Quantum logic gate design and optimization

A method of performing a computational process using a quantum computer includes generating a laser pulse sequence comprising a plurality of laser pulse segments used to perform an entangling gate operation on a first trapped ion and a second trapped ion of a plurality of trapped ions that are aligned in a first direction, each of the trapped ions having two frequency-separated states defining a qubit, and applying the generated laser pulse sequence to the first and second trapped ions. Each of the plurality of laser pulse segments has a pulse shape with ramps formed using a spline at a start and an end of each of the plurality of laser pulse segments.

BACKGROUND

Field

The present disclosure generally relates to a method of generating an entangling gate in an ion trap quantum computer, and more specifically, to a method of optimizing a laser pulse sequence to generate the entangling gate.

Description of the Related Art

In quantum computing, quantum bits or qubits, which are analogous to bits representing a “0” and a “1” in a classical (digital) computer, are required to be prepared, manipulated, and measured (read-out) with near perfect control during a computation process. Imperfect control of the qubits leads to errors that can accumulate over the computation process, limiting the size of a quantum computer that can perform reliable computations.

Among physical systems upon which it is proposed to build large-scale quantum computers, is a group of ions (e.g., charged atoms), which are trapped and suspended in vacuum by electromagnetic fields. The ions have internal hyperfine states which are separated by frequencies in the several GHz range and can be used as the computational states of a qubit (referred to as “qubit states”). These hyperfine states can be controlled using radiation provided from a laser, or sometimes referred to herein as the interaction with laser beams. The ions can be cooled to near their motional ground states using such laser interactions. The ions can also be optically pumped to one of the two hyperfine states with high accuracy (preparation step of qubits), manipulated between the two hyperfine states (single-qubit gate operations) by laser beams, and their internal hyperfine states detected by fluorescence upon application of a resonant laser beam (read-out of qubits). A pair of ions can be controllably entangled (two-qubit gate operations) by qubit-state dependent force using laser pulses that couple the ions to the collective motional modes of a chain of trapped ions, which arise from their Coulombic interaction between the ions. In general, entanglement occurs when pairs or groups of ions (or particles) are generated, interact, or share spatial proximity in ways such that the quantum state of each ion cannot be described independently of the quantum state of the others, even when the ions are separated by a large distance.

As the size of the chain of ions increases, these gate operations are more susceptible to external noise, decoherence, speed limitations, or the like. Therefore, there is a need for a method of optimizing a laser pulse sequence to implement, for instance, the entangling gate which avoids these problems.

SUMMARY

A method of performing a computational process using a quantum computer includes generating a laser pulse sequence comprising a plurality of laser pulse segments used to perform an entangling gate operation on a first trapped ion and a second trapped ion of a plurality of trapped ions that are aligned in a first direction, each of the trapped ions having two frequency-separated states defining a qubit, and applying the generated laser pulse sequence to the first and second trapped ions. Each of the plurality of laser pulse segments has a pulse shape with ramps formed using a spline at a start and an end of each of the plurality of laser pulse segments.

To facilitate understanding, identical reference numerals have been used, where possible, to designate identical elements that are common to the figures. In the figures and the following description, an orthogonal coordinate system including an X-axis, a Y-axis, and a Z-axis is used. The directions represented by the arrows in the drawing are assumed to be positive directions for convenience. It is contemplated that elements disclosed in some embodiments may be beneficially utilized on other implementations without specific recitation.

DETAILED DESCRIPTION

Embodiments described herein are generally related to a method and a system for designing, optimizing and delivering a laser pulse sequence to perform an entangling gate operation between two ions during a quantum computation, and, more specifically, to a laser pulse sequence that increases the fidelity, or the probability that at least two ions are in the intended quantum state(s), of the entangling gate operation between two ions. The optimized laser pulse sequence includes multiple time-segmented laser pulses in which an intensity of at least a portion of each time-segmented laser pulse has a desired intensity level that is gradually ramped at a start and an end of the time-segmented laser pulse with splines (e.g., functions defined piecewise by one or more polynomials or other algebraic expressions), thereby increasing the fidelity of the entangling gate operation.

In quantum computation, any number of desired computational operations can be constructed using one of several known universal gate sets. A universal quantum computer can be built by the use of a universal gate set. The universal gate sets includes a universal gate set, commonly denoted as {R, XX}, which is native to a quantum computing system of trapped ions described herein. Here, the gate R corresponds to manipulation of individual quantum states of trapped ions, and the gate XX corresponds to manipulation of the entanglement of two trapped ions. For those of ordinary skill in the art, it should be clear the gate R can be implemented with near perfect fidelity, while the formation of the gate XX is complex and requires optimization for a given type of trapped ions, number of ions in a chain of trapped ions, and the hardware and environment in which the trapped ions are trapped, to name just a few factors, such that the fidelity of the gate XX is increased and computational errors within a quantum computer are avoided or decreased. In the following discussion, methods of generating and optimizing a laser pulse sequence used to perform computations based on the formation of a gate XX that has an improved fidelity will be described.

General Hardware Configurations

FIG. 1is a schematic partial view of an ion trap quantum computing system, or system100, according to one embodiment. The system100includes a classical (digital) computer102, a system controller104and a quantum processor that includes a group106of trapped ions (i.e., five shown) that extend along the Z-axis. The classical computer102includes a central processing unit (CPU), memory, and support circuits (or I/O). The memory is connected to the CPU, and may be one or more of a readily available memory, such as a read-only memory (ROM), a random access memory (RAM), floppy disk, 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 instructing the CPU. The support circuits (not shown) are also connected to the CPU for supporting the processor in a conventional manner. The support circuits may include conventional cache, power supplies, clock circuits, input/output circuitry, subsystems, and the like.

