to the addressed ion) for entanglement of two ions using radial motion modes and Gaussian beams, with Figures 14a)–c) showing different two-dimensional cross-sectional views and Figure 14d) showing the geometric situation and the coordinate system used in a three-dimensional view. [Figure 15] Figure 15 shows the excitation of the axial center-of-mass mode in a linear ionic crystal, and Figures 15a) to c) show different two-dimensional cross-sections of the same situation. [Figure 16]Figure 16 shows the excitation of the axial stretching mode in a linear ionic crystal, and Figures 16a) to c) show different two-dimensional cross-sections of the same situation. [Figure 17] Figure 17 shows the excitation of in-plane modes in a 2D ionic crystal, with Figures 17a) to c) showing different 2D cross-sectional views, and Figure 17d) showing the geometric situation and the coordinate system used in a 3D view. Detailed explanation

[0012] However, it should be clearly understood that the drawings are for illustrative and explanatory purposes only and are not intended to define any limitations on the subject matter disclosed. Furthermore, it should be noted that identical reference numerals refer to identical or at least functionally equivalent features.

[0013] The following description refers to the accompanying figures, which form part of this disclosure and illustrate specific aspects of the embodiments of the present invention, or specific aspects in which embodiments of the present invention may be used. It is understood that embodiments of the present invention may be used in other aspects and may include structural or logical modifications not shown. Therefore, the following detailed description should not be construed as restrictive, and the scope of the invention is defined by the appended claims.

[0014] Unless otherwise specified, it is understood that the various exemplary embodiments and / or features of the aspects described herein may be combined with each other.

[0015] For the purposes of the following description, terms such as “end,” “top,” “bottom,” “right,” “left,” “vertical,” “horizontal,” “top,” “bottom,” “lateral,” and “vertical,” and their derivatives, refer to the disclosed subject matter as they appear in the drawings. However, unless otherwise explicitly specified, it should be understood that the disclosed subject matter may be subject to various alternative modifications and step sequences. It should also be understood that the specific devices and processes shown in the accompanying drawings and described in the following specification are merely illustrative embodiments or aspects of the disclosed subject matter. Therefore, specific dimensions and other physical characteristics relating to the embodiments or aspects disclosed herein should not be considered limiting unless otherwise stated.

[0016] Any aspect, component, element, structure, action, step, function, or instruction used herein should not be construed as important or essential unless expressly stated otherwise. Furthermore, as used herein, the articles “a” and “an” are intended to include one or more items and may be used interchangeably with “one or more” and “at least one.” Additionally, as used herein, the term “set” is intended to include one or more items (e.g., related items, unrelated items, a combination of related and unrelated items) and may be used interchangeably with “one or more” or “at least one.” When referring to only one item, use the term “one” or similar wording. Furthermore, as used herein, terms such as “has,” “have,” and “having” are intended to be unrestricted terms. Additionally, the phrase “based on” is intended to mean “at least partially based” unless expressly stated otherwise. Quantum computing

[0017] The following briefly introduces some concepts and terms related to the field of quantum computing. However, it should be noted that while the present invention may have many applications in the field of quantum computing and will be described frequently in this context, it relates to a more general field of trapped ion entanglement, which has applications in other fields as well, and is not limited thereto. Single qubit state

[0018] In quantum computing, a qubit represents the fundamental unit of quantum information. Therefore, a qubit is the quantum version of a binary bit in a classical computer. However, while bits can usually only be assumed to have two values, called "0" and "1", a qubit corresponds to a state in a two-state quantum mechanical system. Thus, the state of a qubit |ψ> can be written as a linear combination |ψ>=α|0>+β|1>, where α and β are normalization conditions |α| 2 +|β| 2 |0> and |1> are complex numbers constrained by =1, and represent two states in a two-state quantum mechanical system (since a two-state quantum system is mathematically equivalent to a spin-1 / 2 system, the two ground states are often also represented as |↓> and |↑>). Therefore, qubits are not limited to two values ​​like classical bits, but complex superpositions of the |0> and |1> states can also be assumed. Since the global phase of a quantum state has no observational result, one of the numbers α and β can be chosen to be a real number. As a result, the state of a single qubit is relative to the global phase.

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[0019] In general, the state of two qubits |ψ 12 > can be described as follows.

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[0020] In general, in a system of n qubits, 2 n complex numbers

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[0021] State |ψ 12 > is,

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[0022] On the other hand, an entangled state is a state of two or more qubits that cannot be factored into a product of individual qubit states. In this disclosure, entanglement refers to such a state. However, an entangled state is a bell state.

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[0023] The term quantum logic gate, or simply "quantum gate," refers to the quantum computational version of the logic gate in classical computation, and therefore quantum gates are the building blocks of quantum circuits. Any quantum algorithm corresponds to a unitary operation on the state vector of qubits. Thus, while classical gates perform Boolean functions on bits, quantum gates perform unitary, and especially reversible, operations on one or more qubits. Therefore, an operation performed by a quantum gate on n qubits is equivalent to two operations acting on the state vector of n qubits. n ×2 n It can be described by a unitary matrix. The state vector of n qubits is 2 n Note that it is given by n complex numbers, and not simply by 2n complex numbers.

[0024] Generally, a single-qubit operation performed by a single-qubit gate changes the state of only one qubit, leaving the states of the other qubits unchanged. More specifically, a single-qubit operation performs a unitary operation on the state of a single qubit, independently of the current states of the other qubits. Thus, a single-qubit operation changes the above α and / or β of each qubit by resulting in a complex rotation of the qubit states, which may be visualized as a rotation of the single qubit around the axis of the Bloch sphere.

[0025] When represented by the internal states of an ion, performing a single qubit operation can cause a rotation of the wavefunction of the ion's qubit states. In particular, the ion's motion state may remain unchanged (basically, except for, for example, implementation errors / imperfections). However, the effect of performing a single qubit operation on the ion's internal states other than the two qubit states may not be significant, because the wavefunction of the ion on which the single qubit operation is performed should not normally overlap with the other states (also referred to herein as non-qubit states). Generally, to perform such a rotation on a qubit, an electromagnetic field of appropriate frequency may be applied for an appropriate duration (for example, using a laser beam to a trapped ion).

[0026] An example of a single qubit gate is the Pauli gate (X, Y, Z), which corresponds to the operation of three Pauli matrices on a qubit state, performing a rotation of π radians around the x, y, and z axes of the Bloch sphere, respectively. Note that the Pauli X gate is also called the NOT gate. A further example is the phase shift gate.

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[0027] Single-qubit operations on qubit states can be performed by applying a suitable radiation field, such as a laser beam / pulse. For example, a resonant microwave field can directly couple qubit levels via magnetic dipole interactions, resulting in coherent Rabi oscillations between qubit states. Alternatively or additionally, photostimulated transitions can be used, using two light sources that coherently couple qubit states via excited electronic states.

[0028] A two-qubit operation performed by a two-qubit gate conditionally combines the wave functions of two qubits. This can result in so-called entanglement, which can be used to fully specify the state of n qubits. n This is the reason why a complex number is generally required. An example of a 2-qubit gate is a control gate that acts on the states of two or more qubits. For example, a controlled-not (CNOT) gate acts on two qubits by (i) keeping one of the two qubits, the so-called control qubit, always unchanged, and (ii) performing a NOT operation on the other qubit when the control bit is in the |1> state. That is, the CNOT operation can be described by the following matrix.

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[0029] While single-qubit gates are generally relatively easy to execute (i.e., physically, especially on trapped ions), CNOTs and gates for two or more qubits are typically more demanding and usually slower than single-qubit gates. Similar to single-qubit gates, quantum gates acting on two or more qubits are also typically executed on ion qubits through the interaction of ions with a laser beam, as will be further described below.

[0030] In general, operations involving three or more qubits also exist. However, it can be shown that there exists a set of universal quantum gates that can approximate any unitary matrix, and by extension, any algorithm, with arbitrarily good accuracy. More specifically, a unitary matrix can be approximated as a sequence of operations performed by gates of the universal set (i.e., as a matrix product). The accuracy of the approximation can generally depend on the length of the sequence. In particular, quantum logic operations for any number of qubits can be decomposed into single-qubit operations and two-qubit operations. For example, a widely used set of universal quantum gates for two qubits is the rotation operator R X (θ) = exp(-iXθ / 2), R Y (θ) = exp(-iYθ / 2), R Z (θ) = exp(-iZθ / 2), (X, Y, Z are Pauli gates / matrices) and phase shift gate

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[0031] In general, any two-state quantum mechanical system, or any quantum mechanical system having three or more states, may be used to model a qubit (i.e., may be chosen for physical representation). This specifically means that the quantum system is used to store the quantum information (i.e., quantum states) of the qubit. If the quantum system has three or more states, two of these states may be selected and used to represent the qubit. In particular, a specific degree of freedom of the system (e.g., the spin of an electron) may be used to represent the qubit. Typically, two quantum states that are reliably distinguishable and / or two states with different energies (in other words, different energy levels) are used.

[0032] For example, a qubit may be represented using electron spin, for instance, by designating spin-up and spin-down states (for a given / selected direction) to represent the |0> state and |1> state, respectively. Other examples include nuclear spin states, atomic or nuclear states, nuclear magnetic resonance states, and quantum dots.

[0033] In this disclosure, we consider that two internal states of a trapped ion (i.e., an atom, particle, or molecule with a net charge) typically represent or model a qubit. In this specification, the term “trapped ion” typically refers to an ion confined to a small spatial region, such as a vacuum chamber, by an electromagnetic field. When referring to trapped ions, it should be further noted that these (all) trapped ions may be trapped by / using the same ion trap.

[0034] For example, the ground state of the trapped ion (generally the lowest energy state) may be chosen to represent the |0> state, and the excited state of the trapped ion may be chosen to represent the |1> state of the qubit. However, generally, both the |0> and |1> states may be excited states, i.e., each may be a state other than the ground state. Typically, long-lived excited states, so-called metastable states, are chosen as the |0> and / or |1> states. When a qubit is represented by two internal states of a trapped ion, quantum logic operations can be performed through laser-ion interactions, as will be further explained below.

[0035] It should be further noted that the term "internal state" refers to degrees of freedom other than the ion's kinetic degrees of freedom. The kinetic degrees of freedom pertain to an ion(s), such as the position and momentum of the ion's center of mass, and may be given for the normal modes of (all) trapped ions, as will be further explained below. However, the kinetic degrees of freedom of individual ion components, such as the orbital angular momentum of (valence) electrons, may also be considered to represent internal degrees of freedom. Furthermore, the spin states of electrons and other components of the ion are internal states. Thus, the qubit state may be two different stable electronic quantum states of the ion (in particular, e.g., a hyperfine qubit, a Zeeman ground state qubit, an optical transition qubit, etc.).

[0036] Furthermore, note that in the following, the term "qubit state" refers to the internal state of the trapped ion selected to represent each qubit. An internal state of an ion not used for qubit memory is called a "non-qubit state." In other words, an internal state is either a qubit state or a non-qubit state. Additionally, depending on the context, the term "qubit state" may also refer to the state of a qubit, which can generally be a superposition of two qubit (ground) states.

[0037] Note that in this disclosure, the qubit states of an ion are also referred to as the |↓> state and the |↑> state. This is simply done in consideration of the quantum mechanical description of the spin of a spin-1 / 2 particle such as an electron, which is a well-known example of a two-state quantum system. However, the present invention is not limited to the case where the qubit is represented in a spin-1 / 2 system. As described above, the internal states |↓> and |↑> may be any two different internal states of the (same) trapped ion. The energy levels of the two different internal states of the ion may be different, and / or may be further different by applying an external electric field and / or magnetic field.

[0038] For example, to realize a quantum computer with multiple qubits, multiple ions may be trapped in the same ion trap, and each trapped ion may be used to represent one of the multiple qubits. In general, the ions may be identical (i.e., of the same type, but not identical in the sense that there is only one ion), and the same / corresponding state of each ion may be used to represent the qubit. For example, alkaline earth ions (i.e., beryllium (Be), magnesium (Mg), calcium (Ca), strontium (Sr), barium (Ba), and radium (Ra)) and / or ytterbium (Yb) ions may be trapped and used to represent the qubit. Then, as will be explained in more detail below, the qubit state of the trapped ions may be controlled, manipulated, and / or read out using radiation from one or more laser beams. Ion trap

[0039] The term "ion trap" refers to any device that can be used to trap ions, that is, to spatially confine ions to a specific (small) region. This region may be, but not necessarily, within the ion trap, as in the case of a 2D trap. Some non-limiting examples of ion traps are given below. However, it should be noted that the present invention is not limited to any particular method / device for trapping ions.

[0040] Typically, ion traps use electric and / or magnetic fields to trap ions. For example, ion traps may be Penning traps, Pole traps, three-dimensional (3D) ion traps, and / or linear 3D traps. In this disclosure, the term ion trap refers to an assembly having multiple electrodes, which, when driven, generate an electric field that restricts (traps) the free motion of ions so that ions cannot escape from a specific (preferably small) area in the vicinity of those electrodes. It should be noted that actual ion trap devices / systems may include further mechanical and electrical components such as fixing means, electrical contacts, housings, power supplies, control circuits, and means for cooling ions.

[0041] In this specification, the term "3D trap" refers to all traps except surface traps. A surface trap is a trap in which all electrodes are arranged in the same plane. In particular, a 3D trap may be a Penning trap or a pole trap. Typically, a 3D ion trap has rotational symmetry, such as discrete cylindrical symmetry or continuous cylindrical symmetry.

