Transverse 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] FIELD OF THE INVENTION Embodiments of the present invention relate to the field of entanglement of trapped ions.

[0002] In many technological applications (e.g., quantum computing, quantum simulation, atomic and molecular experiments, spectroscopy, atomic clocks, metrology, etc.), individual ions are trapped and manipulated using laser beams. For example, in the field of quantum computing, trapped ions are used to represent qubits, and quantum computations are performed by illuminating the ions with an appropriate laser beam. A key aspect in such applications, and in particular for performing quantum computations using trapped ions, is the creation of an entangled state of two or more ions that are initially in an entangled state.

[0003] Devices for trapping ions, also called ion traps, typically include multiple electrodes that generate electromagnetic fields to confine ions to a small region of a vacuum chamber. However, the components of such trapping setups limit optical access, making it difficult to address (only) specific ions in the ion / qubit register and / or to address ions from specific directions, especially along the axis of a linear ion column.

[0004] Current approaches to entangle m (two or more) ions / qubits in a register of n trapped ions typically require high-NA (numerical aperture) optical access from at least two different directions simultaneously to reach a focal spot small enough to illuminate only m ions each. 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 lead to technical implementation difficulties, reduce the fidelity of the entanglement operation, increase the cost of the setup, and / or limit the available motional 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 technique for entangling trapped ions.

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

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

[0009] Some embodiments of the present invention provide a method for entangling first and second trapped ions. The method includes using motional modes of the first and second trapped ions to induce a first SDF on the first trapped ion using a first laser beam and to induce 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 step of inducing the first and second SDFs, the first and second laser beams are modulated according to the frequencies of the motional modes to modulate the first and second SDFs, respectively, thereby exciting the motional modes according to the internal states of the first and second trapped ions.

[0010] The details of one or more embodiments are set forth in the accompanying drawings and the description below. Other features, objects, and advantages will become apparent from the description, drawings, and claims. [Brief explanation of the drawings]

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

[0012] It is to be expressly understood, however, that the drawings are for the purposes of illustration and description only and are not intended as a definition of the limits of the disclosed subject matter. Furthermore, it should be noted that like reference signs refer to like or at least functionally equivalent features.

[0013] In the following description, reference is made to the accompanying drawings, which form a part of this disclosure and which show, by way of illustration, certain aspects of embodiments of the present invention or in which embodiments of the present invention may be used. It is understood that embodiments of the present invention may be used in other ways and may include structural or logical changes not shown. Therefore, the following detailed description is not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims.

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

[0015] For purposes of the following description, terms such as "end," "upper," "lower," "right," "left," "vertical," "horizontal," "top," "bottom," "lateral," "longitudinal," and derivatives thereof, refer to the disclosed subject matter as oriented in the drawings. However, it should be understood that the disclosed subject matter may assume various alternative modifications and step sequences unless expressly specified otherwise. It should also be understood that the specific devices and processes illustrated in the accompanying drawings and described in the following specification are merely exemplary embodiments or aspects of the disclosed subject matter. Accordingly, specific dimensions and other physical characteristics related to the embodiments or aspects disclosed herein should not be considered limiting, unless expressly stated otherwise.

[0016] No aspect, component, element, structure, act, step, function, instruction, etc. used herein should be construed as critical or essential unless expressly stated as such. Also, 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." Furthermore, as used herein, the term "set" is intended to include one or more items (e.g., related items, unrelated items, combinations of related and unrelated items, etc.) and may be used interchangeably with "one or more" or "at least one." Where only one item is intended, the term "one" or similar language is used. Also, as used herein, terms such as "has," "have," and "having" are intended to be open-ended terms. Furthermore, the phrase "based on" is intended to mean "based at least in part on," unless expressly stated otherwise. Quantum Computing

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

[0018] In quantum computing, quantum bits, or qubits, represent the basic unit of quantum information. A qubit is thus the quantum equivalent of a binary bit in a classical computer. However, while a bit can only assume two values, usually called "0" and "1", a qubit corresponds to the state of a two-state quantum mechanical system. The state of a qubit |ψ> can therefore be written as a linear combination |ψ>=α|0>+β|1>, where α and β are the normalization conditions |α| 2 +|β| 2 = 1, where |0> and |1> represent the two states of a two-state quantum mechanical system (since two-state quantum systems are mathematically equivalent to spin 1 / 2 systems, the two basis states are often also denoted as |↓> and |↑>). Thus, a qubit is not limited to two values ​​like a classical bit, but can also assume complex superpositions of |0> and |1> states. Since the global phase of a quantum state has no observational consequences, one of the numbers α and β can be chosen to be real. As a result, the state of a single qubit can be expressed as a function of the relative phase

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[0019] In general, the state of two qubits |ψ 12 > may also be written as:

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[0020] In general, for 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 factorized into a product of the individual qubit states. In this disclosure, entanglement refers to such a state. However, an entangled state is not a Bell state.

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

[0024] In general, a single-qubit operation performed by a single-qubit gate changes only the state of a single / one qubit, leaving the state(s) of the other qubit(s) unchanged. More specifically, a single-qubit operation performs a unitary operation on the state of a single qubit, independent of the current states of qubits other than the single qubit. Thus, a single-qubit operation changes the above α and / or β of the respective qubit by effecting a complex rotation of the qubit state, which may be visualized as a rotation about an axis of the Bloch sphere of the single qubit.

[0025] When represented by an internal state of an ion, performing a single qubit operation can result in a rotation of the wave function of the ion's qubit state. In particular, the ion's motional state can remain essentially unchanged (except, for example, for implementation errors / imperfections). However, the effect of performing a single qubit operation on the ion's internal state other than the two qubit states may be less significant, since the wave function of the ion on which the single qubit operation is performed should typically not overlap with said other states (also referred to herein as non-qubit states). In general, to perform such a rotation on a qubit, an electromagnetic field having an appropriate frequency may be applied (e.g., using a laser beam to the trapped ion) for an appropriate duration.

[0026] An example of a single-qubit gate is the Pauli gate (X, Y, Z), which corresponds to the operation of the three Pauli matrices on the 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 an appropriate radiation field, such as a laser beam / pulse. For example, a resonant microwave field can directly couple the qubit levels via magnetic dipole interactions, resulting in coherent Rabi oscillations between the qubit states. Alternatively or additionally, photo-stimulated transitions using two light sources can be used to coherently couple the qubit states via excited electronic states.

[0028] A two-qubit operation performed by a two-qubit gate conditionally couples the wave functions of the two qubits, which can result in so-called entanglement, which requires 2 n This is the reason why complex numbers are generally required. An example of a two-qubit gate is a controlled gate that operates on the states of two or more qubits. For example, a controlled-NOT (CNOT) gate operates on two qubits by (i) always leaving one of the two qubits, the so-called control qubit, untouched, and (ii) performing a NOT operation on the other qubit if 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 typically relatively simple to implement (i.e., physically perform, especially for trapped ions), CNOT and other two- or more-qubit gates are typically more demanding and usually slower than single-qubit gates. Like single-qubit gates, quantum gates operating on two or more qubits are typically performed on ion qubits by the interaction of the ions with a laser beam, as described further below.

[0030] In general, there are also operations involving more than two qubits. However, it can be shown that there exists a set of universal quantum gates that can approximate any unitary matrix, and thus any algorithm, arbitrarily well. More specifically, a unitary matrix can be approximated as a sequence of operations (i.e., as a matrix product) performed by gates of the universal set. The accuracy of the approximation may generally depend on the length of the sequence. In particular, quantum logic operations on any number of qubits can be decomposed into single-qubit 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 gates

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[0031] In general, any two-state quantum mechanical system, or any quantum mechanical system with three or more states, may be used to model (i.e., selected to physically represent) a qubit. This particularly means that the quantum system is used to store the quantum information (i.e., quantum state) 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 particular degree of freedom of the system (e.g., the spin of the 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, the spin of an electron may be used to represent a qubit, e.g., by specifying spin-up and spin-down states (relative to a given / selected direction) to represent |0> and |1> states, respectively. Other examples include nuclear spin states, atomic or nuclear states, nuclear magnetic resonance states, and quantum dots.

[0033] In this disclosure, we consider two internal states of a trapped ion (i.e., an atom, particle, or molecule with a net charge) to typically represent or model a qubit. As used herein, the term "trapped ion" refers to an ion that is confined in a small region of space, such as a vacuum chamber, typically by an electromagnetic field. It is further noted that when referring to trapped ions, these (all) trapped ions may be trapped by / using the same ion trap.

[0034] For example, the ground state (generally the lowest energy state) of the trapped ion may be selected to represent the |0> state, and an excited state of the trapped ion may be selected to represent the |1> state of the qubit. However, in general, 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 selected 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 described further below.

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

[0036] Furthermore, it should be noted that, hereinafter, the term "qubit state" refers to the internal state of the trapped ions selected to represent the respective qubit. The internal states of the ions not used to store the qubit are referred to as "non-qubit states." In other words, an internal state is either a qubit state or a non-qubit state. It should also be noted that, depending on the context, the term "qubit state" can also refer to the state of the qubit, which may generally be a superposition of two qubit (basis) states.

[0037] It should be noted that in this disclosure, the ion's qubit states are also referred to as the |↓> and |↑> states. This is done solely in consideration of the quantum description of the spin of a spin-½ 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 cases where the qubit is represented in a spin-½ system. As explained 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 made different and / or may be made further different by applying external electric and / or magnetic fields.

