Amplitude, Frequency, and Phase Modulation Entanglement Gates for Trapped-Ion Quantum Computers

Abstract

A method for performing an entanglement operation between two trapped ions in a quantum computer, comprising: selecting a gate duration value of a pulse to be applied to a first ion and a second ion in a chain of trapped ions; determining one or more tones of the pulse based on the selected gate duration value and the frequency of the motional mode of the chain of trapped ions, each tone including an amplitude value and a detuning frequency value; generating the pulse having the one or more tones, each tone including the determined amplitude value and the determined detuning frequency value; and applying the generated pulse to the first ion and the second ion for the gate duration value. Each trapped ion has two frequency-separated states that define a qubit, and the motional modes of the chain of trapped ions each have different frequencies.

Description

Technical Field

[0001] The present disclosure generally relates to methods for generating entanglement gates in an ion trap quantum computer, and more particularly, to methods for optimizing pulses to generate entanglement gates. Background Art

[0002] In quantum computing, qubits (quantum bits) are analogous to the bits representing "0" and "1" in a classical (digital) computer and need to be prepared, manipulated, and measured (read out) with near-perfect control during the computing process. Imperfect control of qubits leads to errors that accumulate during the computing process, thus limiting the scale of quantum computers capable of performing reliable computations.

[0003] Among the physical systems proposed for building large-scale quantum computers, there is a chain of ions (e.g., charged atoms) that are trapped by electromagnetic fields and suspended in a vacuum. These ions have internal hyperfine states that are separated by frequencies in the range of several GHz and can be used as the computational states of qubits (referred to as "qubit states"). These hyperfine states can be controlled using radiation provided by lasers, or sometimes referred to in this document as the interaction with laser beams. Using this laser interaction, the ions can be cooled to near their motional ground state. The ions can also be optically pumped with high precision into one of two hyperfine states (preparation of qubits), manipulated between the two hyperfine states by a laser beam (single qubit gate operation), and their internal hyperfine states can be detected by fluorescence when a resonant laser beam is applied (readout of qubits). A pair of ions can be controllably entangled using laser pulses through forces associated with the qubit states (two qubit gate operation), where the laser pulses couple the ions to collective motional modes of the trapped ion chain, which are generated by the Coulomb interaction between the ions. As the scale of the quantum computer increases, the implementation of two qubit gate operations between a pair of ions increases in complexity, and thus the associated errors and the resources required for implementation (such as laser power) increase.

[0004] To increase the scale of a quantum computer that can implement algorithms to solve problems that are otherwise difficult to solve on classical computers, a process is needed to precisely control qubits to perform the desired computational process with minimal resources. Summary of the Invention

[0005] Embodiments of the present disclosure generally relate to a method for performing an entanglement operation between two trapped ions in a quantum computer. The method includes: selecting a gate duration value of a pulse to be applied to a first ion and a second ion in a trapped ion chain; determining one or more tones of the pulse based on the selected gate duration value and the frequency of the motional mode of the trapped ion chain, each tone including an amplitude value and a detuning frequency value; generating a pulse having the one or more tones, each tone including the determined amplitude and the determined detuning frequency value; and applying the generated pulse to the first ion and the second ion for the gate duration value. Each trapped ion has two frequency-separated states that define a qubit, and the motional modes of the trapped ion chain each have different frequencies.

[0006] Embodiments of the present disclosure generally relate to a quantum computing system. The quantum computing system includes: a trapped ion chain, where each trapped ion has two hyperfine states and an excited state that define a qubit; one or more lasers configured to emit a laser beam that is split into two or more non-collinear laser beams provided to each trapped ion, wherein the two or more non-collinear laser beams are configured to cause each of the trapped ions to generate Rabi flopping between the two hyperfine states via the excited state; and a controller configured to select a gate duration value of a pulse to be applied to a first ion and a second ion in the trapped ion chain, determine one or more tones of the pulse based on the selected gate duration value and the frequency of the motional mode of the trapped ion chain, each tone including an amplitude value and a detuning frequency value, generate the pulse having the one or more tones, each tone including the determined amplitude and the determined detuning frequency, and apply the generated pulse to the first ion and the second ion for the gate duration value. Each trapped ion has two frequency-separated states that define a qubit, and the motional modes of the trapped ion chain each have different frequencies.

[0007] Embodiments of the present disclosure generally relate to a method of performing computations using a quantum computer. The method includes a processor in a digital computer executing a software program stored in a non-volatile memory of the digital computer and generating a computation result based on processed quantum information. The executed software program requires performing at least one computation, and performing the at least one computation includes: a processor in the digital computer selecting a quantum algorithm to be implemented on the quantum computer; compiling the selected quantum algorithm into a series of universal logic gates; converting the series of universal logic gates into laser pulses to be applied to a pair of trapped ions within a quantum register; during the process of performing the at least one computation, the processor in the digital computer calculating parameters of the laser pulses to be applied to the pair of trapped ions within the quantum register; generating laser pulses, each laser pulse having a determined amplitude and a determined detuning frequency; applying the generated laser pulses to the pair of trapped ions for a time length equal to the gate duration value; measuring the population of the qubit states of the plurality of trapped ions; and the processor of the digital computer processing quantum information corresponding to the qubit states of the plurality of trapped ions based on the measured population of the qubit states. The quantum computer includes a plurality of trapped ions disposed within a quantum register of the quantum computer, each of the plurality of trapped ions having two frequency-separated states, each frequency-separated state defining a qubit, and the motional modes of the plurality of trapped ions each having a different frequency. Calculating the parameters includes the processor in the digital computer determining an amplitude function and a detuning frequency function of the laser pulses based on information stored in the digital computer regarding the gate duration value and the frequencies of the motional modes of the plurality of trapped ions.

[0008] Embodiments of the present disclosure generally relate to methods for quantum computing systems, the quantum computing systems including: a chain of trapped ions, each trapped ion having two hyperfine states and an excited state that define a qubit; one or more lasers configured to emit laser beams that are split into two or more non-collinear laser beams having a first frequency and a second frequency, the laser beams being provided to a first ion and a second ion in the chain of trapped ions, wherein the two or more non-collinear laser beams are configured to cause Rabi oscillations between each of the two hyperfine states and the excited state of the first ion and the second ion; and a controller including a non-volatile memory storing a plurality of instructions. When executed by a processor, the instructions cause the quantum computing system to: perform an operation including selecting a gate duration value for a pulse to be applied to the first ion and the second ion in the chain of trapped ions, wherein each of the trapped ions has two frequency-separated states that define a qubit, and the motional modes of the chain of trapped ions each have a different frequency; determine one or more tones of the pulse based on the selected gate duration value and the frequencies of the motional modes of the chain of trapped ions, each tone including an amplitude value and a detuning frequency value; generate a pulse having the one or more tones, each tone including the determined amplitude and the determined detuning frequency; and apply the generated pulse to the first ion and the second ion for the gate duration value. BRIEF DESCRIPTION OF THE DRAWINGS

[0009] In a manner enabling a detailed understanding of the above-described features of the present disclosure, the present disclosure briefly summarized above may be described in more specific terms by reference to the embodiments, some of which are illustrated in the drawings. It should be noted, however, that the drawings illustrate only typical embodiments of the present disclosure and should not be considered as limiting its scope, as the present disclosure may admit to other equally effective embodiments.

[0010] Figure 1 is a partial view of an ion trap quantum computer according to one embodiment.

[0011] Figure 2 describes a schematic diagram of an ion trap for confining ions in a chain according to one embodiment.

[0012] Figure 3A 、 Figure 3B and Figure 3C describe several schematic collective transverse motional mode structures of a chain of five trapped ions.

[0013] Figure 4 describes a schematic energy diagram of each ion in a chain of trapped ions according to one embodiment.

[0014] Figure 5Describes the qubit state of an ion, represented as a point on the surface of a Bloch sphere.

[0015] Figure 6A And Figure 6B Describes a schematic diagram of the motional sideband spectrum and motional mode of each ion according to one embodiment.

[0016] Figure 7 Describes a flowchart illustrating a method for generating a power-optimal pulse for performing an XX gate operation on two qubits according to one embodiment.

[0017] Figure 8A Describes an optimal pulse function according to one embodiment.

[0018] Figure 8B Describes the detuning frequency of an optimal pulse according to one embodiment.

[0019] Figure 9 Describes a pulse function according to one embodiment.

[0020] Figure 10 Describes the distortion of an XX gate operation for a chain of five trapped ions according to one embodiment.

[0021] Figure 11 Describes the entanglement interaction obtained through an XX gate operation according to one embodiment.

