Analytical modeling of parametrically-modulated transmon qubits

TL;DR

This paper develops an analytical perturbation theory approach to accurately model parametrically-modulated transmon qubits, aiding the understanding and verification of complex quantum dynamics in superconducting quantum processors.

Contribution

It introduces a systematic perturbation method to derive precise analytic expressions for transmon parameters, specifically applied to parametrically-modulated transmons for entangling gate analysis.

Findings

Derived accurate analytic expressions for transmon parameters.

Applied the model to study parametrically-activated entangling gates.

Enhanced understanding of quantum dynamics in superconducting qubit networks.

Abstract

Building a scalable quantum computer requires developing appropriate models to understand and verify its complex quantum dynamics. We focus on superconducting quantum processors based on transmons for which full numerical simulations are already challenging at the level of qubytes. It is thus highly desirable to develop accurate methods of modeling qubit networks that do not rely solely on numerical computations. Using systematic perturbation theory to large orders in the transmon regime, we derive precise analytic expressions of the transmon parameters. We apply our results to the case of parametrically-modulated transmons to study recently-implemented parametrically-activated entangling gates.

Equations

$${\hat{H}}_F=4E_C{\hat{N}}^2-E_J\cos \hat{\phi }.$$

$$\left[-4E_C\frac{{\partial }^2}{\partial {\phi }^2}-E_J\cos \phi \right]\psi (\phi )=E\psi (\phi ),$$

$$E_n=$$

$${\psi }_n(\phi )=$$

$$\displaystyle-i^{2n+1}\mathcal{M}_{S}\!\left(\frac{E_{n}}{E_{C}},-\frac{E_{J}}{2E_{C}},\frac{\varphi}{2}\right)\bigg{]},$$

$$\hat{\phi }$$

$$\xi =\sqrt{\frac{2E_C}{E_J}}.$$

$${\hat{H}}_F$$

$$\begin{gathered} {\hat{H}}^{(u)}={\omega }_h\sum _{v=0}^u\frac{(-1{)}^u}{2^{u-v+1}(u-v)!} \\ \times \sum _{w=-(v+1)}^{v+1}\frac{{\hat{a}}^{†(v+1+w)}}{(v+1+w)!}\frac{{\hat{a}}^{(v+1-w)}}{(v+1-w)!}, \end{gathered}$$

$$E_n$$

$$E_n^{(p)}=$$

$$|{\psi }_n^{(p)}⟩=$$

$$+$$

$$\omega \simeq$$

$$-$$

$$\eta \simeq$$

$${\hat{H}}_T$$

$$E_{J_{\mathrm{eff}}}$$

$${\varphi }_{\mathrm{eff}}$$

$${\hat{U}}_{\varphi }$$

$${\hat{H}}_T(t)={\hat{U}}_{\mathrm{eigen}}{\hat{U}}_{\varphi }\hat{H}{\hat{U}}_{\varphi }^{†}{\hat{U}}_{\mathrm{eigen}}^{†}$$

$$+i{\hat{U}}_{\mathrm{eigen}}{\dot{\hat{U}}}_{\varphi }{\hat{U}}_{\varphi }^{†}{\hat{U}}_{\mathrm{eigen}}^{†}+i{\dot{\hat{U}}}_{\mathrm{eigen}}{\hat{U}}_{\mathrm{eigen}}^{†}$$

$$=\sum _{n=0}^2{E}_n{\hat{\Pi }}_n-\frac{{\dot{\varphi }}_{\mathrm{eff}}(t)}{2\sqrt{\xi (t)}}[\lambda (t){\hat{\sigma }}_y+\sqrt{2}\Lambda (t){\hat{s}}_y]-\frac{\dot{\xi }(t)}{\xi (t)}\upsilon (t){\hat{S}}_y,$$

