Motion pattern configuration for implementing entangling gates in ion-trap quantum computers

Abstract

A method of performing a computation using a quantum computer, comprising modulating a motional mode structure of a plurality of trapped ions, each of the plurality of trapped ions having two frequency split states defining a qubit; computing a function of detuning frequency and a function of amplitude of a laser pulse that will cause an entangling interaction between a pair of the plurality of trapped ions; and performing a quantum computation in the quantum computer by applying a gate duration on the pair of trapped ions with the laser pulse having the computed function of detuning frequency and the function of amplitude.

Description

TECHNICAL FIELD

[0001] The present invention relates generally to a method of generating entangling gates in an ion-trap quantum computer, and more particularly, to a method of constructing simple laser pulses to generate entangling gates with a motional mode structure configured such that the pulses are practically implementable. BACKGROUND

[0002] In quantum computing, quantum bits or qubits (analogous to bits representing "0" and "1" in classical (digital) computers), need to be prepared, manipulated, and measured (read out) with near-perfect control during the computation process. Imperfect control of qubits causes errors to accumulate during the computation process, limiting the size of quantum computers that can perform reliable computations.

[0003] Among the physical systems proposed to build large-scale quantum computers are chains of ions (e.g., charged atoms) trapped and suspended in a vacuum by electromagnetic fields. These ions have internal hyperfine states that are separated by frequencies in the GHz range that can be used as computational states of the qubits (referred to as "qubit states"). These hyperfine states can be controlled using radiation provided by a laser, or sometimes referred to herein as interaction with a laser beam. With this laser interaction, the ions can be cooled to near their motional ground state. The ions can also be optically pumped with high precision to one of the two hyperfine states (preparation of the qubit), manipulated between the two hyperfine states (single-qubit gate operations) by laser beams, and their internal hyperfine states detected by fluorescence when a resonant laser beam is applied (readout of the qubit). A pair of ions can be controllably entangled (two-qubit gate operations) using laser pulses that couple the ions to collective motional modes of the trapped ion chain that arise from the inter-ion Coulomb interaction.

[0004] However, in large-scale quantum computing, a series of laser pulses are used to perform the computation process, resulting in the complexity of the optical and electronic technology used to modulate these laser pulses. Therefore, there is a need for a procedure to control the qubits in the physical system to perform the desired computation process with reduced complexity. SUMMARY

[0005] Embodiments of the invention provide a method of performing a computation using a quantum computer. The method includes modulating a motional mode structure of a plurality of trapped ions, each of the plurality of trapped ions having two frequency split states defining a qubit; computing a detuning frequency function and an amplitude function of a laser pulse that will cause an entangling interaction between a pair of trapped ions of the plurality of trapped ions; and performing a quantum computation in the quantum computer by applying a gate duration to the pair of trapped ions with the laser pulse having the computed detuning frequency function and amplitude function.

[0006] Embodiments of the invention also provide a non-transitory computer readable medium comprising computer program instructions. When executed by a processor, the computer program instructions cause the processor to compute a detuning frequency function and an amplitude function of a laser pulse that will cause an entangling interaction between a pair of trapped ions of a plurality of trapped ions; and perform a quantum computation in a quantum computer by applying a gate duration to the pair of trapped ions with the laser pulse having the computed detuning frequency function and amplitude function. A motional mode frequency of the plurality of trapped ions is modulated to be an integer multiple of a quotient of 4p divided by the gate duration.

[0007] Embodiments of the invention further provide a quantum computing system. The quantum computing system includes a plurality of trapped ions in an ion trap, each of the plurality of trapped ions having two hyperfine states defining a qubit; and a controller including a non-transitory memory having stored therein a plurality of instructions, which when executed by a processor, cause the quantum computing system to perform operations including computing a detuning frequency function and an amplitude function of a laser pulse that will cause an entangling interaction between a pair of trapped ions of the plurality of trapped ions; and performing a quantum computation in a quantum computer by applying a gate duration to the pair of trapped ions with the laser pulse having the computed detuning frequency function and amplitude function. A confinement potential of the ion trap is modulated such that a motional mode frequency of the plurality of trapped ions is an integer multiple of a quotient of 4p divided by the gate duration. BRIEF DESCRIPTION OF DRAWINGS

[0008] So that the manner in which the above-recited features of the present invention can be understood in detail, a more particular description of the invention, briefly summarized above, can be had by reference to embodiments, some of which are illustrated in the appended drawings. It is to be noted, however, that the appended drawings illustrate only typical embodiments of this invention and are therefore not to be considered limiting of its scope, for the invention can admit to other equally effective embodiments.

[0009] Figure 1 is a partial view of an ion trap quantum computer in accordance with one embodiment.

[0010] Figure 2 A schematic diagram of an ion trap for confining ions in a chain according to one embodiment is depicted.

[0011] Figure 3A Figure 3B Figure 3C A few schematic collective transverse motion mode structures of a chain of trapped ions are depicted.

[0012] Figure 4 A schematic energy diagram of each ion in a chain of trapped ions according to one embodiment is depicted.

[0013] Figure 5 A quantum bit state of an ion represented as a point on a Bloch sphere surface is depicted.

[0014] Figure 6A 6B A schematic diagram of each ion's motion sideband spectrum and motion modes according to one embodiment is depicted.

[0015] Figure 7 A numerically computed topography of the static (DC) voltage V S applied to the trap electrodes according to one embodiment is depicted.

