The present disclosure generally relates to a method of performing a quantum gate operation in an ion trap quantum computer, and more specifically, to a method for reliably performing quantum computation over a long chain of ions that creates an all-to-all qubit connectivity.
Trapped ions represent a promising platform for universal quantum computation, and high-fidelity quantum gate operations have already been demonstrated on short chains of one or two trapped ions. However, further improvements to fidelity of quantum gate operations and the number of trapped ions (i.e., qubits) are necessary to bridge the gap between these early systems of short chains of trapped ions and a commercially viable quantum computer. To this end, several pathways have been proposed and demonstrated in the art, where trapped ions are separated in space at a given time during a quantum program execution. They, however, come at the cost of sparse qubit connectivity between the qubits, since a direct implementation of qubit-to-qubit interaction between an arbitrary pair of qubits is impossible, which is a known source of overhead in performing quantum computation. They also complicate the hardware design, making high-fidelity gate operations more challenging.
Therefore, there is a need for a method for reliably performing quantum computation over a long chain of trapped ions that creates an all-to-all qubit connectivity.
Embodiments of the disclosure include a method of performing a quantum gate operation in an ion trap quantum computing system. The method includes identifying one or more error mechanisms that cause a quantum computational error in a quantum gate operation on a first trapped ion of an ion chain comprising a plurality of trapped ions, wherein the quantum gate operation is performed by applying a first Raman laser beam and a second Raman laser beam that are configured to cause a Raman transition in the first trapped ion in the ion chain and a coupling between the first trapped ion and one or more axial motional modes of the ion chain, computing a first amplitude of the first Raman laser beam, and a second amplitude of the second Raman laser beam such that the effect of the identified one or more error mechanisms is accounted for, and applying the first Raman laser beam having the computed first amplitude and the second Raman laser beam having the computed second amplitude on the first trapped ion to perform the quantum gate operation on the first trapped ion.
Embodiments of the disclosure further include an ion trap quantum computing system. The ion trap quantum computing system includes a quantum processor comprising an ion chain including a plurality of trapped ions, each trapped ion having two hyperfine states, one or more lasers configured to emit a first Raman laser beam and a second Raman laser beam, which is provided to the ion chain in the quantum processor, a classical computer configured to perform operations including identifying one or more error mechanisms that cause a quantum computational error in a quantum gate operation on a first trapped ion of the ion chain, wherein the quantum gate operation is performed by applying the first Raman laser beam and the second Raman laser beam that are configured to cause a Raman transition in the first trapped ion in the ion chain and a coupling between the first trapped ion and one or more axial motional modes of the ion chain, and computing a first amplitude of the first Raman laser beam, and a second amplitude of the second Raman laser beam such that the effect of the identified one or more error mechanisms is accounted for, and a system controller configured to execute a control program to control the one or more lasers to perform operations on the quantum processor, the operations including applying the first Raman laser beam having the computed first amplitude and the second Raman laser beam having the computed second amplitude on the first trapped ion to perform the quantum gate operation on the first trapped ion, and measuring population of qubit states in the quantum processor.
Embodiments of disclosure also include an ion trap quantum computing system. The ion trap quantum computing system includes a classical computer, a quantum processor comprising an ion chain including a plurality of trapped ions, each trapped ion having two hyperfine states, a system controller configured to execute a control program to control the one or more lasers configured to emit a first Raman laser beam and a second Raman laser beam, which is provided to the ion chain in the quantum processor, and non-volatile memory having a number of instructions stored therein which, when executed by one or more processors, causes the ion trap quantum computing system to perform operations including identifying, by the classical computer, one or more error mechanisms that cause a quantum computational error in a quantum gate operation on a first trapped ion of the ion chain, wherein the quantum gate operation is performed by applying the first Raman laser beam and the second Raman laser beam that are configured to cause a Raman transition in the first trapped ion in the ion chain and a coupling between the first trapped ion and one or more axial motional modes of the ion chain, computing, by the classical computer, a first amplitude of the first Raman laser beam, and a second amplitude of the second Raman laser beam such that the effect of the identified one or more error mechanisms is accounted for, applying, by the system controller, the first Raman laser beam having the computed first amplitude and the second Raman laser beam having the computed second amplitude on the first trapped ion to perform the quantum gate operation on the first trapped ion, measuring, by the system controller, population of qubit states in the quantum processor, and outputting, by the classical computer, the measured population of qubit states in the quantum processor.