[0042] Without loss of generality, the axes of symmetry of such cylindrical symmetry are three mutually orthogonal unit vectors.

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[0043] A Penning trap is a trap that uses electrostatic and magnetic fields to trap ions. Typically, only an electrostatic field is used in a Penning trap. In other words, oscillatory and / or alternating fields are not usually used. For example, an axial magnetic field is used to confine charged particles radially.

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[0044] A pole trap refers to a trap that uses an electric field to trap ions. Typically, only an electric field is used to trap ions in a pole trap; in particular, a magnetic field is usually not used. Generally, at least one of the electric fields in a pole trap is alternating (e.g., oscillating), and a pole trap may use both electrostatic and alternating electric fields. For example, the alternating field in a pole trap may be an alternating multipole electric field, especially a quadrupole electric field. Since the voltage switching is often done at radio frequencies, these traps are also called radio frequency (RF) traps.

[0045] A linear 3D trap is a specific type of 3D trap. Typically, in a linear 3D trap, ions are confined radially using an alternating (AC) electric field and axially by a static (DC) potential. Therefore, a linear 3D trap is also generally a (linear) pole trap. Cooling and readout

[0046] The trapped ions may be cooled to approach their ground state by utilizing their interaction with the laser beam. Such cooling of ions may include Doppler cooling and / or sideband cooling. However, the present invention is not limited to any particular cooling technique. Furthermore, the present invention is not limited to cooling the trapped ions to the ground state and / or initializing the trapped ions to the ground state or other specific known ground states. In particular, the implementation of the quantum gates described further below has the advantage that it also works with so-called thermal ions or hot ions, in which case the kinetic state of the trapped ions is not precisely known, not only initially but also during quantum computation. These gates make it possible to perform quantum operations even when the kinetic state of the ions is an unknown superposition of kinetic energy eigenstates. Some cooling may still be performed, for example, to avoid / reduce thermal decoherence resulting from the interaction between the trapped ions and the external environment. The initialization of the qubit state may similarly be performed by laser beam-induced transitions in individual ions.

[0047] Qubits may be read out by irradiating them with a resonant laser beam and detecting their state by fluorescence. Using such a resonant laser beam, rapidly repeating single-photon transitions from one of the qubit levels to a higher excitation level of the ion will emit fluorescent photons, which can be detected if that level is satisfied. If that level is not satisfied, no fluorescent photons will be detected, except for randomly generated dark counts. Exercise mode

[0048] As described above, trapped ions may be trapped by the same ion trap, using the same ion trap, and / or within the same ion trap. In other words, trapped ions may be part of the same "qubit register" or "trapped ion register." For the sake of understanding, we will assume below that all trapped ions are identical (i.e., the same chemical element / isotope, same charge). However, the present invention is not limited to this, as various types of ions can be trapped in ion traps and used in quantum computers. On the other hand, usually all trapped ions have the same charge sign, i.e., each ion has either a positive charge or a negative charge. However, ions do not need to have the same positive / negative charge (although they may).

[0049] In general, trapped ions may be cooled ions confined in free space by an electromagnetic field. If the trapped ions are sufficiently cold, they may form a Coulomb-bonded "ionic crystal" in which each ion has an equilibrium position in which the Coulomb repulsion between the trapped ions balances the external confinement force of the ion trap. For example, the trapped ions may be arranged in a straight line or trapped in a straight line (for example, forming a column / linear chain of trapped ions). In other words, the equilibrium positions of the ions may lie on a straight line. In particular, the trapped ions may be aligned along a trap axis. The trap axis may be one / only axis of symmetry of the trap. In particular, the ions may settle along the axis of the weakest confinement potential. The trapped ions may be arranged in a two-dimensional lattice (for example, their positions may form a two-dimensional mesh).

[0050] When ions are cooled, they can form a sufficiently well-separated quantum system, meaning that the displacement of the ions from their equilibrium position is small, and the motion of the trapped ions is quantized. Since the motion of the ions is coupled by the Coulomb repulsion between them, it can be described by the coupled harmonic oscillation motion of the ions in normal mode, where the term "normal mode" can mean that all ions oscillate at the same frequency and with a fixed phase relationship. Note that terms such as "normal mode," "motion mode," "common motion mode," "oscillation mode," "vibrational mode," "collective oscillation mode," "joint mode," or simply "mode" are used interchangeably.

[0051] Each mode of motion can be described by its respective quantum mechanical harmonic oscillator. Therefore, in the absence of a laser beam implementing quantum gates, a quantum system of N trapped ions can be described by the following Hamiltonian.

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[0052] In general, the motion of ions can be quantized in three directions as shown in the above equation, so there are 3N (orthogonal) normal modes, where N is the number of trapped ions. When the ions are aligned in a straight line, these three directions are two radial directions and one axial direction, with the axial direction being parallel to the line, i.e., along the direction in which the ions are aligned. Note that the radial and axial directions of trapped ions (also called "ion radial direction" and "ion axial direction," respectively) are generally different from the radial and axial directions of a laser beam (also called "beam radial direction" and "beam / propagation direction," respectively). The (ion / beam) radial direction is usually two mutually orthogonal directions perpendicular to each (ion / beam) axial direction (therefore, there can be some ambiguity / selectivity in the definition of radial direction).

[0053] The motion mode is a motion mode common to all trapped ions. In particular, the frequency ω of the motion mode ν This is a characteristic of the system / setup (the whole) including (for example) all trapped ions and traps. They may depend on one or more of the following, for example, the type of ions, the number of ions, the trap design, and especially the strength of the trap's confinement electromagnetic force. The frequency ω of the motion transition ν These typically fall within the frequency range of 300kHz to 5000kHz.

[0054] For example, in the case of a harmonic trap potential, the normal mode (i.e., when N≧2) is the frequency ω where the displacements of the two ions from equilibrium are the same. C,iThis includes three "center of mass modes" (COM) at (i=1,2,3) and three modes in which the magnitude of the displacement amplitude is equal, but ions on different sides of the column are displaced in opposite directions at a given time. The latter modes are also called "stretching modes" (or "breathing modes") in the axial direction and "locking modes" in the radial direction. In general, the frequencies of stretching / locking modes in a particular direction may be related to the frequencies of the COM modes in the same direction, for example, the frequencies of stretching modes are usually ω S =3 1 / 2 ω C Accordingly, it relates to the frequency of the axial COM mode. In particular, when ions are aligned in a straight line, there are a center of mass mode and a stretching mode for each of the two ion radial and ion axial directions (i.e., there is one axial COM mode and one axial stretching mode, and two radial COM modes and two radial locking modes). For brevity and ease of understanding, note that in the explicit examples of this disclosure, a stretching mode or COM mode is usually used to entangle the ions. However, the present invention is not limited thereto, as any mode of motion may be used to entangle the ions in general. In particular, the present invention is not limited to harmonic trap potentials but also applies to anharmonic trap potentials.

[0055] In general, a complete quantum mechanical description of an ion state is, for example, a phonon number eigenstate |n> ν This includes specifying the quantum state of each motion mode. Each state of the internal qubit (for example, the above |ψ for two qubits) 12For >), each motion mode generates a ladder of equidistant energy levels with different phonon numbers. This is illustrated exemplified in Figure 1 for one trapped ion and one motion mode. More specifically, the vertical axis in Figure 1 corresponds to the energy of each state, with higher states in the figure having higher energy levels. The states of other qubits and other motion modes are assumed to be the same for all illustrated states. On the left, the qubit of interest is in the |↓> state, and on the right, it is in the |↑> state. The energy difference between the internal states |↓> and |↑> is, if the motion state of the ion and the states of the other qubits do not change,

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[0056] Entangled quantum gates are a crucial element in the realization of quantum computers. Certain types of physical implementations of entangled gates for trapped ions are known as "geometric quantum gates" or "geometric phase gates." Geometric phase gates are gate implementations for qubits represented by the internal states of each ion trapped by the same trap. These can be made relatively fast compared to other two-ion gates and, furthermore, enable high fidelity because the internal states of the ions do not directly participate in the gate operation. Examples of geometric phase gates include the Mormer-Sorensen gate (MS gate) and the optical shift gate (LS gate), both of which utilize state-dependent forces (SDFs) induced by a laser beam on an addressed ion.

[0057] However, current LS gate implementations typically use two laser beams per ion, i.e., laser light from significantly different directions. The spatial overlap of the two laser beams is used to generate a moving standing wave. The movement of the standing wave corresponds to the modulation of the SDF induced by the standing wave. However, since standing waves cannot be generated by a single laser beam from one direction, the setup for generating standing waves is generally complex. For example, two laser beams that intersect or propagate in opposite directions, or a modulated retroreflection laser beam, are used. Similarly, MS gate schemes typically require induction of SDF by laser beams from different directions to accommodate axial motion modes. Therefore, these approaches typically require simultaneous high NA (numerical aperture) optical access from at least two different directions to reach a focal point small enough to illuminate only specific ions. This can make technical implementation difficult, reduce the fidelity of entanglement operations, increase setup costs, and / or limit the available motion modes.

[0058] In consideration of the above, one embodiment provides a method for entanglement of two or more trapped ions (also called addressed ions). As shown in Figure 2, in S240, a first SDF is induced in a first addressed ion using a first laser beam, and simultaneously, in S240, a second SDF is induced in a second addressed ion (other than the first addressed ion) using a second laser beam. Each SDF acts perpendicular to the propagation direction of the respective laser beam that induces the SDF (with respect to the respective ion). In S220, the first and second SDFs are modulated, respectively, by modulating the first and second laser beams according to the frequency of the motion mode used for entanglement. In this way, motion modes are excited according to the internal states of the first and second trapped ions, which can result in entanglement as will be further described below.

[0059] The entanglement methods described herein may generally be used to entangle two or more ions, i.e., more than two ions, and a 3-qubit gate may also be implemented, for example, as further described below. However, for the sake of brevity and ease of understanding, we will explicitly describe only the case where two ions / qubits are entangled using one mode of motion. In this regard, note that these two ions, also referred to herein as “addressed ions,” can be freely selected from the trapped ions (i.e., they do not need to be adjacent). Let ω1 be the frequency difference between the states of the first qubit (this is the difference between the |1> and |0> states of the first ion).

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[0060] As shown in Figure 2, this method may also include a step S200 for generating first and second laser beams. However, the laser beams may be received from an external source. It should be noted that some or all of the steps may be performed simultaneously, for example, as part of the same step. For example, the modulation of the laser beam S220 may be considered part of the generation step S200, and there may be, for example, one step for generating the modulated laser beam. Furthermore, as will be further explained below when discussing the addressing unit, the modulation of the laser beam may be performed in a single step. In particular, a laser beam addressed to an ion may be obtained by modulating one laser beam, splitting the modulated laser beam into multiple laser beams, and directing them to their respective ions (i.e., directing the laser beams obtained by splitting to different ions). However, the present invention is not limited thereto, for in general each laser beam may be generated and modulated separately from other laser beams, and / or there may be one separate step S240 for inducing an SDF for each addressed ion.

[0061] Furthermore, it should be noted that the laser beam modulation S220 may be performed after the induction of SDF S240 for each ion has begun. In other words, the order of steps S220 and S240 may be reversed. Generally, as will be further explained below, steps S220 and S240 are performed during their respective overlapping periods. In other words, there is a period (i.e., a gate time) during which both steps S220 and S240 are performed simultaneously (for both / all addressed ions).

[0062] According to one embodiment, in accordance with the method described above, an apparatus is provided for controlling first and second laser beams to entangle two or more trapped ions. As shown in Figure 3, the apparatus comprises a circuit 315 configured to simultaneously control (i) a first laser beam 361 for inducing a first SDF 371 in a first addressed ion 381, and (ii) a second laser beam 362 for inducing a second SDF 372 in a second addressed ion 382. Each SDF acts perpendicular to the propagation direction of the respective laser beams that induce the SDF (for example, the circuit 315 is configured to control the laser beams 361 and 362 to induce perpendicular SDFs). The first and second SDFs are modulated, respectively, by modulating the first and second laser beams according to the frequencies of the motion modes used for entanglement (for example, circuit 315 is configured to modulate laser beams 361 and 362 and / or the induced vertical SDF). In this way, motion modes are excited according to the internal states of the first and second trapped ions. Figure 3 further shows that the first trapped ion 381 and the second trapped ion 382 may be aligned on the trap axis 390, and other ions such as ion 383 may also be present on the trap axis 390.

[0063] For example, using a transverse SDF induced by a transverse gradient, only one laser beam is needed per ion, and it may even be possible to deliver all these beams to each ion from the same direction using the same optical system (see addressing unit). As will be further explained below, such a single "laser beam" may contain light of different frequencies (polychromatic light) and / or may be obtained by superimposing / combining multiple laser beams of different beam shapes. These beam components (having different shapes and / or frequencies) may have (slightly) different k-vectors, particularly for practical reasons, and therefore the laser beam addressing the ion may contain beam components with (slightly) different k-vectors. However, even in such cases, because a transverse SDF is used, optical access to the addressed ion is only required from one direction. For example, two or more trapped ions can be addressed using a single optical deflection device (such as one AOM or one AOD) and a single high-NA objective lens. In particular, the above method / apparatus may enable the entanglement of two or more trapped ions by irradiating each of the two or more ions with one (only) laser beam per addressed ion. This can reduce the complexity of the entanglement setup (compared to, for example, when standing waves are used), which is advantageous for all ion trap architectures, especially surface traps where optical access is usually limited. Using a single laser beam can also increase the fidelity of the entanglement operation because of interference-induced instability between the two beam paths.

