Efficient motion pattern characterization for high fidelity confined ion quantum computation

Abstract

A method of using an ion trap quantum computer includes making a first measurement of the bright state population of each ion in an ion chain, each ion coupled to one of the motion patterns of the ion chain while varying a laser coupling frequency, calculating a pattern frequency of the one of the motion patterns based on the bright state population measured in the first measurement, and calculating a pattern frequency of the one of the motion patterns based on the calculated pattern frequency. Measuring the bright state population of each ion in the ion chain for the second time; and calculating a coupling strength of each ion with one of the motion patterns 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 mode frequency of the one of the motion patterns calculated and the non-zero temperature effect of the motion patterns.

Description

[0001] CROSS-REFERENCE TO RELATED APPLICATIONS

[0002] This application claims priority to U.S. Patent Application No. 18 / 202270 filed on May 25, 2023 and U.S. Provisional Application Serial No. 63 / 348,421 filed on June 2, 2022, both of which are incorporated herein by reference in their entirety. Technical Field

[0003] The present disclosure generally relates to a method of performing entanglement gate operations in a trapped ion-based quantum computer, and more particularly to a method of characterizing the motion patterns of an ion chain. Background Art

[0004] In quantum computing, the requirements for scalability include efficient characterization, calibration, and verification of quantum computing systems, as well as high-fidelity initialization, logic operations, and readout of such quantum computing systems. In trapped ion-based quantum computers, quantum information encoded in bulk trapped ions (qubits) is processed through the motion patterns of the ions (e.g., collective vibrations), so efficient and accurate motion pattern characterization can enable efficient and accurate quantum computing.

[0005] Therefore, there is a need for methods and systems that can efficiently and accurately characterize the motion patterns of ion chains. Summary of the invention

[0006] An embodiment of the present disclosure provides a method for using an ion trap quantum computer. The method includes measuring the bright state population of each ion in an ion chain containing multiple ions for a first time within a fixed duration, wherein each ion is coupled to one of the motion modes of the ion chain, while changing the laser coupling frequency used to couple each ion to one of the motion modes; calculating the mode frequency of one of the motion modes based on the frequency that maximizes the bright state population of each ion measured in the first measurement; fitting the maximum bright state population of each ion measured in the first measurement to the bright state population calculated based on the calculated mode frequency of one of the motion modes and the non-zero temperature effect of the motion mode; , to calculate the coupling strength between each ion and one of the motion modes; performing a second measurement of the bright state population of each ion in the ion chain within a fixed duration, each ion being coupled to one of the motion modes to which each ion was not coupled in the first measurement, while fixing the laser coupling frequency used to couple each ion to one of the motion modes; and calculating the coupling strength between each ion and one 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 one of the motion modes and the non-zero temperature effect of the motion mode.

[0007] The embodiment of the present disclosure also provides a method for 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 containing multiple ions within a fixed duration, wherein each ion is coupled to one of the motion modes of the ion chain, while changing the laser coupling frequency used to couple each ion with one of the motion modes; calculating the mode frequency of one of the motion modes based on the frequency that maximizes the bright state population of each ion measured in the first measurement; performing a second measurement of the bright state population of each ion in the ion chain within multiple durations, wherein each ion is coupled to one of the motion modes, while fixing the laser coupling frequency used to couple each ion with one of the motion modes; and calculating the coupling strength of each ion with one 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 one of the motion modes and the non-zero temperature effect of the motion mode.

[0008] The embodiment of the present disclosure also provides a quantum computing system. The quantum computing system includes an ion chain including a plurality of ions, each ion in the ion chain having two hyperfine states defining a quantum bit; a system controller; and a classical computer, including a processor and a non-volatile memory storing a plurality of instructions, which, when executed by the processor, cause the quantum computing system to perform operations including: the system controller performs a first measurement of the bright state population of each ion in the ion chain within a fixed duration, each ion being coupled to one of the motion modes of the ion chain, while changing the laser coupling frequency used to couple each ion to one of the motion modes; the processor performs a first measurement based on the first measurement. The mode frequency of one of the motion modes is calculated based on the frequency at which the bright state population of each ion measured in the first measurement is maximized; the system controller performs a second measurement on the bright state population of each ion in the ion chain, each ion is coupled to one of the motion modes, and the laser coupling frequency used to couple each ion to one of the motion modes is fixed; and the processor calculates 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 one of the motion modes and the non-zero temperature effect of the motion mode. BRIEF DESCRIPTION OF THE DRAWINGS

[0009] In order to understand the above features of the present disclosure in detail, the present disclosure may be described in more detail with reference to embodiments, some of which are shown in the accompanying drawings. However, it should be noted that the accompanying drawings only illustrate typical embodiments of the present disclosure and are therefore not to be considered as limiting the scope thereof, as the present disclosure may allow for other equally effective embodiments.

[0010] Figure 1 is a schematic partial view of a trapped ion quantum computing system according to one embodiment.

[0011] Figure 2 A flow chart depicting a basic method of characterizing Lamb-Dick parameters according to one embodiment.

[0012] Figure 3 A flow chart of an improved method for characterizing Lamb-Dick parameters according to one embodiment is depicted.

[0013] Figure 4 An example of bright state populations at different evolution times according to one embodiment is shown.

[0014] Figure 5A and Figure 5B An example of the time evolution of the average bright state population is shown according to one embodiment.

[0015] Fig. 6A and Figure 6B An example of the mean relative error in estimating the Lamb-Dick parameter according to one embodiment is shown.

[0016] Figure 7 An example of the predicted time evolution of the average bright state population is shown according to one embodiment.

[0017] Fig. 8A shows an S according to an embodiment (0) and M t S t Example of the average relative uncertainty for various values ​​of .

[0018] Figure 8B shows an estimation of n according to one embodiment j,k Example of mean relative error in .

[0019] Figure 8C An example of measuring time according to one embodiment is shown.

[0020] For ease of understanding, the same reference numerals are used to indicate the same elements shared by the accompanying drawings, where possible. In the accompanying drawings and the following description, an orthogonal coordinate system including an X-axis, a Y-axis, and a Z-axis is used. For convenience, it is assumed that the direction indicated by the arrow in the figure is the positive direction. It is conceivable that the elements disclosed in some embodiments can be beneficially used in other embodiments without specific description. DETAILED DESCRIPTION

[0021] As the size of quantum computers increases, it is necessary to efficiently characterize the parameters related to quantum computer systems with high precision to achieve high-fidelity quantum logic. In quantum computers based on trapped ions, the entanglement strength between qubits (trapped ions) is mediated by the motion mode of the ion chain, so it becomes crucial to characterize the coupling strength between each ion and the motion mode (called the Lamb-Dick parameter). The embodiments described herein provide a physical model that accurately predicts the amplitude and sign of the Lamb-Dick parameters when the motion modes are detected in parallel. The embodiments described herein also provide an improved characterization method that shortens the characterization time by more than an order of magnitude compared to conventional methods.

[0022] The overall system capable of performing quantum computing using trapped ions will include a classical (digital) computer, a system controller, and a quantum processor. The classical computer performs support and system control tasks, including selecting the quantum algorithm to be implemented on the quantum using a user interface (such as a graphics processing unit (GPU)), compiling the selected quantum algorithm into a series of general logic gates, converting the series of general logic gates into a series of pairwise entangled gate operations to be applied to the quantum processor, and using a central processing unit (CPU) to calculate the amplitude and tuning frequency of the laser pulses that cause the series of pairwise entangled gate operations. The software program for performing the tasks 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 lasers that manipulate the internal hyperfine states (qubit states) of the trapped ions and acousto-optic modulators that read out the internal hyperfine states (qubit states) of the trapped ions. The system controller receives the calculated amplitude and detuning frequency of the laser pulse from the classical computer at the beginning of running the selected algorithm on the quantum processor, controls various hardware related to controlling any and all aspects of running the selected algorithm in the quantum processor, and returns the reading of the quantum processor (e.g., the population of the quantum bit state of the trapped ion), thereby outputting the result of the quantum calculation to the classical computer at the end of running the algorithm to generate and output the solution of the selected quantum algorithm based on the processing result of the quantum calculation.

