Inductively-shunted transmon qubit for superconducting circuits

Abstract

Techniques for modifying the Josephson potential of a transmon qubit by shunting the transmon with an inductance are described. The inclusion of this inductance may increase the confined potential of the qubit system compared with the conventional transmon, which may lead to a transmon qubit that is stable at much higher drive energies. The inductive shunt may serve the purpose of blocking some or all phase-slips between the electrodes of the qubit. As a result, the inductively shunted transmon may offer an advantage over conventional devices when used for applications involving high energy drives, whilst offering few to no drawbacks in comparison to conventional devices when used at lower drive energies.

Claims

1. A circuit quantum electrodynamics system comprising: a charge qubit, the charge qubit comprising an island of superconducting material coupled to a superconducting reservoir via a Josephson junction and coupled to at least one capacitor; and a series array of Josephson junctions coupled to the charge qubit such that the series array is coupled in parallel to the Josephson junction of the charge qubit and in parallel to the at least one capacitor of the charge qubit.

2. The circuit quantum electrodynamics system of claim 1, wherein each Josephson junction in the series array of Josephson junctions has a Josephson coupling energy K.sub.J′, wherein the Josephson junction of the charge qubit has a Josephson coupling energy K.sub.J, and wherein K.sub.J′ is greater than K.sub.J.

3. The circuit quantum electrodynamics system of claim 1, wherein the series array of Josephson junctions has a combined inductive energy E.sub.L that is between 0.8×E.sub.J and 1.2×E.sub.J, where E.sub.J is the Josephson energy of the Josephson junction of the charge qubit.

4. The circuit quantum electrodynamics system of claim 3, wherein E.sub.L is between 0.9×E.sub.J and 1.1×E.sub.J.

5. The circuit quantum electrodynamics system of claim 1, wherein the Josephson junction of the charge qubit has a Josephson coupling energy K.sub.J, wherein the at least one capacitor of the charge qubit has a charging energy E.sub.C, and wherein the ratio E.sub.J/E.sub.C is at least 5.

6. The circuit quantum electrodynamics system of claim 5, wherein the ratio E.sub.J/E.sub.C is at least 25.

7. The circuit quantum electrodynamics system of claim 1, wherein a dc magnetic flux threaded through an inductive loop formed by the series array of Josephson junctions and the charge qubit is less than 10.sup.−15 weber.

8. The circuit quantum electrodynamics system of claim 1, further comprising an oscillator coupled to the charge qubit and the series array of Josephson junctions.

9. The circuit quantum electrodynamics system of claim 8, wherein the oscillator is a microwave cavity.

10. The circuit quantum electrodynamics system of claim 1, further comprising a transmission line coupled to the charge qubit and the series array of Josephson junctions.

11. The circuit quantum electrodynamics system of claim 1, wherein the series array of Josephson junctions is M Josephson junctions coupled in series, and wherein M is greater or equal to 3 and M is less than or equal to 20.

12. The circuit quantum electrodynamics system of claim 11, wherein M is greater or equal to 5 and M is less than or equal to 10.

13. A circuit quantum electrodynamics system comprising: a charge qubit, the charge qubit comprising an island of superconducting material coupled to a superconducting reservoir via a Josephson junction and coupled to at least one capacitor; and a superinductor coupled in parallel to the Josephson junction of the charge qubit and in parallel to the at least one capacitor of the charge qubit.

14. The circuit quantum electrodynamics system of claim 13, wherein the superinductor has an inductive energy E.sub.L that is between 0.8×E.sub.J and 1.2×E.sub.J, where E.sub.J is the Josephson energy of the Josephson junction of the charge qubit.

15. The circuit quantum electrodynamics system of claim 13, wherein the Josephson junction of the charge qubit has a Josephson coupling energy K.sub.J, wherein the at least one capacitor of the charge qubit has a charging energy E.sub.C, and wherein the ratio E.sub.J/E.sub.C is at least 5.

16. The circuit quantum electrodynamics system of claim 13, wherein a dc magnetic flux threaded through an inductive loop formed by the superinductor and the charge qubit is less than 10.sup.−15 weber.

17. The circuit quantum electrodynamics system of claim 13, wherein the superinductor comprises granular aluminum.

18. A method of operating a circuit quantum electrodynamics (cQED) system, the cQED system comprising an inductively shunted transmon qubit coupled to a transmission line, the method comprising: applying a pump of frequency ω.sub.p to the inductively shunted transmon qubit via the transmission line; and adjusting ω.sub.p, thereby causing a coupling strength between a Floquet mode of the inductively shunted transmon qubit |Φ.sub.n) and a Floquet mode of the inductively shunted transmon qubit |Φ.sub.m> to fall below a selected threshold, wherein n is 0 or 1, and wherein m is between 1 and 20.