[0038] For example, to realize a quantum computer having multiple qubits, multiple ions may be trapped in the same ion trap, with each trapped ion being used to represent one of the multiple qubits. Generally, the ions may be identical (i.e., of the same type, not identical in the sense that there is only one ion), and the same / corresponding state of each ion may be used to represent a 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 qubits. Then, as described in more detail below, the qubit states 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, i.e., to spatially confine ions to a specific (small) region. This region may be within the ion trap, but is not necessarily so, as in the case of 2D traps. 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, the ion trap may be a Penning trap, a Paul trap, a three-dimensional (3D) ion trap, and / or a linear 3D trap. In this disclosure, the term ion trap refers to an assembly having multiple electrodes that, when actuated, generate an electric field that restricts (traps) the freedom of movement of ions, preventing them from escaping a specific (preferably small) region in the vicinity of the electrodes. It should be noted that an actual ion trap device / system may include additional mechanical and electrical components, such as fixation means, electrical contacts, a housing, a power supply, control circuitry, and means for cooling ions.

[0041] As used herein, the term "3D trap" refers to all traps except surface traps, which are traps in which all electrodes are arranged in the same plane. In particular, a 3D trap may be a Penning trap or a Paul trap. Typically, a 3D ion trap has rotational symmetry, for example, discrete cylindrical symmetry or continuous cylindrical symmetry.

[0042] Without loss of generality, the axis of symmetry of such cylindrical symmetry is defined by three mutually orthogonal unit vectors

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[0043] A Penning trap refers to a trap that uses electrostatic and magnetic fields to trap ions. Typically, Penning traps use only electrostatic fields; in other words, they typically do not use oscillating and / or alternating fields. For example, a static axial magnetic field is used to confine charged particles radially.

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[0044] A Paul trap refers to a trap that uses electric fields to trap ions. Typically, Paul traps use only electric fields to trap ions. In particular, magnetic fields are typically not used. Typically, at least one of the electric fields in a Paul trap is alternating (e.g., oscillating), and Paul traps may use both electrostatic and alternating electric fields. For example, the alternating field in a Paul trap may be an alternating multipole field, particularly a quadrupole field. Because the voltage switching is often performed 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. A linear 3D trap is therefore also commonly a (linear) Paul trap. Cooling and Readout

[0046] Trapped ions may be cooled to approach their motional ground state using interaction with a laser beam. Cooling of such 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 trapped ions to the motional ground state and / or initializing trapped ions to the motional ground state or any other specific known ground state. In particular, quantum gate implementations, described further below, have the advantage of working with so-called thermal or hot ions, where the motional state of the trapped ions is not precisely known initially or even during the quantum computation. These gates enable quantum operations to be performed even when the ion's motional state is an unknown superposition of kinetic energy eigenstates. Some cooling may still be performed, such as to avoid / reduce thermal decoherence due to interactions of the trapped ions with the external environment. Initialization of qubit states may similarly be performed by laser-beam-induced transitions in individual ions.

[0047] The qubit may be read out by illuminating it with a resonant laser beam and detecting the qubit state via fluorescence. Using such a resonant laser beam, rapid single-photon transitions from one of the qubit levels to a higher excited level of the ion are used to emit fluorescence photons that can be detected if the level is filled. If the level is not filled, no fluorescence photons are detected except for randomly generated dark counts. Exercise Mode

[0048] As mentioned above, trapped ions may be trapped by, using, and / or within the same ion trap. In other words, trapped ions may be part of the same "qubit register" or "trapped ion register." Hereinafter, for ease of understanding, it will be assumed that all trapped ions are identical (i.e., same chemical element / isotope, same charge). However, the present invention is not limited to this, as various types of ions can generally be trapped in ion traps and used in quantum computers. However, typically all trapped ions have the same charge sign, i.e., each ion has a positive charge or each ion has a negative charge. However, the ions do not have to have the same positive / negative charge (although they may have the same positive / negative charge).

[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-coupled "ion crystal," in which each ion has an equilibrium position where the Coulomb repulsion between the trapped ions balances the external confining force of the ion trap. For example, the trapped ions may be arranged in a line or trapped in a line (e.g., forming a column / linear chain of trapped ions). In other words, the equilibrium positions of the ions may be on a line. In particular, the trapped ions may be aligned on a trapping axis, which may be the trap's only / only axis of symmetry. In particular, the ions may settle along the weakest axis of the confining potential. The trapped ions may be arranged in a two-dimensional lattice of positions (e.g., the positions may form a two-dimensional mesh).

[0050] When ions are cooled, they can form a sufficiently well-isolated quantum system, i.e., their displacement from their equilibrium position is small, and the motion of the trapped ions is quantized. Because the motion of the ions is coupled by their mutual Coulomb repulsion, it can be described by their coupled harmonic oscillatory motion with respect to normal modes, where the term "normal mode" can mean that all ions vibrate at the same frequency and with a fixed phase relationship. Note that terms such as "normal mode," "motional mode," "common motional mode," "oscillation mode," "vibrational mode," "collective vibrational mode," "joint mode," or simply "mode" are used interchangeably.

[0051] Each motional mode can be described by its own quantum mechanical harmonic oscillator. Thus, in the absence of a laser beam implementing a quantum gate, a quantum system of N trapped ions can be described by the following Hamiltonian:

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[0052] In general, ion motion can be quantized into three directions as shown in the above equation, so there are 3N (orthogonal) normal modes, where N is the number of trapped ions. If the ions are aligned in a line, these three directions are two radial directions and one axial direction, with the axial directions 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 directions are typically two mutually orthogonal directions that are orthogonal to the respective (ion / beam) axial directions (thus, there may be some ambiguity / selectivity in the definition of radial direction).

[0053] The motional mode is a motional mode common to all trapped ions. In particular, the frequency of the motional mode ω ν are properties of the system / setup (as a whole), including (e.g., all) trapped ions and the trap. They may depend, for example, on one or more of the type of ion(s), the number of ions, the design of the trap, and especially the strength of the confining electromagnetic force of the trap. The frequency ω of the motional transition ν is typically in the frequency range of 300 kHz to 5000 kHz.

[0054] For example, for a harmonic trapping potential, the normal modes (i.e., for N ≥ 2) have frequencies ω where the displacements of the two ions from equilibrium are the same. C,iThe three "center of mass" modes (COM) at (i=1, 2, 3) and three modes where the displacement amplitudes are 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 rocking modes in the radial direction. In general, the frequency of a stretching / rocking mode in a particular direction may be related to the frequency of the COM mode in that particular direction; for example, the frequency of a stretching mode is typically ω S =3 1 / 2 ω C is related to the frequency of the axial COM mode according to (1). In particular, when ions are aligned, there is a center-of-mass mode and a stretching mode for each of the two ion radial directions and ion axis directions (i.e., there is one axial COM mode and one axial stretching mode, and two radial COM modes and two radial locking modes). It should be noted that for simplicity and ease of understanding, the explicit examples of this disclosure typically use stretching or COM modes to entangle ions. However, the invention is not limited in this regard, as in general any motional mode may be used to entangle ions. In particular, the invention is not limited to harmonic trapping potentials, but also applies to anharmonic trapping potentials.

[0055] In general, a complete quantum mechanical description of the state of an ion can be given by, for example, the phonon number eigenstate |n> ν , which includes the specification of the quantum state of each motional mode with respect to each internal qubit state (e.g., |ψ above for the two-qubit case). 12For a trapped ion (|↓>), each motional mode generates a ladder of equidistant energy levels with a different number of phonons. This is exemplarily shown in FIG. 1 for one trapped ion and one motional mode. More specifically, the vertical axis of FIG. 1 corresponds to the energy of each state, with states higher in the diagram having higher energy levels. The states of the other qubits and other motional modes are assumed to be identical for all states shown. 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 such that, if the ion's motional state and the other qubit states are not changed,

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[0056] Entangled quantum gates are a key element in the realization of quantum computers. A specific type of physical implementation of entangled gates on trapped ions is known as a "geometric quantum gate" or "geometric phase gate." Geometric phase gates are gate implementations for qubits represented by the internal states of each ion trapped by the same trap. They can be relatively fast compared to other two-ion gates and, furthermore, enable high fidelity because the ion's internal state is not directly involved in the gate operation. Geometric phase gates include the Molmer-Sorensen gate (MS gate) and the light-shift gate (LS gate), both of which utilize a state-dependent force (SDF) induced by a laser beam on addressed ions.

[0057] However, current implementations of LS gates typically use two laser beams per ion, i.e., laser beams 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, because a standing wave cannot be generated by a single laser beam from one direction, the setup for generating a standing wave is generally complex. For example, two crossed or counter-propagating laser beams or a modulated retroreflected laser beam are used. Similarly, MS gate methods typically require the SDF to be induced by laser beams from different directions to be able to address 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 spot small enough to irradiate only specific ions. This can lead to technical implementation difficulties, reduced fidelity of entanglement operations, increased setup costs, and / or limited available motion modes.

[0058] In view of the above, one embodiment provides a method for entangling two or more trapped ions (also referred to as addressed ions). As shown in FIG. 2 , in S240, a first SDF is induced in the 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 (on the respective ion) perpendicular to the propagation direction of the respective laser beam that induces that SDF. In S220, the first and second SDFs are modulated by modulating the first and second laser beams according to the frequency of a motional mode used for entanglement. In this manner, motional modes are excited according to the internal states of the first and second trapped ions, which can result in entanglement, as further described below.

[0059] The entanglement method(s) described herein may generally be used to entangle more than two ions, i.e., to entangle more than just two ions with one another; for example, a three-qubit gate may also be implemented, as described further below. However, for simplicity and ease of understanding, we will typically only explicitly discuss the case of entangling two ions / qubits using one mode of motion. In this regard, it should be noted 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, etc.). We denote the frequency difference between the states of the first qubit as ω1 (which is the frequency difference between the |1> and |0> states of the first ion).

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

[0061] Furthermore, it should be noted that the modulation of the laser beam S220 may be performed after the initiation of the SDF S240 for each ion. 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 in respective overlapping time periods. In other words, there is a time period (i.e., gate time) during which both steps S220 and S240 are performed simultaneously (for both / all addressed ions).

