Quantum Simulation of Topological Zero Modes on a 41-Qubit Superconducting Processor

Quantum simulation of different exotic topological phases of quantum matter on a noisy intermediate-scale quantum (NISQ) processor is attracting growing interest. Here, we develop a one-dimensional 43-qubit superconducting quantum processor, named Chuang-tzu, to simulate and characterize emergent topological states. By engineering diagonal Aubry-André-Harper (AAH) models, we experimentally demonstrate the Hofstadter butterfly energy spectrum. Using Floquet engineering, we verify the existence of the topological zero modes in the commensurate off-diagonal AAH models, which have never been experimentally realized before. Remarkably, the qubit number over 40 in our quantum processor is large enough to capture the substantial topological features of a quantum system from its complex band structure, including Dirac points, the energy gap’s closing, the difference between even and odd number of sites, and the distinction between edge and bulk states. Our results establish a versatile hybrid quantum simulation approach to exploring quantum topological systems in the NISQ era.

Figure 1

Device and pulse sequences. (a) Optical micrograph of the 43-qubit quantum chip. (b) Diagonal AAH model simulated by periodically tuning the qubit’s frequency and the pulse sequence for its band structure spectroscopy. (c) Off-diagonal AAH model engineered by Floquet engineering qubit’s frequency and the pulse sequence for its band structure spectroscopy.

Figure 2

Hofstadter butterfly energy spectrum. By engineering various instances of AAH models, the energy spectrum of the Bloch electrons in perpendicular magnetic fields can be measured using band structure spectroscopy [11, 12]. Here we use $N=41$ qubits to simulate the quasiperiodic lattice. (a) Experimentally measured squared FT magnitudes $|{\tilde{\chi }}_j{|}^2$ when choosing a target qubit $Q_j$. (b)–(d) Experimental data of $I_{b_v}\equiv {\sum }_j|{\tilde{\chi }}_j{|}^2$ (b), the summation of the squared FT magnitudes, which is compared with the numerical data by simulating the dynamics of the system (c), and the theoretical prediction (d).

Figure 3

Experimental characterization of the topological zero-energy edge modes in commensurate off-diagonal AAH models for $\pi$ flux ($b_{\lambda }=1/2$). Band structure spectroscopy of off-diagonal AAH models with even number $N=40$ (a) and odd number $N=41$ (b) of sites, which are compared with the theoretical projected band structures (dashed curves). Here $u/(2\pi )=4.78\mathrm{MHz}$ and $\lambda =0.4$. Normalized squared FT magnitudes $|{\tilde{\chi }}_j{|}_{\mathrm{n}.\mathrm{m}.}^2$ when choosing the leftmost qubit ${\mathrm{Q}}_1$ (c) and the rightmost qubit ${\mathrm{Q}}_{40}$ (d) as target qubits with $N=40$. $|{\tilde{\chi }}_j{|}_{\mathrm{n}.\mathrm{m}.}^2$ for boundary qubits ${\mathrm{Q}}_1$ (e) and ${\mathrm{Q}}_{41}$ (f) as target qubits with $N=41$. (g)–(j) Time evolution of the excitation probability $P_j$ during the QWs of a single excitation initially prepared at the boundary qubits ($Q_1$ or $Q_{40}$) for ${\phi }_{\lambda }=0$ (g),(h) and ${\phi }_{\lambda }=\pi$ (i),(j) with $N=40$. (k)–(n) Time evolution of $P_j$ during the QWs of a single excitation initially placed at the boundary qubits ($Q_1$ or $Q_{41}$) for ${\phi }_{\lambda }=0$ (k),(l) and ${\phi }_{\lambda }=\pi$ (m),(n) with $N=41$.

Figure 4

Band structure spectroscopy of generic commensurate off-diagonal AAH models with $N=40$ for $b_{\lambda }=1/4$. (a) Experimental $I_{{\phi }_{\lambda }}$ for $\lambda =\sqrt{2}$ and $u/(2\pi )=2.77\mathrm{MHz}$. The gap between two central bands closes, and no topological edge states between these two bands are observed. (b) Experimental $I_{{\phi }_{\lambda }}$ for $\lambda =1$ and $u/(2\pi )=3.35\mathrm{MHz}$. The two central bands are clearly observed gapped without edge modes in the regimes ${\phi }_{\lambda }\in (-\pi /4,\pi /4)$ and $(3\pi /4,5\pi /4)$. (c) Normalized FT magnitudes of two boundary qubits $(|{\tilde{\chi }}_1{|}^2+|{\tilde{\chi }}_{40}{|}^2{)}_{\mathrm{n}.\mathrm{m}.}$, compared with the theoretical projected band structures (dashed curves). The topologically nontrivial zero-energy modes are observed between two central bands. (d) Four bulk qubits $({\sum }_{j=13,15,26,32}|{\tilde{\chi }}_j{|}^2{)}_{\mathrm{n}.\mathrm{m}.}$, compared to the theoretical projected band structures (dashed curves). Four band crossing points are observed near ${\phi }_{\lambda }=\pi /4$, $3\pi /4$, $5\pi /4$, and $7\pi /4$.