[0064] Furthermore, this may allow for easy access to motion modes perpendicular to the propagation of the laser beam, and consequently, enable a wider variety of interion interactions. Thus, easy access to all motion modes (for example, access to one axial mode and two radial modes of a linear ion chain) may become possible. For example, easy access to the axial mode of a 3D linear pole trap, and / or easy access to specific / arbitrary in-plane modes of a 3D pole trap in a planar crystal may be facilitated.

[0065] Therefore, entanglement of two or more trapped ions becomes easier (unidirectional, easier optical access) and more diverse (axial and radial gates / modes). This simplicity can further improve the compatibility of laser-based entanglement gates with various ion trap designs (e.g., 3D linear pole traps, 3D planar crystal pole traps, 2D surface traps). In particular, this method / apparatus can be used with quantum gates to the first and second qubits, especially LS gates and / or

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[0066] However, it should be noted that the present invention is generally applicable to the generation of entangled states of two or more trapped ions and is not limited to quantum computing. In particular, although usually discussed in the context of quantum computing, the techniques described herein may very commonly be used to entangle trapped ions in / for applications other than quantum computing. Furthermore, it should be noted that whether or not a qubit / trapped ion is entangled after a 2-qubit gate operation generally depends on the initial state of the qubit / trapped ion. In other words, an entangle gate typically generates an entangled state only from a specific initial state, and not from any / all initial states. Therefore, terms such as “entangle gate” and “method for entanglement” refer to a gate / method that generates an entangled state from at least one initially unentangled state of an addressed ion. Circuits and controllers

[0067] In general, circuit 315 may be configured to control the laser beam in any manner described herein, i.e., the circuit may be configured to control the optical elements to obtain any laser beam described herein (e.g., modulation, frequency, beam shape of the laser beam(s)). Similarly, the entanglement method according to the present invention may include any steps that the apparatus (in particular circuit 315) is configured to perform according to a certain embodiment. That is, unless otherwise indicated by the context, the contents described herein may apply to both the entanglement method and the apparatus for entanglement.

[0068] The circuit or processing circuit 315 may be part of the controller, and the controller may include other hardware and / or software. Figure 4 shows an exemplary structure of such a controller 310. As shown, the controller may include the circuit 315 that performs control of a laser beam (e.g., a laser beam unit 320), and may further control further components included in the apparatus for entanglement of ions, memory 410, and optionally a transceiver 440 and a user interface 430. The controller may be, for example, part of a computing device or any other suitable device. The memory 410 may store a program that may be executed by the processing circuit 315 to perform any step of the method described herein. The processing circuit 315 may comprise one or more processors and / or other dedicated or programmable hardware. The transceiver 440 may be wireless and / or configured to receive and / or transmit (control) signals. The controller 310 may further include a user interface 430 for displaying messages or device status, etc., and / or receiving user input. For example, the controller may be configured to receive input for a specific unitary operation to be performed on a trapped ion, determine the characteristics of the laser beam on which the unitary operation is performed (this may be a sequence of laser beams, i.e., the controller may first decompose the received unitary operation into a predetermined sequence of universal gate operations and then execute the sequence), and then control the laser beam unit / laser beam accordingly as described above.

[0069] Bus 401 may interconnect memory, processing circuits, transceivers, and user interfaces. Note that the controller may be implemented by any hardware means other than a general-purpose processor, and can be implemented as a microcontroller, such as by programmable hardware like a field-programmable gate array (FPGA), or as dedicated hardware like an application-specific integrated circuit (ASIC). Any combination of the above hardware and, optionally, software may be used. Laser beam unit

[0070] The apparatus may further comprise a laser beam unit 320, but the present invention is not limited thereto, as the apparatus / circuit may be configured solely to control the laser beam unit 320. It should be further noted that the laser beam unit 320 is not limited to emitting two (different) laser beams 361 and 362. Generally, the laser beam unit may be configured to emit one or more laser beams, and the circuit 315 may be configured to control the two or more laser beams to entangle two or more ions (in particular, if two addressed ions are adjacent and a broad beam is used, laser beams 361 and 362 may be the same laser beam).

[0071] Furthermore, the laser beam unit 320 may generally include one or more lasers 330, such as stabilized diode lasers. However, the present invention is not limited to a specific laser source, and lasers other than diode lasers, such as solid-state crystal lasers, can also be used. There may be one laser beam 340 from one laser 330, from which multiple laser beams may be obtained using a beam splitter included in the optical element 350, or multiple lasers may each generate multiple laser beams 340. The present invention is not limited to a specific scenario. In particular, the laser beam unit 320 does not have to include a laser, and the circuit 315 may be configured to control the laser beam unit 320 receiving light (one or more laser beams) from an external light source.

[0072] Furthermore, the laser beam unit 320 may include optical elements 350 for generating a suitable laser beam having the characteristics described herein. The optical elements 350 may include active optical elements 350 such as acousto-optic modulators (AOMs) and acousto-optic deflectors (AODs), and / or passive optical elements 350 (e.g., lenses, attenuators, phase plates). As a further example, the active optical elements may include spatial-light modulators (SLMs) such as digital micromirror devices and / or liquid crystal devices. The circuit may be configured to control the laser(s) (e.g., the power and / or frequency of the emitted light) and / or active optical elements to implement one or more of the methods described herein. In particular, the circuit 315 may be configured to address a specific ion (i.e., to direct a specific beam to a specific ion such that the ion is located within the laser beam at a specific predetermined radial distance from the laser beam axis), modulate the laser beam, shape the laser beam into a predetermined shape, and / or, in particular, adjust / generate the laser beam to induce a lateral SDF at the location on each ion. Optical element

[0073] As shown in Figure 5, the optical element 350 may include a modulation unit 510, which is configured to perform modulation of the laser beam(s) 340 when controlled by a circuit, as will be further described below. Furthermore, the optical element 350 may include an objective lens 550 configured to focus the first laser beam 361 and the second laser beam 362 to the position of the trapped ions. More specifically, the first laser beam 361 and the second laser beam 362 may pass through at least one common lens 550 within the optical element 350. For example, if the ions are trapped in a linear ion trap, the trap axis 390 through which the ions align may be included in the focal plane 570 of the objective lens 550. Such an objective lens, including at least one common lens through which both laser beams pass, facilitates the excitation and / or entanglement of the trapped ions by one-sided access to the trap that confines the ions.

[0074] In general, some or all of the laser beams may be directed to addressed ions using the same objective lens. This is also shown in Figure 5, which exemplifies the assumption that the laser beam unit includes an addressing unit 520, which may also be controlled by circuit 315. Such an addressing unit facilitates the individual / separate control of the direction of the laser beams to the trapped ions and / or focusing the laser beams to different locations in the region where the ions are confined. This makes it possible to address a specific ion from among several trapped ions (other unaddressed ions may be present, as shown). More specifically, as shown in Figure 5, the first laser beam 361 and the second laser beam 362 may be directed to the first trapped ion 381 and the second trapped ion 382 using the same objective lens 550 by adjusting the angle of incidence of the beams with respect to the main plane 560 of the objective lens 550 (or lens 550). This can make it easier to address a specific ion from among multiple trapped ions, and / or direct a laser beam to a specific location using a single objective lens. In other words, a single high-quality objective lens may be used to direct each beam to its respective ion. Note that if the lens is not a thin lens as shown in Figure 5, the angle of incidence will be the angle with respect to the front principal plane (i.e., the first of the principal planes that the laser beams intersect).

[0075] Alternatively or additionally, the beam may be directed to a desired position by adjusting the inclination angle of the laser beam with respect to the optical axis (usually the rotational symmetry axis of the lens / objective lens). This is shown in Figure 5, which shows the optical axis of the objective lens 550 and the inclination angles of the first and second beams with respect to the optical axis 580 as θ1 and θ2, respectively.

[0076] Figure 6 shows an exemplary optical path for two or more laser beams 340 (note that at the positions indicated by “340”, there may be only one laser beam, either monochromatic or polychromatic). The exemplary setup includes a first AOD 620, a second AOD 640, a 4F relay optical system 630, a magnifying optical system 650, a collimation lens 660, and an objective lens 670. The optical components provide examples of the static (passive) optical components and dynamically controlled (active) optical components described above.

[0077] More specifically, the addressing unit 520 may include, for example, optical elements 620, 630, 640, and 650, where 620 and 640 are actively controlled by the controller 310. The first AOD620 is used to generate two (or more) laser beams, each having a different direction, from a single laser beam 340. Note that the frequencies of the two or more laser beams generated by the first AOD620 may be slightly different and / or shifted relative to the frequency of the laser beam 340. These laser beams obtained by the splitting of the AOD620 are then directed to individual addressed ions. The second AOD640 is used to further increase the deflection and compensate for the frequency offset by the first AOD620. Therefore, if it is not particularly important that the laser beams addressed to different ions have the same frequency (as is the case with the LS gate implementation described below), the AOD620 and the 4F relay optical system 630 may be omitted. Beam shaping may be performed by the magnifying optical system 650 and subsequent optical elements.

[0078] Furthermore, Figure 6 shows the vacuum window 680, which provides optical access to the ion trap and focuses the laser beam on the trap axis 690. The transversality of SDF

[0079] Generally, each SDF is perpendicular to the propagation direction of the laser beam that induces the SDF (i.e., perpendicular to the wave vector). For example, if there are two addressed ions (i.e., a "first" ion and a "second" ion), the SDF induced on the first addressed ion acts perpendicular to the propagation direction of the laser beam that addresses / irradiates the first ion, and the SDF induced on the second addressed ion acts perpendicular to the propagation direction of the laser beam that addresses / irradiates the second ion. In other words, the terms "perpendicular," "transverse," and "transversal" are used interchangeably and refer to the right angle between the SDF and the propagation direction of the laser beam that induces the SDF. In other words, the SDF is perpendicular to the direction of laser beam propagation at the location of the addressed ion (see Figure 3. In Figure 3, the propagation direction of laser beam 361 is perpendicular to SDF 371, and the propagation direction of laser beam 362 is perpendicular to SDF 372).

[0080] It should be noted that a laser beam can (and generally will) induce forces other than the transverse SDF on ions. Such other forces may or may not be state-dependent. Furthermore, such other forces may or may not be perpendicular (for example, they may even be parallel). In other words, the SDF may be the (perpendicular) component of the force(s) induced on ions by the laser beam. The SDF may be the entire perpendicular component of the force(s) induced by the laser beam, or a part of the said perpendicular component. Generation of SDF and lateral gradients

[0081] As described above, a lateral SDF is simultaneously induced for each addressed ion. For example, if there are two addressed ions (i.e., the "first" and "second" ions), the first and second SDFs are simultaneously induced for the first and second addressed ions, respectively. In other words, there is a one-to-one correspondence between SDFs and addressed ions (i.e., an SDF refers to a force acting on a specific ion, not a force field that can act on multiple ions). SDFs acting on different ions may be the same in magnitude and direction (with respect to the corresponding qubit state), or, as shown in the example in Figure 12, SDFs acting on different ions may be the same in magnitude but opposite in direction (with respect to the corresponding qubit state). However, the present invention is not limited to such scenarios, and in general, SDFs for different ions and states may be different. SDFs are induced by directing each laser beam towards each ion so that the ion is "located" (positioned) within the electromagnetic field of the laser beam. In some embodiments, for example, when SDF is induced by a laser beam having a non-constant intensity gradient in the beam radial direction, the specific location of ions in the laser beam (e.g., radial distance from the beam axis) may also be relevant.

[0082] Generally, each SDF is induced by one (e.g., just one or a single) laser beam (e.g., from one direction). Other expressions that may be used interchangeably with “induced” are that the laser beam “causes” or “acts” an SDF. Generally, a different laser beam may be used for each ion (e.g., there may be a one-to-one correspondence between the laser beam and the addressed ion). However, generally, two or more addressed ions may be irradiated with the same laser beam, for example, if two addressed ions are adjacent / neighboring trapped ions. The same laser beam may still induce different SDFs for ions, for example, because the intensity gradient of the laser beam may be position-dependent (e.g., a Gaussian beam).

[0083] In general, (for example, each / any) SDF may be generated by the (non-zero) transverse component of the gradient of the laser beam that induces the SDF. The “transverse” component of the gradient is the component perpendicular to the direction of propagation of the laser beam, i.e., perpendicular to the beam axis. For example, gradient

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[0084] Furthermore, it should be noted that the gradient is the gradient of the laser beam at the location of the addressed ion. Therefore, the SDF acting on the ion is induced by the gradient of the laser beam at the location of the ion. Moreover, the gradient is generally the gradient of the electromagnetic field of the laser beam. In other words, the electromagnetic field of the laser beam may have a gradient component perpendicular to the propagation direction of the laser beam at the location of the trapped ion. In particular, the gradient of the laser beam is (i) The intensity gradient of the electromagnetic field of the laser beam (especially the amplitude gradient of the electric or magnetic field of the laser beam), (ii) The phase gradient of the electromagnetic field of the laser beam, and / or (iii) Polarization gradient of the electromagnetic field of the laser beam That's fine.