[0016] Figure 8 A numerically computed gate power ratio |χ ij / Ω 2 τ 2 | according to one embodiment is depicted.

[0017] Figure 9A A numerically computed pulse function according to one embodiment is depicted. 9B

[0018] A numerically computed residual coupling a for fluctuations δω in the mode frequencies according to one embodiment is depicted. Figure 10

[0019] A flowchart illustrating a method 1100 for performing computations using a quantum computer is depicted. Figure 11 For ease of understanding, the same reference numbers will be used in different drawings to designate the same or similar elements. In the drawings and the following description, an orthogonal coordinate system including an X-axis, a Y-axis, and a Z-axis is used. For convenience, the direction indicated by the arrow in the figures is assumed to be the positive direction. It is contemplated that elements disclosed in some embodiments can be beneficially used in other embodiments without specific recitation.

[0020] DETAILED DESCRIPTION

[0021] ​​​​The embodiments described herein relate generally to methods and systems for constructing and delivering pulses to perform entangling gate operations between two trapped ions during quantum computation, and more particularly, to a method of designing pulses that can be implemented in the described systems while simplifying the system complexity, but still achieving high fidelity of entangling gate operations, or achieving high probability of at least two ions being in predetermined qubit states after performing entangling gate operations between two ions. It should be noted that although the method described herein is for entangling gate operations between two ions, the method can also be used for single qubit operations and entangling operations between more than two ions.

[0022] The overall system capable of quantum computation 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 run using a user interface (such as a graphical processor (GPU)), compiling the selected quantum algorithm into a series of universal logic gates, converting the series of universal logic gates into laser pulses to be applied for the quantum register, and pre-computing parameters to optimize the laser pulses using a central processing unit (CPU). Software programs for performing decomposition and executing quantum algorithm tasks are stored in non-volatile memory within the classical computer. The quantum register includes trapped ions coupled with various hardware, including lasers that manipulate internal hyperfine states (qubit states) of the trapped ions and acousto-optic modulators that read out the internal hyperfine states (qubit states) of the trapped ions. The system controller receives the pre-computed parameters of the pulses from the classical computer at the beginning of running the selected algorithm on the quantum register, controls the various hardware associated with controlling any and all aspects of running the selected algorithm on the quantum register, and returns read values of the quantum register, thereby outputting the results of the quantum computation to the classical computer at the end of running the algorithm.

[0023] The methods and systems described herein include processes for converting logic gates into laser pulses applied to a quantum register, and processes for pre-computing parameters to optimize laser pulses applied to a quantum register and for improving the performance of a quantum computer.

[0024] Among several known sets of universal logic gates through which any quantum algorithm can be decomposed, a set of universal logic gates, generally denoted as {R, XX}, is inherent to the trapped-ion quantum computing systems described herein. Among these, the R gate corresponds to manipulation of individual qubit states of trapped ions, and the XX gate (also referred to as an "entangling gate") corresponds to manipulation of entanglement of two trapped ions. It should be clear to one of ordinary skill in the art that the R gate can be implemented with near perfect fidelity, while the formation of the XX gate is complex and requires optimization for a given type of trapped ion, number of ions in the chain of trapped ions, and the hardware and environment in which the trapped ions are trapped, to name a few factors, to improve fidelity of the XX gate and avoid or reduce computational errors within the quantum computer. In the following discussion, methods for the generation and optimization of pulses for performing computations based on the formation of XX gates with improved fidelity will be described.

[0025] As quantum computers scale in size, the complexity of entangling gate operations for performing quantum computations increases, and the complexity of pulses for performing these entangling gate operations also increases. There can be practical limitations in implementing pulses of increased complexity. The methods and systems described in the present invention modify such pulses so that they can be practically implemented without compromising precise control over the qubits.

[0026] General hardware configuration

[0027] Figure 1 is a partial view of an ion-trap quantum computer or 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 102 of trapped ions (i.e., five are shown) extending along the Z-axis. Each ion in the chain 102 of trapped ions is an ion having a nuclear spin I and an electron spin S, the difference between which 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 of the ions in the chain 102 of trapped ions are of the same species and isotope (e.g., 171 Yb + ). In some other embodiments, the chain 102 of trapped ions includes one or more species or isotopes (e.g., some of the ions are 171 Yb +some other ions are 133 Ba + ). In yet another embodiment, the chain of trapped ions 102 can include various isotopes of the same species (e.g., different isotopes of Yb, different isotopes of Ba). The ions in the chain of trapped ions 102 are individually addressed with separate laser beams.

[0028] 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 read-only memory (ROM), random access memory (RAM), floppy disks, hard disks, or any other form of digital storage. Software instructions, algorithms, and data can be coded and stored within memory for instruction of the CPU. The support circuits (not shown) are also connected to the CPU for supporting the processor in a conventional manner. The support circuits can include conventional cache, power supplies, clock circuits, input / output circuitry, subsystems, and the like.

[0029] An imaging objective 104, such as an objective with a numerical aperture (NA) of, for example, 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 non-collimated Raman laser beam from a laser 108 provided along the X axis performs operations on the ions. A diffraction beamsplitter 110 creates an array of static Raman beams 112 that are individually switched using multi-channel acousto-optic modulators (AOMs) 114, and is configured to selectively act on individual ions. A global Raman laser beam 116 illuminates multiple ions at once. In some embodiments, individual Raman laser beams (not shown) individually illuminate individual ions. A system controller (also referred to as an “RF controller”) 118 controls the AOMs 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 working 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 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 the processor to perform various functions associated with receiving and analyzing data and controlling any and all aspects of the methods and hardware used to create the ion-trap quantum computer system 100 discussed herein.