[0023] I. General Hardware Configuration

[0024] Figure 1 1 is a schematic partial view of a trapped ion quantum computing system 100 or simply system 100 according to one embodiment. System 100 may represent a hybrid quantum classical computing system. System 100 includes a classical (digital) computer 102 and a system controller 104. Figure 1Other components of the system 100 shown are associated with a quantum processor, including an atomic ion chain 106 (i.e., five atomic ions are shown as circles equidistant from each other) that are trapped and form a linear Coulomb crystal extending along the Z axis. Each ion in the ion chain 106 is an ion having a nuclear spin I and an electronic spin S, such that the difference between the nuclear spin I and the electronic spin S is zero, such as a positive ytterbium ion 171 Yb + , positive barium ion 133 Ba + , positive cadmium ion 111 Cd + or 113 Cd + , they all have nuclear spin and 2 S 1 / 2 In some embodiments, all ions in the ion chain 106 are of the same species and isotope (e.g., 171 Yb + In some other embodiments, the ion chain 106 includes one or more species or isotopes (eg, some ions are 171 Yb + , some other ions are 133 Ba + ). In other embodiments, 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 with separate laser beams. The classical computer 102 includes a central processing unit (CPU), a memory, and support circuits (or I / O) (not shown). The memory is connected to the CPU and can be one or more readily available memories, such as a read-only memory (ROM), a random access memory (RAM), a floppy disk, a hard disk, or any other form of local or remote digital memory. Software instructions, algorithms, and data can be encoded and stored in the memory for instructing the CPU. Support circuits (not shown) are also connected to the CPU for supporting the processor in a conventional manner. The support circuits may include conventional caches, power supplies, clock circuits, input / output circuits, subsystems, etc.

[0025] An imaging objective 108, such as one with a numerical aperture (NA) of 0.37, collects the fluorescence of the ions along the Y axis and maps each ion onto a multi-channel photomultiplier tube (PMT) 110 (or some other imaging device) to measure each ion. A Raman laser beam from a laser 112 arranged along the X axis operates on the ions. A diffraction beam splitter 114 produces an array of Raman laser beams 116 that are individually switched using a multi-channel acousto-optic modulator (AOM) 118. The AOM 118 is configured to selectively act on each ion by individually controlling the emission of the Raman laser beam 116. The global Raman laser beam 120 is not confocal with the Raman laser beam 116, and irradiates all ions at once from different directions. In some embodiments, individual Raman laser beams (not shown) may be used, rather than a single global Raman laser beam 120, to irradiate each ion. The system controller (also referred to as "RF controller") 104 controls the AOM 118, thereby controlling the intensity, timing and phase of the laser pulses to be 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 a working memory for various programs and data. The storage unit 128 includes a non-volatile memory, such as a hard disk drive (HDD) or flash memory, which can store various programs even when the power is off. The CPU 122, ROM 124, RAM 126 and storage unit 128 are interconnected by a bus 130. The system controller 104 executes the control program stored in the ROM 124 or the storage unit 128, and uses the RAM 126 as a work area. The control program will include a software application that includes 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.

[0026] II. Trapped Ion Quantum Computing System

[0027] In a trapped ion quantum computing system, such as system 100, two internal states of the atomic ion, such as 2 S 1 / 2 Hyperfine states, which are often used as computational qubit states, are represented by |0> and |1>. The hyperfine ground state (i.e., 2 S 1 / 2The qubit state |0> is represented by the term "low energy state in the hyperfine state" (low energy state in the hyperfine state). In the following, the terms "internal state", "hyperfine state" and "qubit state" are used interchangeably to represent |0> and |1>. In addition, the hyperfine states |0> and |1> can be referred to as "dark state" and "bright state", respectively. 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 a motion ground state without phonon excitation close to any motion mode, and then the qubit state is prepared in the dark state |0> by optical pumping. When many ions are trapped and form a linear Coulomb crystal in the 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 oscillators. The internal and external degrees of freedom of an ion chain consisting of N ions (e.g., the qubit state of a single ion and the collective motion of the ions) can be expressed by the Hamiltonian To describe

[0028]

[0029] in is the carrier frequency (also called “qubit frequency”) of ion j, corresponding to the frequency difference between the two qubit states of ion j, ω k is the mode frequency of motion mode k, is the Pauli-z operator in the qubit space of ion j, and is the annihilation and creation operator of motion mode k.

[0030] Typical laser-induced multi-qubit gate operations between ions, such as The method uses a laser electric field to couple the internal and external degrees of freedom of the ions participating in the ion chain. In the rotating coordinate system, the frequency The interaction Hamiltonian for the classical oscillating electric field at the laser coupling frequency (called the “laser coupling frequency”) can be written as

[0031]

[0032] in is the lifting operator of ion j, Ω j is the qubit Rabi frequency (i.e., the coupling between the two qubits of ion j), φ j is the laser phase, η j,k is the Lamb-Dick parameter that quantifies the coupling strength between ion j and motion mode k. For simplicity, the laser phase can be chosen to be φ j= 0. Typically, N' motion modes out of a total of 3N motion modes (N'<3N) are strongly coupled to the laser, while the contribution of the remaining motion modes to the multi-qubit gate operation is negligible. In order for the multi-qubit gate operation to work faithfully and efficiently, the Lamb-Dick parameters η of these N' motion modes need to be known with high precision j,k and mode frequency ω k .

[0033] II.A Characterization of Lamb-Dick Parameters

[0034] Characterizing the Lamb-Dick parameter η j,k and mode frequency ω k The conventional method for determining these parameters is to use the sideband spectra of the blue sideband (BSB) transitions. To characterize the Lamb-Dick parameter η of mode k of motion on ion j, j,k and the mode frequency ω of motion mode k k , applying laser pulses of fixed duration, while the laser coupling frequency of each pulse The BSB resonant frequency Before applying the laser pulse, initialization of ion j is performed (e.g., laser cooling, such as Doppler cooling or resolved sideband cooling, and a state preparation process, such as optical pumping, to prepare ion j in the dark state |0>). For each scanned laser coupling frequency BSB transition coupling near resonance |0,n> j,k and |1,n+1> j,k , where |a,b> j,k represents the composite state of the qubit state |a> (a∈{0,1}) of ion j and the motional Fock state |b> of motion mode k with phonon number b. At the end of each pulse, the bright state population (i.e., the probability that ion j is in the bright state |1>) is measured, which unusually depends on the Lamb-Dick parameter η j,k and mode frequency ω k The value of .

[0035] Similar to other spectroscopy methods, conventional mode characterization methods are designed to detect the mode frequency ω k When a more accurate and efficient characterization of the Lamb-Dick parameter η is required j,k The embodiments described herein provide improvements over conventional mode characterization methods, particularly because of the need to characterize the Lamb-Dick parameter η j,k N×N' different values ​​of .

[0036] To extract the Lamb-Dick parameter η j,k , the measured data of the bright state population of ion j are fitted to a model (called the “baseline model”, with the superscript “ (0)”), which is commonly used in the case of spanning two composite states |0,0> j,k and |1,1> j,k The approximate interaction Hamiltonian in the subspace of

[0037]

[0038] The time evolution operator can be written as

[0039]

[0040] where * denotes complex conjugation, and

[0041]

[0042] Here, t is the evolution time, are two composite states |0,0> j,k and |1,1> j,k The effective Rabi frequency between is the detuning of the laser coupling frequency starting from the BSB transition as well as Combining (4) and (5), the resulting time evolution operator is then applied to the initial composite state |0,0> j,k , the bright state population is

[0043]

[0044] It is used to fit the measured data of the bright state population of ion j and extract the Lamb-Dick parameter η j,k .

[0045] II.B Improvements on the Baseline Model

[0046] The baseline model is approximate for two main reasons: (i) spectator motion modes (i.e., undetected motion modes) are ignored, and (ii) the motion modes are assumed to be always prepared in the motion ground state. To estimate the bright state population more accurately, the contribution of spectator motion modes due to non-zero diffusion of ion position wave packets and non-resonant BSB transitions as well as the effect of non-zero temperature can be considered. It should also be noted that the conventional mode characterization method using (6) does not reveal the Lamb-Dick parameter η j,k The signs of these qubits relative to each other are crucial to the design and operation of multi-qubit gates.

[0047] Methods according to embodiments described herein are provided to improve conventional pattern characterization methods in the following aspects:

[0048] 1. Parallelization - In a chain of N ions with N′ motion modes strongly coupled to the laser, there are N×N′ different Lamb-Dick parameters η j,k. Characterize N×N′ different Lamb-Dick parameters η j,k It takes O(N 2 ) operations. To support large-scale quantum computers, parallelization is necessary to reduce the complexity to O(N).