19. The method of claim 18, wherein ω.sub.p is selected so that the coupling strength remains below the selected threshold irrespective of the amplitude of the pump.

20. The method of claim 18, wherein the threshold is 0.1 MHz.

21. The method of claim 20, wherein the threshold is 0.01 MHz.

22. The method of claim 18, wherein the inductively shunted transmon qubit has a characteristic frequency ω.sub.q, and wherein ω.sub.p is between 0.5ω.sub.q and 3.0ω.sub.q.

23. The method of claim 22, wherein the inductively shunted transmon qubit has a characteristic frequency ω.sub.q, and wherein ω.sub.p is between 1.4ω.sub.q and 1.8ω.sub.q.

24. The method of claim 18, wherein n is 0 and m is 4.

25. The method of claim 18, wherein n is 1 and m is 5.

26. A circuit quantum electrodynamics system comprising: a charge qubit, the charge qubit comprising a Josephson junction coupled to at least one capacitor; and an inductor coupled in parallel to the Josephson junction of the charge qubit and in parallel to the at least one capacitor of the charge qubit, wherein the inductor has an inductive energy E.sub.L, the Josephson junction has a Josephson energy E.sub.J, and wherein E.sub.L/E.sub.J is at least 3 and no greater than 6.

27. The circuit quantum electrodynamics system of claim 26, wherein the Josephson junction of the charge qubit has a Josephson coupling energy K.sub.J, wherein the at least one capacitor of the charge qubit has a charging energy E.sub.C, and wherein the ratio E.sub.J/E.sub.C is at least 5.

28. The circuit quantum electrodynamics system of claim 26, wherein a dc magnetic flux threaded through an inductive loop formed by the series array of Josephson junctions and the charge qubit is less than 10.sup.−15 weber.

29. The circuit quantum electrodynamics system of claim 26, wherein E.sub.L/E.sub.J is at least 4 and no greater than 5.

Description

BRIEF DESCRIPTION OF DRAWINGS

(1) Various aspects and embodiments will be described with reference to the following figures. It should be appreciated that the figures are not necessarily drawn to scale. In the drawings, each identical or nearly identical component that is illustrated in various figures is represented by a like numeral. For purposes of clarity, not every component may be labeled in every drawing.

(2) FIG. 1A depicts a photon cavity coupled to a transmon qubit, according to some embodiments;

(3) FIG. 1B illustrates a conventional transmon qubit, according to some embodiments;

(4) FIG. 1C illustrates experimental data obtained from a cavity coupled to a transmon qubit as in FIG. 1A, according to some embodiments;

(5) FIG. 2 illustrates the potential of a conventional transmon qubit, according to some embodiments;

(6) FIG. 3 depicts an illustrative inductively shunted transmon, according to some embodiments;

(7) FIG. 4 illustrates the potential of the inductively shunted transmon of FIG. 3, according to some embodiments;

(8) FIGS. 5A-5B depict illustrative implementations of the inductively shunted transmon of FIG. 4, according to some embodiments;

(9) FIG. 6 depicts a physical realization of an inductively shunted transmon, according to some embodiments;

(10) FIG. 7 illustrates the potential of the inductively shunted transmon of FIG. 6, according to some embodiments;

(11) FIG. 8 depicts an experimental configuration in which a pump is applied to an inductively shunted transmon, according to some embodiments;

(12) FIG. 9 depicts quasi-energies of several Floquet states of an inductively shunted transmon, according to some embodiments;

(13) FIG. 10 depicts quasi-energies of Floquet states of an inductively shunted transmon, focusing on a crossing between 0th and 4th states, according to some embodiments;

(14) FIG. 11 depicts the coupling of four different crossings between Floquet states as a function of pump frequency and the normalized AC Stark shift, according to some embodiments; and

(15) FIG. 12 illustrates a relationship between the ratio between the inductive energy and Josephson energy of the inductively shunted transmon and the amplitude of a coupling between 0th and 4th Floquet states, according to some embodiments.

DETAILED DESCRIPTION

(16) As discussed above, quantum information may be stored using quantum bits, referred to as “qubits,” which are typically quantum mechanical systems exhibiting two or more states. Some qubits utilize quantum states that relate to the presence or absence of electrical charge. For instance, a charge qubit may be formed from an island of superconducting material connected via a Josephson junction to a superconducting reservoir. Since the island is superconducting, conduction electrons therein form Cooper pairs and condense to a single ground state. Cooper pairs can tunnel across the Josephson junction but in a manner that exhibits a number of discrete electrostatic energy states within the island. Placing the system into one of these energy states can represent quantum information, and frequently the two lowest energy states are operated as a qubit. When desired, the charge states can be manipulated by applying suitable pulses to the superconducting island.

