Universal gate pulse for two-qubit gates in trapped-ion quantum computers

Abstract

A method for performing at least a portion of the computational process includes the steps of: calculating a pulse function for a pulse to be applied to a pair of first trapped ions in a first ion chain based on a phase space condition, where the phase space condition is derived using equally spaced composite frequencies in a frequency interval that includes a range set by the highest and lowest motional mode frequencies of the first ion chain; generating a pulse based on the calculated pulse function; and applying the generated pulse to each of a pair of second trapped ions in a second ion chain to perform an entanglement gate operation between the pair of second trapped ions in the second ion chain.

Description

[Technical field]

[0001] The present disclosure relates generally to apparatus and methods for performing computations using quantum computers, and more particularly to apparatus and methods for performing computations by performing entanglement gate operations in trapped ion quantum computers. [Background technology]

[0002] As several promising architectures for quantum computing emerge, the size of the quantum computers that can be supported is also increasing. This inevitably increases the complexity of the processes required to build and operate such quantum computers. In trapped ion quantum computers, the two-qubit entangled gates, which are the basic gates on which any logic gate is built, are typically designed and optimized for each pair of qubits individually. Thus, as the number of qubits in a quantum computer increases, the number of qubit pairs grows quadratically, making the design of two-qubit entangled gates for all qubit pairs an increasingly difficult task.

[0003] Therefore, what is needed is a way to design an apparatus and method for performing a two-qubit entanglement gate without increasing complexity as the number of qubits increases. Summary of the Invention [Problem to be solved by the invention]

[0004] An embodiment of the present disclosure includes a method for performing at least a portion of a computational process, the method including the steps of: calculating a pulse function for a pulse to be applied to a pair of first trapped ions in a first ion chain based on a phase space condition, the phase space condition being derived using equally spaced composite frequencies in a frequency interval that includes a range set by the highest and lowest motional mode frequencies of the first ion chain, generating pulses based on the calculated pulse function, and applying the generated pulses to each of a pair of second trapped ions in a second ion chain to perform an entanglement gate operation between the pair of second trapped ions in the second ion chain.

[0005] Embodiments of the present disclosure further include an ion trap quantum computing system. The trapped ion quantum computing system includes a quantum processor including a first ion chain including a plurality of trapped ions, each having two hyperfine states and defining a quantum bit; one or more lasers configured to emit a laser beam that is provided to the plurality of trapped ions in the first ion chain; a classical computer configured to perform operations including calculating a pulse function for a pulse to be applied to a pair of first trapped ions in a second ion chain based on a phase space condition, the phase space condition being derived using equally spaced composite frequencies in a frequency interval that includes a range set by a highest and lowest motional mode frequency of the second ion chain; and generating a pulse based on the calculated pulse function; and a system controller configured to execute a control program to control the one or more lasers to perform operations at the quantum processor including applying the generated pulses to each of the pair of second trapped ions in the first ion chain to perform an entanglement gate operation between the pair of second trapped ions in the first ion chain; and measuring a population of qubit states in the quantum processor, the classical computer further configured to output the measured population of qubit states in the quantum processor.

[0006] An embodiment of the present disclosure further includes a trapped ion quantum computing system including a classical computer, a quantum processor including a first ion chain including a plurality of trapped ions, each having two hyperfine states and defining a qubit, a system controller configured to execute a control program to control one or more lasers to perform operations on the quantum processor, and a non-volatile memory having stored thereon a plurality of instructions that, when executed by the one or more processors, cause the trapped ion quantum computing system to perform the following operations: calculating, by the classical computer, a pulse function for a pulse applied to a pair of first trapped ions in the second ion chain based on a phase space condition; The method includes deriving phase space conditions using equally spaced composite frequencies within a frequency interval that includes a range set by the highest and lowest motional mode frequencies of the second ion chain; generating, by a classical computer, pulses based on the calculated pulse function; applying, by a system controller, the generated pulses to each of a pair of second trapped ions in the first ion chain to perform an entanglement gate operation between the pair of second trapped ions in the first ion chain; measuring, by the system controller, a population of qubit states in the quantum processor; and outputting, by the classical computer, the measured population of qubit states in the quantum processor. [Brief description of the drawings]

[0007] So that the above features of the present disclosure can be understood in detail, a more particular description of the present disclosure briefly summarized above can be made by reference to embodiments, some of which are illustrated in the accompanying drawings. It should be noted, however, that the accompanying drawings illustrate only typical embodiments of the present disclosure and should not be considered as limiting its scope, since the present disclosure may admit of other equally effective embodiments.

