Floquet Quantum Simulation with Superconducting Qubits

We propose a quantum algorithm for simulating spin models based on the periodic modulation of transmon qubits. Using the Floquet theory, we derive an effective time-averaged Hamiltonian, which is of the general $XYZ$ class, different from the isotropic $XY$ Hamiltonian typically realized by the physical setup. As an example, we provide a simple recipe to construct a transverse Ising Hamiltonian in the Floquet basis. For a 1D system, we demonstrate numerically the dynamical simulation of the transverse Ising Hamiltonian and quantum annealing to its ground state. We benchmark the Floquet approach with a digital simulation procedure and demonstrate that it is advantageous for limited resources and finite anharmonicity of the transmons. The described protocol represents a hardware-efficient quantum software and can serve as a simple yet reliable path towards configurable quantum simulators with currently existing superconducting chips.

Figure 1

Sketch of the system. A chain of superconducting transmon-type qubits coupled through isotropic $XY$ coupling $J$. Each qubit is subject to a periodically modulated effective magnetic field ${\mathbf{h}}_j(t)$.

Figure 2

Transverse Ising dynamics. (a) Total normalized magnetization of the $N=4$ chain $M_{\alpha }(t)=⟨\psi (t)|\sum _j^N{\sigma }_j^{\alpha }|\psi (t)⟩/N$. Solid curves show the ideal continuous evolution under the transverse Ising Hamiltonian. Bullets correspond to stroboscopic periods of the Floquet dynamics. We set $J<0$, $h^z/J=3/2$, and $\omega /|J|=50$ and run the simulation for a total of 20 stroboscopic periods. (b) Simulation infidelities. Blue diamonds identify the dynamical overlap of the Floquet evolution with the ideal transverse Ising evolution. The red bullets correspond to digital evolution with $N_{\mathrm{Tr}}=20$ Trotter steps.

Figure 3

Transverse Ising annealing. (a) Fidelity of the simulated state with respect to the ideal target state. The dark red line denotes annealing with the ideal transverse Ising Hamiltonian. The blue thin line corresponds to evolution under the time-dependent Hamiltonian, with blue dots showing state fidelities at accessible stroboscopic times. We assume $J_{\mathrm{sim}}<0$, $h^z/|J_{\mathrm{sim}}|=1$, $t_f=12.5|J_{\mathrm{sim}}{|}^{-1}$, and $\omega /|J|=20$. (b) Final time infidelities $1-F^T$ for ideal continuous, Floquet, and digital evolution shown for different numbers of qubits. Parameters are the same as in (a) with final simulation time $t_f=15|J_{\mathrm{sim}}{|}^{-1}$. The digital evolution corresponds to $N_{\mathrm{Tr}}=20$ Trotter steps. (c) Final infidelity $1-F^c$ of the Floquet simulation as a function of the modulation frequency for $N=4$, 5, 6. (d) Infidelity $1-F^c$ of the digital annealing as a function of the number of Trotter steps for $N=4$, 5, 6 qubits.

Figure 4

Accounting for finite anharmonicity. (a) Sketch of a realistic transmon chain with weakly anharmonic multilevel structure. In each circuit, we account for $\{|0{⟩}_j,|1{⟩}_j,|2{⟩}_j\}$ states. The infidelity of the simulation arises from microwave driving of the $|1{⟩}_j\leftrightarrow |2{⟩}_j$ transition and additional flip-flop coupling. (b) Optimal number of Trotter steps $N_{\mathrm{Tr}}^{(\mathrm{opt})}$ plotted for different values of the gate error. (c) Corresponding optimal infidelity as a function of the gate error. The results are shown for transverse Ising annealing with $t_f=15|J{|}^{-1}$ and $N=4$.

Figure 5

Imperfections. (a) Infidelity of Floquet transverse Ising annealing ($N=4$) with respect to the ideal target state calculated for a fixed anharmonicity and varying frequency. The point of low infidelity defines an optimal frequency for the simulation. (b) Final-state infidelity for transverse Ising annealing for $N=4$, $\omega ={\omega }^{\mathrm{opt}}$, and $t_f=15|J{|}^{-1}$. The anharmonicity $A$ spans the range $A/\omega =19–99$. Horizontal lines show optimized digital protocol infidelities for fixed single-gate error $\epsilon$, with the cutoff (large dots) determined by a minimal gate time $t_{\mathrm{gate}}\ge 35A^{-1}$.

Figure 6

Influence of decay. Final-state infidelity for annealing to the ground state of the transverse Ising model, shown as a function of qubit decay rate $\gamma$. Continuous annealing (magenta curves) and Floquet annealing (blue curves) are considered, with fidelity measured with respect to the ideal target state. The parameters are $N=4$, $\omega /|J|=20$, and $t_f=15|J_{\mathrm{sim}}{|}^{-1}$.

Figure 7

Influence of cross talk. Final-state infidelity for annealing to the ground state of the transverse Ising model as a function of the mean cross-talk coefficient ${\mu }_{\mathbf{c}}$. The calculation is performed for several sets of cross-coefficients drawn from the half-normal distribution with different ${\mu }_{\mathbf{c}}$. The error bar shows one standard deviation of the calculated infidelity of the sample; i.e., it represents the fluctuations expected between different devices. The curve is calculated using 20 sets of the cross-couplings. Other parameters are $\omega /|J|=20$, $t_f=15|J_{\mathrm{sim}}{|}^{-1}$, and $N=4$.

Figure 8

$XYZ$ annealing. (a) Time dependence of the infidelity for the Floquet evolution (blue line) and optimal digital annealing (red line). Here the infidelity is measured with respect to the instantaneous wave function of the continuous annealing. The digital evolution contains $N_{\mathrm{Tr}}=477$ Trotter steps, equal to half the number of stroboscopic periods. (b) Modulation frequency dependence of the final-state infidelity for annealing with $t_f=200|J{|}^{-1}$, measured with respect to an ideal target state. (c) Final infidelity of the digital quantum simulation of the $XYZ$ model as a function of the number of Trotter steps. $F$ is measured with respect to an ideal target state.

Figure 9

Superconducting qubit chain with isotropic $XY$ interaction $J$, a static effective magnetic field in the $z$ direction on the odd sublattice, $h_{\mathrm{odd}}^z$, and a fast time-dependent magnetic field acting on even sublattice sites, ${\mathbf{h}}_{\mathrm{even}}(t)$.

Figure 10

Digital simulation scheme from Ref. [17]. It relies on Trotterization of the isotropic $XY$ model dynamics, where additional single-qubit rotations at every second site effectively eliminate the $YY$ coupling.

Figure 11

Digital simulation scheme of the Heisenberg-type model from Ref. [17]. It relies on the sequential application of bare $XY$ couplings [$\propto {\alpha }_{xy}J({\sigma }_j^x{\sigma }_{j+1}^x+{\sigma }_j^y{\sigma }_{j+1}^y)$] and its $\pi /2$ rotated version corresponding to $XZ$ [$\propto {\alpha }_{xz}J({\sigma }_j^x{\sigma }_{j+1}^x+{\sigma }_j^z{\sigma }_{j+1}^z)$] and $YZ$ [$\propto {\alpha }_{yz}J({\sigma }_j^y{\sigma }_{j+1}^y+{\sigma }_j^z{\sigma }_{j+1}^z)$] interactions. Here ${\alpha }_{xy,xz,yz}$ correspond to dimensionless coefficients which allow tuning the model into the anisotropic $XYZ$ model. We note that the described sequence can be further optimized by combing single-qubit rotations to unity matrices and $z$ rotations. Moreover, the ordering of the terms can be changed to optimize the procedure.