Distortion analysis of trapped-ion quantum computers with imperfect beam geometry

Abstract

A method of performing a quantum gate operation in an ion-trap quantum computing system includes identifying one or more error mechanisms that cause quantum computing errors in a quantum gate operation on a first trapped ion of an ion chain that includes a plurality of trapped ions, wherein the quantum gate operation is performed by applying a first Raman laser beam and a second Raman laser beam; calculating a first amplitude of the first Raman laser beam and a second amplitude of the second Raman laser beam such that effects of the identified one or more error mechanisms are addressed; and applying the first Raman laser beam with the calculated first amplitude and the second Raman laser beam with the calculated second amplitude on the first trapped ion to perform the quantum gate operation on the first trapped ion.

Description

TECHNICAL FIELD

[0001] The present disclosure relates generally to a method of performing quantum gate operations in an ion-trap quantum computer, and more particularly, to a method for reliably performing quantum computations on long ion chains that produce all-to-all quantum bit connectivity. BACKGROUND

[0002] Trapped ions represent a promising universal quantum computing platform, and high-fidelity quantum gate operations have been demonstrated on short chains of one or two trapped ions. However, there is a need to further increase the fidelity of quantum gate operations and the number of trapped ions (i.e., qubits) to bridge the gap between these early trapped-ion short-chain systems and a commercially viable quantum computer. To this end, several approaches have been proposed and demonstrated in the art, in which trapped ions are spatially separated at a given time during quantum program execution. However, they come at the cost of sparse qubit connectivity between qubits, as direct implementation of qubit-to-qubit interactions between any pair of qubits is not possible, which is a known source of overhead for performing quantum computations. They also complicate the hardware design, making high-fidelity gate operations more challenging.

[0003] Therefore, there is a need for a method for reliably performing quantum computations on long chains of trapped ions that produce all-to-all quantum bit connectivity. SUMMARY

[0004] Embodiments of the present disclosure include a method of performing a quantum gate operation in an ion-trap quantum computing system. The method includes identifying one or more error mechanisms that cause quantum computation errors in a quantum gate operation on a first trapped ion of an ion chain comprising a plurality of trapped ions, wherein the quantum gate operation is performed by applying a first Raman laser beam and a second Raman laser beam configured to induce Raman transitions in the first trapped ion of the ion chain and induce coupling between the first trapped ion and one or more axial motional modes of the ion chain; calculating a first amplitude of the first Raman laser beam and a second amplitude of the second Raman laser beam such that an effect of the identified one or more error mechanisms is addressed; and applying the first Raman laser beam with the calculated first amplitude and the second Raman laser beam with the calculated second amplitude on the first trapped ion to perform the quantum gate operation on the first trapped ion.

[0005] Embodiments of the present disclosure also include an ion-trap quantum computing system. The ion-trap quantum computing system includes: a quantum processor including an ion chain, the ion chain including a plurality of trapped ions, each trapped ion having two hyperfine states; one or more lasers configured to emit a first Raman laser beam and a second Raman laser beam, the first Raman laser beam and the second Raman laser beam provided to the ion chain in the quantum processor; a classical computer configured to perform operations including: identifying one or more error mechanisms that cause quantum computing errors in a quantum gate operation on a first trapped ion of the ion chain, wherein the quantum gate operation is performed by applying the first Raman laser beam and the second Raman laser beam, the first Raman laser beam and the second Raman laser beam configured to cause Raman transitions in the first trapped ion in the ion chain and cause coupling between the first trapped ion and one or more axial motional modes of the ion chain; and calculating a first amplitude of the first Raman laser beam and a second amplitude of the second Raman laser beam such that an effect of the identified one or more error mechanisms is resolved; and a system controller configured to execute a control program to control the one or more lasers to perform operations on the quantum processor including: applying the first Raman laser beam with the calculated first amplitude and the second Raman laser beam with the calculated second amplitude on the first trapped ion to perform the quantum gate operation on the first trapped ion; and measuring a population of a quantum bit state in the quantum processor.

[0006] Embodiments of the present disclosure also include an ion-trap quantum computing system. The ion-trap quantum computing system includes: a classical computer; a quantum processor including an ion chain, the ion chain including a plurality of trapped ions, each trapped ion having two hyperfine states; a system controller configured to execute a control program to control one or more lasers configured to emit a first Raman laser beam and a second Raman laser beam, the first Raman laser beam and the second Raman laser beam being provided to the ion chain in the quantum processor; and a non-volatile memory having stored therein a plurality of instructions that, when executed by one or more processors, cause the ion-trap quantum computing system to perform operations comprising: identifying, by the classical computer, one or more error mechanisms that cause quantum computing errors in a quantum gate operation on a first trapped ion of the ion chain, wherein the quantum gate operation is performed by applying the first Raman laser beam and the second Raman laser beam, the first Raman laser beam and the second Raman laser beam being configured to cause Raman transitions in the first trapped ion in the ion chain and to cause coupling between the first trapped ion and one or more axial motional modes of the ion chain; calculating, by the classical computer, a first amplitude of the first Raman laser beam and a second amplitude of the second Raman laser beam such that an effect of the identified one or more error mechanisms is resolved; applying, by the system controller, the first Raman laser beam having the calculated first amplitude and the second Raman laser beam having the calculated second amplitude on the first trapped ion to perform the quantum gate operation on the first trapped ion; measuring, by the system controller, a population of a quantum bit state in the quantum processor; and outputting, by the classical computer, the measured population of the quantum bit state in the quantum processor. BRIEF DESCRIPTION OF DRAWINGS

[0007] In order to enable a detailed understanding of the above-mentioned features of the present disclosure, the present disclosure briefly summarized above can be described more specifically by referring to the embodiments (some of which are illustrated in the accompanying drawings). However, it should be noted that the accompanying drawings only illustrate typical embodiments of the present disclosure, and thus should not be considered as limiting the scope thereof, as the present disclosure can allow other equally effective embodiments.

[0008] Figure 1 is a partial view of an ion-trap quantum computer according to one embodiment.

[0009] Figure 2 shows a schematic diagram of an ion-trap for confining ions in a chain according to one embodiment.

[0010] Figure 3A , Figure 3B and Figure 3CSome schematic structures showing the collective lateral motion pattern of a chain of five trapped ions are presented.

[0011] Figure 4 A schematic energy diagram of each ion in a trapped ion chain according to one embodiment is shown.

[0012] Figure 5 The qubit state of an ion is shown as a point on a Bloch sphere.

[0013] Figure 6A The brightness population as a function of the average phonon number is shown according to one embodiment.

[0014] Figure 6B The ratio of the optimized Rabi rate to the static Rabi rate as a function of the average phonon number, according to one embodiment, is shown.

[0015] Figure 7 The brightness population as a function of the average phonon number is shown according to one embodiment.

[0016] Figure 8 The heating rate of the axial motion mode of a single ion is shown according to one embodiment.

[0017] Figure 9A The brightness population as a function of the average phonon number is shown according to one embodiment.

[0018] Figure 9B The ratio of the optimized Rabi rate to the static Rabi rate as a function of the average phonon number, according to one embodiment, is shown.

[0019] Figure 10 A simulation of the distortion of the final state according to one embodiment is shown.

[0020] For ease of understanding, the same reference numerals are used to denote common elements in the accompanying drawings, where possible. An orthogonal coordinate system including the X, Y, and Z axes is used in the drawings and the following description. For convenience, the direction indicated by the arrows in the figures is assumed to be positive. It is conceivable that elements disclosed in some embodiments may be advantageously used in other embodiments without specific description. Detailed Implementation

[0021] Embodiments described herein relate generally to a method for reliably performing quantum computations on a long chain of trapped ions that produces all-to-all quantum bit connectivity (i.e., all qubits are coupled on both small and large distances). The method includes identifying mechanisms that can cause errors in quantum computations and addressing the identified mechanisms. Embodiments of the present disclosure provided herein provide a description of a generalized Hamiltonian for an ion-trap quantum computer system in which quantum gate operations are implemented on a chain of trapped ions using two-photon Raman transitions. This generalized Hamiltonian framework provides tools to identify mechanisms that cause errors in quantum gate operations, such as misalignment, defocusing, or distortion of Raman laser beams, and shifts of trapped ions from their equilibrium positions caused by ion-laser interactions. The methods described herein for identifying mechanisms of quantum computation errors and addressing the identified mechanisms that cause quantum computation errors, which have been validated by comparing simulated results with experimental results for the fidelity of quantum gate operations based on the methods, are useful in reliably performing quantum computations on long chains of trapped ions (e.g., 100 or more trapped ions). In some embodiments, the method further includes applying a sequence of compensation pulses to increase the fidelity of quantum gate operations.