[0022] For ease of understanding, wherever possible, the same reference numerals are used to denote the same elements common to the drawings. In the drawings and the following description, an orthogonal coordinate system including an X-axis, a Y-axis, and a Z-axis is used. For convenience, the directions indicated by the arrows in the figures are assumed to be the positive directions. It is contemplated that the elements disclosed in some embodiments may be beneficially used in other embodiments without special statement. Detailed Description

[0023] The embodiments described herein generally relate to methods and systems for designing, optimizing, and delivering pulses to perform entanglement gate operations between two ions during quantum computing, and more particularly, to a pulse that increases the fidelity of an entanglement gate operation or the probability that two ions are in an expected qubit state after performing an entanglement gate operation between at least two ions, and further reduces the laser power required to perform the entanglement gate operation.

[0024] The entire system capable of performing quantum computing using trapped ions will include a classical computer, a system controller, and a quantum register. The classical computer performs support and system control tasks, including: selecting a quantum algorithm to run via a user interface using, for example, a graphics processing unit (GPU); compiling the selected quantum algorithm into a series of universal logic gates; converting the series of universal logic gates into laser pulses to apply to the quantum register; and pre-computing parameters to optimize the laser pulses using a central processing unit (CPU). Software programs for performing the tasks of decomposing and executing quantum algorithms are stored in non-volatile memory within the classical computer. The quantum register includes trapped ions coupled to various hardware, including lasers for manipulating the internal hyperfine states (qubit states) of the trapped ions and acousto-optic modulators for reading out the internal hyperfine states (qubit states) of the trapped ions. The system controller receives the pre-computed parameters of the power-optimal pulses from the classical computer when starting to run the selected algorithm on the quantum register, controls the various hardware associated with controlling any and all aspects of running the selected algorithm on the quantum register, and returns the readout of the quantum register, thereby outputting the result of the quantum computation to the classical computer at the end of the algorithm run.

[0025] The methods and systems described herein include a process for converting logic gates into laser pulses to apply to a quantum register, and a process for pre-computing parameters that optimize the laser pulses applied to the quantum register and are used to improve the performance of the quantum computer.

[0026] Among several known sets of universal logic gates through which any quantum algorithm can be decomposed, there is a set of universal logic gates (commonly denoted as {R, XX}) native to the quantum computing system of trapped ions described herein. Here, the R gate corresponds to the manipulation of the individual qubit states of the trapped ions, and the XX gate (also known as the "entanglement gate") corresponds to the manipulation of the entanglement of two trapped ions. For those of ordinary skill in the art, it should be clear that the R gate can be implemented with near-perfect fidelity, while the formation of the XX gate is complex and requires optimization of, among other factors, the given type of trapped ions, the number of ions in the trapped ion chain, and the hardware and environment in which the trapped ions are trapped, so as to improve the fidelity of the XX gate and avoid or reduce computational errors in the quantum computer. In the following discussion, methods for generating and optimizing pulses for performing computations based on the formation of XX gates with improved fidelity will be described.

[0027] As the scale of a quantum computer increases, the complexity of performing entangled gate operations for quantum computing increases, and the complexity of the pulses used to perform these entangled gate operations also increases. The laser power required to implement such complex pulses then increases, so the available laser power can limit the scale of the quantum computer that can be achieved. The methods and systems described in this disclosure simplify the construction of the pulses and further reduce the laser power required to implement the pulses, such that the quantum computer can be scaled up to a larger scale and can perform more complex computational operations. This means performing entangled gates faster with a given power budget. The error commensurate with the delivered laser power will decrease as the required laser power decreases.

[0028] General hardware configuration

[0029] Figure 1 is a partial view of an ion trap quantum computer or system 100 according to one embodiment. System 100 includes a classical (digital) computer 101, a system controller 118, and a quantum register, which is a chain 102 of trapped ions (i.e., the five shown) extending along the Z-axis. Classical computer 101 includes a central processing unit (CPU), a memory, and support circuitry (or I / O). The memory is connected to the CPU and can be one or more of readily available memories such as read-only memory (ROM), random access memory (RAM), floppy disks, hard disks, or any other form of local or remote digital storage. Software instructions, algorithms, and data can be encoded and stored in the memory for instructing the CPU. Support circuitry (not shown) is also connected to the CPU to support the processor in a conventional manner. The support circuitry can include conventional caches, power supplies, clock circuits, input / output circuits, subsystems, and the like.

[0030] Imaging objective lens 104, such as an objective lens with a numerical aperture (NA) of, for example, 0.37, collects fluorescence from ions along the Y-axis and maps each ion onto a multi-channel photomultiplier tube (PMT) 106 for measuring individual ions. A non-collinear Raman laser beam provided along the X-axis from laser 108 performs operations on the ions. Diffraction beam splitter 110 generates an array of static Raman beams 112, which are individually switched using multi-channel acousto-optic modulator (AOM) 114 and are configured to selectively act on individual ions. Global Raman laser beam 116 irradiates all ions simultaneously. System controller (also referred to as "RF controller") 118 controls AOM 114. System controller 118 includes a central processing unit (CPU) 120, read-only memory (ROM) 122, random access memory (RAM) 124, storage unit 126, etc. CPU 120 is the processor of RF controller 118. ROM 122 stores various programs, and RAM 124 is the working memory for various programs and data. Storage unit 126 includes non-volatile memory such as a hard disk drive (HDD) or flash memory and stores various programs even when power is off. CPU 120, ROM 122, RAM 124, and storage unit 126 are interconnected via bus 128. RF controller 118 executes control programs stored in ROM 122 or storage unit 126 and uses RAM 124 as a working area. The control programs will include one or more software applications that include program code (e.g., instructions) that can be executed by the processor to perform various functions associated with receiving and analyzing data and controlling any and all aspects of the methods and hardware for creating the ion trap quantum computer system 100 discussed herein.

[0031] Figure 2 A schematic diagram of an ion trap 200 (also referred to as a Paul trap) for confining ions in chain 102 according to one embodiment is described. A confinement potential is applied by both a static (DC) voltage and a radio frequency (RF) voltage. A static (DC) voltage V is applied to end cap electrodes 210 and 212 S to confine ions along the Z-axis (also referred to as "axial" or "longitudinal"). Due to the Coulomb interaction between ions, the ions in chain 102 are almost uniformly distributed axially. In some embodiments, ion trap 200 includes four hyperbolic electrodes 202, 204, 206, and 208 extending along the Z-axis.

[0032] During operation, a sinusoidal voltage V1 (with amplitude V RF / 2) is applied to a pair of opposing electrodes 202, 204, and a voltage that is in phase with sinusoidal voltage V1 (and amplitude V RF / 2) A sinusoidal voltage V2 with a 180° phase shift drives at a frequency ω RF is applied to another pair of opposing electrodes 206, 208 to generate a quadrupole potential. In some embodiments, the sinusoidal voltage is applied only to a pair of opposing electrodes 202, 204, while the other pair of opposing electrodes 206, 208 is grounded. The quadrupole potential generates an effective binding force for each trapped ion in the X - Y plane (also referred to as the "radial" or "transverse" plane) perpendicular to the Z - axis, and this binding force is proportional to the distance from the saddle point (i.e., the position where the RF electric field vanishes in the axial (Z - direction)). The motion of each ion in the radial direction (i.e., the direction in the X - Y plane) is approximated as a harmonic oscillation (referred to as secular motion) with a restoring force towards the saddle point in the radial direction, and can be modeled separately by spring constants k x and k y as discussed in more detail below. In some embodiments, when the quadrupole potential is symmetric in the radial direction, the spring constants in the radial direction are modeled as equal. However, in some cases, undesirably, the motion of the ions in the radial direction may be distorted due to some asymmetries in the physical trap configuration, small DC patch potentials due to non - uniformities on the electrode surfaces, etc., and due to these and other external distortion sources, the ions may deviate from the center of the saddle point.

[0033] Trapped Ion Configurations and Qubit Information

[0034] Figure 3A 、 Figure 3B and Figure 3C describe several schematic structures of the collective transverse motion modes (also simply referred to as "motion mode structures") of a chain 102 of, for example, five trapped ions. Here, the confinement potential due to the static voltage V S applied to the end - cap electrodes 210 and 212 is weaker compared to the confinement potential in the radial direction. The collective motion modes of the chain 102 of trapped ions in the transverse direction are determined by the combination of the Coulomb interaction between the trapped ions and the confinement potential generated by the ion trap 200. The trapped ions undergo collective transverse motion (referred to as "collective transverse motion mode", "collective motion mode" or simply "motion mode"), and each mode has a different energy (or equivalently, frequency) associated with it. The motion mode with the p - th lowest energy is hereinafter referred to as |n ph > p , where n ph represents the number of motion quanta (in units of energy excitations, called phonons) in the motion mode, and the number of motion modes P in a given transverse direction is equal to the number N of ions trapped in the chain 102. Figures 3A to 3C Schematically shows an embodiment of the different types of collective transverse motion modes that five trapped ions positioned in the chain 102 may undergo. Figure 3Ais the co - motion pattern with the highest energy|n ph > P is a schematic diagram, where P is the number of motion patterns. In the co - motion pattern|n> P all ions oscillate in - phase transversely. Figure 3B is the tilted motion pattern with the second - highest energy|n ph > P-1 is a schematic diagram. In the tilted motion pattern, ions at opposite ends move out - of - phase transversely (i.e., in opposite directions). Figure 3C is a high - order motion pattern|n ph > P-3 is a schematic diagram, whose energy is lower than that of the tilted motion pattern|n ph > P-1 and ions move in a more complex pattern.