[0030] 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 depicted. The confinement potential is imposed by both static (DC) and radio frequency (RF) voltages. A static (DC) voltage V S is applied to endcap electrodes (also referred to as “DC control electrodes”) 210 and 212 to confine ions along the Z-axis (also referred to as the “axial,” “longitudinal,” or “first” direction). Due to the Coulomb interaction between ions, the ions in the chain 102 are nearly uniformly distributed in the axial direction. In some embodiments, the ion trap 200 includes four hyperbolic electrodes (also referred to as “RF rods”) 202, 204, 206, and 208 that extend along the Z-axis.

[0031] During operation, a sinusoidal voltage V1 (having an amplitude V RF / 2) is applied to one pair of opposing electrodes 202, 204, and a sinusoidal voltage V2 (having an amplitude V RF / 2) that is phase-shifted by 180° from the sinusoidal voltage V1 is applied to the other pair of opposing electrodes 206, 208 at a drive frequency ω RF / 2) to the other pair of opposing electrodes 206, 208, thereby creating a quadrupole potential. In some embodiments, the sinusoidal voltage is applied only to one pair of opposing electrodes 202, 204, and the other pair of opposing electrodes 206, 208 is grounded. The quadrupole potential creates an effective confinement force for each trapped ion in the X-Y plane perpendicular to the Z-axis (also referred to as the “radial,” “lateral,” or “second” direction) that is proportional to the distance from the saddle point (i.e., the location in the axial (Z-direction)) where the RF electric field vanishes. The motion of each ion in the radial direction (i.e., the direction in the X-Y plane) is approximately harmonic (referred to as “long-term motion”), with a restoring force in the radial direction toward the saddle point and can be modeled by spring constants k x and k y , respectively, which are discussed in more detail below. In some embodiments, when the quadrupole potential is symmetric in the radial direction, the spring constants in the radial direction are modeled as equal. However, in some undesirable cases, the motion of the ions in the radial direction can be distorted due to some asymmetry in the physical trap configuration, small DC patch potentials due to non-uniformity of the electrode surfaces, etc., and the center of the ion can deviate from the saddle point due to these and other external distortion sources.

[0032] It should be noted that, Figure 2The specific example of a conventional macroscopic ion trap shown in the middle is just one possible example of an ion trap for confining ions, and does not limit the possible configurations, dimensions, etc. of ion traps according to the present application. For example, the ion trap can be a microfabricated surface trap in which the electromagnetic confinement potentials needed to trap ions are formed above the surface of a semiconductor chip. Examples of such surface traps include the Sandia high-optical access (HOA) 2.0 trap and the GTRI / Honeywell ball-grid array (BGA) trap, both of which are known in the art. As is well known, such microfabricated surface traps provide better capabilities to design the geometry of the ion trap and to configure the various parameters of the electromagnetic confinement potentials with high repeatability and manufacturing yield, as compared to macroscopic ion traps that are assembled by hand.

[0033] Trapped ion configurations and qubit information

[0034] Figure 3A 、 Figure 3B and Figure 3C depicts some schematic structures of collective transverse motion modes (also referred to as “motion mode structures”) of, for example, a chain 102 of five trapped ions. Therein, the confinement potential in the transverse direction is weaker than the confinement potential in the radial direction due to the static voltages V S applied to the endcap electrodes 210 and 212. 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 in combination with the confinement potential generated by the ion trap 200. The trapped ions experience collective transverse motion (referred to as “collective transverse motion modes”, “collective motion modes” or simply “motion modes”), wherein each mode has a unique energy (or equivalently frequency) associated with it. The motion mode with the p-th lowest energy (or equivalently mode frequency ω p ) is referred to as |n ph > p where n ph denotes the motional quantum number (in units of energy excitation, referred to as “phonon”) in the 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 collective transverse motion modes that five trapped ions located in the chain 102 can experience are schematically shown. Figure 3A is the common motion mode (also referred to as “center-of-mass (COM) mode”) |n ph > P with the highest energy, wherein P is the number of the mode and the total number of motion modes. In the common motion mode |n ph > P , all ions oscillate in phase in the transverse direction. Figure 3B is the tilt motion mode (simply referred to as “tilt mode”) |nph P-1 schematic diagram of the tilted motion mode. In the tilted motion mode, the ions at both ends move out of phase in the transverse direction (i.e., in opposite directions). Figure 3C is a higher order motion mode (also referred to as the "sawtooth mode") |n ph P-3 schematic diagram of the tilted motion mode |n ph P-1 with lower energy and in which the ions move in a more complex pattern of motion.

[0035] It should be noted that the specific configurations described above are merely one of several possible examples of a trap for confining ions according to the present application, and do not limit the possible configurations, specifications, etc. of the trap according to the present application. 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 the ions to move in the radial direction as a harmonic motion can be a multi-layer trap in which multiple layers of electrodes are stacked and an RF voltage is applied to two diagonally opposite electrodes, or a surface trap in which all the electrodes are located in a single plane on a chip. Furthermore, the trap can be divided into multiple segments, in which adjacent pairs 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 that are 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.