So that the manner in which the above-recited features of the present disclosure can be understood in detail, a more particular description of the disclosure, briefly summarized above, may 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 disclosure and are therefore not to be considered limiting of its scope, for the disclosure may admit to other equally effective embodiments.
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.
Embodiments described herein are generally related to a method for reliably performing quantum computation over a long chain of trapped ions that creates an all-to-all qubit connectivity (i.e., all qubits are coupled over both small and large distances). The method includes identifying mechanisms by which quantum computational errors may incur, and addressing the identified mechanisms. Embodiments of the disclosure provided herein provide a generalized Hamiltonian that describes an ion trap quantum computer system, in which two-photon Raman transitions are used to implement quantum gate operations on a chain of trapped ions. Such a generalized Hamiltonian framework provides tools to identify mechanisms that cause errors in quantum gate operations, such as misalignment, defocus, or distortion of a Raman laser beam, and a displacement of trapped ions from their equilibrium locations due the ion-laser interaction. The method described herein for identifying mechanisms for quantum computational errors and addressing the identified mechanism that caused the quantum computational errors, which has been validated by comparing simulation results of fidelity of quantum gate operations based on the method with experimental results, is useful in reliably performing quantum computation over a long chain of trapped ions (e.g., 100 or more trapped ions). In some embodiments, the method further includes applying compensating pulse sequences to increase fidelity of quantum gate operations.
The methods provided herein enables systematic and quantitative error analysis for a variety of hardware implementations of an ion trap quantum computer system and provides guidance in devising appropriate error mitigation strategies.
The classical computer 101 includes 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 objective 104, 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) 106 for measurement of individual ions. Non-copropagating Raman laser beams from a laser 108, which are provided along the X-axis, perform operations on the ions. A diffractive beam splitter 110 creates an array of static Raman beams 112 that are individually switched using a multi-channel acousto-optic modulator (AOM) 114 and is configured to selectively act on individual ions. A global Raman laser beam 116 illuminates ions at once. In some embodiments, individual Raman laser beams (not shown) each illuminate individual ions. The system controller (also referred to as a “RF controller”) 118 controls the AOM 114. The system controller 118 includes a central processing unit (CPU) 120, a read-only memory (ROM) 122, a random access memory (RAM) 124, a storage unit 126, and the like. The CPU 120 is a processor of the RF controller 118. The ROM 122 stores various programs and the RAM 124 is the working memory for various programs and data. The storage unit 126 includes 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 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 which is stored in the ROM 122 or the storage unit 126 and uses the RAM 124 as a working area. The control program will include one or more software applications that include program code (e.g., instructions) that may be executed by a 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 system 100 discussed herein.
During operation, a sinusoidal voltage V1 (with an amplitude VRF/2) is applied to the opposing pair of electrodes 202, 204 and a sinusoidal voltage V2 with a phase shift of 180° from the sinusoidal voltage V1 (and the amplitude VRF/2) is applied to the other opposing pair of electrodes 206, 208 at a driving frequency ωRF, generating a quadrupole potential. In some embodiments, a sinusoidal voltage is only applied to one opposing pair of the electrodes 202, 204, and the other opposing pair of the electrodes 206, 208 is grounded. The quadrupole potential creates an effective confining force in the Y-Z plane perpendicular to the X-axis (also referred to as a “non-axial direction” or “transverse direction”) for each of the trapped ions, which is proportional to the distance from a saddle point (i.e., a position in the axial direction (X-direction)) at which the RF electric field vanishes. The motion in the radial direction (i.e., direction in the Y-Z 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 ky and kz, respectively. 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.
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.
An individual qubit state of each trapped ion may be manipulated by, for example, a mode-locked laser at 355 nanometers (nm) via the excited 2P1/2 level (denoted as |e). As shown in
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 (Bet, Cat, Sr+, Mg+, and Ba+) or transition metal ions (Zn+, Hg+, Cd+).
