The present invention relates generally to quantum computing, and particularly to control of trapped-ion gates in a quantum computer.
Quantum computers apply principles of quantum physics in solving computational problems and have the potential to perform certain computations far more efficiently than existing digital computers. The basic building block of a quantum computer is the qubit. Quantum computers comprise quantum gates built up from qubits, including single-qubit, two-qubit, and multi-qubit gates.
Trapped-ion systems, in which individual atomic ions serve as qubits, hold promise as a scalable, reliable platform for quantum computing. In a trapped-ion system, the individual atomic ions are trapped by electric fields in an ultra-high vacuum and are cooled to their motional ground states. The internal electronic levels of the ions, as well as the motion of the ions in the trap, are controlled with high precision using lasers, microwaves, or radio-frequency (RF) fields. To perform computations, gates are applied to the input states of the atomic ions by driving fields of the appropriate frequencies, amplitudes and duration.
The term “amplitude,” as used in the present description and the claims, refers to the complex amplitude, which includes both the magnitudes and the phases of the driving fields.
Embodiments of the present invention that are described hereinbelow provide improved trapped-ion gates and methods for driving such gates.
There is therefore provided, in accordance with an embodiment of the invention, a method for quantum computing, which includes providing an array of qubits having an internal transition frequency from a ground state to an excited state. A two-qubit gate, including two of the qubits in the array, is initialized to a first state. The two-qubit gate is switched by applying, for a time sufficient to drive the two-qubit gate to a second state, radiation including simultaneously first upper and lower spectral components, having a first amplitude, in upper and lower displacement sidebands, respectively, of the internal transition frequency, and second upper and lower spectral components, having a second amplitude with a magnitude that is at least 10% of the first amplitude, in upper and lower squeezing sidebands, respectively, of the internal transition frequency.
In a disclosed embodiment, the first state is an unentangled state, and the second state has a target entanglement phase φ≠0.
In some embodiments, the first upper and lower spectral components have first frequencies given by ƒ1=ω±(v+nξ), and the second upper and lower spectral components have second frequencies given by ƒ2=ω±(2v+mξ), wherein ω is the internal transition frequency, v is a phonon frequency of the array of qubits, ξ is a detuning frequency, and m and n are integers. In some of these embodiments, applying the radiation includes applying multiple first upper and lower spectral components having different, respective values of n and multiple second upper and lower spectral components having different, respective values of m. In a disclosed embodiment, the multiple first upper and lower spectral components and multiple second upper and lower spectral components have different respective amplitudes, including at least one positive amplitude and at least one negative amplitude.
In an alternative embodiment, the magnitude of the second amplitude is at least 50% of the first amplitude.
In some embodiments, applying the radiation includes choosing the first and second upper and lower spectral components and the first and second amplitudes so as to increase a fidelity of the two-qubit gate under deviations in a Rabi frequency of the radiation. Additionally or alternatively, applying the radiation includes choosing the first and second upper and lower spectral components and the first and second amplitudes so as to increase a fidelity of the two-qubit gate under deviations in a duration of application of the radiation relative to a switching time of the two-qubit gate.
In some embodiments, providing the array of qubits includes trapping an array of ions in an ion trap, wherein the two-qubit gate includes two of the ions in the array. In one embodiment, the internal transition frequency is an electronic transition frequency, and applying the radiation includes applying laser radiation.
In a disclosed embodiment, the first upper and lower spectral components and the second upper and lower spectral components are all phase-coherent.
Additionally or alternatively, applying the radiation includes choosing the first and second upper and lower spectral components and the first and second amplitudes so as to increase a fidelity of the two-qubit gate under deviations in a phonon frequency and temperature of the array of qubits.
In a disclosed embodiment, applying the radiation includes choosing the first upper and lower spectral components and the second upper and lower spectral components to satisfy constraints C1 through C6 as defined in the specification in Table I.