[0008] [Figure 1] FIG. 1 is a partial schematic diagram of an ion trap quantum computing system according to one embodiment. [Diagram 2] 1 shows a schematic diagram of an ion trap that confines ions in groups according to one embodiment. [Diagram 3] 4 shows a schematic energy diagram of each ion in a group of trapped ions according to one embodiment. [Figure 4A] 4 illustrates an exemplary universal gate pulse characteristic according to one embodiment. [Figure 4B] 4 illustrates an exemplary universal gate pulse characteristic according to one embodiment. [Figure 4C] 4 illustrates an exemplary universal gate pulse characteristic according to one embodiment. [Figure 4D] 4 illustrates an exemplary universal gate pulse characteristic according to one embodiment. [Figure 5A] 1 illustrates the performance of five different exemplary universal gate pulses according to one embodiment. [Figure 5B] 1 illustrates the performance of five different exemplary universal gate pulses according to one embodiment. [Figure 5C] 1 illustrates the performance of five different exemplary universal gate pulses according to one embodiment. [Figure 6] 6 shows a flow chart illustrating a method 600 that may be used to perform at least a portion of the computational process.

[0009] For ease of understanding, the same reference numbers are used, where possible, to designate identical elements common to the figures. The figures and the following description use a Cartesian coordinate system including an X-axis, a Y-axis, and a Z-axis. Directions represented by arrows in the figures are assumed to be positive for convenience. It is believed that elements disclosed in some embodiments may be beneficially utilized in other implementations without specific specification. DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

[0010] The embodiments described herein relate to a pulse shaping method that can be used to universally perform two-qubit entangled gate operations on any pair of ions in an ionic chain of any length. Once this universal gate pulse is designed for a particular pair of ions in an ionic chain, it can be applied to any other pair of ions in the ionic chain.

[0011] An overall system capable of performing quantum computations using trapped ions includes a classical computer, a system controller, and a quantum register. The classical computer performs support and system control tasks including selecting a quantum algorithm to execute 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 laser pulses to apply to the quantum register, and using a central processing unit (CPU) to pre-calculate parameters that optimize the laser pulses. During computational operations initiated on the classical computer, a software program that performs the task of decomposing and executing the quantum algorithm, stored in non-volatile memory within the classical computer, is used to initiate the completion of a complex computational task using the quantum register. The quantum register includes trapped ions coupled with various hardware, including a laser that manipulates the internal hyperfine state (qubit state) of the trapped ion, and an acousto-optic modulator that reads out the internal hyperfine state (qubit state) of the trapped ion. The system controller receives pre-calculated parameters of the pulses from the classical computer when the selected algorithm begins execution on the quantum register, controls the various hardware associated with controlling any and all aspects used to execute the selected algorithm on the quantum register, and returns the read value of the quantum register, thereby outputting the result of the quantum computation to the classical computer when the algorithm finishes execution. The classical computer software then uses the results received from the quantum register to complete the computational operation originally initiated on the classical computer.

[0012] (General hardware configuration) FIG. 1 is a partial schematic diagram 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. Other components of system 100 shown in FIG. 1 relate to a quantum processor that includes a chain 106 of trapped ions (i.e., five chains shown as circles approximately equally spaced from one another) extending along the Z axis. Each ion in the chain 106 of trapped ions has 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, e.g., a positive ytterbium ion. 171 Yb + , positive barium ions 133 Ba + , positive cadmium ions 111 CD + or 113 CD + all of which have nuclear spin I=1 / 2 and 2 S 1 / 2 In some embodiments, all ions in the chain of trapped ions 106 are of the same species and isotope (e.g., 171 Yb + In some other embodiments, the chains of trapped ions 106 include one or more species or isotopes (e.g., some ions are 171 Yb + and some other ions 133 Ba +In yet further embodiments, the chains of trapped ions 106 may include various isotopes of the same species (e.g., different isotopes of Yb, different isotopes of Ba). The ions in the chains of trapped ions 106 are individually processed with separate 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 a readily available memory, either locally or remotely, such as a read-only memory (ROM), a random access memory (RAM), a floppy disk, a hard disk, or any other form of digital storage. Software instructions, algorithms, and data may be coded and stored in the memory to instruct 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, and the like.