(17) One type of charge qubit is the so-called transmon. In terms of the above-described charge qubit, in a transmon the island of superconducting material is coupled to one side of a capacitor in addition to the aforementioned Josephson junction. The transmon is designed to reduce charge noise by increasing the ratio of the Josephson energy of the junction to the charging energy of the capacitor. Increase of this ratio results in reduced sensitivity to charge noise and to the energy levels of the system becoming essentially independent of the amount of electrical charge across the junction. The transmon is described further, for instance, in “Charge-insensitive qubit design derived from the Cooper pair box,” J. Koch et al., Phys. Rev. A 76, 042319 (2007), which is hereby incorporated by reference in its entirety.

(18) The inventors have recognized and appreciated that a number of quantum information processing techniques involve the application of strong drive energies, and have observed experimentally that the coupling of a transmon to other systems does not scale with the drive energy beyond some critical energy. It has been observed that, beyond these drive energies, desirable properties of the transmon, such as its coherence properties, begin to deteriorate. The inventors have proposed that these observations may be a result of the transmon state escaping from a potential confinement imposed by the Josephson junction, resulting in free-particle like states, and/or a result of non-linear resonance(s) of the non-linear system. Irrespective of the underlying cause(s), this property of the transmon limits how it may be used in circuit quantum electrodynamics (cQED) systems by limiting the magnitude of the drives that may be applied to the transmon.

(19) This challenge is illustrated via FIGS. 1A-1C. FIG. 1A depicts a photon cavity 110 coupled to a transmon qubit 130, which is illustrated further in FIG. 1B. The transmon qubit 130 includes a Josephson junction 131 with a phase φ across the junction, and a capacitance 132 coupled in parallel with the junction 131. In general in the system of FIG. 1A there may also be a transmission line present via which energy may be pumped into the qubit and cavity system, although this is not depicted in the figure.

(20) FIG. 1C illustrates experimental data obtained from a cavity coupled to a transmon qubit as in FIG. 1A, wherein the magnitude of energy transmission from the cavity is shown as a shade of grey (darker being greater transmission), as a function of the frequency of the cavity (vertical axis) and the power of the pump applied to the cavity-qubit system measured by a number of photons (horizontal axis). As may be noted, for weaker drives wherein the pump power is less than about 100 photons, the cavity transmission is stable and follows the linear variation of frequency with pump power expected from the AC Stark shift. However, at higher pump powers in region 150, the cavity frequency becomes unstable. A second region of instability 151 occurs at even higher pump powers.

(21) Without wishing to be bound by theory, with respect to instability 150, it is believed the coupling between states of the non-linear transmon system at higher pump powers cause the cavity frequency to be unstable. This instability may be seen in FIG. 1C in region 150. With respect to instability 151, it is believed that the physics of the first few eigenstates of the transmon is dominated by the localization of the superconducting phase across the junction, or equivalently the suppression of phase-slips, which renders the energy level structure approximately harmonic. These desirable features may break down at eigenenergies comparable to the Josephson energy E.sub.J of the Josephson junction, owing to the boundedness of the potential. This breakdown may be seen in FIG. 1C in region 151, where the pump power has grown beyond the confining potential of the transmon.

(22) The transmon potential is illustrated in FIG. 2, according to some embodiments. Line 210 in diagram 200 depicts the transmon potential U normalized by the Josephson energy of the transmon E.sub.J, against the phase difference across the transmon φ. Note that the transmon potential U is periodic with the phase difference φ. Also shown in diagram 200 are the wavefunctions of several energy eigenstates of the transmon qubit, including the ground state E.sub.0, the first excited state E.sub.1, etc. As may be noted from FIG. 2, the transmon potential is unconfining at eigenenergies above the Josephson energy E.sub.J. In addition, the energy eigenstates of the transmon are not completely localized within the potential wells of the transmon potential as can be seen by the variation of some of the eigenfunction wavefunctions within the potential's peaks. These features of the transmon can lead to the aforementioned instabilities at sufficiently high drive energies.

(23) The inventors have recognized and appreciated techniques for modifying the Josephson potential of a transmon by shunting the transmon with an inductance. The inclusion of this inductance has been observed to increase the confined potential of the system compared with the conventional transmon, which may lead to a transmon that is stable at much higher drive energies. The inductive shunt may serve the purpose of blocking some or all phase-slips between the electrodes of the qubit. As a result, the inductively shunted transmon may offer an advantage over conventional devices when used for applications involving high energy drives, whilst offering few to no drawbacks in comparison to conventional devices when used at lower drive energies.

(24) An illustrative inductively shunted transmon is depicted in FIG. 3, according to some embodiments. In the example of FIG. 3, inductively shunted transmon 300 includes a Josephson junction 310, a capacitance 315, and an inductance 320. As shown, each of the Josephson junction 310, capacitance 315, and inductance 320 are coupled in parallel with one another. In the example of FIG. 3, the Josephson junction has a Josephson energy E.sub.J, the capacitance 315 has a charging energy E.sub.C, and the inductance 320 has an inductive energy E.sub.L. In addition, there is an inductive loop formed by the loop containing the inductance 320 and the Josephson junction 310; in the example of FIG. 3, there is a dc magnetic flux through this loop φ.sub.ext.