[0022] The methods provided herein enable systematic and quantitative error analysis of various hardware implementations of ion-trap quantum computer systems and provide guidance for designing appropriate error reduction strategies.

[0023] I. General hardware configuration

[0024] Figure 1 is a partial view of an ion-trap quantum computer system (or simply system) 100 according to one embodiment. The system 100 includes a classical (digital) computer 101, a system controller 118, and a quantum register, which is a chain of trapped ions 102 extending along the Z-axis. In Figure 1 In the example shown, five trapped ions are shown for simplicity of discussion. However, the chain 102 can include more than five trapped ions, such as 100 or more trapped ions. Each ion in the chain 102 of trapped ions is an ion having a nuclear spin I and an electron spin s, such that the difference between the nuclear spin I and the electron spin s is zero, such as a positive ytterbium ion 171 Yb + , a positive barium ion 133 Ba + , a positive cadmium ion 111 Cd + or 113 Cd + , all of which have nuclear spin and 2 S 1 / 2 hyperfine states. In some embodiments, all ions in the chain 102 of trapped ions are of the same species and isotope (e.g., 171 Yb+ ). In some other embodiments, the trapped-ion chain 102 includes one or more species or isotopes (e.g., some ions are 171 Yb + , some other ions are 133 Ba + ). In further embodiments, the trapped-ion chain 102 can include various isotopes of the same species (e.g., different isotopes of Yb, different isotopes of Ba). The ions in the trapped-ion chain 102 are individually addressed with individual laser beams.

[0025] The classical computer 101 includes a central processing unit (CPU), a memory, and support circuits (or I / O). The memory is connected to the CPU, and can be one or more of a readily available memory, such as those found in the art, which can be a read only memory (ROM), a random access memory (RAM), a floppy disk, a hard disk or any other form of digital storage, local or remote. Software instructions, algorithms and data can be coded and stored within the memory for instruction of the CPU. The support circuits can also be connected to the CPU for support of the processor. The support circuits can include cache, power supplies, clock circuits, input / output circuitry, subsystems, and the like.

[0026] An imaging objective 104, such as an objective with a numerical aperture (NA) of 0.37, collects fluorescence from the ions along the Y axis and maps each ion onto a multi-channel photomultiplier tube (PMT) 106 for measurement of the individual ions. A counter-propagating Raman laser beam from a laser 108 is provided along the X axis to perform operations on the ions. A diffractive beamsplitter 110 produces a static array of Raman beams 112 that are individually switched using a multi-channel acousto-optic modulator (AOM) 114 and is configured to selectively act on individual ions. A global Raman laser beam 116 illuminates all of the ions at once. In some embodiments, different Raman laser beams (not shown) each illuminate a different ion. A system controller (also referred to as an “RF controller”) 118 controls the AOM 114. The system controller 118 includes a central processing unit (CPU) 120, a read only memory (ROM) 122, a random access memory (RAM) 124, a storage unit 126, and the like. The CPU 120 is a processor of the RF controller 118. The ROM 122 stores various programs, and the RAM 124 is a work memory of various programs and data. The storage unit 126 includes a non-volatile memory such as a hard disk drive (HDD) or a flash memory, and stores various programs even if power is turned off. The CPU 120, the ROM 122, the RAM 124, and the storage unit 126 are interconnected via a bus 128. The RF controller 118 executes a control program stored in the ROM 122 or the storage unit 126 and uses the RAM 124 as a work area. The control program includes one or more software applications that include program code (e.g., instructions) executable by a processor to perform various functions associated with receiving and analyzing data and controlling any and all aspects of the methods and hardware used to produce the ion-trap quantum computer system 100 discussed herein.

[0027] Figure 2 A schematic diagram of an ion trap 200 (also referred to as a Paul trap) for confining ions in a chain 102 according to one embodiment is shown. Confinement potentials are applied by static (DC) voltages and radio frequency (RF) voltages. A static (DC) voltage V S is applied to the endcap electrodes 210 and 212 to confine ions along the X axis (also referred to as the “axial direction” or “vertical direction”). The ions in the chain 102 are nearly uniformly 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 X axis.

[0028] During operation, a sinusoidal voltage V1 (with amplitude V RF is applied to the endcap electrodes 210 and 212 to confine ions along the X axis (also referred to as the “axial direction” or “vertical direction”). The ions in the chain 102 are nearly uniformly 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 X axis. RF / 2) is applied to one pair of opposing electrodes 202, 204, and has a phase that differs by 180° (and has an amplitude V RF / 2) is applied to the other pair of opposing electrodes 206, 208, resulting in a quadrupole potential. In some embodiments, the sinusoidal voltage is applied only to one pair of opposing electrodes 202, 204, while the other pair of opposing electrodes 206, 208 is grounded. This quadrupole potential generates an effective confinement force for each trapped ion in the Y-Z plane perpendicular to the X-axis (also referred to as the "off-axis direction" or "transverse direction") that is proportional to the distance from the saddle point where the RF electric field vanishes (i.e., the position in the axial direction (X-direction)). The motion of each ion in the radial direction (i.e., the direction in the Y-Z plane) is approximately a harmonic oscillation (referred to as secular motion) with a restoring force in the radial direction towards the saddle point, and can be modeled by spring constants k x and k y , respectively. In some embodiments, when the quadrupole potential is symmetric in the radial direction, the spring constants in the radial direction are modeled to be equal. Unfortunately, however, in some cases, the motion of the ions in the radial direction can be distorted due to certain asymmetries in the physical trap configuration, small DC patch potentials due to non-uniformity of the electrode surfaces, etc., and the ions can be off-center from the saddle point due to these and other external distortion sources.

[0029] II. Trapped Ion Configuration and Qubit Information

[0030] Figure 3A 、 Figure 3B and Figure 3C show several schematic structures of collective transverse motion modes of, for example, a chain 102 of five trapped ions. Here, the confinement potential induced by the static voltage V S is weak compared to the confinement potential in the radial direction. The collective motion modes of the chain 102 of trapped ions in the transverse direction are determined by the Coulomb interactions between the trapped ions combined with the confinement potential generated by the ion trap 200. The trapped ions experience collective transverse motions (simply referred to as "motion modes"), each of which has a unique energy (or equivalently, frequency) associated with it. In the following, we refer to the motion mode with the pth lowest energy as |n> p where n denotes the number of motional quanta (in units of energy, referred to as phonons) in that motion mode, and the number of motion modes P in a given transverse direction is equal to the number N of trapped ions in the chain 102. Figures 3A-3C An example of different types of motion modes that five trapped ions located in the chain 102 can experience are schematically shown. Figure 3A is the collective motion mode (also referred to as "center-of-mass mode") with the highest energyP schematic diagram of a tilt mode |n P in which all ions oscillate in phase in the transverse direction. Figure 3B schematic diagram of a tilt mode |n P-1 in which ions at opposite ends move in opposite directions in the transverse direction. Figure 3C schematic diagram of a higher order mode |n P-3 which has lower energy than the tilt mode |n P-1 and in which ions move in a more complex pattern. It should be clear to one of ordinary skill in the art that similar descriptions apply to axial modes.

[0031] It should be noted that the specific configuration described above is just one of several possible examples of a trap for confining ions according to the present disclosure, and does not limit the possible configurations, specifications, etc. of a trap according to the present disclosure. For example, the geometry of the electrodes is not limited to the hyperbolic electrodes described above. In other examples, a trap that produces an effective electric field that causes ions to move in the radial direction as harmonic oscillations can be a multi-layer trap in which multiple layers of electrodes are stacked and RF voltages are applied to two diagonally opposite electrodes, or a surface trap in which all electrodes are in a single plane on a chip. Furthermore, the trap can be divided into multiple segments, pairs of adjacent segments can be linked by shuttling one or more ions, or coupled by photonic interconnects. The trap can also be an array of individual trapping regions closely arranged with each other on a microfabricated ion trap chip. In some embodiments, the quadrupole potential has a spatially varying DC component in addition to the RF component described above.

