Efficient Evaluation of Motional Mode Characteristics for High-Fidelity Trapped-Ion Quantum Computation

Abstract

A method of using an ion trap quantum computer includes performing a first measurement of the bright state population of each ion in an ion chain, wherein each ion is coupled to one of the motional modes of the ion chain while a laser coupling frequency is varying; calculating a mode frequency of the one motional mode among the motional modes based on the bright state population measured in the first measurement; performing a second measurement of the bright state population of each ion in the ion chain; and calculating a coupling strength between each ion and the one motional mode among the motional modes by fitting the bright state population of each ion measured in the second measurement to a value of the bright state population calculated based on the calculated mode frequency of the one motional mode among the motional modes and a non-zero temperature effect of the motional mode.

Description

Technical Field

[0001] (Cross - Reference to Related Applications) This application claims priority to U.S. Patent Application No. 18 / 202,270, filed May 25, 2023, and U.S. Provisional Application 63 / 348,421, filed Jun. 2, 2022, each of which is incorporated herein by reference in its entirety.

[0002] The present disclosure generally relates to methods of performing entangling gate operations in a trapped - ion - based quantum computer, and more specifically, to methods of characterizing the motional modes of an ion chain.

Background Art

[0003] In quantum computing, the requirements for scalability include efficient characterization, calibration, and verification of a quantum computing system in addition to high - fidelity initialization, logical operations, and readout of the quantum computing system. In a trapped - ion - based quantum computer, the quantum information encoded in trapped ions (qubits) is processed via the motional modes of the ions (e.g., collective vibrations), and thus, efficient and accurate characterization of the motional modes leads to efficient and accurate quantum computing.

[0004] Accordingly, there is a need for methods and systems that enable efficient and accurate characterization of the motional modes of an ion chain.

Summary of the Invention

[0005] Embodiments of the present disclosure provide a method of using an ion trap quantum computer. The method includes performing a first measurement of the bright state population of each ion in an ion chain including a plurality of ions for a fixed duration, wherein each ion is coupled to one of the motional modes of the ion chain, and during which the laser coupling frequency for coupling each ion to the one motional mode among the motional modes varies; calculating the mode frequency of the one motional mode among the motional modes based on the frequency at which the bright state population of each ion measured in the first measurement is maximized; calculating the coupling strength between each ion and the one motional mode among the motional modes by fitting the maximized bright state population of each ion measured in the first measurement to the value of the bright state population calculated based on the calculated mode frequency of the one motional mode among the motional modes and the non-zero temperature effect of the motional mode; performing a second measurement of the bright state population of each ion in the ion chain for a fixed duration, wherein each ion is coupled to one of the motional modes that each ion was not coupled to in the first measurement, and during which the laser coupling frequency for coupling each ion to the one motional mode among the motional modes is fixed; and calculating the coupling strength between each ion and the one motional mode among the motional modes by fitting the bright state population of each ion measured in the second measurement to the value of the bright state population calculated based on the calculated mode frequency of the one motional mode among the motional modes and the non-zero temperature effect of the motional mode.

[0006] Embodiments of the present disclosure also provide a method of using an ion trap quantum computer. The method includes performing a first measurement of the bright state population of each ion in an ion chain including a plurality of ions for a fixed duration, wherein each ion is coupled to one of the motional modes of the ion chain, and during which the laser coupling frequency for coupling each ion to the one of the motional modes varies; calculating the mode frequency of the one of the motional modes based on the frequency at which the bright state population of each ion measured in the first measurement is maximized; performing a second measurement of the bright state population of each ion in the ion chain for a plurality of durations, wherein each ion is coupled to one of the motional modes, and during which the laser coupling frequency for coupling each ion to the one of the motional modes is fixed; and calculating the coupling strength between each ion and the one of the motional modes by fitting the bright state population of each ion measured in the second measurement to a value of the bright state population calculated based on the calculated mode frequency of the one of the motional modes and the non-zero temperature effect of the motional mode.

[0007] Embodiments of the present disclosure further provide a quantum computing system. The quantum computing system includes an ion chain including a plurality of ions, wherein each ion in the ion chain has two hyperfine states defining a qubit, an ion chain, a system controller, and a classical computer including a processor and a non-volatile memory storing a large number of instructions. When the instructions are executed by the processor, the quantum computing system is caused to perform the steps of: performing, by the system controller, a first measurement of the bright state population of each ion in the ion chain for a fixed duration, wherein each ion is coupled to one of the motional modes of the ion chain, and during which the laser coupling frequency for coupling each ion to the one of the motional modes varies; calculating, by the processor, the mode frequency of the one of the motional modes based on the frequency at which the bright state population of each ion measured in the first measurement is maximized; performing, by the system controller, a second measurement of the bright state population of each ion in the ion chain, wherein each ion is coupled to one of the motional modes of the ion chain, and during which the laser coupling frequency for coupling each ion to the one of the motional modes is fixed; and calculating, by the processor, the coupling strength between each ion in the ion chain and one of the motional modes of the ion chain based on the bright state population measured in the first measurement, the bright state population measured in the second measurement, the calculated mode frequency of the one of the motional modes, and the non-zero temperature effect of the motional mode.

Brief Description of the Drawings

[0008] To be able to understand in detail the features listed above of the present disclosure, a more specific description of the present disclosure, briefly outlined above, can be obtained by referring to the embodiments, some of which are shown in the accompanying drawings. However, it should be noted that the accompanying drawings show only typical embodiments of the present disclosure, and thus the present disclosure should not be regarded as limiting the scope of the present disclosure, as the present disclosure may admit other equally effective embodiments.

[0009]

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[0010] For ease of understanding, the same reference numerals are used throughout the drawings to denote the same elements where possible. In the drawings and the following description, an orthogonal coordinate system including the X-axis, Y-axis, and Z-axis is used. For convenience, the directions indicated by the arrows in the figures are taken as the positive directions. It is contemplated that elements disclosed in some embodiments may be beneficially utilized in other implementations without specific recitation.

Best Mode for Carrying Out the Invention

[0011] As the size of a quantum computer increases, parameters related to the quantum computer system need to be characterized with high precision and efficiency to achieve high-fidelity quantum logic. In a trap-ion-based quantum computer, the strength of entanglement between qubits (trap ions) is mediated by the motional modes of the ion chain, and thus it is essential to characterize the coupling strength (referred to as the Lamb-Dicke parameter) between each ion and the motional mode. The embodiments described herein provide a physical model that accurately predicts both the magnitude and sign of the Lamb-Dicke parameter when the motional modes are probed in parallel. The embodiments described herein further provide an improved characterization method that reduces the characterization time by one order of magnitude or more compared to the characterization time of conventional methods.