(25) According to some embodiments, the Josephson energy E.sub.J may be approximately equal to the inductive energy E.sub.L. In some embodiments, the ratio of the inductive energy to the Josephson energy, E.sub.L/E.sub.J, is greater than or equal to 0.5, 0.75, 0.85, 0.9, 0.95, 1.0, 1.5, 2.0, 3.0, 4.0, 5.0, or 6.0. In some embodiments, the ratio of the inductive energy to the Josephson energy, E.sub.L/E.sub.J, is less than or equal to 10.0, 8.0, 6.0, 5.0, 4.0, 3.0, 2.5, 2.0, 1.5, 1.2, 1.1, or 1.0. Any suitable combinations of the above-referenced ranges are also possible (e.g., a ratio E.sub.L/E.sub.J of greater than or equal to 0.75 and less than or equal to 1.5).

(26) According to some embodiments, the Josephson energy E.sub.J may be much larger than the charging energy E.sub.C. In some embodiments, the ratio of the Josephson energy to the charging energy, E.sub.J/E.sub.C, is greater than or equal to 5, 10, 25, 50, 75, 100, 150, or 200. In some embodiments, the ratio of the Josephson energy to the charging energy, E.sub.J/E.sub.C, is less than or equal to 500, 300, 200, 100, 75, 50, 25, or 10. Any suitable combinations of the above-referenced ranges are also possible (e.g., a ratio E.sub.J/E.sub.C of greater than or equal to 25 and less than or equal to 100).

(27) As discussed above, one of the advantageous of the transmon over other charge qubits is a reduced sensitivity to charge noise and to the energy levels of the system becoming essentially independent of the amount of electrical charge across the junction. These advantages scale with the ratio E.sub.J/E.sub.C; as such, a larger value of E.sub.J/E.sub.C may be desirable (although the anharmonicity α of the transmon also decreases with E.sub.J/E.sub.C so these effects are generally balanced by not making the charging energy too large).

(28) In some embodiments, the dc magnetic flux φ.sub.ext is less than or equal to 10.sup.−12 weber, 10.sup.−13 weber, 10.sup.−14 weber, 10.sup.−15 weber, 10.sup.−16 weber, 10.sup.−17 weber, 10.sup.−18 weber, or 10.sup.−19 weber. In some embodiments, a magnetic flux density of the dc magnetic flux φ.sub.ext through the inductive loop formed by the loop containing the inductance 320 and the Josephson junction 310 is less than or equal to 10.sup.−3 tesla, 10.sup.−4 tesla, 10.sup.−5 tesla, 10.sup.−6 tesla, 10.sup.−7 tesla, 10.sup.−8 tesla, 10.sup.−9 tesla, or 10.sup.−10 tesla. Notwithstanding the above, the dc magnetic flux φ.sub.ext may preferably be zero.

(29) According to some embodiments, the inductance 320 may be implemented using one or more linear inductors, and may include geometrical inductors in addition to highly inductive materials sometimes referred to as “superinductors.” In some embodiments, the inductance 320 may comprise one or more Josephson junctions. Further examples of particular implementations of an inductively shunted transmon are discussed further below.

(30) FIG. 4 illustrates the potential of the inductively shunted transmon 300, according to some embodiments. Line 410 in diagram 400 depicts the inductively shunted transmon potential U normalized by the Josephson energy of the Josephson junction E.sub.J, against the phase difference across the inductively shunted transmon φ (note that φ in FIG. 4 is a different quantity from the φ used in FIG. 2 in that the phase in FIG. 4 is non-periodic). Also shown in diagram 400 are the wavefunctions of several energy eigenstates of the inductively shunted transmon qubit, including the ground state E.sub.0, the nineteenth first excited state E.sub.19, etc. In the illustrated example, the Josephson energy E.sub.J is equal to the inductive energy E.sub.L.

(31) In comparison with FIG. 2, which illustrates the conventional transmon, it may be noted that the potential of the inductively shunted transmon is confining at much higher energies than the conventional transmon. As shown in FIG. 2, the potential is unconfined at energies above the Josephson energy E.sub.J, whereas as shown in FIG. 4, the potential is confining up to, and exceeding, 80 times the Josephson energy E.sub.J. As a result, much higher energy eigenstates of the inductively shunted transmon are confined by the potential (e.g., the eigenstate E.sub.99, whereas in FIG. 2 even E.sub.9 has an energy that exceeds the potential well of the conventional transmon). Moreover, the wavefunctions of the inductively shunted transmon shown in FIG. 4 are much more localized in phase than those of the conventional transmon. This feature may better isolate the instabilities at weaker pump energies discussed above in relation to FIGS. 1A-1C.