[0035] It should be noted that the above - mentioned specific configuration is only one of several possible examples of a trap for confining ions according to the present disclosure, and does not limit the possible configurations, specifications, etc. of the trap according to the present disclosure. For example, the geometry of the electrodes is not limited to the above - mentioned hyperbolic electrodes. In other examples, the trap that generates an effective electric field can be a multi - layer trap or a surface trap, where the effective electric field causes ions to move as harmonic oscillations radially. In a multi - layer trap, several electrode layers are stacked and an RF voltage is applied to two diagonally - opposite electrodes. In a surface trap, all electrodes are located in a single plane on a chip. In addition, the trap can be divided into multiple segments, and adjacent pairs can be linked by one or more ion shuttles or coupled by photon interconnections. The trap can also be an array of individual confinement regions arranged close to each other on a micro - fabricated ion - trap chip. In some embodiments, in addition to the above - mentioned RF component, the quadrupole potential also has a spatially - varying DC component.

[0036] Figure 4 Describes a schematic energy diagram 400 for each ion in a trapped ion chain 102 according to an embodiment. In one example, each ion can be a positive ytterbium ion 171 Yb + which has 2 S 1 / 2 hyperfine states (i.e., two electron states), whose energy splitting corresponds to a frequency difference of ω 01 / 2π = 12.642821 GHz (referred to as the "carrier frequency"). The qubit is formed by these two hyperfine states, denoted as |0> and |1>, where the hyperfine ground state is selected (i.e., 2 S 1 / 2The lower energy state in the hyperfine state) is denoted as |0>. Hereinafter, the terms "hyperfine state", "internal hyperfine state", and "qubit" may be used interchangeably to denote |0> and |1>. By known laser cooling methods, such as Doppler cooling or resolved sideband cooling, each ion can be cooled (i.e., the kinetic energy of the ion can be reduced) to the motional ground state |0> p near which there is no phonon excitation in any motional mode p (i.e., n ph = 0), and then the qubit state is prepared in the hyperfine ground state |0> by optical pumping. Here, |0> represents the single qubit state of the trapped ion, and |0> with subscript p p represents the motional ground state of the motional mode p of the trapped ion chain 102.

[0037] The single qubit state of each trapped ion can be manipulated, for example, by a mode-locked laser at 355 nanometers (nm) through the excited 2 P 1 / 2 energy level (denoted as |e>). As Figure 4 shown, the laser beam from the laser can be split into a pair of non-collinear laser beams (the first laser beam with frequency ω1 and the second laser beam with frequency ω2) in a Raman configuration, and is detuned by the single-photon transition detuning frequency Δ = ω1 - ω 0e from the transition frequency ω between |0> and |e>, as 0e shown. The two-photon transition detuning frequency δ includes the amount of energy provided to the trapped ion by the first and second laser beams, and their combination is used to transfer the trapped ion between the hyperfine states |0> and |1>. When the single-photon transition detuning frequency Δ is much larger than the two-photon transition detuning frequency (also simply referred to as the "detuning frequency") δ = ω1 - ω2 - ω Figure 4 shown. The two-photon Rabi frequency Ω(t) causes Rabi oscillations (also called "carrier transitions") between the two hyperfine states |0> and |1>. The intensity (i.e., the absolute value of the amplitude) of the two-photon Rabi frequency Ω(t) is proportional to Ω 01 Ω 0e / 2Δ, where Ω 1e and Ω 0e Ω 1e / 2Δ is proportional to, where Ω 0e and Ω 1eThey are the single-photon Rabi frequencies caused by the first and second laser beams, respectively. Hereinafter, this set of non-collinear laser beams used to manipulate the internal hyperfine states (qubit states) of qubits in a Raman configuration may be referred to as a "composite pulse" or simply as a "pulse", and the time-dependent pattern of the resulting two-photon Rabi frequency Ω(t) may be referred to as the "amplitude" of the pulse or simply as the "pulse", which will be shown and further described below. The detuning frequency δ = ω1 - ω2 - ω 01 may be referred to as the detuning frequency of the composite pulse or the detuning frequency of the pulse. The amplitude of the two-photon Rabi frequency Ω(t) determined by the amplitudes of the first and second laser beams may be referred to as the "amplitude" of the composite pulse.

[0038] It should be noted that the specific atomic species used in the discussions provided herein is only an example of atomic species that have a stable and well-defined two-level energy structure and optically accessible excited states when ionized, and thus is not intended to limit the possible configurations, specifications, etc. of ion trap quantum computers according to the present disclosure. For example, other ion species include alkaline earth metal ions (Be + , Ca + , Sr + , Mg + , and Ba + ) or transition metal ions (Zn + , Hg + , Cd + ).

[0039] Provide Figure 5To help visualize the qubit state of the ion, it is represented as a point on the surface of the Bloch sphere 500, which has an azimuthal angle φ and a polar angle θ. The application of the composite pulse as described above results in Rabi oscillations occurring between the qubit states |0⟩ (represented as the north pole of the Bloch sphere) and |1⟩ (the south pole of the Bloch sphere). Adjusting the duration and amplitude of the composite pulse flips the qubit state from |0⟩ to |1⟩ (i.e., from the north pole to the south pole of the Bloch sphere), or flips the qubit state from |1⟩ to |0⟩ (i.e., from the south pole to the north pole of the Bloch sphere). This application of the composite pulse is called a "π pulse". Additionally, by adjusting the duration and amplitude of the composite pulse, the qubit state |0⟩ can be converted to the superposition state |0⟩ + |1⟩, where the two qubit states |0⟩ and |1⟩ are added and equally weighted in phase (the normalization factor of the superposition state is omitted below without loss of generality), and the qubit state |1⟩ can be converted to the superposition state |0⟩ - |1⟩, where the two qubit states |0⟩ and |1⟩ are equally weighted but have different phases. This application of the composite pulse is called a "π / 2 pulse". More generally, the superposition of two equally weighted qubit states |0⟩ and |1⟩ added together is represented by a point on the equator of the Bloch sphere. For example, the superposition states |0⟩ ± |1⟩ correspond to points on the equator with azimuthal angles φ equal to zero and π respectively. The superposition state corresponding to a point on the equator with azimuthal angle φ is represented as |0⟩ + e iφ |1⟩ (e.g., |0⟩ ± i|1⟩ for φ = ±π / 2). The transformation between two points on the equator (i.e., rotation about the Z-axis on the Bloch sphere) can be achieved by shifting the phase of the composite pulse.

[0040] In an ion trap quantum computer, the motional mode can act as a data bus to mediate entanglement between two qubits, and this entanglement is used to perform an XX gate operation. That is, each of the two qubits is entangled with the motional mode, and then this entanglement is transferred to entanglement between the two qubits by using motional sideband excitation, as described below. Figure 6A and Figure 6B Schematically depicts the view of the motional sideband spectrum of the ions in chain 102 according to one embodiment at a motional mode |n p with frequency ω ph 〉 p . As Figure 6B shown, when the detuning frequency of the composite pulse is zero (i.e., the frequency difference between the first and second laser beams is tuned to the carrier frequency, δ = ω1 - ω2 - ω 01 = 0), simple Rabi oscillations (carrier transitions) occur between the qubit states |0⟩ and |1⟩. When the detuning frequency of the composite pulse is positive (i.e., the frequency difference between the first and second laser beams is tuned to be higher than the carrier frequency, δ = ω1 - ω2 - ω 01= μ > 0, called the blue sideband), the combined qubit motional state |0>|n ph > p and |1>|n ph +1> p undergoes Rabi oscillations (i.e., when the qubit state |0> flips to |1>, a transition occurs from the p-th motional mode with n phonon excitations represented by |n ph > p to the p-th motional mode with (n ph +1) phonon excitations represented by |n p +1> ph ). When the detuning frequency of the composite pulse is negative (i.e., the frequency difference between the first and second laser beams is tuned to be lower than the frequency ω ph > p of the motional mode |n p , δ = ω1 - ω2 - ω 01 = -μ < 0, called the red sideband), the combined qubit motional state |0>|n ph > p and |1>|n ph -1〉 p undergoes Rabi oscillations (i.e., when the qubit state |0> flips to |1>, a transition occurs from the motional mode |n ph > p to the motional mode |n ph -1> p with one less phonon excitation). A π / 2 pulse applied to the blue sideband of the qubit converts the combined qubit motional state |0>|n ph 〉 p to a superposition of |0〉|n ph > p and |1>|n ph +1> p . A π / 2 pulse applied to the red sideband of the qubit converts the combined qubit motion |0>|n ph > p to a superposition of |0>|n ph > p and |1>|n ph -1〉 p . When the two-photon Rabi frequency Ω(t) is less than the detuning frequency δ = ω1 - ω2 - ω 01 = ±μ, the blue sideband transition or the red sideband transition can be selectively driven. Therefore, by applying the correct type of pulse (such as a π / 2 pulse), the qubit can be entangled with the desired motional mode and subsequently with another qubit, resulting in entanglement between the two qubits. In an ion trap quantum computer, entanglement between qubits is required to perform XX gate operations.