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

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

[0064] Furthermore, this may enable easy access to motional modes perpendicular to the laser beam propagation, which may in turn enable a wider variety of ion-ion interactions. Therefore, easy access to all motional modes (e.g., one axial mode and two radial modes of a linear ion chain) may be possible. For example, easy access to the axial modes of a 3D linear Paul trap and / or easy access to specific / any in-plane modes of a 3D Paul trap on a planar crystal may be facilitated.

[0065] Therefore, entangling two or more trapped ions can be easier (unidirectional, easier optical access) and more versatile (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 Paul traps, 3D planar crystalline Paul traps, 2D surface traps). In particular, the method / apparatus can be used to implement quantum gates to the first and second qubits, in particular LS gates and / or

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[0066] It should be noted, however, that the present invention applies generally to the generation of entangled states of two or more trapped ions and is not limited to quantum computing. In particular, while typically discussed in the context of quantum computing, the techniques described herein may be used quite generally to entangle trapped ions within / for applications other than quantum computing. Furthermore, it should be noted that whether a qubit / trapped ion is entangled after the operation of a two-qubit gate generally depends on the initial states of the qubit / trapped ion. In other words, an entanglement gate typically generates entangled states only from specific initial states, rather than from any / all initial states. Accordingly, terms such as "entanglement gate" and "method for entangling" refer to a gate / method that generates an entangled state from at least one initially unentangled state of addressed ions. Circuits and Controllers

[0067] In general, the circuitry 315 may be configured to control a laser beam according to any of the methods described herein, i.e., the circuitry may be configured to control optical elements to obtain any of the laser beams described herein (e.g., modulation of the laser beam(s), frequency, beam shape, etc.). Similarly, an entangling method according to the present invention may include any steps that an apparatus (particularly the circuitry 315) is configured to perform according to an embodiment. That is, what is described herein may apply to both an entangling method and an apparatus for entangling, unless the context indicates otherwise.

[0068] The circuit or processing circuit 315 may be part of a controller, which may include other hardware and / or software. FIG. 4 shows an exemplary structure of such a controller 310. As shown, the controller may include a circuit 315 that performs control of the laser beam (e.g., laser beam unit 320) and may also control further components included in the apparatus for entangling ions, a memory 410, and possibly a transceiver 440 and a user interface 430. The controller may, for example, be (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 of the steps of the methods 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, device status, etc., and / or receiving user input. For example, the controller may be configured to receive input of a particular unitary operation to be performed on trapped ions, determine the characteristics of a laser beam that will perform said unitary operation (which may be a sequence of laser beams, i.e. the controller may first decompose the received unitary operation into a sequence of predetermined universal gate operations and then execute said sequence), and then control the laser beam unit / laser beam accordingly as described above.

[0069] The bus 401 may interconnect the memory, processing circuitry, transceiver, and user interface. It should be noted that the controller may be implemented by any hardware means other than using a general-purpose processor, and may be implemented as a microcontroller, by programmable hardware such as a field programmable gate array (FPGA), or as dedicated hardware such as an application specific integrated circuit (ASIC). Any combination of the above hardware and possibly software may be used. Laser Beam Unit

[0070] The apparatus may further comprise a laser beam unit 320, although the invention is not limited thereto, as the apparatus / circuitry may be configured only to control the laser beam unit 320. It is further noted that the laser beam unit 320 is not limited to emitting two (different) laser beams 361 and 362. In general, the laser beam unit may be configured to emit one or more laser beams, and the circuitry 315 may be configured to control said two or more laser beams to entangle two or more ions (particularly, if two addressed ions are adjacent and a wide beam is used, the laser beams 361 and 362 may be the same laser beam).

[0071] Furthermore, in general, the laser beam unit 320 may include one or more lasers 330, such as, for example, stabilized diode laser(s). However, the present invention is not limited to a specific laser source, and lasers other than diode lasers, such as solid-state crystal lasers, may also be used. There may be one laser beam 340 from one laser 330, from which multiple laser beams may be obtained using beam splitters included in the optical element 350, or multiple laser beams 340 may be generated by multiple lasers, respectively. The present invention is not limited to a specific scenario. In particular, the laser beam unit 320 may not include a laser, and the circuitry 315 may be configured to control the laser beam unit 320 to receive light (one or more laser beams) from an external light source.

[0072] Additionally, 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. Circuitry may be configured to control the laser(s) (e.g., the power and / or frequency of the emitted light) and / or the active optical elements to perform one or more of the methods described herein. In particular, circuitry 315 may be configured to address particular ions (i.e., direct a particular beam at a particular ion so that the ion is located within the laser beam at a particular predetermined radial distance from the laser beam axis), modulate the laser beam, shape the laser beam into a predetermined shape, and / or, among other things, condition / generate the laser beam to induce a transverse SDF at a location on each ion. Optical elements

[0073] 5 , the optical element 350 may include a modulation unit 510 configured to perform modulation of the laser beam(s) 340 when controlled by the circuitry, as described further 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 at the location 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 ions are trapped in a linear ion trap, the trapping axis 390 along which the ions are aligned 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 excitation and / or entanglement of trapped ions by providing single-sided access to the trap that confines the ions.

[0074] In general, some or all of the laser beams may be directed at addressed ions using the same objective lens. This is also shown in FIG. 5 , which exemplarily assumes that the laser beam unit includes an addressing unit 520, which may also be controlled by the circuit 315. Such an addressing unit facilitates individually / separately controlling the direction of the laser beams relative to the trapped ions and / or focusing the laser beams at different positions in the ion-confined region. This may enable addressing a specific ion from among multiple trapped ions (although, as shown, other ions may be present that are not addressed). More specifically, as shown in FIG. 5 , the first laser beam 361 and the second laser beam 362 may be directed at 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 a principal plane 560 of the objective lens 550 (or lens 550). This may facilitate addressing a specific ion among multiple trapped ions and / or directing 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 a respective ion. Note that if the lens is not a thin lens as in Figure 5, the angle of incidence will be relative to the front principal plane (i.e., the first of the principal planes intersected by the laser beam).

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

[0076] 6 shows exemplary optical paths for two or more laser beams 340 (note that only one laser beam of monochromatic or polychromatic light may be present at the location indicated by "340"). The exemplary setup includes a first AOD 620, a second AOD 640, 4F relay optics 630, magnification optics 650, a collimation lens 660, and an objective lens 670. The optical components provide examples of the static (passive) 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 AOD 620 is used to generate two (or more) laser beams, each with a different direction, from the single laser beam 340. Note that the frequency(ies) of the two or more laser beams generated by the first AOD 620 may be slightly different and / or shifted relative to the frequency(ies) of the laser beam 340. These laser beams, obtained by splitting the AOD 620, are then directed to the individual addressed ions. The second AOD 640 is used to further increase the deflection and compensate for the frequency offset(s) caused by the first AOD 620. Therefore, if it is not particularly important that the laser beams addressing different ions have the same frequency (which is the case for the LS gate implementation described below), the AOD 620 and the 4F relay optics 630 may be omitted. Beam shaping may be performed by the magnification optics 650 and subsequent optical elements.

[0078] Additionally, FIG. 6 shows a vacuum window 680 that provides optical access to the ion trap and focuses the laser beam on the trap axis 690 . SDF transversality

[0079] Generally, each SDF is perpendicular to the propagation direction of the laser beam inducing 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 addressing / illuminating the first ion, and the SDF induced on the second addressed ion acts perpendicular to the propagation direction of the laser beam addressing / illuminating 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 inducing the SDF. In other words, the SDF is perpendicular to the propagation direction of the laser beam at the location of the addressed ion (see Figure 3, where 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 the laser beam can (and typically will) induce forces other than the transverse SDF on the ions. Such other forces may generally be state-dependent or state-independent. Furthermore, such other forces may be perpendicular or non-perpendicular (e.g., even parallel). Thus, in other words, the SDF may be the (perpendicular) component of the force(s) induced on the ions by the laser beam. The SDF may be the entire perpendicular component of the force(s) induced by the laser beam, or a portion of said perpendicular component. SDF and lateral gradient generation

[0081] As described above, transverse SDFs are simultaneously induced on each addressed ion. For example, if there are two addressed ions (i.e., a “first” and a “second” ion), first and second SDFs are simultaneously induced on the first and second addressed ions, respectively. In other words, there is a one-to-one correspondence between the SDFs and the addressed ions (i.e., the SDF refers to a force acting on a specific ion, not a force field that may act on multiple ions). The SDFs acting on different ions may be identical in magnitude and direction (relative to the corresponding qubit state), or, as shown in the example of FIG. 12, the SDFs acting on different ions may be identical in magnitude but opposite in direction (relative to the corresponding qubit state). However, the present invention is not limited to such a scenario, and in general, the SDFs for different ions and states may be different. The SDFs are induced by directing respective laser beams at each ion such that the ions are “located” (positioned) within the electromagnetic fields of the laser beams. In some embodiments, the specific position of the ions within the laser beam (e.g., radial distance from the beam axis) may also be relevant, for example, when the SDF is induced by a laser beam with a non-constant intensity gradient across the beam diameter.

[0082] Generally, each SDF is induced by one (e.g., only one or single) laser beam (e.g., from one direction). Other expressions that may be used interchangeably with "induced" are that the laser beam "causes" or "exerts" the 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 ions). However, generally, the same laser beam may be irradiated on two or more addressed ions, for example, if the two addressed ions are adjacent / nearby trapped ions. The same laser beam may still induce different SDFs on the ions, for example, because the intensity gradient of the laser beam may be position-dependent (e.g., a Gaussian beam).