(32) FIGS. 5A-5B depict illustrative implementations of the inductively shunted transmon of FIG. 4, according to some embodiments. In the example of FIG. 5A, inductively shunted transmon 500 comprises a superinductor material 520 as an inductive shunt. Superinductor 520 may comprise, for example, granular aluminum, Nb nanowire, and/or any other material with a high kinetic inductance. Materials with a high kinetic inductance have a high inductance as a result of their material properties, in contrast with geometrical inductors such as coils, which have an inductance as a result of their geometrical shape.

(33) In the example of FIG. 5B, inductively shunted transmon 550 comprises a series array of M Josephson junctions 570 as an inductive shunt. While the M Josephson junctions are non-linear components, together they may approximate a linear inductance while also providing a periodic potential that would not otherwise be present with a typical linear inductor.

(34) In the example of FIG. 5B, each of the M Josephson junctions in array 570 has a Josephson energy E.sub.J′. The Josephson energy E.sub.J′ may be the same as, or may differ from, the Josephson energy E.sub.J of the Josephson junction 560. According to some embodiments, the inductance provided by the series array 570 may be equal to (or may be proportional to)

(35) $\frac{E_{J^{\prime }}}{M}.$
In view of the desirable characteristic that E.sub.L˜E.sub.J and since M>1, therefore, in some embodiments the Josephson energy E.sub.J′ may be greater than E.sub.J.

(36) In some embodiments, the ratio of the Josephson energy E.sub.J′ to the Josephson energy E.sub.J is greater than or equal to 2, 3, 5, 8, 10, 15, or 20. In some embodiments, the ratio of the Josephson energy E.sub.J′ to the Josephson energy E.sub.J is less than or equal to 25, 20, 15, 10, 8, 5, or 3. Any suitable combinations of the above-referenced ranges are also possible (e.g., a ratio E.sub.J′/E.sub.J of greater than or equal to 3 and less than or equal to 10).

(37) In some embodiments, the number M of Josephson junctions in series array 570 is greater than or equal to 2, 3, 4, 5, 8, 10, or 15. In some embodiments, the number M of Josephson junctions in series array 570 is less than or equal to 25, 20, 15, 10, 8, 5, or 3. Any suitable combinations of the above-referenced ranges are also possible (e.g., a number M of Josephson junctions in the series array of greater than or equal to 2 and less than or equal to 5).

(38) FIG. 6 depicts a physical realization of an inductively shunted transmon, according to some embodiments. In the example of FIG. 6, a device shown in photograph 601 comprises an inductively shunted transmon as shown in 602. The inductively shunted transmon includes a conventional transmon 610 shunted by a series array of three Josephson junctions 620 (e.g., equivalent to the inductively shunted transmon 550 with M=3).

(39) FIG. 7 illustrates the potential of the inductively shunted transmon of FIG. 6, according to some embodiments. In contrast to FIG. 4, which illustrates the potential of an inductively shunted transmon with an idealized inductance, FIG. 7 depicts the potential for the particular device of FIG. 5B in which M=3. Line 710 in diagram 700 depicts the inductively shunted transmon potential U normalized by the Josephson energy of the Josephson junction E.sub.J, against the phase difference across the inductively shunted transmon φ. Also shown in diagram 700 are the wavefunctions of several energy eigenstates of the inductively shunted transmon qubit, including the ground state E.sub.0, the fifth first excited state E.sub.5, etc. It may be noted that the potential in FIG. 7 is still confined, although not to the same extent as in FIG. 4. Nonetheless, the implementation illustrated by FIG. 7 still represents a significant improvement over the implementation represented by FIG. 2. For instance, in FIG. 7 the potential is confining up to ten times the Josephson junction E.sub.J and up to the excited state E.sub.28 (compared to the Josephson junction E.sub.J and the excited state E.sub.7, respectively, in the case of FIG. 2). Note also that much greater localization of the wavefunctions within the potential well is achieved in the case of FIG. 7 compared with FIG. 2.

(40) Having now described the structure of the inductively shunted transmon, some features of the device will now be described. As discussed below, the inductively shunted transmon has particularly useful characteristics that enable beneficial operations of a cQED system not otherwise available with a conventional transmon.

(41) FIG. 8 depicts an experimental configuration upon which the subsequent description relies. In FIG. 8, an inductively shunted transmon 800 is pumped via a transmission line, with the pump being described by 2Ω cos ω.sub.pt, where Ω is the amplitude of the pump, ω.sub.p is the frequency of the pump, and t is time. Note that in the example of FIG. 8, the inductive shunt of the transmon 800 is represented by a series array of M Josephson junctions, which each have a Josephson energy βE.sub.J, where E.sub.J is the Josephson energy of the primary Josephson junction of the transmon 810.

