Stark Shift Cancellation

Abstract

Systems and techniques are provided that facilitate Stark shift cancellation. In various embodiments, the system can include a control qubit coupled to a target qubit. In various cases, the control qubit can be driven by a first tone that entangles the control qubit with the target qubit. In various aspects, the control qubit can be further driven by a second tone simultaneously with the first tone. In various cases, the second tone can have an opposite detuning sign to the first tone. In various cases, the first tone can induce a Stark shift in the operating frequency of the control qubit, and the second tone can cancel the Stark shift.

Description

[Technical Field]

[0001] The subject disclosure relates to qubits, and more particularly to canceling Stark shifts in qubits. Summary of the Invention

[0002] The following presents a summary to provide a basic understanding of one or more embodiments of the present invention. This summary is not intended to identify or delineate key or critical elements of any particular embodiment or scope of the claims. Its sole purpose is to present concepts in a simplified form as a prelude to the more detailed description that is presented later. In one or more embodiments described herein, a device, system, computer-implemented method, apparatus, or computer program product, or combination thereof, that can facilitate Stark shift cancellation, is described.

[0003] According to one or more embodiments, a system is provided. The system can include a control qubit coupled to a target qubit. In various aspects, the control qubit can be driven by a first tone, and the first tone can entangle the control qubit with the target qubit. In various cases, the control qubit can be driven by a second tone simultaneously with the first tone. In various cases, the second tone can have an opposite detuning sign to the first tone. In various aspects, the first tone can induce a Stark shift in the operating frequency of the control qubit, and the second tone can cancel the Stark shift.

[0004] According to one or more embodiments, a device is provided. The device can include a control qubit coupled to a target qubit. In various aspects, the control qubit can be driven by a first tone, where the first tone can induce a Stark shift in the operating frequency of the control qubit. In various cases, the control qubit can be driven by a second tone. In various cases, the second tone can cancel the Stark shift. In various cases, the second tone can have an opposite detuning sign to the first tone. In various cases, the second tone can be applied simultaneously with the first tone. In various cases, a first absolute difference between the frequency of the first tone and the operating frequency of the control qubit can be less than the absolute value of the anharmonicity of the control qubit, and a second absolute difference between the frequency of the second tone and the operating frequency of the control qubit can be less than the absolute value of the anharmonicity of the control qubit.

[0005] According to one or more embodiments, an apparatus is provided. The apparatus can include a qubit lattice. In various aspects, a control qubit in the qubit lattice can be driven by a quantum entanglement gate tone. In various cases, the control qubit can be driven by a Stark-shift cancellation tone simultaneously with the quantum entanglement gate tone. In various cases, the Stark-shift cancellation tone can have an opposite detuning sign to the quantum entanglement gate tone.

[0006] According to one or more embodiments, the aforementioned systems, devices, and / or apparatus may be implemented as a method. [Brief explanation of the drawings]

[0007] [Figure 1] FIG. 1 is a block diagram of an example non-limiting system for facilitating Stark shift cancellation, according to one or more embodiments described herein. [Figure 2]1 is an example, non-limiting graph illustrating Stark shift as a function of driving tone frequency, according to one or more embodiments described herein. [Figure 3] 1 is a flowchart of an example of a non-limiting method for facilitating calibration of Stark-shift cancellation tones, according to one or more embodiments described herein. [Figure 4] 1 is a flowchart of an example of a non-limiting method that facilitates performing zero-noise extrapolation using Stark shift cancellation, according to one or more embodiments described herein. [Figure 5] FIG. 1 is a block diagram of an example non-limiting system including a spectator qubit that facilitates Stark shift cancellation, in accordance with one or more embodiments described herein. [Figure 6] FIG. 1 is a block diagram of an example non-limiting system including a qubit lattice to which Stark shift cancellation may be applied, according to one or more embodiments described herein. [Figure 7] 1 is a flowchart of an example of a non-limiting method for facilitating Stark shift cancellation, according to one or more embodiments described herein. [Figure 8] 1 is a flowchart of an example of a non-limiting method for facilitating Stark shift cancellation, according to one or more embodiments described herein. [Figure 9] FIG. 1 is a block diagram of an example non-limiting operating environment in which one or more embodiments described herein may be facilitated. [Figure 10] FIG. 1 illustrates an exemplary, non-limiting cloud computing environment in accordance with one or more embodiments described herein. [Figure 11] FIG. 2 illustrates example, non-limiting abstraction model layers according to one or more embodiments described herein. DETAILED DESCRIPTION OF THE INVENTION

[0008] The following detailed description is merely illustrative and is not intended to limit the embodiments or the application and / or uses of the embodiments, nor is it intended to be bound by any information expressed or implied in the preceding Background or Summary sections or in the Detailed Description section.

[0009] One or more embodiments will now be described with reference to the drawings, wherein like reference numerals are used to refer to like elements throughout. In the following, for purposes of explanation, numerous specific details are set forth in order to provide a more thorough understanding of one or more embodiments. It will be apparent, however, that in various instances, one or more embodiments may be practiced without these specific details.

[0010] The subject disclosure relates to quantum computing using quantum circuits. Quantum computing employs quantum physics, rather than transistor-based binary digital technology, to encode and process information. Quantum computing devices may employ quantum bits (also referred to as qubits), which operate according to the laws of quantum physics and can exhibit phenomena such as superposition and entanglement. The quantum physics principle of superposition allows a quantum bit to be partially in a state that simultaneously represents both a value of "1" and a value of "0." The quantum physics principle of entanglement allows quantum bits to be correlated with each other such that their combined state cannot be resolved into the individual quantum bit states. For example, the state of a first quantum bit may depend on the state of a second quantum bit. Thus, quantum circuits may employ quantum bits that encode and process information in ways that may be quite different from transistor-based binary digital technology.

[0011] There are various types of qubits. One type of qubit is a frequency or flux tunable qubit, which may comprise or be associated with a superconducting quantum interference device (SQUID) loop, which may include a set of Josephson junctions to facilitate the performance of the SQUID loop's functions. The frequency of a flux tunable qubit may be adjusted or tuned by varying the magnetic flux passing through the SQUID loop. Another type of qubit may be a fixed frequency qubit, which may comprise a single Josephson junction rather than a SQUID loop. In contrast to a frequency or flux tunable qubit, whose frequency may be adjusted by varying the magnetic flux passing through the SQUID loop, the frequency of a fixed frequency qubit may be fixed and typically cannot be easily or simply changed. The frequency of a fixed frequency qubit is generally determined probabilistically by the details of the fabrication process utilized to produce the fixed frequency qubit. Fixed frequency qubits have the advantage that they do not require magnetic flux lines and therefore require fewer wiring or other circuits or components in the cryostat, and the control electronics for the operation of such qubits are less than frequency or flux tunable qubits, and are less susceptible to magnetic flux noise.

[0012] In the field of quantum computing, quantum entanglement gates (e.g., CNOT gates, controlled phase gates) can be applied to a control qubit by exposing the control qubit (e.g., by driving the control qubit) to a microwave tone having a frequency that matches the operating frequency (e.g., transition frequency) of a target qubit coupled to the control qubit. Such a microwave tone can be referred to as a non-resonant tone because the frequency of the microwave tone does not match the operating frequency of the control qubit. Furthermore, because a microwave tone can be applied to the control qubit and the frequency of the microwave tone can match the operating frequency of the target qubit, the microwave tone can transform the states of both the control qubit and the target qubit, and such transformation can be determined by the amplitude and / or duration of the microwave tone. For example, a given amplitude and / or duration can cause the microwave tone to perform a given gate (e.g., a CNOT gate) on the control qubit and the target qubit, while a different amplitude and / or duration can cause the microwave tone to perform a different gate (e.g., a controlled phase gate) on the control qubit and the target qubit. In either case, such microwave tones can entangle the control qubit with the target qubit, but they can also change the operating frequency of the control qubit, a change known as the Stark shift. In other words, driving the control qubit with a non-resonant tone can shift the operating frequency of the control qubit either upward or downward.

[0013] Stark shifts can be undesirable for a variety of reasons. For example, Stark shifts can cause dynamic frequency collisions. In other words, when a control qubit experiences a Stark shift, the operating frequency of the control qubit may shift to a new value, which may be equal to and / or within any suitable threshold margin of the operating frequencies of adjacent qubits and / or observed qubits. In such cases, the shifted operating frequency of the control qubit may be too close to the operating frequencies of adjacent qubits and / or observed qubits, meaning that the control qubit and adjacent qubits / observed qubits may interfere with each other or may be indistinguishable for quantum computing purposes.

[0014] As another example, Stark shifts can disrupt zero-noise extrapolation. As those skilled in the art will recognize, zero-noise extrapolation is an error mitigation technique that involves obtaining multiple measurements of the resulting quantum state (e.g., after applying one or more quantum gates of interest) while varying the noise level so that the zero-noise limit can be identified by extrapolation. In practice, controllably varying the noise level of a quantum gate can be extremely difficult. Assuming time-invariant noise, it has been shown that stretching a quantum gate in time by a certain factor (e.g., increasing the duration of the microwave tone that executes such a quantum gate by a certain factor and appropriately recalibrating the microwave tone amplitude) is equivalent to amplifying the noise associated with that quantum gate by the same factor. Unfortunately, however, the Stark shift experienced by a control qubit is a function not only of the drive frequency but also of the drive amplitude. Thus, if we perform zero-noise extrapolation on the control qubit by repeatedly stretching the quantum gate in time and correspondingly rescaling the gate's drive amplitude, the control qubit may experience a different Stark shift at each iteration (e.g., by the rescaled amplitude at each iteration), and correspondingly, the control qubit's relaxation time may be significantly different at each iteration. In other words, the presence of a Stark shift may violate the assumption of time-invariant noise, and zero-noise extrapolation may yield highly inaccurate results. This problem may be particularly severe for superconducting qubits interacting with defect two-level systems.

[0015] Therefore, systems and / or techniques that address one or more of these technical problems are desirable.