[0049] 2. Accuracy - To characterize the Lamb-Dick parameter η with high accuracy j,k , it is necessary to consider the coupling effect between ion j and other motion modes k′≠k. The coupling is caused by the non-zero diffusion of the ion position wave packet and the non-resonant BSB transition.

[0050] 3. Sign problem - need to distinguish the Lamb-Dick parameter η j,k The relative sign of , while in (6), the bright state population depends only on the Lamb-Dick parameter η j,k The magnitude of , not its sign.

[0051] 4. Efficiency - incorrect mode frequency ω k And the shot noise will lead to the Lamb-Dick parameter η j,k To reduce the uncertainty, longer measurement times are required.

[0052] These improvements can be achieved through the following goals:

[0053] Objective 1: To find an effective model to better characterize the bright-state population dynamics of ions undergoing BSB transitions.

[0054] Objective 2: Explore the parameters that can distinguish the Lamb-Dick parameter η j,k The methods and corresponding models have positive and negative signs relative to each other.

[0055] Objective 3: Find a more efficient parallel method that achieves a Lamb-Dick parameter η below the target value j,k Allow for minimal measurement time while estimating uncertainties.

[0056] III. Improved Model

[0057] This section discusses various improved models for predicting the bright state population of ions, all of which undergo BSB transitions in parallel. These models are more accurate than the conventionally used baseline model in (6) in predicting the bright state population of ions, and thus characterize the Lamb–Dick parameter η j,k and mode frequency ω k Section III.A discusses three effects occurring in parallel BSB transitions that are not considered in the baseline model. Section III.B introduces a total of five models, which gradually consider the effects discussed in Section III.A and their combinations, culminating in the most complex model.

[0058] III.A Effect

[0059] This section discusses three effects in the parallel BSB transition of ions. Considering these effects in the model can more accurately characterize the Lamb-Dick parameter η j,k .

[0060] (a) Non-zero temperature

[0061] Even with the most sophisticated cooling techniques, it is unlikely that the motion mode is in the absolute ground state. Therefore, the baseline model described in (4)-(6) is generalized to an initial state with an arbitrary number of phonons n. Assume that except for |0,n> j,k and |1,n+1> j,k The composite states other than these two composite states do not affect the BSB transition, so the two composite states |0,n> j,k and |1,n+1> j,k The Rabi frequency between is given by

[0062]

[0063] in is a generalized Laguerre polynomial.

[0064] As described below, this generalized Rabi frequency can be used to evaluate the bright state population of BSB transitions that undergo non-zero temperature. For example, is defined as when the initial composite state is |0,n> j,k The bright state population of ion j at time t is obtained by using the generalized Rabi frequency in (6) Alternative effective Rabi frequency And obtained.

[0065] (b) Debye-Waller (DW) effect

[0066] The propagation of the ion position wave packets associated with each mode manifests itself as a reduction in the Rabi frequency, which is widely known as the DW effect. Even when the motional modes are cooled to their motional ground state, the DW effect due to the zero-point fluctuations remains.

[0067] When the motion mode k is detected by ion j, the DW effect caused by the spectator motion mode k′≠k leads to two composite states |0,n k > j,k and |1,n k +1> j,k The Rabi frequency reduction between

[0068]

[0069] in is the initial phonon number n of motion mode k'(k'∈{1,2,..,N'}) k′ The vector of The initial number of phonons is n k′ The average DW reduction factor of the audience motion pattern k′ when .

[0070] For efficient characterization, each of the N ions is used to probe a given motion mode in parallel, and this is repeated N' times with different permutations of the motion mode to probe the Lamb-Dick parameter η j,k In this case, each spectator motion mode k′ is also probed by another ion j′(k′), so the number of phonons of motion mode k′ is n k′ and n k′ +1. Therefore, the average DW reduction factor becomes

[0071]

[0072] where α, β ≥ 0 (α + β = 1) means that ion j′ (k′) and motion mode k′ are in the composite state | 0, n k′ > j′(k′),k′ and |1,n k′ +1> j′(k′),k′ The probability of

[0073]

[0074] in is a Laguerre polynomial.

[0075] When the motion mode k′ is resonantly probed for a sufficiently long evolution time, the number of phonons in the motion mode k′ can be approximated to n in half the time. k′ and the other half of the time k′ +1. An exception is when ion j′(k′) is located at a node of motion mode k′ (η j′(k′),k′ ≈0), and the BSB transition of ion j′(k′) with respect to motion mode k′ is not expected to occur. Then, in (9), the approximation can be applied

[0076]

[0077] , where ∈ η is a discriminator used to determine whether j′(k′) is located at a node of motion pattern k′, usually chosen to be a small number (≈10 -4 ).

[0078] Using (8)-(11), equation (6) can be further generalized by using the reduced Rabi frequency Alternative effective Rabi frequency A non-zero initial number of phonons is allowed for all modes of motion. is the bright state population of ion j undergoing a parallel BSB transition, where initially all ions are in the dark state |0> and the number of phonons in motion mode k' is The k'th element n k′ .

[0079] (c) Cross-mode coupling

[0080] When ion j probes motion mode k, non-resonant BSB transitions with other motion modes k'≠k also occur. The effect of other motion modes on the qubit state is called cross-mode coupling. Although it can be achieved by using a detuning frequency much smaller than Δ j,k′ Rabi frequency Ω j,k To reduce cross-mode coupling, a smaller Rabi frequency results in a slower BSB transition. Therefore, there is a trade-off between reducing the error due to cross-mode coupling and performing shorter characterization measurements.

[0081] Cross-mode coupling can in principle be included in a model that simulates the time evolution of the entire Hamiltonian for N ions and N′ modes of motion. However, the simulation time grows exponentially with the number of ions N. Therefore, a more realistic approach is to include only nearest-neighbor modes of motion and the ions that probe them in the simulation, limiting the size of the simulated system to a maximum of three ions and three modes of motion.

[0082] III.B Five Improved Models

[0083] In the following, five models for the bright state population of ions undergoing parallel BSB transitions are discussed. These five models are improved from the baseline model in (6).

[0084] (a) Model 1: Debye-Waller (DW) effect

[0085] Model 1 takes into account the DW effect while still assuming that the temperature is zero. When the initial state is |0,0> j,k When the average bright state population Given by the following formula

[0086]

[0087] Here is the bright state From (6), we get: The Rabi frequency reduced by (8) Alternative.

[0088] It should be noted that the reduced Rabi frequency Not only depends on the Lamb-Dick parameter η j,k, but also depends on other Lamb-Dick parameters η j,k′ Model 1 is an improvement over the baseline model because it emphasizes the impact of other motion modes k′≠k on the bright state population of the ion probing motion mode k, while taking into account that all motion modes are probed in parallel.

[0089] (b) Model 2: non-zero temperature

[0090] In addition to the DW effect considered in Model 1, Model 2 also takes into account the non-zero temperature effect. Allowing multiple different initial phonon numbers, among which is the average number of phonons at non-zero temperature, and the average bright state population Can be

[0091]

[0092] in The bright state population is found in (6) The effective Rabi frequency By reducing the Rabi frequency For simplicity, the thermal distribution of all motion modes is assumed to have the same average phonon number It is just that the generalization to arbitrary distributions is direct.

[0093] The summands in (13) are The sum is calculated, that is, for some threshold probability p th ,satisfy In the example described in this paper, for the number of ions n≤7, the threshold probability is set to p th =10 -4 . For bright state population Each evaluation of is parallelizable, so the regular computation time does not necessarily increase with In other examples, From the distribution Random sampling in Because all the things to consider becomes very large. In this case, the accuracy of the distribution is determined by the threshold probability p th The sampling accuracy is determined by the number of samples extracted.

[0094] (c) Model 3: Time-varying DW (TDDW) effect

[0095] Model 3 further takes into account the time dependence of the DW reduction factor. This is because for each motion mode k detected by ion j, the spectator motion mode k′≠k is also detected by another ion j′(k′)≠j, and its phonon number changes with time n at the time of detection. k′ and n k′ +1. The TDDW reduction factor is given by

[0096]

[0097] in and The ion j′(k′) and the motion mode k′ are in the composite state |0, n respectively at time t. k′ > j′(k′),k′ and |1,n k′ +1> j′(k′),k′ Here, It can be calculated using (6), where the effective Rabi frequency is The reduced Rabi frequency given by (7) Alternative.