[0012] An overall system capable of performing quantum computing using trapped ions includes a classical (digital) computer, a system controller, and a quantum processor. The classical computer performs support and system control tasks including selecting a quantum algorithm to be implemented quantumly using a user interface such as 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 a series of pairwise entanglement gate operations for application to the quantum processor, and calculating the amplitude and detuning frequency of laser pulses that cause a series of pairwise entanglement gate operations using a central processing unit (CPU). A software program for performing the task of decomposing and executing the quantum algorithm is stored in non-volatile memory within the classical computer. The quantum processor includes trapped ions coupled to various hardware including a laser for manipulating the internal hyperfine states (qubit states) of the trapped ions and an acousto-optic modulator for reading out the internal hyperfine states (qubit states) of the trapped ions. The system controller receives the calculated amplitude and detuning frequency of the laser pulses from the classical computer at the start of executing the selected algorithm on the quantum processor, controls various hardware related to controlling any and all aspects used to execute the selected algorithm on the quantum processor, and at the end of execution of the algorithm, reads out the quantum processor (e.g., the population of the qubit states of the trapped ions), thus returning the output of the result of the quantum calculation to the classical computer, and generating and outputting a solution to the selected quantum algorithm based on the processing result of the quantum calculation.

[0013] I. General Hardware Configuration FIG. 1 is a schematic partial view of a trapped ion quantum computing system 100, or simply system 100, according to one embodiment. System 100 can represent a hybrid quantum-classical computing system. System 100 includes a classical (digital) computer 102 and a system controller 104. Other components of system 100 shown in FIG. 1 are associated with a quantum processor, which includes a chain 106 of atomic ions (i.e., five shown as circles at approximately equal intervals from each other) that are trapped to form a linear Coulomb crystal extending along the Z-axis. Each ion in ion chain 106 is an ion having a nuclear spin I and an electron spin S such that the difference between the nuclear spin I and the electron spin S is zero, for example, a positive ytterbium ion 171 Yb + , a positive barium ion 133 Ba + , a positive cadmium ion 111 Cd + , 113 Cd + , and all of them have a nuclear spin I = 1 / 2 and 2 S 1 / 2 in the hyperfine state. In some embodiments, all ions in ion chain 106 are of the same species and isotope (e.g., 171 Yb + ). In some other embodiments, ion chain 106 includes one or more species or isotopes (e.g., some ions are 171 Yb + and some other ions are 133 Ba +It includes). In a further embodiment, the ion chain 106 may include various isotopes of the same species (e.g., different isotopes of Yb, different isotopes of Ba). The ions in the ion chain 106 are individually addressed by individual laser beams. The classical computer 102 includes a central processing unit (CPU), memory, and support circuits (or I / O) (not shown). The memory is connected to the CPU and may be one or more of readily available memories such as read-only memory (ROM), random access memory (RAM), floppy (registered trademark) disk, hard disk, or any other form of local or remote digital storage. Software instructions, algorithms, and data can be encoded and stored in the memory to instruct the CPU. Support circuits (not shown) are also connected to the CPU to support the processor in a conventional manner. The support circuits may include conventional cache, power supply, clock circuit, input / output circuit, subsystem, etc.

[0014] The imaging objective lens 108, for example, an objective lens having a numerical aperture (NA) of 0.37, collects fluorescence from ions along the Y-axis and maps each ion onto a multi-channel photomultiplier tube (PMT) 110 (or some other imaging device) for measurement of individual ions. The Raman laser beam from the laser 112 provided along the X-axis performs operations on the ions. The diffraction beam splitter 114 creates an array of Raman laser beams 116 that are individually switched using the multi-channel acousto-optic modulator (AOM) 118. The AOM 118 is configured to selectively act on individual ions by individually controlling the emission of the Raman laser beams 116. The global Raman laser beam 120 does not co-propagate with the Raman laser beams 116 and irradiates all the ions at once from a different direction. In some embodiments, instead of a single global Raman laser beam 120, individual Raman laser beams (not shown) can be used to irradiate individual ions respectively. The system controller (also referred to as the “RF controller”) 104 controls the AOM 118 and thus controls the intensity, timing, and phase of the laser pulses applied to the trapped ions in the ion chain 106. The CPU 122 is the processor of the system controller 104. The ROM 124 stores various programs, and the RAM 126 is the working memory for various programs and data. The storage unit 128 includes non-volatile memory such as a hard disk drive (HDD) or flash memory and stores various programs even when the power is turned off. The CPU 122, ROM 124, RAM 126, and storage unit 128 are interconnected via a bus 130. The system controller 104 executes a control program stored in the ROM 124 or the storage unit 128 and uses the RAM 126 as a working area. The control program includes software applications containing program code that can be executed by the CPU 122 to perform various functions related to receiving and analyzing data and controlling any and all aspects of the methods and hardware used to implement and operate the trapped ion quantum computing system 100 discussed herein.

[0015] II. Trap Ion Quantum Computer System In a trap ion quantum computing system such as system 100, two internal states of an atomic ion, 2 S 1 / 2 such as hyperfine states, are typically used as computational qubit states denoted as │0> and │1>. The hyperfine ground state (i.e., 2 S 1 / 2 the lower energy state of the hyperfine state) can be selected to represent the qubit state │0>. Hereinafter, the terms "internal state", "hyperfine state", and "qubit state" can be used interchangeably to represent │0> and │1>. Furthermore, the hyperfine states │0> and │1> can be referred to as "dark state" and "bright state", respectively. Each ion is cooled to near the motional ground state for any motional mode without phonon excitation (i.e., the motional energy of the ion may be reduced) by a known laser cooling method such as Doppler cooling or resolved sideband cooling, and then the qubit state can be prepared in the dark state │0> by optical pumping. When multiple ions are trapped and form a linear Coulomb crystal as in ion chain 106, the external motion of the ions (e.g., the collective motion of the ions) can be quantized and approximated as a set of coupled quantum harmonic oscillators. The internal and external degrees of freedom of an ion chain consisting of N ions (e.g., the qubit states of individual ions and the collective motion of the ions) can be described by the Hamiltonian

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[0016] Typical laser-induced multiqubit gate operations between ions, such as the Mølmer-Sørensen method, use the laser electric field to couple the internal and external degrees of freedom of the participating ions in the ion chain. The Hamiltonian [Number] The frequency [Number] (referred to as the "laser coupling frequency") that couples the qubit states of ion j in the N-ion chain in the rotating frame with respect to the Hamiltonian [Number] can be written as [Number] is the raising operator of ion j, and Ω j is the qubit state Rabi frequency (i.e., the coupling between two qubit states of ion j), and φ j is the laser phase, and η j,k is the Lamb-Dicke parameter that quantifies the coupling strength between ion j and the motion mode k. For the sake of simplicity, the laser phase is φ jcan be selected as =0. Typically, out of a total of 3N motional modes, N' (N'<3N) motional modes are strongly coupled to the laser, while the remaining motional modes contribute negligibly to the multiqubit gate operation. For the multiqubit gate operation to function faithfully and efficiently, the Rabi parameters η j,k and the mode frequency ω k of these N' motional modes need to be known with high precision.