[0036] Figure 4 depicts 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, with a difference of zero between the nuclear spin I and the electron spin S. 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 (referred to as the "carrier frequency") of ω 01 / 2π = 12.642812 GHz. In other examples, each ion can be a positive barium ion 133 Ba + , a positive cadmium ion 111 Cd + , or 113 Cd + , all of which have a nuclear spin and 2 S 1 / 2 hyperfine states. The qubit is constructed from the two hyperfine states, denoted as |0> and |1>, with the hyperfine ground state (i.e., 2 S​​​1 / 2 |0> is represented by a low-energy state in a hyperfine state. In the following text, the terms "hyperfine state," "internal hyperfine state," and "qubit" are used interchangeably to denote |0> and |1>. Each ion can be cooled (i.e., its kinetic energy can be reduced) to a phonon-free ground state |0> close to any motion mode p by known laser cooling methods, such as Doppler cooling or resolved sideband cooling. p (that is, n) ph =0), and then a quantum bit state is prepared in the hyperfine ground state |0> by optical pumping. Here, |0> represents the individual quantum bit state of the trapped ion, while |0> with subscript p p The ground state represents the motion mode p of the trapped ion chain 102.

[0037] The individual qubit state of each trapped ion can be generated by, for example, a 355 nm mode-locked laser via excitation. 2 P 1 / 2 Manipulated by energy levels (denoted as |e>). Figure 4 As shown, the laser beam from the laser can be split into a pair of non-co-propagating laser beams (a first laser beam with frequency ω1 and a second laser beam with frequency ω2) in a Raman configuration, and with respect to the transition frequency ω between |0> and |e>. 0e With the single-photon transition detuning frequency Δ=ω1-ω 0e 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, when combined, are used to transfer the trapped ion between the hyperfine states |0> and |1>. 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") δ = ω1 - ω2 - ω 01 (Hereinafter expressed as ±μ, where μ is a positive value), single-photon Rabi frequency Ω 0e (t) and Ω 1e (t) (which is time-dependent and determined by the amplitude and phase of the first and second laser beams, with Rabi oscillations occurring at this frequency between states |0> and |e> and between states |1> and |e>, respectively), and the spontaneous emission rate from the excited state |e>, induces Rabi oscillations (called "carrier transitions") between two hyperfine states |0> and |1> at the two-photon Rabi frequency Ω(t). The intensity (i.e., the absolute value of the amplitude) of the two-photon Rabi frequency Ω(t) is related to Ω. 0e Ω 1e / 2Δ is proportional, where Ω 0e and Ω 1esingle-photon Rabi frequency due to the first and second laser beams, respectively. In the following, this set of non-collinear laser beams in a Raman configuration for manipulating the internal hyperfine states (qubit states) of the quantum bit can be referred to as a "composite pulse" or simply as a "pulse", and the resulting time-dependent pattern of the two-photon Rabi frequency Ω(t) can be referred to as the "amplitude" of the pulse or simply as the "pulse", which will be explained and further described below. The detuning frequency δ = ω1- ω2- ω 01 The detuning frequency of the composite pulse or the detuning frequency of the pulse can be referred to. The amplitude of the two-photon Rabi frequency Ω(t) determined by the amplitudes of the first and second laser beams can be referred to as the "amplitude" of the composite pulse.

[0038] 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 stable and well-defined two-level energy structure upon ionization and has optically accessible excited states, and thus is not intended to limit the possible configurations, specifications, etc. of ion-trap quantum computers according to the present application. For example, other ion species include alkaline earth ions (Be + , Ca + , Sr + , Mg + , Ba + ) or transition metal ions (Zn + , Hg + , Cd + ).

[0039] Provided Figure 5To aid in visualizing the qubit states of an ion, they are represented as points on the surface of a Bloch sphere 500 with an azimuth angle φ and a polar angle θ. The application of the composite pulse described above results in Rabi oscillations between the qubit states |0> (represented as the north pole of the Bloch sphere) and |1> (the south pole of the Bloch sphere). Adjusting the duration and amplitude of the composite pulse flips the qubit state from |0> to |1> (i.e., from the north pole to the south pole of the Bloch sphere), or from |1> to |0> (i.e., from the south pole to the north pole of the Bloch sphere). This application of the composite pulse is called a “π-pulse”. Furthermore, by adjusting the duration and amplitude of the composite pulse, the qubit state |0> can be converted into a superposition state |0>+|1>, where the two qubit states |0> and |1> are added with equal weight and in phase (for convenience, the normalization factor of the superposition state is omitted below), and the qubit state |1> can be converted into a superposition state |0>-|1>, where the two qubit states |0> and |1> are added with equal weight and out of phase. This application of the composite pulse is called the “π / 2-pulse”. More generally, the superposition of two qubit states |0> and |1> with equal weight is represented by a point located on the equator of the Bloch sphere. For example, the superposition state |0>±|1> corresponds to points with an azimuth angle φ of zero and π above the equator, respectively. The superposition state corresponding to a point with an azimuth angle φ on the equator is represented as |0>+e iφ |1> (for example, for φ=±π / 2, it is |0>±i|1>). The transformation between two points on the equator (i.e., rotation about the Z-axis on the Bloch sphere) can be achieved by shifting the phase of the composite pulse.