[0098] Now, to evaluate the bright state population considering the TDDW effect This time-varying DW reduction factor replaces the average DW reduction factor in (8), which makes the reduced Rabi frequency Therefore, the time evolution from 0 to t can be divided into short time steps and the time evolution operators in (4) and (5) are applied, while updating the reduced Rabi frequency at each time step To solve the bright state population As shown in (13), the bright state population and the number of phonons The weighted average of provides the average bright state population

[0099] (d) Model 4: Nearest neighbor (NN).

[0100] Model 4 considers the NN motion patterns of the detected motion patterns and their assigned ions. In other words, the subspace of the detected motion pattern k, its NN motion patterns k-1 and k+1 (where the motion patterns are ordered with increasing pattern frequency) and their assigned ions j(k), j(k-1) and j(k+1) (two ions, two motion patterns of k=1 and N′) is considered.

[0101] The interaction Hamiltonian describing the subspace is

[0102]

[0103] Where J = {j(k-1), j(k), j(k+1)}, K = {k-1, k, k+1}. The initial composite state is By taking the matrix elements corresponding to the resonant transitions and evaluating the time evolution operator of this Hamiltonian from time 0 to t, we obtain the average bright state population As shown in (13).

[0104] In (15), evaluating the time evolution operator for the three-ion, three-mode Hamiltonian takes longer than simply evaluating trigonometric functions and polynomials as in previous models. However, this model includes the NN motion modes, so its accuracy is less affected by cross-mode coupling. It should be noted that it correctly captures the quantum interference between qubit states and motion modes beyond the single-ion single-mode model. The predicted bright state population has an influence on the Lamb-Dick parameter η j,k Relative to the Lamb-Dick parameter η j,k±1 The sign of is sensitive.

[0105] (e) Model 5: TDDW+NN.

[0106] Model 5 introduces the TDDW effect discussed in (c) Model 3 into the NN model in (d) Model 4. This is achieved by replacing the average DW reduction factor in (15) with the TDDW factor in (14).

[0107] IV. Methods

[0108] In this section, the Lamb-Dick parameter η is characterized according to the embodiments described herein. j,k and mode frequency ω k There are two methods, namely the “basic method” and the “improved method”. For different methods, the sensitivity of the measured bright state population of N ion qubits undergoing BSB transitions to the Lamb-Dick parameter may be different. Since the Lamb-Dick parameter η j,k There are N × N' different values, so parallelization of the measurements becomes necessary. The basic method discussed below is a modified version of the conventional mode characterization method for parallel detection of the Lamb-Dick parameter η j,k The improved method can determine the Lamb-Dick parameter η more accurately and quickly. j,k .

[0109] Since the method described in this paper is to characterize the Lamb-Dick parameter η with high accuracy j,k The methods are designed for this purpose, so a rough estimate of the Lamb-Dick parameter is assumed before performing these methods. j,k Estimated values ​​and mode frequencies ω within a few kHz k An estimate may be sufficient.

[0110] IV.A Basic Methods

[0111] Figure 2 A flow chart is depicted illustrating the characterization of the Lamb-Dick parameter η j,k The basic method 200 of this paper is to quantify the coupling strength between ion j and motion mode k. Here, the N ions in the ion chain are labeled j and the N' motion modes of the ion chain that are strongly coupled to the laser are labeled k. Therefore, the Lamb-Dick parameter η to be determined is j,k The number is N×N′.

[0112] The basic method 200 includes two steps. The first step in block 210 includes measuring the mode frequencies ω of all N' motion modes by using a frequency scanning measurement of N ions. k , and simultaneously determine N×N' Lamb-Dick parameters In the typical case, the number of ions N is equal to the number of motion patterns N′, so all N′ motion patterns are assigned an ion to be detected. If the number of motion patterns N′ is greater than the number of ions N, repeat the first step rounds, so that all N′ motion modes are assigned an ion to be detected in at least one round. , together determine the mode frequency ω of motion mode k k and sports mode k and ion j i Lamb-Dick parameter of (k) Here, Represents the smallest integer greater than or equal to the independent variable, j i (k) represents the ions used to detect motion mode k in the i-th round.

[0113] The second step in block 220 involves determining the remaining (N-1)×N′ Lamb-Dick parameters Repeat the second step wheel.

[0114] Specifically, the i-th round of block 210 begins with sub-block 212, where each ion j (=1, 2, ..., N) is assigned to one of the probe motion modes k (=1, 2, ..., N'), including motion modes that have not been detected in previous rounds. The ions assigned to the detection motion mode k in the i-th round are denoted as j i (k) Among the N' motion modes that are strongly coupled to the laser, in sub-block 212, N ions are allocated to detect the N motion modes, and no ions are allocated to detect the (N'-N) motion modes.

[0115] The i-th round of block 210 continues to sub-block 214, where 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 ion j i (k) is the hyperfine ground state. Each ion j i (k) can be initialized (i.e., cooled to reduce the kinetic energy of the ion) by known laser cooling methods such as Doppler cooling or resolved sideband cooling, bringing it close to the ground state of motion without any phonon-excited motion modes, and then optically pumping the qubit state prepared in the hyperfine ground state |0〉.

[0116] The i-th round of block 210 continues to sub-block 216, where at a fixed time τ (0) Perform the blue sideband (BSB) transition for each ion j i (k) The bright state population P j,k (t) frequency scanning measurement. Each ion j i (k)(=1,2,…,N) is excited by the laser pulse, and the laser coupling frequency At the expected BSB resonant frequency The frequency of each ion j is measured. i (k) The bright state population P j,k (t), where is ion j i (k) is the qubit frequency, ω k is the estimated value of the pattern frequency of motion pattern k. Since at a fixed time τ (0) The bright state population P j,k (t) at BSB transition frequency Detuned frequency The maximum value is reached when it is zero Therefore the mode frequency ω k is measured to make the bright state population P j,k (t) minus ion j i (k) The qubit frequency (Right now ) maximizes the laser coupling frequency

[0117] The i-th round of block 210 continues to sub-block 218, where for each ion j i Lamb-Dick parameter of (k) and the assigned motion mode k. Lamb-Dick parameter The maximum bright state population measured in sub-block 216 can be Fit to the average bright state population derived using any of the above models 1-5 to calculate.

[0118] In each round In the example, sub-blocks 212-218 are executed in parallel for N ions, and the Rounds are repeated until all N′ motion patterns are detected.

[0119] It should be noted that in order to accurately measure the mode frequency ω k , the mode allocation in sub-block 212 Fixed time τ (0) and the qubit state Rabi frequency Ω j All need to be carefully chosen so that the bright state population at zero detuning frequency Big enough.

[0120] The i-th round of block 220 begins with sub-block 222, where each ion j (=1, 2, ..., N) is assigned to one of the motion modes k. In sub-block 222, the ions are assigned to different motion mode arrangements (e.g., different combinations of ions j and motion mode k). The ions assigned to motion mode k in the i-th round are denoted as j i (k).

[0121] The i-th round of block 220 continues to sub-block 224, where each ion j i (k)(=1, 2, . . . , N) is initialized to the dark state |0>. This initialization of ions is the same as sub-block 214.

[0122] The i-th round of block 220 continues to sub-block 226, where at a fixed time τ (0) Measure the blue sideband (BSB) transition of each ion j i (k) The bright state population P j,k (t). No frequency scan measurement is performed in sub-block 226. Each ion j i (k)(=1,2,…,N) is excited by the laser pulse, and the laser coupling frequency Pin to The mode frequency ω k Known from the first step in block 210 .

[0123] The i-th round of block 220 continues to sub-block 228, where for each ion j i Lamb-Dick parameter of (k) and the assigned motion mode k. Lamb-Dick parameter The maximum bright state population measured in sub-block 216 can be Fit to the average bright state population derived using any of the above models 1-5 to calculate.

[0124] In each round In the example, sub-blocks 222-228 are executed in parallel for N ions, thereby determining N′×(N-1) remaining Lambdic parameters In order to determine the sub-blocks 222-228, repeat Rounds are repeated until all remaining N′×(N-1) Lamb-Dick parameters are determined.

[0125] IV.B Improved Methods

[0126] Figure 3 A flow chart is shown to illustrate the characterization of the Lamb-Dick parameter η j,k The improved method 300 quantifies the coupling strength between ion j and motion mode k. Here, the N ions in the ion chain are also labeled j, and the N′ motion modes of the ion chain that are strongly coupled to the laser are labeled k. Therefore, the Lamb-Dick parameter η to be determined is j,k It is N×N′.