In an ion trap quantum computer, the motional modes may act as a data bus to mediate entanglement between two qubits. 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. For those of ordinary skill in the art, it should be clear this is achieved by considering each qubit's individual evolution under the action of the detuned laser pulses either simultaneously or sequentially, depending on the entangling gate protocol of choice. Thus, it suffices to investigate an ion-beam interaction of an ion in an ion chain to provide a detailed, complete description of the two-qubit entangling operator for a protocol of any choice.
An ion trap quantum computer system, such as the system 100, that includes two non-copropagating Raman laser beams, such as the Raman beams 112, for selectively accessing individual ions in the chain 102 may be described by an effective Hamiltonian H. The effective Hamiltonian H describes coupling between the motional degrees of freedom (i.e., motional modes of the chain 102) and the internal degree of freedom (i.e., the internal hyperfine states {|↓, |↑>}) of the ions in the chain 102 through the Raman beams. This effective Hamiltonian H can be separated into two parts as
where H0 describes the internal and the motional degrees of freedom for the ions in the chain 102 independently and H describes the light-matter interaction that couples the internal and the motional degrees of freedom for the ions in the chain 102. Treating the laser field of the Raman beams classically, while treating the rest of the system 100 quantum mechanically, together with the dipole approximation, the interaction Hamiltonian H1 can simply be written as
where {right arrow over (E)} is the electric field and {right arrow over (d)}k is the dipole operator of the kth ion out of N total number of ions in the chain 102. For the independent Hamiltonian H0, the ions are considered to be confined in a Paul trap, such as the ion trap 200 and aligned in the chain 102. With the approximation of the motional modes as harmonic oscillations as described above and consideration of an effective two-level system for the internal degrees of freedom of each ion as a qubit, the Hamiltonian H0 reduces to
where ℏ is the reduced Planck constant, ωkqbt is the effective qubit angular frequency for the kth ion, {circumflex over (σ)}kα with α=x, y, or z is the Pauli matrix along the α-axis, ωp is the normal motional mode frequency of the pth normal motional mode with Fock state creation and annihilation (also referred to as “ladder”) operators âp† and âp.
Hereinafter, only two Raman beams (b=1, 2) that drive qubit transitions on a particular ion k are considered, and the ion index k is dropped wherever contextually clear for simplicity. The electric field near the ion is given by
where the individual electric field {right arrow over (E)}b of the Raman beam (b=1 or 2) can be written as
where {circumflex over (∈)}b is the polarization vector, ωb is the angular frequency, Eb and Φp are real functions of the ion position in each of the beam propagation coordinates {right arrow over (r)}b, and h. c. denotes the Hermitian conjugate. After the standard adiabatic elimination of the excited internal states of the ion, the individual summand Hl,k of the interaction Hamiltonian Hl in (2) can be approximated as
where
The coupling of the Raman beam to the motional modes of the ion is embedded in the {right arrow over (r)}b dependent terms in (6). To rewrite them in terms of the normal motional mode operators âp and âp†, the ion position with respect to each Raman beam is first written as
where {right arrow over (r)}b(0) is the equilibrium ion position in the beam coordinate {xb, yb, zb} and êα
where ζp(0)=√{square root over (ℏ/2mωp)} is the spread of the zero-point wavefunction of mode p with the mass m of the ion and vpα
In this section, the Gaussian beam, a prototypical example used widely in the trapped-ion quantum computing community, is considered to derive a suite of expressions required for the generalized Hamiltonian. Specifically, in Sec. IV.A, useful notations for a Gaussian profile of a coherent beam are defined. In Sec. IV.B, using the notations defined in Sec. IV.A, a suite of series expansions that explicitly depend on the non-ideal parameters, including the electric field expression, are derived. In Sec. IV.C, the ways in which noise may now couple into the system are described and a strategy to use the derived expressions described herein for an efficient and systematic error analysis in practice is provided.