[0040] In ion trap quantum computers, motion modes can serve as a data bus to mediate entanglement between two qubits, which is then used to perform XX-gate operations. That is, each of the two qubits is entangled with a motion mode, and this entanglement is then transferred to the entanglement between the two qubits using motion sideband excitations, as described below. Figure 6A and Figure 6B The schematic depiction illustrates a mode with a mode frequency ω according to one embodiment. p Movement patterns | n ph > p A view of the motion sideband spectrum of ions in chain 102. (See image.) Figure 6B As shown, when the detuning frequency of the composite pulse is zero (i.e., the frequency difference between the first and second laser beams is tuned to the carrier frequency, δ=ω1-ω2-ω), 01 When |ω1| = 0, a simple Rabi oscillation (carrier transition) occurs between the qubit states |0> and |1>. When the detuning frequency of the composite pulse is positive (i.e., the frequency difference between the first and second laser beams is tuned to be higher than the carrier frequency, δ = ω1 - ω2 - ω1), a simple Rabi oscillation (carrier transition) occurs between the qubit states |0> and |1>. 01= μ > 0, called the "blue sideband") Rabi oscillations occur between the combined qubit motional states |0> |n ph p and |1> |n ph +1> between the first and second laser beams is tuned to be lower than the carrier frequency by the mode frequency ω ph p , a transition from the pth motional mode with n phonon excitations represented as |n ph +1> p to the pth motional mode with (n ph +1) phonon excitations represented as |n p +1>occurs). When the detuning frequency of the composite pulse is negative (i.e., the frequency difference between the first and second laser beams is tuned to be lower than the carrier frequency by the mode frequency ω ph p , δ = ω1- ω2- ω 01 = - μ < 0, called the "red sideband") Rabi oscillations occur between the combined qubit motional states |0> |n ph p and |1> |n ph -1> p (i.e., a transition from the motional mode |n ph p to the motional mode |n ph -1>with one less phonon excitation occurs). A π / 2 pulse acting on the qubit on the blue sideband converts the combined qubit motional state |0> |n p ph p to the superposition |0> |n ph p and |1> |n ph +1> p . A π / 2 pulse acting on the qubit on the red sideband converts the combined qubit motional |0> |n ph p to the superposition |0> |n ph p and |1> |n ph -1> p . When the two-photon Rabi frequency Ω(t) is much smaller than the detuning frequency δ = ω1- ω2- ω 01 = ± μ, either the blue sideband transition or the red sideband transition can be selectively driven. Thus, by applying the appropriate type of pulse, such as a π / 2 pulse, the qubit can be entangled with the desired motional mode, which can then be entangled with another qubit, resulting in entanglement between the two qubits. In ion-trap quantum computers, performing XX gate operations requires entanglement between qubits.​​​​​​​​

[0041] By controlling and / or guiding the transformation of the combined qubit states as described above, XX-gate operations can be performed on two qubits (the i-th and j-th qubits). In general, the XX-gate operation (with maximum entanglement) transforms the two-qubit states as follows: |0> i |0> j ,|0> i |1> j ,|1> i |0> j and |1> i |1> j :

[0042]

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

[0044] Therefore, applying a π / 2 pulse on the blue sideband to the i-th qubit and a π / 2 pulse on the red sideband to the j-th qubit can combine the two qubits and the motion mode into a state |0>. i |0> j |n ph > p Convert to |0> i |0> j |n ph > p and |1> i |1> j |n ph > p The two qubits are now in an entangled state due to the superposition of their components. It should be clear to those skilled in the art that the number of phonon excitations relative to the initial number of phonon excitations, n, can be removed using a sufficiently complex pulse sequence. ph Two qubit states entangled in different motion modes (i.e., |1> i |0> j |n ph +1> p and |0> i |1> j |n ph -1> p Therefore, the combined state of the two qubits and motion modes after the XX gate operation can be considered unentangled, because at the end of the XX gate operation, the number of initial phonon excitations n in the p-th motion mode is reduced. ph The state remains unchanged. Therefore, the following description generally describes the state of the qubit before and after the XX gate operation, without including motion modes.

[0045] More generally, the combined state of the i-th and j-th qubits transformed by applying a composite pulse on the sideband with an amplitude function Ω(t) and a detuning frequency function μ(t) of duration τ (called the "gate duration") can be determined based on the entanglement interaction χ. i，j (τ) is described as follows:

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

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

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

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

[0050] wherein,

[0051]

[0052] is the Lamb-Dicke parameter quantifying the coupling strength between the i-th (j-th) ion and the p-th motional mode having a mode frequency ω p is the pulse function defined as g(t) = Ω(t) sin(ψ(t)), ψ(t) is the cumulative phase function (also simply referred to as "phase function") of the pulse ψ0is the initial phase, which in the following for simplicity without loss of generality can be set to zero (0), and P is the number of motional modes (equal to the number of ions N in the chain 102).

[0053] Construction of pulses for entangling gate operations

[0054] The entanglement between the two qubits (trapped ions) described above can be used to perform an XX gate operation. The XX gate operation (XX gate) together with single qubit operations (R gates) form a universal set of gates {R, XX} that can be used to build a quantum computer to perform a desired computational process. In constructing a pulse for delivery to the chain of trapped ions 102 to perform an XX gate operation between two trapped ions (e.g., the ithand jthtrapped ions) in the chain 102, the amplitude function Ω(t) and the detuning frequency function μ(t) of the pulse are adjusted as control parameters by imposing the following gate requirements to ensure that the pulse performs the intended XX gate operation.

[0055] Gate Requirement 1: First, in order to perform an XX gate operation XX(θ ij (0 < θ ij ≤ π / 2) with the desired rotation angle θ ij , the entangling interaction χ ij generated by the pulse between the ithand jthtrapped ions must be equal to the rotation angle θ ij / 4. A full entangling XX gate requires θ ij = π / 2. The first requirement is also referred to as the non-zero entangling interaction requirement.