[0085] It should be noted that a laser beam may have two or more non-zero gradients among the above gradients (i) to (iii) that contribute to the generation of the SDF. In other words, the SDF may generally be generated by one or more vertical components among the gradients (i) to (iii). Furthermore, it should be noted that, for example, in the case of phase gradients and / or polarization gradients, the SDF may also be induced by the orbital angular momentum (OAM) of the laser beam.

[0086] More specifically, location

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[0087] In summary, SDF may be applied to ions using the gradient of the laser beam (e.g., intensity, phase, and / or polarization gradient). By extension, SDF can be used for entanglement generation in combination with appropriate modulation, as will be further explained below. Beam shape

[0088] For example, a laser beam with an intensity / amplitude gradient may be realized by using a laser beam having a specific shape. For example, the shape of each (one or more) laser beam used to induce an SDF may be a Gaussian beam shape, a supergaussian beam shape, a Laguerre-Gaussian beam shape, and / or a Hermitian-Gaussian beam shape (or beam gradient). These beam profiles have a transverse amplitude (and intensity) gradient that is not constant in the beam radial / transverse direction. This makes it possible to adjust the gradient (and thus the SDF) by changing the position and / or direction of the laser beam relative to an addressed ion, as will be further described below. Note that the terms “beamform,” “beam shape,” and “beam profile” are used interchangeably in this disclosure.

[0089] Specifically, the electric field of a Gaussian-shaped laser beam is

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[0090] As can be seen from the above equation, using a high numerical aperture (NA) optical system can increase the intensity gradient in the transverse direction r. In other words, the intensity gradient can be increased by using a focused laser. This can facilitate the addressing of individual and / or specific trapped ions (i.e., induce a force(s) only on the addressed ion(s) and not on other trapped ions). This can make it possible to implement an entangle gate that acts only on a subset of m ions in a qubit register of n>m ions. A larger intensity gradient can lead to a larger SDF, which can enable faster gate times.

[0091] In general, a (new) beam shape may be generated by superimposing beams of different shapes. In other words, a laser beam addressed to an ion may contain two or more beam components with different beam shapes (e.g., electromagnetic field components), for example, two or more spatially distinct beam components. Alternatively or additionally (and especially when the beam components have the same beam shape), the beam shapes of the beam components may be displaced relative to each other (e.g., radially in the laser beam). Furthermore, alternatively or additionally, the beam components may have different polarizations, different polarization gradients, different phases, different phase gradients, and / or electric or magnetic fields of different directions at the location of the addressed ion. Thus, a laser beam may contain two or more beam components having different electromagnetic field characteristics at the location of the addressed ion.

[0092] This may make it possible to generate special beam shapes from "standard" beam shapes. Generally, one, two or more, or each of the beam components may be Hermitian Gaussian beams (for example, each may have a beam shape of a Hermitian Gaussian mode). For example, as shown in Figure 8a, two Gaussian beams (for example, TEM 00 The beams may overlap with each other.

[0093] Hermitian Gaussian mode is typically used in TEM lm This is written as follows, where TEM stands for transverse electro-magnetic. Hermitian Gaussian beam TEM propagating in the z direction. lm The time-independent portion of the electric field can be described as follows:

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[0094] In this regard, note that the black arrows in Figures 8a), 8b), and 8c) indicate the polarization and electric field magnitude of each beam component. Therefore, for example in Figure 8a), the two beam components 801 and 802 are linearly polarized in the y direction. Furthermore, Figure 8a) shows that the addressed ion 803 is at the center of the two laser beam components, but is displaced to the right of the axis of beam component 801 and to the left of the axis of beam component 802. In other words, ion 803 is located midway between the axis of beam component 801 (for example, at (x1,y1,0)) and the axis of beam component 802 (for example, at (x2,y2,0)) (for example, (x1 / 2+x2 / 2,y1 / 2+y2 / 2,0)). Therefore, the two beam components 801 and 802 induce SDF in opposite directions relative to the ion (i.e., in the direction toward or away from the respective beam component axes). Therefore, the intensities of beam components 801 and 802 may be modulated at the same frequency but with a phase shift of 180°. Using this setting, the two laser beam components may be used to enhance the SDF to the ion (more specifically, to increase the average excitation (over gate time) of the motion modes due to the SDF). More specifically, a laser beam may generally contain two beam components that are the same shape but displaced from each other in the radial direction of the laser beam. Such a laser beam may (i) be directed towards an addressed ion such that the two laser beam components are displaced in opposite directions (radially) with respect to the ion, and (ii) be intensity-modulated with a phase shift of 180° relative to each other (otherwise, the intensity modulations may be identical).

[0095] As another example, as shown in Figure 8b), a laser beam addressing an ion may contain two beam components having different beam shapes. Generally, the axes of these two beam components may or may not share the same beam axis (i.e., the beam components are not radially displaced relative to each other).

[0096] For example, one of the two components is a Gaussian beam shape (e.g., TEM)00 The beam shape may be (for example, TEM), and the other beam component may have a Hermitian Gaussian beam shape (for example, TEM) 10 The beam shape may be as shown in Figure 8b). The ions are at the center of the laser beam axis, and the laser beam axis is not only identical to the axis of the Gaussian beam component 812, but also identical to the axis of the Hermitian Gaussian beam component 811. In this configuration, these laser beams can propagate in either the same direction (co-propagating) or in the opposite direction (counter-propagating).

[0097] Figure 9a) shows the first beam component 811 (TEM) when both beams are modulated in the same phase at the same time. 10 Electric field E (dashed line) 1,0 The y-dependence of (y is the radial distance from the beam axis, such as the trap axis), the second beam component 812 (TEM) 00 Electric field E (dotted line) 0,0 The y-dependence of the ion and the resulting sum of electric fields corresponding to the electric field of the laser beam addressing the ions (solid line) are shown. As shown in Figure 8b), both electric fields of the beam components are polarized in the y direction. The fact that the electric field of the first beam component is negative with respect to negative y (see Figure 9a) means that the electric field is directed in the opposite direction to the y direction. Therefore, with respect to negative y, the electric fields of the first and second components cancel each other out.

[0098] TEM 00 and TEM 10 The electric fields of the individual beam components 812 and 811 corresponding to the Hermitian Gaussian beam shape can be described, respectively (at x=z=0, and therefore, for example, along the trap axis) as follows:

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[0099] In this case, the intensities of the two beam components may be modulated together (especially without a phase offset at the same frequency, as in the example of FIG. 8a). For example, the intensity of a laser beam (as a whole) including two beam components 811 and 812 may be modulated. However, it is also possible to modulate the two beam components at different frequencies, which results in a phase difference between the two beam components that changes over time. This means that the relative sign between the electric fields of the two components at a given position changes over time. In that case, since the intensity maximum of the beam overlap oscillates periodically between the position on the right side and the position on the left side of the ion, the sign of the intensity gradient at the position of the ion changes (periodically) over time. For example, beam component 811 can be modulated at frequency ω Mod and beam component 812 can be modulated at 2ωMod It can be modulated at t = π / ω Mod The magnitude π = Δω per second Mod Accumulation of the relative phase of t occurs. Therefore, the position of the maximum intensity varies periodically between the left and right sides of the ion at frequency ω Mod Using this, the motion mode can be continuously excited or de-excited, so the gate time can be shortened compared to the case of a single Gaussian beam component where the SDF acts only in one direction.

[0100] Assuming that the ion is at the center of the laser beam component, the intensity and intensity gradient of the laser beam including the two beam components can be estimated as follows. [Number]

[0101] In particular, when only the intensity and thus the electric field amplitude E1 of the first beam component 811 (TEM 10 ) are increased, the intensity gradient (and thus the SDF) increases, but the intensity at the ion position remains constant. Usually, in conventional implementations of laser-based entangled gates, there is a scattering limit that limits the fidelity of the entangled gate because the velocity of the scattered photons (which induces decoherence) and the strength of the SDF depend on the same electric field amplitude. In the proposed method, due to the overlap of the two laser beams, this dependence can be separated as described, so the scattering limit can be significantly improved.

[0102] Another possible example is shown in Fig. 8c), and the configuration of the laser beam is the same as in Fig. 8b) except for the direction of polarization of the first beam component. As shown in Fig. 8c), the linear polarization of the first Hermite-Gaussian beam TEM 10 component 813 is in a direction orthogonal to the linear polarization of the Gaussian beam TEM 00 component 814. Therefore, in general, two beam components having different shapes may be linearly polarized in mutually orthogonal radial directions.

[0103] In Figure 8c), when the electric fields of beam components 813 and 814 are modulated at the same frequency but with a phase difference of 90° or 270° from each other (resulting in the intensity of one beam component being zero when the intensity of the other beam component is at its maximum, and vice versa), two circularly polarized components are generated when the two beam components 813 and 814 are superimposed. Specifically, a first circularly polarized component is generated where the right-hand portion of beam component 813 (the portion where the arrow points upward in Figure 8c) overlaps with beam component 814, and a second circularly polarized component is generated where the left-hand portion of beam component 813 (the portion where the arrow points downward in Figure 8c) overlaps with beam component 814. These circularly polarized components are in phase and are oppositely polarized (also known as left / right rotation, or sigma plus / minus). Therefore, for example, in Figure 8c), sigma minus polarization may exist to the left of the ion, and sigma plus polarization may exist to the right of the ion. Such polarization gradients can produce (or be used to implement) SDFs that act in opposite directions on the two qubit states of an ion (for example, the qubit states interact with the two circularly polarized beams in different ways). For this configuration to work, the phases of the two beam components do not need to be perfectly out of phase (in general, it may be sufficient if the two beam components are modulated with shifted phases (or not modulated with the same phase, i.e., if there is a phase difference)).

[0104] In both Figures 8b) and 8c), the ion position is TEM 10 Because the field is zero, a small AC Stark shift may be possible. If the AC Stark shifts of the two qubit states of an addressed ion are different, an (undesirable) rotation of the qubit state of the addressed ion may occur, which can be reduced in this way. Furthermore, in the configuration shown in Figure 8c), by selecting the appropriate transition, it may be possible to make the AC Stark shifts identical (or similar) for both qubit states of the addressed ion. In this way, the differential Stark shift can be small or zero, and by extension, the rotation of the ion's qubit state can be reduced or avoided.

[0105] This approach, which uses multiple beam components of different shapes, may be combined with SDF modulation using polychromatic light. For example, the intensity gradient resulting from the overlap of two beam components can be modulated by inducing a frequency difference between the two beam components (corresponding to a detuned motion mode frequency), as described below.

[0106] Hermitian Gaussian beams (including Gaussian beams) and Laguerre Gaussian beams may be generated / formed, for example, using a spatial light modulator (SLM) and / or phase plate, which may be included in the optical elements 350 of the laser beam unit 320. However, the present invention is not limited to any beam shape. For example, the beamform may have a constant gradient in the radial / transverse direction with respect to the propagation direction, which may reduce the dependence of the SDF on the precise position of irradiated ions in the laser beam and simplify the technical implementation. SDF state dependent

[0107] The term "state-dependent force" (SDF), in the context of trapped ions, typically refers to a force that depends on the internal state of the trapped ion on which the SDF acts, such as a spin-dependent force. More specifically, an SDF induced by a laser beam depends on the state of the qubit represented by the ion. In other words, the SDF differs depending on the two qubit states of the addressed ion. More specifically, an SDF induced by a laser beam may act differently on a trapped ion when it is in the |0> state and when it is in the |1> state. Since a trapped ion can be in a superposition state of |0> and |1> states |ψ>=α|0>+β|1>, the SDF may act differently on the |0> and |1> components of the trapped ion's state simultaneously. In other words, the SDF may be a quantum mechanical superposition of (two) forces acting on the qubit states |0> and |1>, respectively. For example, if the ion's qubit states have different spins, the SDF may depend on the spin of the trapped ion. On the other hand, SDFs may generally be i) independent of the motion state of the trapped ions (this is often only true for first-order forces), and ii) independent of the internal states of other trapped ions. Generally, the strength and / or direction of the force may depend on the internal state. In particular, SDFs may have opposite directions and / or the same magnitude for two qubit states. For example, an SDF may be a dipole force, especially an optical dipole force.

[0108] Generally, approaches to making forces state-dependent may depend on the ion, and in particular on the internal state of the ion used to represent the qubit. Typically, when implementing an LS gate, a laser beam is used to (non-resonantly) excite transitions between qubit states and excited (non-qubit) states and / or induce a dipole force (i.e., the SDF may be a dipole force). Generally, unlike MS gates, LS gates are based on photon elastic scattering, i.e., scattering / blinding where only the direction of the photons changes due to the scattering process, and the wavelength does not change. Therefore, the frequency of the laser beam is usually detuned significantly from dipole transitions, i.e., transitions from the qubit state used for coupling to the excited non-qubit state(s).

[0109] More specifically, the energy level E of an atom / ion changes due to an electric field that changes over time. i Shift E i →E i +ΔE i This induces what is known as the "AC Stark" or "optical shift." As can be seen from the following results obtained from the second-order perturbation theory concerning the non-degenerate energy eigenstates |i> and |j> of the ions, the shift is proportional to the intensity of the laser beam.

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[0110] In this specification, the term "detuning" generally refers to the frequency difference / shift of the laser frequency relative to the quantum system frequency. In the case of detuning discussed herein, the laser frequency is the electromagnetic field oscillation frequency ω L Therefore, the frequency of the quantum system is the frequency (corresponding to the energy difference) of the transition from the qubit state to the excited (non-qubit) state. However, as will be further explained below, the term detuning refers to the frequency ω at which the laser beam is modulated. Mod and the frequency ω of the motion mode M It is also used for the frequency difference δ between the system and the system. Therefore, this term generally refers to the difference in laser frequency with respect to some (resonant) frequency given by the system of trapped ions. Laser frequencies lower than the quantum system frequency are called red-detuned, and lasers tuned higher than the quantum system frequency are called blue-detuned. Generally, detuning is considered small in a sense. In particular, the detuning with respect to the frequency from which the laser is detuned is smaller than the frequency difference of the laser with respect to almost all other frequencies.