[0032] Figure 4 shows a schematic energy diagram 400 of each ion in the chain of trapped ions 102 according to one embodiment. Each ion in the chain of trapped ions 102 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 can be a positive ytterbium ion 171 Yb + with a nuclear spin and 2 S 1 / 2 hyperfine states (i.e., two electron states) with an energy splitting corresponding to a frequency difference ω 01 / 2π = 12.642821 GHz (referred to as the “carrier frequency”). In other examples, each ion can be a positive barium ion 133 Ba + , a positive cadmium ion 111 Cd + or 113 Cd+ They all possess nuclear spin. as well as 2 S 1 / 2 Hyperfine states. A qubit is formed by two hyperfine states, denoted as |↓> and |↑>. In the following text, the terms "hyperfine state," "internal hyperfine state," and "qubit" are used interchangeably to denote |↓> and |↑>. Each ion can be cooled (i.e., its kinetic energy can be reduced) to a motion ground state |0> close to any motion mode p without phonon excitations (i.e., n=0) using known laser cooling methods such as Doppler cooling or analytical sideband cooling. p Then, the quantum bit state is prepared in the hyperfine ground state |↓> by optical pumping.

[0033] The state of a single qubit for each trapped ion can be generated by, for example, a 355-nanometer (nm) mode-locked laser through excitation. 2 P 1 / 2 Manipulated by energy levels (denoted as |e>). Figure 4 As shown, the laser beam from this laser can be split into a pair of non-co-propagating laser beams in a Raman configuration (a first laser beam with frequency ω1 and a second laser beam with frequency ω2), and with respect to the transition frequency ω between |↓> and |e>. ↓e With the single-photon transition detuning frequency Δ=ω1-ω ↓e Disharmony, such as Figure 4 As shown. The two-photon transition detuning frequency δ includes the amount of energy supplied to the trapped ion by the first and second laser beams, which, upon binding, is used to transfer the trapped ion between the hyperfine states |↓> and |↑>. When the single-photon transition detuning frequency Δ is much greater than the two-photon transition detuning frequency (also simply referred to as the "detuning frequency") Δω=ω1-ω2-ω ↓↑ (Hereinafter expressed as ±μ, where μ is a positive value), single-photon Rabi frequency Ω ↓e (t) and Ω ↑e (t)(These are time-dependent and determined by the amplitude and phase of the first and second laser beams at the single-photon Rabi frequency Ω) ↓e (t) and Ω ↑e Rabi oscillations occur between states |↓> and |e> and between |↑> and |e>, respectively, and at the spontaneous emission rate from the excited state |e>, Rabi oscillations (called "carrier transitions") occur between the two hyperfine states |↓> and |↑> at the two-photon Rabi frequency (also known as the "Rabi rate") Ω(t). The intensity (i.e., the absolute value of the amplitude) of the two-photon Rabi frequency Ω(t) is related to Ω ↓e Ω ↑e / 2Δω is proportional, where Ω ↓e and Ω ↑e|Ω(t)| = |a1(t)| + |a2(t)|, respectively, are the single-photon Rabi frequencies induced by the first and second laser beams, respectively. In the following, this set of counter-propagating laser beams for Raman configurations that manipulate the internal hyperfine states (qubit states) of the qubit can be referred to as “counter-propagating pulses”. When the propagation is opposite to each other, the set of laser beams can be referred to as “counter-propagating pulses”. It should be clear to one of ordinary skill in the art that related methods can be used to get transitions between |↓> and |↑> using co-propagating (hereinafter referred to as “co-propagating pulses”). The time-dependent pattern of the resulting two-photon Rabi frequency Ω(t) can be referred to as the “amplitude” of the pulses for any propagation setup, as will be illustrated and further described in the following, as long as the context is clear. The detuning frequency Δω = ω1- ω2- ω ↓↑ The detuning frequency of the composite pulse or the detuning frequency of the pulses can be referred to as the “amplitude” of the composite pulse.

[0034] It should be noted that the specific atomic species used in the discussion provided herein is merely one example of an atomic species that has a stable and well-defined two-level energy structure when ionized and whose excited states are optically accessible, and thus is not intended to limit the possible configurations, specifications, etc. of ion-trap quantum computers according to the present disclosure. For example, other ion species include alkaline earth ions (Be + , Ca + , Sr + , Mg + , and Ba + ) or transition metal ions (Zn + , Hg + , Cd + ).

[0035] Figure 5A quantum bit state, represented as a point on the surface of a Bloch sphere 500 with azimuthal angle φ and polar angle θ, is provided to assist in visualizing the ion. As noted above, application of a composite pulse results in Rabi oscillations between the quantum bit states |↓> (represented as the north pole of the Bloch sphere) and |↑> (the south pole of the Bloch sphere). Adjusting the duration and amplitude of the composite pulse causes the quantum bit state to flip from |↓> to |↑> (i.e., from the north pole to the south pole of the Bloch sphere), or from |↑> to |↓> (i.e., from the south pole to the north pole of the Bloch sphere). This application of a composite pulse is referred to as a "π-pulse." Furthermore, by adjusting the duration and amplitude of the composite pulse, the quantum bit state |↓> can be converted to the superposition state |↓> + |↑>, in which the two quantum bit states |↓> and |↑> are added with equal weight in phase (hereinafter, without loss of generality, the normalization factor of the superposition state is omitted), and the quantum bit state |↑> can be converted to the superposition state |↓> - |↑>, in which the two quantum bit states |↓> and |↑> are added with equal weight out of phase. This application of a composite pulse is referred to as a "π / 2-pulse." More generally, a superposition of two quantum bit states |↓> and |↑> that are added with equal weight is represented by a point on the equator of the Bloch sphere. For example, the superposition states |↓> ± |↑> correspond to points on the equator with azimuthal angle φ of zero and π, respectively. A superposition state corresponding to a point on the equator with azimuthal angle φ is represented as |↓> + e iφ |↑>(e.g., for φ = ±π / 2, |↓> ± i |↑>). A transformation between two points on the equator (i.e., a rotation about the Z-axis on the Bloch sphere) can be achieved by shifting the phase of the composite pulse.

[0036] In an ion-trap quantum computer, the motional mode can act as a data bus to mediate entanglement between two quantum bits. That is, each of the two quantum bits is entangled with the motional mode, and then this entanglement is transferred to entanglement between the two quantum bits by using the motional sideband excitation. It should be clear to one of ordinary skill in the art that this is achieved by considering the individual evolution of each quantum bit under the action of a detuned laser pulse, either simultaneously or sequentially, depending on the entanglement gate protocol chosen. Thus, studying the ion-beam interaction of an ion in an ion chain suffices to provide a detailed, complete description of the two-qubit entanglement operator for any chosen protocol.

[0037] III. OVERVIEW OF ION-TRAP QUANTUM COMPUTER SYSTEM

[0038] An ion-trap quantum computer system, such as system 100, which includes two counter-propagating Raman beams, such as Raman beams 112, for selectively accessing individual ions in chain 102, can be described by an effective Hamiltonian H. The effective Hamiltonian H describes the coupling between the motional degrees of freedom (i.e., the motional modes of chain 102) and the internal degrees of freedom (i.e., the internal hyperfine states {|↓>, |↑>}) of the ions in chain 102 via the Raman beams. This effective Hamiltonian H can be split into two parts

[0039] H = H0+ H I , (1)

[0040] where H0independently describes the internal degrees of freedom and the motional degrees of freedom of the ions in chain 102, H I describes the light-matter interaction that couples the internal degrees of freedom and the motional degrees of freedom of the ions in chain 102. By treating the laser field of the Raman beams with classical methods, while treating the rest of system 100 with quantum mechanical methods along with a dipole approximation, the interaction Hamiltonian H I can be written as

[0041]

[0042] where is the electric field, is the dipole operator of the kth ion of the total N ions in chain 102. For the independent Hamiltonian H0, the ions are considered to be confined in a Paul trap, such as ion trap 200, and arranged in chain 102. By approximating the motional modes as harmonic oscillations as described above, and considering the effective two-level system of the internal degrees of freedom of each ion as a qubit, Hamiltonian H0is reduced to

[0043]

[0044] where is the reduced Planck constant, is the effective qubit angular frequency of the kth ion, (α = x, y, or z) is the Pauli matrix along the a-axis, ω p is the normal mode frequency of the pth normal mode with Fock state creation and annihilation (also referred to as “ladder”) operators and .