An imaging objective108, such as an objective lens with a numerical aperture (NA), for example, of 0.37, collects fluorescence along the Y-axis from the ions and maps each ion onto a multi-channel photo-multiplier tube (PMT)110for measurement of individual ions. Non-copropagating Raman laser beams from a laser112, which are provided along the X-axis, perform operations on the ions. A diffractive beam splitter114creates an array of static Raman beams116that are individually switched using a multi-channel acousto-optic modulator (AOM)118and is configured to selectively act on individual ions. A global Raman laser beam120illuminates all ions at once. The system controller (also referred to as a “RF controller”)104controls the AOM118and thus controls laser pulses to be applied to trapped ions in the group106of trapped ions. The system controller104includes a central processing unit (CPU)122, a read-only memory (ROM)124, a random access memory (RAM)126, a storage unit128, and the like. The CPU122is a processor of the system controller104. The ROM124stores various programs and the RAM126is the working memory for various programs and data. The storage unit128includes a nonvolatile memory, such as a hard disk drive (HDD) or a flash memory, and stores various programs even if power is turned off. The CPU122, the ROM124, the RAM126, and the storage unit128are interconnected via a bus130. The system controller104executes a control program which is stored in the ROM124or the storage unit128and uses the RAM126as a working area. The control program will include software applications that include program code that may be executed by processor in order to perform various functionalities 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 system100discussed herein.

FIG. 2depicts a schematic view of an ion trap200(also referred to as a Paul trap) for confining ions in the group106according to one embodiment. The confining potential is exerted by both static (DC) voltage and radio frequency (RF) voltages. A static (DC) voltage Vs is applied to end-cap electrodes210and212to confine the ions along the Z-axis (also referred to as an “axial direction” or a “longitudinal direction”). The ions in the group106are nearly evenly distributed in the axial direction due to the Coulomb interaction between the ions. In some embodiments, the ion trap200includes four hyperbolically-shaped electrodes202,204,206, and208extending along the Z-axis.

During operation, a sinusoidal voltage V1(with an amplitude VRF/2) is applied to an opposing pair of the electrodes202,204and a sinusoidal voltage V2with a phase shift of 180° from the sinusoidal voltage V1(and the amplitude VRF/2) is applied to the other opposing pair of the electrodes206,208at a driving frequency ωRF, generating a quadrupole potential. In some embodiments, a sinusoidal voltage is only applied to one opposing pair of the electrodes202,204, and the other opposing pair206,208is grounded. The quadrupole potential creates an effective confining force in the X-Y plane perpendicular to the Z-axis (also referred to as a “radial direction” or “transverse direction”) for each of the trapped ions, which is proportional to a distance from a saddle point (i.e., a position in the axial direction (Z-direction)) at which the RF electric field vanishes. The motion in the radial direction (i.e., direction in the X-Y plane) of each ion is approximated as a harmonic oscillation (referred to as secular motion) with a restoring force towards the saddle point in the radial direction and can be modeled by spring constants kxand ky, respectively, as is discussed in greater detail below. In some embodiments, the spring constants in the radial direction are modeled as equal when the quadrupole potential is symmetric in the radial direction. However, undesirably in some cases, the motion of the ions in the radial direction may be distorted due to some asymmetry in the physical trap configuration, a small DC patch potential due to inhomogeneity of a surface of the electrodes, or the like and due to these and other external sources of distortion the ions may lie off-center from the saddle points.

Trapped Ion Configuration and Quantum Bit Information

FIGS. 3A, 3B, and 3Cdepict a few schematic structures of collective transverse motional modes (also referred to simply as “motional mode structures”) of a chain102of five trapped ions, for example. Here, the confining potential due to a static voltage Vs applied to the end-cap electrodes210and212is weaker compared to the confining potential in the radial direction. The collective motional modes of the chain102of trapped ions in the transverse direction are determined by the Coulomb interaction between the trapped ions combined with the confining potentials generated by the ion trap200. The trapped ions undergo collective transversal motions (referred to as “collective transverse motional modes,” “collective motional modes,” or simply “motional modes”), where each mode has a distinct energy associated with it. A motional mode having the m-th lowest energy is hereinafter referred to as |nm, where n denotes the number of motional quanta (in units of energy excitation, referred to as phonons) in the motional mode, and the number of motional modes in a given transverse direction is equal to the number of trapped ions N in the chain102.FIGS. 3A-3Cschematically illustrates examples of different types of collective transverse motional modes that may be experienced by five trapped ions that are positioned in a chain102.FIG. 3Ais a schematic view of a common motional mode |nNhaving the highest energy, where N is the number of trapped ions in the chain102. In the common motional mode |nN, all ions oscillate in phase in the transverse direction.FIG. 3Bis a schematic view of a tilt motional mode |nN−1which has the second highest energy. In the tilt motional mode, ions on opposite ends move out of phase in the transverse direction (i.e., in opposite directions).FIG. 3Cis a schematic view of a higher-order motional mode |nN−3which has a lower energy than that of the tilt motional mode |nN−1, and in which the ions move in a more complicated mode pattern.

It should be noted that the particular configuration described above is just one among several possible examples of a trap for confining ions according to the present disclosure and does not limit the possible configurations, specifications, or the like of traps 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 generates an effective electric field causing the motion of the ions in the radial direction as harmonic oscillations may be a multi-layer trap in which several electrode layers are stacked and an RF voltage is applied to two diagonally opposite electrodes, or a surface trap in which all electrodes are located in a single plane on a chip. Furthermore, a trap may be divided into multiple segments, adjacent pairs of which may be linked by shuttling one or more ions, or coupled by photon interconnects. A trap may also be an array of individual trapping regions arranged closely to each other on a micro-fabricated ion trap chip. In some embodiments, the quadrupole potential has a spatially varying DC component in addition to the RF component described above.

FIG. 4depicts a schematic energy diagram400of each ion in the chain102of trapped ions according to one embodiment. In one example, each ion may be a positive Ytterbium ion,171Yb+, which has the2S1/2hyperfine states (i.e., two electronic states) with an energy split corresponding to a frequency difference (referred to as a “carrier frequency”) of ω01/2π=12.642821 GHz. A qubit is formed with the two hyperfine states, denoted as |0and |1, where the hyperfine ground state (i.e., the lower energy state of the2S1/2hyperfine states) is chosen to represent |0. Hereinafter, the terms “hyperfine states,” “internal hyperfine states,” and “qubits” may be interchangeably used to represent |0and |1. Each ion may be cooled (i.e., kinetic energy of the ion may be reduced) to near the motional ground state |0mfor any motional mode m with no phonon excitation (i.e., nph=0) by known laser cooling methods, such as Doppler cooling or resolved sideband cooling, and then the qubit state prepared in the hyperfine ground state |0by optical pumping. Here, |0represents the individual qubit state of a trapped ion whereas |0mwith the subscript m denotes the motional ground state for a motional mode m of a group106of trapped ions.