$${\hat{\sigma }}_{+}$$

$${\hat{s}}_{+}$$

$${\hat{S}}_{+}$$

$$\hat{N}$$

$$\lambda$$

$$\Lambda$$

$$\upsilon$$

$${\hat{H}}_C$$

$$g$$

$$g_{22}$$

$$\delta {\omega }_{1,2}$$

$$\delta {\eta }_{1,2}$$

$$\chi$$

$$\begin{gathered} {\hat{H}}_{\mathrm{M}}(t)=\sum _{n=0}^2\left[E_{F_n}{\hat{\Pi }}_{F_n}+E_{T_n}(t){\hat{\Pi }}_{T_n}\right] \\ +g(t)[{\lambda }_F{\hat{\sigma }}_{y_F}+\sqrt{2}{\Lambda }_F{\hat{s}}_{y_F}][{\lambda }_T(t){\hat{\sigma }}_{y_T}+\sqrt{2}{\Lambda }_T(t){\hat{s}}_{y_T}] \\ -\frac{{\dot{\varphi }}_{\mathrm{eff}}(t)}{2\sqrt{{\xi }_T(t)}}{\lambda }_T(t){\hat{\sigma }}_{y_T}-\frac{{\dot{\varphi }}_{\mathrm{eff}}(t)}{\sqrt{2{\xi }_T(t)}}{\Lambda }_T(t){\hat{s}}_{y_T}-\frac{{\dot{\xi }}_T(t)}{{\xi }_T(t)}\upsilon (t){\hat{S}}_{y_T}. \end{gathered}$$

Full text

I Introduction

Understanding the complex dynamics of a quantum machine requires the accurate modeling of the individual building blocks: interacting sets of qubits. Precise understanding is crucial to design, manipulate, optimize, and verify the machine. In the field of superconducting quantum computers, the transmon Koch et al. (2007); Barends et al. (2013) is currently widely used as qubit Kelly et al. (2015); Barends et al. (2016); Roushan et al. (2017); O’Malley et al. (2016); Kandala et al. (2017); Ristè et al. (2017); Takita et al. (2016); IBM Quantum Experience (2016); Langford et al. ; Versluis et al. (2017); Song et al. (2017); Salathé et al. (2015); Hacohen-Gourgy et al. (2016); Wendin (2017); Caldwell et al. (2017); Reagor et al. (2017); Otterbach et al. (2017) or more general-purpose quantum devices Naik et al. ; Ofek et al. (2016); Bretheau et al. (2015). Transmons, weakly non-linear oscillators derived from a Cooper-pair box, are made from a Josephson tunnel junction shunted by a capacitance. The transmon regime corresponds to a large Josephson energy compared to the charging energy — it is a compromise between a large anharmonicity and a weak sensitivity to charge noise. The coherence and gate times of transmons in quantum computing experiments have been steadily improving over the last several years, and transmons are now one of the leading candidates to an architecture that can meet the stringent requirements of fault-tolerant quantum computing Aliferis et al. (2006); Fowler et al. (2012).

Although analytical expressions for the behavior of non-interacting transmons are well understood, the accurate description for the behavior of interacting transmons requires the diagonalization of coupled systems (i.e., the charge basis description of the transmons with charge dipole interactions). Numerical diagonalization of these systems quickly becomes intractable because a large number of basis states are necessary to obtain high accuracy even for non-interacting transmons. A more efficient approach is to use analytical expressions of transmon energies and states. Exact diagonalization of the Cooper-pair box Hamiltonian is achieved with Mathieu functions Cottet (2002); Blais et al. (2007); Koch et al. (2007), but manipulating them can be cumbersome. An alternative is to consider controlled approximations, such as the approximate diagonalization via standard perturbation theory, which is widely used in quantum mechanics Messiah (1961). For transmons, the natural small parameter is the ratio of the charging energy of the Cooper-pair box to the Josephson energy of the junction, as this parameter is typically below $2\,\%$.

In this paper, we revisit transmon qubit theory by utilizing a systematic perturbation theory to model interacting transmons at sub-kHz accuracy with respect to numerical diagonalization. The resulting analytical expressions are particularly useful to estimate crosstalk in the dispersive regime. We then use these analytic expressions to model the parametric control of transmon qubits to realize two-qubit gates. In our proposal for entangling gates a fixed-frequency transmon is capacitively coupled to a tunable transmon and its flux bias is modulated so as to compensate for the detuning between the two two-qubit transitions of interest. Our gates are thus similar to other proposals for parametric gates Bertet et al. (2006); Niskanen et al. (2006, 2007); Beaudoin et al. (2012); Strand et al. (2013); Royer et al. (2017); Roy et al. (2017); McKay et al. (2016); Naik et al. , but distinct in the sense that our proposal directly modulates the qubit used for computation instead of using couplers to mediate the interaction. Our theory has been already used to predict parametric iSWAP and controlled Z gates that have been successfully realized on 2-qubit Caldwell et al. (2017), 8-qubit Reagor et al. (2017), and 19-qubit processors Otterbach et al. (2017).