[0045] In the following, only two Raman beams (b = 1, 2) are considered that drive the qubit transition on a particular ion k, and for simplicity, the ion index k is omitted where clear from the context. The electric field in the vicinity of an ion is given by

[0046]

[0047] where the individual electric fields of the Raman beams (b = 1 or 2) can be written as

[0048]

[0049] where is the polarization vector, ω b is the angular frequency, E b and Φ b are real functions of the ion position in each beam propagation coordinate h.c. stands for Hermitian conjugate. After standard adiabatic elimination of the excited internal states of the ion, the individual addends H I,k of the interaction Hamiltonian H I in (2) can be approximated as

[0050]

[0051] where is the effective dipole constant, Δω is the effective two-photon transition detuning from the Raman transition of the quantum bit transition, is the quantum bit spin operator, which depends on the details of the Raman transition scheme. Here, without loss of generality, we assume that ω1- ω2= ω qbt + Δω, or in other words, the transition from the quantum bit state |↓> to |↑> requires the absorption of a photon from Raman beam 1 and the emission of a photon from Raman beam 2.

[0052] The coupling of the Raman beams to the ion motional modes is embedded in the subscript of (6). In order to rewrite them in terms of the normal motional mode operators and we first write the ion position with respect to each Raman beam as

[0053]

[0054] where is the equilibrium ion position in the beam coordinates {x b , y b , z b}, and is the unit vector in the direction of the axis α b = x b , y b , or z b . Then, the terms and can be written in terms of the ion drift (i.e., the ion motion from the equilibrium ion position ζ αb as Expand nearby. Finally, operators based on normal motion patterns can be used. and ion drift Quantization and rewriting as

[0055]

[0056] in It is the expansion of the zero-point wavefunction of mode p of an ion with mass m. These are the inverse elements of the motion pattern vector matrix. Then, for a given experimental environment, this can be achieved by extending... To fully explain ion-laser interactions in an appropriate form.

[0057] IV. Application to Elliptic Astigmatic Gaussian Beams

[0058] In this section, we consider a Gaussian beam (a prototype example widely used in the trapped-ion quantum computing community) to derive the set of expressions needed for the generalized Hamiltonian. Specifically, in Section IV.A, useful notation for the Gaussian distribution of the coherent beam is defined. In Section IV.B, using the notation defined in Section IV.A, a set of series expressions explicitly dependent on nonideal parameters, including the electric field expression, is derived. In Section IV.C, the ways in which noise can now be coupled into the system are described, and strategies are provided for efficient and systematic error analysis using the derived expressions described herein in practice.

[0059] A. Elliptic astigmatic Gaussian beam

[0060] In the following text, a Gaussian beam is considered to have... A specific form of simple, diffuse elliptical shape, because this specific form can describe a non-symmetrical Gaussian beam due to a certain experimental setup. Beam amplitude E b and phase angle Φ b Ion positions can be used Written as

[0061]

[0062]

[0063] Where it is assumed that the beam is along y b - Axial propagation, and the two principal axes along the x-axis b - and z b -Axis propagation. Here, P b It is the power of the beam, k b =2π / λ b It is a wavelength of λ b wave vector, φ b It is possible to follow yb - A constant phase at the origin of any chosen axis. y is defined by the following formula. b The two principal semi-axes of the light spot ellipse and

[0064]

[0065] in In focus Along α b -The waist of the axis, It is by Given the Rayleigh range. Radius of curvature. Given by the following formula

[0066]

[0067] Where η b It is the Gouy phase, that is,

[0068]

[0069] B. Expansion of the electric field

[0070] Space-related items It can be expanded from the focal point of the beam. The positions of the equilibrium ions are represented as... Ion drift is expressed as The y-distance between the equilibrium ion position and the x and z foci is defined as The scheme considered here, in which ions do not escape from each focus of the corresponding principal axis outside the Rayleigh range, can then be concisely written as: In this scheme, the beam amplitude E in (9) b and phase angle The foci of (10), (11) and (12) about the two principal axes can be included. and Expand, expand into

[0071]

[0072]

[0073] and

[0074]

[0075]

[0076] Among them, space-related items Split into functions A and B as defined by the following formula:

[0077]

[0078]

[0079]

[0080]

[0081]

[0082] where denotes binomial coefficients, denotes the top function, p0may be or may be or q0may be or may be or

[0083]

[0084]

[0085]

[0086]

[0087]

[0088] is dimensionless. In (8), β b , and are quantized and used to evaluate the A and B functions defined in the operator expression (15) according to the normal mode of motion and collect all the non-operator coefficients into the c-coefficients. It should be noted that the c-coefficients are inversely proportional to the square root of their corresponding mode of motion frequency. Furthermore, it should be noted that terms are traditionally used to formulate two-qubit entangling gates, such as the protocol or the Cirac-Zoller protocol, which are well known in the art. It is sometimes easier to keep the exponential form of because its exponent has only first order terms of the ladder operator of the mode of motion.

[0089] It should also be noted that the generalized Hamiltonian framework detailed in this section and shown above is fully general for hardware imperfections, such as beam shape, beam imperfections (e.g. astigmatism), ion position, etc. It enables quantum hardware designers to directly assess the impact of various experimental imperfections on quantum computing fidelity by reducing one or more error mechanisms identified by the generalized Hamiltonian framework. The generalized Hamiltonian framework described herein serves as a diagnostic tool to help designers locate the main sources of quantum computing errors, which is crucial for developing reliable quantum computers.

[0090] C. Error mechanisms and analysis strategies

[0091] By inspecting the above expansion, there are four general mechanisms that can lead to quantum computing errors through the spatial dependence of the Raman beams (b = 1 or 2) that can be identified using a classical computer, such as classical computer 101. The beam amplitudes E b and phase angles Φ b (b = 1 or 2) of the Raman beams are computed using a classical computer, such as classical computer 101, to address the impact of the identified error mechanisms (e.g., minimize or eliminate the impact of the identified error mechanisms). The first error mechanism is related to any misalignment, defocusing, or distortion of the Raman beams (b = 1 or 2) at the ions that is not addressed and that leads to quantum computing errors. The second error mechanism is related to stray fields in the vicinity of the ions that are not compensated for or addressed, which can also lead to quantum computing errors. Through the first and second error mechanisms, quantum computing errors propagate through all parameters in (16). Specifically, they affect the non-operator terms in the definition of the scaled position parameters or or zero-point spread matrix elements of the inverse mode vector matrix beam waist or Rayleigh range The third error mechanism is related to so-called harmonic terms that do not change the motion space, i.e., they have equal number of and operators. and / or of any even total power will contain resonant terms. Except for the trivial case of the constant term, all other terms depend on the occupation of the motional Fock space. Thus, any imprecise control or misinformation about the ion’s motional degrees of freedom can lead to quantum computing errors in the manipulation of the ion qubit. A classical example of this mechanism is the well-known Debye-Waller effect. In some embodiments, errors caused by the third error mechanism can be suppressed by a parameter related to the motional modes, which in turn can be adjusted by tuning the confining potential and / or other parameters of the ion trap, such as ion trap 200. In some embodiments, pulse sequencing methods can additionally or alternatively be used to suppress errors caused by the third error mechanism. The fourth error mechanism is related to ion drift in the phase space of the ion during the quantum gate operation induced by the subsumed terms. When such ion drift occurs and the ion does not return to its initial position in the phase space after the quantum gate operation is completed, it can lead to undesired, persistent entanglement between the internal and motional degrees of freedom of the ion, which is difficult to correct. This effect can be partially suppressed by reducing the and corresponding coefficients of the subsumed terms, sufficient detuning from any motional sideband resonance, and / or by actively shaping the pulses used to perform the quantum gate operation. In some embodiments, errors caused by the fourth error mechanism can be suppressed by sufficient detuning in the MHz range from any motional sideband resonance within reasonable laser power. In some embodiments, active pulse shaping can additionally or alternatively be used to suppress errors caused by the fourth error mechanism. The coefficients of the non-resonant terms related to the fourth error mechanism are time integrals of the pulse functions, so there exist pulse shapes that eliminate or reduce said coefficients. and corresponding coefficients of the subsumed terms, sufficient detuning from any motional sideband resonance, and / or by actively shaping the pulses used to perform the quantum gate operation. In some embodiments, errors caused by the fourth error mechanism can be suppressed by sufficient detuning in the MHz range from any motional sideband resonance within reasonable laser power. In some embodiments, active pulse shaping can additionally or alternatively be used to suppress errors caused by the fourth error mechanism. The coefficients of the non-resonant terms related to the fourth error mechanism are time integrals of the pulse functions, so there exist pulse shapes that eliminate or reduce said coefficients.

