Floquet Analysis of Frequency Collisions

Abstract

Implementation of high-fidelity gate operations on integrated-qubit systems is of vital importance for fault-tolerant quantum computation. Qubit frequency allocation is an essential part of improving control fidelity. A metric for qubit frequency allocation, frequency collision, has been proposed on simple systems of only a few qubits driven by a mono-modal microwave drive. However, frequency allocation for quantum processors for more advanced purposes, such as quantum error correction, needs further investigation. In this study, we propose a Floquet analysis of frequency collisions. The key to our proposed method is a reinterpretation of frequency collisions as an unintended degeneracy of Floquet states, which allows a collision analysis on more complex systems with many qubits driven by multi-modal microwave drives. Although the Floquet state is defined in an infinite-dimensional Hilbert space, we develop algorithms, based on operation perturbation theory, to truncate the Hilbert space down to the optimal computational complexity. In particular, we show that the computational complexity of the collision analysis for a sparse qubit lattice is linear with the number of qubits. Finally, we demonstrate our proposed method on Cross-Resonance based experimental protocols. We first study the Cross-Resonance gate in an isolated three-qubit system, where the effectiveness of our method is verified by comparing it with previous studies. We next consider the more complex problem of syndrome extraction in the heavy-hexagon code. Our proposed method advances our understanding of quantum control for quantum processors and contributes to their improved design and control.

Figures (12)

• Figure 1: Concept of our Floquet-based collision analysis. The left figure shows a general quantum control schematic. The qubits align in a planar lattice, and multiple microwaves are simultaneously irradiated. Some of the microwaves can be resonant with each other. Using Floquet theory and applying the Fourier transform (FT) to the Schrodinger equation corresponding to the left figure, we obtain a Floquet Hamiltonian and a corresponding energy level diagram shown on the right. The black bold and purple dotted lines represent Floquet states and their couplings, respectively. If the Floquet Hamiltonian satisfies the strongly dispersive condition in the operation basis described in \ref{['sec:floquet_collision']}, we can control our system sufficiently. On the other hand, if Floquet states are degenerate, as shown in the right figure, Floquet collision occurs and the control fidelity deteriorates significantly.
• Figure 2: (Top) A schematic for anti-crossing between Floquet states $\ket{\psi_{A,B}}$. The top figure shows how Floquet quasi-excitation energy changes while sweeping a system parameter. We can find that an anti-crossing occurs near a degenerate point of two Floquet states, where Floquet the quasi-excitation energy acquires an energy gap proportional to an effective coupling strength between the Floquet states. (Bottom) Dependence of collision angle $\theta_{\mathrm{AB}}$ between the Floquet states $\ket{\psi_{\mathrm{A},\mathrm{B}}}$ while sweeping system parameter. The collision angle becomes $\pi/2$ at the degenerate point and close to $0$ under a strong-dispersive condition.
• Figure 3: (a) Graphical interpretation of the $k$th-order perturbations acting on the Floquet states $\ket{\psi_{\mathrm{A},\mathrm{B}}}$. A loop of length $k$ around Floquet states $\ket{\psi_{\mathrm{A},\mathrm{B}}}$ and a walk of length $k$ between them on the Floquet Hamiltonian correspond to $k$th-order Floquet quasi-excitation energy shifts and effective coupling, respectively. To verify the $k$th-order Floquet collision, we compare the Floquet quasi-excitation energy detuning, accounting for up to $k$th-order energy shifts, to the $k$th-order effective coupling strength. Such loops and walks are contained in a region within distance $d\leq \lfloor{k/2\rfloor}$ of each state, such that we can use the Floquet subspace Hamiltonian corresponding to this region to verify the $k$th-order Floquet collision with finite computational cost. All Floquet collisions up to the $k$th-order caused on the Floquet state $\ket{\psi_{\mathrm{A}}}$ include only Floquet collisions with the Floquet states within distance $d\leq k$. Thus, all Floquet collisions caused on $\ket{\psi_{\mathrm{A}}}$ can be calculated with the Floquet subspace Hamiltonian corresponding to the region within distance $d\leq \lfloor{3k/2\rfloor}$ from $\ket{\psi_{\mathrm{A}}}$ in the Floquet Hamiltonian. (b) The circuit and the corresponding low-excitation Floquet energy level diagram of two isolated transmon qubits irradiated by independent microwave drives. Even though there is no qubit-qubit interaction, there is a walk of distance $2$ between states $\ket{0,0;0,0}$ and $\ket{1,1;-1,-1}$ mediated by $\ket{0,1;0,-1}$ and $\ket{1,0;-1,0}$ on the Floquet Hamiltonian. However, the effective couplings between them, mediated by two coupling walks, always cancel each other out.
• Figure 4: Floquet energy level diagram of the CR Hamiltonian (\ref{['eq:cr_hamiltonian']}). For simplicity, the CR drive frequency is set to the bare target qubit frequency $\omega_t$ . Also, the $\mathrm{BZ}$ index of the Floquet states is omitted. The gray regions represent the distance from the computational states $\qty{\ket{g\pm},\ket{e\pm}}$, respectively, with darker shades further away. Although the $\ket{x\pm}$ states appear to be degenerate for $x\in[g,e,f,h,i\cdots]$, they are in fact detuned by a second-order energy shift originating from the CR interaction, preserving the strong-dispersive condition.
• Figure 5: Numerical simulations of \ref{['collision_finder_wo_structure']} for a two-qubit CR system while sweeping the control-target detuning $\Delta_{ct}/2\pi$ from $-1$ to $1$ GHz. Qubits anharmonicity and qubit-qubit coupling are set to $-330$ MHz and $3.8$ MHz, respectively. The top panel shows the Floquet quasi-excitation energies in the Floquet subspace generated for searching up to the second-order Floquet collisions. As a function of detuning, certain Floquet quasi-excitation energies shift and anti-cross with each other. The middle and lower panels represent the maximum collision angles between the Floquet state pairs corresponding to first- and second-order Floquet collisions, respectively. The vertical lines and their labels represent the frequency collisions shown in \ref{['tab:collision_freq']}.