[0041] By controlling and / or guiding the conversion of the combined qubit motion state as described above, an XX gate operation can be performed on two qubits (the i-th and j-th qubits). Generally, the XX gate operation (with maximum entanglement) converts the two-qubit state |0> as follows: i |0> j 、|0> i |1> j 、|1> i |0> j 、and |1> i |1> j :

[0042]

[0043] For example, when both qubits (the i-th and j-th qubits) are initially in the hyperfine ground state |0> (denoted as |0> i |0> j ), and a π / 2 pulse on the blue sideband is subsequently applied to the i-th qubit, the combined state of the i-th qubit and the motion mode |0> i |n ph > p is converted to the superposition of |0> i |n ph > p and |1> i |n ph +1> p . Thus, the combined state of these two qubits and the motion mode is converted to the superposition of |0> i |0> j |n ph > p and |1> i |0> j |n ph +1> p . When a π / 2 pulse on the red sideband is applied to the j-th qubit, the combined state of the j-th qubit and the motion mode |0> j |n ph > p is converted to the superposition of |0> j |n ph > p and |1> j |n ph -1> p , and the combined state |0> j |n ph +1〉 p is converted to |0> j |n ph +1>p and |1> j |n ph > p of superposition.

[0044] Therefore, applying a π / 2 pulse on the blue sideband to the i-th qubit and a π / 2 pulse on the red sideband to the j-th qubit can convert these two qubits and the motion mode |0> i |0〉 j |n ph 〉 p The combined state is converted to |0〉 i |0〉 j |n ph 〉 p and|1〉 i |1> j |n ph 〉 p The two qubits are now in an entangled state. It should be clear to those skilled in the art that the motion mode (whose number of phonon excitations is different from the initial number of phonon excitations n) is ph ) two entangled quantum bit states (i.e., |1〉 i |0> j |n ph +1> p and|0> i |1> j |n ph -1> p ) can be removed by a sufficiently complex pulse sequence, so that at the end of the XX gate operation the initial number n of phonon excitations in the p-th motion mode ph When it remains unchanged, it can be considered that after the XX gate operation, the combined state of the two qubits and the motion mode is disentangled. Therefore, the following will generally describe the qubit state before and after the XX gate operation, excluding the motion mode.

[0045] More generally, the combined state of the i-th and j-th qubits switched by applying a composite pulse with amplitude Ω(t) and detuned frequency μ(t) on the sidebands with duration τ (called the “gate duration”) can be expressed by the entanglement interaction χ i,j (τ) is described as follows:

[0046] |0> i |0> j →cos(2χ i,j (τ))|0> i |0> j -isin(2χ i,j (τ))|1> i |1> j

[0047] |0> i |1> j →cos(2χ i,j (τ))|0> i |1> j -isin(2χ i,j (τ))|1> i |0> j

[0048] |1> i |0> j →-isin(2χ i,j (τ))|0〉 i |1> j +cos(2χ i,j (τ))|1> i |0> j

[0049] |1> i |1〉 j →-isin(2χ i,j (τ))|0〉 i |0〉 j +cos(2χ i,j (τ))|1> i |1〉 j

[0050] where

[0051] η i,p is the Lamb-Dicke parameter that quantifies the coupling strength between the i-th ion and the p-th motional mode with frequency ω p and ψ(t) is the cumulative phase of the pulse, ψ0 is the initial phase, which can be assumed to be zero (0) in the following without loss of generality for simplicity, and P is the number of motional modes (equal to the number N of ions in the chain 102). Construction of the pulse for entanglement gate operation

[0052] The entanglement interaction between the above two qubits can be used to perform the XX gate operation. The XX gate operation (XX gate) together with single qubit operations (R gates) forms a universal set of gates {R, XX}, which can be used to construct a quantum computer configured to perform a desired computational process. To perform the XX gate operation between the i-th and j-th qubits, a pulse is constructed that satisfies the condition χ

[0053] (τ) = θ i,j (0 < θ i,j (0 < θ i,j≤ π / 8) (i.e., the entanglement interaction χ i,j (τ) has the desired value θ i,j , called the condition for non - zero entanglement interaction) pulse, and apply this pulse to the i - th and j - th qubits. When θ i,j = π / 8, the transformation of the combined state of the above - mentioned i - th and j - th qubits corresponds to an XX - gate operation with maximum entanglement. The amplitude Ω(t) and detuning frequency μ(t) of this pulse are control parameters, and these control parameters can be adjusted to ensure non - zero tunable entanglement of the i - th and j - th qubits, so as to perform the desired XX - gate operation on the i - th and j - th qubits. In the examples described below, the same pulse is applied to both the i - th and j - th qubits. However, in some embodiments, different pulses are applied to the i - th and j - th qubits.

[0054] The control parameters of the pulse, the amplitude and detuning frequency functions, must also satisfy the condition that the trapped ion displaced from the initial position returns to the initial position when the motion mode is excited by the transported pulse. The l - th qubit (l = i, j) in the superposition state |0〉±|1〉 is displaced during the gate duration τ due to the excitation of the p - th motion mode and follows the trajectory ±α l,p (t′) in the phase space (position and momentum) of the p - th motion mode. The trajectory is determined by the amplitude Ω(t) and the cumulative phase of the pulse, where g(t) is a pulse function defined as g(t)=Ω(t)sin(ψ(t)). Therefore, for the chain 102 of N trapped ions, in addition to the condition for non - zero entanglement interaction χ i,j (τ)=θ i,j (0 < θ i,j ≤ π / 8), the condition α l,p (τ)=0 (l = i, j) (i.e., the trajectory α l,p (τ) must be closed, called the condition for the trapped ion to return to its original position and momentum values (or the closure of the phase - space trajectory)) must also be imposed on all P motion modes (p = 1, 2,…, P).

[0055] The control parameters of the pulse, the amplitude Ω(t) and the detuning frequency μ(t), are also adjusted so that the resulting pulse is power - optimal, where the required laser power is minimized (called the condition for minimizing power). Since the required laser power is inversely proportional to the gate duration τ, if the gate duration τ is fixed, the power - optimal pulse achieves the XX - gate operation with the minimum power requirement, or if the laser power budget is fixed, the XX - gate operation is achieved with the shortest gate duration τ.

[0056] In some embodiments, the amplitude Ω(t) and the detuning frequency μ(t) are chosen to be symmetric or antisymmetric in time about the midpoint t = τ / 2 of the gate duration, i.e., In the examples described below, for simplicity, the amplitude Ω(t) and the detuning frequency μ(t) are chosen to be symmetric (Ω (+) (t) and μ (+) (t)), and may be referred to as Ω(t) and μ(t) without the subscript (+). At the symmetric detuning frequency μ(t), the cumulative phase ψ(t) is antisymmetric, i.e., The condition for the trapped ion to return to its original position and momentum values can be rewritten in terms of the antisymmetric component g (-) (t) of the pulse function g(t) (hereinafter also referred to as the "negative parity pulse function" or simply the "pulse function") as

[0057] where M pn is defined as

[0058]

[0059] In this paper, in terms of a complete basis, such as the Fourier-sine basis, the condition on the gate duration τ using the basis functions can equivalently be written in matrix form as where M is the P×N pn coefficient matrix of M A , is the N n Fourier coefficient vector of A A . In the calculation of the pulse function g (-) (t), the number of basis functions N A is chosen to be greater than the number of motion modes P and large enough to achieve convergence. Thus, there are N0(=N A -P) non-trivial (i.e., at least one Fourier coefficient A n is non-zero) Fourier coefficient vectors that satisfy the condition for the trapped ion to return to its original position and momentum values.