[0083] In general, (e.g., each / any) SDF may be generated by a (non-zero) transverse component of the gradient of the laser beam inducing said SDF. A "transverse" component of the gradient is a component that is orthogonal to the direction of propagation of the laser beam, i.e., orthogonal to the beam axis. For example, a gradient

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[0084] It should be further noted that the gradient is the gradient of the laser beam at the location of the addressed ion. Therefore, the SDF acting on an ion is induced by the gradient of the laser beam at the location of said ion. Furthermore, 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 said 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 (in particular the gradient of the amplitude of the electric or magnetic field of the laser beam); (ii) a phase gradient in the electromagnetic field of the laser beam, and / or (iii) Polarization gradient of the electromagnetic field of the laser beam may be.

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

[0086] More specifically, the location

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[0087] In summary, laser beam gradients (e.g., intensity, phase, and / or polarization gradients) may be used to impose SDFs on ions, which in turn, together with appropriate modulation, etc., can be used to create entanglement, as described further below. Beam Shape

[0088] For example, a laser beam with an intensity / amplitude gradient may be achieved by using a laser beam with a specific shape. For example, the shape of the laser beam(s) used to induce the SDF may have a Gaussian beam shape, a super-Gaussian beam shape, a Laguerre-Gaussian beam shape, and / or a Hermite-Gaussian beam shape (or beam gradient). These beam profiles have a transverse amplitude (and intensity) gradient that is not constant in the beam radial / lateral direction. This may allow the gradient (and therefore the SDF) to be adjusted by changing the position and / or direction of the laser beam relative to the addressed ions, as further described below. It should also be noted that the terms "beam form," "beam shape," and "beam profile" are used interchangeably in this disclosure.

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

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[0090] As can be seen from the above equation, the use of high numerical aperture (NA) optics can increase the intensity gradient in the transverse direction r. In other words, a focused laser may be used to increase the intensity gradient. This may facilitate addressing individual and / or specific trapped ions (i.e., inducing force(s) only on the addressed ion(s) and not on other trapped ions). This may make it possible to implement entanglement gates that act on only a subset of m ions in a qubit register of n > m ions. A larger intensity gradient may result in a larger SDF, allowing for faster gate times.

[0091] In general, (new) beam shapes may be generated by superimposing beams of different shapes. In other words, a laser beam that addresses ions may include two or more beam components (e.g., electromagnetic field components) with different beam shapes, e.g., two or more spatially distinct beam components. Alternatively or additionally (and particularly 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 of the laser beam). Further, alternatively or additionally, the beam components may have different polarizations, different polarization gradients, different phases, different phase gradients, and / or different directions of electric or magnetic fields at the location of the addressed ions. Thus, a laser beam may include two or more beam components with different electromagnetic field properties at the location of the addressed ions.

[0092] This may allow for the generation of special beam shapes from "standard" beam shapes. In general, one, two or more, or each of the beam components may be a Hermite-Gaussian beam (e.g., each may have a Hermite-Gaussian mode beam shape). For example, as shown in FIG. 8a, two Gaussian beams (e.g., TEM 00 beams) may overlap each other.

[0093] Hermite-Gaussian modes are typically used in TEM lm where TEM stands for transverse electro-magnetic. Hermite-Gaussian beam TEM propagating in the z direction lm The time-independent part of the electric field in can be written as

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[0094] In this regard, it should be noted that the black arrows in Figures 8a) as well as 8b) and 8c) indicate the polarization and electric field magnitude of the respective beam components. Thus, for example, in Figure 8a), two beam components 801 and 802 are linearly polarized in the y-direction. Furthermore, Figure 8a) shows that the addressed ion 803 is centered between the two laser beam components but displaced to the right from the axis of beam component 801 and to the left from the axis of beam component 802. In other words, the ion 803 is midway (e.g., at (x1, y1, 0)) between the axis of beam component 801 (e.g., at (x1, y1, 0)) and the axis of beam component 802 (e.g., at (x2, y2, 0)). Thus, the two beam components 801 and 802 induce SDFs on the ion in opposite directions (i.e., 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 180° out of phase with each other. Using this setup, two laser beam components may be used to enhance the SDF on the ions (more specifically, to increase the average excitation (over the gate time) of motional modes due to the SDF). More specifically, a laser beam may generally include two beam components of the same shape but displaced from each other in the radial direction of the laser beam. Such a laser beam may (i) be directed at an addressed ion such that the two laser beam components are displaced in opposite directions (radially) relative to the ion, and (ii) be intensity modulated 180° out of phase with each other (otherwise, the intensity modulations may be identical).

[0095] As another example, and this is shown in Figure 8b), the laser beam that addresses the ions may include two beam components with different beam shapes. In general, the axes of these two beam components may have the same beam axis (i.e., the beam components are not radially displaced relative to each other), or may be radially displaced relative to each other.

[0096] For example, one of the two components may have a Gaussian beam shape (e.g., TEM00 beam shape), and the other beam component may have a Hermite-Gaussian beam shape (e.g., TEM 10 As shown in Figure 8b), the ion is centered on the laser beam axis, which is not only coincident with the axis of the Gaussian beam component 812 but also with the axis of the Hermite-Gaussian beam component 811. In this configuration, the laser beams can be either co-propagating or counter-propagating.

[0097] 9a) shows the first beam component 811 (TEM 10 , dashed line) 1,0 (y is the radial distance from the beam axis, e.g., the trap axis), the second beam component 812 (TEM 00 , dotted line) of the electric field E 0,0 Figure 8 shows the y-dependence of the electric field of the beam component, as well as the resulting sum of the electric fields (solid lines) corresponding to the electric field of the laser beam addressing the ions. As shown in Figure 8b, the electric fields of both beam components are polarized in the y-direction. The fact that the electric field of the first beam component is negative for negative y (see Figure 9a) means that the electric field points in the opposite direction to the y-direction. Therefore, for negative y, the electric fields of the first and second components cancel each other.

[0098] TEM 00 and TEM 10 The electric fields of the individual beam components 812 and 811, corresponding to a Hermite-Gaussian beam shape, respectively, can be written (at x=z=0, thus, for example, along the trapping axis) as:

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[0099] In this case, the intensities of the two beam components may be modulated together (in particular at the same frequency and without phase offset, as in the example of FIG. 8a). For example, the intensity of the (total) laser beam comprising the two beam components 811 and 812 may be modulated. However, it is also possible to modulate the two beam components with different frequencies, which results in a phase difference between the two beam components that varies 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, the sign of the intensity gradient at the position of the ion changes (periodically) over time, since the intensity maximum of the beam superposition oscillates periodically between positions to the right and left of the ion. For example, beam component 811 is modulated with a frequency ω Mod and beam component 812 can be modulated by 2ωMod can be modulated by t=π / ω Mod Magnitude π=Δω per second Mod There is an accumulation of relative phases of t. Therefore, the position of the intensity maximum is determined by the frequency ω Mod This allows for continuous excitation and de-excitation of motional modes, which can reduce gate times compared to the case of a single Gaussian beam component where the SDF acts in only 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 containing the two beam components can be estimated as follows:

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[0101] In particular, the first beam component 811 (TEM 10 Increasing only the intensity of the electric field E1, and thus the electric field amplitude E1, increases the intensity gradient (and thus the SDF), while the intensity at the ion's location remains constant. Typically, prior art implementations of laser-based entanglement gates suffer from a scattering limit that limits the fidelity of the entanglement gate, since the velocity of the scattered photons (which induces decoherence) and the strength of the SDF depend on the same electric field amplitude. In the presented method, with the overlap of two laser beams, this dependence can be decoupled as described, thereby significantly improving the scattering limit.

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

[0103] In FIG. 8c, if the electric fields of beam components 813 and 814 are modulated at the same frequency but 90° or 270° out of phase with each other (so that the intensity of one beam component is at its maximum when the intensity of the other beam component is zero, and vice versa), then superposition of the two beam components 813 and 814 produces two circularly polarized components: a first circularly polarized component where the right portion of beam component 813 (indicated by the arrow pointing up in FIG. 8c) overlaps with beam component 814, and a second circularly polarized component where the left portion of beam component 813 (indicated by the arrow pointing down in FIG. 8c) overlaps with beam component 814. These circularly polarized components are in phase and have opposite circular polarizations (also known as left / right rotation, or sigma plus / minus). Thus, for example, in FIG. 8c, there may be sigma minus polarization on the left side of the ion and sigma plus polarization on the right side of the ion. Such polarization gradients can create (or be used to implement) SDFs that act in opposite directions on the two qubit states of the ion (e.g., the qubit states interact differently with the two circularly polarized light). The two beam components do not need to be perfectly out of phase for this configuration to work (although generally, it can be sufficient if the two beam components are modulated out of phase (or not modulated in the same phase, i.e., a phase difference).

[0104] In both Figure 8b) and Figure 8c), the TEM at the position of the ion 10 Since the field is zero, a small AC Stark shift may be possible. Different AC Stark shifts for the two qubit states of the addressed ion may result in (undesirable) rotation of the qubit state of the addressed ion, which may be reduced in this manner. Furthermore, in the configuration shown in FIG. 8c), by choosing appropriate transitions, 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 may be small or zero, which in turn may reduce or avoid rotation of the qubit state of the ion.

[0105] This approach of using 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 (corresponding to the motional mode frequency plus detuning) between the two beam components, as described below.

[0106] Hermite-Gaussian beams (including Gaussian beams) and Laguerre-Gaussian beams may be generated / shaped using, for example, a spatial light modulator (SLM) and / or a 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 radial / lateral gradient with respect to the propagation direction, which may reduce the dependence of the SDF on the exact position of the irradiated ions within the laser beam and simplify the technical implementation. State dependency of SDF

[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 it acts, e.g., a spin-dependent force. More specifically, the 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, the SDF induced by a laser beam may act differently on the trapped ion when it is in the |0> state and when it is in the |1> state. Because the trapped ion can be in a superposition state |ψ>=α|0>+β|1> of |0> and |1> states, the SDF may simultaneously act differently on the |0> and |1> components of the trapped ion's state. 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, the SDF may depend on the spin of the trapped ion if the ion's qubit states have different spins. On the other hand, the SDF may generally be i) independent of the motional state of the trapped ion (this is often true only to first order) and ii) independent of the internal states of other trapped ions. In general, the strength and / or direction of the force may depend on the internal states. In particular, the SDF may have opposite directions and / or the same magnitude for the two qubit states. For example, the SDF may be a dipole force, in particular an optical dipole force.

