Synthesizing Five-Body Interaction in a Superconducting Quantum Circuit

Synthesizing many-body interaction Hamiltonians is a central task in quantum simulation. However, it is challenging to synthesize Hamiltonians that have more than two spins in a single term. Here we synthesize $m$-body spin-exchange Hamiltonians with $m$ up to 5 in a superconducting quantum circuit by simultaneously exciting multiple independent qubits with time-energy correlated photons generated from a qudit. The dynamic evolution of the $m$-body interaction is governed by the Rabi oscillation between two $m$-spin states, in which the states of each spin are different. We demonstrate the scalability of our approach by comparing the influence of noises on the three-, four- and five-body interaction and building a many-body Mach-Zehnder interferometer which potentially has a Heisenberg-limit sensitivity. This study paves a way for quantum simulation involving many-body interaction Hamiltonians such as lattice gauge theories in quantum circuits.

Figure 1

(a) Device schematic and image illustrating the five frequency-tunable transmon circuits labeled from $Q_0$ to $Q_4$, with $Q_0$ being surrounded by $Q_1$ to $Q_4$. Each circuit $Q_j$ has its own flux bias line ${\mathrm{Z}}_j$ for fast frequency tuning, microwave line $X{Y}_j$ for SU(2) spin rotation, and readout resonator ${\mathrm{RR}}_j$ that couples to a common transmission line TL for dispersive readout of $Q_j$’s state. The dots in the image are bumps for the flip-chip process. Inset: the structure of the circuits in the dashed line area including $Q_0$ and $Q_1$. (b) Energy configurations of the qudit and four qubits for the five-body interaction. The lines denote the levels with distances in a vertical direction proportional to their energy differences. The double arrows denote the allowed transitions with the black ones denoting the major contributing paths.

Figure 2

(a) Sequence diagram for observing the dynamic evolution. After preparing the system to the initial state $|01111⟩$ by applying $X_{\pi }$ rotation (a $\pi$ rotation around $x$ axis) to $Q_j$ with $j=1–4$, we quickly tune the transition frequencies of all qubits to activate the five-body interaction. After a specific time $\tau$, the occupational probabilities of the system for different computational states are measured. (b) The experimentally measured occupational probabilities of $|c_1(\tau ){|}^2$ for $|01111⟩$ (blue dots) and $|c_2(\tau ){|}^2$ for $|40000⟩$ (red dots) for different interaction times $\tau$. Error bars represent statistical errors. Lines are the results obtained by numerical simulation, where five and three levels are considered for $Q_0$ and other qubits, respectively. (c) The effect of noises on the evolution under the $m$-body spin-exchange Hamiltonian. The low-frequency noises are simulated by random detunings between the two states in the interaction. Except for the added noises, the sequence diagrams are similar to the one in (a) but for different $m$’s. The populations of the two relevant states are measured as functions of time. The noises have a destructive effect on the coherent tunneling between the two spin configurations, which is demonstrated by the diminishing oscillation contrast when we increase $m$ from 3 to 5.

Figure 3

(a) Sequence diagram for detecting the magnetic field leveraging the $m$-body interaction, with $m=5$ here as an example. The magnetic field is synthesized by applying to each transmon circuit a square $Z$ pulse to offset its transition frequency between states $|0⟩$ and $|m-1⟩$ ($|1⟩$) for $Q_0$ (other qubits) by an amount of $-{\mathrm{\Delta }}_B$ (${\mathrm{\Delta }}_B$) for a time ${\tau }_B$. The dynamical phase accumulated during this process can be detected by sandwiching itself between two $m$-body interaction operations with a fixed time ${\tau }_{\mathrm{I}}\sim 2\pi /8{\lambda }_m$. (b) The experimentally measured occupational probabilities of $|c_1({\tau }_B){|}^2$ (blue dots) and $|c_2({\tau }_B){|}^2$ (red dots) for different time ${\tau }_B$. Lines are the fitting results. The fitted oscillation frequencies for $m=3$, 4, 5 are 14.9, 19.5, and 24.2 MHz respectively, agreeing well with the ratio of $3:4:5$.