[0056] Gate Requirement 2: Second, the laser power required to implement the pulse can be minimized so that the constructed pulse is at an optimal power. The second requirement is also referred to as the requirement that the laser pulse has the lowest peak power.

[0057] Gate Requirement 3: Third, all trapped ions in the chain 102 that are displaced from their initial positions when the motional mode is excited by the delivered pulse must return to the initial positions at the end of the XX gate operation. The third requirement is also referred to as the ion mode decoupling requirement, where the trapped ions excited by the motional mode return to their original positions and momentum values.

[0058] However, a series of laser pulses constructed as described above for performing a computational process in a large-scale quantum computation presents a complex technical problem for the optics and electronics used to modulate such laser pulses. In the embodiments described herein, a new degree of freedom is introduced in the design and fabrication of the ion trap (e.g., ion trap 102) used to confine the ions to compensate for some of the added technical complexity in the optics and electronics. Thus, it is desirable to design and fabricate the ion trap to modulate the confining potential of the ion trap to configure the mode frequencies ω p In some embodiments, the pulse is chosen to have certain symmetries, such as the phase of the pulse being inverted at the middle point of the gate duration of the pulse, to make the gate operation more robust. Thus, in constructing a pulse for performing an XX gate operation, the following configuration conditions are imposed.

[0059] Condition 1: The motional mode structure is modulated so that the mode frequencies ωp is an integer multiple of 4π / τ, the error δk p is small, where τ is the gate duration, k p is a positive integer (i.e., ω p = 4(k p + δk p )π / τ).

[0060] Condition 2: The pulse function g(t) is chosen to have symmetry g(t + τ / 2) = -g(t) for t ∈ [0, τ / 2]. That is, the pulse is a combination of two consecutive pulse segments that are identical in shape (i.e., same detuning frequency μ(t) and same amplitude function Ω(t)) but opposite in phase.

[0061] Condition 3: In one example, the pulse is further simplified by requiring that the amplitude function Ω(t) of the pulse is constant Ω during the gate duration τ and the detuning frequency function μ(t) of the pulse is an integer multiple of 2π / τ μ(t) = 2lπ / τ, where l is a positive integer. That is, for t ∈ [0, τ / 2], the pulse function is described as Thus, the constant amplitude Ω of the pulse and the positive integer l that determines the detuning frequency μ of the pulse are control parameters to be adjusted when constructing a pulse for performing an XX gate operation.

[0062] As mentioned above, the gate requirement 1 (also referred to as the non-zero entanglement interaction requirement) requires that the entanglement interaction χ ij between the i-th and j-th trapped ions produced by the pulse has a rotation angle value θ ij (0 < θ ij ≤ π / 2).

[0063] By imposing the conditions 1, 2, and 3, the entanglement interaction χ ij can be simplified as

[0064]

[0065] where p l satisfies

[0066] The gate requirement 2 (also referred to as the requirement that the laser pulse has the lowest peak power) requires that, for a given set of Lamb-Dicke parameters and mode frequencies ω p , the positive integer l is chosen such that the constant amplitude Ω is the lowest value that satisfies the non-zero entanglement interaction requirement. That is, the positive integer l is chosen such that the gate power ratio defined by the dimensionless quantity |χ ij / Ω 2 τ 2 | is maximized. Once this positive integer l is chosen, χ ij = θij / 4, calculate the constant amplitude Ω.

[0067] Gate requirement 3 (also known as the ion mode decoupling requirement) requires that the trapped ion excited by the pulse return to its initial position. After the pulse is applied for the gate duration τ, the residual coupling α between the p-th motion mode and the i-th and j-th trapped ions is determined by the Lamb-Dicke parameter. The pulse function g(t) of the pulse and the mode frequency ω of the p-th motion mode. p To determine, and requiring that it be less than the error budget ∈, such that the displacements of the i-th and j-th trapped ions in phase space from their initial positions are restricted to .

[0068]

[0069] Where φ p It is the initial phase associated with the p-th motion mode at time t=0, T p It is the temperature of the p-th motion mode. It is the reduced Planck constant, k B It is the Boltzmann constant.

[0070] By applying conditions 1, 2, and 3, the residual coupling α can be simplified to

[0071]

[0072] For simplicity without loss of generality, we assume that the motion modes have been adequately cooled using known laser cooling methods, such as Doppler cooling or resolved sideband cooling, resulting in an average phonon number. Less than And the initial phase φ associated with the p-th motion mode p If the value is zero, then the residual coupling α is restricted to...

[0073]

[0074] because

[0075] At the mode frequency ω p Error limit δk p In very small cases (i.e., |δk) p |<<1), the upper bound of the residual coupling α can be extended to the error δk p A series of terms of different orders are shown below:

[0076]

[0077] in

[0078]

[0079] It should be noted that for a given error budget ε of the residual coupling α, if the motion pattern p does not satisfy k p = l / 2, then the pulse with positive integer l being even is more robust against the error δk p of the required pattern frequency ω p than the pulse with positive integer l being odd.