(42) According to some embodiments, in the illustrative experimental setup, the frequency ω.sub.p may be close to twice that of the qubit frequency ω.sub.q. As one non-limiting example, ω.sub.q/2π=4860 MHz and ω.sub.p/2π=9230 MHz. In some embodiments, the ratio ω.sub.p/ω.sub.q may be equal to or greater than 1.0, 1.2, 1.4, 1.6, 1.8, or 2.0. In some embodiments, the ratio ω.sub.p/ω.sub.q may be less than or equal to 2.2, 1.8, 1.6, 1.4, 1.2, or 1.0. Any suitable combinations of the above-referenced ranges are also possible (e.g., a ratio ω.sub.p/ω.sub.q of between 1.4 and 1.8).

(43) For the experimental configuration of FIG. 8, the Hamiltonian of the system may be written as:

(44) $\begin{array}{cc}\begin{array}{c}\hat{H}\left(t\right)\approx {\hat{H}}_{\mathrm{qubit}}+{\hat{H}}_{\mathrm{pump}}\\ \approx 4E_C{\hat{N}}^2-E_j\cos \hat{\phi }+\frac{1}{2}{E}_L{\hat{\phi }}^2+2\mathrm{\hslash \Omega }\hat{N}\cos {\omega }_{p}t\end{array}& \left(1\right)\end{array}$

(45) where {circumflex over (N)} is the transmon mode operator corresponding to the number of Cooper pairs across the junction, and {circumflex over (φ)} represents the dimensionless flux operator across the Josephson junction 810.

(46) Based on the Floquet theorem, solutions to the time-dependent Schrodinger equation are given by:
Ψ.sub.n(t)=e.sup.−∈.sup.n.sup.t/ℏΦ.sub.n(t)
where

(47) ${\Phi }_n\left(t+\frac{2\pi }{{\omega }_p}\right)={\Phi }_n\left(t\right)-\frac{{\mathrm{\hslash \omega }}_p}{2}\le {ϵ}_n\le \frac{{\mathrm{\hslash \omega }}_p}{2}\left(\mathrm{mod}{\mathrm{\hslash \omega }}_p\right)$

(48) Hereafter, Ψ.sub.n(t) are referred to as Floquet states, Φ.sub.n(t) as Floquet modes, and ∈.sub.n as quasi-energies, with each of these quantities having the same mode index n. As noted above, the quasi-energies are invariant under translation by multiples of ℏω.sub.p.

(49) The quasi-energies of the Floquet states can be plotted, as shown in FIG. 9, which illustrates the quasi-energies for the states with index 0, 1 and 2 (where Ψ.sub.n is also referred to hereafter as |n; e.g., Ψ.sub.0 as |0, Ψ.sub.1 as |1, etc.). In FIG. 9, the quasi-energy is normalized by the pump frequency ℏω.sub.p so that the illustrated quasi-energies span a vertical width of 1. Since the quasi-energies are invariant under translations of multiple of ℏω.sub.p as noted above, the quasi-energies ∈.sub.n are plotted mod ℏω.sub.p so all values occupy the same illustrated range of values.

(50) In the example of FIG. 9, the horizontal axis is expressed as Δ.sub.01.sup.ac, the AC Stark shift between the |0 and |1 states, normalized by the anharmonicity α. The anharmonicity of the inductively shunted transmon is the difference between the energy gap between the first two energy levels and the energy gap between the second two energy levels: α=E.sub.12−E.sub.01=(E.sub.2−E.sub.1)−(E.sub.1−E.sub.0). The representation in FIG. 9 based on

(51) $\frac{{\Delta }_{01}^{\mathrm{ac}}}{\alpha }$
expresses the pump strength in terms of the equivalent excitation that an operation performed in a time 1/α would generate. As an alternative, the value

(52) $\frac{{\Delta }_{01}^{\mathrm{ac}}}{\alpha }$
may also be expressed as

(53) ${\left(\frac{\Omega }{{\omega }_p}\right)}^2.$

(54) Additional Floquet states may be plotted in the same manner, as shown in FIG. 10. As illustrated by FIG. 10, some of the quasi-energies cross one another for particular values of

(55) $\frac{{\Delta }_{01}^{\mathrm{ac}}}{\alpha }.$
In particular, the |0 and |4 states cross in FIG. 10, details of which are shown in the circular inset. At this crossing, the pump may cause a transition between these Floquet states from the energy of two pump photons with an amplitude Ω.sub.0,4. This crossing and others like it represent instabilities in the dynamical state of the system and it is therefore desirable to avoid producing these instabilities. One challenge with avoiding the instability, however, is that the space of these states is very dense, as may be appreciated by FIG. 10, which only shows a select few states.