This paper is organized as follows. We start by presenting the perturbation theory for a single transmon qubit in Sec. II. We then consider the case of tunable transmons in Sec. III. We treat the capacitive coupling of transmons in Sec. IV. We use the results to study the physics of parametrically-activated entangling gates in Sec. V.

II Fixed-frequency transmons

The circuit of a fixed-frequency transmon consists of a Josephson junction shunted by a capacitance, as depicted in Fig. 1 (left), and is governed by the Hamiltonian,

[TABLE]

The two conjugate quantum variables here are the Cooper pair number operator $\hat{N}$ and the superconducting phase difference $\hat{\varphi}$ satisfying the commutation rule $[\hat{\varphi},\hat{N}]=i$. The Schrödinger equation for the transmon in the phase representation,

[TABLE]

can be solved exactly in terms of Mathieu functions Cottet (2002); Blais et al. (2007); Koch et al. (2007),

[TABLE]

with $\mathcal{M}_{A}$ the Mathieu characteristic value, $\mathcal{M}_{C}$ the even Mathieu function, $\mathcal{M}_{S}$ the odd Mathieu function, and $\mu_{n}=(-1)^{n+1}n+[n\ \mathrm{mod}\ 2]$ the indexes for the eigenenergies. The structure of the solution indicates that transmons are described with the dimensionless parameter $E_{J}/(2E_{C})$ and the characteristic energy $E_{C}$. The transmon wavefunction in Eq. (4) is expressed in the phase representation. Going to the charge representation requires Fourier transforming the Mathieu functions, a calculation that turns out to be rather cumbersome.

The commutation relation between $\hat{\varphi}$ and $\hat{N}$ allows us to express these conjugate variables as the two quadratures of a bosonic field $\hat{a}$ (characterized by $[\hat{a},\hat{a}^{\dagger}]=1$),

[TABLE]

The positive real number $\xi$ is related to the zero point fluctuations, $\sqrt{\langle\hat{\varphi}^{2}\rangle}=\sqrt{2\xi}$ and $\sqrt{\langle\hat{N}^{2}\rangle}=1/\sqrt{2\xi}$. We then express Hamiltonian Eq. (1) in terms of $\hat{a},\hat{a}^{\dagger}$ and diagonalize the quadratic part; It is characterized by the plasma frequency $\omega_{h}=\sqrt{8E_{C}E_{J}}$ and solved by assigning the dimensionless parameter $\xi$ to

[TABLE]

In what follows, $\xi$ will be the small parameter of the perturbation theory (typically, $\xi<0.2$). The transmon Hamiltonian is then expressed as a function of the bosonic field, normal ordered and written as a Taylor series in $\xi$,

[TABLE]

with $\hat{H}^{(0)}=\omega_{h}\,\hat{a}^{\dagger}\hat{a}$ and, for $u\geq 1$,

[TABLE]

expressed in units of $\omega_{h}$.

The eigenenergies $E_{n}$ and eigenstates $|\psi_{n}\rangle$ are then obtained using perturbation theory Messiah (1961) in $\xi$,

[TABLE]

by solving the Schrödinger equation at each order in $\xi$. The unperturbed system is a harmonic oscillator of frequency $\omega_{h}$, the energies are $E_{n}^{(0)}=n\omega_{h}$ and the corresponding eigenstates are the Fock states $|\psi_{n}^{(0)}\rangle=|n\rangle$. The eigenenergies and eigenstates at order $\xi^{p}$, for $p\geq 1$, are obtained by recurrence,

[TABLE]

To compute the eigenenergies and eigenstates of level $n$ at order $p$, Fock states $|0\rangle\to|n+4p\rangle$ are required because terms such as $\hat{a}^{{\dagger}2(q+1)}$ and $\hat{a}^{2(q+1)}$ are involved. In particular at each new order the Hilbert space is extended by the action of $\hat{a}^{{\dagger}4}$ and $\hat{a}^{4}$. The eigenstates $|\psi_{n}\rangle$ derived by recurrence are not normalized. The normalized states $|\Psi_{n}\rangle$ are easily obtained via $|\Psi_{n}\rangle=|\psi_{n}\rangle/\sqrt{\langle\psi_{n}|\psi_{n}\rangle}$. The diagonalization transformation is represented by the operator $\hat{U}_{\mathrm{eigen}}$ that transforms the $n+4p$ Fock states into the $n$ eigenstates.