[0016] Various embodiments of the present invention may address one or more of these technical problems. Specifically, various embodiments of the present invention may provide systems and / or techniques that may facilitate Stark shift cancellation. In particular, the inventors have recognized that the direction of the Stark shift experienced by a control qubit when driven by a microwave tone may depend, in various cases, on the sign of the detuning of the microwave tone. That is, if the detuning of the microwave tone is positive, the control qubit may experience a Stark shift in one direction (e.g., the operating frequency of the control qubit may increase or decrease). In contrast, if the detuning of the microwave tone is instead negative, the control qubit may experience a Stark shift in the opposite direction (e.g., if the operating frequency of the control qubit increases with a positive detuning sign, the operating frequency of the control qubit may decrease with a negative detuning sign, and if the operating frequency of the control qubit decreases with a positive detuning sign, the operating frequency of the control qubit may increase with a negative detuning sign). Thus, the inventors have realized that the Stark shift experienced by a control qubit being driven by a first microwave tone can be canceled out or nullified by simultaneously driving the control qubit with a second microwave tone having an opposite detuning sign to the first microwave tone. In other words, the first microwave tone can increase or decrease the operating frequency of the control qubit, and the second microwave tone can decrease or increase the shifted operating frequency of the control qubit back. As a result, the change in the operating frequency of the control qubit can be zero or less than any suitable threshold. In other words, the inventors of the various embodiments described herein have devised techniques that can eliminate or significantly reduce the Stark shift. Therefore, dynamic frequency collisions caused by the Stark shift can be similarly eliminated or reduced, and zero-noise extrapolation, which is hindered by the Stark shift, can be performed more accurately.

[0017] In particular, various embodiments described herein may include a control qubit coupled to a target qubit. In various aspects, the control qubit may be any suitable type of quantum computing structure and / or architecture capable of exhibiting qubit behavior (e.g., capable of superposition of two basis states). For example, the control qubit may be a superconducting qubit comprised of one or more Josephson junctions (e.g., a phase qubit, a charge qubit, a flux qubit, a transmon qubit). As another example, the control qubit may be a spin qubit and / or a quantum dot. Similarly, the target qubit may be any suitable quantum computing structure and / or architecture capable of exhibiting qubit behavior (e.g., a superconducting qubit, a spin qubit, and / or a combination thereof). In any case, the control qubit may have and / or exhibit a particular operating frequency (e.g., a particular transition frequency), and the target qubit may have and / or exhibit a different operating frequency (e.g., a different transition frequency).

[0018] In various embodiments, the control qubit may be coupled to the target qubit by any suitable quantum computing coupler. For example, in some cases, the quantum computing coupler may include one or more superconductor cables and / or superconductor wires that directly couple the control qubit to the target qubit. As another example, the quantum computing coupler may include one or more superconductor cables and / or superconductor wires that capacitively couple the control qubit to the target qubit. As yet another example, the quantum computing coupler may include any suitable combination of the foregoing.

[0019] In various cases, the control qubit can be driven by or exposed to a first microwave tone, or both. More specifically, as those skilled in the art will recognize, any suitable electronic control circuit (e.g., a waveform generator) can be coupled to the control qubit by a first drive line (e.g., a first superconductor cable / wire), the electronic control circuit can generate the first microwave tone, and the first drive line can carry the first microwave tone from the electronic control circuit to the control qubit. In various cases, the first microwave tone can be a quantum entanglement tone. In other words, the first microwave tone can perform a desired quantum entanglement gate (e.g., a CNOT gate, a controlled phase gate) on the control qubit and the target qubit. To facilitate such entanglement, the frequency of the first microwave tone may be equal to (and / or within any suitable threshold margin of) the operating frequency of the target qubit, and the amplitude and / or duration of the first microwave tone may be calibrated to any suitable magnitude to perform the desired quantum entanglement gate. Therefore, if the operating frequency of the target qubit is lower than the operating frequency of the control qubit, the first microwave tone may be considered to have a negative detuning sign (e.g., the frequency of the first microwave tone may be equal to the operating frequency of the target qubit, and thus the difference between the frequency of the first microwave tone and the operating frequency of the control qubit may be a negative number). In contrast, if the operating frequency of the target qubit is higher than the operating frequency of the control qubit, the first microwave tone may be considered to have a positive detuning sign (e.g., the frequency of the first microwave tone may be equal to the operating frequency of the target qubit, and thus the difference between the frequency of the first microwave tone and the operating frequency of the control qubit may be a positive number). In either case, the control qubit may experience a first Stark shift when exposed to the first microwave tone, and the direction of the first Stark shift may depend on the detuning sign of the first microwave tone.

[0020] In various embodiments, the control qubit can be further driven by or exposed to a second microwave tone, or both. More specifically, as those skilled in the art will recognize, any suitable electronic control circuitry (e.g., a waveform generator) can be coupled to the control qubit via a second drive line (e.g., a second superconductor cable / wire), where the electronic control circuitry can generate the second microwave tone, and the second drive line can carry the second microwave tone from the electronic control circuitry to the control qubit. In various cases, the second microwave tone can be applied to the control qubit simultaneously with the first microwave tone (and / or within any suitable threshold time interval of the first microwave tone). In various cases, the second microwave tone can be considered a Stark-shift canceling tone. In other words, the second microwave tone can cancel, nullify, or cancel, or a combination thereof, the first Stark shift caused by the first microwave tone, as described herein. To facilitate such cancellation, nullification, or cancellation, or a combination thereof, the second microwave tone may have a detuning sign opposite to the detuning sign of the first microwave tone. Thus, if the detuning sign of the first microwave tone is negative (e.g., the frequency of the first microwave tone is lower than the operating frequency of the control qubit), the detuning sign of the second microwave tone may be positive (e.g., the frequency of the second microwave tone may be set higher than the operating frequency of the control qubit). On the other hand, if the detuning sign of the first microwave tone is positive (e.g., the frequency of the first microwave tone is higher than the operating frequency of the control qubit), the detuning sign of the second microwave tone may be negative (e.g., the frequency of the second microwave tone may be set lower than the operating frequency of the control qubit). Thus, in various cases, the control qubit, when exposed to the second microwave tone, may experience a second Stark shift that is opposite to the first Stark shift induced by the first microwave tone.In other words, a first microwave tone may shift the operating frequency of the control qubit in one direction (e.g., an increase), and a second microwave tone may shift the operating frequency of the control qubit in the opposite direction (e.g., a decrease). Thus, using appropriate amplitude calibration, the magnitude of the first Stark shift can be made equal to (and / or within any appropriate threshold margin of) the magnitude of the second Stark shift, such that the second Stark shift cancels the first Stark shift. That is, applying the second microwave tone simultaneously with the first microwave tone will result in the control qubit experiencing zero change in operating frequency, as if it were not experiencing a Stark shift at all.

[0021] Therefore, the various embodiments described herein may be considered systems and / or techniques that can cancel or significantly reduce the magnitude of the Stark shift, thereby avoiding or reducing dynamic frequency collisions that may be caused by the Stark shift, and enabling more accurate zero-noise extrapolation that is hindered by the Stark shift.

[0022] Various embodiments of the present invention may be employed to use hardware and / or software to solve what is, in effect, a highly technical problem (e.g., facilitating Stark shift cancellation) that is not abstract, not merely a law of nature, not merely a natural phenomenon, and not implementable as a set of mental acts by humans. Rather, various embodiments described herein include tangible quantum computing structures / architectures and / or techniques related to tangible quantum computing structures / architectures that can be implemented to reduce, mitigate, or ameliorate the technical problem of the Stark shift. Indeed, as noted above, the Stark shift can significantly hinder the performance of zero-noise extrapolation (e.g., qubit interactions with two-level quantum systems with defects near the frequency can result in variable relaxation times, and such relaxation times can vary significantly with the Stark shift). Furthermore, as noted above, the Stark shift can cause dynamic frequency collisions between the control qubit and one or more neighboring / observed qubits. Unfortunately, existing systems / technologies do not offer any mitigation for the Stark shift problem.

[0023] In contrast, various embodiments described herein can address the Stark shift issue. Specifically, the systems / techniques described herein can include driving a control qubit with a first microwave tone that entangles the control qubit with a target qubit coupled to the control qubit, while simultaneously driving the control qubit with a second microwave tone. In various instances, the second microwave tone can have a detuning sign opposite to that of the first microwave tone. Thus, even if the first microwave tone shifts the operating frequency of the control qubit in an increasing or decreasing direction, the second microwave tone can shift the operating frequency of the control qubit in the opposite direction, resulting in a net zero change in the operating frequency of the control qubit. In other words, the Stark shift induced by the first microwave tone can be considered to be canceled and / or cancelled by the second microwave tone. Because the various embodiments described herein can mitigate the Stark shift issue, such embodiments certainly constitute a concrete and tangible technical improvement in the field of quantum computing.

[0024] It should also be emphasized that the various embodiments described herein are not directed to merely transient signals and / or propagating waveforms. As described herein, various embodiments of the present invention can solve the technical problem of the Stark shift, which refers to a change in the operating frequency of a control qubit when the control qubit is driven by a non-resonant signal / tone. As described herein, the inventors' devised solution to this technical problem involves simultaneously driving the control qubit with another non-resonant signal / tone having an opposite detuning sign. Thus, various embodiments of the present invention cannot be intelligently explained without discussing / explaining such microwave tones / signals and / or the control parameters of the microwave tones / signals (e.g., frequency, duration, amplitude). Despite such discussion / explaining microwave tones / signals, various embodiments of the present invention are not limited to such microwave tones / signals and include many in addition thereto. Rather, such embodiments are directed to specific, tangible, non-transient quantum computing structures / architectures that generate and / or respond to such microwave tones / signals (e.g., a control qubit coupled to a target qubit that may experience a Stark shift when driven with a first tone. The Stark shift can be eliminated by simultaneously driving the control qubit with a second tone having an opposite detuning sign to the first tone).

[0025] Moreover, various embodiments of the present invention may control tangible hardware-based and / or software-based devices based on the disclosed teachings. For example, the operating frequency of tangible qubits (e.g., superconducting qubits comprised of Josephson junctions) that may be included in embodiments of the present invention may be protected from Stark shifts by the teachings described herein.

[0026] It should be appreciated that the figures and disclosure herein illustrate non-limiting examples of various embodiments of the present invention.

[0027] FIG. 1 illustrates a block diagram of an example non-limiting system 100 that can facilitate Stark shift cancellation in accordance with one or more embodiments described herein.

[0028] In various embodiments, system 100 may include a control qubit 102 and a target qubit 104. In various aspects, control qubit 102 may be any suitable quantum computing structure and / or architecture capable of exhibiting qubit behavior. For example, control qubit 102 may be comprised of one or more Josephson junctions and may be a superconducting qubit, such as a flux qubit, a charge qubit, a phase qubit, a transmon qubit, or any suitable variation or combination thereof. As another example, control qubit 102 may be a spin qubit. As another example, control qubit 102 may be a quantum dot. In any event, control qubit 102 may exhibit an initial operating frequency having any suitable amplitude. For purposes of illustration, the initial operating frequency of control qubit 102 may be ω control_initial It can be expressed as:

[0029] In various embodiments, target qubit 104 can be any suitable quantum computing structure and / or architecture capable of exhibiting qubit behavior. For example, target qubit 104 is comprised of one or more Josephson junctions and can be a superconducting qubit, such as a flux qubit, a charge qubit, a phase qubit, a transmon qubit, or any suitable variation or combination thereof. As another example, target qubit 104 can be a spin qubit. As another example, target qubit 104 can be a quantum dot. In any event, target qubit 104 can exhibit an operating frequency having any suitable amplitude. For purposes of illustration, the operating frequency of target qubit 104 can be ω targetIn various embodiments, the initial operating frequency of the control qubit 102 may be different from the operating frequency of the target qubit 104. That is, ω control_initial ≠ω target is.