[0111] The denominator in the above equation means that detuning from a particular frequency implies that the corresponding transition dominates the energy shift of the qubit state. The energy shift and the dipole force may be approximately determined based on the transition(s) to the excited state(s) selected by the detuning. Since the energy shift is proportional to the intensity of the laser beam (the interaction Hamiltonian H1 is proportional to the magnitude of the electric field), spatially varying intensity corresponds to a potential energy field, a so-called dipole potential. As a result, the (electric) dipole force acting on the resulting ion is proportional to the intensity gradient of the electric field. Further, since the interaction Hamiltonian H1 may depend on the polarization characteristics of the laser beam, the matrix element <j│H1│i> may depend on the polarization characteristics of the laser beam for a given state |i> and <j|, and this may be used to implement an SDF based on the polarization / phase gradient of the laser beam.

[0112] A first example of such a technique is shown in FIG. 10. Let the frequency difference between the qubit states |↓> and |↑> of the addressed qubit be represented as ω0, and let the frequency of the laser beam directed at the addressed ion be ω L and let the frequency of the motional mode used be ω M As shown in FIG. 10, the coupling is implemented using the excited state |e> (i.e., the internal state of the addressed ion different from the qubit state). More specifically, the frequency ω L of the electromagnetic field oscillation that induces the SDF is detuned from both of the transition frequencies ω e,↓ and ω e,↑ where ω e,↓ is the frequency difference between |↓> and the excited state |e>, and ω e,↑ is the frequency difference between |↑> and the excited state |e> (thus, ω e,↓ = ω e,↑ + ω0). In FIG. 10, the detuning for the transition from |↓> to |e> is represented as Δ, i.e., ω L = ω e,↓It is +Δ. Note that Figure 10 shows a negative detuning Δ<0. Typically, in this example, the detuning is large compared to the energy difference between the qubit states, |Δ|≫ω0, but is still small enough to be negligible for the other excited states (for example, the detuning for the other excited states must remain much larger than the detuning for the selected excited state |e>). This makes the detuning of transitions from both qubit states approximately equal. Typically, an excited state |e> is selected where the transition from the qubit state to the excited state is an electric dipole transition. The dipole force dependence of the qubit states may be realized by appropriate tuning / setting and / or modulation of the polarization component of the laser beam. For example, 9 Be + Using ions, the hyperfine ground state can be used as a qubit state, i.e., |↓>=|S 1 / 2 (F=2,m F =-2)> and |↑>=|S 1 / 2 (F=1,m F =-1)>, 2 P 1 / 2 and / or 2 P 3 / 2 The state can be used as an excited state (multiple states are possible).

[0113] A second example of this technology is shown in Figure 11, where the laser beam frequency ω L This differs from the previous example in that it is detuned symmetrically from the two transitions to the excited state. More specifically, the laser frequency is red-detuned (or negatively detuned, i.e., ω) with respect to the transition from the |↓> state to the excited state|e>. L <ω e,↓ (and), that is, ω L =ω e,↓ It is -ω0 / 2, and is blue-detuned (or positively detuned, i.e., ω) for the transition from the |↑> state to the excited state |e>. L >ω e,↓ (and), that is, ω L =ω e,↑It is +ω0 / 2. Such detuning (one red, the other blue) generates an SDF acting in opposite directions on the two qubit states. Since we are exemplarily assuming that the detuning is symmetric, the SDF has the same magnitude (absolute size) for both qubit states. This technique is also possible when the induced Stark shift is state-independent within the limit of a large detuning |Δ|≫ω0, for example, when the energy splitting of the qubit states is linearly independent of the change in the magnetic field (e.g., the same magnetic quantum number m F It can also be used in qubit states that have a clock state. For example, 43 Ca + When using ions, |↓>=|S 1 / 2 (F=4,m F =0)> and |↑>=|S 1 / 2 (F=3,m F We can use |e>=|D as the qubit state, where |e>=|D 5 / 2 > can be used as an excited state. Specific examples of optical qubits (multiple qubits are possible)

[0114] The first and second examples of this technology described above are, for example, D 5 / 2 40 Ca + , 88 Sr + , 138 Ba + , and 226 Ra + This also applies to optical transition qubits such as |↓>=|S. For example, a specific example of an optical qubit is a qubit called |↓>=|S 1 / 2 (m j =+1 / 2)> and |↑>=|D 5 / 2 (m j According to =+3 / 2)> 40 Ca + It may be encoded as follows.

[0115] In the case of a laser beam with a wavelength of 532 nm, the main contribution to the Stark shift is: (i) The |↓> state is caused by two dipole transitions |↓>→|e1> and |↓>→|e2> with wavelengths of 393 nm and 397 nm, respectively. In other words, the energy difference between the excited state |e1> and the |↓> state is

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[0116] laser beam frequency f L =2πω L =c / (532nm) is (generally, "ω" is the angular frequency and the symbol "f" refers to the corresponding normal frequency, which is related by ω=2πf), |e1>From the transition, Δf1=2π(ω L -ω e1,↓ ) = c / (532nm) - c / (393nm) ≈ -200 THz is detuned, |e2>From the transition, Δf2=2π(ω L -ω e2,↓ ) = c / (532nm) - c / (397nm) ≈ -192THz is detuned, |e3>From the transition, Δf3=2π(ω L -ω e3,↓ ) = c / (532nm) - c / (854nm) ≈ 212 THz is detuned.

[0117] In this example, these large detunings make the transitions insensitive to the polarization of the laser beam, allowing, for example, the use of a linearly polarized laser beam.

[0118] Selecting an excited state |e> with a large frequency difference from the qubit state corresponds to a large laser frequency ω corresponding to a large momentum transfer or a short wavelength (e.g., 200-2000 nm) photon. L It should be noted that this leads to the possibility of faster entangled gating compared to when longer wavelength electromagnetic radiation is used.

[0119] Further details of the exemplary techniques described above can be found in [1] and [2] (first example), [3] (second example), and [4] (third example). Finally, it should be noted that two or more excited states may be used to induce a state-dependent (dipole) force. In particular, there may be multiple excited states with energies close to each other, all of which contribute significantly to the dipole force. In general, the present invention is not limited to any particular method for inducing an SDF. SDF modulation

[0120] Generally, the geometric phase gates implemented herein allow coupling between internal qubit states via external motion states by applying an electromagnetic field (e.g., by a suitable laser beam / pulse). That is, qubits are stored in the internal states of each ion, and the motion degrees of freedom may be utilized as a "quantum bus" to entangle the internal qubit degrees of freedom. The laser beam that induces an SDF in an addressed ion couples / excites the motion mode in a state-dependent manner. The present invention differs from established implementations of MS gates and LS gates, in particular, with respect to the method by which the SDF is generated, and the direction of the SDF (or at least one component) relative to the direction of the (single) laser beam generating the SDF.

[0121] As will be explained in more detail below, (one or more) normal-mode (quantum) harmonic oscillators corresponding to the collective quantization motion of Coulomb-coupled ionic crystals are used for processing and transferring quantum information. Generally, a specific motion mode (e.g., one) may be selected and used to perform entanglement operations of the qubit states of two or more trapped ions. For brevity, the use of this single motion mode will be explicitly described, but the present invention is not limited thereto, and entanglement may be transmitted using multiple motion modes. The frequency of this mode used, i.e., the “motion mode used”, is ω M This is expressed as follows, and the subscript is frequently omitted when referring to the mode of motion used (i.e., |n> M (Not |n>). Other or most other motion modes do not excite other motion modes. Appropriate laser frequency ω L and / or modulation frequency ω Mod This can be ignored by selecting [a specific mode]. However, in larger ionic crystals, it may be difficult to use only one mode (excitation / de-excitation). The mode of motion used can be any of the ion's modes, and it should be further noted that it is also possible to couple to a mode of motion in the direction of the laser beam propagation, for example, by directing the laser beam towards the ion so that the addressed ion is not at the focal point of the laser beam.

[0122] Generally, the states of two or more qubits can be entangled by using the motion modes of a trapped ion and modulating the SDF (e.g., periodically) according to the frequency of the motion mode to non-resonantly excite the mode. The SDF may also be modulated by (periodically) changing the intensity / phase / polarization gradient at the location of an addressed ion. This approach, which will be further described below, is also known as an optical shift (LS) gate and details can be found in [1]-[4]. Displacement and geometric phase of a harmonic oscillator

[0123] The LS gate is based on the coherent displacement in phase space of the harmonic oscillator in the mode of use. The displacement is induced by the SDF, which depends on the state of the qubit. More specifically, it resonates with the oscillator, i.e., at frequency ω Mod The frequency ω of the quantum harmonic oscillator M The classical force

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[0124] Multiple sequential displacements may be performed to move the harmonic oscillator back to its original state (i.e., the total / cumulative displacement due to the displacement is zero) within a closed loop. Such a path / trajectory may be, for example, a circle or a polygon, but is referred to herein as a loop. When the state is moved back to its original position in phase space, the resulting geometric phase φ is

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[0125] Since SDF depends on the state of the addressed qubits, a state-dependent phase shift is obtained. In other words, SDF conditionally excites the usable motion mode depending on the state of all the addressed qubits. That is, the amount of excitation may depend on the state of the qubits, and / or whether the mode is excited at all may depend on the state of the qubits. This conditional excitation can be used to perform entanglement operations on ions using the state dependence of the force.

[0126] As an example, Figure 12 shows how an axial center of mass (COM) mode, in which all ions (including unaddressed ions) vibrate together at regular intervals, can be conditionally excited. In Figure 12, and as shown in the three-dimensional view of Figure 12e), we assume that the ions are aligned in a straight line in the y direction, and therefore the y direction is the ion axis direction. We assume that the propagation direction of both laser beams is in the z direction, and therefore the z direction is the beam direction of both beams. Figures 10a) to 10d) show xy cross-sections at z=0, illustrating how the SDF induced by the laser beams acts on the |00>, |01>, |10>, and |11> states, respectively. For both laser beams, we assume that i) the direction of the induced SDF is toward the center of each laser beam in the |0> state, and ii) the direction away from the center of each laser beam in the |1> state. Therefore, the SDF is in opposite directions with respect to the two qubit states. For example, a laser beam may have a Gaussian beam profile in which the intensity is maximum along the beam axis. Thus, the intensity gradient may point to the center of the laser beam, the force acting on |0> may be in the direction of the gradient, and the force acting on |1> may be in the opposite direction to the gradient. For example, the |0> and |1> states may undergo negative and positive energy shifts (see AC / optical shifts above) with respect to their energy levels, respectively, due to the laser beam.

[0127] More specifically, as can be seen from the figure, the first addressed ion 381 is located to the right of the beam axis of the first laser beam, and the second addressed ion 382 is located to the left of the beam axis of that laser beam. As a result, as shown in Figure 12a), the direction 1201 of the SDF acting on the first addressed ion in the |0> state is opposite to the direction 1202 of the SDF acting on the second addressed ion in the |0> state, so the forces cancel each other out in the |00> state, and the COM mode is not excited when the addressed qubit is in the |00> state. This means that the |00> state is not changed by the laser beam, i.e., |00>→|00> is performed by irradiating with the laser beam over the gate time. Furthermore, as shown in Figure 12b), the direction 1201 of the SDF acting on the first addressed ion in the |0> state is the same direction as the direction 1212 of the SDF acting on the second addressed ion in the |1> state. Therefore, the COM mode is excited when the addressed qubit is in the |01> state. Thus, the |01> state acquires phase φ from interaction with the laser beam, i.e., by irradiating with the laser beam over the gate time, |01>→e iφ The result is |01>. Similarly, as shown in Figure 12c), when the qubit is in the |10> state, the SDF acts in the same direction, so the |10> state also acquires phase φ, i.e., |10>→e iφ This results in |10>. Finally, as shown in Figure 12d), the direction 1211 of the SDF acting on the first addressed ion in the |1> state is opposite to the direction 1212 of the SDF acting on the second addressed ion in the |01> state. Therefore, the forces cancel each other out in the |11> state, and the COM mode is not excited when the addressed qubit is in the |1> state, i.e., |11>→|11>.

[0128] As shown in Figure 12, the LS gate can generally be used to implement the following unitary operators ({|00>,|01>,|10>,|11>} basis).

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[0129] Such an LS gate is equivalent to a universally controlled π-phase gate, and this gate is,

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[0130] For example, by initializing with a state such as |00> (not fully entangled) and performing a single qubit operation on each qubit, we can achieve a state that is still not fully entangled.

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[0131] Note that the modulation of the SDF (i.e., step S220) is performed while the SDF is induced in the ions. In other words, the laser beam acts on the ions with a modulated SDF for the duration of the gate time. Modulation refers to the change in each quantity over time.

[0132] More specifically, the term "gate time" T or "gate duration" T refers to the time / duration during which a laser beam(s) is modulated and (simultaneously) irradiates an addressed ion. Therefore, the gate time may also be the duration during which the SDF is applied / induced and modulated, i.e., the length of time during which the SDF interacts with the ion. In particular, the laser beam may be a laser pulse having a gate time duration. Outside the gate time, the amplitude of the laser pulse / beam may be zero, or the laser beam may not irradiate an addressed ion outside the gate time.