[0060] The condition for non-zero entanglement interaction can be rewritten in terms of the pulse function g (-) (t) as

[0061]

[0062]

[0063] where D nm is defined as

[0064]

[0065] Or equivalently defined in matrix form as where D is the N nm ×N A coefficient matrix of A , and is the transpose vector of

[0066] The power minimization condition corresponds to the minimization of the power function

[0067]

[0068] This is the absolute square value averaged by the pulse function g (-) (t) over the gate duration τ. Therefore, by calculating the linear combination of the Fourier coefficient vectors the power-optimal pulse can be constructed, where the coefficient Λ is determined such that the conditions for non-zero entanglement interaction and power minimization are satisfied. α

[0069] Therefore, the amplitude Ω(t) and detuning frequency μ(t) of the pulse for performing the XX gate can be based on the Fourier coefficients A n (n = 1, 2,..., N A ) (i.e., the frequency components of the pulse function g (-) (t)) or the equivalent Fourier coefficient vector of the pulse function g (-) (t) to calculate. It should be noted that the conditions for the trapped ions to return to their original position and momentum values and the conditions for non-zero entanglement interaction are in the form of linear algebra expressed by the Fourier coefficient vector . Therefore, the Fourier coefficients A n that satisfy these conditions and the power minimization condition can be calculated by known linear algebra calculation methods without approximation or iteration.

[0070] In the embodiments described herein, the amplitude Ω(t) and detuning frequency μ(t) are chosen to be symmetric about the midpoint t = τ / 2 of the pulse, so the pulse function g(t) only includes the antisymmetric component, g (-) (t). Then the pulse function g (-) (t) is expanded in the Fourier basis (using the antisymmetric basis function sin(2πnt / τ) due to antisymmetry). Since the Fourier basis is a complete set, the pulse function g n with the Fourier coefficients A (-) determined to satisfy the power minimization condition(t) is guaranteed to be the most power-optimal of all possible pulses that satisfy the conditions of the trapped ion returning to its original position and momentum and non-zero entanglement interaction. In addition, the pulse function g (-) The expansion of (t) corresponds to the construction of the pulse in the frequency domain (frequency is 2πn / τ), so the pulse function g (-) The pulses constructed by (t) can be directly realized by a multi-tone laser (i.e., a laser beam with multiple tones, each with a different amplitude and a corresponding amplitude). That is, each frequency is 2πn / τ and the amplitude is A n (n=1,2,…,N A ) and the laser beam phase is fixed N A The XX gate operation can be directly performed with a single tone laser beam. The pulse function can be expanded over the gate duration using any function that forms a complete set or an incomplete set. However, when the pulse function is expanded with an incomplete set, the pulse function g calculated by the above method cannot be guaranteed to be (-) (t) is power-optimal.

[0071] It should be noted that the above-mentioned specific exemplary embodiments are only some possible examples of the method of constructing a pulse function according to the present disclosure, and do not limit the possible configurations, specifications, etc. of the method of constructing a pulse function. For example, the symmetry of the amplitude Ω(t) and the detuning frequency μ(t) may be selected to be antisymmetric (having negative parity) or to have mixed symmetry (having mixed parity) based on convenience related to the configuration, specifications, etc. of the system 100. However, by appropriate selection of the symmetry of the amplitude Ω(t) and the detuning frequency μ(t) and / or echo techniques, imposing symmetry on the amplitude Ω(t) and the detuning frequency μ(t) may eliminate errors in external parameters, such as the Lamb-Dicke parameter η i,p Or the impulse function g (-) The relative offset of (t).

[0072] Stabilization of mode frequency fluctuations and calibration of XX gates

[0073] When constructing the power-optimal pulse, due to the application of the coefficient vector The linear algebraic form of the conditions on can be used to calculate the Fourier coefficient vector without significantly increasing Additional conditions can be added without increasing the complexity. For example, while keeping all the conditions in linear algebraic form, additional conditions for external errors such as the frequency ω of the motion mode can be imposed. p and fluctuations in laser beam intensity, stable pulse conditions. In an ion trap quantum computer or system 100, the frequency of the motion pattern may fluctuate due to stray electric fields, accumulated charge in the ion trap 200 caused by photoionization or temperature fluctuations. Typically, the frequency ω of the motion pattern is ω over a time span of several minutes. pDrift, the offset is Δω p / (2π) ≈ 1 kHz. When the frequency of the motion mode drifts to ω p +Δω p it no longer satisfies the conditions of non-zero entanglement interaction based on the motion mode frequency ω p , the trapped ions return to their original position and momentum values, and minimizing power, resulting in a reduced fidelity of the XX gate operation. It is known that the distortion 1 - F of the XX gate operation between the i-th and j-th qubits at zero temperature of the motion mode phonons is given by . This indicates that by requiring α l,p (l = i, j) to be stable up to the k-th order with respect to the change Δω p in ω p , the XX gate operation can be stabilized against the drift of the frequency ω p of the motion mode,

[0074]

[0075] (l = 1, 2, …, N, p = 1, 2, …, P, k = 1, 2, …, K)

[0076] (referred to as k-th order stability), where K is the maximum desired degree of stability. The pulse calculated by requiring this stability condition can perform an XX gate operation that is resilient to noise (i.e., the drift of the frequency ω p of the motion mode).

[0077] Since the entanglement interaction χ i,j (τ) is related to the frequency ω p of the motion mode, the fluctuations of the frequency ω p of the motion mode may also affect the value of the entanglement interaction χ i,j (τ). That is, the resulting entanglement interaction χ i,j (τ) can have a value different from the expected value θ i,j set under the condition of non-zero entanglement interaction. Therefore, in some embodiments, the condition for stabilizing the pulse against the fluctuations of the frequency ω p of the motion mode can also require that the ω i,j component of the entanglement interaction χ p with respect to ω p is stable up to the k-th order with respect to the change Δω p in ω

[0078]

[0079] (k-th order stability).

[0080] The intensity of the laser beam and the Lamb-Dicke parameter η i,p fluctuations can also affect the entanglement interaction χ i,j (τ) values, since the entanglement interaction χ i,j (τ) is related to the amplitude A of each tone n (n = 1, 2, …, N A ). That is, the resulting entanglement interaction χ i,j (τ) can have a value different from the expected value θ set under the condition of non-zero entanglement interaction i,j . Therefore, in some embodiments, known broadband pulse sequences commonly applicable to single-qubit gate operations, such as Solovay-Kitaev (SK) sequences and Suzuki-Trotter sequences, can be applied to the trapped ions in chain 102 to mitigate errors in the entanglement interaction χ i,j (τ) with respect to, for example, the Lamb-Dicke parameter η i,p offsets. The same technique can be used to stabilize the entanglement interaction χ i,j (τ) against any error source that disturbs the value of the entanglement interaction χ i,j (τ).

[0081] As an alternative or supplement to stabilizing the entanglement interaction χ i,j (τ), the resulting entanglement interaction χ i,j (τ) can be calibrated to the expected value θ i,j by modifying the pulse amplitude Ω(t).

[0082] Demodulation of the pulse

[0083] To apply power-optimal and error-resilient pulses on the i-th and j-th qubits, it is necessary to demodulate the amplitude Ω(t) and detuning frequency μ(t) of the power-optimal pulse from the calculated pulse function g (-) (t) = Ω(t)sin(ψ(t)) (i.e., extract the amplitude Ω(t) and detuning frequency μ(t) and convert the pulse function g (-) (t) into a pulse with a series of time-dependent pulse segments for a single laser beam), where is the phase accumulated due to the detuning frequency μ(t). If the demodulation process is performed with a fixed detuning frequency (i.e., μ(t) = μ0), the resulting pulse is an amplitude-modulated (AM) pulse, where the amplitude Ω(t) is modulated. If the demodulation process is performed with a fixed amplitude (i.e., Ω(t) = Ω0), the resulting pulse is a phase-modulated (PM) pulse, where the phase ψ(t) is modulated. If the phase ψ(t) is achieved by modulating the detuning frequency μ(t), the resulting pulse is a frequency-modulated (FM) pulse. The demodulation process can be performed with any combination of modulation of the amplitude Ω(t), the phase ψ(t) (and thus the detuning frequency μ(t)), and the frequency by conventional demodulation methods known in the field of signal processing to construct a power-optimal pulse.

[0084] The first step of an exemplary demodulation process is to find the zeros of the pulse function g (-) (t) = Ω(t)sin(ψ(t)) at t = ζ j (j = 0, 1, …, N z -1) (i.e., g(ζ j ) = 0). Here, N z is the total number of zeros of the pulse function g (-) (t). The amplitude Ω(t) can be chosen such that the amplitude Ω(t) has no zeros. Thus, when sin(ψ(t)) is zero, the pulse function g (-) (t) is zero (i.e., sin(ψ(ζ j )) = 0). Due to the properties of the sine function, sin(ψ(ζ j )) = 0 when ψ(ζ z ) = jπ (j = 0, 1, …, N j -1), including the zeros at the start and end of the gate duration τ of the pulse (i.e., t = ζ0 = 0 and ).