An individual qubit state of each trapped ion may be manipulated by, for example, a mode-locked laser at 355 nanometers (nm) via the excited2P1/2level (denoted as |e). As shown inFIG. 4, a laser beam from the laser may be split into a pair of non-copropagating laser beams (a first laser beam with frequency ω1and a second laser beam with frequency ω2) in the Raman configuration, and detuned by a one-photon transition detuning frequency Δ=ω1-ωewith respect to the transition frequency ω0ebetween |0and |e, as illustrated inFIG. 4. A two-photon transition detuning frequency δ includes adjusting the amount of energy that is provided to the trapped ion by the first and second laser beams, which when combined is used to cause the trapped ion to transfer between the hyperfine states |0and |1. When the one-photon transition detuning frequency Δ is much larger than a two-photon transition detuning frequency (also referred to simply as “detuning frequency”) δ=ω1−ω2−ω01(hereinafter denoted as ±μ, μ being a positive value), single-photon Rabi frequencies Ω0e(t) and Ω1e(t) (which are time-dependent, and are determined by amplitudes and phases of the first and second laser beams), at which Rabi flopping between states |0and |eand between states |1and |erespectively occur, and a spontaneous emission rate from the excited state |e, Rabi flopping between the two hyperfine states |0and |1(referred to as a “carrier transition”) is induced at the two-photon Rabi frequency Ω(t). The two-photon Rabi frequency Ω(t) has an intensity (i.e., absolute value of amplitude) that is proportional to Ω0eΩ1e/2Δ, where Ω0eand Ω1eare the single-photon Rabi frequencies due to the first and second laser beams, respectively. Hereinafter, this set of counter-propagating laser beams in the Raman configuration to manipulate internal hyperfine states of qubits (qubit states) may 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) may be referred to as an “amplitude” of a pulse or simply as a “pulse,” which are illustrated and further described in conjunction withFIGS. 7A and 8Abelow. The detuning frequency δ=ω1−ω2−ω01may be referred to as detuning frequency of the composite pulse or detuning frequency of the pulse. The amplitude of the two-photon Rabi frequency Ω(t), which is determined by amplitudes of the first and second laser beams, may be referred to as an “amplitude” of the composite pulse.

It should be noted that the particular atomic species used in the discussion provided herein is just one example of atomic species which has stable and well-defined two-level energy structures when ionized and an excited state that is optically accessible, and thus is not intended to limit the possible configurations, specifications, or the like of an ion trap quantum computer according to the present disclosure. For example, other ion species include alkaline earth metal ions (Be+, Ca+, Sr+, Mg+, and Ba+) or transition metal ions (Zn+, Hg+, Cd+).

FIG. 5is provided to help visualize a qubit state of an ion is represented as a point on a surface of the Bloch sphere500with an azimuthal angle ϕ and a polar angle θ. Application of the composite pulse as described above, causes Rabi flopping between the qubit state |0(represented as the north pole of the Bloch sphere) and |1(the south pole of the Bloch sphere) to occur. Adjusting time duration and amplitudes of the composite pulse flips the qubit state from |0to |1(i.e., from the north pole to the south pole of the Bloch sphere), or the qubit state from |1to |0(i.e., from the south pole to the north pole of the Bloch sphere). This application of the composite pulse is referred to as a “π-pulse”. Further, by adjusting time duration and amplitudes of the composite pulse, the qubit state |0may be transformed to a superposition state |0+|1, where the two qubit states |0and |1are added and equally-weighted in-phase (a normalization factor of the superposition state is omitted hereinafter without loss of generality) and the qubit state |1to a superposition state |0−|1, where the two qubit states |0and |1are added equally-weighted but out of phase. This application of the composite pulse is referred to as a “π/2-pulse”. More generally, a superposition of the two qubits states |0and |1that are added and equally-weighted is represented by a point that lies on the equator of the Bloch sphere. For example, the superposition states |0±|1correspond to points on the equator with the azimuthal angle ϕ being zero and π, respectively. The superposition states that correspond to points on the equator with the azimuthal angle ϕ are denoted as |0+eiϕ|1(e.g., |0±i|1for ϕ=±π/2). Transformation between two points on the equator (i.e., a rotation about the Z-axis on the Bloch sphere) can be implemented by shifting phases of the composite pulse.

In an ion trap quantum computer, the motional modes may act as a data bus to mediate entanglement between two qubits and this entanglement is used to perform an XX gate operation. That is, each of the two qubits is entangled with the motional modes, and then the entanglement is transferred to an entanglement between the two qubits by using motional sideband excitations, as described below.FIGS. 6A and 6Bschematically depict views of a motional sideband spectrum for an ion in the group106in a motional mode |nphMhaving frequency ωmaccording to one embodiment. As illustrated inFIG. 6B, when the detuning frequency of the composite pulse is zero (i.e., a frequency difference between the first and second laser beams is tuned to the carrier frequency, δ=ω1−ω2−ω01=0), simple Rabi flopping between the qubit states |0and |1(carrier transition) occurs. When the detuning frequency of the composite pulse is positive (i.e., the frequency difference between the first and second laser beams is tuned higher than the carrier frequency, δ=ω1−ω2−ω01=μ>0, referred to as a blue sideband), Rabi flopping between combined qubit-motional states |0|nphmand |1|nph+1moccurs (i.e., a transition from the m-th motional mode with n-phonon excitations denoted by |nphmto the m-th motional mode with (nph+1)-phonon excitations denoted by |nph+1moccurs when the qubit state |0flips to |1). When the detuning frequency of the composite pulse is negative (i.e., the frequency difference between the first and second laser beams is tuned lower than the carrier frequency by the frequency ωmof the motional mode |nphm, δ=ω1−ω2−ω01=−μ<0, referred to as a red sideband), Rabi flopping between combined qubit-motional states |0|nphmand |1|nph−1moccurs (i.e., a transition from the motional mode |nphmto the motional mode |nph−1mwith one less phonon excitations occurs when the qubit state |0flips to |1). A π/2-pulse on the blue sideband applied to a qubit transforms the combined qubit-motional state |0|nphminto a superposition of |0|nphmand |1|nph+1m. A π/2-pulse on the red sideband applied to a qubit transforms the combined qubit-motional |0|nphminto a superposition of |1|nphmand |1|nph−1m. When the two-photon Rabi frequency Ω(t) is smaller as compared to the detuning frequency δ=ω1−ω2−ω01=±μ, the blue sideband transition or the red sideband transition may be selectively driven. Thus, qubit states of a qubit can be entangled with a desired motional mode by applying the right type of pulse, such as a π/2-pulse, which can be subsequently entangled with another qubit, leading to an entanglement between the two qubits that is needed to perform an XX-gate operation in an ion trap quantum computer.