[0030] In some cases, control qubit 102 may be the same type of qubit as target qubit 104 (e.g., both may be transmon qubits). In other cases, control qubit 102 may be a different type of qubit than target qubit 104 (e.g., one may be a transmon qubit and the other may be a phase qubit).

[0031] In various embodiments, the control qubit 102 may be coupled to the target qubit 104 via a coupler 106. In various cases, the coupler 106 may be any suitable quantum computing structure and / or architecture capable of electrically coupling two qubits. For example, the coupler 106 may be one or more superconducting wires that physically and / or directly attach the control qubit 102 to the target qubit 104. As another example, the coupler 106 may be one or more superconducting wires that capacitively attach the control qubit 102 to the target qubit 104. As another example, the coupler 106 may be one or more microwave resonators that physically and / or capacitively couple the control qubit 102 to the target qubit 104. In either case, the coupler 106 may electromagnetically couple the control qubit 102 to the target qubit 104 so as to facilitate entanglement between the control qubit 102 and the target qubit 104. 1 depicts coupler 106 as a straight, single path, this is merely a non-limiting example for ease of illustration. Those skilled in the art will recognize that coupler 106 may exhibit any suitable shape, size, dimensions, or number of paths, or combinations thereof.

[0032] 1 , those skilled in the art will recognize that control qubit 102, target qubit 104, and coupler 106 can be assembled or fabricated on any suitable quantum computing substrate. For example, control qubit 102, target qubit 104, and coupler 106 can be assembled or fabricated on a silicon wafer. Moreover, those skilled in the art will recognize that control qubit 102, target qubit 104, and coupler 106 can be assembled or fabricated by any suitable micro- and / or nano-fabrication techniques, such as photolithography, deposition, and / or double-angle evaporation.

[0033] In various embodiments, the control qubit 102 may be driven by a quantum entanglement gate tone 108. In various aspects, the quantum entanglement gate tone 108 may be any suitable microwave signal and / or microwave waveform that can entangle the control qubit 102 with the target qubit 104. In various instances, the quantum entanglement gate tone 108 may be driven by a signal having a magnitude such as ω entangle In various cases, the quantum entanglement gate tone 108 may have any suitable frequency, expressed as Ω entangle 1, the quantum entanglement gate tone 108 may have any suitable duration. To facilitate entanglement between the control qubit 102 and the target qubit 104, the frequency of the quantum entanglement gate tone 108 may be equal to (and / or, if not equal, within any suitable threshold margin of) the operating frequency of the target qubit 104. That is, ω entangle =ω target is.

[0034] As will be appreciated by those skilled in the art, the amplitude and / or duration of quantum entanglement gating tone 108 may be controllably adjusted and / or modulated to control the type of entanglement promoted by quantum entanglement gating tone 108. For example, in some cases, the amplitude and / or duration of quantum entanglement gating tone 108 may be set to some particular value such that quantum entanglement gating tone 108 performs a CNOT gate on control qubit 102 and target qubit 104. In other cases, the amplitude and / or duration of quantum entanglement gating tone 108 may be set to some other particular value such that quantum entanglement gating tone 108 performs a controlled phase gate on control qubit 102 and target qubit 104. Thus, any suitable type of quantum entanglement operation may be achieved by controllably adjusting the amplitude and / or duration of quantum entanglement gating tone 108.

[0035] 1 , those skilled in the art will recognize that there can be any suitable electronic control circuitry (e.g., an arbitrary waveform generator) coupled to control qubit 102 through a first drive line (e.g., any suitable superconducting wire). Thus, in various embodiments, such electronic control circuitry can generate quantum entanglement gate tone 108, and the first drive line can transmit and / or carry quantum entanglement gate tone 108 to control qubit 102.

[0036] In various embodiments, control qubit 102 may experience a Stark shift due to entanglement gate tones 108. In other words, when control qubit 102 is exposed to entanglement gate tones 108, its operating frequency may shift from an initial operating frequency ω control_initial From ω control_shifted In particular, this Stark shift can be approximately defined by the following equation:

number

[0037] Specifically, in various embodiments, control qubit 102 may be further driven by Stark-shift cancellation tone 110. In various aspects, Stark-shift cancellation tone 110 may be any suitable microwave signal and / or microwave waveform that can cancel, nullify, or offset the Stark shift caused by quantum entanglement gate tone 108. In various instances, Stark-shift cancellation tone 110 may be driven by a frequency band that is equal to or greater than ω.Stark In various cases, the Stark-shifted cancellation tone 110 may have any suitable frequency, expressed as Ω. Stark 1, Stark-shifted cancellation tone 110 may have any suitable amplitude, represented as . Although not explicitly shown in FIG. 1, Stark-shifted cancellation tone 110 may have any suitable duration. Although not explicitly shown in FIG. 1, those skilled in the art will recognize that there may be any suitable electronic control circuitry (e.g., an arbitrary waveform generator) coupled to control qubit 102 by a second drive line (e.g., any suitable superconducting wire). Thus, in various embodiments, such electronic control circuitry may generate Stark-shifted cancellation tone 110, and the second drive line may transmit and / or carry Stark-shifted cancellation tone 110 to control qubit 102.

[0038] A Stark shift cancellation tone 110 may be applied to the control qubit 102 simultaneously with (and / or within any suitable threshold time window of) the entanglement gating tone 108 to facilitate cancellation, nullification, and / or counter-action of the Stark shift induced by the entanglement gating tone 108. Moreover, the frequency of the Stark shift cancellation tone 110 may be set to any suitable value such that the sign of the detuning of the Stark shift cancellation tone 110 is opposite to the sign of the detuning of the entanglement gating tone 108. That is, ω entangle and ω control_initial Given ω Stark The value of (ω Stark -ω control_initial )(ω entangle -ω control_initial )<0, and (ω Stark -ω control_initial )=Δ Stark denotes the detuning of the Stark-shift cancellation tone 110, and (ω entangle -ω control_initial )=Δ entangle represents the detuning of the quantum entanglement gate tone 108.

[0039] More specifically, in various embodiments, when the detuning sign of the Stark-shifted cancellation tones 110 is opposite to the detuning sign of the quantum entanglement gating tones 108 (e.g., Δ Stark Δ entangle <0), the operating frequency of the control qubit 102 can be shifted by the Stark-shift cancellation tone 110 in a direction opposite to that shifted by the entanglement gate tone 108. In other words, the entanglement gate tone 108 is control_initial ω control_shifted and the Stark-shifted cancellation tone 110 can be considered to be shifted to ω control_shifted ω control_initial Thus, in various cases, when Stark-shift cancellation tones 110 are applied to the control qubit 102 simultaneously with the entanglement gate tones 108, there may be no net change in the operating frequency of the control qubit 102.

[0040] Moreover, the Stark-shift cancellation tones 110 may have an opposite detuning sign to the entanglement gate tones 108, so that the frequency of the Stark-shift cancellation tones 110 is not equal to the frequency of the entanglement gate tones 108 (e.g., ω Stark ≠ω entangle ), so (for example, ω entangle =ω target Therefore, the Stark-shift cancellation tone 110 can suppress unwanted entanglement between the control qubit 102 and the target qubit 104 (e.g., the Stark-shift cancellation tone 110 can be considered a non-resonant drive whose frequency does not match the operating frequency of the target qubit 104).

[0041] For further clarity as to how Stark shift cancellation tones 110 cancel and / or negate the Stark shift caused by quantum entanglement gate tones 108, consider FIG.

[0042] 2 illustrates an example, non-limiting graph 200 showing the Stark shift as a function of drive tone frequency, in accordance with one or more embodiments described herein. In other words, graph 200 may be viewed as a plot of the Stark shift that the control qubit 102 experiences in response to a drive tone. Such a plot may be given by the following equation:

number

[0043] As shown, graph 200 exhibits an asymptote at 4.7 GHz and another asymptote at 5 GHz, where the asymptote at 5 GHz is f(ω drive ) denominator term Δ drive That is, when a particular drive tone has a frequency of 5 GHz in this non-limiting example, the detuning Δ drive can be zero. Moreover, the asymptote at 4.7 GHz is drive ) denominator term (α control -Δ drive ), i.e., when a particular driving tone has a frequency of 4.7 GHz in this non-limiting example, the quantity (α control -Δ drive) can be zero.

[0044] As shown in this non-limiting example, when a particular drive tone has a frequency between 4.7 GHz and 5 GHz, the value of the Stark shift experienced by the control qubit 102 is positive (e.g., 4.7 GHz < ω drive <5GHz, f(ω drive )>0) (e.g., the operating frequency of the control qubit 102 may increase), which corresponds to a negative detuning sign (e.g., such frequency is lower than the operating frequency of 5 GHz). On the other hand, as shown in this non-limiting example, when a particular drive tone has a frequency higher than 5 GHz, the value of the Stark shift experienced by the control qubit 102 may be negative (e.g., 5 GHz<ω drive When f(ω drive )<0) (eg, the operating frequency of the control qubit 102 may decrease), which corresponds to a positive detuning sign (eg, such frequency is higher than the operating frequency of 5 GHz).

[0045] Thus, there may be various drive frequencies that can induce a positive Stark shift in the operating frequency of control qubit 102, and other drive frequencies that can induce a negative Stark shift in the operating frequency of control qubit 102. Thus, if quantum entanglement gate tone 108 induces a positive Stark shift in the operating frequency of control qubit 102, the frequency of Stark shift cancellation tone 110 may be selected such that Stark shift cancellation tone 110 induces a negative Stark shift in the operating frequency of control qubit 102, resulting in zero (or very small) net change in the operating frequency of control qubit 102. Similarly, if quantum entanglement gating tone 108 induces a negative Stark shift in the operating frequency of control qubit 102, the frequency of Stark shift cancellation tone 110 may be selected such that Stark shift cancellation tone 110 induces a positive Stark shift in the operating frequency of control qubit 102, resulting in zero (or very small) net change in the operating frequency of control qubit 102.