There is also provided, in accordance with an embodiment of the invention, a system for quantum computing, including an array of qubits having and internal transition frequency from a ground state to an excited state. A radiation source is configured to apply to the qubits in the array radiation including simultaneously first upper and lower spectral components, having a first amplitude, in upper and lower displacement sidebands, respectively, of the internal transition frequency, and second upper and lower spectral components, having a second amplitude with a magnitude that is at least 10% of the first amplitude, in upper and lower squeezing sidebands, respectively, of the internal transition frequency. A controller is configured to initialize a two-qubit gate, including two of the qubits in the array, to a first state and to switch the two-qubit gate by driving the radiation source to apply the radiation including the first upper and lower spectral components at the first amplitude and second upper and lower spectral components at the second amplitude for a time sufficient to drive the two-qubit gate to a second state.
There is additionally provided, in accordance with an embodiment of the invention, a method for quantum computing, which includes providing an array of qubits having an internal transition frequency from a ground state to an excited state. A two-qubit gate, including two of the qubits in the array, is initialized to a first state. The two-qubit gate is switched by applying, for a time sufficient to drive the two-qubit gate to a second state, radiation including simultaneously first spectral components w1(t) in upper and lower displacement sidebands of the internal transition frequency and second spectral components w2(t) in upper and lower squeezing sidebands, respectively, of the internal transition frequency. The first and second spectral components have respective frequencies and amplitudes satisfying constraints C1 through C6 as defined in the specification in Table I.
In a disclosed embodiment, the first state is an unentangled state, and the second state has a target entanglement phase φ≠0, and the target entanglement phase is φ=−π/2 for full entanglement.
There is further provided, in accordance with an embodiment of the invention, a method for quantum computing, which includes providing an array of qubits having an internal transition frequency from a ground state to an excited state. A multi-qubit gate, including three or more of the qubits in the array, is initialized to a first state. The multi-qubit gate is switched by applying, for a time sufficient to drive the multi-qubit gate to a second state, radiation including simultaneously first upper and lower spectral components, having a first amplitude, in upper and lower displacement sidebands, respectively, of the internal transition frequency, and second upper and lower spectral components, having a second amplitude with a magnitude that is at least 10% of the first amplitude, in upper and lower squeezing sidebands, respectively, of the internal transition frequency.
In some embodiments, applying the radiation includes choosing the first and second upper and lower spectral components and the first and second amplitudes so as to increase a fidelity of the multi-qubit gate under deviations in an operating parameter of the multi-qubit gate.
The present invention will be more fully understood from the following detailed description of the embodiments thereof, taken together with the drawings in which:
In quantum computing, fidelity (F) is a key indicator of the performance of a given quantum gate. The fidelity is a statistical measure of the difference between the states of a practical, physical gate and of an ideal gate following a state switching operation. For fault-tolerant quantum computing on a practical scale, the gate infidelity (1-F) should be very small, for example no greater than 10−4. Achieving this level of fidelity in two-qubit gates and multi-qubit systems generally remains a major challenge.
Multi-qubit entanglement gates are crucial to quantum computing, as they are an essential part of a universal gate set. In trapped-ion systems, entanglement gates are typically generated by driving the ions with electromagnetic fields that create phonon-mediated qubit-qubit interactions. One scheme that can be used for this purpose is the Mølmer-Sørensen (MS) gate, in which the two-qubit gate is switched by applying laser radiation simultaneously to the upper and lower displacement sidebands of an internal transition frequency of the ions. The “displacement sidebands” are defined as frequency bands detuned above and below the center frequency of the internal transition by a phonon frequency corresponding to the normal vibrational modes of the array of ions in the ion trap. The upper and lower sidebands are also referred to as the “blue” and “red” sidebands.
The MS gate is thus switched by applying laser radiation to the trapped ions at frequencies ƒ1=ω±(v+ξ), wherein ω is the internal transition frequency, v is a phonon frequency of the array of ions in the ion trap, and ξ is a detuning frequency, which is much smaller than v. In the description below, for the sake of clarity and concreteness, the transition frequency ω is assumed to be the frequency of a suitable electronic transition in a trapped-ion system, which is excited using laser radiation. The principles of the present invention may alternatively be applied, mutatis mutandis, to quantum gates applied to other trapped-ion transitions, such as transitions between spin states of a Zeeman-split manifold, or between states in hyperfine-split manifolds, as well as to other types of quantum gates, in which case microwave or radiofrequency excitation may be used.