By controlling and/or directing transformations of the combined qubit-motional states as described above, an XX-gate operation may be performed on two qubits (i-th and j-th qubits). In general, the XX-gate operation (with maximal entanglement) respectively transforms two-qubit states |0i0j, |0i|1j, |1i|0j, and |1i|1jas follows:
|0i|0j→|0i|0j−i|1i|1j
|0i|1j→|0i|1j−i|1i|0j
|1i|0j→−i|0i|1j+i|1i|0j
|1i|1j→−i|0i|0j+i|1i|1j.
For example, when the two qubits (i-th and j-th qubits) are both initially in the hyperfine ground state |0(denoted as |0|0j) and subsequently 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 motional mode |0i|nphmis transformed into a superposition of |0i|nphmand |1i|nph+1m, and thus the combined state of the two qubits and the motional mode is transformed into a superposition of |0i|0j|nphmand |1i|0j|nph+1m. When a π/2-pulse on the red sideband is applied to the j-th qubit, the combined state of the j-th qubit and the motional mode |0j|nphmis transformed to a superposition of |0j|nphmand |1j|nph−1mand the combined state |0j|nph+1mis transformed into a superposition of |0j|nph+1mand |1j|nphm.

Thus, applications of a rπ/2-pulse on the blue sideband on the i-th qubit and a π/2-pulse on the red sideband on the j-th qubit may transform the combined state of the two qubits and the motional mode |0|0j|nphminto a superposition of |0i|0j|nphmand |1i|1j|nphm, the two qubits now being in an entangled state. For those of ordinary skill in the art, it should be clear that two-qubit states that are entangled with motional mode having a different number of phonon excitations from the initial number of phonon excitations nph(i.e., |1i|0j|nph+1mand |0i|1j|nph−1m) can be removed by a sufficiently complex pulse sequence, and thus the combined state of the two qubits and the motional mode after the XX-gate operation may be considered disentangled as the initial number of phonon excitations nphin the m-th motional mode stays unchanged at the end of the XX-gate operation. Thus, qubit states before and after the XX-gate operation will be described below generally without including the motional modes.

More generally, the combined state of i-th and j-th qubits transformed by the application of the composite pulse on the sidebands for duration τ (referred to as a “gate duration”) can be described in terms of an entangling interaction χi,j(τ) as follows:
|0i|0j→cos(χi,j(τ))|0i|0j−isin(χi,j(τ)|1i|1j
|0i|1j→cos(χi,j(τ))|0i1|j−isin(χi,j(τ))|1i|0j
|1i|0j→−isin(χi,j(τ))|0i|1j+cos(χi,j(τ))|1i|0j
|1i|1j→−isin(χi,j(τ))|0i|0j+cos(χi,j(τ))|1i|1j
where,

The control parameters, the pulse sequences Ωi(t) and Ωj(t), must also satisfy conditions that the trapped ions that are displaced from their initial positions as the motional modes are excited by the delivery of the pulse sequences Ωi(t) and Ωj(t) to the trapped ions return to the initial positions. The l-th ion in a superposition state |0±|1(l=i,j) is displaced due to the excitation of the m-th motional mode during the gate duration τ and follows the trajectories ±αl,m(τ) in phase space (position and momentum) of the m-th motional mode, where αl,m(τ)=iηl,m∫0τΩl(t) sin(μt) e−iωmtdt. Thus, for the chain102of N trapped ions, the condition αl,m(τ)=0 (l=i, j) (referred also as “closing of phase space”) must be imposed for all the N motional modes, in addition to the condition 0<χi,j(τ)≤π/4. These conditions may be satisfied by evenly partitioning the gate duration τ into Ns(where 1<Ns) segments (s=1, 2, . . . , Ns) of equal duration and varying the intensity of the pulse sequence Ωi(t) in each segment. That is, the pulse sequence Ωi(t) is divided into Nspulse segments, each having an intensity Ωsand a duration. The intensities Ωs(s=1, 2, . . . , N) of the pulse segments are then determined such that the closing of phase spaces (αl,m(τ)=0) are satisfied to a high numeric precision and the non-zero entangling interaction (0<|χi,j(τ)|≤π/4) are satisfied.

Entangling Gate Operations

The non-zero entangling interaction between two qubits described above can be used to perform an XX-gate operation. The XX-gate operation (gate XX) along with single-qubit operation (the gate R) forms a universal gate set {R, XX} that can be used to build a quantum computer that is configured to perform desired computational processes.FIG. 7Adepicts a segmented pulse sequence Ω(t) for the gate duration τ (e.g., ˜145 μs) that may initially be used to cause an XX-gate operation on second and fourth ions of a chain102of seven trapped ions. In the example depicted inFIG. 7A, the pulse sequence Ω(t) for all seven trapped ions in the chain102is the same and is divided into nine stepwise pulse segments, each having a different intensity Ωs(s=1, 2, . . . , 9). In this example, the intensities Ωs(t) (s=1, 2, . . . , 9) of the pulse segments are determined such that all of the conditions 0<χi,j(τ)≤π/4 and αl,m(τ)=0 (l=i, j) are satisfied. The segmented pulse sequence Ω(t) may be generated by combining the pulse segments having the determined intensities Ωs(t) (s=1, 2, . . . , 9).