[0046] Specifically, as seen in graph 200, when the absolute value of the detuning of the quantum entanglement gate tone 108 is less than the absolute value of the anharmonicity of the control qubit 102 (e.g., |Δ entangle |<|α control |), the frequency of Stark-shift cancellation tones 110 is selected such that (1) Stark-shift cancellation tones 110 have the opposite detuning sign as entanglement gate tones 108, and (2) the absolute value of the detuning of Stark-shift cancellation tones 110 is less than the absolute value of the anharmonicity of control qubit 102 (e.g., |Δ Stark |<|α control By choosing the frequency of the Stark shift cancellation tone 110 so that |Δ entangle |<|α control ω becomes | entangle Given ωentangle There exists at least one frequency with the opposite detuning sign to ω entangle can cancel or offset, or both, the Stark shift caused by ω Stark can be selected as

[0047] For example, assume that quantum entanglement gating tone 108 has a frequency given by number 202. In such a case, quantum entanglement gating tone 108 may be considered to have a negative detuning value (e.g., number 202 is less than the operating frequency of 5 GHz). Moreover, in such a case, quantum entanglement gating tone 108 may be considered to have a detuning value whose amplitude is less than the absolute value of the anharmonicity of control qubit 102 (e.g., number 202 is less than 300 MHz away from the operating frequency of 5 GHz). Furthermore, in such a case, quantum entanglement gating tone 108 may be considered to induce a positive Stark shift in the operating frequency of control qubit 102 (e.g., graph 200 is greater than zero at number 202).

[0048] Given these, there exists at least one frequency that can cancel and / or offset the positive Stark shift caused by number 202. Number 204 can be one such frequency. As shown, the frequency represented by number 204 has a positive detuning sign that is opposite in sign to the detuning of number 202 (e.g., number 204 is greater than the operating frequency of 5 GHz). Furthermore, as shown, number 204 can be considered to have a detuning whose amplitude is less than the absolute value of the anharmonicity of control qubit 102 (e.g., number 202 is less than 300 MHz away from the operating frequency of 5 GHz). Moreover, as shown, number 204 can induce a negative Stark shift in the operating frequency of control qubit 102 (e.g., graph 200 is less than zero at number 204). Furthermore, as shown, the magnitude of the negative Stark shift caused by number 204 can be equal to (and / or within any suitable margin of) the magnitude of the positive Stark shift caused by number 202.

[0049] Thus, if entanglement gating tones 108 have a frequency corresponding to number 202, then Stark shift cancellation tones 110 may have a frequency corresponding to number 204. In such a case, Stark shift cancellation tones 110 may be considered to be capable of inducing a negative Stark shift that cancels and / or offsets the positive Stark shift induced by entanglement gating tones 108. Such cancellation and / or counteraction occurs when the absolute value of the detuning of Stark shift cancellation tones 110 differs from the absolute value of the detuning of entanglement gating tones 108 (e.g., |Δ Stark |≠|Δ entangle Note that this can also be accelerated in the |) case.

[0050] In essence, graph 200 shows that whenever the absolute value of the detuning of entanglement gating tone 108 is less than the absolute value of the anharmonicity of control qubit 202, there exists a frequency with the opposite detuning sign to that of entanglement gating tone 108 that can cancel and / or cancel the Stark shift caused by entanglement gating tone 108. Such a frequency can be selected as the frequency for Stark shift cancellation tone 110.

[0051] In various embodiments, any suitable calibration technique may be implemented to generate the quantum entanglement gate tones 108 and / or the Stark-shift cancellation tones 110. For example, as one skilled in the art will recognize, the frequency of the quantum entanglement gate tones 108 may be set equal to the operating frequency of the target qubit 104 (e.g., to promote entanglement), and the amplitude and / or duration of the quantum entanglement gate tones 108 may be calibrated to any suitable value (e.g., to promote any desired transformation performed by the quantum entanglement gate). Furthermore, as one skilled in the art will further recognize, the frequency of Stark shift cancellation tone 110 may be set to any suitable value that results in Stark shift cancellation tone 110 having an opposite detuning sign to entanglement gating tone 108, and the amplitude and / or duration of Stark shift cancellation tone 110 may be experimentally swept to a particular amplitude and / or duration identified as canceling the Stark shift caused by entanglement gating tone 108. This will be further discussed with reference to FIG.

[0052] FIG. 3 is a flow diagram of an example of a non-limiting method that can facilitate calibration of Stark-shift cancellation tones according to one or more embodiments described herein.

[0053] In various embodiments, act 302 includes setting an initial operating frequency ω 0_initialfor a control qubit q (e.g., 102) coupled to a target qubit q (e.g., 104) with a given frequency ω drive and a given amplitude Ω drive The method may include selecting a quantum entanglement gate tone (e.g., 108) having

[0054] In various embodiments, act 304 may be performed by adjusting any suitable amplitude Ω. Stark and a frequency ω that yields another tone with the opposite detuning sign relative to the entanglement gate tone. Stark In other words, the frequency ω Stark is (ω Stark -ω 0_initial )(ω drive -ω 0_initial )<0.

[0055] In various cases, act 306 can include simultaneously applying the entanglement gate tone and another tone to the control qubit.

[0056] In various cases, act 308 may determine (e.g., by Ramsey interferometry) the current operating frequency ω of the control qubit. 0_current This can include measuring

[0057] In various embodiments, act 310 includes: 0_current -ω 0_initial That is, act 310 may include determining whether |ω is less than any suitable threshold. 0_current -ω 0_initial It may be determined whether | is close enough to zero. If it is not close enough, the method 300 may proceed to act 312. If it is close enough, the method 300 may proceed to act 314.

[0058] In various instances, act 312 may involve adjusting the amplitude Ω of another tone. StarkThus, method 300 can return to act 306.

[0059] In various cases, act 314 may include terminating method 300. In other words, the other tone may be considered an appropriately calibrated Stark-shift cancellation tone in act 314. Although not explicitly shown in FIG. 3 , act 314 may further include performing any appropriate fine calibration on the entanglement gating tone and / or the other tone (e.g., to correct for any coherence errors for the entanglement gating tone that may have been introduced by the introduction of the other tone).

[0060] In any case, Acts 306-312 are |ω 0_current -ω 0_initial is sufficiently close to zero. As one skilled in the art will recognize, such repetitions / repetitions may be considered an amplitude sweep.

[0061] In various aspects, method 300 may be viewed as illustrating how to tune a given quantum entanglement gate tone to have a net zero Stark shift.

[0062] This disclosure has thus far described how Stark-shift cancellation tones 110 may be implemented to cancel and / or nullify the Stark shift induced by quantum entanglement gate tones 108 (e.g., as shown in FIGS. 1-2 ) and how Stark-shift cancellation tones 110 may be calibrated (e.g., as shown in FIG. 3 ). Because Stark-shift cancellation tones 110 can cancel the Stark shift in the operating frequency of control qubit 102, zero-noise extrapolation may be accurately applied to control qubit 102, and control qubit 102 can avoid dynamic frequency collisions with neighboring qubits. These benefits are further described with reference to FIGS. 4-6 .

[0063] Consider zero-noise extrapolation first. As previously discussed, and as those skilled in the art will recognize, zero-noise extrapolation is an error mitigation technique that involves repeatedly executing a quantum gate with gradually increasing and decreasing levels of noise to identify the zero-noise limit that the quantum gate approaches. In practice, such varying noise levels may be achieved by repeatedly scaling the quantum gate in time by a set of scaling factors (e.g., a set of scalars) and repeatedly calibrating the amplitude of the quantum gate at each iteration. Unfortunately, the accuracy of zero-noise extrapolation can be hindered by Stark shifts. Therefore, in various embodiments, Stark shift cancellation may be implemented to improve the accuracy of zero-noise extrapolation. More specifically, whenever a quantum gate is scaled by a scaling factor, Stark shift cancellation tones corresponding to the scaled quantum gate may be identified / calibrated and applied simultaneously with the scaled quantum gate. This is further explained with reference to FIG. 4.

[0064] FIG. 4 illustrates a flow diagram of an example of a non-limiting method 400 that can facilitate performing zero-noise extrapolation using Stark shift cancellation in accordance with one or more embodiments described herein.

[0065] In various embodiments, act 402 may include selecting a quantum entanglement gate tone (e.g., 108) having a given amplitude, a given frequency, or a given duration, or a combination thereof, for a control qubit (e.g., 102) coupled to a target qubit (e.g., 104).

[0066] In various aspects, act 404 can include identifying a set of scaling factors (eg, a set of scalars) for performing zero-noise extrapolation on the quantum entanglement gate tones.

[0067] In various instances, act 406 may include initializing an empty set of Stark-shift cancellation tones.

[0068] In various cases, act 408 may include determining whether all of the scale factors in the set of scale factors have been addressed (e.g., determining whether the respective Stark-shift cancellation tones have been identified / calibrated for each of the set of scale factors). If not, method 400 may proceed to act 410. If addressed, method 400 may proceed to act 418.

[0069] In various aspects, act 410 may include selecting a scale factor within the set of scale factors that has not yet been addressed (e.g., selecting a scale factor for which a corresponding Stark-shift cancellation tone has not yet been identified / calibrated).

[0070] In various instances, act 412 may include applying a selected stretching factor to the quantum entanglement gate tone, resulting in a stretched quantum entanglement gate tone. As one skilled in the art will recognize, this may include locking a given frequency of the quantum entanglement gate tone to the operating frequency of the target qubit, stretching a given duration of the quantum entanglement gate tone by the stretching factor, or adjusting a given amplitude of the quantum entanglement gate tone to maintain the transformation performed by the quantum entanglement gate tone after stretching by the stretching factor, or a combination thereof.

[0071] In various cases, act 414 may include calibrating a novel tone (e.g., 110) that offsets the Stark shift in the operating frequency of the control qubit caused by the stretched entanglement gate tone. In various cases, such calibration may be facilitated as described with reference to FIG.

[0072] In various aspects, act 416 may include adding the new tone to the set of Stark-shift cancellation tones. In various cases, method 400 may return to act 408.

[0073] In various instances, act 418 may include performing zero-noise extrapolation based on the set of warping factors and the set of Stark-shift cancellation tones, which may include applying the set of warping factors to the entanglement gate tones to produce a set of warped entanglement gate tones, driving a control qubit sequentially, repeatedly, or both with the set of warped entanglement gate tones, and for each warped entanglement gate tone (e.g., for each repetition), simultaneously driving the control qubit with a corresponding Stark-shift cancellation tone from the set of Stark-shift cancellation tones.

[0074] In addition to improving the accuracy of the zero-noise extrapolation, to prevent dynamic frequency collisions, the Stark shift cancellation described herein can be implemented as shown in Figures 5-6.