[0013] An imaging objective 108, for example a 0.37 numerical aperture (NA) objective, collects fluorescence from the ions along the Y-axis and maps each ion to a multi-channel photomultiplier tube (PMT) 110 for measuring the individual ions. A Raman laser beam from a laser 112, provided along the X-axis, performs operations on the ions. A diffractive beam splitter 114 creates 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 individual ions by individually controlling the emission of the Raman laser beams 116. A global Raman laser beam 120, which does not co-propagate with the Raman laser beam 116, irradiates all ions at once from different directions. In some embodiments, individual Raman laser beams (not shown) can be used to irradiate individual ions rather than a single global Raman laser beam 120. The system controller (also called "RF controller") 104 controls the intensity, timing and phase of the laser pulses applied to the trapped ions in the chain 106 of trapped ions to control the AOM 118. 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 the various programs and data. The storage unit 128 includes a non-volatile memory, such as a hard disk drive (HDD) or a flash memory, and stores the various programs even when the power is turned off. The CPU 122, the ROM 124, the RAM 126 and the storage unit 128 are interconnected via a bus 130. The system controller 104 executes a control program that is 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 that include 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 described herein.

[0014] 2 shows a schematic diagram of an ion trap 200 (also called a Paul trap) that confines ions in chains 106, according to one embodiment. The confinement potential is applied by both a static (DC) voltage and a radio frequency (RF) voltage. The static (DC) voltage V S is applied to the end cap electrodes 210 and 212 to confine ions along the Z axis (also called the "axial" or "longitudinal" direction). The ions in the chains 106 are approximately evenly distributed in the axial direction due to Coulomb interactions between the ions. In some embodiments, the ion trap 200 includes four hyperbolic shaped electrodes 202, 204, 206, and 208 that extend along the Z axis.

[0015] During operation, (amplitude V RF / 2) sinusoidal voltage V 1 is applied to a pair of opposing electrodes 202, 204, and a sinusoidal voltage V 1 A phase shift of 180° from (and amplitude V RF / 2) sinusoidal voltage V 2 is the driving frequency ω RF to generate a quadrupole potential. In some embodiments, a sinusoidal voltage is applied to only one pair of opposing electrodes 202, 204, and the other pair of opposing electrodes 206, 208 is grounded. The quadrupole potential generates an effective confinement force in the XY plane perpendicular to the Z axis (also called the "radial" or "transverse") for each trapped ion, which is proportional to the distance from the saddle point (i.e., the location in the axial (Z) direction) where the RF field vanishes. The radial (i.e., in the XY plane) motion of each ion is approximated as a harmonic oscillation (called secular motion) with a restoring force toward the radial saddle point, each with a spring constant k x and k yIn some embodiments, the radial spring constants are modeled as equal when the quadrupole potential is radially symmetric. However, if undesired, the radial ion motion may be distorted due to some asymmetry in the physical trapping configuration, small DC patch potentials due to non-uniformities on the surface of the electrodes, etc., and these and other external sources of distortion may cause the ion to de-center from the saddle point.

[0016] A different type of trap, not shown, is a microfabricated trapping chip, which uses an approach similar to that described above to hold or confine ions or atoms in place on the surface of the microfabricated trapping chip. A laser beam, such as the Raman laser beam described above, can be directed at the ions or atoms just above the surface.