[0085] The second step of the demodulation process is to calculate the detuning frequency μ(t) based on the zeros of the pulse function g (-) (t). In some embodiments, the detuning frequency μ(t) is approximated as a constant value between adjacent zeros of the pulse function g (-) (t) (i.e., for ζ j-1 < t < ζ j , j = 1, 2, …, N z -1), μ(t) ≈ μ j ). Since the phase ψ(t) is accumulated due to the detuning frequency μ(t), as in , thus the phase difference between t = ζ j and t = ζ j-1 is Thus t = ζj-1 and t = ζ j The detuning frequency μ j between them is determined to be μ j = π / (ζ j - ζ j-1 ). The third step of the demodulation process is to calculate the amplitude Ω(t). The pulse function g (-) (t) = Ω(t)sin(ψ(t)) at t = ζ j The time derivative at this point is

[0086] g′(ζ j ) = Ω′(ζ j )sin(ψ(ζ j )) + Ω(ζ j )cos(ψ(ζ j ))ψ′(ζ j ) = (-1) j Ω(ζ j )μ(ζ j ),

[0087] where ψ(ζ j ) = jπ and Therefore, using the calculated time derivative of the pulse function (i.e., ), the amplitude Ω(t) at t = ζ j is calculated as Ω(ζ j ) = (-1) j g′(ζ j ) / μ(ζ j ).

[0088] In some embodiments, a set of calculated detuning frequencies μ j (j = 1, 2,.., N z - 1) are interpolated using a spline (e.g., a function piecewise defined by one or more polynomials or other algebraic expressions), and the interpolated value of the detuning frequency μ(t) is used for μ(ζ j ) to calculate the amplitude Ω(ζ j ). In some embodiments, μ(ζ j ) is (i) μ j , (ii) μ j+1 , or (iii) (μ j + μ j+1 ) / 2 is used as μ(ζ j ) to calculate the amplitude Ω(ζ j ).

[0089] In some embodiments, a set of calculated amplitudes Ω(ζ j ) are also interpolated using a spline to calculate the time-dependent amplitude Ω(t).

[0090] If the phase modulation (PM) pulse is demodulated, a set of calculated phases ψ(ζ j ) can be used with spline interpolation to calculate the time-dependent phase ψ(t).

[0091] Method for generating power-optimal pulses to perform XX gate operations

[0092] Figure 7 Described is a flowchart according to an embodiment, which shows a method 700 for generating power-optimal pulses for performing an XX gate operation on two ions (the i-th and j-th ions) of a chain 102 of N trapped ions. In this example, the chain 102 of N trapped ions is a quantum register. The software program and controller within the classical computer described above are used to determine and control the generation of the power-optimal pulses and their delivery to the two ions within the quantum register during the execution of method 700.

[0093] In block 702, the frequencies ω of the motional modes (p = 1, 2, …, P) can be directly measured from the ion trap quantum computer system 100 p such that the motional mode structure can be calculated based on the measured frequencies ω p The Lamb-Dicke parameter η is determined by the motional mode structure, the photon momentum of the laser beam driving the motional sideband transitions, the ion mass, and the measured frequency ω p For example, a value of the gate duration τ is selected based on the number N of trapped ions in the chain 102, the available laser power, etc., to be used as an input parameter for generating the power-optimal pulses. i,p

[0094] In blocks 704 - 708, based on the selected value of the gate duration and the measured frequency ω p the amplitude Ω(t) and the detuning frequency μ(t) of the power-optimal pulses are calculated such that the conditions of the trapped ions returning to their original position and momentum values, non-zero entanglement interaction, and minimum power are satisfied.

[0095] In block 704, the Fourier coefficients A of the pulse function n (n = 1, 2, …, N A ) are calculated such that the condition of the trapped ions returning to their original position and momentum values is satisfied. There is a non-trivial set N0 (= N - P) of Fourier coefficients that satisfy the condition of the trapped ions returning to their original position and momentum values A .

[0096] In block 706, a linear combination of the Fourier coefficients is calculated ​To satisfy non-zero entanglement interaction

[0097] In block 708, the demodulated calculated pulse function g (-) (t) is used to calculate the amplitude Ω(t) and the detuning frequency μ(t) through g (-) (t)=Ω(t)sin(ψ(t)), where the cumulative phase is related to the detuning frequency μ(t). To demodulate the calculated pulse function g (-) (t), first, find the zeros of the calculated pulse function g (-) (t) at t = ζ j (j = 0, 1, …, N z -1) (i.e., g(ζ j ) = 0), where N z is the number of zeros of the calculated pulse function g (-) (t). At the start and end of the gate duration τ of the pulse (t = ζ0 = 0 and the calculated pulse function g (-) (t) is zero. Second, calculate the detuning frequency μ(t) based on the zeros of the pulse function g (-) (t). In some embodiments, the detuning frequency μ(t) is approximated as a constant value between adjacent zeros of the pulse function g (-) (t) (i.e., for ζ j-1 < t < ζ j , j = 1, 2, …, N z -1, μ(t) ≈ μ j ), and each value μ j is calculated as μ j = π / (ζ j - ζ j-1 ). Third, based on the calculated detuning frequency μ(t) and the time derivative g′(t) of the calculated pulse function g (-) (t), calculate the amplitude Ω(t) as Ω(ζ j ) = (-1) j g′(ζ j ) / μ(ζ j ). In some embodiments, a set of calculated detuning frequencies μ j (j = 1, 2,.., N z -1) are interpolated using a spline (e.g., a function piecewise defined by one or more polynomials or other algebraic expressions), and the interpolated value of the detuning frequency μ(t) is used as μ(ζ j ) for calculating the amplitude Ω(ζ j ). In some embodiments, μ(ζ j ) is (i) μj , (ii) μ j+1 , or (iii) (μ j + μ j+1 ) / 2 is used as μ(ζ j ) to calculate the amplitude Ω(ζ j ). In some embodiments, a set of calculated amplitudes Ω(ζ j ) is also interpolated with a spline to calculate the time-dependent amplitude Ω(t).

[0098] If a demodulation process is performed on the phase modulation (PM) pulse, a set of calculated phases ψ(ζ j ) can be interpolated with a spline to calculate the time-dependent phase ψ(t).

[0099] In block 710, a pulse having the determined amplitude Ω(t) and detuning frequency μ(t) is generated by adjusting the amplitude and frequency of the laser. The generated pulse is applied to two qubits (the i-th and j-th qubits) of the chain 102 of N trapped ions to perform an XX gate operation on the two qubits. The pulses generated in blocks 702 - 710 are optimal because the required laser power is minimized.

[0100] Applying the generated pulse to the two qubits during block 710 implements the XX gate operation in a series of universal gate {R, XX} operations into which the selected quantum algorithm is decomposed. All XX gate operations (XX gates) in the series of universal gate {R, XX} operations, along with single-qubit operations (R gates), are implemented by the above method 700 to run the selected quantum algorithm. At the end of running the selected quantum algorithm, the population of the qubit states (trapped ions) in the quantum register (the chain 102 of trapped ions) is measured (read out) by the imaging objective lens 104 and mapped onto the PMT 106, so that the quantum computing result within the selected quantum algorithm can be determined and provided as an input to a classical computer (e.g., a digital computer). Then, the classical computer can use the result of the quantum computation to perform the required activities or obtain the solution to problems that typically cannot be determined by a classical computer alone or within a reasonable time. Problems that are known to be difficult or impossible to solve by today's conventional computers (i.e., classical computers) and can be solved by using the results obtained from the performed quantum computation can include, but are not limited to, simulating the properties of complex molecules and materials, factoring large integers.

[0101] Embodiments

[0102] In the following text, embodiments of power-optimal pulses generated according to the above method 700 are shown. In Embodiment 1, the amplitude Ω(t) and the detuning frequency μ(t) are the control parameters that are adjusted. In Embodiment 2, only the amplitude Ω(t) is the control parameter that is adjusted.

[0103] Example 1

[0104] Figure 8A An embodiment of the power-optimal pulse function g (-) (t) is described, which is used to cause an XX gate operation (with maximum entanglement, θ i,j = π / 8) on the first and third ions of a chain 102 of five trapped ions. In this embodiment, the amplitude Ω(t) and the detuning frequency μ(t) of the pulse are the control parameters, which are determined to satisfy the conditions that the trapped ions return to their original position and momentum values, a non-zero entanglement interaction (θ i,j = π / 8), and minimum power. The gate duration τ is 80 μs. The frequency ω p of the p-th motional mode of the chain 102 and the Lamb-Dicke parameter η i,p of the i-th ion and the p-th motional mode are listed in Table I and Table II, respectively. When determining the power-optimal pulse function g (-) (t), N A = 1000 is used.