[0075] 5 shows a block diagram of an example non-limiting system 500 including a spectator qubit that can facilitate Stark shift cancellation in accordance with one or more embodiments described herein. As shown, system 500 can include the same components as system 100 and can further include a spectator qubit 502.

[0076] In various embodiments, spectator qubit 502 can be any suitable quantum computing structure and / or architecture capable of exhibiting qubit behavior. For example, spectator qubit 502 is comprised of one or more Josephson junctions and can be a superconducting qubit, such as a flux qubit, a charge qubit, a phase qubit, a transmon qubit, or any suitable variation or combination thereof. As another example, spectator qubit 502 can be a spin qubit. As another example, spectator qubit 502 can be a quantum dot. In any event, spectator qubit 502 can exhibit an operating frequency having any suitable amplitude. For purposes of illustration, the operating frequency of spectator qubit 502 can be ω spectator In various aspects, the operating frequency of the spectator qubit 502 may be different from the operating frequency of the control qubit 102, or different from the operating frequency of the target qubit 104, or both, to avoid frequency collisions. spectator ≠ω control_initial And ω spectator ≠ω target is.

[0077] In some cases, spectator qubit 502 may be the same type of qubit as control qubit 102 and / or target qubit 104. In other cases, spectator qubit 502 may be a different type of qubit than control qubit 102 and / or target qubit 104.

[0078] In various embodiments, the spectator qubit 502 may be coupled to the control qubit 102 by a coupler 504. In various cases, the coupler 504 may be any suitable quantum computing structure and / or architecture capable of electrically coupling two qubits. For example, the coupler 504 may be one or more superconducting wires that physically and / or directly attach the spectator qubit 502 to the control qubit 102. As another example, the coupler 504 may be one or more superconducting wires that capacitively attach the spectator qubit 502 to the control qubit 102. As another example, the coupler 504 may be one or more microwave resonators that physically and / or capacitively couple the spectator qubit 502 to the control qubit 102. In either case, coupler 504 may electromagnetically couple spectator qubit 502 to control qubit 102 so as to facilitate entanglement between spectator qubit 502 and control qubit 102. While Figure 5 depicts coupler 504 as a straight, single path, this is merely a non-limiting example for ease of illustration. Those skilled in the art will recognize that coupler 504 may exhibit any suitable shape, size, dimensions, or number of paths, or combinations thereof.

[0079] Although not explicitly shown in FIG. 5, those skilled in the art will recognize that spectator qubit 502 may be fabricated and / or manufactured on the same quantum substrate (e.g., a silicon wafer) as control qubit 102 and target qubit 104 using any suitable micro- and / or nano-fabrication techniques.

[0080] In various embodiments, ω spectator =ω control_shiftedAssume that . In other words, when quantum entanglement gate tone 108 is applied to control qubit 102 in the absence of Stark shift cancellation tone 110, a dynamic frequency collision may occur between control qubit 102 and spectator qubit 502. In other words, quantum entanglement gate tone 108 may cause the operating frequency of control qubit 102 to equal (or become too close to) the operating frequency of spectator qubit 502. However, when quantum entanglement gate tone 108 is applied to control qubit 102 simultaneously with Stark shift cancellation tone 110, the net change in the operating frequency of control qubit 102 may be zero (or very small), meaning that the operating frequency of control qubit 102 can be prevented from equalizing (or becoming too close to) the operating frequency of spectator qubit 502. In this way, Stark shift cancellation may be implemented to prevent dynamic frequency collision.

[0081] FIG. 6 shows a block diagram of an example non-limiting system including a qubit lattice to which Stark shift cancellation may be applied, according to one or more embodiments described herein.

[0082] As shown, there may be a qubit lattice 600. In various embodiments, qubit lattice 600 may include any suitable number of qubits of any suitable type fabricated on any suitable quantum computing substrate (not shown). In the non-limiting example of Figure 6, qubit lattice 600 may include nine qubits: qubit 602, qubit 604, qubit 606, qubit 608, qubit 610, qubit 612, qubit 614, qubit 616, and qubit 618.

[0083] In various embodiments, each qubit in the qubit lattice may exhibit a corresponding operating frequency. For example, qubit 602 exhibits an operating frequency ω 602 and qubit 604 can have an operating frequency ω 604and qubit 606 can have an operating frequency ω 606 and qubit 608 can have an operating frequency ω 608 and qubit 610 can have an operating frequency ω 610 and qubit 612 can have an operating frequency ω 612 and qubit 614 can have an operating frequency ω 614 and qubit 616 can have an operating frequency ω 616 and qubit 618 can have an operating frequency ω 618 can have:

[0084] In various cases, each qubit in qubit lattice 600 may have a different operating frequency than any of its nearest neighbors. For example, qubit 610 may be considered to have four nearest neighbors: qubit 604, qubit 608, qubit 612, and qubit 616. Thus, ω 610 The amplitude of ω 604 , ω 608 , ω 612 , and ω 616 As another example, qubit 614 may have two nearest neighbors, qubit 608 and qubit 616. Thus, ω 614 The amplitude of ω 608 Yaω 616 As another example, qubit 604 may have three nearest neighbors: qubit 602, qubit 606, and qubit 610. Thus, ω 604 The amplitude of ω 602 , ω 606 , and ω 610 may differ from those in

[0085] In various embodiments, Stark shift cancellation may be implemented in qubit lattice 600 to prevent dynamic frequency collisions. For example, assume it is desired to treat qubit 610 as the control qubit and qubit 612 as the target qubit. In such a case, qubit 610 would have a frequency offset of ω 612 (and / or ω 612 qubit 610 may be driven by a quantum entanglement gate tone having a frequency (e.g., within any suitable threshold margin of ω ) at the operating frequency of qubit 610. Moreover, qubit 610 may be simultaneously driven by a Stark-shift cancellation tone having an opposite detuning sign relative to the quantum entanglement gate tone (e.g., such Stark-shift cancellation tone may be calibrated as described above with reference to FIG. 3). Thus, the quantum entanglement gate tone can induce the desired entanglement between qubit 610 and qubit 612, and the Stark-shift cancellation tone can be driven by a quantum entanglement gate tone having a frequency (e.g., within any suitable threshold margin of ω ) at the operating frequency of qubit 610. 610 ) from experiencing a net shift. Thus, the Stark-shift cancellation tones prevent the operating frequency of qubit 610 from clashing with the operating frequency of qubit 610's nearest neighbors (e.g., ω 610 But, ω 604 , ω 608 , ω 612 , or ω 616 (preventing the robot from getting too close to the target).

[0086] As another example, suppose it is desired to treat qubit 608 as the control qubit and qubit 614 as the target qubit. In such a case, qubit 608 has a ω 614 (and / or ω 614qubit 608 may be driven by a quantum entanglement gate tone having a frequency (e.g., within any suitable threshold margin of ω ) at the operating frequency of qubit 608. Moreover, qubit 608 may be simultaneously driven by a Stark-shift cancellation tone having an opposite detuning sign relative to the quantum entanglement gate tone (e.g., such Stark-shift cancellation tone may be calibrated as described above with reference to FIG. 3). Thus, the quantum entanglement gate tone can induce the desired entanglement between qubit 608 and qubit 614, and the Stark-shift cancellation tone can be driven by a quantum entanglement gate tone having a frequency (e.g., within any suitable threshold margin of ω ) at the operating frequency of qubit 608. 608 ) from experiencing a net shift. Thus, the Stark-shift cancellation tones prevent the operating frequencies of qubit 608 from clashing with the operating frequency of qubit 608's nearest neighbors (e.g., ω 608 But, ω 602 , ω 610 , or ω 614 (preventing the robot from getting too close to the target).

[0087] In this manner, to prevent dynamic frequency collisions in qubit lattice 600 by suppressing the Stark shifts induced by the entanglement gate tones, for a given pair of qubits in qubit lattice 600, one control qubit and the other target qubit, and for a given entanglement gate desired to be applied to such a pair of qubits, a unique Stark-shift cancellation tone can be calibrated and applied to the control qubit simultaneously with a given entanglement gate tone. In some cases, unique and / or distinct frequencies may be used for each such Stark-shift cancellation tone. However, in other cases, a common frequency may be used for each such Stark-shift cancellation tone, with each such Stark-shift cancellation tone having a unique amplitude and / or duration. For example, if all of the control qubits in qubit lattice 600 have higher operating frequencies than their desired target qubits, then a single common frequency may be selected for each of the Stark-shift cancellation tones that is higher than the highest operating frequency in qubit lattice 600. On the other hand, if all of the control qubits in qubit lattice 600 have lower operating frequencies than their desired target qubits, then a single common frequency may be selected for each of the Stark-shift cancellation tones that is lower than the lowest operating frequency in qubit lattice 600. In either case, the common frequency may ensure that each Stark-shift cancellation tone has an opposite detuning sign relative to its corresponding entanglement gate tone.

[0088] Although qubit lattice 600 in Figure 6 is shown as having a regular, linear arrangement, this is merely a non-limiting example. In various cases, qubit lattice 600 may be arranged in any suitable manner as desired.

[0089] 7-8 illustrate example flow diagrams of non-limiting methods 700 and 800 that can facilitate Stark shift cancellation according to one or more embodiments described herein.

[0090] Consider first method 700. In various embodiments, act 702 can include applying a first tone (e.g., 108) to a control qubit (e.g., 102) coupled to a target qubit (e.g., 104). In various cases, the first tone can entangle the control qubit with the target qubit.

[0091] In various aspects, act 704 may include applying a second tone (e.g., 110) to the control qubit simultaneously with the first tone. In various cases, the second tone may have an opposite detuning sign to the first tone.

[0092] Although not explicitly shown in Figure 7, the first tone induces a Stark shift in the operating frequency of the control qubit (e.g., ω control_initial ω control_shifted ), and the second tone cancels the Stark shift (for example, ω control_shifted ω control_initial (return to ).

[0093] Although not explicitly shown in FIG. 7, the method 700 may further include tuning the frequency of the second tone (e.g., ω ) according to an inverse detuning sign (e.g., as described with reference to FIG. 3 ). Stark ) is fixed, and the amplitude of the second tone (say Ω Stark ) calibrating the second tone by sweeping the second tone until the Stark shift in the operating frequency of the control qubit is nullified.

[0094] Although not explicitly shown in FIG. 7, a spectator qubit (e.g., 502) may be coupled to the control qubit, and the second tone may be coupled to the operating frequency of the spectator qubit (e.g., ω spectator) and the operating frequency of the control qubit can be prevented from dynamically clashing.