[0133] In general, the gate time may be predetermined, i.e., the gate time may be a predetermined time / duration. In particular, the gate time may be determined such that, after the gate time, the qubit is completely de-entangled from the state of the harmonic oscillator of the motion mode used. More specifically, the internal degrees of freedom used to represent the addressed qubit are de-entangled from the motion degrees of freedom of the qubit's state. For example, after the gate time, the wave function of the addressed ion may be in its original motion state (i.e., the initial state at the start of the gate time).

[0134] More specifically, after modulation over gate time, (i) the internal state of the addressed ion (e.g., the qubit state) may not be entangled with the state of the motion mode, and / or (ii) the total (cumulative) displacement of the harmonic oscillator in position-momentum space for the motion mode used may be zero, where the total displacement is the total displacement due to the modulation of the SDF over gate time.

[0135] For example, when determining the gate time (in advance), a cost function that depends on the gate time and gate fidelity (i.e., the fidelity of the entanglement operation performed) may be minimized. More specifically, shorter gate times may reduce the impact of the phase instability of the laser involved on the gate fidelity. Shorter gate times also have the advantage of faster quantum computation, which can be particularly important when performing extended quantum computations involving a large number of consecutive gate operations. However, gate times typically depend on the strength of the SDF, and consequently the intensity of the laser beam, and arbitrarily increasing these can lead to instability, technical problems, and ultimately, a decrease in fidelity. Periodic modulation and detuning δ

[0136] In general, a cycle in the phase space of a harmonic oscillator is a classical force

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[0137] For example, SDF modulates the modulation frequency ω during the gate time. Mod =ω M It may also be modulated with +δ. In this regard, frequency ω M Modulation frequency ω Mod The (usually small) detuning δ is the phase

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[0138] When the oscillator is periodically driven through a circular orbit in phase space (i.e., ω Mod =ω M +δ), the driving force is due to the detuning δ and the frequency ω of the operating mode. M The system becomes asynchronous but resynchronizes after a duration of 2π / δ. Therefore, the detuning δ determines how quickly the oscillator rotates in phase space. Thus, reducing the detuning parameter δ may allow for an increase in gate velocity. Over a duration of 2π / δ, the motion state is displaced along a circular path in phase space, returning to its original point in phase space after a time of 2π / δ, resulting in a geometric phase equal to the enclosed area of ​​phase space, which can be determined according to the following equation (see [2]).

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[0139] In general, the loop may be traversed multiple times. More specifically, the gate time may be given as T = 2πn / δ, where n is an integer greater than zero and δ is the frequency ω of the motion mode used. MThis is a detuning process. By selecting the laser beam intensity (the dipole force is proportional to the intensity) and the detuning δ, a π / 2 phase shift can be obtained for a specific state of an addressed qubit over a gate time T, for example, as in the example above.

[0140] In general, a larger detuning δ can shorten the time it takes for the oscillator to return to its original state. This can also shorten the gate time (i.e., speed up the gate) because the gate time becomes an integer multiple of the time required to execute one cycle in phase space. More specifically, a harmonic oscillator may be driven through multiple loops / cycles, which allows for the accumulation of state-dependent geometric phases from all cycles to implement a particular quantum gate. However, if the detuning δ is too large, it may couple with other motion modes, which can be undesirable. Therefore, the detuning δ may generally be chosen to be small compared to the motion mode being used, and / or small compared to the frequency difference between the motion mode being used and other (arbitrary) motion modes.

[0141] Generally, the frequency δ may be determined by optimizing the trade-off between gate time and gate fidelity (or between gate time and gate error), usually under the constraint that the qubit state is not entangled with the degrees of freedom of motion after the gate time T = 2πn / δ, and / or that the total acquired phase φ = nφ0 obtained over the gate time T has a specific / predetermined value given by the implemented quantum gate. (For example, it may be predetermined before the entanglement operation is performed.) Thus, this determination may involve determining an appropriate integer n that corresponds to the number of cycles in phase space and on which the gate time also depends. Furthermore, the term "gate fidelity" refers to the fidelity of the entanglement operation performed on the ion addressed by the laser beam. Generally, this is a measure of how well the intended unitary operation on the qubit is actually realized by the physical implementation. As can be seen from the above equation for the acquired phase, as the force increases, and consequently the intensity of the laser beam increases, the gate becomes faster, but if the beam intensity is too high, the fidelity may decrease.

[0142] Furthermore, a large detuning δ means a shorter gate time and a larger displacement in phase space. This can lead to an even larger ion wave packet size, potentially breaking the Ramdicke approximation based on a wave packet size small compared to the laser wavelength; that is, the coupling between the optical field and ion motion will no longer be a good linear approximation. These and other considerations may introduce further constraints or trade-offs in the selection / determination of δ. SDF Modulation Method

[0143] The SDF may be modulated by inducing an SDF by modulating the transverse gradient of the laser beam at the location of each addressed ion. The SDF acting on all addressed ions, in particular all transverse gradients at the locations of the addressed ions, may be modulated in the same way (e.g., the same frequency and the same phase). In particular, the gradient may be directly / actively modulated at a frequency (or more frequencies) close to one or more motion modes. This makes it possible to use only one laser beam from one direction without using standing waves for modulation. In particular, the SDF induced by the laser beam (one, each, or all) at the locations of the ions addressed by the laser beam, 1) Modulating the intensity and / or amplitude of the electromagnetic field of a laser beam, 2) Changing the position and / or direction of the laser beam relative to the trapped ions, 3) Including light of different frequencies in the laser beam, and / or 4) Modulating the polarization and / or phase of the laser beam. It is used with modulation.

[0144] It should be noted that each of the methods 1) to 4) modulates the SDF at the location of each trapped ion by modulating the gradient of the laser beam at said location. The present invention is not limited to any of these methods, and it should be noted that these methods may be combined by modulating the SDF using two or more of these methods simultaneously. Furthermore, in general, SDFs acting on different ions may be modulated by the same(or more) modulation method or by different(or more) modulation methods. Method 1) Modulation of intensity and / or amplitude

[0145] In general, intensity / phase / polarization gradient modulation can be achieved by modulating the intensity of a laser beam. For example, the frequency, phase, and / or amplitude of individual beams can be modulated using one or more AOMs and / or one or more AODs, for example, AOMs in a double-pass configuration. In Figure 6, for example, amplitude and / or phase modulation can be performed using AOD620 and / or AOD640. Alternatively or additionally, laser beam modulation can also be performed using an AOM (not shown in Figure 6) placed, for example, before the first AOD620 in Figure 6.

[0146] However, the present invention is not limited to a specific method for modulating intensity, etc. Therefore, a modulated laser beam may have a modulated gradient that induces a modulated SDF acting on an addressed ion. For example, when two ions are addressed by two laser beams respectively (for example, as shown in Figure 12), the electromagnetic fields of the first and second laser beams

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[0147] As explained above, dipole force

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[0148] In general, modulation of intensity / phase / polarization gradients can be achieved by modulating (e.g., changing) the position of the laser beam relative to the position of the ions. For example, the laser beam may have a gradient that is not constant in the beam radial direction, and / or radial distance from the beam axis, as in the case of a Gaussian beam.

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[0149] The intensity gradient at the ion's position may be altered by changing the ion's position relative to the beam axis (i.e., by changing the x and / or y coordinates). Since the position is relative to the beam axis, this may be achieved by moving the laser beam axis relative to the addressed ion's position. In this way, the intensity gradient at the addressed ion's position can be varied, and the resulting SDF can be modulated. In particular, by changing the laser beam's position relative to the ion's position, radial gradients can be used to excite radial or axial motion modes (or both). This advantage also applies to planar crystals in 3D pole traps, or to more advanced trap topologies having surface ion traps. Method 3) Modulation using a multicolor laser beam

[0150] In general, modulation of intensity / phase / polarization gradient can be achieved by including light of multiple different frequencies in a laser beam. In other words, a laser beam may contain two or more beam components of different frequencies.

[0151] For example, by using light of two frequencies, a periodically changing intensity gradient at the location of an addressed ion can be realized. The beat notes of the two frequencies result in effective intensity modulation. More specifically, the laser beam is affected by the electric field amplitude.

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[0152] wave vector

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[0153] For example, a monochromatic laser beam may be split into two beams using a splitter, and at least one of the acquired laser beams may be shifted using, for example, an acousto-optic modulator (AOM), and the shifted laser beams may be fed into the same fiber (this results in a single laser beam that induces SDF in ions).

[0154] Alternatively, an AOM can be used to simultaneously split a monochromatic laser beam into multiple laser beams having different frequencies and directions (i.e., splitting and shifting in a single step). In this way, mainly three laser beams may be generated, one having the same direction and frequency as the original monochromatic laser beam, and two laser beams being first-order diffraction beams. A multichromatic laser beam can then be obtained by feeding two or more of these three laser beams into the same fiber.

[0155] As a further possibility, an electro-optic modulator (EOM) can be used to obtain a laser beam with two sidebands from a monochromatic laser beam (in a single step). Method 4) - Phase and / or polarization modulation

[0156] In general, phase / polarization modulation can be achieved by modulating the phase / polarization of a laser beam. More specifically, if the electric field of the laser beam is described as follows,

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[0157] As mentioned above, using lateral SDF can facilitate easy access to various motion modes in different directions. Axial and / or radial modes

[0158] Figure 13 shows an exemplary beam geometry for entanglement of two ions in a linear ion chain using axial motion modes. As shown in Figure 13d), which reproduces Figure 12e), the labeling of axes relative to the laser direction and ion axes, and the definition of the coordinate origin are identical to the example in Figure 12. The two ions are addressed by two Gaussian laser beams propagating in the Z direction, i.e., the darker the region of the laser beam, the higher the intensity of the laser beam. To excite axial modes (as described above, the specific axial modes excited by the beam depend on the modulation of the SDF), an SDF in the ion axis direction (y direction) is required. Since the laser beam is Gaussian distributed, the intensity gradient used to induce the SDF is directed toward the center of the beam axis. Therefore, the laser beam is directed toward the ions such that the ions are displaced from their respective beam axes in the y direction. As shown in Figure 13a) showing the xy cross-section at z=0, Figure 13b) showing the yz cross-section at x=0, and Figure 13c) showing the xz cross-section at x=-a, in the example of Figure 13, the displacement of both addressed ions from the beam axis (which is y=-a for the left addressed ion) is a negative displacement in the y direction (the addressed ion is to the left of the beam axis).

[0159] Figure 14 shows an exemplary beam geometry for entanglement of two ions in a linear ion chain using radial motion modes. The laser beam is directed at the ions such that the ions are displaced from their respective beam axes in the x-direction (otherwise the situation may be the same as in Figures 13 and / or 12, as shown in Figure 14d). Displacement in the x-direction means that the intensity gradient at the ion's position is in the x-direction, which is one of the radial directions relative to the ion axis (y-direction). Thus, the induced SDF may also be in the x-direction, and motion modes in the x-direction can be excited and deexcited by the SDF. As shown in Figure 14a) showing the xy cross-section at z=0, Figure 14b) showing the yz cross-section at x=0, and Figure 14c) showing the xz cross-section at x=-a, in the example of Figure 13, the displacement of both addressed ions from the beam axis (which is y=-a for the left addressed ion) is a negative displacement in the x-direction (the addressed ion is to the left of the beam axis).

[0160] In general, multiple interaction modes may be used simultaneously to entangle different pairs of ions at the same time. For example, four SDFs may be induced simultaneously in four (e.g., mutually distinct) ions. Each of the four SDFs may be induced using four laser beams.

[0161] The first motion mode may be excited in accordance with the internal states of two ions acting on the two SDFs by modulating two of the four SDFs according to the frequency of the first motion mode. The second motion mode may be excited in accordance with the internal states of two ions acting on the other two SDFs by modulating the other two of the four SDFs according to the frequency of a second motion mode (different from the first motion mode). Note that the first and second motion modes are different motion modes of all four ions. The two motion modes may both be axial modes, both be radial modes, or one may be a radial mode and the other an axial mode.

[0162] More ion pairs may be entangled by adding additional beams to additional ions and adjusting the laser beam parameters accordingly. Center of Mass (COM) mode and stretching mode

[0163] Lateral SDF can facilitate coupling to different motion modes in the same direction, such as different axial modes.

[0164] Figures 15 and 16 show excitation of the axial COM mode and axial stretching mode, respectively, using the same beam geometry as described in Figure 13. However, the situation in Figure 15 differs from that in Figure 13 in that the left addressed ion 381 is to the right of the beam axis in the y-direction (similar to the situation in Figure 13, the right addressed ion 382 is to the left of the beam axis in the y-direction). Furthermore, the situation in Figure 16 differs from that in Figure 13 in that both addressed ions are to the right of the beam axis in the y-direction.

[0165] More specifically, for excitation of the axial COM mode, Figure 15a) shows the xy cross-section at z=0, Figure 15b) shows the yz cross-section at x=0, and Figure 15c) shows the xz cross-section at x=-a. Similarly, for excitation of the axial stretching mode, Figure 16a) shows the xy cross-section at z=0, Figure 16b) shows the yz cross-section at x=0, and Figure 16c) shows the xz cross-section at x=-a.