The MS gate, however, is sensitive to the amplitude of the laser field and exhibits a degradation of fidelity that is quadratic in field intensity noise (or, equivalently, in variations in the Rabi frequency Ω, which is proportional to the vector electric field amplitude of the laser radiation). Thus, even small deviations in the laser intensity or in the beam-pointing accuracy or polarization can cause the infidelity to rise to unacceptable levels. Other sources of infidelity can include deviations in the phonon frequency of the ion array, heating of the phonon mode, and inaccuracy in the duration over which the laser radiation is applied in switching the gate relative to the ideal switching time.
Embodiments of the present invention that are described herein address these problems by applying phase-coherent laser radiation to the two-qubit gate with substantial intensity at frequencies in upper and lower squeezing sidebands of the internal atomic transition frequency, in addition to the phase-coherent radiation at the displacement sidebands frequencies. The “squeezing sidebands” are defined as frequency bands displaced above and below the internal atomic transition frequency by twice the phonon frequency of the normal vibrational modes of the array of ions in the ion trap, i.e., at frequencies ƒ2=ω±(2v+mξ), wherein m is an integer (which may be positive or negative). The amplitude of the phase-coherent laser radiation in the squeezing sidebands has a magnitude that is at least 10% of the amplitude of the laser radiation in the displacement sidebands, and may be 50% or more of the amplitude of the laser radiation in the displacement sidebands.
In some embodiments, the laser source may irradiate the ions in the two-qubit gate with phase-coherent multiple frequency components in the squeezing sidebands, having different values of the integer parameter m and different, respective amplitudes. The laser irradiation may have multiple phase-coherent frequency components in the displacement sidebands, as well. The frequencies and amplitudes of these phase-coherent frequency components, which may be positive or negative, are optimized using a protocol comprising constraints that maximize the fidelity of the gate in the face of variations in the amplitude of the laser radiation and other noise factors. By appropriate choice of frequencies, phases, and amplitudes, the fidelity of the gate can be made robust against variations in the laser amplitude. For example, infidelity can be made to scale as the inverse of the fourth power of deviations in the Rabi frequency (1−F˜Ω−4) or with other functional scaling depending on the frequencies, phases, and amplitudes of the frequency components. Additionally or alternatively, in similar fashion, the fidelity can be made robust against variations in the phonon frequency Av and in the switching time AT.
Although the embodiments described below relate specifically to two-qubit gates, the principles of the present invention may be extended to create robust multi-qubit gates with three or more qubits. Like the two-qubit gates described herein, these multi-qubit gates use spin-dependent forces for coupling internal qubit-spin states to motion, and they are susceptible to loss of fidelity due to deviations in the Rabi frequency and other parameters. In alternative embodiments of the present invention, the robustness of such multi-qubit gates is enhanced by application of phase-coherent electromagnetic fields with appropriate amplitudes and detuning in upper and lower displacement sidebands and upper and lower squeezing sidebands. As gates with three or more qubits have multiple vibrational modes that can be used to couple the internal states of the qubits, the displacement and squeezing sidebands may be chosen according to the phonon frequencies of any of the normal vibrational modes.
Furthermore, although the embodiments described herein relate specifically to trapped-ion quantum computing systems, the principles of the present invention may alternatively be applied, mutatis mutandis, to arrays of qubits of other sorts, such as superconducting (SC) qubits or an array of trapped neutral atoms inside an optical cavity. In trapped-ion systems, the qubits exploit the internal electronic degrees of freedom of the trapped ions (such as the electronic spins), while interactions between qubits are mediated by bosons based on the harmonic phonon modes. SC qubits exploit transmons, for example, and their interactions are mediated by bosons based on photonic excitation of a waveguide resonator to which all the SC qubits are coupled. Similar Gates built from these SC qubits are known as Resonator-Induced Phase (RIP) gates. In an alternative embodiment of the present invention, RIP qubits of this sort can similarly be switched by combined excitation of displacement and squeezing sidebands defined by the resonator modes, with excitation amplitudes chosen so that the gate fidelity is robust against variations in the Rabi frequency.