[0017] 3 shows a schematic energy diagram 300 of each ion in the chain of trapped ions 106 according to one embodiment. Each ion in the chain of trapped ions 106 is an ion with 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. In one example, each ion has an energy distribution ω 01 The nuclear spins I=1 / 2 and I=2π=12.642812 GHz have an energy splitting corresponding to a frequency difference (called the "carrier frequency") 2 S 1 / 2 Positive ytterbium ions with hyperfine states. 171 Yb + In another example, the ions may all have nuclear spin I=1 / 2 and 2 S 1 / 2 Positive barium ion with hyperfine states. 133 Ba + , positive cadmium ions 111 CD + or 113 CD + The quantum bit may be formed of two hyperfine states, represented by |0> and |1>, and the hyperfine ground state (i.e. 2 S 1 / 2A low energy state among the hyperfine states is chosen to represent |0>. Hereinafter, the terms "hyperfine state", "internal hyperfine state" and "qubit" may be used interchangeably to represent |0> and |1>. Each ion can be cooled to near the motional ground state of any motional mode without phonon excitation (i.e., the kinetic energy of the ion may be lowered) by known laser cooling methods such as Doppler cooling or resolved sideband cooling, and then the qubit state can be prepared in the hyperfine ground state |0> by optical pumping.

[0018] The individual qubit states of each trapped ion can be, for example, excited 2 P 1 / 2 The laser beam from the laser can be operated by a mode-locked laser at 355 nanometers (nm) through the Raman level (denoted |e>). As shown in Figure 3, the laser beam from the laser is split into a pair of non-copropagating laser beams (with frequencies ω 1 and a first laser beam having a frequency ω 2 ) with a transition frequency ω between |0> and |e> as explained in FIG. 3. 0e With respect to the one-photon transition detuning frequency Δ=ω 1 -ω 0e The two-photon transition detuning frequency δ involves adjusting the amount of energy provided to the trapped ion by the first and second laser beams, which, when used in combination, move the trapped ion between hyperfine states |0> and |1>. The one-photon transition detuning frequency Δ can be adjusted to the two-photon transition detuning frequency (also simply called the "detuning frequency") δ=ω 1 -ω 2 -ω 01 (hereafter denoted as ±μ, where μ is a positive value), the single-photon Rabi frequency Ω at which Rabi oscillations occur between states |0> and |e>, and between states |1> and |e>, respectively. 0e (t) and Ω 1eThe two-photon Rabi frequency Ω(t) (which is time-dependent and determined by the amplitudes and phases of the first and second laser beams), as well as the spontaneous emission rate from the excited state |e>, the Rabi oscillations between the two hyperfine states |0> and |1> (called "carrier transitions") are induced at the two-photon Rabi frequency Ω(t). The two-photon Rabi frequency Ω(t) is given by Ω 0e Ω 1e / 2Δ, where Ω 0e and Omega 1e are the single-photon Rabi frequencies of the first and second laser beams, respectively. Hereinafter, this set of non-copropagating laser beams in a Raman configuration for manipulating the internal hyperfine state (qubit state) of a qubit may be referred to as a "composite pulse" or simply a "pulse", and the resulting time-dependent pattern of the two-photon Rabi frequency Ω(t) may be referred to as the "amplitude" of the pulse or simply a "pulse", which are illustrated and further described below. Detuning frequency δ=ω 1 -ω 2 -ω 01 may be referred to as the detuning frequency of the composite pulse or the detuning frequency of the pulse. The amplitude of the two-photon Rabi frequency Ω(t), determined by the amplitudes of the first and second laser beams, may be referred to as the “amplitude” of the composite pulse.

[0019] It should be noted that the specific atomic species used in the description provided herein is merely an example of an atomic species that has a stable and well-defined two-level energy structure and an optically accessible excited state when ionized, and is not intended to limit the possible configurations, specifications, etc. of the trapped ion quantum computer according to the present disclosure. For example, other ion species include alkaline earth metal ions (Be + , Ca + , Sr + , Mg + and Ba + ) or transition metal ions (Zn + , Hg + , Cd + ).