Hereinafter, the Gaussian beam is considered to have an elliptical shape with simple astigmatism of a particular form of Ebe±iΦ
where the beam is assumed to propagate along the yb-axis and the two principal axes are along the xb- and the zb-axes. Here, Pb is the power of the beam, kb=2π/λb is the wavevector with λb the wavelength, and ϕb is a constant phase at the origin which can be chosen arbitrarily along the y-axis. The two principal semi axes, wx
where wα
where ηb is the Gouy phase, i.e.,
The spatially dependent terms Ebe±iϕ
where the spatially dependent terms Ebe±iϕ
where: () denotes a binomial coefficient, ┌•┐ denotes the ceiling function, p0 may be λx
are dimensionless. In (8), βb, γα
It should also be noted that the generalized Hamiltonian framework detailed in this section, and shown above, is entirely general with respect to hardware imperfections, such as beam shapes, beam imperfection (e.g., astigmatism), ion positions, or the like. It enables quantum hardware designers to straightforwardly assess the impact of a variety of experimental imperfections on the quantum computational fidelity by reducing one or more error mechanisms identified by the generalized Hamiltonian framework. The generalized Hamiltonian framework described herein serves as a diagnostic tool that aids the designers to locate the major sources of quantum computational errors, critical for developing reliable quantum computers.
By examining the expansions described above, four general mechanisms that may cause a quantum computational error through the spatially dependent terms Ebe±iϕ
The third error mechanism is related to the so-called resonant terms that do not change the motional space, i.e., they have equal numbers of âp and âp† operators. Any even total power of {circumflex over (β)}b, {circumflex over (γ)}αb, and/or {circumflex over (λ)}α
In practice, it is cumbersome to directly use the expressions of A and B functions in (15). A proper and justifiable truncation of the power series in (15) becomes an important task for an approximate yet effective error analysis. It should be noted that each function in (15) can be written in the form of a summation of operators, ΣijÔij, where each operator is in the form of cij{circumflex over (p)}1iq1j. cij here is a complex constant and coo is always non-zero. The task then is reduced to neglecting some of the operators if their contribution to the Hamiltonian is small. To quantify the contribution, the operator norm ∥Ôij∥ is used. The norm is evaluated in a large but finite number of motional degrees of freedom, truncated such that realistic motional-space dynamics can be adequately captured within. In the next section, the error analysis is performed for a realistic situation and a concrete example is provided.
It should be noted the power series of the non-operator terms in (15) can be rewritten in a more compact way by examining terms with ascending power of {circumflex over (p)}1. Doing so renders evaluating the size of the individual coefficients of the powers of {circumflex over (p)}1 more straightforward. The results for the first three orders are shown in Sec. VIII.
V. Parallel Raman Beam geometry
In this section, an approximate Hamiltonian is derived from the generalized Hamiltonian using realistic parameters. In one example, a concrete analysis based on (15) for a realistic set of Raman beam parameters relevant to contemporary trapped-ion quantum computing architectures is provided. In Sec. V.A, sizes of the parameters commensurate to a contemporary trapped-ion quantum computer are specified. In Sec. V.B, the error analysis strategy laid out in Sec. IV.C is applied and a simplified version of the evolution operator that approximately describes the quantum state evolution is described. In Sec. V.C, the approximate evolution operator is applied and in particular the significance of the axial motional mode temperature in determining fidelity of quantum gate operations when using tightly focused Raman beams is shown.
It is emphasized that the way heating of axial motional mode affect fidelity of a quantum gate operation is similar to the way Debye-Waller effect impact fidelity of a quantum gate operation, i.e., the Rabi frequency for driving the internal degrees of freedom of a trapped ion depends on the number of phonons in the axial motional mode. Therefore, any distribution of motional mode with a non-zero width (i.e., a given motional mode having a probability distribution of the number of phonons that is not peaked at a particular number of phonons with probability 1) directly translates to a distribution in the Rabi frequency with a corresponding non-zero width that decoheres the quantum gate operation. This point is briefly discussed towards the end of Sec. V.C.