An electronic qubit control and computation processor 32 drives radiation source 28 to direct additional beams toward the trapped ions in order to perform quantum computational operations and then read out the computational results. Typically, the results are read out by tuning a laser beam to an absorption line of one of the qubit states and then measuring the resulting fluorescent emission using an optical detector 30. Processor 32 receives the result of the computation and drives laser source 28 to perform additional computational steps in accordance with the algorithm being implemented.
The cooled ions 40 are held in a linear array by the electromagnetic fields within trap 24. Coulomb repulsion between ions 40 and trapping fields determine the equilibrium distance between the ions and also determines the phonon frequency v of the vibrational modes of motion of ions 40 in the array. These vibrational modes give rise to vibrational sidebands of the optical transition frequencies between the states of ions 40. Absorption of a photon in one of these sideband frequencies causes the array of ions to vibrate while driving the internal transition of the absorbing ion, thus transferring energy to the normal modes of the ion array and entangling the internal and motional states of the ions by a spin-dependent force. This process is described below in greater detail with reference to
To operate the two-qubit gate, an excitation laser 46 (or multiple lasers) coherently irradiates ions 40 on appropriate sidebands of a selected internal transition of the ions. The beam of laser 46 is modulated, for example by a suitable acousto-optic modulator, to coherently include frequency components in both the displacement sidebands (ƒ1=ω±(v+ξ)) and the squeezing sidebands (ƒ2=ω±(2v+mξ)), as defined above. The amplitudes of the frequency components are chosen, using a protocol that is described further hereinbelow, so as to optimize the robustness of the two-qubit gate against various sources of noise, including (but not limited to) variations in the Rabi frequency of laser 46. Laser 46 is operated in this manner for a gate time T that is suitable for driving the two-qubit gate from its initial state to an entangled target state.
After completion of the computational cycle, a readout laser 48 reads the state of the gate or gates among ions 40. Readout laser 48 is tuned to an absorption band of one of the states of the ions in the gate. Absorption of the laser radiation by the ions in the appropriate state gives rise to fluorescence, which is measured by optical detector 30 (
Each of the ions in gate 50 has an electronic ground state |g> and an internal excited state |e>, for example corresponding respectively to selected Zeeman spin levels within the |4S1/2> and |3D5/2> manifolds of Ca+. Thus, two-qubit gate 50 has a ground state 52 |gg,n>, in which both ions are in the respective ground states, and an excited state 54 |ee,n>, in which both ions are in the respective excited states, wherein n is the phonon number. Transitions between ground state 52 and excited state 54 take place by two-photon interaction through intermediate states 56 and 58, identified as |ge> and |eg>. Each of the intermediate states is split into multiple motional states that are separated by the phonon frequency v, for example |ge,n+2>, |ge,n+1>, |ge,n>, |ge,n−1>, and |ge,n−2>.
In the pictured embodiment, laser 46 (
Although
Furthermore, although the examples described herein use spectral components only in the displacement and squeezing sidebands, the principles of the present invention may be extended to create multi-qubit gates with coherent excitation in additional motional sidebands, for example third- and/or fourth-order sidebands f3=ω±(3v+pξ) and f4=ω±(4v+qξ), as well as sidebands in other motional modes. Such multi-band schemes are considered to be within the scope of the present invention.
The detuning ξ is a function of the Lamb-Dicke parameter n and the Rabi frequency Ω. In the Mølmer-Sørensen gate, in which only displacement sidebands are used, ξ=2√{square root over (n)}ηΩ, wherein n is an integer. The Lamb-Dicke parameter is given by the ratio of the width of the wave function of the motional states of the ions to the laser wavelength:
Here k is the laser wave number, and m is the ion mass.