[0092] In practice, it is complicated to directly use the A and B functions in (15). A proper and reasonable truncation of the power series in (15) becomes an important task for an approximate but effective error analysis. It should be noted that each function in (15) can be written in the form of an operator sum where each operator has the form where c ij is a complex constant, c 00 is always non-zero. If the contribution of certain operators to the Hamiltonian is small, this task is simplified to neglect them. To quantify this contribution, the operator norm is used. This norm is evaluated over a large but finite number of motional degrees of freedom and is truncated such that the real motional space dynamics can be sufficiently captured therein. In the next section, error analysis is performed for practical cases and specific examples are provided.

[0093] It should be noted that the power series of the non-operator terms in (15) can be rewritten in a more compact way by inspecting the terms with ascending powers of This makes the size of the individual coefficients of the powers of more direct. The results for the first three orders are shown in Section VIII.

[0094] V. Parallel Raman beam geometry

[0095] In this section, the approximate Hamiltonian is derived from the general Hamiltonian using realistic parameters. In one example, a specific analysis of a set of realistic Raman beam parameters relevant to contemporary trapped-ion quantum computing architectures is provided based on (15). In Section V.A, the size of the parameters comparable to contemporary trapped-ion quantum computers is specified. In Section V.B, the error analysis strategy presented in Section IV.C is applied, and a simplified version of the evolution operator that approximates the evolution of the quantum state is described. In Section V.C, the approximate evolution operator is applied, and it is shown in particular that the axial motional mode temperature is important in determining the fidelity of quantum gate operations when tightly focused Raman beams are used.

[0096] It is important to emphasize that the heating of the axial motional mode affects the fidelity of the quantum gate operations in a similar way as the Debye-Waller effect affects the fidelity of the quantum gate operations, i.e., the Rabi frequency driving the internal degree of freedom of the trapped ion depends on the phonon number in the axial motional mode. Therefore, any distribution of the motional mode with a non-zero width (i.e., the peak of the probability distribution of the phonon number of a given motional mode is not at a specific phonon number with probability 1) directly translates into a distribution in the corresponding non-zero width of the Rabi frequency that de-coheres the quantum gate operations. This point will be briefly discussed at the end of Section V.C.

[0097] A. Parameter specifications

[0098] Assume that the chain of trapped ions is addressed by an array of parallel propagating Raman beams that are capable of driving transitions between |↓> and |↑>, tightly focused along the chain axis to enable individual addressing of the trapped ions along the chain. The normal motional modes of the ion chain depend on the dominant projection of its mode vector and include an axial motional mode predominantly along (in the X-direction), a horizontal motional mode along (in the Y-direction), and a vertical motional mode along (in the Z-direction). The coordinate system used herein is defined with respect to the axis of the Raman beams, assuming that the Raman beams propagate along (in the Y-direction) transverse to the axis of the ion chain (in the X-direction), and appear to have a loose dimension along (in the Z-direction) and a tight dimension along An elliptical Gaussian distribution with tight size in the X direction. It is assumed that the equilibrium position of each ion is located near the focal point of each Raman beam so that and

[0099] For quantitative analysis, consider the wavelength λ = 355 nm and the beam waist of the Raman beam to be greater than or approximately ~1 μm (~5 μm), which can be in 171 Yb + ions found in trapped-ion quantum computers. The Rayleigh range of the two principal axes is given by and It is assumed that the alignment error in and is less than 100 nm, and the focusing error in is bounded by 10% of the respective Rayleigh range. Thus, it is satisfied that and Similarly, the alignment of a given normal mode with the dominant principal axis of the Raman beam is quantified by an error parameter ε, which is defined as the maximum relative ion drift of the motion mode vector matrix element scaled by a factor of in the direction of the non-dominant principal axis. The size of the accidental projection of each mode vector along is bounded as shown in Table I. For the considered system, it is assumed that the error parameter ε is less than 0.05. The normal mode frequency ω p / 2π takes about 3 MHz in the horizontal direction, 2.5 MHz in the vertical direction, and between about 150 kHz and about 2 MHz in the axial direction, depending on the number of ions, inter-chain spacing, and DC potential. The resulting size of the c-coefficients appearing in (16) is summarized in Table II.

[0100] Table I: Alignment of the principal axes of the Gaussian beam with the motion mode vectors of different mode groups.

[0101]

[0102] Table II: Evaluation of the size of the c-coefficients in (16) using actual experimental parameters and conditions.

[0103]

[0104] To cool Yb + ions, Doppler cooling to the 2 S 1 / 2 to 2 P 1 / 2 transition is used. The motion mode temperature after Doppler cooling is given by the average phonon number at the Doppler limit Γ = 2π x 19.6 MHz is the excitation 2 P 1 / 2 natural linewidth of the state. Thus, the number of phonons at the Doppler limit is about 4 phonons for the non-axial motion modes and can be about 5 to about 70 phonons for the axial motion modes, depending on the motion mode frequency. For the horizontal motion modes, in the case of the counter-propagating setup, a sideband cooling sequence is applied, which consists of a coherent red sideband pulse followed by optical pumping. This always cools the horizontal motion modes to an average phonon number

[0105] B. Hamiltonian approximation

[0106] The power series truncation strategy proposed in Section IV.C is performed using the actual parameters, such as the parameter values detailed above. To this end, the degree of truncation in the motion space is determined. For the non-axial directions, the initial temperature of the motion modes, i.e., the horizontal motion modes and the vertical motion modes, is assumed to be the motion mode temperature at the Doppler limit, since the motion modes in the non-axial directions are not easily heated, i.e., the number of phonons is not easily increased. Note that the heating of the motion modes in the non-axial directions is more difficult since their vibration frequencies are larger than those of the axial motion modes and also since the electric field that causes the heating needs to obey the open boundary conditions in the axial direction. For the axial direction, the motion modes are easily heated, and the effect of the heating of the axial motion modes on the fidelity after a period of heating is discussed. Thus, about 10 2 phonons of the non-axial motion modes, i.e., the horizontal motion modes and the vertical motion modes, in the non-axial directions and about 10 4 phonons of the axial motion modes in the axial direction are considered, assuming that each motion mode in a given direction is heated more or less uniformly. However, it should be noted that there can be a dominant motion mode in each direction that is heated the most, while the remaining motion modes are not easily heated. To account for this situation, about 10 2 x N phonons of the dominant non-axial motion modes and about 10 4 x N phonons of the dominant axial motion modes are also considered. When determining which operator terms to omit from the Hamiltonian , the uniform heating of all motion modes and the heating of the dominant motion modes in each direction are considered. In designing the effective Hamiltonian, only when the contributions from in both cases are less than 10 -2 is the term omitted from the Hamiltonian As a specific example, assume that the number of ions in the chain 102, N, is N < 50.

[0107] The expression in (15) can now be approximated according to the strategy presented in section IV.C and the parameters specified in section V.A. Only the terms with a magnitude larger than 10 -2 of the A and B functions are

[0108]

[0109]

[0110]

[0111]

[0112]

[0113]

[0114] where It should be noted that the A2function in (17) can be further approximated in case it is used for z b directions, the second A2function used in (13), can be further approximated. In particular, function can be truncated to b due to the larger beam waist along the z direction and the lower temperature of the vertical motion mode. and

[0115] The simplified A and B functions are inserted into the amplitude and phase functions in (13) and (14), respectively, and then the simplified amplitude and phase functions are inserted into the interaction Hamiltonian H I in (6), the interaction Hamiltonian H I is

[0116]

[0117] where and All non-operator terms are collected into the Rabi rate Ω0and the phase Ψ0, which are defined as

[0118]

[0119]

[0120]

[0121] and

[0122]

[0123]

[0124] In order to achieve (18), This is used for both co- and counter-propagating beams, meaning the Debye-Waller effect is negligible. In a co-propagating setting, the local coordinate systems of the beams are mostly aligned, and when two... When the Hamiltonian is substituted into the function, the resulting term is... Because it was eliminated The operator is very close to the identity operation. More specifically, using the system information given above, it can be visually shown that... Less than a predetermined error threshold, for example, 10 -2 In the backpropagation settings, The contribution to the Hamiltonian is significant unless the horizontal motion mode is sufficiently cooled to make n H <<1. Since sideband cooling was used to cool the horizontal motion mode in all experiments using the backpropagation setup to suppress the Debye-Waller effect, it can be assumed that... It is less than the predetermined error threshold, and therefore can be ignored as in the case of same-direction propagation.