[0108] In general, approaches to making the force 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 the transition between the qubit state and the excited (non-qubit) state and / or induce a dipole force (i.e., the SDF may be a dipole force). In general, unlike MS gates, LS gates are based on photon elastic scattering, i.e., scattering / deflection where only the photon's direction, but not its wavelength, is changed by the scattering process. Therefore, the frequency of the laser beam is typically significantly detuned from the dipole transition, i.e., the transition from the qubit state used for coupling to the excited non-qubit state(s).

[0109] More specifically, a time-varying electric field changes the energy level E of an atom / ion. i Shift E i →E i +ΔE i , a so-called "AC Stark" or "optical shift" is induced. This shift is proportional to the intensity of the laser beam, as can be seen from the following result obtained from second-order perturbation theory for the non-degenerate energy eigenstates |i>, |j> of the ion:

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[0110] It should be noted that, as used herein, the term "detuning" generally refers to the frequency difference / shift of the laser frequency relative to the frequency of the quantum system. For the detuning discussed herein, the laser frequency is the electromagnetic field oscillation frequency ω L and 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 explained further below, the term detuning refers to the frequency ω at which the laser beam is modulated. Mod and the frequency of the motion mode ω M The term is also used to refer to the frequency difference δ between the quantum system and the laser. Thus, the term generally refers to the difference in frequency of the laser relative to some (resonant) frequency provided by the system of trapped ions. A laser frequency lower than the frequency of the quantum system is called red-detuned, and a laser tuned higher than the frequency of the quantum system is called blue-detuned. In general, the detuning is considered to be small in some sense. In particular, the detuning relative to the frequency from which the laser is detuned is smaller than the frequency difference of the laser relative to most or all other frequencies.

[0111] The denominator in the above equation means that detuning from a specific 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), a 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. Furthermore, 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 denoted 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 denoted as Δ, i.e., ω L = ω e,↓+Δ. Note that Figure 10 shows a negative detuning Δ<0. Typically, in this example, the detuning is chosen to be large compared to the energy difference between the qubit states, |Δ|≫ω0, but still small enough that the other excited states can still be neglected (e.g., the detuning for the other excited states needs to remain much larger than the detuning for the selected excited state |e>). This ensures that the detuning for the transitions from both qubit states is approximately equal. Typically, an excited state |e> is chosen such that the transition from the qubit state to the excited state is an electric dipole transition. The dependence of the dipole force on the qubit state may be achieved by appropriate adjustment / setting and / or modulation of the polarization components of the laser beam. For example, 9 Be + With ions, the hyperfine ground state can be used as the 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 the excited state(s).

[0113] A second example of this technique is shown in Figure 11, which shows the effect of a laser beam with frequency ω L This differs from the previous example in that the two transitions to the excited state are symmetrically detuned. More specifically, the laser frequency is red-detuned (or negatively detuned) for the transition from the |↓> state to the excited state |e>, i.e., ω L <ω e,↓ ), i.e., ω L =ω e,↓ −ω0 / 2, and is blue detuned (or positively detuned) for the transition from the |↑> state to the excited state |e>, i.e., ω L >ω e,↓ ), i.e., ω L =ω e,↑+ω0 / 2. Such a detuning (one red, one blue) generates an SDF acting in opposite directions on the two qubit states. We exemplarily assume that the detuning is symmetric, so the SDF has the same magnitude (absolute size) for both qubit states. This technique is useful even when the induced Stark shift is state-independent within the limit of large detuning |Δ|≫ω0, e.g., when the energy splitting of the qubit states is linearly independent of changes in the magnetic field (e.g., for the same magnetic quantum number m F For example, 43 Ca + With Ion, |↓>=|S 1 / 2 (F=4,m F =0)> and |↑>=|S 1 / 2 (F=3,m F =0)> can be used as the qubit state, and |e>=|D 5 / 2 > can be used as an excited state. Specific examples of optical qubit(s)

[0114] The above first and second examples of this technique are, for example, 5 / 2 40 Ca + , 88 Sr + , 138 Ba + , and 226 Ra + For example, as a specific example of an optical qubit, the qubit is one with |↓>=|S 1 / 2 (m j =+1 / 2)> and |↑>=|D 5 / 2 (m j =+3 / 2)> 40 Ca + may be encoded as

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

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

[0117] In this example, these large detunings make the transition insensitive to the polarization of the laser beam, eg, a linearly polarized laser beam may be used.

[0118] Selecting an excited state |e> with a large frequency difference from the qubit state allows for a large laser frequency ω corresponding to a large momentum transfer or a short wavelength (e.g., 200-2000 nm) photon. L Note that this leads to a faster entanglement gate compared to when longer wavelength electromagnetic radiation is used.

[0119] Further details of the above exemplary techniques are available in [1] and [2] (first example), [3] (second example), and [4] (third example). Finally, it should be noted that more than one excited state may be used to induce a state-dependent (dipole) force. In particular, there may be multiple excited states close in energy, all of which contribute significantly to the dipole force. In general, the present invention is not limited to any particular scheme for inducing SDF. SDF Modulation

[0120] In general, the geometric phase gates implemented herein enable coupling between internal qubit states via external motional states by applying an electromagnetic field (e.g., by an appropriate laser beam / pulse). That is, qubits are stored in the internal states of the respective ions, and the motional degrees of freedom may be exploited as a "quantum bus" for entangling the internal qubit degrees of freedom. The laser beam that induces an SDF in the addressed ions couples / excites the used motional modes in a state-dependent manner. The present invention differs from established implementations of MS and LS gates, particularly with regard to the way the SDF is generated and the direction of the SDF (or at least one component) relative to the direction of the (single) laser beam that generates the SDF.

[0121] As will be explained in more detail below, (quantum) harmonic oscillators of normal modes, corresponding to the collective quantized motion of Coulomb-coupled ionic crystals, are used to process and transfer quantum information. In general, a particular motional mode (e.g., one) may be selected and used to perform entanglement manipulation of the qubit states of two or more trapped ions. For simplicity, the use of this single motional mode is explicitly described, but the invention is not so limited, and multiple motional modes may be used to transfer entanglement. The frequency of this mode used, i.e., the "used motional mode," is denoted by ω. M and the subscript is often omitted when referring to the mode of motion used (i.e., |n> M (rather than |n>). Other or most other modes of motion are not excited by the appropriate laser frequency ω L and / or modulation frequency ω Mod However, in larger ionic crystals it may be difficult to use (excite / de-excite) only one mode. It is further noted that the motional mode used may be any of the modes of the ions, and in particular it is also possible to couple to a motional mode that is in the propagation direction of the laser beam, for example by directing the laser beam at the ions so that the addressed ions are not in the focus of the laser beam.

[0122] In general, the motional modes of trapped ions can be used to non-resonantly excite the modes by modulating (e.g., periodically varying) the SDF according to the frequency of the motional modes to entangle the states of two or more qubits. The SDF may also be modulated by (periodically) varying the intensity / phase / polarization gradient at the location of the addressed ions. This approach, described further below, is also known as a light-shift (LS) gate, and details are available in [1]-[4]. Displacement and geometric phase of the harmonic oscillator

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

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[0124] Multiple successive displacements may be performed to move the harmonic oscillator back to its original state in a closed path (i.e., the total / cumulative displacement due to the displacements is zero). Such a path / trajectory may be, for example, a circle or a polygon, and is also 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] Because the SDF depends on the state of the addressed qubit, a state-dependent phase shift is obtained. In other words, the SDF conditionally excites the desired motional mode depending on the state of all addressed qubits. That is, the amount of excitation may depend on the state of the qubit, and / or whether the mode is excited at all may depend on the state of the qubit. This conditional excitation can be used to perform entanglement operations on ions using the state dependence of the force.

[0126] As an example of this, Figure 12 shows how an axial center-of-mass (COM) mode can be conditionally excited, in which all ions (including unaddressed ions) oscillate together at regular intervals. In Figure 12, and as shown in the three-dimensional view in Figure 12e), we assume that the ions are aligned in the y direction, which is the ion axis direction. We assume that both laser beams propagate in the z direction, which is the beam direction for both beams. Figures 10a)–10d) show xy cross-sectional views 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 induced SDF direction is toward the center of the respective laser beam for the |0> state, and ii) it is away from the center of the respective laser beam for the |1> state. Therefore, the SDFs are in opposite directions for the two qubit states. For example, the laser beam may have a Gaussian beam profile with a maximum intensity at the beam axis. Thus, an intensity gradient may point toward the center of the laser beam, and the force acting on |0> may be in the direction of the gradient, while the force acting on |1> may be in the opposite direction to the gradient. For example, the |0> and |1> states may experience negative and positive energy shifts (see AC / optical shifts above) in their energy levels due to the laser beam, respectively.

[0127] More specifically, as can be seen, 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 the laser beam. As a result, as shown in Figure 12a), the direction of the SDF 1201 acting on the first addressed ion in the |0> state is opposite to the direction of the SDF 1202 acting on the second addressed ion in the |0> state. Therefore, in the |00> state, the forces cancel out, 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 the laser beam for the gate time. Furthermore, as shown in Figure 12b), the direction of the SDF 1201 acting on the first addressed ion in the |0> state is the same as the direction of the SDF 1212 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. The |01> state therefore acquires a phase φ from its interaction with the laser beam, i.e., by illuminating the laser beam for the gate time, |01> → e iφ 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 a phase φ, i.e., |10> → e iφ |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, in the |11> state, the forces cancel and no COM mode is excited when the addressed qubit is in the |1> state, i.e., |11> → |11>.