In the example that is described hereinbelow, laser 46 drives a two-qubit trapped-ion gate with a beam having an amplitude W(t)=w1(t)+w2(t), wherein w1(t) consists of frequency components f1 in the displacement sidebands, while w2(t) consists of frequency components f2 in the squeezing sidebands. The laser beam, which may comprise a single beam that addresses the ions globally or multiple local beams addressing the ions individually, has a total Rabi frequency Ω and is applied to the gate for a time T. As explained above, the functional forms and parameters of w1(t) and w2(t) are chosen so that the final state of the two-qubit gate, at time T, varies slowly with deviations in the Rabi frequency ΔΩ, as well as with deviations in the phonon frequency Δv, in the temperature, and in the gate switching time ΔT. One protocol for this purpose is presented below. Various other mathematical methods may alternatively be applied in optimizing w1(t) and w2(t). All such methods are considered to be within the scope of the present invention. For convenience in defining the protocol, we define the parameter r(t):
We also use the following notational shorthand (for arbitrary functions f and g):
We define the following phase functions corresponding to the interaction between the ions in the two-qubit gate:
At the gate time t=T, there should be no residual displacement or squeezing and no rotation of the spins of the ions. The entanglement at time T is expected to reach a target entanglement phase φ≠0, for example φ≠−π/2 for full entanglement, after starting from zero entanglement at t=0. The choice of frequency components should also be such that the entanglement phase is robust against variations in the Rabi frequency Ω. These requirements together impose the constraints listed below in Table I on w1(t) and w2(t):
The final constraint (C6) implies that the leading-order term in the variation of the entanglement phase with small changes in Ω is zero. C6 can alternatively be generalized to a series of constraints with increasing orders of differentiation, ∂Ωn.
The constraints above do not uniquely define w1(t) and w2(t), and various solutions can be developed to provide robustness against different types of noise. All such solutions are considered to be within the scope of the present invention. In other words, after reading the present disclosure, the person of ordinary skill in the art will be able readily to verify whether or not a given choice of w1(t) and w2(t) satisfies the constraints in Table I, as well as to apply analytical and computational methods that are known in the art to find other solutions.
One particular family of solutions to the constraints in Table I is presented hereinbelow, assuming that φ=−π/2. To satisfy the integral constraints (C1, C2, and C3), the integrands are composed of non-zero multiples of the gate rate (detuning frequency) ξ=2π/T, giving a set of detuned frequencies in each motional sideband. The following waveforms will satisfy the first four constraints:
Here w1(t) comprises odd harmonics with coefficients a2n+1, while r(t) comprises even harmonics with coefficients s2n.
Various choices of the coefficients a2n+1 and s2n can satisfy constraints C5 and C6. A simple solution, with relatively few harmonics, is the following:
Numerical optimization gives the coefficient values a=0.3608·ξ and s=0.7820. In this example, the amplitudes of the spectral components of the laser field in the squeezing sidebands have magnitudes greater than those of the spectral components in the displacement sidebands.
The Rabi frequency that is required to drive the two-qubit gate in accordance with this spectral scheme is Ωrobust≈(3/η+6/η2)ξ. This frequency value is greater than the Rabi frequency of a conventional Mølmer-Sørensen gate, for example, meaning that laser 46 operates at higher intensity in the present scheme in order to drive the squeezing sidebands. In practical applications of quantum computing, however, the limit on computer performance is typically fidelity, rather than laser power.
Although the example presented above relates to transitions from an initial ground state with unentangled phase (φ=0) to a target excited state with entanglement phase φ≠0 (and specifically φ=−π/2), the techniques described herein are likewise applicable to substantially any state transition of the two-qubit gate, including transitions from entangled to unentangled states.
Similar improvements in fidelity can be demonstrated in the face of errors in gate timing δT, errors in phonon frequency δv, and motional mode heating, as well as susceptibility of the infidelity to finite ion temperature.
The embodiments described above are cited by way of example, and the present invention is not limited to what has been particularly shown and described hereinabove. Rather, the scope of the present invention includes both combinations and subcombinations of the various features described hereinabove, as well as variations and modifications thereof which would occur to persons skilled in the art upon reading the foregoing description and which are not disclosed in the prior art.
This application claims the benefit of U.S. Provisional Patent Application 63/287,105, filed Dec. 8, 2021, which is incorporated herein by reference.
Filing Document | Filing Date | Country | Kind |
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PCT/IB2022/061873 | 12/7/2022 | WO |
Number | Date | Country | |
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63287105 | Dec 2021 | US |