[0017] II.A Characterization of the Rabi Parameter Conventional methods for characterizing these parameters (the Rabi parameter η j,k and the mode frequency ω k ) are sideband spectroscopy using blue-sideband (BSB) transitions. To characterize the Rabi parameter η j,k for the motional mode k of ion j and the mode frequency ω k of the motional mode k, while the laser coupling frequency

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[0018] Similar to other spectroscopic approaches, the conventional mode characterization method is designed to probe the mode frequency ω k The embodiments described herein provide an improvement over the conventional mode characterization method when a more accurate and efficient characterization of the Rabi parameter η j,k is required, since there are N×N’ different values for the Rabi parameter η j,k to be characterized.

[0019] To extract the Rabi parameter η j,k the measurement data of the bright state population of ion j is fitted to a model (referred to as the "baseline model" denoted by the superscript (0)), which conventionally uses the approximate interaction Hamiltonian

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[0020] Here, t is the evolution time, and

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[0021] II.B Improvements to the Baseline Model The baseline model is an approximation for two main reasons. (i) The spectator motional mode (i.e., the unprobed motional mode) is ignored, and (ii) it is assumed that the motional modes are always prepared in the motional ground state. For a more accurate estimation of the bright state population, the contribution of the spectator motional mode due to the non-zero spread of the ion's position wave packet and the off-resonant BSB transition, as well as the effect of non-zero temperature, can also be taken into account. Also, note that the conventional mode characteristic evaluation method using (6) does not reveal the relative sign of the Lamb-Dicke parameter η j,k which is important for multi-qubit gate design and operation.

[0022] The method according to the embodiments described herein is provided to improve the conventional mode characteristic evaluation method with respect to the following aspects. 1. Parallelization: There are N×N' different Lamb-Dicke parameters η j,k existing in an N-ion chain having N' motional modes that are strongly coupled to the laser. To characterize each of the N×N' different Lamb-Dicke parameters η j,k one by one, O(N 2 ) operations are required. To support large-scale quantum computers, parallelization is necessary to reduce the complexity to O(N). 2. Accuracy: To characterize the Lamb-Dicke parameter η j,k with high accuracy, it is necessary to consider the effect of the coupling to other motional modes k'≠k of ion j. The coupling arises from both the non-zero spread of the ion's position wave packet and the non-resonant BSB transition. 3. Sign problem: It is necessary to distinguish the relative signs of the Lamb-Dicke parameters η j,k However, in (6), in the bright state population, it depends only on the magnitude of the Lamb-Dicke parameter η j,k and does not depend on its sign. 4. Efficiency: When the mode frequency ω k and shot noise are inaccurate, the characterization of the Lamb-Dicke parameter η j,k will also be inaccurate. To reduce the uncertainty, a fairly long measurement time is required.

[0023] Such improvements can be achieved by the following objectives. Objective 1: Find an effective model to better characterize the dynamics of the bright state population of ions undergoing BSB transitions. Objective 2: Explore a method and corresponding model that can distinguish the signs of the Lamb-Dicke parameters η j,k from each other. Objective 3: Find a more efficient parallelization method that minimizes the measurement time while suppressing the uncertainty in estimating the Lamb-Dicke parameter η j,k below the target value.

[0024] III. Improved Model This section discusses various improved models that predict the bright state population of ions that all undergo BSB transitions in parallel. These models predict the bright state population of the ions and thereby the Lamb-Dicke parameter η j,k and the mode frequency ω k are more accurate than the baseline model conventionally used in (6) when characterizing them. Section III.A discusses three effects that occur in parallel BSB transitions not considered in the baseline model. Section III.B introduces a total of five models, considering step by step the effects discussed in Section III.A and their combinations until reaching the most refined model.

[0025] III.A Effects This section discusses three effects in the parallel BSB transitions of ions. By considering these effects in the model, a more accurate characterization of the Lamb-Dicke parameter η j,k becomes possible.

[0026] (a) Non-zero temperature Even after using the most refined cooling techniques, the motional mode is unlikely to be in the absolute motional ground state. Therefore, the baseline model described in (4) - (6) is generalized to an initial state with an arbitrary number of phonons n. The Rabi frequency between the two composite states |0,n> j,k and |1,n + 1> j,k is given by, assuming that composite states other than these two do not affect the BSB transition,

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[0027] This generalized Rabi frequency

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[0028] (b) The Debye-Waller (DW) effect The spread of the position wave packet of the ions associated with each mode appears as a decrease in the Rabi frequency, widely known as the DW effect. Even when the motional mode is cooled to the motional ground state, the DW effect due to zero-point fluctuations persists.

[0029] When the motional mode k is being probed through ion j, the DW effect due to spectator motional modes k’≠k is [Number] given by the reduction in the Rabi frequency between the two composite states |0,n k > j,k and |1,n k +1> j,k and, where [Number] is the vector of the initial phonon number n k’ of the motional mode k’ (k’∈{1,2,…,N’}) and [Number] is the average DW reduction coefficient of the spectator motion mode k' with the initial phonon number n k’ For efficient characterization, each of the N ions is used to probe the assigned motion modes in parallel, which is repeated N' times in different permutations of the motion modes to probe all N×N' values of the Lamb-Dicke parameter η

[0030] For efficient characterization, each of the N ions is used to probe the assigned motion modes in parallel, which is repeated N' times in different permutations of the motion modes to probe all N×N' values of the Lamb-Dicke parameter η j,k In this case, each spectator motion mode k' is also probed through a different ion j'(k'), and thus the phonon number of the motion mode k' fluctuates between n k’ and n k’ +1. Therefore, the average DW reduction coefficient is

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[0031] When the motion mode k' is resonantly probed over a sufficiently long interrogation time, the phonon number of the motion mode k' can be approximated as n for the first half of the time k’ and n k’ +1 for the second half. The exception is when the ion j'(k') is at a node of the motion mode

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[0032] Using (8) - (11), Equation (6) can be further generalized to allow non - zero initial phonon numbers for all motion modes by replacing the effective Rabi frequency [Number] with a reduced Rabi frequency [Number] . The resulting [Number] is the bright - state population of ion j that undergoes parallel BSB transitions. Initially, all ions are in the dark state, and the phonon number k’ of the motion mode is [Number] the k’ - th element of k’ n.

[0033] (c) Cross - mode coupling When ion j probes motion mode k, non - resonant BSB transitions with other motion modes k’≠k also occur. The resulting effect of other motion modes on the qubit state is called cross - mode coupling. Cross - mode coupling has a Rabi frequency Ω j,k’ that is much smaller than the detuning frequency Δ j,kcan be reduced by using it, but the smaller the Rabi frequency, the slower the BSB transition becomes. Therefore, there is a trade-off between reducing the error due to cross-mode coupling and shortening the characteristic measurement.

[0034] In principle, cross-mode coupling can be included in a model that simulates the time evolution of the entire Hamiltonian of N ions and N' motional modes. However, the simulation time increases exponentially with the number of ions N. Therefore, a more realistic approach is to limit the simulated system size to a maximum of three ions and three motional modes by including only the nearest-neighbor motional modes in the simulation and the ions that probe them.