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

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[0129] Such an LS gate is equivalent to a universal controlled π-phase gate, which is

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[0130] For example, by initializing to, say, the (not fully entangled) |00> state and performing single-qubit operations on each qubit, we can obtain the (not yet fully entangled) state

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

[0132] More specifically, the term "gate time" T or "gate duration" T refers to the time / duration during which the laser beam(s) are modulated and irradiated (simultaneously) to the addressed ions. Thus, the gate time may 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 ions. In particular, the laser beam may be a laser pulse having a duration of the gate time. The amplitude of the laser pulse / beam outside the gate time may be zero, or the laser beam may not be irradiated to the addressed ions outside the gate time.

[0133] In general, the gate time may be predetermined, i.e., the gate time may be of a predetermined time / duration. In particular, the gate time may be determined such that after the gate time, the qubit is fully detangled from the harmonic oscillator state of the used motional mode. More specifically, the internal degrees of freedom used to represent the addressed qubit are detangled from the motional 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 motional state (i.e., its initial state at the start of the gate time).

[0134] More specifically, after modulation over the gate time, (i) the internal state (e.g., qubit state) of the addressed ion may be unentangled with the state of the motional mode, and / or (ii) the total (cumulative) displacement in position-momentum space of the harmonic oscillator of the used motional mode may be zero, the total displacement being the total displacement due to modulation of the SDF over the gate time.

[0135] For example, when determining the gate time (in advance), a cost function that depends on the gate time and the gate fidelity (i.e., the fidelity of the entanglement operation performed) may be minimized. More specifically, a shorter gate time can reduce the impact of phase instabilities of the involved lasers on the gate fidelity. A short gate time also has the advantage of speeding up quantum computation, which may be particularly important when extended quantum computations involving a large number of successive gate operations are performed. However, the gate time usually depends on the strength of the SDF and thus also on the intensity of the laser beam, and arbitrarily increasing it may cause instabilities, technical problems, and ultimately reduce fidelity. Periodic modulation and detuning δ

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

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[0137] For example, the SDF is a modulation frequency ω Mod =ω M +δ. In this regard, the frequency ω M Modulation frequency ω from Mod The (usually small) detuning δ of

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[0138] If an oscillator is driven periodically through a circular orbit in phase space (i.e., ω Mod =ω M +δ), the driving force is due to the detuning δ at the frequency ω M and resynchronizes after a duration of 2π / δ. The detuning δ therefore determines how fast the oscillator rotates through phase space. Therefore, decreasing the detuning parameter δ may allow for an increase in gate speed. Over a duration of 2π / δ, the motional state is displaced along a circular path in phase space, returning to its original point in phase space after a time 2π / δ, resulting in a geometric phase equal to the enclosed phase space area, which can be determined according to the following equation (see [2]):

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[0139] In general, the loop may be passed through 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 being used. MBy choosing the intensity of the laser beam (the dipole force is proportional to the intensity) and the detuning δ, a π / 2 phase shift can be obtained for a particular state of the addressed qubit over the gate time T, 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., make the gate faster) because the gate time is an integer multiple of the time required to execute one closed path in phase space. More specifically, a harmonic oscillator can be driven through multiple loops / cycles, allowing the accumulation of state-dependent geometric phases from all cycles to implement a particular quantum gate. However, too large a detuning δ can result in coupling with other motional modes, which may be undesirable. Therefore, the detuning δ can generally be selected to be small compared to the motional mode of interest and / or small compared to the frequency difference between the motional mode of interest and any other motional mode.

[0141] In general, the frequency δ may be determined (e.g., predetermined before the entanglement operation is performed) by optimizing the tradeoff between gate time and gate fidelity (or between gate time and gate error), typically subject to the constraints that the qubit state is not entangled with motional degrees of freedom after a 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 imposed by the implemented quantum gate. Thus, this determination may involve determining an appropriate integer n corresponding to the number of cycles in phase space, on which the gate time also depends. Furthermore, the term “gate fidelity” refers to the fidelity of the entanglement operation performed on the ions addressed by the laser beam. In general, 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 obtained phase, a larger force, and therefore a larger laser beam intensity, results in a faster gate, but too much beam intensity may reduce the fidelity.

[0142] Furthermore, a larger detuning δ implies a shorter gate time and a larger displacement in phase space. This may further increase the size of the ion wave packet, which may break down the Lamb-Dicke approximation, which is based on a wave packet size small compared to the laser wavelength, i.e., the coupling between the optical field and the ion motion is no longer a good linear approximation. This and other considerations may impose further constraints or trade-offs on the choice / determination of δ. SDF modulation method

[0143] The SDF may be modulated by modulating the transverse gradient of the laser beam at the position of each addressed ion to induce an SDF. The SDFs acting on all addressed ions, in particular all transverse gradients at the positions of the addressed ions, may be modulated in the same way (e.g., with the same frequency and the same phase). In particular, the gradients may be directly / actively modulated at a frequency (or frequencies) close to one or more motional 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 a laser beam (one, each, or all) at the positions of the ions addressed by said laser beam may be modulated in the same way (e.g., with the same frequency and the same phase). 1) modulating the intensity and / or amplitude of the electromagnetic field of the laser beam; 2) changing the position and / or direction of the laser beam relative to the trapped ions; 3) Including different frequencies of light in the laser beam; and / or 4) Modulating the polarization and / or phase of the laser beam It may be modulated by

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

[0145] In general, intensity / phase / polarization gradient modulation can be achieved by modulating the intensity of the laser beam. For example, the frequency, phase, and / or amplitude of the individual beams can be modulated using one or more AOMs and / or one or more AODs, e.g., AOMs in a double-pass configuration. In FIG. 6, for example, amplitude and / or phase modulation can be performed using AOD 620 and / or AOD 640. Alternatively or additionally, laser beam modulation can also be performed in FIG. 6 using an AOM (not shown in FIG. 6) positioned, for example, before the first AOD 620.

[0146] However, the present invention is not limited to a particular method of modulating intensity, etc. Thus, a modulated laser beam may have a modulated gradient that induces a modulated SDF acting on the addressed ions. For example, if two ions are addressed by two laser beams, respectively (e.g., as in FIG. 12), the electromagnetic fields of the first and second laser beams may be

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[0147] As explained above, the dipole force

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[0148] In general, modulation of the intensity / phase / polarization gradient can be achieved by modulating (e.g., varying) the position of the laser beam relative to the position of the ion. For example, the laser beam may have a non-constant gradient along the beam diameter and / or may vary with 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 changed by changing the ion's position relative to the beam axis (i.e., by changing the x and / or y coordinates). Because 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 may be varied, and the resulting SDF can be modulated. In particular, by changing the laser beam position relative to the ion's position, the transverse gradient can be used to excite radial or axial motional modes (or both). This advantage also applies to more advanced trap topologies with planar crystals in 3D Paul traps or surface ion traps. Method 3) - Modulation using a polychromatic laser beam

[0150] In general, intensity / phase / polarization gradient modulation can be achieved by including multiple different frequencies of light 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 two frequencies of light, a periodically varying intensity gradient at the location of the addressed ion can be achieved. The beat notes of the two frequencies result in a substantial intensity modulation. More specifically, when a laser beam modulates the electric field amplitude

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[0152] wave vector

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[0153] For example, two beams may be obtained by splitting a monochromatic laser beam using a splitter, shifting the frequency of at least one of the obtained laser beams using, for example, an acousto-optic modulator (AOM), and feeding the laser beams into the same fiber after the shift (this results in one laser beam inducing an SDF in the ions).

[0154] Alternatively, an AOM can be used to simultaneously split (i.e., split and shift in one step) a monochromatic laser beam into multiple laser beams with different frequencies and directions. In this way, three main laser beams may be generated: one laser beam with the same direction and frequency as the original monochromatic laser beam, and two laser beams of first diffraction orders. A polychromatic laser beam can then be obtained by feeding two or more of these three laser beams into the same fiber.

[0155] A further possibility is to use an electro-optic modulator (EOM) to obtain (in one step) a laser beam with two sidebands from a monochromatic laser beam. 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 a laser beam is written as:

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[0157] As mentioned above, the use of lateral SDFs can facilitate easy access to various directional modes of motion. Axial and / or radial modes

[0158] Figure 13 shows an example beam geometry for entangling two ions in a linear ion chain using axial motion modes. As shown in Figure 13d), a reproduction of Figure 12e), the axis labeling and definition of the coordinate origin relative to the laser direction and ion axis are identical to the example in Figure 12. Two ions are addressed with two Gaussian laser beams propagating in the z direction, i.e., darker areas of the laser beam indicate higher laser beam intensity. To excite axial modes (as explained above, the specific axial modes excited by a beam depend on the modulation of the SDF), an SDF is required along the ion axis (y direction). Because the laser beam is Gaussian, the intensity gradient used to induce the SDF is directed toward the center of the beam axis. Therefore, the laser beams are directed at the ions so that they 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] FIG. 14 shows an exemplary beam geometry for entangling two ions of a linear ion chain using radial motion modes. A laser beam is directed at the ions so that they are displaced from their respective beam axes in the x direction (the situation may otherwise be similar to that of FIG. 13 and / or FIG. 12, as shown in FIG. 14d). Displacement in the x direction means that the intensity gradient at the ion's position is in the x direction, which is a radial direction relative to the ion axis (y direction). Therefore, the induced SDF may be in the x direction, and x-directional motion modes can be excited and de-excited by the SDF. As shown in FIG. 14a, which shows an xy cross section at z=0, FIG. 14b, which shows a yz cross section at x=0, and FIG. 14c, which shows an xz cross section at x=-a, in the example of FIG. 13, the displacement of both addressed ions from the beam axis (which becomes 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 reciprocal modes may be used simultaneously to entangle different pairs of ions simultaneously. For example, four SDFs may be simultaneously induced on four (e.g., mutually distinct) ions, respectively. The four SDFs may be induced using four laser beams, respectively.