[0127] The improved method 300 further comprises two steps. The first step in block 310 is a frequency scanning measurement to calculate the mode frequencies ω of all N′ motion modes using N ions. k , as in the first step of block 210 of the basic method 200. However, in the first step of block 310, the Lamb-Dick parameter η is not calculated. j,k Repeat step 1 rounds, so that all N′ motion modes are assigned an ion for detection in at least one round.

[0128] The second step in block 320 is to populate the bright state P j,k (t) to calculate the Lamb-Dick parameter η j,k The second step is repeated for N′ rounds.

[0129] Specifically, the i-th round of block 310 Starting with sub-block 312, each ion j (=1, 2, ..., N) is assigned to one of the detected motion modes k (=1, 2, ..., N'), including motion modes not detected in previous rounds. Sub-block 312 is the same as sub-block 212 of basic method 200.

[0130] The i-th round of block 310 continues to sub-block 314, 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 sub-block 214 of the basic method 200.

[0131] The i-th round of block 310 continues to sub-block 316, where at a fixed time τ (0) Perform the blue sideband (BSB) transition for each ion j i(k) The bright state population P j,k (t) frequency scanning measurement. Each ion j i (k)(=1,2,…,N) is excited by the laser pulse, and the laser coupling frequency At the expected BSB resonant frequency The frequency of each ion j is measured. i (k) The bright state population P j,k (t), where is ion j i (k) is the qubit frequency, ω k is the estimated value of the pattern frequency of motion pattern k. Since at a fixed time τ (0) The bright state population P j,k (t) at BSB transition frequency Detuned frequency The maximum value is zero. Therefore the mode frequency ω k is measured to make the bright state population P j,k (t) minus ion j i (k) The qubit frequency (Right now ) maximizes the laser coupling frequency Sub-block 316 is identical to sub-block 216 of basic method 200 .

[0132] In each round In the example, sub-blocks 312-316 are executed in parallel for N ions, and the Rounds are repeated until all N′ motion patterns are detected.

[0133] The i-th round of block 320 begins with sub-block 322, where ions j (=1, 2, ..., N) are each assigned to one of motion modes k (=1, 2, ..., N'). In sub-block 322, ions are assigned to different permutations of motion modes (e.g., different combinations of ions j and motion mode k). The ions assigned to detect motion mode k in the i-th round are denoted as j i (k).

[0134] The i-th round of block 320 continues to sub-block 324, 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 sub-block 224 of the basic method 200.

[0135] The i-th round of block 320 continues to sub-block 326, in which the laser coupling frequency is fixed. For each ion j in the blue sideband (BSB) transition i (k) The bright state population P j,k(t) Perform time scan measurement. Each ion j i (k)(=1,2,…,N) is excited by the laser pulse, and the laser coupling frequency Fixed (Therefore at a fixed detuned frequency Δ j,k ), and at different evolution times Measure each ion j i (k) The bright state population P j,k (t), where is ion j i (k) is the qubit frequency, mode frequency ω k Known from the first step in block 310 .

[0136] The i-th round of block 320 continues to sub-block 328, where each ion j is calculated i Lamb-Dick parameter of (k) and the assigned motion mode k. Lamb-Dick parameter The bright state population P can be measured by j,k (t) is fitted to the average bright state population derived using any of the above models 1-5 to calculate.

[0137] Figure 4 It shows that the perfect resonance (Δ j,k = 0) The bright state population of BSB transition j,k In this example, the number of ions N is set equal to the number of motion modes that are strongly coupled to the laser (N = N′ = 5), and the qubit state Rabi frequency is chosen to be Each mode k is detected by ion j(k) = k. t = 20 with equidistant time stamps (recording the bright state population P j,k (t) evolution time) records the bright state population P j,k (t), and then fit to determine η j,k The error bars show the shot noise for 1000 shots.

[0138] In each round (i=1, ..N′), sub-blocks 322-328 are executed in parallel for N ions, and N′ rounds are repeated to exclusively pair the N ions with the N′ motion modes and determine all N′×N Lamb-Dick parameters

[0139] IV.C Measurement time scale

[0140] In the above method, the trapped ion quantum computer undergoes a cycle of ion cooling, qubit state preparation, BSB transition and ion bright state population measurement. The time scales of cooling, state preparation and measurement can be on the order of 10ms, 10μs and 100μs respectively. BSB migration requires milliseconds because the qubit state Rabi frequency needs to be small enough to suppress cross-mode coupling.

[0141] In the case where the number of ions N is equal to the number of motion modes N' that are strongly coupled to the laser (N'=N), which corresponds to a common laser alignment setting, the Lamb-Dick parameter η is characterized according to the basic method 200 j,k and mode frequency ω k The total time required T (0) Then for

[0142]

[0143] in is the number of detuned frequencies considered in the first step in block 210, S (0) is the number of scatters per data point, is the BSB transition time τ (0) The cycle time is (0) These values ​​are indicated for the basic method 200. Characterization of the Lamb-Dick parameter η according to the improved method j,k and mode frequency ω k The total time T required is

[0144]

[0145] Among them, M Δ (M t ) is the number of detuned frequencies (time stamps) in the frequency (time) sweep, s Δ (S t ) is the number of shots per frequency (time) scan, and Includes BSB transition time The number of particles measured in each frequency (time) scan.

[0146] The lower limits of the above parameters are given by the Lamb-Dick parameter η j,k The target accuracy of the measurement determines the minimum required (M Δ ) and τ (0) (τ Δ ) is the mode frequency ω k The upper limit of the uncertainty in the medium is determined by the Lamb-Dick parameter η j,k The target accuracy required.

[0147] In the basic method 200, when the mode frequency ω k When the uncertainty of is large, the Lamb-Dick parameter η j,k The uncertainty will also become larger, because both parameters directly affect the bright state population This can be combined with the improved method 300 j,k -estimation process. Compared with the single-valued bright state population in the baseline case Differently, in the time scan measurement of the improved method 300, the bright state population is measured at different evolution times t. When fitting time series data to the model, the Lamb-Dick parameter η can be estimated in a differentiable way j,k and detuning frequency Δ j,k , that is, the Lamb-Dick parameter η j,k Only affects the bright state population The frequency of oscillation, while the detuned frequency Δ j,k Affects its frequency and amplitude, such as Figure 5A and 5B When estimating the Lamb-Dick parameters with a certain accuracy, this separation of the signals of the different parameters to be estimated allows, for example, the mode frequency ω k Compared with the basic method 200, the Lamb-Dick parameter η j,k The same accuracy is targeted which in turn leads to significantly shorter frequency sweep measurement times.

[0148] Figure 5A and Figure 5B The laser coupling frequencies for the BSB transition are shown The Lamb-Dick parameter η 1,1 and detuning frequency Δ 1,1 The average bright state population in the BSB transitions of different values ​​of Example of the time evolution of . The Rabi frequency Ω1 of the qubit state of ion 1 is chosen to be Ω1 = 2π × 10 kHz. The bold lines are η 1,1 =0.0119×1 and Δ 1,1 =0Hz. 1,1 Only affects the oscillation frequency, while Δ 1,1 This allows for more accurate measurements of η in the presence of uncertainty in the mode frequency. j,k It should be noted that when the average bright state population When the average bright state population is close to 0.5, rather than close to 0 or 1, For η 1,1 The improved method 300 uses the entire average bright state population that always includes points close to 0.5. In contrast, in the basic method 200 of measuring N Lamb-Dick parameters in parallel, it is difficult to find the pulse length τ (0) , so that the average bright state population of all N qubits is Therefore, it is expected that the improved method 300 will result in η j,k The average uncertainty is small.

[0149] Fitting the measured bright state populations to Models 1–5 is a nontrivial task because the average bright state population Not only depends on the Lamb-Dick parameter η j,k , and also depends on other Lamb-Dick parameters of the spectator motion pattern, including the nearest neighbor motion pattern. A simple approach is to fully fit the average bright state population where all N×N′ Lamb-Dick parameters η j′,k′ are all fitted parameters. However, for large N, fitting N × N′ parameters at once requires too long a conventional computation time to be practical. Therefore, the fitting routine may be a multi-iteration process, with η from an initial guess or from previous fitting iterations being used in the model. j′,k′ The fitting routine is highly parallelized, so the runtime of the computational part of the characterization method can be scaled with a large number of ions N.