[0035] III.B Five improved models Hereinafter, five models of the bright state population of ions undergoing parallel BSB transitions will be discussed. These five models are improvements from the baseline model of (6).

[0036] (a) Model 1: Debye-Waller (DW) effect Model 1 takes into account the DW effect while still assuming zero temperature. The average bright state population [Number] is given by j,k when the initial state is |0,0> [Number] where the bright state population [Number] is obtained by (6), and the effective Rabi frequency [Number] in (8) is the reduced Rabi frequency

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[0037] Reduced Rabi frequency

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[0038] (b) Model 2: Non-zero temperature Model 2 takes into account the non-zero temperature effect in addition to the DW effect considered in Model 1. By allowing a plurality of different initial phonon numbers in the distribution function

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[0039] The sum in (13) is over a finite number of th for

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[0040] (c) Model 3: Time-Dependent DW (TDDW) Effect Model 3 further considers the time-dependence of the DW reduction coefficient. This is because for each motion mode k probed through ion j, the spectator motion mode k’≠k is also probed through another ion j’(k’)≠j, and the number of phonons varies between n k’ and n k’ +1 as it is probed over time. The TDDW reduction coefficient is

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[0041] Here, the bright state population considering the TDDW effect

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[0042] (d) Model 4: Nearest Neighbor (NN) Model 4 takes into account the NN motion modes of the probed motion mode and their assigned ions. In other words, consider the subspace of the probed motion mode k, its NN motion modes k - 1 and k + 1 (the motion modes are arranged in ascending order of mode frequency), and their assigned ions j(k), j(k - 1), and j(k + 1) (two ions, two motion modes for k = 1 and N').

[0043] The interaction Hamiltonian that describes the partial space is [Number] wherein, [Number] is. The initial composite state is [Number] By taking the matrix element corresponding to the resonant transition and evaluating the time-evolution operator of this Hamiltonian from time 0 to t, the average bright state population [Number] is obtained as in (13).

[0044] Evaluating the time-evolution operator of the three-ion three-mode Hamiltonian in (15) takes a considerably longer time than simply evaluating trigonometric functions and polynomials as in conventional models. However, since this model includes NN motional modes, its accuracy is less affected by cross-mode coupling. It should be noted that this appropriately captures the quantum interference between the qubit state and the motional mode beyond the single-ion single-mode model. The predicted bright state population is sensitive to the sign of the Rabi parameter η j,k±1 with respect to the Rabi parameter η j,k of.

[0045] (e) Model 5: TDDW+NN Model 5 incorporates the TDDW effect discussed in model 3 (c) into the NN model n of model 4 (d). This is done by replacing the average DW reduction coefficient in (15) with the TDDW coefficient in (14).

[0046] IV. Method In this section, the Rabi parameter η according to the embodiments described hereinj,k and the mode frequency ω k Provide two methods, a "basic method" and an "improved method", for characterizing. The measured bright state population of the N-ion qubit undergoing the BSB transition can have different sensitivities to the Rabi Dicke parameters for different methods. The Rabi Dicke parameter η j,k has N×N' different values, so parallelization of the measurements is required. Conventional mode characterization methods are mainly designed only to probe the mode frequency ω k . The basic method discussed below is a modified version of the conventional mode characterization method for probing the values of the Rabi Dicke parameter η j,k in parallel. The improved method can determine the Rabi Dicke parameter η j,k more accurately and quickly.

[0047] The methods described herein are designed to characterize the Rabi Dicke parameter η j,k with high precision, so a rough estimate of the Rabi Dicke parameter is assumed before implementing this method. It is sufficient if the estimate of η j,k is within one digit and the estimate of the mode frequency ω k is within a few kHz.

[0048] IV.A Basic method FIG. 2 is a flowchart showing a basic method 200 for characterizing the Rabi Dicke parameter η j,k , and this method quantifies the coupling strength between ion j and motion mode k. Here, the N ions in the ion chain are labeled with j, and the N' motion modes of the ion chain that are strongly coupled to the laser are labeled with k. Therefore, the number of Rabi Dicke parameters η j,k to be determined is N×N'.

[0049] The basic method 200 includes two steps. The first step in block 210 is to use N ions to measure all the mode frequencies ω of the N' motion modes by frequency sweep measurement kmeasuring, and simultaneously determining N' of N×N' Lamb-Dicke parameters [Number] including. In a typical case, the number N of ions is equal to the number N' of motional modes, and thus, each of the N' motional modes is assigned an ion to be probed. When the number N' of motional modes is greater than the number N of ions, this first step is repeated in rounds such that each of the N' motional modes is assigned an ion to be probed in at least one round. [Number] The rounds are repeated. In the i-th round [Number] the mode frequency ω of motional mode k k and the Lamb-Dicke parameter i for motional mode k and ion j [Number] are both determined. Here [Number] represents the smallest integer greater than or equal to the argument, and j i (k) represents the ion used to probe motional mode k in the i-th round.

[0050] The second step of block 220 includes determining the remaining (N - 1)×N' Lamb-Dicke parameters [Number] The second step [Number] is repeated in rounds.

[0051] Specifically, the i-th round of block 210 starts at sub-block 212, and ion j (= 1, 2, …, N) is respectively assigned to one of the probe motion modes k (= 1, 2, …, N’) including the motion modes not probed in the previous round. The ion assigned to probe motion mode k in the i-th round is denoted as j i (k). Among the N’ motion modes strongly coupled to the laser, in sub-block 212, N ions are assigned to probe N motion modes, and no ions are assigned to probe the (N’ - N) motion modes.

[0052] The i-th round of block 210 proceeds to sub-block 214, and each ion j i (k) (= 1, 2, …, N) is initialized to the dark state │0>. In the example described herein, the dark state │0> is the hyperfine ground state of ion j i (k). Each ion j i (k) is initialized to near the motional ground state of any motion mode without phonon excitation (i.e., cooled so that the motional energy of the ion is reduced) by a known laser cooling method such as Doppler cooling or resolved sideband cooling, and then may be initialized to the qubit state prepared in the hyperfine ground state │0> by optical pumping.

[0053] The i-th round of block 210 proceeds to sub-block 216, and the frequency-scan measurement of the bright-state population P i (t) of each ion j j,k (k) in the blue-sideband (BSB) transition is performed for a fixed time τ (0) . Each ion j i (k) (= 1, 2, …, N) has a laser coupling frequency

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[0054] The i-th round of block 210 proceeds to sub-block 218, and the Rabi parameter i for each ion j

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[0055] Sub-blocks 212 - 218 are executed in parallel for N ions in each round

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[0056] To accurately measure the mode frequency ω k the mode assignment in sub-block 212

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[0057] The i-th round of block 220 starts from sub-block 222, where ions j (= 1, 2, …, N) are each assigned to one of the motion modes k. In sub-block 222, the ions are assigned to different permutations of the motion modes (e.g., different combinations of ion j and motion mode k). The ion assigned to motion mode k in the i-th round is represented as j i (k).