[0125] (18) The interacting Hamiltonian H I It can be readily used, for example, in Monte Carlo simulations to evaluate noise sources ranging from beam misalignment and instability to noise at ion positions, as well as the effect of heating on fidelity of motion modes in single-qubit gate operations. It can also be readily incorporated into two-qubit Hamiltonians to evaluate errors in two-qubit gate operations. Note that when the two beams are perfectly aligned with each other, (18) is reduced to a single summation with only one Hermite polynomial term in each addend.

[0126] C. Temperature effect of axial motion mode

[0127] Here, the approximate Hamiltonian expression in (18) is tested by studying the effect of heating of the axial motion mode on fidelity. It is assumed that only one axial motion mode, such as the centroid (COM) axial motion mode, has a dominant behavior in determining the motion mode temperature, thus omitting the motion mode index number p. Next, two practical cases concerning the waist and alignment of the beam are considered.

[0128] In the first case, representing a unidirectional propagation setup, consider those with the same waistband. Two closely focused beams. As a good approximation, we can assume they are perfectly aligned, i.e. and The sum over m in (18) then simplifies to a single term due to the sum rule for products of two Hermite polynomials. The interaction Hamiltonian H I can then be transformed with U0= exp(-iH0t / h) to obtain

[0129]

[0130] where

[0131]

[0132]

[0133] is given by the effective beam waist The parameter η is proportional to the response of the ions to the recoil provided by the optical beam. The parameter ξ is related to the alignment. Both are scaling parameters (ratios) related to the effective beam waist.

[0134] In the second case, representing a counter-propagating setup, consider the case where one of the Raman beams is narrowly focused and individually addressed, while the other Raman beam is very loosely focused and capable of addressing long ion chains. The beam waist of the loosely focused global addressing beam exceeds 100 μιη, which allows us to truncate any terms with m > 0. Thus, the interaction Hamiltonian H I is also of the form in (21). The only difference is that the effective beam waist here is given by the beam waist of the narrowly focused beam .

[0135] For single-qubit gate operations, the Raman transition is taken at the qubit frequency, i.e. Δω = 0. Then any terms with an imbalance of and are off-resonant, and thus suppressed. Neglecting these rapidly rotating couplings, the evolution operator for a single-qubit gate pulse of constant power with duration t sqg can be written as

[0136]

[0137] where |n> is the Fock state in the axial motional mode space, I is the identity operator in the qubit space, and Θ n is defined as

[0138]

[0139] Using

[0140]

[0141] where denotes the Gauss hypergeometric function. Equation (24) explicitly shows how the Rabi frequency for driving the spin degree of freedom depends on the phonon number in the axial motional mode. Thus, a distribution of phonon numbers in the axial motional mode with non-zero width leads to a distribution of Rabi frequencies with corresponding non-zero width, which in turn leads to decoherence of the quantum gate operation.

[0142] It should be noted that the convergence of (24) depends heavily on η and n. For example, for perfect alignment, i.e. ξ→0, with η = 0.01 and n = 2000, m = 4 is required to achieve convergence to the third significant digit. To achieve the same accuracy, m = 11 is required for η = 0.02 and n = 2000, and m = 92 for η = 0.02 and n = 20000. To alleviate some of the convergence problems, one can obtain Thus (24) can be simplified to

[0143]

[0144] Once proper care is taken for the convergence, (26) can be inserted directly in (23) to evaluate the influence of the axial motional mode temperature on the single-qubit gate operation fidelity for different initial states and measurement schemes. This analysis is described in more detail in the next section in connection with experimental results.

[0145] VI. Examples

[0146] In this section, the theoretical results are compared with experimental results. In particular, the influence of the high-temperature axial motional mode in the presence of a tightly focused Raman laser beam is investigated. The experimental setup used to obtain the results described herein comprises a 171 Yb + ion chain, where the axial chain spacing can be controlled by adjusting the voltages of several DC electrodes on the trap. Quantum gate operations are performed by Raman transitions induced by two 355 nm Gaussian beams. State initialization follows a Doppler cooling sequence, where the initial motional mode temperature is cooled to the Doppler limit. Using a counter-propagating setup, the horizontal motional modes are further cooled to This reduces to the effective Hamiltonian of (3), where the qubit states {|↓>, |↑>} are the |F = 0, m F = 0> and |F = 1, m F = 0> hyperfine levels of the ground electronic state, respectively. High-fidelity states are prepared by optical pumping to |0〉 at the beginning of each experiment, and measurements are performed by spatially resolved state-dependent fluorescence detection.

[0147] A. Measurement of the axial motional mode temperature effect

[0148] Following the theoretical analysis shown in Section V, the following steps are performed to probe the experimental apparatus: (A) the quantum state is initialized to where |0> is the qubit state vector, and

[0149]

[0150] is the density operator of the axial motion mode thermal state at time t, with average Fock state occupation number (B) a time delay At is applied, such that the axial motion mode is heated to a higher and the quantum state becomes p0(At); (C) a single-qubit gate operation is implemented, whose unitary matrix is given by (23), where (D) the final state is measured. These steps are repeated to sample the probability of the measured final state being in |↑>. The measurement projection is denoted by where is the identity operator in the motion space, based on Section V.C, the probability of a positive measurement outcome (called the “bright population”) P ↑ is given by

[0151]

[0152] where is the average phonon number after heating by the time delay At. The time delay At used in the examples described herein is on the order of milliseconds, which is much larger than the duration of the single-qubit gate operation, which is on the order of 10 to 100 ps. Therefore, the heating during the gate operation is ignored. Here the bright population P ↑ is a direct measure of the final state fidelity, and thus a good proxy measure of the fidelity of the single-qubit gate operation. The same set of steps is repeated for multiple values of At for each experimental setup of different beam arrangements, chain lengths, and axial motion mode frequencies.

[0153] Figure 6A shows the bright population P1 as a function of the average phonon number obtained from expression (28) for an axial motion mode frequency of 2p x 153 kHz. Specifically, the bright population P ↑ is optimized with respect to the Rabi rate Q0, while assuming an initial average phonon number of which is comparable to the Doppler limit of the axial motion mode. As shown in (9) and (19), the beam amplitudes E b and the phase angle F b(b = 1 or 2) to adjust the Rabi rate Ω0. The beam amplitude E of the Raman beam can be adjusted using a system controller, for example by tightening or loosening the focus of the beam b and the phase angle Φ b (b = 1 or 2). Once a particular Rabi rate Ω0, hereafter referred to as static Rabi rate is obtained, the bright population P can be plotted as a function of the average phonon number ↑ For comparison, the bright population P ↑ is measured as a function of the time delay Δt experimentally using a static Rabi rate calibrated without any delay, and the time delay Δt is mapped to the average phonon number according to the constant heating rate model, i.e. The experiments are performed on a single ion confined in a harmonic trap, where the axial motional mode frequency is adjusted to 2π x 153 kHz by varying the voltage of the DC electrodes of the ion trap. A co-propagating beam setup with a beam waist of is used. The experimental results shown by the circles 602 and the simulated results shown by the dashed line 604 are based on the static Rabi rate based approach, where the Rabi rate The agreement between the experimental results and the simulated results demonstrates the influence of the heating of the axial motional mode on the fidelity of the quantum gate operation. The experimental results shown by the diamonds 606 and the simulated results shown by the solid line 608 are obtained at the same heating rate and initial average phonon number is obtained with respect to the Rabi rate Ω0for each time delay Δt mapped to the average phonon number as described before. The optimal Rabi rate obtained according to this approach is denoted by

[0154] B. Improvement of quantum gate fidelity

[0155] The bright population P ↑ can be easily improved by the static Rabi rate based approach to improve the fidelity of the quantum gate operation. As discussed in the static Rabi rate based approach, the Rabi rate Ω0is assumed to be obtained for an initial average phonon number Theoretically, the bright population P ↑ can be maximized with respect to the Rabi rate Ω0for any average phonon number If the bright population P ↑ allows to be individually optimized for different values of the average phonon number , then a greater bright population P↑ The value of . Figure 6B Show Figure 6A The ratio of the optimized Rabi rate to the static Rabi rate in the experimental and simulation results shown. It can be observed that, based on the average phonon number Adjusting the Rabi ratio Ω0 can improve the fidelity of quantum gate operations. Diamond 610 represents experimental results, and solid line 612 represents simulation results. Therefore, for a given known initial motion mode temperature and heating rate... It can predict the optimal Rabi rate for any quantum gate operation embedded in a quantum circuit without explicit calibration. This improves the overall fidelity of quantum circuits.