[0095] Although not explicitly shown in FIG. 7, the first tone may be one of multiple time-warped tones that are sequentially applied to the control qubit during a zero-noise extrapolation procedure (e.g., as described with reference to FIG. 4).

[0096] Consider now Figure 8. In various embodiments, act 802 can include applying a first tone (e.g., 108) to a control qubit (e.g., 102) coupled to a target qubit (e.g., 104). In various cases, the first tone induces a Stark shift in the operating frequency of the control qubit (e.g., ω control_initial ω control_shifted There is a possibility that the

[0097] In various aspects, act 804 can include applying a second tone (e.g., 110) to the control qubit. In various cases, the second tone cancels the Stark shift (e.g., ω control_shifted ω control_initial (return to ).

[0098] Although not explicitly shown in FIG. 8, the second tone may have an opposite detuning sign to the first tone.

[0099] Although not explicitly shown in FIG. 8, the second tone may be applied simultaneously with the first tone.

[0100] Although not explicitly shown in FIG. 8 , a first absolute difference between the frequency of the first tone and the operating frequency of the control qubit (e.g., the absolute value of the first tone detuning |Δ entangle |) is the absolute value of the anharmonicity of the control qubit (say |α control |), and a second absolute difference between the frequency of the second tone and the operating frequency of the control qubit (e.g., the absolute value of the detuning of the second tone |Δ Stark|) may be less than the absolute value of the anharmonicity of the control qubit.

[0101] Although not explicitly shown in FIG. 8, the second tone may prevent the operating frequency of the control qubit from colliding with the operating frequency of an adjacent qubit (eg, 502).

[0102] Various embodiments described herein include systems and / or techniques for facilitating Stark shift cancellation. Specifically, whenever an entanglement gate tone is applied to a control qubit, a Stark shift cancellation tone may be simultaneously applied to the control qubit to cancel and / or cancel the Stark shift induced by the entanglement gate tone. Specifically, this may be facilitated by setting the frequency of the Stark shift cancellation tone so that it has an opposite detuning sign relative to the entanglement gate tone (e.g., a positive Stark shift cancellation tone detuning and a negative entanglement gate tone detuning, or a negative Stark shift cancellation tone detuning and a positive entanglement gate tone detuning). Thus, for any desired entanglement gate tone, the corresponding Stark shift cancellation tone may be calibrated to tune the entanglement gate tone to a net zero Stark shift. Such Stark shift cancellation can prevent dynamic frequency collisions and enable more accurate performance of zero-noise extrapolation. Thus, the various embodiments described herein certainly constitute concrete and tangible technical improvements in the field of quantum computing.

[0103] Those skilled in the art will recognize that the disclosure herein describes non-limiting examples of various embodiments of the subject innovation. Various portions of the disclosure herein utilize the term "each" for ease of description and / or explanation when discussing various embodiments of the subject innovation. Those skilled in the art will recognize that such use of the term "each" is a non-limiting example. In other words, when the disclosure herein provides a description that applies to "each" of several particular objects or components, it should be understood that this is a non-limiting example of various embodiments of the subject innovation, and it should be further understood that in various other embodiments of the subject innovation, such a description may apply to fewer than "each" of the particular objects and / or components.

[0104] To provide additional context for the various embodiments described herein, Figure 9 and the following discussion are intended to provide a brief general description of a suitable computing environment 900 in which various embodiments of the embodiments described herein may be implemented. While the embodiments are described above in the general context of computer-executable instructions operating on one or more computers, those skilled in the art will recognize that they may also be implemented in combination with other program modules, or as a combination of hardware and software, or both.

[0105] Generally, program modules include routines, programs, components that perform particular tasks or implement particular abstract data types, such as data structures. Furthermore, those skilled in the art will recognize that the methods of the present invention can be practiced with other computer system configurations, including single-processor or multi-processor computer systems, minicomputers, mainframe computers, Internet of Things (IoT) devices, distributed computing systems, and personal computers, portable computing devices, microprocessor-based or programmable consumer electronics devices, and the like, each of which can be operatively coupled to one or more associated devices.

[0106] The illustrated embodiments herein may also be practiced in distributed computing environments where certain tasks are performed by remote processing devices that are linked through a communications network, and program modules may be located in both local and remote memory storage devices.

[0107] Computing devices generally include a variety of media, which may include computer-readable storage media, machine-readable storage media, and / or communication media; these terms are used interchangeably herein as follows: A computer-readable storage medium or machine-readable storage medium may be any available storage medium, removable or non-removable, including both volatile and nonvolatile media, that can be accessed by a computer. By way of example and not limitation, a computer-readable storage medium or machine-readable storage medium may be implemented in connection with any method or technology for storing information, such as computer-readable or machine-readable instructions, program modules, structured or unstructured data, etc.

[0108] A computer-readable storage medium may include, but is not limited to, random access memory (RAM), read-only memory (ROM), electrically erasable programmable ROM (EEPROM), flash memory or other storage technology, compact disc-based read-only memory (CD-ROM), digital versatile disc (DVD), Blu-ray disc (BD) or other optical disk storage, magnetic cassette, magnetic tape, magnetic disk storage or other magnetic storage device, solid-state drive or other solid-state storage device, or other tangible and / or non-transitory medium that may be used to store the desired information. In this regard, the terms "tangible" or "non-transitory" as applied herein as modifiers to storage, memory, or computer-readable medium should be understood to exclude only the propagating transitory signal itself, and do not waive any right to all standard storage, memory, or computer-readable medium that is not the propagating transitory signal itself.

[0109] The computer-readable storage medium may be accessed by one or more local or remote computing devices for various operations on the information stored on the medium, for example, by access requests, queries or other data retrieval protocols.

[0110] Communication media typically embodies computer-readable instructions, data structures, program modules, or other structured or unstructured data in a data signal, such as a modulated data signal, such as a carrier wave or other transmission means, and includes any information delivery or transmission media. The term "modulated data signal" or signal refers to a signal that has one or more of its characteristics set or changed in such a manner as to encode information in one or more signals. By way of example, and not limitation, communication media includes wired media such as a wired network or direct-wired connection, and wireless media such as acoustic, RF, infrared and other wireless media.

[0111] Referring again to Figure 9, an exemplary environment 900 for implementing various embodiments of the aspects described herein includes a computer 902 including a processing unit 904, a system memory 906, and a system bus 908. The system bus 908 couples system components including the system memory 906 to the processing unit 904. The processing unit 904 can be any of a variety of commercially available processors. Dual microprocessors and other multi-processor architectures can also be employed as the processing unit 904.

[0112] The system bus 908 may be any of several types of bus structures capable of further interconnecting peripheral and local buses to a memory bus (with or without a memory controller) using any of a variety of commercially available bus architectures. The system memory 906 includes ROM 910 and RAM 912. The basic input / output system (BIOS), which may be stored in non-volatile memory such as ROM, erasable programmable read-only memory (EPROM), or EEPROM, contains the basic routines that help transfer information to and from the internal elements of the computer 902, such as during start-up. The RAM 912 may also include high-speed RAM, such as static RAM, for caching data.

[0113] The computer 902 further includes an internal hard disk drive (HDD) 914 (e.g., EIDE, SATA), one or more external storage devices 916 (e.g., a magnetic floppy disk drive (FDD), a memory stick or flash drive reader, a memory card reader, etc.), and a drive 920, such as a solid state drive, optical disk drive, etc., that can read from and write to a disk 922, such as a CD-ROM disk, DVD, BD, etc. Alternatively, if a solid state drive is included, the disk 922 would not be included unless it were separate. The internal HDD 914 is shown as being located internal to the computer 902, but could be configured for external use in a suitable chassis (not shown). Additionally, although not shown in the environment 900, a solid state drive (SSD) could be used in addition to or in place of the HDD 914. HDD 914, external storage device 916, and drive 920 may be connected to system bus 908 by HDD interface 924, external storage device interface 926, and drive interface 928, respectively. Interface 924 for external drive implementations may include at least one or both of Universal Serial Bus (USB) and Institute of Electrical and Electronics Engineers (IEEE) 1394 interface technologies. Other external drive connection technologies are within the contemplation of the embodiments described herein.

[0114] The drives and their associated computer-readable storage media provide non-volatile storage of data, data structures, computer-executable instructions, and the like. For computer 902, the drives and storage media accommodate storage of any data in a suitable digital format. While the above description of computer-readable storage media refers to each type of storage device, those skilled in the art will recognize that other types of computer-readable storage media, whether existing or developed in the future, may also be used in the exemplary operating environment, and that any such storage media may contain computer-executable instructions for performing the methods described herein.

[0115] The drives and RAM 912 may store a number of program modules, including an operating system 930, one or more application programs 932, other program modules 934, and program data 936. All or portions of the operating system, applications, modules, and / or data may also be cached in RAM 912. The systems and methods described herein may be implemented using various commercially available operating systems or combinations of operating systems.

[0116] Computer 902 may optionally include emulation technology. For example, a hypervisor (not shown) or other intermediary may emulate the hardware environment of operating system 930, and the emulated hardware may optionally differ from the hardware depicted in FIG. 9 . In such an embodiment, operating system 930 may comprise one of multiple virtual machines (VMs) hosted on computer 902. Additionally, operating system 930 may provide a runtime environment, such as the Java® Runtime Environment or the .NET Framework, for application 932. A runtime environment is a consistent execution environment that allows application 932 to run on any operating system that includes the runtime environment. Similarly, operating system 930 may support containers, which are lightweight, standalone, executable software packages that include, for example, code, runtime, system tools, system libraries, and application configuration, and application 932 may take the form of a container.

[0117] Additionally, computer 902 can enable a security module, such as a trusted processing module (TPM). For example, with a TPM, a boot component can hash the next boot component and verify that the result matches a protected value before loading the next boot component. This process can occur at any layer of computer 902's code execution stack, such as at the application execution level or the operating system (OS) kernel level, thereby enabling security at any level of code execution.

[0118] A user may enter commands and information into the computer 902 through one or more wired or wireless input devices, such as a keyboard 938, a touch screen 940, and a pointing device such as a mouse 942. Other input devices (not shown) may include a microphone, an infrared (IR) remote control, a radio frequency (RF) remote control, or other remote control, a joystick, a virtual reality controller and / or headset, a game pad, a stylus pen, an image input device such as a camera, a gesture sensor input device, a visual motion sensor input device, an emotion or facial detection device, a biometric input device such as a fingerprint or iris scanner, etc. These and other input devices are often connected to the processing unit 904 through an input device interface 944 that is couplable to the system bus 908, but may also be connected by other interfaces, such as a parallel port, an IEEE 1394 serial port, a game port, a USB port, an IR interface, a BLUETOOTH® interface, etc.