[0166] In Figure 15, the SDF (white arrow) is modulated near the COM mode frequency, so modes other than axial COM are not excited. In Figure 16, the SDF is modulated near the axial stretching mode frequency, so modes other than the axial stretching mode are not excited. In both figures, if the two ions are in different internal qubit states, the two addressed ions accumulate a net phase shift. If the two ions are in the same internal qubit state, no net phase shift is obtained.

[0167] Therefore, when ions are in different states, the SDF in Figure 15 acts in the same direction, exciting or de-exciting the COM mode oscillations shown by the black arrows in Figures 13a) and 13b). More specifically, the force 1501 acting on the first ion when it is in the |0> state is in the same direction as the force 1512 acting on the second ion when it is in the |1> state, and the force 1511 acting on the first ion when it is in the |1> state is in the same direction as the force 1502 acting on the second ion when it is in the |0> state.

[0168] To excite or deexcite stretching mode oscillations for different qubit states, as shown by the black arrows in Figures 16a) and 16b), the SDF (white arrow) in Figure 16 acts in opposite directions when the ions are in different states. More specifically, the force 1601 acting on the first ion when it is in the |0> state is in the opposite direction to the force 1612 acting on the second ion when it is in the |1> state, and the force 1611 acting on the first ion when it is in the |1> state is in the opposite direction to the force 1602 acting on the second ion when it is in the |0> state. Modes of motion of planar ionic crystals

[0169] As already stated, the present invention may be applied not only to linear ion rows but also, in general, to entangle two or more ions of any ionic crystal having a common / joint motion mode. In particular, a variety of beam steering options may be used to excite a specific (freely selectable) motion mode of a planar ionic crystal, which can be very difficult with conventional methods because the optical access is further restricted by the geometric shape of the planar ionic crystal when entanglement of only specific ions.

[0170] This is shown in Figure 17, which illustrates the motion modes of ions in a 2D ionic crystal. More specifically, Figure 17d) shows the resting / equilibrium positions of the ions, along with the 3D coordinate system used. As can be seen from the figure, the ions are arranged in a zigzag pattern within the plane. Furthermore, for the aforementioned 2D ionic crystal, Figure 17a) shows the xy cross-section at z=0, Figure 17b) shows the yz cross-section at x=0, and Figure 17c) shows the xz cross-section at x=-a.

[0171] As can be seen in Figures 17 and 17b), the arrows indicate the direction and amplitude of the vibration of each ion according to the motion mode, but the motion modes of such 2D ionic crystals are not generally limited to one dimension. Note that the arrows in Figure 17b) are projections onto the zy-plane. In other words, a single motion mode can correspond to the vibration of an ion in different directions and with different amplitudes. However, the motion modes of a planar ionic crystal can be used to excite and entangle the ion in the same way as the motion modes of a linear ionic crystal, that is, by directing a laser beam modulated at a frequency detuned from the motion mode towards an addressed ion. Further aspects

[0172] The embodiments and exemplary implementations described above illustrate several non-limiting examples. It is understood that various modifications may be made without departing from the subject matter of the claims. For example, modifications may be made to adapt the embodiments to new systems and scenarios without departing from the central concepts described herein. Furthermore, specific pulse sequences that reduce crosstalk errors may be used in combination. In addition, any step of the method described herein may be included as code instructions in a program that may be executed by one or more processors.

[0173] In summary, the disclosure provides embodiments for entanglement of two or more trapped ions. For this purpose, a common mode of motion of the two or more trapped ions is used, which is conditionally excited and / or deexcited depending on the internal state of the two or more trapped ions. The common mode of motion is conditionally excited / deexcited by inducing a respective vertical state-dependent force (SDF) in each of the two or more trapped ions, which is modulated according to the frequency of the mode of motion. More specifically, each vertical SDF is induced by a laser beam and acts perpendicular to the propagation direction of the laser beam that induces the vertical SDF. The SDF may be modulated by (i) modulating the intensity and / or amplitude of the electromagnetic field of a first laser beam, (ii) changing the position and / or direction of the first laser beam relative to the first trapped ions, and / or (iii) including light of different frequencies in the laser beam.

[0174] According to the first aspect, A method is provided for entanglement of a first trapped ion and a second trapped ion. The method utilizes the modes of motion of the first trapped ion and the second trapped ion (for example, for the entanglement), (i) The step of inducing a first state-dependent force on the first trapped ion using a first laser beam, (ii) The step of using a second laser beam to induce a second state-dependent force on the second trapped ion. The steps for inducing the first state-dependent force and the second state-dependent force are performed simultaneously. The first state-dependent force acts perpendicular to the propagation direction of the first laser beam, The second state-dependent force acts perpendicular to the propagation direction of the second laser beam, In the step of inducing the first state-dependent force and the second state-dependent force, the first laser beam and the second laser beam are modulated according to the frequency of the motion mode, thereby modulating the first state-dependent force and the second state-dependent force, respectively, and exciting the motion mode according to the internal state of the first trapped ion and the second trapped ion. Furthermore, (a) in modulating the first state-dependent force, the first state-dependent force is such that at the position of the first trapped ion, (ai) Modulating the intensity and / or amplitude of the electromagnetic field of the first laser beam, (a.ii) Changing the position and / or direction of the first laser beam relative to the first trapped ion, and / or (a.iii) Modulated by including light of different frequencies in the laser beam, and / or (b) In modulating the second state-dependent force, the second state-dependent force is such that at the position of the second trapped ion, (bi) Modulating the intensity and / or amplitude of the electromagnetic field of the second laser beam, (b.ii) Changing the position and / or direction of the second laser beam relative to the second trapped ion, and / or (b.iii) Modulated by including light of different frequencies in the laser beam.

[0175] According to the second embodiment provided in addition to the first embodiment, (i) In the step of inducing the first state-dependent force, the electromagnetic field of the first laser beam has a gradient component perpendicular to the propagation direction of the first laser beam at the location of the first trapped ion, wherein the gradient component of the first laser beam is an intensity gradient component, a phase gradient component, and / or a polarization gradient component, and / or (ii) In the step of inducing the second state-dependent force, the electromagnetic field of the second laser beam has a gradient component perpendicular to the propagation direction of the second laser beam at the position of the second trapped ion, wherein the gradient component of the second laser beam is an intensity gradient component, a phase gradient component, and / or a polarization gradient component.

[0176] According to the third aspect provided in addition to the first or second aspect, In the step of inducing the first state-dependent force, the first state-dependent force is modulated for the duration of the gate time. In the step of inducing the second state-dependent force, the second state-dependent force is modulated for the duration of the gate time. The gate time, after the modulation over the gate time, (i) The internal states of the first trapped ion and the second trapped ion are not entangled with the state of the motion mode, and / or (ii) A predetermined time during which the total displacement of the harmonic oscillator in the motion mode is zero, and the total displacement is the total displacement due to the modulation of the first state-dependent force and the second state-dependent force over the gate time.

[0177] According to the fourth aspect provided in addition to the third aspect, (i) The gate time is given as T = 2πn / δ, where n is an integer greater than zero and δ is a predetermined frequency. (ii) In the step of inducing the first state-dependent force, the first state-dependent force modulates at a modulation frequency ω during the gate time. Mod =ω M Modulated with +δ, (iii) In the step of inducing the second state-dependent force, the second state-dependent force modulates at a modulation frequency ω during the gate time. Mod =ω M Modulated with +δ, (iv)ω M This is the frequency of the aforementioned motion mode.

[0178] According to the fifth aspect provided in addition to the fourth aspect, The frequency δ is predetermined by optimizing the trade-off between the gate time T and the gate fidelity, wherein the gate fidelity is the fidelity of the entanglement of the first trapped ion and the second trapped ion.

[0179] According to the sixth aspect provided in addition to any one of the first to fifth aspects, The first laser beam and / or the second laser beam have a Gaussian beam shape, a supergaussian beam shape, a Laguerre-Gaussian beam shape, or a Hermitian-Gaussian beam shape.

[0180] According to the seventh aspect provided in addition to any one of the first to fifth aspects, (i) The first laser beam has two or more beam components having different beam shapes, and / or (ii) The second laser beam has two or more beam components having different beam shapes.

[0181] According to the eighth aspect provided in addition to the seventh aspect, (i) The first beam component of the two or more beam components of the first laser beam has a Gaussian beam shape, the second beam component of the two or more beam components of the first laser beam has a Hermitian Gaussian beam shape, and / or (ii) The second beam component of the second laser beam has a Gaussian beam shape, and the second beam component of the second laser beam has a Hermitian Gaussian beam shape.

[0182] According to the ninth aspect provided in addition to any one of the first to eighth aspects, The method, simultaneously with the step of inducing the first state-dependent force and the second state-dependent force, (i) Using a third laser beam to induce a third state-dependent force on a third trapped ion, (ii) The step of using a fourth laser beam to induce a fourth state-dependent force on a fourth trapped ion. (a) The third state-dependent force acts perpendicular to the propagation direction of the third laser beam, (b) The fourth state-dependent force acts perpendicular to the propagation direction of the fourth laser beam, (c) In the step of inducing the third state-dependent force and the fourth state-dependent force, the third laser beam and the fourth laser beam are modulated according to the frequencies of the second motion modes of the first trapped ion, the second trapped ion, the third trapped ion, and the fourth trapped ion, respectively, thereby modulating the third state-dependent force and the fourth state-dependent force, and exciting the second motion mode according to the internal state of the third trapped ion and the fourth trapped ion.

[0183] According to the tenth aspect provided in addition to any one of the first to ninth aspects, The first trapped ion and the second trapped ion are ions from a plurality of ions trapped in a straight line or in a two-dimensional lattice.

[0184] According to the eleventh aspect provided in addition to any one of the first to tenth aspects, The first laser beam and the second laser beam are directed towards the first trapped ion and the second trapped ion by adjusting the angle of incidence of the beams with respect to the principal plane of the objective lens, using the same objective lens.

[0185] According to a twelfth aspect provided in addition to any one of the first to eleventh aspects, (i) The first state-dependent force depends on the internal state of the first trapped ion, and / or (ii) The second state-dependent force depends on the internal state of the second trapped ion.

[0186] According to the 13th aspect provided in addition to any one of the 1st to 12th aspects, (i) The two internal states of the first trapped ion are used to model the first qubit, (ii) The two internal states of the second trapped ion are used to model the second qubit, (iii) The first state-dependent force differs depending on the two internal states of the first trapped ion, (iv) The second state-dependent force differs depending on the two internal states of the second trapped ion. According to the 14th aspect, instead of (i) to (iv) of the 14th aspect, or in addition to (i) to (iv) of the 14th aspect, the method provides quantum gates for the first qubit and the second qubit, particularly optical shift LS gates and / or σ gates. z σ z Implement the gate.

[0187] According to a fourteenth aspect, an apparatus is provided for controlling a first laser beam and a second laser beam to entangle the first trapped ions and the second trapped ions using the motion modes of the first and second trapped ions. The apparatus is (i) A first laser beam for inducing a first state-dependent force on the first trapped ion, (ii) A second laser beam for inducing a second state-dependent force on the second trapped ion and It includes a circuit configured to control both simultaneously. The first state-dependent force acts perpendicular to the propagation direction of the first laser beam, and the second state-dependent force acts perpendicular to the propagation direction of the second laser beam. The circuit modulates the first and second laser beams in accordance with the frequency of the motion mode during the induction of the first and second state-dependent forces (for example, during the induction), thereby exciting the motion mode in accordance with the internal states of the first and second trapped ions. Furthermore, the circuit, (a) In the modulation of the first state-dependent force, the first state-dependent force is expressed at the position of the first trapped ion. (ai) Modulating the intensity and / or amplitude of the electromagnetic field of the first laser beam, (a.ii) Changing the position and / or direction of the first laser beam relative to the first trapped ion, and / or (a.iii) Modulation by including light of different frequencies in the laser beam, and / or (b) In the modulation of the second state-dependent force, the second state-dependent force is expressed at the position of the second trapped ion. (bi) Modulating the intensity and / or amplitude of the electromagnetic field of the second laser beam, (b.ii) Changing the position and / or direction of the second laser beam relative to the second trapped ion, and / or (b.iii) The laser beam is configured to be modulated by including light of different frequencies.

[0188] According to the 15th aspect provided in addition to the 14th aspect, The aforementioned circuit is (i) In the induction of the first state-dependent force, the electromagnetic field of the first laser beam has a gradient component perpendicular to the propagation direction of the first laser beam at the location of the first trapped ion, and the gradient component of the first laser beam is an intensity gradient component, a phase gradient component, and / or a polarization gradient component, and / or (ii) In the induction of the second state-dependent force, the electromagnetic field of the second laser beam has a gradient component perpendicular to the propagation direction of the second laser beam at the location of the second trapped ion, and the gradient components of the second first laser beam are an intensity gradient component, a phase gradient component, and / or a polarization gradient component.

[0189] According to the 16th aspect provided in addition to the 14th or 15th aspect, The aforementioned circuit, (i) In the induction of the first state-dependent force, the first state-dependent force is modulated for the duration of the gate time, (ii) In the induction of the second state-dependent force, the second state-dependent force is configured to be modulated over the gate time. The gate time is configured to be modulated over the gate time, (a) The internal states of the first trapped ion and the second trapped ion are not entangled with the state of the motion mode, and / or (b) A predetermined time during which the total displacement of the harmonic oscillator in the motion mode is zero, and the total displacement is the total displacement due to the modulation of the first state-dependent force and the second state-dependent force over the gate time.