The expression of the diagonalization operator for the first 5 transmon eigenstates at $5^{\mathrm{th}}$ order in $\xi$ is provided in Appendix A. In the following we consider transmons as three-level systems, characterized by their frequency $\omega=E_{1}-E_{0}$ and anharmonicity $\eta=(E_{1}-E_{0})-(E_{2}-E_{1})$, positive by definition. At $5^{\mathrm{th}}$ order, the transmon frequency and anharmonicity read

[TABLE]

The first five orders are identical to the available asymptotic expansion of Mathieu functions for large arguments Abramowitz and Stegun (1964). Higher order expansions are easily obtained through these recursive expressions, and as an example, we report $25^{\mathrm{th}}$ order expansions in Appendix B. The accuracy of the perturbation theory with respect to the numerical diagonalization of the Hamiltonian of a transmon qubit is plotted in Fig. 2.

III Frequency-tunable transmons

The transmon frequency can be tuned with an external magnetic field $\phi_{\mathrm{ext}}$ by replacing the Josephson junction with a SQUID, see Fig. 1 (right). The tunable-transmon Hamiltonian is then that of a split Cooper-pair box,

[TABLE]

where $E_{J_{1}}$ and $E_{J_{2}}$ are the Josephson energies of the SQUID loop. The Hamiltonian can be recast as an effective single-junction transmon,

[TABLE]

with the flux-dependent effective Josephson energy and offset phase,

[TABLE]

The expression of the effective Josephson energy reveals “sweet spots” for flux noise, points of operation that are insensitive at first order to fluctuations of flux bias. Sweet spots are located at the extrema of the energy spectrum, e.g. at $\phi_{\mathrm{ext}}=0,\pi$. The phase is localized around the offset phase $\phi_{\mathrm{eff}}$; it can be removed with the displacement operator

[TABLE]

giving the same form as the fixed-frequency Hamiltonian Eq. (1), $\hat{H}_{T}=4E_{C}\hat{N}^{2}-E_{J_{\mathrm{eff}}}\cos\hat{\varphi}$. The charge operator is invariant under this unitary transformation — $U_{\phi}$ will not affect, e.g., the capacitive coupling interaction. The perturbation theory developed for a fixed-frequency transmon can be applied to the effective Hamiltonian with the small parameter $\xi=\sqrt{2E_{C}/E_{J_{\mathrm{eff}}}}$. The accuracy of the perturbation theory with respect to the numerical diagonalization of the Hamiltonian of a tunable transmon qubit is plotted in Fig. 3.

For time-dependent flux biases, $\phi_{\mathrm{ext}}(t)$, the unitary transformation for phase displacement $\hat{U}_{\phi}(t)$ and diagonalization $\hat{U}_{\mathrm{eigen}}$ generates nonadiabatic terms,

[TABLE]

where we note $\hat{\Pi}_{n}=|\Psi_{n}\rangle\langle\Psi_{n}|$ the projector on transmon eigenstate $|\Psi_{n}\rangle$. The nonadiabatic Hamiltonian acts as a drive between the different transmon levels. The ladder operators between the three transitions are

[TABLE]

The parameters $\lambda$, $\Lambda$ are the weights of the charge number operator in the three-level transmon eigenbasis,

[TABLE]

and $\upsilon$ comes from $\dot{\hat{U}}_{\mathrm{eigen}}\hat{U}_{\mathrm{eigen}}^{\dagger}$. Keeping the first five terms in the expansion, we get,

[TABLE]

Expressions at higher orders are given in Appendix B.

IV Capacitively-coupled transmons

The capacitive coupling of two Cooper-pair boxes, as depicted in Fig. 1, generates a charge-charge interaction through the coupling capacitance, $g_{C}\hat{N}_{1}\hat{N}_{2}$. From Fig. 1, $g_{C}=\frac{(2e)^{2}C}{C_{F}C_{T}}$. To treat the capacitive coupling in the transmon eigenbasis, we use Eq. (24) to obtain the coupling Hamiltonian,

[TABLE]

and define the coupling strengths,

[TABLE]

For detunings much larger than the coupling, $|\omega_{1}-\omega_{2}|\gg g$, the transverse coupling gives rise to state-dependent frequency shifts. At lowest order in the small parameter $g/|\omega_{1}-\omega_{2}|$, the frequencies and anharmonicities are modified as follows, $\omega_{1,2}\to\omega_{1,2}+\delta\omega_{1,2}$ and $\eta_{1,2}\to\eta_{1,2}+\delta\eta_{1,2}$ with

[TABLE]

The interaction in the dispersive regime is of the form $\hat{H}_{\mathrm{disp}}=\chi|11\rangle\langle 11|$ with the dispersive shift O’Malley et al. (2015),

[TABLE]

The dispersive parameters are compared to the numerical diagonalization of the coupled Hamiltonian Eq. (28) in Fig. 4.