FIG. 7Bdepicts simulated results of instantaneous and time-averaged populations of an off-resonant carrier excitation of the second and fourth ions of the chain102of seven trapped ions from the qubit state |0to |1during a XX-gate operation performed by the segmented pulse sequence Ω(t) of nine stepwise pulse segments depicted inFIG. 7A, based on a value of the detuning frequency μ chosen to be between the frequencies ω3and ω4of the third and fourth motional modes. InFIG. 7B, the off-resonant carrier excitation701of the second and fourth ions undesirably varies around a non-zero value and the time-averaged off-resonant carrier excitation702does not return to zero during different stages of the pulse sequence and at the end of the XX-gate operation.

Alternately,FIG. 8Adepicts an improved segmented pulse sequence Ω(t) over the gate duration τ that is created and applied after performing the optimization steps described below to perform an XX gate operation for second and fourth ions of a chain102of seven trapped ions according to embodiments of the disclosure described herein. In the example depicted inFIG. 8A, the same pulse sequence Ω(t) is applied on second and fourth ions in the chain102of seven trapped ions and is divided into nine pulse segments. InFIG. 8A, each pulse segment has an intensity Ωs(s=1, 2, . . . , 9) and has a pulse shape with ramps formed using a spline at a start and an end of the pulse segment in the form of sine-squared functions, for example. That is, the ramp intensity between adjacent pulse segments having intensities Ωs(starting at a time tS) and ΩS+1(starting at a time tS+τ/(2Ns+1)) is given by (ΩS+1−ΩS)×sin2(π(t−tS)/2tR), where tRis a ramp duration. The intensities Ωs(t) (s=1, 2, . . . , 9) of the stepwise pulse segments are determined such that all the conditions 0<χi,j(τ)≤π/4 and αl,m(τ)=0 (l=i, j) are satisfied. The ramped, segmented pulse sequence Ω(t) may be generated by combining the pulse segments having the determined intensities with ramps at the start and the end thereof.

FIG. 8Bdepicts improved results of instantaneous and time-averaged populations of an off-resonant carrier excitation of second and fourth ions of the chain102between qubit state |0to |1during the XX-gate operation performed by the improved segmented pulse sequence Ω(t) depicted inFIG. 8A. InFIG. 8B, the off-resonant carrier excitation801is made to oscillate about zero, which ensures that the time-averaged off-resonant carrier excitation802of the second and fourth ions of the chain102stays close to zero throughout and returns to zero at the end of the XX-gate operation. InFIG. 8C, as a result of the delivery of the improved segmented pulse sequence Ω(t) illustrated inFIG. 8A, the phase space trajectories αl,m(τ) (in arbitrary unit) of the m-th motional mode (m=1, 2, . . . , 7) are formed, which are graphically depicted therein. Each of the phase space trajectories α1-α7shown inFIG. 8C, separately illustrate the trajectories of each of the seven motional modes of the chain102of seven trapped ions, in the phase space (i.e., position in the horizontal axis and momentum in the vertical axis), where the origin illustrates the initial positions of the ions before the pulse sequence Ω(t) has been applied. As shown inFIG. 8C, the phase space trajectories αl,m(τ) of some of the motional modes, such as the phase space trajectories α4and α5, do not return to the origins at the end of the XX-gate operation, causing displacement of the ions. This is because the original segmented pulse sequence Ω(t) as shown inFIG. 7Ais slightly modified due to the addition of the splines as shown inFIG. 8Aand hence does not close all phase space trajectories.

It is thus useful to include the effect of splines while solving for optimized segmented pulse sequence Ω(t) such that phase space trajectories of all motional mode return to the origins at the end of the XX gate operation. InFIG. 8Ddepicts phase space trajectories αl,m(T) (in arbitrary unit) in phase space of the m-th motional mode (m=1, 2 . . . , 7) due to this further optimized segmented pulse sequence Ω(t) (not shown). InFIG. 8D, the phase space trajectories αl,m(τ) return to the origins at the end of the XX-gate operation (i.e., there is zero residual excitation of the motional modes at the end of the XX-gate operation). The improved segmented pulse sequence Ω(t) therefore results in the reduction of the residual off-resonant carrier excitation as shown inFIG. 8Bdue to the use of the ramps, while maintaining residual motional excitation at the end of the gate minimum as shown inFIG. 8D, thereby improving the fidelity of the XX-gate operation.

FIG. 9depicts a flowchart illustrating a method900that includes various steps performed that are used to generate an improved or optimized segmented pulse sequence Ω(t) to perform a XX-gate operation on two qubits (i-th and j-th qubits) of a chain102of N trapped ions according to one embodiment.

In block902, values for the gate duration τ, the detuning frequency μ, the ramp duration tR, and Lamb-Dicke parameter for the i-th ion and m-th motional mode ηi,mare selected as input parameters. The gate duration gate duration τ and the detuning frequency μ are chosen as described below in conjunction withFIG. 15. The Lamb-Dicke parameters are computed as described below in conjunction withFIG. 11. The motional mode frequencies (ωm, m=0, 1, . . . , N−1) are measured directly from the ion trap quantum computer system100whereas the motional mode structures are computed as described in conjunction withFIGS. 10, 11, and 12A-12C. The Lamb-Dicke parameters ηi,mare determined by the motional mode structures, photon momentum of the laser beams that drive the motional sideband transitions, the ion mass, and the transverse motional mode frequency ωm. The segmented pulse sequence Ω(t) includes NSpulse segments, where NSis iteratively chosen based on the number of ions N for a desired gate duration τ and required overall intensity of the entangling interaction to perform the XX-gate operation. In one example described herein, each pulse segment of the pulse sequence Ω(t) has an equal length τS(=τ/NS) and a intensity Ωs(s=1, 2, . . . , Ns) with ramps at a start and an end thereof using splines with the ramp duration tR. However, in some embodiments, pulse segments of the segmented pulse sequence Ω(t) may have varying lengths in time.