[0150] It should be noted that although this section discusses more accurate and efficient estimation of the Lamb-Dick parameters, the method described in this paper can also be easily used for better mode frequency estimation. For example, The bright state population of the ions measured under the condition of γ is fitted into models 1-5, which can more accurately estimate the mode frequency ω. k .

[0151] V. Example

[0152] In this section, we demonstrate by example that the three goals of efficient pattern characterization described in Section II can be achieved by the improved model and method described in this paper. More specifically, we show that (i) j,k , (ii) demonstrate that Model 4 can distinguish the relative signs of the Lamb-Dick parameters, and (iii) for a given η j,k To estimate the target accuracy, the improved method 300 requires significantly shorter feature measurement time than the basic method 200 .

[0153] For numerical experiments, a numerical simulation of the parallel BSB transition measurement is performed. The BSB Hamiltonian in the interaction picture is given by

[0154]

[0155] in is found in (2). In the example shown in Section V, the number of motion modes N' is set equal to the number of ions N in the ion chain (N' = N), which is consistent with typical laser alignment. The implicit time evolution operator applies to all initial states where ion j′(k′) is the ion assigned to mode k′, and the phonon number vector satisfy As mentioned above. For all motion modes, the average phonon number is set to And the threshold probability is set to 10 -4 (p th =10 -4 ). At time t, the composite state of the qubit state of ion j(k) and the motion Fock state of motion mode k is projected onto the qubit subspace, and the bright state population of all motion modes k is generated. Finally, the weighted average of the bright state population is calculated as in (13): It was then fitted to the models discussed previously to test the accuracy of each model. Note that the Hilbert space dimension grows exponentially with the number of ions N, so the model was tested with N=7.

[0156] VA Accuracy

[0157] First, the performance of the baseline model and Models 1–5 in adequately capturing the qubit population evolution is compared using numerical simulations of bright state populations. Here, as an example, all ions are assumed to have the same qubit state Rabi frequency. Drive at the same time. t = 20 with equidistant time stamps τ m Record the bright state population. The maximum time stamp is selected as Where <Δk x > is projected onto the mode direction and <ω mode >The wave vector difference between the two Raman laser beams, <ω mode > is a rough estimate of the mode frequency, so the longest BSB transition relative to the center-of-mass mode takes about five Rabi half-cycles for all N and Ω0.

[0158] Fig. 6A and Figure 6B The estimated Lamb-Dick parameter η obtained using various models is shown separately j,kAn example of the average relative error of , where the parameters are fixed to 2π × 2 kHz for the qubit state Rabi frequency Ω0 and the number of ions N with qubit state Rabi frequency Ω0. The labels are arranged in order of magnitude for the baseline and models 1–5 described in Section III. The relative error here is defined as in are the estimated Lamb-Dick parameters from the fit to the model. j,k <10 -4 ) corresponds to the Lamb-Dick parameter. For N≤5, the error is η j,k All N 2 The average value is taken. For N>5, the error is the N values ​​η measured in parallel. j＝k,k In general, Models 1-5 have better estimates of η compared to the baseline model. j,k In particular, only with the improved model can the accuracy of the model with a size less than 10 -3 Considering both the DW effect and the non-zero temperature effect of the audience motion mode significantly reduces the error, especially when Ω0 is small and the cross-mode coupling is not dominant.

[0159] Models 2-5 show power-law behavior, with relative errors and It should be noted that, taking the perturbation mechanism as an example, the Rabi frequency Ω j,k ∝Ω0 is much smaller than the detuning frequency Δ of the undetected motion mode k'≠k of ion j j,k′ The observed power law is reminiscent of the dominance of cross-mode coupling errors in this regime.

[0160] It can be seen that incorporating the NN motion mode into the model can reduce the error of cross-mode coupling. For the case of N < 5, models 4 and 5 have noticeably smaller errors relative to models 2 and 3. However, for longer ion chains, the errors are not much different. For example, at η j,k±1 Less than η j,k±2 In the case of k±2, the mode k±2 has no significant effect on the measurement η j,k The effect of the error can be comparable or greater than the effect of the NN mode k±1. For this case, the NN model can be modified to include the mode with a significant effect, but at the expense of longer fitting calculation time.

[0161] The model that includes the TDDW effect achieves the highest accuracy. Figure 6B In Figure 2, when N = 7, the errors of models 3 and 5 are 2.5 times smaller than those of models 2 and 4. The TDDW effect may be more important for characterizing the Lamb-Dick parameters with higher accuracy in longer ion chains.

[0162] It should be noted that the physical distance between neighboring ions is assumed to be fixed. Therefore, as the number of ions N increases, the spacing between mode frequencies decreases, which leads to more severe cross-mode coupling at fixed qubit states Rabi frequencies.

[0163] VB positive and negative sign problem

[0164] Lamb-Dick parameter η j,k The sign of the Lamb-Dick parameter relative to other Lamb-Dick parameters determines the gate pulse design on many trapped ion quantum computers, thus directly affecting the quantum computation fidelity. Unfortunately, conventional mode characterization methods cannot distinguish the Lamb-Dick parameter η from the j,k The sign of η is determined by the qubit population, since the qubit population is independent of the sign in the baseline model in (6). It is shown here that the Lamb-Dick parameter η can be distinguished using the NN model (Model 4) j,k The positive and negative sign of .

[0165] First, in order to distinguish η using the BSB transition j,k The sign of η is such that multiple ions need to be considered, because the Lamb-Dick parameter η is only used to describe the relative motion between different ions. j,k It should also be noted that for single mode, for different η j,k The ions move in different relative directions, but the qubit population undergoes exactly the same evolution. Only when at least two ions and two modes are considered simultaneously, η j,k The sign of determines whether the symmetry of the two ions participating in one mode is the same or opposite to that of the other mode, and this difference affects the qubit population.

[0166] By irradiating both ions with the same two-tone beam, driving the two ions to couple to two different modes in parallel, where each tone is resonant with the respective mode frequency, BSB causes transitions to both modes to occur simultaneously on both ions. The predicted evolutions, one with the same symmetry and the other with the opposite symmetry, allow the determination of the Lamb-Dick parameter η directly from the resulting signal of the measurement. j,k What kind of symmetry, that is, the positive and negative signs are correct.

[0167] Figure 7 The average bright state population Example of the predicted time evolution of where, for N = 5, the Lamb-Dick parameter η 1,1 = ±0.0119 relative to η 1,2 =0.0335, η 2,1 = -0.0521 and η 2,2 =-0.0705. The first and second ions are coupled to two laser frequencies. The two frequencies resonate with the first and second modes at frequencies ω1 and ω2, respectively. The qubit state Rabi frequencies of the first and second ions are 2π×30kHz and 2π×9kHz, respectively, thus roughly matching |0,0> 1,1 and |1,1> 1,1 Between and |0,0> 1,2 and |1,1> 1,2 The resulting Rabi frequency between .

[0168] It can be seen that when the Lamb-Dick parameter η 1,1 = ±0.0119, the population curves are clearly discernible and accurately predicted by the NN model (Model 4). This shows that the Lamb-Dick parameter η can be reliably distinguished by simultaneously inducing all four possible BSB transitions between the two ions and the two modes when the parameters are carefully selected and the observed evolution is compared with the evolution predicted by the NN model. j,k The positive and negative sign of .

[0169] VC Characterization Measurement Time

[0170] The characterization measurement times of the basic method 200 and the improved method 300 given by (16) and (17), respectively, depend on the following parameters: (i) and M in the improved method 300 Δ , the number of detuned frequencies scanned in the frequency scan, (ii) S in the basic method 200 (0) Whenever applicable, these parameters (i)-(iii) will be minimized while estimating the Lamb-Dick parameter η j,k It should be noted that the target accuracy achieved is mainly affected by shot noise and other parameters (such as ω k ) is hindered by uncertainty.

[0171] In order to be consistent with Section VA, M t Fixed at M t =20, Where N = 5. In addition, in order to compare the total measurement time based on the same footnote, for the improved method, S Δ Set to S Δ =M t S t , and compare this value with the The basic method of S (0) Therefore, the nodes that can be rotated for the basic method 200 are Ω0, S (0) and For the improved method 300, Ω0, S t 、MΔ and τ Δ .