V Parametrically-activated entangling gates

We consider two capacitively coupled transmon qubits, the first at fixed frequency and the second tunable. The flux bias pulse is modulated; it renders the Hamiltonian time dependent via $E_{J_{\mathrm{eff}}}(t)$ and $\phi_{\mathrm{eff}}(t)$. We note the parameters $\xi_{F}=\sqrt{2E_{C_{F}}/E_{J_{F}}}$ and $\xi_{T}(t)=\sqrt{2E_{C_{T}}/E_{J_{\mathrm{eff}}}(t)}$. The coupling in the quadrature basis, $g(t)=g_{C}/(4\sqrt{\xi_{F}\xi_{T}(t)})$, is expressed in terms of the capacitive coupling $g_{C}$. The system Hamiltonian in the transmon basis is then

[TABLE]

We specify the modulation of the flux bias pulse,

[TABLE]

which oscillates around the parking flux $\overline{\phi}_{p}$ at the modulation frequency $\omega_{p}$ and amplitude $\widetilde{\phi}_{p}$. The Hamiltonian is time dependent via $\xi_{T}(t)$, therefore via $\cos[\phi_{\mathrm{ext}}(t)]$; The general Fourier series of such time-dependent parameters, e.g. the frequency $\omega_{T}(t)$, reads

[TABLE]

The Fourier coefficients are defined as $\bm{\omega}_{k}=\frac{\omega_{p}}{\pi(1+\delta_{k,0})}\int_{0}^{\frac{2\pi}{\omega_{p}}}\mathrm{d}t\,\cos[k(\omega_{p}t+\theta_{p})]\omega_{T}(t)$. An analytical expression of the Fourier components is provided in Appendix C. At the flux sweet spots, where $\cos\overline{\phi}_{p}=0$ like at a maximum $\overline{\phi}_{p}=0$ or a minimum $\overline{\phi}_{p}=\pi$ of the energy spectrum, the odd Fourier coefficients vanish, $\bm{\omega}_{2k+1}=0$. The leading behavior at small modulation amplitudes is then $\omega_{T}(t)\approx\bm{\omega}_{0}+\bm{\omega}_{2}\cos[2(\omega_{p}t+\theta_{p})]$: a shifted frequency $\bm{\omega}_{0}$ corresponding to the averaged value of $\omega_{T}(t)$ and an oscillation of amplitude $\bm{\omega}_{2}$ at $2\omega_{p}$. This upconversion from the flux modulation at $\omega_{p}$ to the parameter modulation at $2\omega_{p}$ comes from the shape of the spectrum around these parking points. At an extremum, the slope vanishes and periodic excursions on the curvature doubles the frequency. In the following, we consider parking at a flux sweet spot and use the notation $\overline{\omega}\equiv\bm{\omega}_{0}$, $\widetilde{\omega}\equiv\bm{\omega}_{2}$. The time evolution of the frequency is plotted in Fig. 5 (a) as a function of time over a period of modulation $2\pi/\omega_{p}$ for a large modulation amplitude, $\widetilde{\phi}_{p}=2\pi$. Starting from $\phi_{\mathrm{ext}}=0$, the flux goes through the minimum at $\phi_{\mathrm{ext}}=\pi$ and bounces off at the maximum at $\phi_{\mathrm{ext}}=2\pi$, then comes back to the starting point before passing through the other minimum at $\phi_{\mathrm{ext}}=-\pi$ and reaching the maximum at $\phi_{\mathrm{ext}}=-2\pi$. The temporal evolution of the anharmonicity is plotted in Fig. 5 (b). The Fourier series with the 50 first harmonics is an accurate description of the frequency and anharmonicity for such large modulation amplitudes [see Fig. 5 (c)].