It is assumed that a chain of trapped ions is addressed by an array of Raman beams propagating in parallel, capable of driving transitions between |↓ and |↑, tightly focused along the chain axis, to achieve individual addressability of trapped ions along the chain. The normal motional modes of the ion chain depending on the dominant projection of their mode vector include axial motional modes that are predominantly along {circumflex over (x)}b (in the X direction), horizontal motional modes that are along ŷb (in the Y direction), and vertical motional modes that are along {circumflex over (z)}b (in the Z direction). The coordinate systems used herein are defined with respect to the axes of the Raman beams, which are assumed to propagate along ŷb (in the Y direction) transverse to the ion chain axis (in the X direction), and exhibit an elliptical Gaussian profile with a loose dimension along {circumflex over (z)}b (in the Z direction) and a tight dimension along {circumflex over (x)}b) in the X direction). The equilibrium position of each ion is assumed to reside near the focal point of each Raman beam, such that yα
For a quantitative analysis, Raman beams with a wavelength/=355 nm and waists wx
≤10 μm and
≤200 μm. It is assumed the alignment errors in |xb(0)| and |zb(0)| are less than 100 nm, and the focusing error in |yα
indicates data missing or illegible when filed
× {square root over (N)}
× {square root over (N)}
× {square root over (N)}
× {square root over (N)}
× {square root over (N)}
indicates data missing or illegible when filed
For cooling of the Yb+ ions, Doppler cooling on the 2S1/2 to 2P1/2 transition is utilized. The motional mode temperatures after the Doppler cooling are given by the average number of phonons at the Doppler limit
By use of realistic parameters, such as the parameter values detailed above, the power-series truncation strategy laid out in Sec. IV.C is performed. To do so, the extent of truncation in the motional space is determined. For the non-axial directions, the initial temperature of the motional modes (i.e., horizontal motional modes and vertical motional modes) is assumed to be the motional mode temperature at the Doppler limit, since the motional modes in the non-axial directions do not easily heat (i.e., the number of phonons does not easily increase). It is noted that the heating of motional modes in the non-axial direction is harder because their vibrational frequencies are larger than axial motional modes and also because electric field that lead to heating need to respect boundary conditions which are open in the axial direction. For the axial direction, the motional modes readily heat, and the fidelity impact from heating of axial motional mode after a time period of heating is discussed. Thus, about 102 phonons for non-axial motional modes (i.e., horizontal motional modes and vertical motional modes) in the non-axial directions and about 104 phonons for axial motional modes in the axial direction are considered, assuming each and every motional mode for a given direction heat more or less evenly. However, it should be noted that it is possible that there could be a dominant motional mode per direction that heats the most while the rest of the motional modes do not readily heat. To account for such a case, about 102×N phonons for the dominant non-axial motional modes and about 104×N phonons for the dominant axial motional modes are also considered. When determining which operator terms Ôij to drop from the Hamiltonian, both even heating of all motional modes and heating of a dominant motional mode for each direction are considered. In devising the effective Hamiltonian, Ôij is dropped from the Hamiltonian only if the fractional contribution from Ô is less than 10−2 in both cases. It is assumed the number N of ions in the chain 102 be N≤50 for concreteness.
The expressions in (15) may now be approximated according to the strategy outlined in Sec. IV.C with the parameters specified in Sec. V.A. Keeping only the terms with the size of the fractional contribution larger than 10−2, the A and B functions are
where s0=1/√{square root over (1+p02)}. It should be noted that the A2 function in (17) may be further approximated in the case where it is used for zb direction, as in the second A2 function used in (13). Specifically, A2(λz
Inserting the simplified A and B functions to the amplitude and phase functions in (13) and (14), respectively, then inserting the simplified amplitude and phase functions to the interaction Hamiltonian H1 in (6), the interaction Hamiltonian H1 is
with {circumflex over (γ)}λ,x
To arrive at (18), ∥ei({circumflex over (β)}
The interaction Hamiltonian H1 in (18) can readily be used to assess fidelity impacts of noise sources ranging from beam misalignment and instability, to noise on the ion positions, as well as heating of motional modes in a single-qubit gate operation using, for instance, a Monte-Carlo type simulation. It can also be easily incorporated into a two-qubit Hamiltonian to evaluate errors in a two-qubit gate operation. Note that when the two beams are perfectly aligned with each other, (18) reduces to a single summation with only one Hermite polynomial term in each summand.
Here, the approximate Hamiltonian expression in (18) is put to test by investigating the fidelity impact of heating of axial motional modes. It is assumed that only one axial motional mode, for instance the center-of-mass (COM) axial motional mode, has a dominant behavior in determining the motional mode temperature, thus dropping the motional mode index p. Next, two realistic situations regarding the beam waist and the beam alignment are considered.