[0080] It should also be noted that the pulses constructed in the methods described herein can be readily adapted to suppress crosstalk errors produced by spectator ions that are located close to the ions on which the XX gate operation is intended to be performed. The XX gate XX(θ ij ) can be implemented by a pulse satisfying condition 2 as a combination of two XX gates XX(θ ij / 2 ) and a single-qubit gate. Due to conditions 1 and 2, the entanglement rotation angle accumulated in the first half of the gate duration τ is already the same as the entanglement rotation angle accumulated in the second half of the gate duration τ, i.e.,

[0081]

[0082] By flipping the phase on the pulse as in condition 2, and leaving the phase of the spectator ions unchanged, the crosstalk will be suppressed.

[0083] Pattern configuration

[0084] In the embodiments described herein, the ion trap used to confine the ions is designed such that condition 1 is satisfied (i.e., the pattern frequency ω p is an integer multiple of 4π / τ, ω p = 4(k p + δk p )π / τ, with a small error δk p , where k p is a positive integer, and τ is the gate duration). That is, the static (DC) voltages V S and radio frequency (RF) voltages applied to the control electrodes of the ion trap 200 to produce the confining potential are adjusted such that the resulting motion pattern structure satisfies condition 1. Figure 7 A numerically computed topography of the static (DC) voltages V S applied to a plurality of electrodes of an exemplary Sandia High Optical Access (HOA) 2.0 trap 700, which confines three ions 702, 704, and 706, is shown such that the resulting motion pattern structure satisfies condition 1. In this exemplary ion trap 700, the pulses to be constructed for performing the XX gate operation are chosen to have a gate duration τ = 69.466 μβ, and the three motion patterns, i.e., the center-of-mass (COM) pattern, the tilt pattern, and the zigzag pattern, have positive integers k p= 97, 95, and 92 mode frequencies ω p In the example HOA 2.0 trap 700, a radio frequency (RF) voltage at a frequency of 50.6 MHz is further applied to the trap electrodes with an amplitude of 289.71 V. Figure 7 the static (DC) voltage V S traps three ions 702, 704, and 706, which are linearly arranged in a chain at a distance of about 71 μm above the surface of the ion trap 700 with a spacing of about 4.3 μm. The static (DC) voltage V S produces a slightly higher radial confinement on the central ion 704 than on the end ions 702 and 704, resulting in a mode frequency ω p for the tilt mode, resulting in a mode frequency ω p for the sawtooth mode, resulting in a mode frequency ω p These mode frequencies ω p slightly deviate from the mode frequencies ω p = 97, 95, and 92, respectively, corresponding to positive integers k p = 97, 95, and 92, respectively, corresponding to positive integers k p = 97, 95, and 92, respectively, corresponding to positive integers k

[0085] Figure 8 depicts in Figure 7 In the example shown in ij / Ω 2 τ 2 | 800. If it is even, then the gate power ratio |χ ij / Ω 2 τ 2 | is maximized (gate requirement 2) when the positive integer l is chosen to be the positive integer 192, while if the positive integer l is odd, then the gate power ratio |χ

[0086] Figure 9A and Figure 9B depicts in Figure 7 In the example shown in Figure 9B The pulse function g(t) 904 for l = 192 in depicts a sharp point at the midpoint of the gate duration τ = 69.466 μs, which isFigure 9A The cusp in the pulse function g(t)902 with l=193 is absent. However, the cusp in the pulse function g(t)904 with l=192 can be effectively removed by applying an appropriate single-qubit gate at the cusp.

[0087] Figure 10 Depicting in Figure 7 In the example shown, the mode frequency ω p The residual coupling α is a function of the fluctuation δω in the motion pattern. For simplicity, all three motion modes are chosen to have the same frequency fluctuation δω. The residual coupling α1002 at l=193 and the residual coupling α1004 at l=192 both include non-zero values ​​at the frequency fluctuation δω=0, which is only due to the static error in the motion mode configuration (i.e., the mode frequency ω). p The mode frequency ω that precisely satisfies condition 1 p This is due to deviations in the value of the residual coupling α, and also includes larger non-zero values ​​at frequency fluctuations δω, which increase due to hardware errors or noise. As mentioned above, this is due to the dependence of residual coupling α on the even number l. Rather than dependence on odd numbers l The residual coupling 1004 for (even numbers) is much smaller than the residual coupling 1002 for l = 193 (odd numbers). It should be noted that, as in the examples described herein, for any motion mode, when l ≠ 2k... p In this case, even numbers l have an advantage over odd numbers l. A compromise is that a pulse with l = 192 requires 30% more power than one with l = 193. While fully considering hardware limitations, a choice should be made between even and odd numbers l based on the specific circumstances.

[0088] Figure 11 A flowchart illustrating a method 1100 for performing computations using a quantum computer is shown.

[0089] In block 1110, the motion mode structure of multiple trapped ions is modulated. Each of the multiple trapped ions has two frequency-separated states that define a qubit.

[0090] In block 1120, the detuning frequency function and amplitude function of the laser pulse used to induce entanglement interaction between a pair of trapped ions among the plurality of trapped ions are calculated.

[0091] In block 1130, quantum computation is performed in a quantum computer by applying a gate duration to the pair of trapped ions using laser pulses having calculated detuning frequency and amplitude functions.

[0092] As described above, when generating pulses for performing entanglement gate operations between two trapped ions in a chain, control parameters (the detuning frequency function and amplitude function of the pulses) are determined to satisfy the gate requirements. Thus, the motion mode structure of the trapped ions in the chain is configured such that the constructed pulses are simple and can be practically implemented in speed- and bandwidth-constrained hardware. When designing ion traps to confine ions and thus modulate the motion mode structure in a desired manner, electrodes of the ion traps can be constructed to provide the desired confinement potential. With appropriate technological advancements, the desired confinement potential can be induced using an increased number of electrodes and electrodes of various shapes. The embodiments described herein provide a method for redistributing the technological complexity that must be overcome to build practical large-scale quantum computers.