[0020] (Tangled Gate Operation) In a trapped ion quantum computer, the motional modes of the chain of trapped ions 106, generated by the Coulomb interaction between the ions, can serve as a data bus to mediate the entanglement between two qubit ions (ith ion and jth ion) in the chain of trapped ions 106, which is used to perform XX gate operations between the two ions. That is, each of the two ions is entangled with a motional mode, and this entanglement is transferred to the entanglement between the two ions by using motional sideband excitation, as known in the art. Specifically, the combined state of the two ions (ith ion and jth ion) is transformed by applying a laser pulse consisting of a composite pulse of the motional sidebands to the two ions for a gate duration τ, which is shaped for a gate duration τ such that the transformation of the combined state obeys the intended XX gate operation (referred to as the "gate angle condition"), but the state of the trapped ions remaining in the chain 106 remains unchanged at the end of the gate duration τ (referred to as the "phase space condition"). Hereinafter, both the gate angle condition and the phase space condition are referred to as the "gating conditions". Such techniques for shaping laser pulses are usually based on amplitude, frequency or phase modulation of the laser pulse over the gate duration τ. Among these techniques, an accurate and computationally efficient pulse shaping technique, referred to herein as the exact amplitude frequency modulation (E-AMFM) method, is proposed, which provides laser pulse shaping that performs exact XX gating with 100% fidelity (i.e., the gating conditions are exactly satisfied). The E-AMFM method furthermore takes into account the motional mode frequency drift Δω pFeatures such as active stabilization of the fidelity of XX gating to a desired degree K with respect to (hereinafter referred to as "stabilization conditions"), and power optimization of the laser pulse (hereinafter referred to as "power optimization conditions"). Note that the number of gating conditions increases with increasing number of ions in the chains 106 of trapped ions and / or the number of active stabilization conditions equal to the degree K of active stabilization. Thus, for longer chains 106 of trapped ions (i.e., more ions in the chains 106) and / or a higher degree K of active stabilization, the laser pulse needs to be shaped such that more conditions need to be satisfied to perform accurate XX gating, which reduces the degrees of freedom (i.e., possible values ​​of amplitude and phase) in selecting the power-optimized pulse shape. In the examples described below, the same laser pulse is applied to both the ith ion and the jth ion. However, in some embodiments, different laser pulses are applied to the ith ion and the jth ion.

[0021] The pulse shaping method will be explained in more detail below. First, the amplitude modulation and detuning modulation of the laser pulse are called the amplitude function Ω(t) and the detuning frequency function μ(t), respectively. Furthermore,

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[0022] Regarding the transformation of the combined state of two quantum bits (the i-th quantum bit and the j-th quantum bit) by applying a laser pulse,

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[0023] The above phase space condition requires that trapped ions in the chain 106, displaced from their initial positions as the motional modes are excited by the delivery of a laser pulse, return to their initial positions. The l-th trapped ion in the chain 106, which is in the superposition state |0>±|1>, is displaced by the excitation of the p-th motional mode during the gate duration τ, and follows the phase space (position and momentum) trajectory of the p-th motional mode:

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[0024] Motional mode frequency drift Δω p The stabilization condition of order K for

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[0025] Alternatively, the phase space condition and the stabilization condition of order K are both

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[0026] These conditions can be expressed in matrix form as

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[0027] The gate angle condition is the entanglement interaction χ generated between the i-th and j-th ions by the laser pulse. ij is the desired value θ ij We need to have the XX gate operation with maximum entanglement.

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[0028] (Universal Gate Pulse) In the embodiments described herein, a universal gate pulse is devised that approximately satisfies the above phase space condition, but can be applied to any pair of ions in an ion chain of any length.

[0029] The pulse function of the universal gate pulse (also called the "universal pulse function") that induces XX gate operation between pairs of ions, g (u) In constructing (t), the above phase space condition and stabilization condition are replaced by approximate phase space condition and approximate stabilization condition, respectively, and the frequency ω p (also simply called "motion mode frequencies") are equally spaced composite frequencies (also called "grid frequencies") within the frequency interval of the motion mode frequencies.

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[0030] Specifically, the grid frequency

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[0031] Universal Pulse Function

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[0032] Motional mode frequency drift Δω p The approximate stabilization condition of order K for the motion mode frequency ω p The grid frequency

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[0033] To satisfy the stabilization condition, the vector

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[0034] Furthermore, the null space vector

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[0035] The gate angle conditions are:

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[0036] The power optimization condition is the control parameter vector

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[0037] Those skilled in the art will recognize the universal pulse function g (u) (t) is the ion chain motion mode frequency ω p It will be understood that the universal pulse function g (u) (t) is the motion mode frequency ω p As long as ω is within a preset frequency interval, the motion mode ω of the ion chain p The universal pulse function g (u) Once (t) is calculated to induce XX gate operation between a particular pair of ions in an ion chain having a particular number of ions (i.e., a particular length), it can be used to induce XX gate operation between any pair of ions in an ion chain of any length.