To highlight the parametrically-activated coupling, we go into the interaction picture with the unitary,

[TABLE]

To proceed, we note the energy difference $\omega_{F,T_{ij}}=E_{F,T_{j}}-E_{F,T_{i}}$ and the time-integral $\varpi_{T_{ij}}(t)=\int_{0}^{t}\mathrm{d}t^{\prime}\,\omega_{T_{ij}}(t^{\prime})$. The result Eq. (61) is then used to find the Fourier series of the Hamiltonian in the interaction picture,

[TABLE]

with the notation $|FT\rangle$ and where the frequency difference $\Delta$ and sum $\Sigma$ are,

[TABLE]

Each line of the Hamiltonian Eq. (37) is a coupling between a pair of two-qubit states that can be brought into resonance with the right modulation frequency $\omega_{p}$. The rate of the corresponding Rabi oscillations is set by the effective coupling strength, and hence the modulation amplitude. The capacitive coupling thus gives access to a large variety of two-qubit gates: iSWAP with $|10\rangle\langle 01|$ at $\omega_{p}=\frac{1}{2}|\Delta|$; controlled Z with $|11\rangle\langle 20|$ (CZ20) at $\omega_{p}=\frac{1}{2}|\Delta-\eta_{F}|$ and $|11\rangle\langle 02|$ (CZ02) at $\omega_{p}=\frac{1}{2}|\Delta+\overline{\eta}_{T}|$; Bell-Rabi with $|00\rangle\langle 11|$ at $\omega_{p}=\frac{1}{2}\Sigma$. The flux modulation then completely controls the activation and rate of the gates. The effective couplings read,

[TABLE]

with the leading term equal to $g_{ij}^{(n)}\simeq\bar{g}_{ij}\mathrm{J}_{n}\!\left(\frac{\bm{[\omega_{j\textbf{-}1,j}]}_{2}}{2\omega_{p}}\right)$, valid at small modulation amplitudes. In Eq. (39) we have taken into account only the averaged value of the couplings, $\bar{g}$, since the modulation amplitude of the couplings, $\tilde{g}$, is typically small. The leading correction to the above expression for $g_{ij}^{(n)}$ reads $-\frac{1}{2}\widetilde{g}_{ij}\{\mathrm{J}_{n-1}[\widetilde{\omega}_{T_{j-1,j}}/(2\omega_{p})]+\mathrm{J}_{n+1}[\widetilde{\omega}_{T_{j-1,j}}/(2\omega_{p})]\}$. The time-independent parameters $\bar{g}$ and $\tilde{g}$ are obtained from the couplings

[TABLE]

The phases are $\beta_{ij_{n}}=2n\theta_{p}-\sum_{k=1}^{\infty}\frac{{\bm{[\omega_{ij}]}}_{2k}}{2k\omega_{p}}\sin(2k\theta_{p})$. The effective drives are given in Appendix D. For modulation frequencies well below qubit frequencies and close to half the frequency detuning $\Delta$, a rotating wave approximation allows us to consider only the first three lines of Eq. (37).

The modulation frequencies that activate the entangling gates iSWAP, CZ02, and CZ20 are plotted in Fig. 6 (a) as a function of the modulation amplitude. The corresponding renormalization coefficient of the coupling is plotted in Fig. 6 (b), it corresponds to $(g_{11}^{(1)}/\bar{g}_{11},g_{12}^{(1)}/\bar{g}_{12},g_{21}^{(1)}/\bar{g}_{21})$ for (iSWAP,CZ02,CZ20), respectively. Local maxima of the renormalization coefficient achieve $0.62,0.73,0.41$ in this example, they constitute the optimal points of operation for fast gates (vertical lines).

We have focused our study so far on the coherent dynamics of transmons under flux modulation. Modulating the flux also changes the dissipative properties of tunable transmons, in particular dephasing due to flux noise Didier et al. (2015). Indeed, even if the transmon is parked at a flux sweet spot where the qubit is flux-noise insensitive at leading order, when modulated the flux bias explores regions of higher dephasing rate thereby reducing the effective dephasing time Caldwell et al. (2017). The physics of dephasing under flux modulation is the subject of ongoing research.

VI Conclusion and perspectives

We have developed an accurate analytical model for coupled transmons in the presence of flux pulse modulation. Our approach provides efficient and precise simulations of large-scale transmon-based quantum processors, which would be otherwise intractable with numerical diagonalization. The tools developed in this work are useful for designing large-scale processors, optimizing design parameters and predicting the performances of quantum operations. This model has already been successfully used in recent experiments on parametrically-activated two-qubit gates Caldwell et al. (2017); Reagor et al. (2017); Otterbach et al. (2017).