[0058] The i-th round of block 220 proceeds to sub-block 224, where each ion j i (k) (= 1, 2, …, N) is initialized to the dark state │0>. This initialization of the ions is the same as that in sub-block 214.

[0059] The i-th round of block 220 proceeds to sub-block 226, where the population P i (t) of the bright state of each ion j j,k (k) in the blue sideband (BSB) transition is measured for a fixed time τ (0) . Frequency scanning measurement is not performed in sub-block 226. Each ion j i (k) (= 1, 2, …, N) is excited by a laser pulse while the laser coupling frequency

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[0060] The i-th round of block 220 proceeds to sub-block 228, where the Rabi parameter i for each ion j

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[0061] Each round [Number] in, sub-blocks 222 to 228 are executed in parallel for N ions, and thus, N of the remaining N'×(N - 1) Lamb-Dicke parameters [Number] are determined. To determine, sub-blocks 222 to 228 are [Number] repeated until all of the remaining N'×(N - 1) Lamb-Dicke parameters [Number] rounds are repeated

[0062] IV.B Improvement method FIG. 3 is a flowchart showing an improvement method 300 for characterizing the Lamb-Dicke parameter η j,k which quantifies the coupling strength between ion j and motion mode k. Again, the N ions in the ion chain are labeled with j, and the N' motion modes of the ion chain that strongly couple to the laser are labeled with k. Thus, the number of Lamb-Dicke parameters η j,k to be determined is N×N'

[0063] Improvement method 300 also includes two steps. The first step of block 310 is a frequency scanning measurement that uses N ions to calculate all the mode frequencies ω of N' motional modes, similar to the first step of block 210 of the basic method 200. However, in the first step of block 310, the Lamb-Dicke parameter η k is not calculated. The first step is repeated for j,k rounds such that each of the N' motional modes is assigned ions for probing in at least one round. [Number] rounds.

[0064] The second step of block 320 is a time scanning measurement of the bright state population P j,k (t) for calculating the Lamb-Dicke parameter η. The second step is repeated for N' rounds. j,k Specifically, the i-th round of block 310 starts with sub-block 312, and each ion j (= 1, 2,..., N) is assigned to one of the probe motional modes k (= 1, 2,..., N') that include motional modes not probed in the previous round. Sub-block 312 is the same as sub-block 212 of the basic method 200.

[0065] The i-th round of block 310 proceeds to sub-block 314, and each ion j [Number] (k) (= 1, 2,..., N) is initialized to the dark state │0>. This initialization of the ion is the same as that of sub-block 214 of the basic method 200.

[0066] The i-th round of block 310 proceeds to sub-block 316, and each ion j in the blue sideband (BSB) transition i (k) (= 1, 2,..., N) is initialized to the dark state │0>. This initialization of the ion is the same as that of sub-block 214 of the basic method 200.

[0067] The i-th round of block 310 proceeds to sub-block 316, and each ion j in the blue sideband (BSB) transitioni Bright state population P of (k) j,k Frequency sweep measurement of (t) is performed for a fixed time τ (0) for each ion j i (k)(=1,2,…,N) is the laser coupling frequency

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[0068] Sub-blocks 312 to 316 are executed in parallel for N ions in each round

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[0069] The i-th round of block 320 starts at sub-block 322, and ion j (= 1, 2,..., N) is respectively assigned to one of the motional modes k (= 1, 2,..., N'). In sub-block 322, the ions are assigned to different permutations of the motional modes (e.g., different combinations of ion j and motional mode k). The ion assigned to probe motional mode k in the i-th round is denoted as j i (k).

[0070] The i-th round of block 320 proceeds to sub-block 324, and each ion j i (k) (= 1, 2,..., N) is initialized to the dark state │0>. This initialization of the ion is the same as that of sub-block 224 of the basic method 200.

[0071] The i-th round of block 320 proceeds to sub-block 326, and the time-scan measurement of the bright-state population P i (t) of each ion j j,k in the blue-sideband (BSB) transition is performed at a fixed laser coupling frequency

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[0072] In the i-th round of block 320, it proceeds to sub-block 328, and the Rabi parameter i for each ion j

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[0073] Figure 4 shows the bright state population Pj,k (t) undergoes a BSB transition that resonates completely in parallel at various evolution times (Δ j,k = 0). In this example, the number of ions N is set to be equal to the number of motional modes N' strongly coupled to the laser (N = N' = 5), the qubit state Rabi frequency is

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[0074] In each round (i = 1, …, N'), sub - blocks 322 - 328 are executed in parallel for N ions, repeated for N' rounds, covering all N ions paired with N' motional modes, and all N'×N Lamb - Dicke parameters

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[0075] IV.C Measurement Time Scale In the above - mentioned method, the trapped - ion quantum computer goes through cycles of ion cooling, qubit state preparation, BSB transition, and measurement of the bright state population of ions. The time scales for cooling, state preparation, and measurement may be on the order of 10 ms, 10 μs, and 100 μs, respectively. Since the BSB transition requires a time on the order of milliseconds because the Rabi frequency of the qubit state needs to be made small enough to suppress cross - mode coupling.

[0076] When the number N of ions is equal to the number N’ of motional modes strongly coupled to the laser (N’ = N), this corresponds to the laser alignment setting commonly used, and the Rabi parameter η j,k and the mode frequency ω k The total time T required to characterize (0) is

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[0077] The lower limit of the above parameters is the Rabi parameter η j,kis determined by the target accuracy in the measurement. In particular, the minimum

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[0078] In the basic method 200, when the uncertainty in the mode frequency ω k is large, the uncertainty in the Lamb-Dicke parameter η j,k also becomes large, which is because both parameters directly affect the bright state population

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[0079] FIGS. 5A and 5B respectively show the Lamb-Dicke parameter η 1,1 from the BSB transition, and the detuning frequency Δ

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[0080] Fitting the measured bright state population to models 1-5 is not an easy task. The reason is that the average bright state population

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[0081] Although this section has discussed a more accurate and efficient estimation of the Lamb-Dicke parameter, it should be noted that the methods described herein can also be easily used for better mode frequency estimation. For example, various laser coupling frequencies

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[0082] V. EXAMPLES In this section, by showing examples, it is demonstrated that the three objectives of the efficient mode characteristic evaluation described in Section II can be achieved using the improved models and methods described herein. More specifically, (i) a comparison of the accuracy of Models 1-5 with the baseline model when measuring the Lamb-Dicke parameter η j,k , (ii) demonstration that Model 4 can distinguish the relative sign of the Lamb-Dicke parameter η j,k , and (iii) showing the requirement that for a given target accuracy in η j,k estimation, the improved method 300 significantly reduces the characteristic evaluation measurement time compared to the basic method 200.