[0190] According to the 17th aspect provided in addition to the 16th aspect, The aforementioned circuit, (i) In the induction of the first state-dependent force, the first state-dependent force is modulated at a frequency ω during the gate time. Mod =ω M Modulated with +δ, (ii) In the induction of the second state-dependent force, the second state-dependent force is modulated at a frequency ω during the gate time. Mod =ω M Modulated with +δ It is configured such that the gate time is given as T = 2πn / δ, where n is an integer greater than zero, δ is a predetermined frequency, and ω M This is the frequency of the aforementioned motion mode.

[0191] According to the 18th aspect provided in addition to the 17th aspect, The frequency δ is predetermined by optimizing the trade-off between the gate time T and the gate fidelity, wherein the gate fidelity is the fidelity of the entanglement of the first trapped ion and the second trapped ion.

[0192] According to the 19th aspect provided in addition to any one of the 14th to 18th aspects, The first laser beam and / or the second laser beam have a Gaussian beam shape, a supergaussian beam shape, a Laguerre-Gaussian beam shape, or a Hermitian-Gaussian beam shape. In particular, the circuit may be configured to control the first laser beam and / or the second laser beam to have a Gaussian beam shape, a supergaussian beam shape, a Laguerre-Gaussian beam shape, or a Hermitian-Gaussian beam shape (for example, by controlling an optical element).

[0193] According to the 20th aspect provided in addition to any one of the 14th to 18th aspects, (i) The first laser beam has two or more beam components having different beam shapes, and / or (ii) The second laser beam has two or more beam components having different beam shapes.

[0194] According to a 21st aspect provided in addition to the 20th aspect, (i) the first beam component of the two or more beam components of the first laser beam has a Gaussian beam shape, the second beam component of the two or more beam components of the first laser beam has a Hermitian Gaussian beam shape, and / or (ii) the second beam component of the two or more beam components of the second laser beam has a Gaussian beam shape, and the second beam component of the two or more beam components of the second laser beam has a Hermitian Gaussian beam shape.

[0195] In particular, in the 20th and 21st embodiments, The circuit may be configured to control the first laser beam and / or the second laser beam to have such beam components.

[0196] According to the 22nd aspect provided in addition to any one of the 14th to 21st aspects, The circuit controls the first and second laser beams in such a way that it induces the first and second state-dependent forces, (i) Control the third laser beam to induce a third state-dependent force on the third trapped ion, (ii) The fourth laser beam is configured to control the fourth trapped ion to induce a fourth state-dependent force. (a) The third state-dependent force acts perpendicular to the propagation direction of the third laser beam, (b) The fourth state-dependent force acts perpendicular to the propagation direction of the fourth laser beam, The aforementioned circuit, (c) In the induction of the third state-dependent force and the fourth state-dependent force (for example, during the induction), the third laser beam and the fourth laser beam are modulated according to the frequencies of the second motion modes of the first trapped ion, the second trapped ion, the third trapped ion and the fourth trapped ion, thereby modulating the third state-dependent force and the fourth state-dependent force, respectively, and exciting the second motion mode according to the internal state of the third trapped ion and the fourth trapped ion.

[0197] According to the 23rd aspect provided in addition to any one of the 14th to 22nd aspects, The first trapped ion and the second trapped ion are ions from a plurality of ions trapped in a straight line or in a two-dimensional lattice.

[0198] According to the 24th aspect provided in addition to any one of the 14th to 23rd aspects, The circuit is configured to direct the first and second trapped ions towards the first and second trapped ions by adjusting the angle of incidence of the beams with respect to the main plane of the objective lens, using the same objective lens.

[0199] According to the 25th aspect provided in addition to any one of the 14th to 24th aspects, (i) The first state-dependent force depends on the internal state of the first trapped ion, and / or (ii) The second state-dependent force depends on the internal state of the second trapped ion.

[0200] According to the 26th aspect provided in addition to any one of the 14th to 25th aspects, (i) The two internal states of the first trapped ion are used to model the first qubit, (ii) The two internal states of the second trapped ion are used to model the second qubit, (iii) The first state-dependent force differs depending on the two internal states of the first trapped ion, (iv) The second state-dependent force differs depending on the two internal states of the second trapped ion. According to the 28th aspect, instead of (i) to (iv) of the 28th aspect, or in addition to (i) to (iv) of the 28th aspect, The circuit provides quantum gates for the first qubit and the second qubit, particularly optical shift LS gates and / or σ gates. z σ z It is configured to implement a gate.

Claims

1. A method for entanglement of a first trapped ion (381) and a second trapped ion (382) using the motion modes of a first trapped ion (381) and a second trapped ion (382), Step (S240) of inducing a first state-dependent force (371) on the first trapped ion (381) using a first laser beam (361), Step (S240) of inducing a second state-dependent force (372) on the second trapped ion (382) using a second laser beam (362), Includes, The step (S240) that induces the first state-dependent force (371) and the second state-dependent force (372) is performed simultaneously. The first state-dependent force (371) acts perpendicular to the propagation direction of the first laser beam (361), The second state-dependent force (372) acts perpendicular to the propagation direction of the second laser beam (362), In the step of inducing the first state-dependent force (371) and the second state-dependent force (372), the first laser beam and the second laser beam (362) are modulated according to the frequency of the motion mode to modulate the first state-dependent force (371) and the second state-dependent force (372), respectively (S220), thereby exciting the motion mode according to the internal state of the first trapped ion (381) and the second trapped ion (382). In modulating the first state-dependent force (371) (S220), the first state-dependent force (371) is such that at the position of the first trapped ion (381), Modulating the intensity and / or amplitude of the electromagnetic field of the first laser beam (361), Changing the position and / or direction of the first laser beam (361) relative to the first trapped ion (381), and / or Including light of different frequencies in the laser beam, Modulated by, and / or In modulating the second state-dependent force (372) (S220), the second state-dependent force (372) is such that at the position of the second trapped ion (3z2), Modulating the intensity and / or amplitude of the electromagnetic field of the second laser beam (362), Changing the position and / or direction of the second laser beam (362) relative to the second trapped ion (382), and / or Including light of different frequencies in the laser beam, Modulated by, In the step of inducing the first state-dependent force (371), the electromagnetic field of the first laser beam (361) has a gradient component perpendicular to the propagation direction of the first laser beam (361) at the position of the first trapped ion (381), The gradient components of the first laser beam (361) are an intensity gradient component, a phase gradient component, and / or a polarization gradient component. The first laser beam (361) includes a first beam component (811) having a first beam shape and a second beam component (812) having a second beam shape. The first beam shape is, Unlike the second beam shape described above, The electric field is zero at the position of the first trapped ion (381), The gradient component of the first laser beam at the position of the first trapped ion (381) increases with the intensity of the first beam component (811). A method characterized by the following features.

2. In the step of inducing the second state-dependent force (372), the electromagnetic field of the second laser beam (362) has a gradient component perpendicular to the propagation direction of the second laser beam (362) at the position of the second trapped ion (382), and the gradient component of the second laser beam (362) is an intensity gradient component, a phase gradient component, and / or a polarization gradient component. The method according to claim 1.

3. In the step of inducing the first state-dependent force (371), the first state-dependent force (371) is modulated for the duration of the gate time. In the step of inducing the second state-dependent force (372), the second state-dependent force (372) is modulated during the gate time. The gate time, after the modulation over the gate time, The internal states of the first trapped ion (381) and the second trapped ion (382) are not entangled with the state of the motion mode, and / or The total displacement in the position-momentum space of the harmonic oscillator in the motion mode is zero, and the total displacement is the total displacement due to the modulation of the first state-dependent force (371) and the second state-dependent force (372) over the gate time. The method according to claim 1, wherein the time is predetermined.

4. The gate time is given as T = 2πn / δ, where n is an integer greater than zero and δ is a predetermined frequency. In the step of inducing the first state-dependent force (371), the first state-dependent force (371) modulates at a modulation frequency ω during the gate time. Ｍｏｄ = ω Ｍ Modulated with +δ, In the step of inducing the second state-dependent force (372), the second state-dependent force (372) modulates at a modulation frequency ω during the gate time. Ｍｏｄ = ω Ｍ Modulated with +δ, ω Ｍ The frequency of the aforementioned motion mode is The method according to claim 3.

5. The aforementioned frequency δ is The integer n is determined based on the implemented quantum gate, Minimizing the cost function which depends on the gate time T and gate fidelity, wherein the gate fidelity is the fidelity of the entanglement of the first trapped ion (381) and the second trapped ion (382), The method according to claim 4, as predetermined by [the relevant method].

6. The first laser beam and / or the second laser beam (362) have a Gaussian beam shape, a supergaussian beam shape, a Laguerre-Gaussian beam shape, or a Hermitian-Gaussian beam shape. The method according to claim 1.

7. The first laser beam (361) has two or more beam components having different beam shapes and / or beam shapes that are displaced relative to each other, and / or The second laser beam (362) has two or more beam components having different beam shapes and / or beam shapes that are displaced relative to each other. The method according to claim 1.

8. The first beam component of the first laser beam (361) has a Gaussian beam shape, the second beam component of the first laser beam (361) has a Hermitian Gaussian beam shape, and / or The second beam component of the second laser beam (362) has a Gaussian beam shape, and the second beam component of the second laser beam (362) has a Hermitian Gaussian beam shape. The method according to claim 7.

9. The method, simultaneously with the step of inducing the first state-dependent force (371) and the second state-dependent force (372), The steps include using a third laser beam to induce a third state-dependent force on a third trapped ion, The steps include using a fourth laser beam to induce a fourth state-dependent force on a fourth trapped ion, It further includes, The third state-dependent force acts perpendicular to the propagation direction of the third laser beam, The fourth state-dependent force acts perpendicular to the propagation direction of the fourth laser beam, In the step of inducing the third state-dependent force and the fourth state-dependent force, the third laser beam and the fourth laser beam are modulated according to the frequencies of the second motion modes of the first trapped ion (381), the second trapped ion (382), the third trapped ion, and the fourth trapped ion, thereby modulating the third state-dependent force and the fourth state-dependent force, respectively, and exciting the second motion mode according to the internal state of the third trapped ion and the fourth trapped ion. The method according to claim 1.

10. The first trapped ion (381) and the second trapped ion (382) are ions among a plurality of ions trapped in a straight line or in a two-dimensional lattice. The method according to claim 1.

11. The first laser beam (361) and the second laser beam (362) are directed towards the first trapped ion (381) and the second trapped ion (382) by adjusting the angle of incidence of the beams with respect to the main plane of the objective lens, using the same objective lens. The method according to claim 1.

12. The first state-dependent force (371) depends on the internal state of the first trapped ion (381), and / or The second state-dependent force (372) depends on the internal state of the second trapped ion (382). The method according to claim 1.

13. The two internal states of the first trapped ion (381) are used to model the first qubit. The two internal states of the second trapped ion (382) are used to model the second qubit. The first state-dependent force (371) differs depending on the two internal states of the first trapped ion (381), The second state-dependent force (372) varies depending on the two internal states of the second trapped ion (382), and / or The method involves quantum gates for the first qubit and the second qubit, particularly optical shift LS gates and / or σ gates. ｚ σ ｚ Implement the gate. The method according to any one of claims 1 to 12.

14. An apparatus for controlling a first laser beam (361) and a second laser beam (362) to entangle the first trapped ion (381) and the second trapped ion (382) using the motion modes of the first trapped ion (381) and the second trapped ion (382), A first laser beam (361) for inducing a first state-dependent force (371) in the first trapped ion (381), A second laser beam (362) for inducing a second state-dependent force (372) on the second trapped ion (382) and The circuit (315) is configured to control the following simultaneously: The first state-dependent force (371) acts perpendicular to the propagation direction of the first laser beam (361), The second state-dependent force (372) acts perpendicular to the propagation direction of the second laser beam (362), In the induction of the first state-dependent force (371) and the second state-dependent force (372), the first laser beam and the second laser beam (362) are modulated according to the frequency of the motion mode, thereby modulating the first state-dependent force (371) and the second state-dependent force (372), respectively, and the motion mode is excited according to the internal state of the first trapped ion (381) and the second trapped ion (382). The circuit (315) In the modulation of the first state-dependent force (371), the first state-dependent force (371) is modulated by modulating the intensity and / or amplitude of the electromagnetic field of the first laser beam (361) at the position of the first trapped ion (381), changing the position and / or direction of the first laser beam (361) relative to the first trapped ion (381), and / or including light of different frequencies in the laser beam, and / or In the modulation of the second state-dependent force (372), the second state-dependent force (372) is modulated by modulating the intensity and / or amplitude of the electromagnetic field of the second laser beam (362) at the position of the second trapped ion (382), changing the position and / or direction of the second laser beam (362) relative to the second trapped ion (382), and / or including light of a different frequency in the laser beam. It is configured in such a way, In the induction of the first state-dependent force (371), the electromagnetic field of the first laser beam (361) has a gradient component perpendicular to the propagation direction of the first laser beam (361) at the position of the first trapped ion (381), The gradient components of the first laser beam (361) are an intensity gradient component, a phase gradient component, and / or a polarization gradient component. The first laser beam (361) includes a first beam component (811) having a first beam shape and a second beam component (812) having a second beam shape. The first beam shape is, Unlike the second beam shape described above, The electric field is zero at the position of the first trapped ion (381), The gradient component of the first laser beam at the position of the first trapped ion (381) increases with the intensity of the first beam component (811). An apparatus characterized by the following features.

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