[0105] Table I

[0106]

[0107] Table II

[0108]

[0109] Figure 8A The power-optimal pulse function g (-) (t) = Ω(t)sin(ψ(t)) shown in can be decomposed into a slow-varying envelope function 802 corresponding to the amplitude Ω(t) of the pulse (and a fast-varying oscillation 804 corresponding to sin(ψ(t)), where the cumulative phase is related to the detuning frequency μ(t). Figure 8B The detuning frequency μ(t) 806 that causes the fast-varying oscillation 804 is shown in. The power-optimal pulse for performing an XX gate operation between the first ion and the third ion of a chain 102 of five trapped ions can be constructed based on the amplitude Ω(t) 802 and the detuning frequency μ(t) 806 of the pulse. As described herein, the method of constructing a pulse to be applied to a pair of ions to perform an XX gate operation on the pair of ions includes: (i) determining the pulse function g (-) , (ii) determining the detuning frequency that causes the determined pulse function g (-)the amplitude Ω(t) and detuning frequency μ(t), and (iii) constructing a pulse having the determined amplitude Ω(t) and detuning frequency μ(t).

[0110] Example 2

[0111] Figure 9 describes an embodiment of the power-optimal pulse function g (-) (t) that is used to effect an XX gate operation (with maximum entanglement, θ i,j = π / 8) on the first and third ions of a chain 102 of five trapped ions. In this embodiment, the pulse is divided into 11 step-pulse segments, and the detuning frequency μ(t) is fixed at μ0 / (2π) = 2.42 MHz, and only the amplitude of the pulse (i.e., the series of amplitudes of the step-pulse segments) is the control parameter, which is determined such that the conditions of the trapped ions returning to their original position and momentum values, non-zero entanglement interaction (θ i,j = π / 8), and minimum power are satisfied. The gate duration τ selected in this embodiment is 76.45 μs. The frequencies ω p of the p-th motional mode of the chain 102 and the Lamb-Dicke parameters η i,p of the i-th ion and the p-th motional mode are listed in Tables I and II, respectively.

[0112] It has been found that the laser power required to construct the pulse (with the detuning frequency μ(t) fixed) in Example 2 is about 30% higher than the laser power required to construct the pulse in combination with Example 1 Figure 8A and Figure 8B (where the detuning frequency μ(t) is the control parameter to be adjusted). This is expected because the pulse with a fixed detuning frequency lacks the additional degree of freedom associated with the detuning frequency μ(t), which can be modulated to minimize the required laser power. In addition, the determined pulse amplitude Ω(t) has a sharp change in the required laser power (related to the absolute value of the pulse amplitude) at the transitions between adjacent step-pulse segments, which can lead to ringing and a phenomenon called the Gibbs phenomenon, and may result in a reduction in the fidelity of the XX gate operation in practice. The pulse constructed in Example 1 eliminates this sharp change and thus results in an increase in the fidelity of the XX gate operation.

[0113] Figure 10 describes an embodiment of the distortion of the XX gate operation performed by applying the pulse constructed in Example 1. In this embodiment, the frequencies ω p of the p-th motional mode (p = 1, 2,..., 5) are equally offset by Δω1. When the conditions for the trapped ions used to determine the pulse to return to their original position and momentum values and non-zero entanglement interaction are related to the frequency ω p of the p-th motional mode and thus to the frequency ωp When the offset is sensitive, the distortion of the XX gate operation without stabilization 1002 increases significantly with the increase in the change Δω1 of the frequency ω of the motion pattern. Therefore, the pulses determined using an inaccurate frequency ω p (i.e., a frequency different from the actual frequency) result in inaccurate XX gate operations (i.e., different from the desired XX gate operations). For small changes Δω1 (up to 3 kHz) in the frequency of the motion pattern, the distortion of the XX gate operation with first-order stabilization 1004 applied will remain low. That is, in the case of having first-order stabilization, the XX gate operation can be robust to frequency fluctuations of the motion pattern. However, in the case of having stabilization, the required laser power increases, for example, up to 40% in the embodiments described herein. Therefore, there is a trade-off between the degree of stabilization and the optimization of the required laser power. p (i.e., different from the desired entanglement interaction θ

[0114] Figure 11 Describes an embodiment of the resulting entanglement interaction χ i,j (τ) obtained by applying the pulses constructed as described above in Example 1. In an embodiment where the frequencies ω of the p-th motion pattern (p = 1, 2,..., 5) are equally offset by Δω1, the resulting entanglement interaction χ p (τ) changes as shown in 1102 in i,j . In an embodiment where the frequencies ω of the p-th motion pattern (p = 1, 2,..., 5) are selected to be independently offset, the resulting entanglement interaction χ Figure 11 (τ) changes with the change in the offset Δω1 of the frequency ω1 of the motion pattern p = 1, as shown in 1104 in p . The inaccurate value of the resulting entanglement interaction (i.e., the resulting entanglement interaction is different from the expected value θ i,j ) results in inaccurate XX gate operations (i.e., different from the desired XX gate operations), increasing the distortion of the XX gate operation. By calibrating the value of the entanglement interaction, the distortion of the XX gate operation can be reduced. In the case where there are fluctuations in the frequency ω of the motion pattern, by changing the amplitude of the pulse Ω(t), the resulting entanglement interaction χ Figure 11 (τ) can be calibrated to the desired value θ i,j . p There is a fluctuation, and by changing the amplitude of the pulse Ω(t), the resulting entanglement interaction χ i,j (τ) can be calibrated to the required value θ i,j .

[0115] As described above, when generating pulses to perform an entanglement gate operation between two qubits, the control parameters (amplitude and detuning frequency of the pulses) are determined such that the required laser power is minimized, and if the gate duration is fixed, the resulting pulses can be applied to the two qubits with the minimum laser power requirement, or if the laser power budget is fixed, the resulting pulses can be applied to the two qubits with the shortest gate duration. Additionally, the pulses can be configured such that the entanglement gate operation can be stabilized against external errors, resulting in an improved fidelity of the entanglement gate operation.

[0116] Additionally, determining the control parameters includes solving a set of linear equations. Thus, the determination of the control parameters and subsequent construction of the power-optimal pulses can be performed in an efficient manner to perform the desired XX gate operation. The XX gate operation is performed on other ion pairs using different pulses to run the required quantum algorithm on the quantum register. At the end of running the required quantum algorithm, the population of the qubit states (trapped ions) within the quantum register is measured (read out), so that the quantum computing result using the required quantum algorithm can be determined and provided to the classical computer for obtaining solutions to problems that the classical computer may not be able to solve.

[0117] Furthermore, the frequency components of the power-optimal pulses are also calculated. Thus, such power-optimal pulses can be directly implemented by a multi-tone laser having multiple frequencies.

[0118] While the foregoing relates to specific embodiments, other and further embodiments can be designed without departing from its basic scope, and its scope is determined by the appended claims.

Claims

1. A method for performing an entanglement operation between two trapped ions in a quantum computer, the method comprising: selecting a gate duration value of a pulse of a laser beam to be applied to a first ion and a second ion in a chain of trapped ions, wherein, each trapped ion has two frequency-separated states defining a qubit, and the motional modes of the chain of the trapped ions each have different frequencies; calculating Fourier coefficients of a pulse function of the pulse based on the selected gate duration value and the frequencies of the motional modes of the chain of the trapped ions; demodulating the pulse function of the pulse having the calculated Fourier coefficients to calculate an amplitude and a detuning frequency value of the pulse; and performing the entanglement operation between the first ion and the second ion by applying the pulse having the calculated amplitude and the calculated detuning frequency value to the first ion and the second ion for the gate duration value.

2. The method according to claim 1, the method further comprising: selecting the calculated Fourier coefficients of the pulse function based on the condition that the trapped ions return to their original position and momentum values before demodulating the pulse function of the pulse.

3. The method according to claim 2, the method further comprising: selecting the calculated Fourier coefficients of the pulse function before demodulating the pulse function of the pulse such that the condition that the trapped ions return to their original position and momentum values is stable up to the first order with respect to a drift of the frequency of the motional mode of the chain.

4. The method according to claim 2, the method further comprising: selecting the calculated Fourier coefficients of the pulse function based on the condition of a non-zero entanglement interaction between the first ion and the second ion before demodulating the pulse function of the pulse.

5. The method according to claim 4, wherein The non-zero entanglement interaction value between the first ion and the second ion is between zero and π / 8.