[0038] The universal pulses calculated according to the methods specified herein exhibit relatively small power variations (scale factors) across different qubit pairs and a small kinetic mode frequency drift, Δω pMoreover, the universal pulse works well for different numbers of trapped ions, which is not the case for the non-universal pulse.

[0039] (Example of Universal Gate Pulse) Examples of universal gate pulses are described herein to evaluate the performance of the universal gate pulses, specifically, the power requirements of the universal gate pulses and the motional mode frequency drift Δω p The robustness of the XX gate operation, which is performed by applying a universal gate pulse to

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[0040] 4A, 4B, 4C, and 4D show exemplary universal gate pulse characteristics. In FIG. 4A, 4B, and 4C, the universal pulse function g (u) (t) is N A basis functions

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[0041] 5A, 5B, and 5C show the performance of five different exemplary universal gate pulses. In these examples, the universal pulse function is N A basis functions

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[0042] Universal gate pulse p1 was calculated for one ion pair i=0,j=0 in the 2-ion chain. Universal gate pulses p2 and p3 were calculated for one ion pair i=0,j=0 and one ion pair i=48,j=49 in the 50-ion chain, respectively. Universal gate pulses p4 and p5 were calculated for one ion pair i=0,j=0 and one ion pair i=98,j=99 in the 100-ion chain, respectively.

[0043] 5A, 5B, and 5C show the RMS Rabi frequency, α-infidelity, ε, and α 10 -4 Smaller motion mode frequency drift Δω p , and the χ-infidelity ε χ 10 -4 Smaller motion mode frequency drift Δω p 5A, 5B, and 5C show that regardless of the pair of ions for which the universal gate pulse was originally calculated, the performance of the universal pulse is nearly identical between pairs of ions at different positions in the ion chain.

[0044] FIG. 6 shows a flow chart illustrating a method 600 that may be used to perform at least a portion of the computational process.

[0045] At block 610, a pulse function for a pulse applied to a first pair of trapped ions in the first ion chain is calculated based on a phase space condition, which is derived using equally spaced composite frequencies in a frequency interval that includes a range set by the highest and lowest motional mode frequencies of the first ion chain.

[0046] At block 620, a pulse is generated based on the calculated pulse function.

[0047] In block 630, the generated pulse is applied to each of the pair of second trapped ions in the second ion chain to perform an entanglement gate operation between the pair of second trapped ions in the second ion chain.

[0048] In the embodiments described herein, a pulse shaping method is described that can be used to universally perform two-qubit entanglement gate operations on any pair of ions in an ion chain of any length. Once this universal gate pulse is designed for a particular pair of ions in an ion chain, it can be applied to other pairs of ions in an ion chain of any length. Thus, such a universal gate pulse significantly reduces the design complexity of two-qubit entanglement gates for all pairs of ions, thereby reducing the overall computational complexity in large-scale trapped ion quantum computers.

[0049] While the forgoing is directed to specific embodiments, other and further embodiments may be devised without departing from the basic scope thereof, which is determined by the following claims.

Claims

1. 1. A method for performing at least a portion of a computational process, comprising: calculating a pulse function for a pulse applied to a first pair of trapped ions in a first ion chain based on a phase space condition, the phase space condition being derived using equally spaced composite frequencies in a frequency interval that includes a range set by the highest and lowest motional mode frequencies of the first ion chain; generating the pulse based on the calculated pulse function; applying the generated pulse to each of a pair of second trapped ions in a second ion chain to perform an entanglement gate operation between the pair of second trapped ions in the second ion chain; A method comprising:

2. The method of claim 1 , wherein a number of trapped ions in the second ion chain is different from a number of trapped ions in the first ion chain.

3. The method of claim 1 , wherein the second pair of trapped ions is the same as the first pair of trapped ions.

4. The method of claim 1 , wherein the second pair of trapped ions is different from the first pair of trapped ions.

5. The calculation of the pulse function is further performed based on a stabilization condition; The method of claim 1 , wherein the stabilization condition is derived using the equally spaced synthesis frequencies.