(56) Nonetheless, the pump frequency may be tuned to reduce (and in some cases eliminate) the strength of particular Floquet state couplings. In some cases, one or more particular couplings may be targeted for this purpose since not all of these instabilities have the same coupling strength; indeed, certain crossings may produce greater instabilities than other crossings. For instance, FIG. 11 depicts an example of tuning the pump frequency ω.sub.p such that the coupling strength of a particular crossing is sufficiently low, irrespective of the value of

(57) $\frac{{\Delta }_{01}^{\mathrm{ac}}}{\alpha }.$

(58) FIG. 11 plots the coupling strength (represented by the thickness of the corresponding lines) of four different crossings as a function of pump frequency ω.sub.p (normalized by the qubit frequency ω.sub.q) and the normalized AC Stark shift

(59) $\frac{{\Delta }_{01}^{\mathrm{ac}}}{\alpha }.$
The crossings are written as {tilde over (ω)}.sub.a,b/n, for a crossing between Floquet states a and b, which is produced by n pump photons. As noted above, the crossing between the |0 and |4 states is the result of two pump photons, and so is written as {tilde over (ω)}.sub.0,4/2.

(60) In the example of FIG. 11, the pump frequency may be tuned so that the coupling strength of each of the depicted four crossings is below 0.1 MHz, below 0.01 MHz, or below 0.001 MHz. As shown by the vertical dashed arrow, none of the crossings intersect for a pump frequency of 1.6 times the qubit frequency, irrespective of the value of

(61) 0$\frac{{\Delta }_{01}^{\mathrm{ac}}}{\alpha }.$
As such, at least the depicted instabilities may be mitigated or avoided.

(62) According to some embodiments, a pump of frequency ω.sub.p may be applied to an inductively shunted transmon (e.g., via a transmission line), wherein ω.sub.p may be selected so as to reduce or eliminate a coupling strength between a Floquet mode of the inductively shunted transmon qubit |Φ.sub.n and a Floquet mode of the inductively shunted transmon qubit |Φ.sub.m below a threshold value. In some embodiments, n=0 and m=4; or n=0 and m=5; or n=0 and m=6; or n=0 and m=7; or n=0 and m=8; or n=1 and m=5. In some embodiments, the threshold value is greater than or equal to 0.001 MHz, 0.01 MHz, 0.1 MHz, 1 MHz, or 10 MHz. In some embodiments, the threshold value is less than or equal to 10 MHz, 5 MHz, 1 MHz, 0.1 MHz, 0.05 MHz, 0.01 MHz, or 0.005 MHz. Any suitable combinations of the above-referenced ranges are also possible (e.g., a threshold value of greater than or equal to 0.005 MHz and less than or equal to 0.01 MHz, a threshold value equal to 0.01 MHz, a threshold value equal to 0.005 MHz, etc.). In some cases, the frequency ω.sub.p may be selected to produce the above result; in other cases, a pump of frequency ω.sub.p may be applied and the value of ω.sub.p adjusted until the above result is produced.

(63) It may be noted that the above-discussed and depicted structure of the crossings between Floquet states is particular to the inductively shunted transmon. For a conventional transmon, for instance, it may not be feasible to choose a pump frequency so as to reduce the coupling strength of these crossings in the manner shown in FIG. 11 due to greater degeneracy between the crossings. As such, the ability to tune the pump frequency so as to reduce or eliminate the coupling strength between the most problematic Floquet state couplings may be provided by the inductively shunted transmon, representing an additional advantage of the inductively shunted transmon beyond those discussed above.

(64) In addition, the inductively shunted transmon provides a means to tune its behavior that a conventional transmon does not have. In particular, in a conventional transmon the values of E.sub.J and E.sub.C represent the only available ways to tune the behavior of the qubit. In an inductively shunted transmon, however, the inductive energy E.sub.L can also be tuned.

(65) Considering an analysis of the inductively shunted transmon, for instance, by perturbatively expanding cos {circumflex over (φ)} in Equation (1) above in the absence of the pump:

(66) $\frac{H}{\hslash }\approx {\omega }_a{a}^{†}a-\frac{\alpha }{12}{\left(a+a^{†}\right)}^4+\frac{\beta \left(r\right)}{120}{\left(a+a^{†}\right)}^6$

(67) Where a.sup.† and a are creation and annihilation operators, respectively, and:
ℏω.sub.a=√{square root over (8E.sub.CE.sub.J(1+r))}
ℏα=E.sub.C/(1+r)
r=E.sub.L/E.sub.J

(68) As a result of the above, the ratio E.sub.L/E.sub.J can provide a useful way to tune the system, whereas a conventional transmon lacks this tuning ‘knob’. That is, in a conventional transmon, E.sub.L=0 so r≡0 and so the ability to tune the parameter r simply does not exist.