In the first situation, which is representative of the co-propagating setup, two tightly focused beams with identical waists wx
wxeff is the effective waist given by wxf/√{square root over (2)}. The parameter η is proportional to the response of the ion to the kick provided by the beam. The parameter ξ has to do with alignment. Both are scaled parameters (ratios) with respect to the effective beam waist.
In the second situation, which is representative of the counter-propagating setup, one of the Raman beams is considered to be narrowly focused and individually addressing while the other to be very loosely focused and capable of addressing a long ion chain. The loosely focused, global addressing beam has a waist of more than 100 μm which allows us to truncate any {circumflex over (γ)}λ,xm term with m>0. Thus, the interaction Hamiltonian H1 is again of the form in (21). The only difference is that the effective waist here is given by waist wxf of the narrowly focused beam.
For a single-qubit gate operation, the Raman transition is derived at the qubit frequency, i.e., Δω=0. Then any term with imbalanced numbers of â and ↠are off-resonant and thus suppressed. Neglecting these fast-rotating couplings, the evolution operator of a single-qubit gate pulse that has a constant power of a time duration of tsqg can be simply written as
where |n is a Fock state in the axial motional mode space, σΨ
using
where 21(a, b; c; z) denotes a Gaussian hypergeometric function. Equation (24) explicitly shows how the Rabi frequency for driving the spin degree of freedom depends on the number of phonons in the axial motional mode. Thus, a distribution of the number of phonons in the axial motional mode with a non-zero width results in a distribution of the Rabi frequency with a corresponding non-zero width which in turn induces decoherence to the quantum gate operation.
It should be noted that the convergence of (24) greatly depends on n and n. For instance, for perfect alignment, i.e., ξ→0, with n=0.01 and n=2000, m=4 is needed to achieve convergence to the third significant digit. To achieve the same accuracy, m=11 is needed for n=0.02 and n=2000, and m=92 for n=0.02 and n=20000. To mitigate some of the convergence issue, if it is assumed ξ→0, 2m=(−1)m(2m)!/m! can be obtained and thus (24) can be simplified as
Once a proper care for convergence is taken, it is straightforward to insert (26) in (23) to evaluate the effect of the axial motional mode temperature on the fidelity of a single-qubit gate operation for different initial states and measurement schemes. Such an analysis is described in more detail in the next section in conjunction with the experimental results.
In this section, the theoretical results are compared with the experimental results. Specifically, the impact of high-temperature axial motional modes in the presence of tightly focused Raman laser beams is investigated. An experimental apparatus used to obtain the results described herein includes a chain of 171Yb+ ions in a surface-electrode ion trap, where the axial chain spacing can be controlled by adjusting the voltages on several DC electrodes on the trap. Quantum gate operations are performed via Raman transitions induced by two 355 nm Gaussian beams. The state initialization follows a Doppler cooling sequence, where the initial motional mode temperature is cooled to the Doppler limit. When using the counter-propagating setup, the horizontal motional modes are further cooled to
Following the theoretical analysis shown in Sec. V, the following steps were performed to probe the experimental apparatus: (A) the quantum state is initialized to ρ0(0)=|↓<↑|ρT(0), where |0 is a qubit state vector and
is the density operator of a thermal state of the axial motional mode at time t with an average Fock state occupation number
where
Improvement in the bright population P↓ hence fidelity of a quantum gate operation over the static Rabi rate-based method may readily be achieved by the following. As discussed in the static Rabi rate-based method, it is assumed the Rabi rate Ω0 to be that obtained for the initial average number of phonons
Further improvement in fidelity of a quantum gate operation over the axial motional mode heating may be achieved by raising the axial motional mode frequency through the following two mechanisms. Firstly, increasing the motional mode frequency ωA decreases η in (22), which in turn reduces the widths of the distribution of θn with respect to a specific distribution of n and lessens its decoherence effect on the quantum gate operation.