[0161] Two of the four SDFs may be modulated according to the frequency of a first motional mode to excite a first motional mode according to the internal states of the two ions acted upon by the two SDFs. Two of the four SDFs may be modulated according to the frequency of a second motional mode (different from the first motional mode) to excite a second motional mode according to the internal states of the two ions acted upon by the other two SDFs. Note that the first and second motional modes are different motional modes for all four ions. The two motional modes may generally 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 for additional ions and adjusting the laser beam parameters accordingly. Center of Mass (COM) Mode and Stretch Mode

[0163] The transverse SDF may facilitate coupling to different motion modes in the same direction, for example, different axial modes.

[0164] Figures 15 and 16 illustrate the excitation of axial COM and axial stretching modes, respectively, using a beam geometry similar to that described in Figure 13. However, the situation in Figure 15 differs from 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). The situation in Figure 16 also differs from Figure 13 in that in Figure 16 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, Fig. 15a) shows the xy cross section at z = 0, Fig. 15b) shows the yz cross section at x = 0, and Fig. 15c) shows the xz cross section at x = -a. Similarly, for excitation of the axial expansion / contraction mode, Fig. 16a) shows the xy cross section at z = 0, Fig. 16b) shows the yz cross section at x = 0, and Fig. 16c) shows the xz cross section at x = -a.

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

[0167] 15 act in the same direction when the ions are in different states to excite or de-excite COM mode vibrations, as indicated by the black arrows in Figures 13a) and 13b). More specifically, force 1501 acting on a first ion when the first ion is in the |0> state is in the same direction as force 1512 acting on a second ion when the second ion is in the |1> state, and force 1511 acting on a first ion when the first ion is in the |1> state is in the same direction as force 1502 acting on a second ion when the second ion is in the |0> state.

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

[0169] As already mentioned, the present invention is not only applicable to linear ion columns, but may also be applied in general to entangle two or more ions of any ionic crystal with a common / joint mode of motion. In particular, various beam steering options can be used to excite specific (freely selectable) modes of motion in planar ionic crystals, which can be very difficult with conventional methods because the geometry of planar ionic crystals further restricts optical access when entangling only specific ions.

[0170] This is illustrated in Figure 17, which shows the motion modes of ions in a 2D ionic crystal. More specifically, Figure 17d) shows the rest / equilibrium positions of the ions along with the three-dimensional coordinate system used. As can be seen, the ions are arranged according to a zigzag pattern in the plane. Furthermore, for said 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 each ion's vibration depending on the motional mode, but the motional modes of such 2D ionic crystals are generally not limited to one dimension. Note that the arrows in Figure 17b are projections onto the zy plane. In other words, a single motional mode can correspond to ion vibrations in different directions and with different amplitudes. However, the motional modes of planar ionic crystals can be excited and used to entangle ions in the same way as those of linear ionic crystals, i.e., by directing a laser beam modulated at a frequency detuned from the motional mode at the addressed ion. Further Aspects

[0172] The above-described embodiments and exemplary implementations represent some non-limiting examples. It is understood that various modifications may be made without departing from the claimed subject matter. For example, modifications may be made to adapt the examples 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. Furthermore, any of the steps of the methods described herein may be included as code instructions in a program that may be executed by one or more processors.

[0173] To summarize the above, 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.

[0174] According to a first aspect, A method for entangling first and second trapped ions is provided, the method comprising using modes of motion of the first and second trapped ions (e.g., for the entanglement): (i) inducing a first state-dependent force on the first trapped ion using a first laser beam; (ii) using a second laser beam to induce a second state-dependent force on the second trapped ion. the steps of inducing the first state-dependent force and the second state-dependent force are performed simultaneously; the first state-dependent force acts perpendicular to a direction of propagation of the first laser beam; the second state-dependent force acts perpendicular to a direction of propagation 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 motional mode to modulate the first state-dependent force and the second state-dependent force, respectively, thereby exciting the motional mode according to the internal states of the first trapped ion and the second trapped ion. and (a) modulating the first state-dependent force, wherein the first state-dependent force at the location of the first trapped ion: (ai) modulating the intensity and / or amplitude of the electromagnetic field of said first laser beam; (a.ii) varying 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) modulating the second state-dependent force such that the second state-dependent force at the location of the second trapped ion: (bi) modulating the intensity and / or amplitude of the electromagnetic field of the second laser beam; (b.ii) varying the position and / or direction of the second laser beam relative to the second trapped ions; and / or (b.iii) Modulation by including light of different frequencies in the laser beam.

[0175] According to a second aspect provided in addition to the first aspect, (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 direction of propagation of the first laser beam at the location of the first trapped ion, the gradient component of the first laser beam being 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, the gradient component of the second laser beam being an intensity gradient component, a phase gradient component, and / or a polarization gradient component.

[0176] According to a 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 a gate time; In the step of inducing the second state-dependent force, the second state-dependent force is modulated during the gate time. The gate time is, 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 motional mode; and / or (ii) a predetermined time during which the total displacement in position-momentum space of the harmonic oscillator of the motion mode is zero, the total displacement being the total displacement due to the modulation of the first state-dependent force and the second state-dependent force over the gating time.

[0177] According to a 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) inducing the first state-dependent force, the first state-dependent force has a modulation frequency ω Mod =ω M modulated by +δ, (iii) inducing the second state-dependent force, the second state-dependent force has a modulation frequency ω Mod =ω M modulated by +δ, (iv)ω M is the frequency of the motion mode.

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

[0179] According to a 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 super-Gaussian beam shape, a Laguerre-Gaussian beam shape, or a Hermite-Gaussian beam shape.

[0180] According to a 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 with different beam shapes; and / or (ii) the second laser beam has two or more beam components with different beam shapes;

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

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

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

[0184] According to an 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 at the first trapped ions and the second trapped ions using the same objective lens by adjusting the angles of incidence of the beams with respect to a major plane of the 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 a thirteenth aspect provided in addition to any one of the first to twelfth aspects, (i) two internal states of the first trapped ion are used to model a first qubit; (ii) two internal states of the second trapped ion are used to model a 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 a fourteenth aspect, instead of (i) to (iv) of the fourteenth aspect, or in addition to (i) to (iv) of the fourteenth aspect, the method further comprises the step of: providing quantum gates for the first qubit and the second qubit, in particular optically shifted LS gates and / or σ z σ z Implement a gate.

[0187] According to a fourteenth aspect, there is provided an apparatus for controlling a first laser beam and a second laser beam to entangle a first trapped ion and a second trapped ion using modes of motion of the first trapped ion and the second trapped ion, the apparatus comprising: (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 and a circuit configured to simultaneously control the 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, and the circuit modulates the first laser beam and the second laser beam in response to a frequency of the motional mode in inducing (e.g., during / during) the first state-dependent force and the second state-dependent force, thereby exciting the motional mode in response to an internal state of the first trapped ion and the second trapped ion. Furthermore, the circuit (a) modulating the first state-dependent force by modulating the first state-dependent force at a position of the first trapped ion; (ai) modulating the intensity and / or amplitude of the electromagnetic field of said first laser beam; (a.ii) varying the position and / or direction of the first laser beam relative to the first trapped ion; and / or (a.iii) modulating the laser beam by including light of different frequencies; and / or (b) modulating the second state-dependent force by modulating the second state-dependent force at a position of the second trapped ion; (bi) modulating the intensity and / or amplitude of the electromagnetic field of the second laser beam; (b.ii) varying the position and / or direction of the second laser beam relative to the second trapped ions; and / or (b.iii) by including light of different frequencies in said laser beam.

[0188] According to a fifteenth aspect provided in addition to the fourteenth aspect, The circuit comprises: (i) inducing the first state-dependent force, the electromagnetic field of the first laser beam has a gradient component perpendicular to the direction of propagation of the first laser beam at the location of the first trapped ion, the gradient component of the first laser beam being an intensity gradient component, a phase gradient component, and / or a polarization gradient component; and / or (ii) inducing the second state-dependent force, the electromagnetic field of the second laser beam has a gradient component perpendicular to the direction of propagation of the second laser beam at the location of the second trapped ion, the gradient component of the second first laser beam being an intensity gradient component, a phase gradient component, and / or a polarization gradient component.

[0189] According to a sixteenth aspect provided in addition to the fourteenth or fifteenth aspect, The circuit (i) inducing the first state-dependent force, modulating the first state-dependent force for a gating time; (ii) the inducing of the second state-dependent force is configured to modulate the second state-dependent force for the gate time, wherein the gate time is configured to, after the modulation 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 motional mode; and / or (b) a predetermined time during which the total displacement in position-momentum space of the harmonic oscillator of the motion mode is zero, the total displacement being the total displacement due to the modulation of the first state-dependent force and the second state-dependent force over the gating time.

[0190] According to a seventeenth aspect provided in addition to the sixteenth aspect, The circuit (i) inducing the first state-dependent force, the first state-dependent force being modulated at a modulation frequency ω Mod =ω M +δ modulates, (ii) inducing the second state-dependent force, the second state-dependent force being modulated at a modulation frequency ω Mod =ω M Modulate with +δ wherein the gate time is given as T=2πn / δ, n is an integer greater than zero, δ is a predetermined frequency, and ω M is the frequency of the motion mode.

[0191] According to an eighteenth aspect provided in addition to the seventeenth aspect, The frequency δ is predetermined by optimizing a trade-off between the gate time T and gate fidelity, where the gate fidelity is the fidelity of the entanglement of the first trapped ion and the second trapped ion.

[0192] According to a 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 super-Gaussian beam shape, a Laguerre-Gaussian beam shape, or a Hermite-Gaussian beam shape. In particular, the circuitry may be configured to control (e.g., by controlling optical elements) the first laser beam and / or the second laser beam to have a Gaussian beam shape, a super-Gaussian beam shape, a Laguerre-Gaussian beam shape, or a Hermite-Gaussian beam shape.