In block904, a value of the entangling interaction χi,j(τ) for i-th and j-th qubits and the values of the phase space trajectories αl,m(τ) for l-th ion (l=i, j) and the m-th motional mode (m=0, 1, . . . , N−1) are computed based on the values for the gate duration τ, the detuning frequency μ, the ramp duration tR, the frequency ωmof the m-th motional mode, the number of pulse segments NS, and the Lamb-Dicke parameters ηi,m, ηj,mselected as input parameters in block902. The value of the entangling interaction is,

χi,j⁡(τ)=2⁢∑m⁢ηi,m⁢ηj,m⁢∫0τ⁢∫0t′⁢Ωi⁡(t)⁢Ωj⁡(t′)⁢sin⁡(μ⁢⁢t)⁢sin⁡(μ⁢⁢t′)⁢sin⁡[ωm⁡(t′-t)]⁢⁢dtdt′
for i-th and j-th qubits, and the values of the phase space trajectories are

In block906, a set of N equations linear in terms of the intensities αs(s=1, 2, . . . , Ns) of the pulse segments, is generated by requiring that the computed values αl,n(τ) equal zero for the l-th ion (l=i, j) and the m−th motional mode (m=0, 1, . . . , N−1).

In block908, the set of linear equations are solved to obtain multiple sets of solutions for the intensities Ωs(s=1, 2, . . . , Ns). Among the sets of solution, a single set of solution for the intensities Ωs(s=1, 2, . . . , Ns) that yields the highest XX-gate fidelity is chosen.

In block910, the pulse sequence Ω(t) is generated by combining the pulse segments having the determined intensities Ωs(s=1, 2, . . . , Ns) and ramps using splines at a start and an end of each pulse segment in the form of sine-squared function with the ramp duration tRselected in block902. This generated pulse sequence Ω(t) is applied to the i-th and j-th qubits to perform a XX-gate operation on the i-th and j-th qubits.

Motional Mode Corrections

FIG. 10depicts an example chain102of seven ions confined in the radial direction by a quadrupole potential generated by the ion trap200. The quadrupole potential generated by the ion trap200and applied to ions in the chain102has an RF component and a DC component. The transverse restoring force due to the RF component of the quadrupole potential and the DC component of the quadrupole potential are modeled by RF spring constants kRFand DC spring constants kirespectively, for i-th ion (i=1, 2, . . . , 7) as illustrated with solid arrows inFIG. 10.

In the ion trap quantum computer system100, there can be discrepancies in the actual longitudinal distribution of the ions from the ideal longitudinal distribution (in which the ions are nearly equally spaced), due to stray electric fields along with the locally varying DC quadrupole potential along the chain102. Thus, there can be discrepancies in the actual motional mode structures from the ideal motional mode structures (based on the ideal longitudinal distribution of the ions), and thus lead to degraded fidelity of the XX-gate if a segmented pulse sequence Ω(t) to perform the XX-gate is generated based on the ideal longitudinal distribution. Thus, in some embodiments described herein, the actual longitudinal distribution of the ions and the motional mode frequencies ωm(m=0, 1, . . . , N−1) are measured in the ion trap quantum computer system100. The actual motional mode structures and the Lamb-Dicke parameter ηi,n, are computed based on the measured actual longitudinal distribution of the ions and the measured motional mode frequencies ωm(m=0, 1, . . . , N−1). These values are selected as input parameters in block902of the method900in generating an improved segmented pulse sequence Ω(t) to perform a XX-gate operation.

FIG. 11depicts a flowchart illustrating a method1100that includes various steps performed that are used to generate and determine actual motional mode structures of a chain102of N trapped ions according to one embodiment.

In block1102, the positions of the ions in the longitudinal direction (denoted as z1, z2, . . . , z7inFIG. 10) are precisely measured by collecting generated fluorescence along the Y-axis from the ions to image them on a camera or the PMT106using the imaging objective104.

In block1104, based on the measured positions of the ions in the longitudinal direction, a set of spring constants kifor i-th ion (i=1, 2, . . . N) in the chain102is generated, where the set of spring constants is used to model the restoring force in the transverse direction on the i-th ion. In generating a set of spring constants kifor i-th ion (i=1, 2, . . . N), the motion of each ion in the transverse direction is approximated as a harmonic oscillation under the influence of the RF and DC spring constants kRFand ki, respectively. The DC spring constants kiare computed based on the actual positions of the ions measured in block1102.

In block1106, based on the measured spacing of the ions (i.e., the distance between the measured positions of adjacent ions) in the longitudinal (axial) direction, the strength of the Coulomb interaction between ions is computed.

In block1108, actual motional mode structures are generated based on the set of DC spring constants kifor i-th ion (i=1, 2, . . . N) generated in block1104, the strength of the Coulomb interaction generated in block1108, and the RF spring constant kRFthat is uniform along the ion chain.

In block1110, the Lamb-Dicke parameter ηi,m(m=0, 1, . . . , N−1) are subsequently computed based on the actual motional mode structures. These Lamb-Dicke parameters are selected as input parameters in block902of the method900for generating an improved segmented pulse sequence Ω(t) to perform a XX-gate operation with improved fidelity.

FIGS. 12A and 12Bdepict an example of motional mode structures of a chain102of seven trapped ions corrected by the method1100described above.FIGS. 12A and 12Brespectively depict ideal motional mode structures in dotted lines (i.e., the motional mode structures based on the ideal longitudinal distribution of the ions and the RF and DC spring constants that are uniform along the chain) and actual motional mode structures (i.e., the motional mode structures corrected by the method1100) in solid lines for the 4th and 5th motional modes, respectively. The vertical axes represent displacement amplitudes of the seven ions in the chain102in the transverse direction.FIG. 12Cdepicts a comparison of the pulse sequence Ωi(t) generated to perform a XX-gate operation obtained using the ideal motional mode structures and the actual motional mode structures formed by using the motional mode corrections determined by the steps found in method1100. The pulse sequence Ωi(t) generated or altered based on the actual motional mode structures improves fidelity of the XX-gate operation.