It is noted that the motional model described herein can in fact serve as a convenient tool in experiments to extract the heating rate of the axial motional mode for a single ion or for an ion chain if its COM axial motional mode heats much faster than the rest of the motional modes. To obtain an accurate estimate, the static and optimized P↓ should be measured at different time delay Δt along with the optimal Rabi rate Q0opt. The experimental measurement of the bright distribution P1 as well as the ratio Ω0opt/Ω0st can be fitted to the theoretical predictions by adjusting the initial temperature
The same experiment on a single ion with axial motional mode frequency ωA was repeated for several higher values from 2π×184 kHz up to 2π×513 kHz. Using the fitting method described above, n is extracted as a function of the axial motional mode frequency, shown in
The method of measuring motional mode temperature for a single ion described herein complements the method using sideband spectroscopy, in the way that, while sideband spectroscopy works for modes with mode vector projection along the beam propagation direction, the method described herein works for motional mode with mode vector projection perpendicular to the beam propagation direction. It should be further noted that, while the examples shown herein are for relatively large numbers of phonons, it is straightforward to extend the method described herein to lower numbers of phonons. This may be achieved by reducing state-preparation and measurement error and single-qubit gate error, as well as increasing n through reducing the effective beam waist wxeff.
In some embodiments, narrow band compensating pulse sequences, which are a combination of single-qubit gate operations, that amplify amplitude errors in the qubit space can also be employed to increase the sensitivity of P↓ to the heating rate. It should be noted for an ion chain with more than one ion, the sensitivity is reduced due to the fact that n is generally proportional to 1/√{square root over (N)}. In this section, experimental demonstration of the efficacy of such compensating composite pulse sequences, such as SK1 pulse sequence and the Tycko three-pulse sequence (see below for detail) in mitigating the axial-temperature driven error is discussed below. Specifically, experimental measurement of the bright populations P1 as a function of an average number of phonons n is shown, along with simulations.
A single-qubit gate operation (θ, ϕ) that rotates a Bloch vector by θ about the rotation axis on the equator of the Bloch sphere with polar angle ¢ may be parametrized as
Then, the SK1 pulse sequence SK1(π, 0) is given by
where ψ=arccos (−¼). The Tycko three-pulse sequence Tycko(π, 0) is given by
In practice, (2π, ψ) is implemented in (31) by executing (π, ψ) twice, and similarly for (2π,−ψ).
From
From the generalized Hamiltonian framework, it is straightforward to show that fidelity of a quantum gate operation improves rapidly when decreasing the motional mode temperature.
It should be noted additional terms in the interaction Hamiltonian can now be systematically included in descending order of their contribution towards infidelity of a quantum gate operation to help achieve high fidelity trapped-ion quantum computing. For example, the Debye-Waller effect inducing terms B0± can be included. Higher order terms in the B1± function can be considered as well that originated from the Gouy phase. It should be noted the latter will manifest themselves as a small correction to the Debye-Waller effect. These terms will induce decoherence, if the temperature of the motional modes of an ion chain is high and/or the misalignment between the ion and its addressing beam is large. The Hamiltonian framework disclosed herein analytically captures these effects accurately and provides quantitative methodologies to characterize their impact on fidelity of a quantum gate operation.
Although the effect on single-qubit gate operations, pertaining the coupling to the axial motional modes is mainly described, similar derivation and analysis can readily be extended to two-qubit gate operations. In fact, most of the conclusion including mitigation strategies and techniques for single-qubit gates hold analogous and similar counterparts for two-qubit gate operations. Therefore, in some embodiments, one or more of the mitigation strategies and techniques described herein are used in two-qubit gate operations.
In this disclosure, a general Hamiltonian capable of pinpointing the sources of infidelity in trapped-ion quantum computers with a long chain is provided. By carefully analyzing the Hamiltonian with realistic beam geometry and parameters, quantum computational errors incurred due to alignment and focus have been identified. The generalized Hamiltonian framework described herein is versatile, precisely laying out all terms of importance according to the user defined quality requirement for any trapped-ion quantum computing platform. It is expected the results described herein can enable a hardware engineer to make informed decisions when designing a quantum computer.
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.
This application claims priority to U.S. Provisional Patent Application Ser. No. 63/083,469, filed on Sep. 25, 2020, which is incorporated by reference herein.
Number | Date | Country | |
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63083469 | Sep 2020 | US |