[0156] The fidelity of quantum gate operation for heating along the axial motion mode can be further improved by increasing the frequency of the axial motion mode through the following two mechanisms. First, increase the motion mode frequency ω. A Reducing η in (22) further reduces Θ for a specific distribution of n. n The distribution width is reduced, and its decoherent effect on quantum gate operations is decreased. Figure 7 The average phonon number is shown for various axial motion mode frequencies ranging from 2π×153kHz to 2π×513kHz. The brightness population of the function P ↑ Solid lines 702, 704, 706, 708, 710, and 712 represent the bright population P at axial motion mode frequencies of 2π×153kHz, 2π×184kHz, 2π×217kHz, 2π×256kHz, 2π×362kHz, and 2π×513kHz, respectively, obtained by the method based on optimized Rabi rates. ↑ The dashed lines 714, 716, 718, 720, 722, and 724 represent the brightness population P at axial motion mode frequencies of 2π×153kHz, 2π×184kHz, 2π×217kHz, 2π×256kHz, 2π×362kHz, and 2π×513kHz, respectively, obtained by the static Rabi-based method. ↑ It can be observed that the brightness population P obtained by the method based on the optimized Rabi rate and the method based on the static Rabi rate... ↑ For higher axial motion mode frequencies All aspects decay relatively slowly. For a given average phonon number... The increase in the factor R of the axial motion mode frequency is approximately translated into a decrease in the fidelity of the quantum gate operation with respect to R, because 1-P ↑ With η 2to zeroth order. Second, in most cases, increasing the axial motional mode frequency decreases the heating rate associated with the motional mode, thereby increasing the overall fidelity of any quantum circuit with a depth greater than 1. This can be seen from Figure 7 where the P ↑ substitutes the fidelity. Different axial motional frequencies show different decay behavior of P ↑ .

[0157] C. Heating rate probing

[0158] Note that the motional modes described herein can actually serve as a convenient tool to extract the heating rate of the axial motional mode of a single ion or ion chain in an experiment, if its COM axial motional mode heats much faster than the other motional modes. To obtain an accurate estimate, one should measure the static and optimized P ↑ at different time delays At and optimal Rabi rates . The experimental measurement of the brightness distribution P1and the ratio can be fitted to the theoretical predictions by adjusting the initial temperature and the heating rate as fitting parameters. In some embodiments, the initial temperature is fixed to the motional mode temperature at the Doppler limit to reduce the number of fitting parameters. To account for all other dephasing mechanisms that do not depend on the motional mode temperature but lead to a decrease of the brightness distribution P ↑ , an additional fitting parameter dP ↑ is included, such that the final form of the fitting function is as follows

[0159]

[0160]

[0161]

[0162] The same experiment was repeated for a single ion with an axial motional mode frequency co A from 2p x 184 kHz to 2p x 513 kHz. The heating rate as a function of the axial motional mode frequency was extracted using the fitting method described above. As Figure 8 shown. Figure 8 shows the extracted heating rate of the axial motional mode of a single ion for the eight different axial motional mode frequencies shown as circles 802. The solid line 804 is the best fit function of the form . The heating rate was fitted to an inverse power law of the motional mode frequency, and yields

[0163] The methods described herein for measuring the motional mode temperature of a single ion complement the methods using sideband spectroscopy in that sideband spectroscopy acts on modes with mode vectors projected along the direction of beam propagation, while the methods described herein act on motional modes with mode vectors projected perpendicular to the direction of beam propagation. It should be further noted that while the examples shown herein are for relatively large phonon numbers, it is straightforward to extend the methods described herein to lower phonon numbers. This can be done by reducing state preparation and measurement errors and single-qubit gate errors, as well as by reducing the effective beam waist to increase η.

[0164] D. Compensating pulse sequences

[0165] In some embodiments, a narrow-band compensating pulse sequence (which is a combination of single-qubit gate operations) that places an amplitude error in the qubit space can also be used to increase P ↑ sensitivity to the heating rate. It should be noted that for a chain of ions with more than one ion, this sensitivity decreases as η is typically proportional to In this section, experimental demonstrations of the effectiveness of such compensating composite pulse sequences, such as the SK1 pulse sequence and the Tycko three-pulse sequence (see below for details), in mitigating axial temperature-driven errors are discussed below. Specifically, experimental measurements of the bright population P1 as a function of the average phonon number are shown along with simulations.

[0166] A single-qubit gate operation that rotates the Bloch vector by θ about a rotation axis with polar angle φ on the equator of the Bloch sphere can be parameterized as

[0167]

[0168] The SK1 pulse sequence is then given by

[0169]

[0170] where ψ = arccos(-1 / 4). The Tycko three-pulse sequence is then given by

[0171]

[0172] In practice, the term in (31) is implemented by performing two For is implemented similarly.

[0173] Figure 9A ​It is shown that, in addition to the above SK1 pulse sequence and Tycko three-pulse sequence, for static Rabi rates and optimized Rabi rates The bright population P as a function of the average phonon number ↑ The experiment was performed on the middle ion of a chain of 25 ions with a COM axial motional mode with a mode frequency of A = 2π x 148 kHz, which heats up the fastest. The counter-propagating setup was used, with individually addressed narrow focused beams having a beam waist of 0.87(2) pm along the x-axis, while the globally addressed beam had a beam waist of ~200 pm. Sideband cooling of the horizontal motional mode was implemented prior to state preparation. The dots 902 and the dashed line 904 are the experimental results and the simulated results, respectively, for static Rabi rates The dots 906 and the solid line 908 are the experimental results and the simulated results, respectively, for optimized Rabi rates The experimental results shown by the dots 910 and the simulated results shown by the line 912 demonstrate that the Tycko three-pulse sequence effectively mitigates the decay of the bright population P ↑ The experimental results shown by the dots 914 and the simulated results shown by the line 916 demonstrate that the SK1 pulse sequence also effectively mitigates the decay of the bright population P ↑ It can be observed that the compensating pulse sequences (i.e., the Tycko three-pulse sequence and the SK1 pulse sequence) provide better performance than the optimized Rabi rate-based method and the static Rabi rate-based method without compensating pulse sequences. The simulated results for the SK1 or Tycko three-pulse sequence used in the examples described herein exhibit a system phase error of 0.4 radian / gate.

[0174] Figure 9B The ratio as a function of the average phonon number The diamonds 918 are the experimental results, and the solid line 920 is the simulated results. The error caused by heating of the axial thermal mode is essentially an amplitude error of the quantum gate unitary matrix, which can be mitigated by compensating pulse sequences designed for the target amplitude error.

[0175] From Figure 9A and Figure 9BAs can be seen, these pulse sequences offer further improvements in quantum gate operation fidelity compared to the optimized Rabi rate method, with the Tycko three-pulse sequence being particularly notable. Note that, to achieve consistency between the experimental and simulation results for the pulse sequences, it is assumed that the phase error, which can be attributed to poorly calibrated optical offsets and qubit frequency errors, gradually increases by 0.4 radians per gate. Such phase errors affect the Ming population P of a single Rabi pulse. ↑ There is no impact, but the efficiency of the compensated pulse sequence is reduced. As the basis gates constituting the SK1 or Tycko three-pulse sequences, better calibrated single-qubit gate operations will further improve P... ↑ .

[0176] VII. Hardware Design Strategy

[0177] The generalized Hamiltonian framework intuitively shows that the fidelity of quantum gate operations increases rapidly as the temperature of motion modes decreases. Figure 10 Showing as the average phonon number The intermediate ion in the chain of 25 ions of the function of 1-P ↑ The simulation represents the fidelity of the final state. In this example, only the decorrelation effect of the COM axial motion mode at a frequency of 2π × 148 kHz is considered. Solid line 1002 represents the method based on the optimized Rabi rate, dotted line 1004 represents the SK1 pulse sequence case, and dashed line 1006 represents the Tycko three-pulse sequence case. (Check) Figure 10 Considering hardware design strategies, reducing the heating rate of the ion trap itself by a moderate amount will significantly improve the fidelity of quantum gate operations, as this will prevent heating in axial motion modes. An effective sympathetic cooling scheme during quantum circuit execution can also be considered, thereby reducing the heating rate throughout the quantum computing runtime. Maintaining this level is acceptable. Increasing the frequency of axial motion modes by reducing the ion spacing or using optical tweezers is also a feasible way to improve the fidelity of quantum gate operation, as this helps to reduce the size of η and can lower the heating rate. It should be noted that increasing the beam waist of the individually addressed Raman beam will directly reduce η, thereby reducing unwanted decoherence. Finally, by using the compensation pulse sequence described herein, fidelity can be significantly improved. Figure 10 Additional simulation data is shown, illustrating the expected distortion 1-P1 for SK1 and Tycko three-pulse sequences. Clearly, the compensated pulse sequence can significantly increase the acceptable heating rate for successful quantum gate operation with high fidelity. The range.