Moreover, our approach can be straightforwardly extended to the calculation of more than three transmon states to have more accurate expressions of frequency shifts. This is particularly important for finding the optimal regimes of operation of the controlled Z and iSWAP gates.

Developing an accurate model for dissipation in superconducting qubits is a natural extension of this work. The Keldysh formalism of Green’s functions is a powerful framework particularly well suited for nonlinear open quantum systems Kamenev (2011); Lemonde et al. (2013); Müller and Stace (2017). The perturbative expansion of the transmon parameters can then be calculated in presence of a dissipative bath, allowing to take into account the relaxation and dephasing rates of the quantum machine in its optimization.

Acknowledgments

We acknowledge fruitful daily collaborations with the experimental teams of Rigetti Computing.

Appendix A Diagonalization operator

We provide the explicite expression of the diagonalization operator $\hat{U}_{\mathrm{eigen}}$ at $5^{\mathrm{th}}$ order in $\xi$ of the first 5 transmon eigenstates from the first 25 Fock states.

[TABLE]

with (we write the conjugate operators for convenience),

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Appendix B Transmon parameters at $25^{\mathrm{th}}$ order in $\xi$

Transmon frequency.

[TABLE]

Transmon anharmonicity.

[TABLE]

Weight of the charge operator $\hat{N}$ on $\hat{\sigma}_{y}$.

[TABLE]

Weight of the charge operator $\hat{N}$ on $\hat{s}_{y}$.

[TABLE]

Weight of the nonadiabatic term.

[TABLE]

Appendix C Fourier series

A parameter $f(t)$ that depends on time via $\xi_{T}(t)$ can be written as $f(t)=F[\Xi(\cos[\phi_{\mathrm{ext}}(t)])]$ where $\Xi=\xi_{T}^{-4}$ is linear in $\cos[\phi_{\mathrm{ext}}(t)]$. The external flux is modulated at the frequency $\omega_{p}$, $\phi_{\mathrm{ext}}(t)=\overline{\phi}_{p}+\widetilde{\phi}_{p}\cos(\omega_{p}t+\theta_{p})$. We use the identity $e^{iy\sin x}=\sum_{n\in\mathbb{Z}}\mathrm{J}_{n}(y)e^{inx}$ to calculate the general Fourier series of $f$ in harmonics $\bm{f}_{k}$ of the modulation frequency $\omega_{p}$,

[TABLE]

with $\delta_{x,y}$ the Kronecker delta function, $\mathrm{fl}$ the floor function, $\overline{\Xi}=(E_{J_{1}}^{2}+E_{J_{2}}^{2})/(4E_{C}^{2})$ and $\widetilde{\Xi}=E_{J_{1}}E_{J_{2}}/(2E_{C}^{2})$. At flux sweet spots, $\overline{\phi}_{p}=m\pi$, $m\in\mathbb{Z}$, the odd Fourier coefficients vanish, $\mathbf{f}_{2k+1}=0$.

To further simplify the Fourier coefficients, we use the expression of $f$ obtained from perturbation theory $f=\sum_{p\in\mathbb{Z}}f^{(p)}\xi^{p}$. The sum on $p$ starts at $p=-1$ for $\omega$ because $\omega_{h}=4E_{C}/\xi$, at $p=0$ for $\eta$. The substitution yields,

[TABLE]

with $\mathcal{X}=\widetilde{\Xi}/\overline{\Xi}$, $\bar{\xi}=\overline{\Xi}^{-\frac{1}{4}}$, ${}_{2}\mathrm{F}_{1}$ the hypergeometric function, R_{n,p}=\left\{\begin{array}[]{ll}0&\mathrm{if}\ p=0\ \mathrm{and}\ n>0\\ \frac{\Gamma(n+\frac{p}{4})}{\Gamma(\frac{p}{4})}&\mathrm{else}\end{array}\right. and $\Gamma$ the gamma function.

We then calculate the Fourier series of $e^{i\int_{0}^{t}\mathrm{d}t^{\prime}f(t^{\prime})}$,

[TABLE]

with the phase $\displaystyle\Theta_{p}=\sum_{k=1}^{\infty}\frac{\bm{f}_{k}}{k\omega_{p}}\sin(k\theta_{p})$.

Appendix D Effective drives

The expression of the effective drives in Eq. (37) is

[TABLE]

with

[TABLE]