[0119] A monitor 946 or other type of display device may also be connected to the system bus 908 via an interface, such as a video adapter 948. In addition to the monitor 946, computers typically include other peripheral output devices (not shown), such as speakers, printers, etc.

[0120] Computer 902 can operate in a networked environment using logical connections via wired and / or wireless communication to one or more remote computers, such as remote computer 950. The remote computer 950 may be a workstation, a server computer, a router, a personal computer, a portable computer, a microprocessor-based entertainment appliance, a peer device or other common network node, and typically includes many or all of the elements described relative to computer 902, although for purposes of brevity, only memory / storage 952 is shown. The depicted logical connections include wired / wireless connectivity to a local area network (LAN) 954 and / or to larger networks, such as a wide area network (WAN) 956. Such LAN and WAN networking environments are commonplace in offices and businesses and can facilitate enterprise-wide computer networks, such as intranets, and connect to global communication networks, such as the Internet.

[0121] When used in a LAN networking environment, the computer 902 may be connected to the local network 954 via a wired and / or wireless communication network interface or adapter 958. The adapter 958 may facilitate wired or wireless communication to the LAN 954, which may also include a wireless access point (AP) arranged to communicate with the adapter 958 in a wireless mode.

[0122] When used in a WAN networking environment, the computer 902 may include a modem 960 or may be connected to a communications server on the WAN 956 via other means for establishing communications across the WAN 956, such as via the Internet. The modem 960, a wired or wireless device, may be internal or external and connected to the system bus 908 via the input device interface 944. In a networked environment, program modules may be represented relative to the computer 902, or portions thereof, or may be stored in the remote memory / storage device 952. It will be appreciated that the network connections shown are exemplary and other means of establishing a communications link between computers may be used.

[0123] When used in a LAN or WAN networking environment, computer 902 can access a cloud storage system or other network-based storage system, such as a networked virtual machine, that provides one or more aspects of storage or information processing in addition to, or instead of, the external storage device 916 described above. Generally, the connection between computer 902 and the cloud storage system can be established over LAN 954 or WAN 956, for example, by adapter 958 or modem 960, respectively. Upon connecting computer 902 to the associated cloud storage system, external storage interface 926, with the assistance of adapter 958 or modem 960, or both, manages the storage provided by the cloud storage system in the same way as other types of external storage. For example, external storage interface 926 can be configured to provide access to cloud storage sources as if the cloud storage sources were physically connected to computer 902.

[0124] The computer 902 may be operable to communicate with any wireless device or entity operatively arranged for wireless communication, such as, for example, a printer, a scanner, a desktop computer and / or a portable computer, a personal digital assistant, a communications satellite, any equipment or location associated with a radio-detectable tag (e.g., a kiosk, a newsstand, a store shelf, etc.), and a telephone. This may include Wireless Fidelity (Wi-Fi) and BLUETOOTH® wireless technologies. Thus, communication may be in a predefined structure, similar to a traditional network, or simply ad-hoc communication between at least two devices.

[0125] Referring now to FIG. 10 , an illustrative cloud computing environment 1000 is depicted. As shown, the cloud computing environment 1000 includes one or more cloud computing nodes 1002, which may communicate with local computing devices used by cloud consumers, such as, for example, a personal digital assistant (PDA) or mobile phone 1004, a desktop computer 1006, a laptop computer 1008, or an automotive computing system 1010, or combinations thereof. The nodes 1002 may communicate with each other. The nodes 1002 may be physically or virtually grouped into one or more networks (not shown), such as a private, community, public, or hybrid cloud as described above, or combinations thereof. This enables the cloud computing environment 1000 to provide infrastructure, platform, and / or software as a service, eliminating the need for cloud consumers to maintain resources on their local computing devices. It will be understood that the types of computing devices 1004-1010 shown in FIG. 10 are intended to be illustrative only, and that the compute node 1002 and cloud computing environment 1000 can communicate with any type of computerized device over any type of network and / or network-addressable connection (e.g., using a web browser).

[0126] Referring now to Figure 11, a set of functional abstract layers provided by cloud computing environment 1000 (Figure 10) is shown. Repeated descriptions of similar elements employed in other embodiments described herein are omitted for the sake of brevity. It should be understood in advance that the components, layers, and functions shown in Figure 11 are intended to be illustrative only and are not intended to limit embodiments of the present invention. As depicted, the following layers and corresponding functions are provided:

[0127] Hardware and software layer 1102 includes hardware and software components. Examples of hardware components include mainframe 1104, RISC (reduced instruction set computer) architecture-based server 1106, server 1108, blade server 1110, storage device 1112, and network and networking components 1114. In some embodiments, software components include network application server software 1116 and database software 1118.

[0128] The virtualization layer 1120 provides an abstraction layer from which examples of virtual entities such as virtual networks 1126 including virtual servers 1122, virtual storage devices 1124, virtual private networks, virtual applications and operating systems 1128, and virtual clients 1130 may be provided.

[0129] In one example, management layer 1132 may provide the functions described below. Resource provisioning 1134 provides dynamic procurement of computing and other resources utilized to execute tasks within the cloud computing environment. Metering and pricing 1136 provides cost tracking as resources are utilized within the cloud computing environment and billing or invoicing for the consumption of these resources. In one example, these resources may include application software licenses. Security provides identity verification for cloud consumers and tasks, as well as protection of data and other resources. User portal 1138 provides access to the cloud computing environment for consumers and system administrators. Service level management 1140 provides allocation and management of cloud computing resources so that required service levels are met. Service level agreement (SLA) planning and fulfillment 1142 provides proactive provisioning and procurement of cloud computing resources in anticipation of future requirements according to SLAs.

[0130] Workload tier 1144 provides examples of functionality for which a cloud computing environment may be utilized. Examples of workloads and functionality that may be provided from workload tier 1144 include mapping and navigation 1146, software development and lifecycle management 1148, virtual classroom instruction delivery 1150, data analytics processing 1152, transaction processing 1154, and differentially private federated learning processing 1156. Various embodiments of the present invention utilize the cloud computing environment described with reference to Figures 10 and 11 to perform one or more differentially private federated learning processes according to various embodiments described herein.

[0131] The present invention may be a system, method, apparatus, or computer program product, or any combination thereof, at any possible level of technical detail. A computer program product may include a computer-readable storage medium having computer-readable program instructions for causing a processor to perform aspects of the present invention. A computer-readable storage medium may be a tangible device capable of holding or storing instructions for use by an instruction execution device. A computer-readable storage medium may be, for example, but not limited to, an electronic storage device, a magnetic storage device, an optical storage device, an electromagnetic storage device, a semiconductor storage device, or any suitable combination thereof. A non-exhaustive list of more specific examples of computer-readable storage media may include portable computer diskettes, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable ROM (EPROM or flash memory), static random access memory (SRAM), portable compact disc-based read-only memory (CD-ROM), digital versatile disks (DVDs), memory sticks, floppy disks, mechanically encoded devices such as punch cards or grooved studs containing instructions, and any suitable combination thereof. As used herein, computer-readable storage media itself should not be construed as a transitory signal such as an electric wave or other freely propagating electromagnetic wave, an electromagnetic wave propagated through a waveguide or other transmission medium (e.g., a light pulse passing through a fiber optic cable), or an electrical signal transmitted through an electrical wire.

[0132] The computer-readable program instructions described herein may be downloaded from a computer-readable storage medium to a respective computing device / processing unit or an external computer or external storage device via a network, such as the Internet, a local area network, a wide area network, or a wireless network, or a combination thereof. The network may comprise copper transmission cables, optical fiber transmissions, wireless transmissions, routers, firewalls, switches, gateway computers, or edge servers, or a combination thereof. A network adapter card or network interface in each computing device / processing unit receives the computer-readable program instructions from the network and forwards them for storage in the respective computing device / processing unit's internal computer-readable storage medium. The computer-readable program instructions for carrying out the operations of the present invention may be either assembler instructions, instruction set architecture (ISA) instructions, machine instructions, machine-dependent instructions, microcode, firmware instructions, state setting data, configuration data for integrated circuit design, or source code or object code written in any combination of one or more programming languages, including object-oriented programming languages ​​such as Smalltalk, C++, and procedural programming languages ​​such as the "C" programming language or similar programming languages. The computer-readable program instructions may execute entirely on the user's computer, as a stand-alone software package, partially on the user's computer, partially on the user's computer and partially on a remote computer, or entirely on a remote computer or server. In the latter scenario, the remote computer may be connected to the user's computer via any type of network, including a local area network (LAN) or a wide area network (WAN), or may be connected to an external computer (e.g., via the Internet using an Internet Service Provider).In some embodiments, electronic circuitry including, for example, a programmable logic circuit, a field programmable gate array (FPGA), or a programmable logic array (PLA) may execute computer readable program instructions with state information of the computer readable program instructions personalizing the electronic circuitry to perform aspects of the present invention.

[0133] Aspects of the present invention are described herein with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be embodied by computer-readable program instructions. These computer-readable program instructions can be provided to a processor of a general-purpose computer, a special-purpose computer, or another programmable data processing apparatus to produce a machine, such that the instructions, executed by the processor of the computer or other programmable data processing apparatus, generate means for performing the functions / acts specified in the blocks of the flowchart illustrations and / or block diagrams. These computer-readable program instructions can also be stored on a computer-readable storage medium that can instruct a computer, programmable data processing apparatus, or other apparatus, or combination thereof, to function in a particular way, such that the computer-readable storage medium comprising the instructions comprises an article of manufacture containing instructions that implement aspects of the functions / acts specified in the blocks of the flowchart illustrations and / or block diagrams. The computer-readable program instructions may also be loaded into a computer, other programmable data processing apparatus, or other device such that the instructions, which execute on the computer, other programmable data processing apparatus, or other device, perform a series of operational acts on the computer, other programmable apparatus, or other device to perform the functions / acts specified in the blocks of the flowcharts and / or block diagrams, to create a computer-implemented process.

[0134] The flowcharts and block diagrams illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present invention. In this regard, each block in a flowchart or block diagram may represent a module, segment, or portion of instructions, including one or more executable instructions for implementing a specified logical function. In some alternative implementations, the functions indicated in the blocks may occur out of the order indicated in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or sometimes in the reverse order, depending on the functionality involved. It will also be noted that each block in the block diagrams and / or flowcharts, and combinations of blocks in the block diagrams and / or flowcharts, may be implemented by a special-purpose hardware-based system that performs the specified functions or acts or combinations of special-purpose hardware and computer instructions.