[0178] It should be noted that the additional terms in the interaction Hamiltonian can now be systematically included in descending order of their contribution to the fidelity of quantum gate operations, to help achieve high-fidelity trapped-ion quantum computation. For example, Debye-Waller effect induced terms The higher order terms in the function can also be considered to originate from the Gouy phase. It should be noted that the latter would itself appear as a small correction to the Debye-Waller effect. These terms would lead to decoherence if the temperature of the motional modes of the ion chain is high and / or the misalignment between the ions and their addressing beams is large. The Hamiltonian framework disclosed herein accurately captures these effects analytically and provides a quantitative method to characterize their impact on the fidelity of quantum gate operations.

[0179] While the impact on single-qubit gate operations related to coupling with axial motional modes is primarily described, similar derivations and analyses can be readily extended to two-qubit gate operations. In fact, most of the conclusions, including mitigation strategies and techniques for single-qubit gates, have analogous and similar counterparts for two-qubit gate operations. Thus, in some embodiments, one or more of the mitigation strategies and techniques described herein can be used for two-qubit gate operations.

[0180] In the present disclosure, a general Hamiltonian is provided that is capable of precisely determining the sources of distortions in a trapped-ion quantum computer with long chains. By carefully analyzing the Hamiltonian with realistic beam geometries and parameters, quantum computation errors caused by alignment and focusing have been identified. The generalized Hamiltonian framework described herein is universal, accurately listing all important terms according to user-defined quality requirements for any trapped-ion quantum computing platform. It is expected that the results described herein can enable hardware engineers to make informed decisions when designing quantum computers.

[0181] While the foregoing is directed to particular embodiments, other and further embodiments can be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.

Claims

1. A method for performing quantum gate operations in an ion trap quantum computing system, comprising: One or more error mechanisms are identified by a classical computer, which cause quantum computing errors in a quantum gate operation performed on a first trapped ion of an ion chain comprising a plurality of trapped ions, wherein the quantum gate operation is performed by applying a first Raman laser beam and a second Raman laser beam, the first Raman laser beam and the second Raman laser beam being configured to induce Raman transitions in the first trapped ion in the ion chain and to induce coupling between the first trapped ion and one or more axial motion modes of the ion chain; The temperature of the axial motion mode of the ion chain is measured by the system controller; Based on the measured temperature of the ion chain, the classical computer calculates the first amplitude of the first Raman laser beam and the second amplitude of the second Raman laser beam to reduce errors caused by one or more identified error mechanisms; The system controller applies a first Raman laser beam with a calculated first amplitude and a second Raman laser beam with a calculated second amplitude to the first trapped ion to perform the quantum gate operation on the first trapped ion; as well as Adjust the limiting potential generated by the ion trap in which the plurality of trapped ions are confined.

2. The method according to claim 1, further comprising: An SK1 pulse sequence is applied to the first trapped ion, wherein the SK1 pulse sequence includes multiple single-qubit gate operations performed on the first trapped ion.

3. The method according to claim 1, further comprising: A Tycko three-pulse sequence is applied to the first trapped ion, wherein the Tycko three-pulse sequence includes multiple single-qubit operations performed on the first trapped ion.

4. The method according to claim 1, further comprising: A first Raman laser beam with a calculated first amplitude and a second Raman laser beam with a calculated second amplitude are applied to the second trapped ion in the ion chain. The quantum gate operation is performed on the first trapped ion and the second trapped ion.

5. An ion trap quantum computing system, comprising: A quantum processor comprising an ion chain, the ion chain comprising a plurality of trapped ions, each trapped ion having two hyperfine states; One or more lasers configured to emit a first Raman laser beam and a second Raman laser beam, the first Raman laser beam and the second Raman laser beam being provided to the ion chain in the quantum processor; Classic computer and system controller, configured to perform operations including: The classical computer identifies one or more error mechanisms that cause quantum computing errors in a quantum gate operation performed on a first trapped ion in the ion chain, wherein the quantum gate operation is performed by applying a first Raman laser beam and a second Raman laser beam, the first Raman laser beam and the second Raman laser beam being configured to induce Raman transitions in the first trapped ion in the ion chain and to induce coupling between the first trapped ion and one or more axial motion modes of the ion chain; The temperature of the axial motion mode of the ion chain is measured by the system controller; Based on the measured temperature of the ion chain, the classical computer calculates the first amplitude of the first Raman laser beam and the second amplitude of the second Raman laser beam to reduce errors caused by one or more identified error mechanisms; The system controller applies a first Raman laser beam with a calculated first amplitude and a second Raman laser beam with a calculated second amplitude to the first trapped ion to perform the quantum gate operation on the first trapped ion; Adjust the confinement potential generated by the ion trap in which the plurality of trapped ions are confined; Measuring the population of the qubit states in the quantum processor; and The population of the qubit states in the quantum processor, as measured by the output of the classical computer.

6. The ion trap quantum computing system according to claim 5, further comprising: An SK1 pulse sequence is applied to the first trapped ion, wherein the SK1 pulse sequence includes multiple single-qubit gate operations performed on the first trapped ion.

7. The ion trap quantum computing system according to claim 5, further comprising: A Tycko three-pulse sequence is applied to the first trapped ion, wherein the Tycko three-pulse sequence includes multiple single-qubit operations on the first trapped ion.

8. The ion trap quantum computing system according to claim 5, further comprising: A first Raman laser beam with a calculated first amplitude and a second Raman laser beam with a calculated second amplitude are applied to the second trapped ion in the ion chain. The quantum gate operation is performed on the first trapped ion and the second trapped ion.

9. An ion trap quantum computing system, comprising: Classic computer; A quantum processor comprising an ion chain, the ion chain comprising a plurality of trapped ions, each trapped ion having two hyperfine states; A system controller is configured to execute a control program to control one or more lasers, the one or more lasers being configured to emit a first Raman laser beam and a second Raman laser beam, the first Raman laser beam and the second Raman laser beam being provided to the ion chain in the quantum processor; as well as A non-volatile memory having a plurality of instructions stored therein, which, when executed by one or more processors, cause the ion trap quantum computing system to perform operations, including: The classical computer identifies one or more error mechanisms that cause quantum computing errors in a quantum gate operation performed on a first trapped ion in the ion chain, wherein the quantum gate operation is performed by applying a first Raman laser beam and a second Raman laser beam, the first Raman laser beam and the second Raman laser beam being configured to induce Raman transitions in the first trapped ion in the ion chain and to induce coupling between the first trapped ion and one or more axial motion modes of the ion chain; The temperature of the axial motion mode of the ion chain is measured by the system controller; Based on the measured temperature of the ion chain, the classical computer calculates the first amplitude of the first Raman laser beam and the second amplitude of the second Raman laser beam to suppress errors caused by one or more identified error mechanisms. The system controller applies a first Raman laser beam with a calculated first amplitude and a second Raman laser beam with a calculated second amplitude to the first trapped ion to perform the quantum gate operation on the first trapped ion; Adjust the confinement potential generated by the ion trap in which the plurality of trapped ions are confined; The population of the qubit states in the quantum processor is measured by the system controller; and The population of the qubit states in the quantum processor, as measured by the output of the classical computer.

10. The ion trap quantum computing system according to claim 9, further comprising: An SK1 pulse sequence is applied to the first trapped ion, wherein the SK1 pulse sequence includes multiple single-qubit gate operations performed on the first trapped ion.

11. The ion trap quantum computing system according to claim 9, further comprising: A Tycko three-pulse sequence is applied to the first trapped ion, wherein the Tycko three-pulse sequence includes multiple single-qubit operations performed on the first trapped ion.

12. The ion trap quantum computing system according to claim 9, further comprising: A first Raman laser beam with a calculated first amplitude and a second Raman laser beam with a calculated second amplitude are applied to the second trapped ion in the ion chain. The quantum gate operation is performed on the first trapped ion and the second trapped ion.

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