[0083] To conduct numerical tests, the parallel BSB transition measurements are numerically simulated. The BSB Hamiltonian in the interaction image is

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[0084] V.A Accuracy First, use the numerical simulation of the bright state population to compare the performance of the baseline model and Models 1-5 in capturing the qubit population evolution appropriately. Here, as an example, assume that all ions are driven simultaneously at the same qubit state Rabi frequency

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[0085] Figures 6A and 6B show examples of the average relative error in estimating the Lamb-Dicke parameter η j,k when using various models as a function of the qubit state Rabi frequency Ω0 and the number of ions N with the qubit state Rabi frequency Ω0 fixed at 2π×2 kHz, respectively. The labels are in the order of the baseline and Models 1-5 described in Section III. Here, the relative error is defined as

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[0086] Models 2 - 5 show power - law behavior, and the relative error is

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[0087] is smaller than η j,k±1 the influence of modes k±2 on the measurement error of η j,k±2 becomes equal to or greater than the influence of the NN mode k±1. In such cases, the NN model can be modified to include the modes with large influence at the cost of increasing the fitting calculation time. j,k

[0088] ​Models with the included TDDW effect achieve the highest accuracy. For example, in Figure 6B, when N = 7, the errors of Models 3 and 5 are smaller than those of Models 2 and 4, being 1 / 2.5. The TDDW effect may be more important for characterizing the Lamb-Dicke parameter with higher accuracy in longer ion chains.

[0089] Note that it is assumed that the physical distance between adjacent ions is fixed. Thus, as the number of ions N increases, the interval between mode frequencies decreases, which means that when the qubit state Rabi frequency is fixed, the cross-mode coupling becomes more severe.

[0090] V.B Sign Problem The sign of the Lamb-Dicke parameter η with respect to other Lamb-Dicke parameters j,k directly affects the quantum computing fidelity to determine the gate pulse design on many trapped-ion quantum computers. Unfortunately, the conventional mode characterization method cannot distinguish the sign of the Lamb-Dicke parameter η j,k because the qubit population does not depend on the sign in the baseline model in (6). Here, it is shown that the sign of the Lamb-Dicke parameter η j,k can be distinguished using the NN model (Model 4).

[0091] First, to distinguish the sign of η j,k using the BSB transition, it is necessary to consider multiple ions, because the sign of the Lamb-Dicke parameter η j,k is clearly defined only when the relative motion between different ions is explained. In a single mode, it should also be noted that when the sign of η j,k is different, the ions have different relative motion directions, but the qubit population undergoes exactly the same evolution. The sign of η j,k is determined by whether the symmetry of the participation of two ions in one mode is the same as or opposite to the symmetry in the other mode, and that difference affects the qubit population.

[0092] By irradiating two ions with the same two-tone beam in which each tone resonates at its respective mode frequency and driving the two ions to couple in parallel into two different modes, the BSB transitions to the two modes occur simultaneously on the two ions. The predicted evolutions will have the same symmetry on one hand and the opposite symmetry on the other, and will be significantly different from each other. This makes it possible to determine which symmetry, and thus the sign of the Lamb-Dicke parameter η j,k is directly correct from the signal generated by the measurement.

[0093] Figure 7 shows an example of the predicted time evolution of the average bright state population

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[0094] The Lamb-Dicke parameter η 1,1When = ±0.0119, the population curves are clearly distinguishable and can be accurately predicted by the NN model (Model 4). This shows that when the parameters are carefully selected and the observed evolutions are compared with the evolutions predicted by the NN model, by simultaneously inducing all four possible BSB transitions between two ions and two modes, the sign of the Lamb-Dicke parameter η j,k can be reliably distinguished.

[0095] V.C Characteristic Evaluation Measurement Time (16) and (17) respectively give the characteristic evaluation measurement times of the basic method 200 and the improved method 300, which are the following parameters: (i)

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[0096] To be consistent with Section V.A, M t is fixed at M t = 20,

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[0097] First, calculate the number of shots S j,k of the basic method 200 and the number of shots S (0) of the improvement method 300 required to reduce the uncertainty of the Lamb-Dicke parameter η t . Here, assuming that the model state population has a completely known mode frequency ω k , use model 2 to fit the uncertainty given by the combination of photon and phonon shot noise. Here, Ω0 = 2π × 10 kHz is used, but the effect of shot noise does not significantly affect Ω0.

[0098] Figure 8A shows an example of the average relative uncertainty for various values of S (0) and M t S t . The uncertainty is proportional to the reciprocal of the square root of the number of shots. When S (0) = M t S t , the improvement method always results in a smaller uncertainty of η j,k than the basic method. As explained in Section IV, the improvement method is such that the qubit population has ηj,k including points that are maximally sensitive to the value of

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[0100] ​

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[0101] Figure 8B shows an example of the average relative error in estimating η j,k as a function of Ω0. Multiple Δ j,k values are considered. Using this figure, η j,kWhen a predetermined target accuracy is given in the measurement, the values of Ω0 and δω that satisfy the target accuracy can be determined. For example, if it is desired that the relative uncertainty is lower than 10 k , a reasonable choice of the basic method 200 {improved method 300} is Ω0 / 2π = 7 {10} kHz and δω -3 / 2π = 12 {100} Hz, which is marked as k

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[0102] Here, using all the parameters of the determined method, a comparison is made between the characteristic evaluation measurement times of the basic method 200 and the improved method 300 given by (16) and (17). As a specific example, assume that the times for cooling, state preparation, and state detection are 4 ms, 100 μs, and 150 μs, respectively, and add these times to the BSB transition time to obtain the cycle time per shot. Table 1 shows the sets of parameters for the two methods. Overall, when estimating η j,k for a 5-ion chain, 10 -3To achieve the relative measurement uncertainty of the order of, for the characterization measurement time, in the case of the improvement method, T = 586 seconds, which is shorter than T (0) = 1.11×10 4 seconds and is about 1 / 19. The savings in the improvement method results from fewer shots and lower accuracy in the frequency scan.

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[0103] Finally, to distinguish the advantage that even with fewer shots and lower frequency scan accuracy, Figure 8C shows examples of the measurement times of the two methods for various values of δω k . This emphasizes that at the mode frequency, allowing a larger uncertainty δω k can significantly shorten the characterization measurement time of the improvement method.

[0104] VI Trade-off between the basic method and the improvement method The problem of efficient motion mode characterization at high precision leads to optimization over multiple parameters correlated by various trade-offs. For example, using a smaller laser power (and thus a smaller Ω0) reduces the error due to cross-mode coupling, but at the cost of a longer BSB transition time and the need for better frequency scan accuracy.

[0105] The choice of method and model can also be seen in the context of trade-offs. For example, in a parallelized method, the complexity is O(N 2) is reduced to O(N), but at the cost of introducing additional considerations into the model, such as the DW effect (precisely time-dependent) from other modes probed in parallel. Generally, more accurate models can be used at the expense of longer conventional computation times. To exploit this trade-off, a highly parallelized and efficient algorithm for the fitting routine is explored so that the conventional computational part of this method can be executed relatively quickly, which is especially applicable to long ion chains where the computation tends to be slow.