[0172] First, the calculation needs to reach the Lamb-Dick parameter η j,k Basic method with small uncertainty of 200 shot number S (0) and the particle number S of the improved method 300 t Here, the simulated bright state population is fitted using Model 2, where the mode frequency ω is assumed to be k Perfect knowledge of Ω0 gives the uncertainty due to the combination of photon and phonon shot noise. Ω0 = 2π × 10 kHz was used here, except that the effect of shot noise is not significantly affected by Ω0.

[0173] Fig. 8A Shows S (0) and M t S t Examples of average relative uncertainty for different values. The uncertainty is proportional to the inverse of the square root of the number of shots. (0) =M t S t When the improved method is in η j,k As explained in Section IV, the improved method measures the entire curve, which includes the qubit population versus η j,k Compared with the basic method, this makes η j,k The uncertainty is smaller because at the fixed timestamp τ (0) The measurement under the condition cannot make all the population pairs η of N qubits j,k In particular, to keep the average uncertainty below 10 -3 , Basic Method 200 {Improved Method 300} requires S (0) =3×10 4 {M t s t =10 4}, marked as ▲{★}.

[0174] Next, we calculate the qubit state Rabi frequency Ω0, which determines the BSB transition time τ (0) {τ i}, and the frequency scanning parameters of the basic method 200 {improved method 300} These parameters are measured within the target accuracy η j,k Model 2 is used again with different values ​​of Ω0 and detuning frequency Δ j,k The bright state population is fitted, but this time without assuming Δ j,k Here, we use Fig. 8Abut assumes that the measured qubit probabilities are correct and there is no shot noise.

[0175] The qubit population depends on |Δ j,k / Ω0| 2 , up to the leading order in (6), so as Ω0 increases, due to the non-zero Δ j,k However, when Ω0 is too large, the error due to cross-mode coupling becomes dominant. From this tug-of-war, it can be assumed that Ω0 and δω k Parameter set. Parameter set Ω0 and δω k Allows η to be measured with a specified target accuracy j,k , where δω k Yes |Δ j,k |, or the maximum allowed uncertainty in the mode frequency. For the basic method, δω k There is an additional constraint that when μ j =ω k and μ j =ω k +δω k When, time τ (0) The difference between the qubit populations at τ exceeds the lower limit of the sum of shot noise. For the improved method, the BSB transition time τ of the frequency sweep Δ is unconstrained, so δω k Directly determine τ based on conditions similar to those above Δ In addition, δω k Detuning the frequency and M Δ The number is determined as , where δω k,先验 is the mode frequency range assumed a priori.

[0176] Figure 8B The estimated η is shown j,k As an example of the average relative error as a function of Ω0. Considering multiple Δ j,k Using this graph, when η j,k When a predetermined target accuracy is provided in the measurement, Ω0 and δω that meet the target accuracy can be determined k For example, if you want a relative uncertainty lower than 10 -3 , then the reasonable choice of basic method 200 {improved method 300} is Ω0 / 2π=7{10}kHz, and δω k / 2π=12{100}Hz, marked as ▲{★}. As explained in Section IV, the improved method fits the entire Compared with the basic method of fitting the population at a single time stamp, this method has a different ηj,k and Δ j,k This has the unique effect of allowing more accurate measurements in the presence of large detuned frequencies. k The selected value of τ leads to Δ =0.57ms. In addition, δω of the basic method 200 {improved method 300} k Given , where the prior width δω is assumed k,先验 =2π×1kHz.

[0177] Now, after determining all the parameters of the methods, the characterization measurement times of the basic method 200 and the improved method 300 given in (16) and (17) are compared. As a specific example, it is assumed that the cooling, state preparation and state detection times are 4 ms, 100 μs and 150 μs respectively, which are added to the BSB transition time to produce the cycle time of each shot. Table 1 shows the parameter sets for the two methods. In general, in order to estimate η for the five-ion chain j,k Reach 10 -3 The relative measurement uncertainty of the improved method is T = 586s, which is shorter than the basic method. (0) =1.11×10 4 s is about 19 times shorter. The savings of the improved method come from allowing less shot and lower frequency scanning precision.

[0178]

[0179] Table 1: Parameters of the basic method 200 (left) and the improved method 300 (right), achieving η for the five-ion chain j,k The relative uncertainty is 10 -3 Magnitude. <·> is i=1,..M t According to (16) and (17), the characterization measurement time of the basic method 200 and the improved method 300 are T (0) =1.11×10 4 s and T = 586s.

[0180] Finally, to distinguish the advantages of less shot and lower frequency scanning accuracy, Figure 8C Two methods are shown at different δω k This emphasizes that it is possible to allow a larger uncertainty δω in the mode frequency. k Characterization measurement time for improved methods can be significantly reduced.

[0181] VI. Trade-offs between Basic and Improved Methods

[0182] The problem of high-precision and efficient motion mode characterization comes down to the optimization of multiple parameters associated with various trade-offs. For example, using smaller laser power (and therefore smaller Ω0) can reduce the error due to cross-mode coupling, but at the expense of longer BSB transition times and better frequency scanning accuracy.

[0183] The choice of method and model can also be considered in terms of trade-offs. For example, parallelization methods increase the complexity from O(N 2 ) to O(N), but at the expense of introducing additional considerations in the model, such as DW effects of other modes tested in parallel, and, to be precise, time-varying. In general, a more accurate model can be used at the expense of longer routine computation time. To exploit this trade-off, a highly parallel and efficient fitting routine algorithm could be explored that performs the routine computational part of the method relatively quickly, especially for long ion chains where computations tend to be slow.

[0184] Another important trade-off associated with trapped ions is the spacing between mode frequencies and the physical distance between neighboring ions. The smaller the distance between neighboring ions, the larger the spacing between mode frequencies, which reduces the cross-mode coupling effect and increases the measurement of η. j,k The smaller the error, the smaller the Figure 6B The exponential growth of the error N is shown, Figure 6B A fixed distance between neighboring ions is assumed. However, a smaller inter-ion distance also results in larger optical crosstalk, since the laser beam width cannot be arbitrarily small.

[0185] In the embodiments described herein, methods for motion pattern characterization are provided. In particular, these methods are based on effective physical models that more accurately describe the dynamics of ions in an ion chain and the motion pattern of the ion chain than conventional physical models, thereby enabling accurate and efficient characterization of the motion pattern. Compared to conventional methods, the methods described herein use time scan measurements that allow for faster and more accurate characterization of the motion pattern, and have parallelism because the motion pattern is detected by multiple ions simultaneously to more accurately characterize the motion pattern.

[0186] Appendices A, B, C and D are attached, and all contents thereof are considered to be part of this application and are therefore incorporated into this application.

[0187] While the foregoing is directed to particular embodiments, other and further embodiments may be devised without departing from the basic scope thereof, the scope of which is determined by the following claims.

Claims

1. A method for using an ion trap quantum computer, comprising: performing a first measurement of a bright state population of each ion in an ion chain comprising a plurality of ions, each ion coupled to one of the motion modes of the ion chain, for a fixed duration while varying a laser coupling frequency for coupling each ion to one of the motion modes; calculating a mode frequency of one of the modes of motion based on a frequency that maximizes the bright state population of each ion measured in the first measurement; calculating the coupling strength of each ion to one of 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 one of the motion modes and a non-zero temperature effect of the motion mode; performing a second measurement of the bright state population of each ion in the ion chain for a fixed duration, each ion coupled to one of the motion modes to which said each ion was not coupled in the first measurement, while fixing the laser coupling frequency for coupling said each ion to said one of the motion modes; and The coupling strength of each ion to one of the motion modes is calculated 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 one of the motion modes and the non-zero temperature effect of the motion mode.

2. The method according to claim 1, further comprising: selecting, by a processor in a digital computer, a quantum algorithm to be implemented on the plurality of ions; Compiling the selected quantum algorithm into a series of universal logic gates by the processor in the digital computer; converting, by said processor in said digital computer, said series of general logic gates into a series of pairwise entangled gate operations for application to said plurality of ions in an ion chain; The processor in the digital computer calculates the amplitude and detuning frequency of the laser pulses causing the series of paired entanglement gate operations based on the calculated coupling strength between the motion mode and the ions; applying, by a system controller, a laser pulse having the calculated amplitude and detuned frequency to the plurality of ions in the ion chain; measuring, by the system controller, a population of qubit states of the plurality of ions in the ion chain; and processing, by the processor in the digital computer, quantum information corresponding to qubit states of the plurality of ions in the ion chain based on the measured population of the qubit states; and The processor in the digital computer generates and outputs a solution to the selected quantum algorithm based on the processing result of the quantum calculation.