[0193] According to a twentieth aspect provided in addition to any one of the fourteenth to eighteenth aspects, (i) the first laser beam has two or more beam components with different beam shapes; and / or (ii) the second laser beam has two or more beam components with different beam shapes;

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

[0195] In particular, in the twentieth and twenty-first aspects, The circuitry may be configured to control the first laser beam and / or the second laser beam to have such beam components.

[0196] According to a 22nd aspect provided in addition to any one of the 14th to 21st aspects, the circuitry simultaneously with the controlling of the first laser beam and the second laser beam to induce the first state-dependent force and the second state-dependent force; (i) controlling a third laser beam to induce a third state-dependent force on a third trapped ion; (ii) controlling the fourth laser beam to induce a fourth state-dependent force on the fourth trapped ion; (a) the third state-dependent force acts perpendicular to a direction of propagation of the third laser beam; (b) the fourth state-dependent force acts perpendicular to a direction of propagation of the fourth laser beam; The circuit (c) in inducing the third state-dependent force and the fourth state-dependent force (e.g., during / during the inducing), modulating the third laser beam and the fourth laser beam according to frequencies of second motional modes of the first trapped ion, the second trapped ion, the third trapped ion, and the fourth trapped ion to modulate the third state-dependent force and the fourth state-dependent force, respectively, thereby exciting the second motional modes according to internal states of the third trapped ion and the fourth trapped ion.

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

[0198] According to a 24th aspect provided in addition to any one of the 14th to 23rd aspects, The circuitry is configured to direct the first laser beam and the second laser beam at the first trapped ions and the second trapped ions using the same objective lens by adjusting the angles of incidence of the beams with respect to a major plane of the objective lens.

[0199] According to a 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 a 26th aspect provided in addition to any one of the 14th to 25th aspects, (i) two internal states of the first trapped ion are used to model a first qubit; (ii) two internal states of the second trapped ion are used to model a 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 or in addition to (i) to (iv) of the 28th aspect, The circuit includes quantum gates for the first qubit and the second qubit, in particular optical shift LS gates and / or σ z σ z The gate is configured to implement the gate.

Claims

1. 1. A method for entangling a first trapped ion (381) and a second trapped ion (382) using modes of motion of the first trapped ion (381) and the second trapped ion (382), comprising: Inducing (S240) a first state-dependent force (371) on the first trapped ion (381) using a first laser beam (361); Inducing (S240) a second state-dependent force (372) on the second trapped ion (382) using a second laser beam (362); Including, the step (S240) of inducing 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 direction of propagation of the second laser beam (362); inducing the first state-dependent force (371) and the second state-dependent force (372), modulating the first laser beam (362) and the second laser beam (362) in accordance with a frequency of the motional mode to modulate the first state-dependent force (371) and the second state-dependent force (372), respectively (S220), thereby exciting the motional mode in accordance with an internal state of the first trapped ion (381) and the second trapped ion (382); In the modulating (S220) of the first state-dependent force (371), the first state-dependent force (371) at the position of the first trapped ion (381) modulating the intensity and / or amplitude of the electromagnetic field of said first laser beam (361); Varying the position and / or direction of the first laser beam (361) relative to the first trapped ions (381); and / or including different frequencies of light in said laser beam; and / or modulated by In the modulating (S220) of the second state-dependent force (372), the second state-dependent force (372) at the position of the second trapped ion (3z2) is: modulating the intensity and / or amplitude of the electromagnetic field of said second laser beam (362); Varying the position and / or direction of the second laser beam (362) relative to the second trapped ions (382); and / or including different frequencies of light in said laser beam; is modulated by In the step of inducing the first state-dependent force (371), an electromagnetic field of the first laser beam (361) has a gradient component perpendicular to the direction of propagation of the first laser beam (361) at the position of the first trapped ion (381); the gradient component of the first laser beam (361) is 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: different from the second beam shape, the electric field is zero at the location of the first trapped ion (381); The gradient component of the first laser beam at the location of the first trapped ion (381) increases with the intensity of the first beam component (811). A method characterized by:

2. In the step of inducing the second state-dependent force (372), an 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), the gradient component of the second laser beam (362) being an intensity gradient component, a phase gradient component, and / or a polarization gradient component. The method of claim 1.

3. In the step of inducing the first state-dependent force (371), the first state-dependent force (371) is modulated during a 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 is, 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 motional mode; and / or a total displacement in position-momentum space of the harmonic oscillator of the motion mode is zero, the total displacement being the total displacement due to the modulation of the first state-dependent force (371) and the second state-dependent force (372) over the gating time; 3. The method of claim 1 or 2, wherein the predetermined time period is

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) is modulated at a modulation frequency ω Ｍｏｄ =ω Ｍ +δ, In the step of inducing the second state-dependent force (372), the second state-dependent force (372) is modulated at a modulation frequency ω during the gate time. Ｍｏｄ =ω Ｍ +δ, ω Ｍ is the frequency of the motion mode, The method of claim 3.

5. The frequency δ is determining the integer n based on an implemented quantum gate; minimizing a cost function that depends on the gate time T and gate fidelity, where the gate fidelity is the fidelity of the entanglement of the first trapped ion (381) and the second trapped ion (382); The method of claim 4 , wherein the predetermined value is determined by:

6. the first laser beam and / or the second laser beam (362) have a Gaussian beam shape, a Super-Gaussian beam shape, a Laguerre-Gaussian beam shape, or a Hermite-Gaussian beam shape; The method according to any one of claims 1 to 5.

7. the first laser beam (361) has two or more beam components with different beam shapes and / or with beam shapes displaced with respect to each other; and / or the second laser beam (362) has two or more beam components with different beam shapes and / or with beam shapes displaced relative to each other; The method according to any one of claims 1 to 5.

8. a first beam component of the two or more beam components of the first laser beam (361) has a Gaussian beam shape and a second beam component of the two or more beam components of the first laser beam (361) has a Hermite-Gaussian beam shape; and / or a second beam component of the two or more beam components of the second laser beam (362) having a Gaussian beam shape; and a second beam component of the two or more beam components of the second laser beam (362) having a Hermite-Gaussian beam shape. The method of claim 7.

9. Simultaneously with the step of inducing the first state-dependent force (371) and the second state-dependent force (372), the method further comprises: Inducing a third state-dependent force on the third trapped ion using a third laser beam; inducing a fourth state-dependent force on the fourth trapped ion using a fourth laser beam; further comprising the third state-dependent force acts perpendicular to a direction of propagation of the third laser beam; the fourth state-dependent force acts perpendicular to a direction of propagation of the fourth laser beam; and in the step of inducing the third state-dependent force and the fourth state-dependent force, modulating the third laser beam and the fourth laser beam in accordance with frequencies of second motional modes of the first trapped ion (381), the second trapped ion (382), the third trapped ion, and the fourth trapped ion to modulate the third state-dependent force and the fourth state-dependent force, respectively, thereby exciting the second motional modes in accordance with internal states of the third trapped ion and the fourth trapped ion. The method according to any one of claims 1 to 8.

10. the first trapped ion (381) and the second trapped ion (382) are ions among a plurality of ions trapped at positions in a line or a two-dimensional lattice; The method according to any one of claims 1 to 9.

11. the first laser beam (361) and the second laser beam (362) are directed at the first trapped ion (381) and the second trapped ion (382) using the same objective lens by adjusting the angle of incidence of the beams with respect to a major plane of the objective lens; The method according to any one of claims 1 to 10.

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 an internal state of the second trapped ion (382); The method according to any one of claims 1 to 11.

13. two internal states of the first trapped ion (381) are used to model a first qubit; two internal states of the second trapped ion (382) are used to model a 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) differs depending on the two internal states of the second trapped ion (382); and / or The method further comprises the step of: providing quantum gates for the first qubit and the second qubit, in particular optically shifted LS gates and / or σ ｚ σ ｚ Implementing gates, The method according to any one of claims 1 to 12.

14. 1. An apparatus for controlling a first laser beam (361) and a second laser beam (362) to entangle a first trapped ion (381) and a second trapped ion (382) using modes of motion of the first trapped ion (381) and the second trapped ion (382), comprising: a first laser beam (361) for inducing a first state-dependent force (371) on the first trapped ion (381); a second laser beam (362) for inducing a second state-dependent force (372) on the second trapped ion (382); a circuit (315) configured to simultaneously control 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 direction of propagation of the second laser beam (362); wherein the induction of the first state-dependent force (371) and the second state-dependent force (372) is performed by modulating the first laser beam (362) and the second laser beam (362) in accordance with a frequency of the motional mode, thereby modulating the first state-dependent force (371) and the second state-dependent force (372), respectively, thereby exciting the motional mode in accordance with an internal state of the first trapped ion (381) and the second trapped ion (382); The circuit (315) modulating the first state-dependent force (371) by modulating the intensity and / or amplitude of the electromagnetic field of the first laser beam (361) at the location of the first trapped ion (381), by changing the position and / or direction of the first laser beam (361) relative to the first trapped ion (381), and / or by including light of a different frequency in the laser beam; and / or modulating the second state-dependent force (372) by modulating the intensity and / or amplitude of the electromagnetic field of the second laser beam (362) at the location of the second trapped ion (382), by changing the position and / or direction of the second laser beam (362) relative to the second trapped ion (382), and / or by including light of a different frequency in the laser beam. It is configured as follows: inducing the first state-dependent force (371), an electromagnetic field of the first laser beam (361) having a gradient component perpendicular to the direction of propagation of the first laser beam (361) at the location of the first trapped ion (381); the gradient component of the first laser beam (361) is 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: different from the second beam shape, the electric field is zero at the location of the first trapped ion (381); The gradient component of the first laser beam at the location of the first trapped ion (381) increases with the intensity of the first beam component (811). An apparatus characterized in that

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