[0106] Another important trade-off related to trapped ions is the spacing between mode frequencies versus the physical distance between adjacent ions. As the distance between adjacent ions becomes smaller, the spacing between mode frequencies becomes larger, which reduces the cross-mode coupling effect, thereby making it possible to make the error smaller when measuring η j,k . This can mitigate the exponential increase in error with N when assuming that the distance between adjacent ions is fixed, as shown in FIG. 6B. However, as the inter-ion distance becomes smaller, optical crosstalk increases because the laser beam width cannot be arbitrarily reduced.

[0107] Embodiments described herein provide a method for evaluating the characteristics of motion modes. In particular, the method is based on the dynamics of ions in an ion chain and an effective physical model that more accurately describes the motion modes of the ion chain than conventional physical models, thereby enabling accurate and efficient characterization of the motion modes. The method described herein utilizes time-scan measurements that enable faster and more accurate characterization of motion modes compared to conventional methods, and the parallelism in which motion modes are simultaneously probed by multiple ions for faster and more accurate characterization of motion modes.

[0108] Appendices A, B, C, and D are attached, and their contents are all considered part of this application and are therefore incorporated herein.

[0109] Although the above is directed to specific embodiments, other additional embodiments can be devised without departing from the basic scope, which is determined by the following claims.

Claims

1. A method of using an ion trap quantum computer, comprising: performing a first measurement of the bright state population of each ion in an ion chain containing a plurality of ions for a fixed duration, wherein each ion is coupled to one of the motional modes of the ion chain, and during which the laser coupling frequency for coupling each ion to the one motional mode among the motional modes of the ion chain varies; calculating the mode frequency of the one motional mode among the motional modes based on the frequency at which the bright state population of each ion measured in the first measurement is maximized; calculating the coupling strength between each ion and the one motional mode among the motional modes by fitting the maximized bright state population of each ion measured in the first measurement to the value of the bright state population calculated based on the calculated mode frequency of the one motional mode among the motional modes and the non-zero temperature effect of the motional mode; performing a second measurement of the bright state population of each ion in the ion chain for a fixed duration, wherein each ion is coupled to one of the motional modes that each ion was not coupled to in the first measurement, and during which the laser coupling frequency for coupling each ion to the one motional mode among the motional modes is fixed; calculating the coupling strength between each ion and the one motional mode among the motional modes by fitting the bright state population of each ion measured in the second measurement to the value of the bright state population calculated based on the calculated mode frequency of the one motional mode among the motional modes and the non-zero temperature effect of the motional mode; A method comprising the above steps.

2. selecting, by a processor in a digital computer, a quantum algorithm to be implemented on the plurality of ions; compiling, by the processor in the digital computer, the selected quantum algorithm into a series of universal logic gates; converting, by the processor in the digital computer, the series of universal logic gates into a series of pairwise entanglement gate operations and applying them to the plurality of ions in the ion chain. The step of calculating the amplitude and detuning frequency of the laser pulse by the processor in the digital computer, and generating the series of paired entanglement gate operations based on the calculated motion mode and the binding strength between the ions; The step of applying the laser pulse having the calculated amplitude and detuning frequency to the plurality of ions in the ion chain by a system controller; The step of measuring the population of the qubit states of the plurality of ions in the ion chain by the system controller; The step of processing the quantum information corresponding to the qubit states of the plurality of ions in the ion chain based on the measured population of the qubit states by the processor in the digital computer; The step of generating and outputting a solution to the selected quantum algorithm based on the processing result of the quantum calculation by the processor in the digital computer; The method according to claim 1, further comprising:

3. The step of initializing each ion in the ion chain to the hyperfine ground state of each ion before the first measurement and the second measurement of each ion The method according to claim 1, further comprising:

4. The method according to claim 1, wherein the first measurements of all the ions in the ion chain are performed simultaneously.

5. The method according to claim 1, wherein the second measurements of all the ions in the ion chain are performed simultaneously.

6. The method according to claim 1, wherein the calculation of the binding strength between the ions in the ion chain and the motion mode of the ion chain is further based on the Debye-Waller effect of the ions.

7. The method according to claim 1, wherein the calculation of the binding strength between the ions in the ion chain and the motion mode of the ion chain is further based on the cross-mode coupling effect of the motion mode of the ion chain.

8. A method of using an ion trap quantum computer, comprising: The step of performing, by a system controller, a first measurement of the bright state population of each ion in an ion chain including a plurality of ions for a fixed duration, wherein each ion is coupled to one motion mode of the motion modes of the ion chain, during which the laser coupling frequency for coupling each ion and the one motion mode of the motion modes varies; A step of calculating a mode frequency of the one motion mode among the motion modes based on a frequency at which the bright state population of each ion measured in the first measurement becomes maximum by a processor in the digital computer; A step of calculating a coupling strength between each ion and the one motion mode among the motion modes by fitting the maximum bright state population of each ion measured in the first measurement to a value of the bright state population calculated based on the calculated mode frequency of the one motion mode among the motion modes and a non-zero temperature effect of the motion mode by a processor in the digital computer; A step of performing a second measurement of the bright state population of each ion in the ion chain for a fixed duration by the system controller, wherein each ion is coupled to one motion mode among the motion modes to which each ion is not coupled in the first measurement, and during which a laser coupling frequency for coupling each ion and the one motion mode among the motion modes is fixed; A step of calculating a coupling strength between each ion and the one motion mode among the motion modes by fitting the bright state population of each ion measured in the second measurement to a value of the bright state population calculated based on the calculated mode frequency of the one motion mode among the motion modes and a non-zero temperature effect of the motion mode by a processor in the digital computer; A step of selecting a quantum algorithm to be implemented for the plurality of ions by a processor in the digital computer; A step of compiling the selected quantum algorithm into a series of universal logic gates by a processor in the digital computer; A step of converting the series of universal logic gates into a series of pairwise entanglement gate operations by a processor in the digital computer and applying them to the plurality of ions in the ion chain; The step of calculating the amplitude and detuning frequency of the laser pulse by the processor in the digital computer, and generating the series of paired entanglement gate operations based on the calculated motion mode and the coupling strength between the ions; The step of applying the laser pulse having the calculated amplitude and detuning frequency to the plurality of ions in the ion chain by the system controller; The step of measuring the population of the qubit states of the plurality of ions in the ion chain by the system controller; The step of processing the quantum information corresponding to the qubit states of the plurality of ions in the ion chain based on the measured population of the qubit states by the processor in the digital computer; The step of generating and outputting a solution to the selected quantum algorithm based on the processing result of the quantum calculation by the processor in the digital computer; A method comprising the above steps.

9. A method of using an ion trap quantum computer, comprising: Executing a first measurement of the bright state population of each ion in an ion chain including a plurality of ions for a fixed duration, wherein each ion is coupled to one of the motion modes of the ion chain, and during which the laser coupling frequency for coupling each ion to the one motion mode of the motion modes varies; Calculating the mode frequency of the one motion mode of the motion modes based on the frequency at which the bright state population of each ion measured in the first measurement is maximized; Executing a second measurement of the bright state population of each ion in the ion chain for a plurality of durations, wherein each ion is coupled to one of the motion modes of the motion modes, and during which the laser coupling frequency for coupling each ion to the one motion mode of the motion modes is fixed; Calculating the coupling strength between each ion and the one motion mode of the motion modes by fitting the bright state population of each ion measured in the second measurement to the value of the bright state population calculated based on the calculated mode frequency of the one motion mode of the motion modes and the non-zero temperature effect of the motion mode; A method comprising the above steps.