(69) Considering the pump applied to the inductively shunted transmon as before yields:

(70) $\frac{H}{\hslash }\approx {\omega }_a{a}^{†}a-\frac{\alpha }{12}{\left(a+a^{†}\right)}^4+\frac{\beta \left(r\right)}{120}{\left(a+a^{†}\right)}^6-2i\mathrm{\Omega cos}{\omega }_{p}t\left(a-a^{†}\right)$

(71) This result may be used to determine the coupling strength of certain crossings between Floquet states as function of the tunable parameters of the inductively shunted transmon. For instance, considering the crossing between |0 and |4 Floquet states as shown in FIG. 10 and discussed above, the following may be derived:

(72) $\frac{H}{\hslash }\approx \delta a^{†}a-\frac{\alpha }{2}{a}^{{†}^2}{a}^2+\frac{\beta \left(r\right)}{6}{a}^{{†}^3}{a}^3+{\Omega }_{0,4}\left(r\right)a^{{†}^4}+h.c.$

(73) where ω.sub.p=2(ω.sub.a−δ), and Ω.sub.0,4 is the amplitude of the coupling strength between the |0 and |4 Floquet states. The following relationship may further be determined:

(74) $\frac{{\Omega }_{0,4}}{\alpha }\propto \frac{{\Delta }_{\mathrm{ac}}}{\alpha }.Math.\frac{\alpha }{{\omega }_a}\left(4.5-r\right)$

(75) This relationship is plotted in FIG. 12, showing the value of r against Ω.sub.0,4 (solid line). As may be noted, for a particular value of r, the amplitude of the coupling strength between the |0 and |4 Floquet states is reduced to zero. As a result, through appropriate selection of the ratio between the inductive energy and Josephson energy of the inductively shunted transmon, the undesirable instability produced by the coupling between |0 and |4 Floquet states may be mitigated or removed.

(76) It will be appreciated that the above is an illustrative analysis and that in practice the value of r that reduces Ω.sub.0,4 to zero may not be exactly 4.5. However, the general technique of selecting r based on an undesirable coupling may nonetheless be applicable. As such, when fabricating an inductively shunted transmon the relative values of E.sub.L and E.sub.J may be selected to produce a desired effect on the amplitude of the coupling strength of one or more pairs of Floquet states.

(77) Having thus described several aspects of at least one embodiment of this invention, it is to be appreciated that various alterations, modifications, and improvements will readily occur to those skilled in the art.

(78) Such alterations, modifications, and improvements are intended to be part of this disclosure, and are intended to be within the spirit and scope of the invention. Further, though advantages of the present invention are indicated, it should be appreciated that not every embodiment of the technology described herein will include every described advantage. Some embodiments may not implement any features described as advantageous herein and in some instances one or more of the described features may be implemented to achieve further embodiments. Accordingly, the foregoing description and drawings are by way of example only.

(79) Various aspects of the present invention may be used alone, in combination, or in a variety of arrangements not specifically discussed in the embodiments described in the foregoing and is therefore not limited in its application to the details and arrangement of components set forth in the foregoing description or illustrated in the drawings. For example, aspects described in one embodiment may be combined in any manner with aspects described in other embodiments.

(80) Also, the invention may be embodied as a method, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

(81) Use of ordinal terms such as “first,” “second,” “third,” etc., in the claims to modify a claim element does not by itself connote any priority, precedence, or order of one claim element over another or the temporal order in which acts of a method are performed, but are used merely as labels to distinguish one claim element having a certain name from another element having a same name (but for use of the ordinal term) to distinguish the claim elements.

(82) The terms “approximately” and “about” may be used to mean within ±20% of a target value in some embodiments, within ±10% of a target value in some embodiments, within ±5% of a target value in some embodiments, and yet within ±2% of a target value in some embodiments. The terms “approximately” and “about” may include the target value. The term “substantially equal” may be used to refer to values that are within ±20% of one another in some embodiments, within ±10% of one another in some embodiments, within ±5% of one another in some embodiments, and yet within ±2% of one another in some embodiments.

(83) The term “substantially” may be used to refer to values that are within ±20% of a comparative measure in some embodiments, within ±10% in some embodiments, within ±5% in some embodiments, and yet within ±2% in some embodiments. For example, a first direction that is “substantially” perpendicular to a second direction may refer to a first direction that is within ±20% of making a 90° angle with the second direction in some embodiments, within ±10% of making a 90° angle with the second direction in some embodiments, within ±5% of making a 90° angle with the second direction in some embodiments, and yet within ±2% of making a 90° angle with the second direction in some embodiments.

(84) Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use of “including,” “comprising,” or “having,” “containing,” “involving,” and variations thereof herein, is meant to encompass the items listed thereafter and equivalents thereof as well as additional items.