6. The method of claim 1 , wherein the calculation of the pulse function is further performed based on a gate angle condition and a power optimization condition.

7. The method of claim 1 , wherein the pulse function is decomposed using basis functions, and computing the pulse function comprises computing coefficients of the basis functions.

8. An ion trap quantum computing system, comprising: a quantum processor including a first chain of ions including a plurality of trapped ions each having two hyperfine states and defining a quantum bit; one or more lasers configured to emit a laser beam that is provided to the plurality of trapped ions in the first ion chain; A classical computer, calculating a pulse function for a pulse applied to a first pair of trapped ions in a second ion chain based on a phase space condition, the phase space condition being derived using equally spaced synthetic frequencies in a frequency interval that includes a range set by the highest and lowest motional mode frequencies of the second ion chain; generating the pulse based on the calculated pulse function; a classical computer configured to perform operations including: a system controller executing a control program to control the one or more lasers to cause the quantum processor to: applying the generated pulse to each of a pair of second trapped ions in the first ion chain to perform an entanglement gate operation between the pair of second trapped ions in the first ion chain; measuring a population of qubit states in the quantum processor; a system controller configured to perform operations including Including, The classical computer is further configured to output a population of measured qubit states in the quantum processor.

9. The trapped ion quantum computing system of claim 8 , wherein a number of trapped ions in the second ion chain is different from a number of trapped ions in the first ion chain.

10. The trap ion quantum computing system of claim 8 , wherein the second pair of trapped ions is the same as the first pair of trapped ions.

11. The trapped ion quantum computing system of claim 8 , wherein the second pair of trapped ions is different from the first pair of trapped ions.

12. The calculation of the pulse function is further performed based on a stabilization condition; The ion trap quantum computing system of claim 8 , wherein the stabilization condition is derived using the equally spaced composite frequencies.

13. The ion trap quantum computing system of claim 8 , wherein the calculation of the pulse function is further performed based on a gate angle condition and a power optimization condition.

14. The ion trap quantum computing system of claim 8 , wherein the pulse function is decomposed using basis functions, and computing the pulse function includes computing coefficients of the basis functions.

15. An ion trap quantum computing system, comprising: Classical computers and a quantum processor including a first chain of ions including a plurality of trapped ions each having two hyperfine states and defining a quantum bit; a system controller configured to execute a control program to control one or more lasers to perform operations on the quantum processor; a non-volatile memory that, when executed by one or more processors, provides the ion trap quantum computing system with: calculating, by the classical computer, a pulse function for a pulse applied to a first pair of trapped ions in a second ion chain based on a phase space condition, the phase space condition being derived using equally spaced synthetic frequencies in a frequency interval that includes a range set by the highest and lowest motional mode frequencies of the second ion chain; generating, by the classical computer, the pulse based on the calculated pulse function; applying the generated pulses to each of a pair of second trapped ions in the first ion chain to perform an entanglement gate operation between the pair of second trapped ions in the first ion chain; measuring, by the system controller, a population of qubit states in the quantum processor; outputting, by the classical computer, a population of measured qubit states in the quantum processor; a non-volatile memory having stored therein a plurality of instructions for performing operations including 1. An ion trap type quantum computing system comprising:

16. The trapped ion quantum computing system of claim 15 , wherein a number of trapped ions in the second ion chain is different than a number of trapped ions in the first ion chain.

17. The trapped ion quantum computing system of claim 15 , wherein the second pair of trapped ions is the same as the first pair of trapped ions.

18. The trapped ion quantum computing system of claim 15 , wherein the second pair of trapped ions is different from the first pair of trapped ions.

19. The calculation of the pulse function is further performed based on a stabilization condition; The ion trap quantum computing system of claim 15 , wherein the stabilization condition is derived using the equally spaced composite frequencies.

20. The ion trap quantum computing system of claim 15 , wherein the calculation of the pulse function is further performed based on a gate angle condition.

21. The ion trap quantum computing system of claim 15 , wherein the calculation of the pulse function is further performed based on a power optimization condition.

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