3. The method according to claim 1, further comprising: Prior to the first measurement and the second measurement of each ion, each ion in the ion chain is initialized to a hyperfine ground state of each ion.

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

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

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

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

8. A method of using an ion trap quantum computer, comprising: performing, by a system controller, a first measurement of a bright state population of each ion in an ion chain comprising a plurality of ions, each ion coupled to one of the motion modes of the ion chain, for a fixed duration while varying a laser coupling frequency for coupling each ion to one of the motion modes; calculating, by a processor in a digital computer, a mode frequency of one of the modes of motion based on a frequency that maximizes the bright state population of each ion measured in the first measurement; calculating, by the processor in the digital computer, the coupling strength of each ion to one of 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 of the motion modes and a non-zero temperature effect of the motion mode; performing, by the system controller, a second measurement of the bright state population of each ion in the ion chain for a fixed duration, each ion coupled to one of the motion modes to which each ion was not coupled in the first measurement, while fixing a laser coupling frequency for coupling each ion to one of the motion modes; calculating, by the processor in the digital computer, the coupling strength of each ion to one of 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 of the motion modes and a non-zero temperature effect of the motion mode; selecting, by the processor in the digital computer, a quantum algorithm to be implemented on the plurality of ions; Compiling the selected quantum algorithm into a series of universal logic gates by the processor in the digital computer; converting, by said processor in said digital computer, said series of general logic gates into a series of pairwise entangled gate operations for application to said plurality of ions in said ion chain; The processor in the digital computer calculates the amplitude and detuning frequency of the laser pulses causing the series of paired entanglement gate operations based on the calculated coupling strength between the motion mode and the ions; applying, by a system controller, a laser pulse having the calculated amplitude and detuned frequency to the plurality of ions in the ion chain; measuring, by the system controller, a population of qubit states of the plurality of ions in the ion chain; and processing, by the processor in the digital computer, quantum information corresponding to qubit states of the plurality of ions in the ion chain based on the measured population of the qubit states; and The processor in the digital computer generates and outputs a solution to the selected quantum algorithm based on the processing result of the quantum calculation.

9. A method of using an ion trap quantum computer, comprising: performing a first measurement of a bright state population of each ion in an ion chain comprising a plurality of ions, each ion coupled to one of the motion modes of the ion chain, for a fixed duration while varying a laser coupling frequency for coupling each ion to one of the motion modes; calculating a mode frequency of one of the modes of motion based on a frequency that maximizes the bright state population of each ion measured in the first measurement; performing a second measurement of the bright state population of each ion in the ion chain at a plurality of durations, each ion coupled to one of the motion modes, while fixing a laser coupling frequency for coupling each ion to one of the motion modes; and The coupling strength of each ion to one of the motion modes is calculated 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 one of the motion modes and the non-zero temperature effect of the motion mode.

10. The method according to claim 9, further comprising: selecting, by a processor in a digital computer, a quantum algorithm to be implemented on the plurality of ions; Compiling the selected quantum algorithm into a series of universal logic gates by the processor in the digital computer; converting, by said processor in said digital computer, said series of general logic gates into a series of pairwise entangled gate operations for application to said plurality of ions in said ion chain; The processor in the digital computer calculates the amplitude and detuning frequency of the laser pulses causing the series of paired entanglement gate operations based on the calculated coupling strength between the motion mode and the ions; applying, by a system controller, a laser pulse having the calculated amplitude and detuned frequency to the plurality of ions in the ion chain; measuring, by the system controller, a population of qubit states of the plurality of ions in the ion chain; and processing, by the processor in the digital computer, quantum information corresponding to qubit states of the plurality of ions in the ion chain based on the measured population of the qubit states; and The processor in the digital computer generates and outputs a solution to the selected quantum algorithm based on the processing result of the quantum calculation.

11. The method according to claim 9, further comprising: Prior to the first measurement and the second measurement of each ion, each ion in the ion chain is initialized to the hyperfine ground state of each ion.

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

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

14. The method according to claim 9, wherein The calculation of the coupling strength between the ions in the ion chain and the motion mode of the ion chain is also based on the Debye-Waller effect of the ions.

15. The method according to claim 9, wherein The calculation of the coupling strength between the ions in the ion chain and the motion modes of the ion chain is also based on the cross-mode coupling effect of the motion modes of the ion chain.

16. A method of using an ion trap quantum computer, comprising: performing, by a system controller, a first measurement of a bright state population of each ion in an ion chain comprising a plurality of ions, each ion coupled to one of the motion modes of the ion chain, for a fixed duration while varying a laser coupling frequency for coupling each ion to one of the motion modes; calculating, by a processor in a digital computer, a mode frequency of one of the modes of motion based on a frequency that maximizes the bright state population of each ion measured in the first measurement; performing, by a system controller, a second measurement of the bright state population of each ion in the ion chain at a plurality of durations, each ion coupled to one of the motion modes, while fixing a laser coupling frequency for coupling each ion to one of the motion modes; calculating, by the processor in the digital computer, the coupling strength of each ion to one of 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 of the motion modes and a non-zero temperature effect of the motion mode; selecting, by the processor in the digital computer, a quantum algorithm to be implemented on the plurality of ions; Compiling the selected quantum algorithm into a series of universal logic gates by the processor in the digital computer; converting, by said processor in said digital computer, said series of general logic gates into a series of pairwise entangled gate operations for application to said plurality of ions in said ion chain; The processor in the digital computer calculates the amplitude and detuning frequency of the laser pulses causing the series of paired entanglement gate operations based on the calculated coupling strength between the motion mode and the ions; applying, by a system controller, a laser pulse having the calculated amplitude and detuned frequency to the plurality of ions in the ion chain; measuring, by the system controller, a population of qubit states of the plurality of ions in the ion chain; and processing, by the processor in the digital computer, quantum information corresponding to qubit states of the plurality of ions in the ion chain based on the measured population of the qubit states; and The processor in the digital computer generates and outputs a solution to the selected quantum algorithm based on the processing result of the quantum calculation.

17. A quantum computing system comprising: an ion chain comprising a plurality of ions, each ion in the ion chain having two hyperfine states defining a qubit; system controller; and A classical computer includes a processor and a non-volatile memory storing a plurality of instructions, which, when executed by the processor, cause the quantum computing system to perform operations including: performing, by the system controller, a first measurement of the bright state population of each ion in the ion chain for a fixed duration, each ion coupled to one of the motion modes of the ion chain, while varying a laser coupling frequency for coupling each ion to one of the motion modes; calculating, by the processor, a mode frequency of one of the modes of motion based on a frequency that maximizes a bright state population for each ion measured in a first measurement; performing, by the system controller, a second measurement of the bright state population of each ion in the ion chain, each ion coupled to one of the motion modes, while fixing a laser coupling frequency for coupling each ion to one of the motion modes; and The processor calculates 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 one of the motion modes, and the non-zero temperature effect of the motion mode.

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

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

20. The quantum computing system of claim 17, wherein the operations further comprise: selecting, by a processor in a digital computer, a quantum algorithm to be implemented on the plurality of ions; Compiling the selected quantum algorithm into a series of universal logic gates by the processor in the digital computer; converting, by said processor in said digital computer, said series of general logic gates into a series of pairwise entangled gate operations for application to said plurality of ions in said ion chain; The processor in the digital computer calculates the amplitude and detuning frequency of the laser pulses causing the series of paired entanglement gate operations based on the calculated coupling strength between the motion mode and the ions; applying, by a system controller, a laser pulse having the calculated amplitude and detuned frequency to the plurality of ions in the ion chain; measuring, by the system controller, a population of qubit states of the plurality of ions in the ion chain; and processing, by the processor in the digital computer, quantum information corresponding to qubit states of the plurality of ions in the ion chain based on the measured population of the qubit states; and The processor in the digital computer generates and outputs a solution to the selected quantum algorithm based on the processing result of the quantum calculation.

21. The quantum computing system of claim 17, wherein the operations further comprise: Prior to the first measurement and the second measurement of each ion, each ion in the ion chain is initialized to the hyperfine ground state of each ion.

22. The quantum computing system of claim 17, wherein The calculation of the coupling strength between the ions in the ion chain and the motion modes of the ion chain is also based on the Debye-Waller effect of the ions and the cross-mode coupling effect of the motion modes of the ion chain.

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