[0135] While the subject matter has been described above in the general context of computer-executable instructions for a computer program product executing on one or more computers, those skilled in the art will recognize that the present disclosure may also be implemented in combination with other program modules. Generally, program modules include routines, programs, and components that perform particular tasks or implement particular abstract data types, such as data structures. Those skilled in the art will also recognize that the computer-implemented methods of the present invention may be implemented with other computer system configurations, including single-processor or multiprocessor computer systems, minicomputing devices, mainframe computers, and other computer systems, including computers, portable computing devices (e.g., PDAs, phones), microprocessor-based or programmable consumer or industrial electronic circuits, and the like. The illustrated embodiments may also be implemented in distributed computing environments where tasks are performed by remote processing devices linked through a communications network. However, some, if not all, of the embodiments of the present disclosure may be implemented on stand-alone computers. In a distributed computing environment, program modules may be located in both local and remote memory storage devices.

[0136] As used herein, terms such as "component," "system," "platform," and "interface" can refer to and / or include a computer-related entity or usable machine-related entity having one or more specific functions. The entities disclosed herein can be hardware, a combination of hardware and software, software, or software in execution. For example, a component can be, but is not limited to, a process running on a processor, a processor, an object, an executable, a thread of execution, a program, or a computer, or any combination thereof. By way of example, both an application running on a server and the server can be a component. One or more components can reside within a process and / or thread of execution, and components can be localized on one computer, distributed among two or more computers, or both. In another example, each component can execute from various computer-readable media having various data structures. Components can communicate with local and / or remote processes, such as by pursuance of signals carrying one or more data packets (e.g., data from one component interacting with another component in a local or distributed system, and / or interacting with other systems via signals across a network such as the Internet). As another example, a component may be a device having a particular function provided by mechanical parts operated by electrical or electronic circuitry that is operated by a software or firmware application executed by a processor, in which case the processor may be internal or external to the device and execute at least a portion of the software or firmware application.As another example, a component may be a device that provides a particular function through electronic components without mechanical components, where the electronic components may include a processor or other means for executing software or firmware that at least partially provides the functionality of the electronic component. In one aspect, a component may be emulated by a virtual machine, for example, within a cloud computing system.

[0137] Additionally, the term "or" is intended to mean an inclusive "or" rather than an exclusive "or." That is, "X employs A or B" is intended to mean any of the natural inclusive permutations unless otherwise specified or clear from the context. That is, X employs A, X employs B, and X employs both A and B all satisfy "X employs A or B." Furthermore, the articles "a" and "an," as used in this specification and the accompanying drawings, should generally be construed to mean "one or more" unless otherwise specified, unless the context clearly indicates a singular reference. As used herein, the words "example" and / or "exemplary" are used to mean serving as an example, instance, or illustration. For the avoidance of doubt, the subject matter disclosed herein is not limited by such examples. Additionally, any aspect or design described herein as "example" and / or "exemplary" is not to be construed as necessarily preferred or advantageous over other aspects or designs, nor is it meant to exclude equivalent exemplary structures and techniques known to those skilled in the art.

[0138] As employed herein, the term "processor" may refer to substantially any computing processing unit or computing processing device, including, but not limited to, a single-core processor, a single processor with software multithreading execution capabilities, a multi-core processor, a multi-core processor with software multithreading execution capabilities, a multi-core processor with hardware multithreading techniques, a parallel platform, and a parallel platform with distributed shared memory. Additionally, a processor may refer to an integrated circuit designed to perform the functions described herein, an application-specific integrated circuit (ASIC), a digital signal processor (DSP), a field-programmable gate array (FPGA), a programmable logic controller (PLC), a complex programmable logic device (CPLD), discrete gate or transistor logic, discrete hardware components, or any combination thereof. Furthermore, a processor may utilize, but not be limited to, nanoscale architectures such as molecular or quantum dot-based transistors, switches, and gates to optimize space utilization or improve performance of user equipment. A processor may also be implemented as a combination of computing processing units. In this disclosure, terms such as "memory," "save," "data storage," "database," and substantially any other information storage component associated with the operation or functionality of a component are utilized to refer to a "memory component," an entity embodied in a "memory," or a component comprising a memory. It should be recognized that the memory and / or memory components described herein can be either volatile memory or non-volatile memory, or can include both volatile and non-volatile memory.By way of example, and not limitation, non-volatile memory may include read-only memory (ROM), programmable ROM (PROM), erasable programmable read-only memory (EPROM), electrically erasable programmable ROM (EEPROM), flash memory, or non-volatile random access memory (RAM) (e.g., ferroelectric RAM (FeRAM)). Volatile memory may include RAM, which may act as external cache memory, for example. By way of example, and not limitation, RAM is available in many forms, such as synchronous RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), double data rate SDRAM (DDR SDRAM), enhanced SDRAM (ESDRAM), Synchlink DRAM (SLDRAM), direct Rambus RAM (DRRAM), direct Rambus dynamic RAM (DRDRAM), and Rambus dynamic RAM (RDRAM). Additionally, the disclosed memory components of the systems or computer-implemented methods herein are intended to include, but are not limited to, other suitable types of memory.

[0139] The above description includes merely exemplary systems and computer-implemented methods. Of course, it is not possible to describe every conceivable combination of components or computer-implemented methods for purposes of describing this disclosure, but one of ordinary skill in the art will recognize that many further combinations and permutations of the present disclosure are possible. Moreover, to the extent that terms such as "including," "having," "possessing," and the like are used in the detailed description, claims, appendices, and drawings, such terms are intended to be inclusive, to be interpreted similarly to the term "comprising" when employed as a transitional word in a claim.

[0140] While the description of various embodiments has been presented for purposes of illustration, it is not intended to be exhaustive or limited to the disclosed embodiments. Many modifications and variations will be apparent to those skilled in the art without departing from the scope of the described embodiments. The terms used herein have been selected to best explain the principles of the embodiments, practical applications, or technical improvements over commercially available technologies, or to enable others skilled in the art to understand the embodiments disclosed herein.

Claims

1. a control qubit coupled to a target qubit, the control qubit being driven by a first tone that entangles the control qubit with the target qubit; The control qubit is driven by a second tone simultaneously with the first tone, the second tone having an opposite detuning sign to the first tone.

2. 2. The system of claim 1, wherein the first tone induces a Stark shift in the operating frequency of the control qubit, and the second tone cancels the Stark shift.

3. 3. The system of claim 2, wherein the second tone is calibrated by fixing the frequency of the second tone according to the opposite detuning code and sweeping the amplitude of the second tone until the Stark shift in the operating frequency of the control qubit is nullified.

4. 3. The system of claim 2, further comprising: a spectator qubit coupled to the control qubit, wherein the second tone prevents the operating frequency of the control qubit from dynamically colliding with an operating frequency of the spectator qubit.

5. 10. The system of claim 1, wherein the first tone is one of a plurality of time-warped tones that are sequentially applied to the control qubit during a zero-noise extrapolation procedure.

6. applying a first tone to a control qubit coupled to a target qubit, the first tone entangling the control qubit with the target qubit; applying a second tone to the control qubit simultaneously with the first tone, the second tone having an opposite detuning sign to the first tone; A method comprising:

7. 7. The method of claim 6, wherein the first tone induces a Stark shift in the operating frequency of the control qubit, and the second tone cancels the Stark shift.

8. 8. The method of claim 7, further comprising calibrating the second tone by fixing the frequency of the second tone according to the opposite detuning code and sweeping the amplitude of the second tone until the Stark shift in the operating frequency of the control qubit is nullified.

9. 8. The method of claim 7 , wherein a spectator qubit is coupled to the control qubit, and the second tone prevents the operating frequency of the control qubit from dynamically colliding with the operating frequency of the spectator qubit.

10. 7. The method of claim 6, wherein the first tone is one of a plurality of time-warped tones that are sequentially applied to the control qubit during a zero-noise extrapolation procedure.

11. a control qubit coupled to a target qubit, the control qubit being driven by a first tone that causes a Stark shift in the operating frequency of the control qubit; a control qubit further driven by a second tone that cancels the Stark shift; 1. A device comprising:

12. The device of claim 11 , wherein the second tone has an opposite detuning sign to the first tone.

13. The device of claim 12 , wherein the second tone is applied simultaneously with the first tone.

14. 14. The device of claim 13, wherein a first absolute difference between the frequency of the first tone and the operating frequency of the control qubit is less than an absolute value of an anharmonicity of the control qubit, and a second absolute difference between the frequency of the second tone and the operating frequency of the control qubit is less than the absolute value of the anharmonicity of the control qubit.

15. 12. The device of claim 11, wherein the second tone prevents the operating frequency of the control qubit from colliding with the operating frequency of an adjacent qubit.

16. applying a first tone to a control qubit coupled to a target qubit, the first tone inducing a Stark shift in the operating frequency of the control qubit; applying a second tone to the control qubit that cancels the Stark shift; and A method comprising:

17. 17. The method of claim 16, wherein the second tone has an opposite detuning sign to the first tone.

18. The method of claim 17 , wherein the second tone is applied simultaneously with the first tone.

19. 19. The method of claim 18, wherein a first absolute difference between the frequency of the first tone and the operating frequency of the control qubit is less than an absolute value of an anharmonicity of the control qubit, and a second absolute difference between the frequency of the second tone and the operating frequency of the control qubit is less than the absolute value of the anharmonicity of the control qubit.

20. 17. The method of claim 16, wherein the second tone prevents the operating frequency of the control qubit from colliding with the operating frequency of an adjacent qubit.

21. 1. A device comprising a qubit lattice, comprising: a control qubit in the qubit lattice is driven by a quantum entanglement gate tone; the control qubit is driven simultaneously with the entanglement gate tone by a Stark-shifted cancellation tone having an opposite detuning sign to the entanglement gate tone; Device.

22. 22. The apparatus of claim 21 , wherein the entanglement gating tone increases an operating frequency of the control qubit and the Stark shift cancellation tone decreases the operating frequency of the control qubit.

23. 22. The apparatus of claim 21 , wherein the entanglement gating tone decreases an operating frequency of the control qubit and the Stark shift cancellation tone increases the operating frequency of the control qubit.

24. 22. The apparatus of claim 21 , wherein the frequency of the Stark-shifted cancellation tones is higher than the highest operating frequency in the qubit lattice, thereby preventing the operating frequency of the control qubit from colliding with the operating frequency of a spectator qubit in the qubit lattice.

25. 22. The apparatus of claim 21 , wherein the frequency of the Stark-shifted cancellation tones is lower than the lowest operating frequency in the qubit lattice, thereby preventing the operating frequency of the control qubit from colliding with the operating frequency of a spectator qubit in the qubit lattice.

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