10. selecting, by a processor within the digital computer, a quantum algorithm to be implemented on the plurality of ions; compiling, by the processor within the digital computer, the selected quantum algorithm into a series of universal logic gates; converting, by the processor within the digital computer, the series of universal logic gates into a series of pairwise entanglement gate operations and applying the operations to the plurality of ions within the ion chain; calculating, by the processor within the digital computer, the amplitude and detuning frequency of laser pulses to cause the series of pairwise entanglement gate operations based on the calculated motional modes and the coupling strength between the ions; applying, by a system controller, the laser pulses having the calculated amplitude and detuning frequency to the plurality of ions within the ion chain; measuring, by the system controller, a population of qubit states of the plurality of ions within the ion chain; processing, by the processor within the digital computer, quantum information corresponding to the qubit states of the plurality of ions within the ion chain based on the measured population of qubit states; generating and outputting, by the processor within the digital computer, a solution to the selected quantum algorithm based on the processing result of the quantum calculation; The method according to claim 9, further comprising.

11. initializing each ion within the ion chain to its hyperfine ground state prior to the first measurement and the second measurement of each ion The method according to claim 9, further comprising.

12. The method according to claim 9, wherein the first measurement of all the ions within the ion chain is performed simultaneously.

13. The method according to claim 9, wherein the second measurement of all the ions within the ion chain is performed simultaneously.

14. The method according to claim 9, wherein the calculation of the coupling strength between the ions within the ion chain and the motional modes of the ion chain is further based on the Debye-Waller effect of the ions.

15. The calculation of the binding strength between the ions in the ion chain and the motion mode of the ion chain is further based on the cross-mode binding effect of the motion mode of the ion chain, according to the method of claim 9. [

16. ] A method of using an ion trap quantum computer, comprising: Executing, by a system controller, a first measurement of the bright state population of each ion in an ion chain including a plurality of ions for a fixed duration, wherein each ion is coupled to one of the motion modes of the ion chain, and during which the laser coupling frequency for coupling each ion to the one motion mode of the motion modes varies; Calculating, by a processor in a digital computer, the mode frequency of the one motion mode of the motion modes based on the frequency at which the bright state population of each ion measured in the first measurement is maximized; Executing, by the system controller, a second measurement of the bright state population of each ion in the ion chain for a plurality of durations, wherein each ion is coupled to one of the motion modes of the motion modes, and during which the laser coupling frequency for coupling each ion to the one motion mode of the motion modes is fixed; Calculating, by the processor in the digital computer, the binding strength between each ion and the one motion mode of the motion modes by fitting the bright state population of each ion measured in the second measurement to the value of the bright state population calculated based on the calculated mode frequency of the one motion mode of the motion modes and the non-zero temperature effect of the motion mode; Selecting, by the processor in the digital computer, a quantum algorithm to be implemented for the plurality of ions; Compiling, by the processor in the digital computer, the selected quantum algorithm into a series of universal logic gates; Converting, by the processor in the digital computer, the series of universal logic gates into a series of pairwise entanglement gate operations and applying them to the plurality of ions in the ion chain. The step of calculating the amplitude and detuning frequency of a laser pulse by the processor in the digital computer, and generating the series of paired entanglement gate operations based on the calculated binding strength between the motion mode and the ion; The step of applying the laser pulse having the calculated amplitude and detuning frequency by a system controller to the plurality of ions in the ion chain; The step of measuring, by the system controller, the population of the qubit states of the plurality of ions in the ion chain; The step of processing, by the processor in the digital computer, the quantum information corresponding to the qubit states of the plurality of ions in the ion chain based on the measured population of the qubit states; The step of generating and outputting, by the processor in the digital computer, a solution to the selected quantum algorithm based on the processing result of the quantum calculation; A method comprising the above steps. [

17. ] A quantum computing system, comprising: An ion chain including a plurality of ions, wherein each ion in the ion chain has two hyperfine states defining a qubit; A system controller; A classical computer including a processor and a non-volatile memory storing a number of instructions; When the instructions are executed by the processor, the quantum computing system is caused to: The step of performing, by the system controller, a first measurement of the bright state population of each ion in the ion chain with a fixed duration, wherein each ion is coupled to one of the motion modes of the ion chain, and during which the laser coupling frequency for coupling each ion to the one of the motion modes varies; The step of calculating, by the processor, the mode frequency of the one of the motion modes based on the frequency at which the bright state population of each ion measured in the first measurement is maximized; A step of performing a second measurement of the bright state population of each ion in the ion chain by the system controller, wherein each ion is coupled to one of the motion modes, and during that time, the laser coupling frequency for coupling each ion to the one motion mode of the motion modes is fixed. A step of calculating, by the processor, the coupling strength between each ion in the ion chain and one of the motion modes of the ion chain based on the bright state population measured in the first measurement, the bright state population measured in the second measurement, the calculated mode frequency of the one motion mode of the motion modes, and the non-zero temperature effect of the motion mode. A quantum computing system that executes an operation including the above.

18. The quantum computing system according to claim 17, wherein the second measurement is performed for a fixed duration.

19. The quantum computing system according to claim 17, wherein the second measurement is performed for a plurality of durations.

20. The operation includes A step of selecting a quantum algorithm to be implemented for the plurality of ions by a processor in a digital computer. A step of compiling the selected quantum algorithm into a series of universal logic gates by the processor in the digital computer. A step of converting the series of universal logic gates into a series of pairwise entanglement gate operations by the processor in the digital computer and applying them to the plurality of ions in the ion chain. A step of calculating the amplitude and detuning frequency of a laser pulse by the processor in the digital computer to generate the series of pairwise entanglement gate operations based on the calculated coupling strength between the motion mode and the ion. A step of applying the laser pulse having the calculated amplitude and detuning frequency to the plurality of ions in the ion chain by a system controller. A step of measuring the population of the qubit states of the plurality of ions in the ion chain by the system controller. processing, by the processor in the digital computer, quantum information corresponding to the qubit states of the plurality of ions in the ion chain based on the measured population of the qubit states; generating and outputting, by the processor in the digital computer, a solution to the selected quantum algorithm based on the processing result of the quantum calculation; The quantum computing system according to claim 17, further comprising: **Claim 21** The operation includes: initializing each ion in the ion chain to the hyperfine ground state of each ion before the first measurement and the second measurement of each ion; The quantum computing system according to claim 17, further comprising: **Claim 22** The calculation of the coupling strength between the ions in the ion chain and the motion mode of the ion chain is further based on at least one of the Debye-Waller effect of the ions and the cross-mode coupling effect of the motion mode of the ion chain. The quantum computing system according to claim 17.

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