[0093] While the foregoing relates to specific embodiments, other and further embodiments may be devised without departing from its basic scope, the scope of which is defined by the appended claims.

Claims

1. A method for performing computation using a quantum computer, the method comprising: The motion mode structure of the multiple trapped ions is modulated by adjusting the constraint potential of the ion trap that traps the multiple trapped ions. Each of the multiple trapped ions has two frequency-separated states that define a qubit. Calculate the detuning frequency function and amplitude function of the laser pulse that would cause entanglement interaction between a pair of trapped ions among the plurality of trapped ions; as well as Quantum computation is performed in the quantum computer by applying a gate duration to the pair of trapped ions using laser pulses having calculated detuning frequency and amplitude functions.

2. The method according to claim 1, wherein the modulation of the motion mode structure includes adjusting the constraint potential such that the motion mode frequencies of the plurality of trapped ions are integer multiples of the quotient of 4π divided by the gate duration.

3. The method according to claim 1, wherein the laser pulse comprises two consecutive pulse segments having the same detuning frequency function and the same amplitude function but opposite phases.

4. The method according to claim 1, wherein The amplitude function of the laser pulse is constant during the gate duration, and The detuning frequency function of the laser pulse is an integer multiple of the quotient of 2π divided by the gate duration.

5. The method according to claim 1, wherein the calculation of the detuning frequency function and amplitude function of the laser pulse is based on a first gate requirement, namely, requiring non-zero entanglement interaction.

6. The method of claim 5, wherein the calculation of the detuning frequency function and amplitude function of the laser pulse is based on a second gate requirement, namely, requiring the laser pulse to have a minimum peak power.

7. The method according to claim 6, wherein the calculation of the detuning frequency function and amplitude function of the laser pulse is based on a third gate requirement, namely, requiring the ion modes of the plurality of trapped ions to be decoupled.

8. A non-volatile computer-readable medium comprising computer program instructions, which, when executed by a processor, cause the processor to: Calculate the detuning frequency and amplitude functions of the laser pulse that would induce entanglement interactions between a pair of trapped ions in a group of multiple trapped ions; and Quantum computation is performed in a quantum computer by applying a gate duration to a pair of trapped ions using laser pulses having calculated detuning frequency and amplitude functions. The motion mode frequencies of the plurality of trapped ions are modulated to be integer multiples of the quotient of 4π divided by the gate duration.

9. The non-volatile computer-readable medium of claim 8, wherein the laser pulse comprises two consecutive pulse segments having the same detuning frequency function and the same amplitude function but opposite phases.

10. The non-volatile computer-readable medium according to claim 8, wherein... The amplitude function of the laser pulse is constant during the gate duration, and The detuning frequency function of the laser pulse is an integer multiple of the quotient of 2π divided by the gate duration.

11. The non-volatile computer-readable medium of claim 8, wherein the calculation of the detuning frequency function and amplitude function of the laser pulse is based on a first gate requirement, namely, requiring non-zero entanglement interaction.

12. The non-volatile computer-readable medium of claim 11, wherein the calculation of the detuning frequency function and amplitude function of the laser pulse is based on a second gate requirement, namely, requiring the laser pulse to have a minimum peak power.

13. The non-volatile computer-readable medium of claim 12, wherein the calculation of the detuning frequency function and amplitude function of the laser pulse is based on a third gate requirement, namely, requiring the ion modes of the plurality of trapped ions to be decoupled.

14. A quantum computing system, comprising: Multiple trapped ions in an ion trap, each of which has two hyperfine states that define a qubit; as well as A controller, comprising non-volatile memory having a plurality of instructions stored therein, which, when executed by a processor, cause the quantum computing system to perform operations, including: Calculate the detuning frequency function and amplitude function of the laser pulse that would cause entanglement interaction between a pair of trapped ions among the plurality of trapped ions; as well as Quantum computation is performed in a quantum computer by applying a gate duration to a pair of trapped ions using laser pulses having calculated detuning frequency and amplitude functions. The confinement potential of the ion trap is modulated such that the motion mode frequency of the plurality of trapped ions is an integer multiple of the quotient of 4π divided by the gate duration.

15. The quantum computing system according to claim 14, wherein Each trapped ion is an ion with nuclear spin and electron spin, and the difference between the nuclear spin and the electron spin is zero.

16. The quantum computing system of claim 14, wherein the laser pulse comprises two consecutive pulse segments having the same detuning frequency function and the same amplitude function but opposite phases.

17. The quantum computing system of claim 14, wherein... The amplitude function of the laser pulse is constant during the gate duration, and The detuning frequency function of the laser pulse is 2π divided by an integer multiple of the gate duration.

18. The quantum computing system of claim 14, wherein the calculation of the detuning frequency function and amplitude function of the laser pulse is based on a first gate requirement, namely, requiring non-zero entanglement interaction.

19. The quantum computing system of claim 18, wherein the calculation of the detuning frequency function and amplitude function of the laser pulse is based on a second gate requirement, namely, requiring the laser pulse to have a minimum peak power.

20. The quantum computing system of claim 19, wherein the calculation of the detuning frequency function and amplitude function of the laser pulse is based on a third gate requirement, namely, requiring the ion modes of the plurality of trapped ions to be decoupled.

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