6. The method according to claim 4, the method further comprising: selecting the calculated Fourier coefficients of the pulse function based on the condition of stabilizing the resulting entanglement interaction between the first ion and the second ion with respect to a drift of the frequency of the motional mode of the chain before demodulating the pulse function of the pulse.

7. The method according to claim 5, the method further comprising: modifying the amplitude before applying the pulse to calibrate the resulting entanglement interaction to the non-zero entanglement interaction value in the case of fluctuations in the intensity of the laser beam.

8. The method according to claim 4, the method further comprising applying a broadband pulse sequence to all trapped ions in the chain of trapped ions to stabilize the resulting entanglement interaction in the case of fluctuations in the coupling strength between the first ion and the second ion and the motional mode.

9. The method according to claim 4, the method further comprising: selecting the calculated Fourier coefficients of the pulse function based on minimizing the power of the laser beam provided to the first ion and the second ion during the pulse before demodulating the pulse function of the pulse.

10. The method according to claim 1, the method further comprising: Before demodulating the pulse function of the pulse, selecting the calculated Fourier coefficients of the pulse function based on the condition of non - zero entangled interaction between the first ion and the second ion.

11. The method according to claim 10, wherein, The non - zero entangled value interaction between the first ion and the second ion is between zero and π / 8.

12. The method according to claim 10, the method further comprising: Before demodulating the pulse function of the pulse, also selecting the calculated Fourier coefficients of the pulse function based on stabilizing the condition of the resulting entangled interaction between the first ion and the second ion against the drift of the frequency of the motion mode for the chain.

13. The method according to claim 11, wherein, The method further comprising: Before applying the pulse, modifying the amplitude to calibrate the resulting entangled interaction value to the non - zero entangled interaction value in the case of fluctuations in the intensity of the laser beam.

14. The method according to claim 10, the method further comprising applying a broadband pulse sequence to all trapped ions in the chain of trapped ions to stabilize the resulting entangled interaction value in the case of fluctuations in the coupling strength between the first ion and the second ion and the motion mode.

15. The method according to claim 10, the method further comprising: Before demodulating the pulse function of the pulse, selecting the calculated Fourier coefficients of the pulse function based on minimizing the power of the laser beam provided to the first ion and the second ion during the pulse.

16. The method according to claim 1, wherein, The pulse is symmetric about the mid - point of the gate duration value.

17. The method according to claim 1, wherein, The pulse is antisymmetric about the mid - point of the gate duration value.

18. The method according to claim 1, wherein Demodulating the pulse function of the pulse includes: Converting the pulse function with the calculated Fourier coefficients into a series of time - dependent pulse segments, each time - dependent pulse segment having a different amplitude value, a different detuning value, and the phase of the laser beam.

19. The method according to claim 18, wherein Interpolating the time - dependent pulse segments with a spline.

20. The method according to claim 1, the method further comprising: Executing, by a processor in a digital computer, a software program stored in the non - volatile memory of the digital computer, wherein the executed software program requires performing at least one calculation, and performing the at least one calculation includes: Selecting, by the processor in the digital computer, a quantum algorithm to be implemented on the chain of trapped ions; Compiling the selected quantum algorithm into a series of universal logic gates; Converting the series of universal logic gates into pulses to be applied to an ion pair in the chain of trapped ions; Measuring the population of the qubit states of the ions in the chain of trapped ions; and Processing, by the processor of the digital computer, quantum information corresponding to the qubit states of the ions in the chain of trapped ions based on the measured population of the qubit states; and Generating a calculation result based on the processed quantum information.

21. A quantum computing system, comprising: A chain of trapped ions, each trapped ion having two hyperfine states and an excited state that define a qubit. One or more lasers configured to emit a laser beam, the laser beam being split into two or more non-collinear laser beams, the two or more non-collinear laser beams being provided to each trapped ion, wherein the two or more non-collinear laser beams are configured to cause each trapped ion to perform Rabi oscillations between the two hyperfine states via the excited state, and A controller including a non-volatile memory storing a plurality of instructions, which when executed by a processor cause the quantum computing system to perform operations, the operations including: Selecting a gate duration value for pulses of the laser beam to be applied to a first ion and a second ion in a chain of trapped ions, wherein, Each trapped ion has two frequency-separated states defining a qubit, and The motional modes of the chain of trapped ions each have different frequencies; Calculating Fourier coefficients of the pulse function of the pulse based on the selected gate duration value and the frequencies of the motional modes of the chain of trapped ions; Demodulating the pulse function of the pulse having the calculated Fourier coefficients to calculate the amplitude and detuning frequency value of the pulse; and Performing an entanglement interaction between the first ion and the second ion by applying the pulse having the calculated amplitude and the calculated detuning frequency value to the first ion and the second ion for the gate duration value.

22. The quantum computing system according to claim 21, wherein, Each trapped ion has 2 S 1 / 2 hyperfine states of 171 Yb + and The laser is a mode-locked laser at 355 nm.

23. The quantum computing system according to claim 21, wherein, The operations further include: Before demodulating the pulse function of the pulse, selecting the calculated Fourier coefficients of the pulse function based on the condition that the trapped ion returns to its original position and momentum values.

24. The quantum computing system according to claim 23, wherein, The operations further include: Before demodulating the pulse function of the pulse, selecting the calculated Fourier coefficients of the pulse function such that the condition that the trapped ion returns to its original position and momentum values is stable up to the first order with respect to the drift of the frequency of the motional mode of the chain.

25. The quantum computing system according to claim 23, wherein, The operations further include: Before demodulating the pulse function of the pulse, selecting the calculated Fourier coefficients of the pulse function based on the condition of a non-zero entanglement interaction between the first ion and the second ion.

26. The quantum computing system according to claim 25, wherein, The non-zero entanglement interaction value between the first ion and the second ion is between zero and π / 8.

27. The quantum computing system according to claim 25, wherein The operations further include: Before demodulating the pulse function of the pulse, further selecting the calculated Fourier coefficients of the pulse function based on the condition of stabilizing the resulting entanglement interaction between the first ion and the second ion with respect to the drift of the frequency of the motional mode of the chain.

28. The quantum computing system according to claim 26, further including: Before applying the pulse, modifying the amplitude to calibrate the resulting entanglement interaction value to the non-zero entanglement interaction value in the presence of fluctuations in the intensity of the laser beam.

29. The quantum computing system according to claim 25, wherein, Applying a broadband pulse sequence to all trapped ions in the chain of trapped ions to stabilize the resulting entanglement interaction value in the presence of fluctuations in the coupling strength between the first ion and the second ion and the motional mode.

30. The quantum computing system according to claim 25, wherein, The operations further include: Before demodulating the pulse function of the pulse, the calculated Fourier coefficients of the pulse function are also selected based on minimizing the power of the laser beam provided to the first ion and the second ion during the pulse.

31. The quantum computing system according to claim 21, wherein, The operation further includes: Before demodulating the pulse function of the pulse, the calculated Fourier coefficients of the pulse function are selected based on the condition of non-zero entangled interaction between the first ion and the second ion.

32. The quantum computing system according to claim 31, wherein The value of the non-zero entangled interaction between the first ion and the second ion is between zero and π / 8.

33. The quantum computing system according to claim 31, wherein, The operation further includes: Before demodulating the pulse function of the pulse, the calculated Fourier coefficients of the pulse function are selected based on the condition of stabilizing the resulting entangled interaction between the first ion and the second ion against drift in the frequency of the motional mode of the chain.

34. The quantum computing system according to claim 32, wherein the operation further includes: Modifying the gate duration value to calibrate the resulting entangled interaction value to the non-zero entangled interaction value in the presence of fluctuations in the intensity of the laser beam.

35. The quantum computing system according to claim 31, wherein Applying a broadband pulse sequence to all trapped ions in the chain of trapped ions to calibrate the resulting entangled interaction value in the presence of fluctuations in the coupling strength between the first ion and the second ion and the motional mode.

36. The quantum computing system according to claim 31, wherein, The operation further includes: Before demodulating the pulse function of the pulse, the calculated Fourier coefficients of the pulse function are also selected based on minimizing the power of the laser beam provided to the first ion and the second ion during the pulse.

37. The quantum computing system according to claim 21, wherein, The pulse is symmetric about the midpoint of the gate duration value.

38. The quantum computing system according to claim 21, wherein, The pulse is antisymmetric about the midpoint of the gate duration value.

39. The quantum computing system according to claim 21, wherein Applying the pulse further includes: Converting the pulse function with the calculated Fourier coefficients into a series of time-dependent pulse segments, each time-dependent pulse segment having a different amplitude value, a different detuning value, and the phase of the laser beam.

40. The quantum computing system according to claim 39, wherein, Interpolating the time-dependent pulse segments with a spline.

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