Calibration of XX-Gate

Single- and two-qubit gates operations are driven by the counter-propagating laser beams in the Raman configuration as shown inFIG. 4, where the two-photon transition detuning frequency δ of the laser beams is either tuned to resonance with the carrier transition (δ=0) or close to a motional mode frequency ωm(δ=±+μ, where μ is close to ωm). In both cases, phases of the laser beams are imprinted on the qubits that are driven by the laser beams during the gate operations. However, due to differences between chosen laser beam parameters for the single-qubit and two-qubit gate operations, such as differences in the optical quality of the laser beams, and differences in which the internal atomic structure are used in the Raman configuration, the imprinted phase might be different for different ions and/or different gate operations. This is a source of error in quantum computation and can be removed by characterizing offsets in the imprinted phases brought about by the single-qubit and two-qubit gate operation at each qubit. In some embodiments, the differences in the imprinted phases are removed by first preforming a single-qubit gate operation on i-th and j-th ions and imprinting single-qubit gate phases ϕiand ϕjon the i-th and j-th ions, respectively, and subsequently adjusting a two-qubit gate phase such that the two-qubit gate phase matches the single-qubit gate phases. The single-qubit phases ϕiand ϕjcan also be considered as phase offsets between a single-qubit and two-qubit gate operations for the i-th and j-th ions, respectively.

FIG. 13depicts a flowchart illustrating a method1300that includes various steps performed that are used to determine the phase offsets ϕiand ϕj(i.e., rotations about the Z-axis on the Bloch sphere) of i-th and j-th qubits and use this information to calibrate the imprinted phase of the XX-gate operation (i.e., adjust one of the phase offsets ϕiand ϕj) on the i-th and j-th qubits according to one embodiment. The method1300is described in conjunction withFIGS. 14A to 14D.FIG. 14Adepicts a schematic view of a gate sequence for determining phase offsets. InFIG. 14A, the XX gate operation having phase offsets ϕiand ϕjis denoted as a box1402and applied to i-th and j-th qubits, respectively.

In block1302, i-th and j-th qubits are prepared so that they are both in the hyperfine ground state |0l|0jby known laser cooling methods, such as Doppler cooling or resolved sideband cooling, and optical pumping.

In block1304, a π/2 pulse with a phase +ϕ1404and a π/2 pulse with a phase −ϕ1406are applied to both qubits before and after the application of the XX-gate operation (with χi,j(τ)=π/4)1402, respectively.

In block1306, the population of the hyperfine ground state |0is measured for by scanning the phase ϕ.FIG. 14Bdepicts an example of the population of the i-th and j-th qubits in the hyperfine ground state |0. A value of the phase ϕ at which the population of the i-th qubit in the hyperfine ground state |0has a maximum value corresponds to the phase offset ϕiof the i-th qubit. The phase offset ϕjof the j-th qubit is determined similarly. Thus, the XX-gate operation is correctly determined to transform two-qubit states |0i|0j, |0i|1j, |1i|0j, and |1i|1jas follows:
|0l|0j→|0l|0j−iei(ϕi+ϕj)|1l|1j
|0i|1j→|0i|1j−iei(ϕi−ϕj)|1i|0j
|1i|0j→−ie−i(ϕi−ϕj)|0i|1j+|1i|0j
|1i|1j→−ie−i(ϕi+ϕj)|0i|0j+|1i|1j.

After phase offsets ϕiand ϕjof the i-th and j-th qubits are determined, the phases of the laser beams that drive the XX-gate operation are adjusted accordingly such that ϕiand ϕjare both zero.

In block1308, a phase adjusted XX-gate (with ϕi=0 and ϕj=0 and absolute value of the entangling interaction, also referred to as a geometric phase, |χi,j(τ)|=π/4) is applied in a gate sequence as shown inFIG. 14C. This sequence is used to measure parity oscillations of the two-qubit state when an XX-gate operation1408is applied on the i-th and j-th qubits followed by analysis pulses1410, which are π/2 pulses with an analysis phase ϕaThe parity (i.e., population of the combined states |0i|0jand |1i|1jminus the populations in states |0i|1jand |1i|0j) is measured as the analysis phase ϕais scanned. As depicted inFIG. 14D, the parity oscillates as the analysis phase ϕais varied between 0 and 2π (normalized to 1 inFIGS. 14B and 14D). When the geometric phase χi,j(τ) (referred to simply as χ) is positive, the parity increases as the analysis shift ϕais increased from zero. When the geometric phase χ is negative, the parity decreases as the analysis phase ϕais increased from zero.

In block1310, the calibration for the phase offset ϕjis adjusted based on the parity measured in block1308. If the geometric phase χ is negative, the calibration for the phase offset ϕjis replaced by that for the phase offset ϕj+π (i.e., j-th qubit is rotated about the Z-axis on the Bloch sphere by π). If the geometric phase χ is positive, the calibration for the phase offset ϕjis kept unchanged. The calibration for the phase offset ϕiof the other ion i is kept unchanged.

In block1312, steps of blocks1302to1308are repeated for all pairs of qubits such that an XX-gate operation for each pair of qubits has a positive geometric phase χ. For example, in a chain of seven trapped ions, steps of blocks1302to1308are repeated for all 21 pairs of qubits. Thus, the XX-gate operation for each pair of qubits can be calibrated (i.e. the imprinted phase offsets are adjusted) to improve the fidelity of the XX-gate operation.

Optimization of Detuning and Gate Duration

FIG. 15depicts a flowchart illustrating a method1500that includes various steps performed that are used to optimize the gate duration τ and the detuning μ according to one embodiment.

In block1502, a segmented pulse sequence Ω(t) to perform a XX-gate operation is generated by the method900for a given value of detuning μ, based on values for the gate duration τ and the ramp duration tRselected in block902, the actual motional mode structures generated by the method1100, and experimentally measured mode frequencies ωm(m=1, 2, . . . , N). For optimizing the segmented pulse sequence Ω(t), an average infidelity of an XX-gate operation (i.e., a discrepancy between the XX-gate gate operation implemented by the segmented pulse sequence Ω(t) and the ideal XX-gate operation) is computed over a small variation or collective drift in the motional mode frequencies. That is, the detuning μ is varied within a small range, the infidelity of the XX-gate operation is computed for each value of the detuning μ, and the computed infidelity of the XX-gate operation is averaged over the range that the detuning μ is varied. This variation of the infidelity of the XX-gate operation is referred to as the detuning induced infidelity. The computation of the detuning induced infidelity of the XX-gate operation is repeated for values of the detuning μ in a frequency range which spans all or some the motional mode frequencies ωm(m=1, 2, . . . , N).

In block1504, values of the detuning parameter μ, at which the infidelity has a minimum value and the determined intensities Ωsof the pulse segments have minimum values (thus minimizing the required laser powers), are chosen as the optimal values for the detuning μ. These values for the detuning μ thus chosen yield the segmented pulse sequence Ω(t) that are both robust against a variation of the detuning μ as well as require lower laser intensities.

FIG. 16Adepicts variations of the determined intensities Ωsof the pulse segments for a chain102of thirteen trapped ions as the detuning μ is scanned in a frequency range between the frequencies ω3and ω11of the third and eleventh motional modes.FIG. 16Bdepicts the computed detuning induced infidelity of the XX-gate operation as the detuning μ is varied within a small range (typically of the order of 1 kHz). At the detuning μ of approximate values of 2.698 MHz, 2.72 MHz, 2.78 MHz, and 2.995 MHz (denoted by “χ” inFIG. 16B), the detuning induced infidelity and the determined intensities Ωsare at their minimum values, and thus one of these values for the detuning μ is chosen as the optimal value of the detuning μ with the corresponding pulse sequence Ω(t) as the optimized pulse sequence Ω(t) to perform a high fidelity XX-gate operation.

In block1506, the optimized pulse sequence Ω(t) is filtered such that the corresponding value of the detuning μ is within a useful range based on noise properties of the motional modes.FIG. 17shows an example of the simulated frequency shifts of the motional mode frequencies of all 13 transverse motional modes in a chain102of 13 trapped ions subject to a stray electric field of 4V/m along the chain102. Based on the simulated frequency shifts, it is evident that the lowest two motional modes suffer from a frequency shift which can introduce substantial error in the XX-gate operation at the detuning μ close to the motional mode frequencies ω1and ω2. Therefore, the optimized pulse sequence Ω(t) to perform an XX-gate operation is chosen for implementation only when the detuning μ is larger than the motional mode frequency ω3. This determines the lower bound of the detuning μ for generating the optimized pulse sequence Ω(t) as reflected inFIGS. 16A and 16B. Additionally, the optimized pulse sequence Ω(t) is filtered such that the XX-gate operation is made insensitive to heating of the motional modes by excluding the motional modes that are measured to have high rates of heating. In an example of a chain of thirteen qubits, the heating rate on the center of mass mode which is the 13-th motional mode (i.e., the common motional mode, in general the N-th motional mode of a chain of N ions) is measured to be relatively high in comparison to other motional modes. In order to exclude the utilization of this motional mode in an XX-gate operation a range in which the detuning μ for generating an optimized pulse sequence Ω(t) is set to have an upper limit at the motional mode frequency ω11where the tilt motional mode frequency ω12is also excluded as it is close to the common motional mode frequency ω13. By choosing the detuning μ in a range between the lower limit and the upper limit, the optimized pulse sequence Ω(t) can perform a XX-gate operation that is insensitive to the motional mode heating and the motional mode frequency shifts due to stray electric fields.

In block1508, an optimal value of the gate duration τ is determined by iterating steps in blocks1502to1506. A value of the gate duration τ in generating the pulse sequence Ω(t) to perform an XX-gate operation while minimizing required intensities ΩSof the pulse segments and the detuning induced infidelity of the XX gate operation. As a starting point the gate duration is set to be τ=α/Δωm, where β is a gate duration factor andΔωmis an average difference between adjacent motional mode frequencies within the range of the detuning μ as determined in block1506. It has been observed that while reducing the gate duration factor β generally reduces the detuning induced infidelity at optimal values of the detuning μ (seeFIG. 16A), reducing the gate duration factor β also increases the determined intensities ΩSof the pulse segments required to achieve maximal entanglement where χi,j(τ)=π/4. The increased intensities ΩSof the pulse segments due to shortened gate duration τ may require higher intensities of the laser beams to perform a XX-gate operation. Thus, there is a trade-off between a reduction of the detuning induced infidelity and a reduction in the required intensities of the laser beams which ultimately determines the optimal value of the gate duration τOPTthrough multiple iterations of steps in blocks1502-1506. It should be noted that the trend of the detuning induced infidelity and the intensities ΩSof the pulse segments with respect to the gate duration τ may not always be monotonic as described above and can be complicated thereby requiring sophisticated learning algorithms to determine both the optimal gate duration τOPTand the number of pulse segments NSin the pulse sequence Ω(t) to perform a XX-gate operation, through iterations of steps in blocks1502to1506and may also include the optimization of the number of pulse segments NS.

A segmented pulse sequence generated as described above may perform an entangling operation between two qubits with improved fidelity. In generating such improved pulse sequence to perform an entangling gate operation, input parameters (the gate duration τ, the detuning μ) are optimized, the motional modes frequencies (ωm) are accurately measured, the motional mode structures are determined. Based on the optimized input parameters, measured motional modes frequencies, and the generated motional mode structures, intensities of each of pulse segments of the segmented pulse sequence are determined, and the resulting segmented pulse sequence can applied to perform an XX-gate operation with improved fidelity.

Additionally, each of the pulse segments in the improved segmented pulse sequence has a pulse shape with ramps at a start and an end thereof, which is used to reduce infidelity from off-resonant carrier excitations. Calibration of the XX-gate operation to adjust imprinted phase shifts in each qubit further improves the fidelity of the XX-gate operation.

While the foregoing is directed to specific embodiments, other and further embodiments may be devised without departing from the basic scope thereof, and the scope thereof is determined by the claims that follow.