Hybrid Analog-Digital Simulation of the Abelian Higgs model

Abstract

To investigate gauge theories with near-term quantum computers warrants exploration of nontraditional quantum simulators to find resource-efficient simulation protocols and ultimately access exotic features of different field theories, including unexplored regimes of the QCD phase diagram. In this work, using superconducting transmon qutrit processors, we formulate and implement a pulse-based, three-level, hybrid analog-digital simulation protocol of the (1+1) dimensional Abelian Higgs model (AHM) on two sites. Alongside this approach, we experimentally realize a gate-based implementation of the same model. Using the natural mapping of the three-level truncation of the transmon Hilbert space to the spin-1 truncated AHM, we observe real time dynamics of AHM field observables, which are analogous to electric field operators, with both protocols. For the analog-digital protocol, we engineer a Floquet simulation with a combination of local analog drives, driven modification of the natural interaction Hamiltonian of the two transmons, and dynamical decoupling pulses. For the digital protocol, we use a state-of-the-art qutrit processor to implement a Trotterized simulation of the model incorporating advanced error mitigation techniques. We further discuss the scalability of the two approaches, and their potential to be extended to the simulation of other model Hamiltonians. Our experiments demonstrate a viable platform for future studies of spin-1 and SU(3) based gauge theory models on current and near-term transmon qutrit processors.

Figures (14)

• Figure 1: (a) Lattice field theory representation of the Abelian Higgs model (AHM) in (1+1) dimensions. Vertices represent matter fields and edges represent gauge links in the discretized spacetime. In this work we consider a time-slice that is two plaquettes wide. (b) In the appropriate limits and approximations (see Ref. Bazavov:2015kka), the (1+1)D AHM can be mapped to a spin-1 chain, where the spin states correspond to the plaquette quantum numbers. We illustrate the two spin-1 particles we simulate in this experiment along with the shape of their interaction Hamiltonian $H_\mathrm{AHM,int}\propto L_z\!\otimes\! L_z$. (c) Our experimental platform consists of a pair of transmon qutrits, whose native cross-Kerr interaction $H_\mathrm{ck}$ is diagonal but differs greatly in form from the target interaction. By applying microwave drives to the two transmons, we may approximate evolution under the model Hamiltonian in two distinct ways. (d) An analog-digital hybrid approach to engineering the evolution of the target Hamiltonian. Here we alternate between evolution under a Hamiltonian engineered with analog drives and approximately instantaneous local $\pi_{02}$ gates, which transpose the states $\ket{0}$ and $\ket{2}$ on each transmon individually. (e) The fully digital approach. Here a sequence of digital gates drawn from a universal set is repeated to effectively Trotterize the time evolution. Definitions of the gates are given in Sec. \ref{['sec:gates']}. For both protocols, a sketch of the trajectory of the state is shown. In the analog-digital simulation, the evolution can deviate substantially from the ideal trajectory (dashed line) but returns after a full Floquet cycle up to the error in the Magnus expansion. In the digital simulation, the evolution is closer to the exact solution throughout the evolution, and approximates it up to Trotter error at the end of each Trotter step.
• Figure 2: Single qutrit analog experiment. (a) Protocol for performing a single-qutrit analog simulation of the Abelian Higgs model. Drives on both transitions of the qutrit follow the same envelope $\Omega_{01}(t)=\lambda\Omega_{12}(t)=\Omega(t)=\Omega_0 A_T(t)$ and have equal and opposite detunings $\Delta_1(t) = -\Delta_2(t) =\Delta(t)=\Delta_0 A_T(t)$. Each data point in \ref{['fig:analog_data_single']} is measured by setting a pulse duration $T$ and reading out qutrit eigenstate populations at the end of the pulse. The ramps are a fixed fraction $r=0.1$ of the total duration, producing a stretching of the pulse for increasing $T$. The dimensionless envelope function $A_T(t)$ is normalized to have an area equal to its duration $T$; see Appendix \ref{['app:rotating_frame']} for details. (b) Expectation values of field operators $L_z$ (top) and $L_z^2$ (bottom) for a single qutrit under analog evolution starting in initial transmon state $\ket{n=2}$, with $\kappa/2\pi=\chi/2\pi=1$, $\beta=0$. The scale frequency was chosen to be $t_s^{-1}=\qty{1}{\MHz}$ and converts between physical device time (upper axis) and dimensionless model time (lower axis). Dotted lines show exact Schrödinger equation evolution under the model Hamiltonian, solid lines are master equation simulations assuming the independently measured dephasing and relaxation times of Qutrit B in Table \ref{['tab:analog_coherence']}. Uncertainties are estimated from shot noise $\sigma\sim 1/\sqrt{N_\mathrm{shots}}$ where $N_\mathrm{shots}=3000$.
• Figure 3: Engineering the $L_z\!\otimes\! L_z$ interaction. (a) Experimentally measured interaction Hamiltonian terms from Eq. \ref{['eq:Hck_zzDecomp']} for the undriven (bare) two-transmon system. (b) Abstract circuit schematic of the interaction rate measurement. The JAZZ block consists of two periods of free evolution of duration $T$, with a $\pi$ pulse on both transmons between them garbow_bilinear_1982takita_experimental_2017. The subspaces of the $\pi$ and $\frac{\pi}{2}$ pulses are determined by which matrix element of the interaction Hamiltonian is being targeted, as is explained in depth in Appendices \ref{['app:interaction_rates']} and \ref{['app:calibration_sequences']}. (c)--(d) Interaction rates and measurement circuit schematic when averaging with a $\ket{0}\leftrightarrow\ket{2}$ swap. We observe that the rates for the odd-parity terms, $z_{12}$ and $z_{21}$ are effectively eliminated by this dynamical decoupling sequence, while the even-parity terms remain unchanged. The circuit now involves four periods of free evolution with duration $T/2$. (e)--(f) Interaction rates and circuit schematic with both averaging and Stark drives applied. Applying Stark drives at the appropriate frequency and amplitude nearly eliminates the driven $\tilde{z}_{22}$ term, leaving the modified $\tilde{z}_{11}$ as the only appreciable interaction rate in the Hamiltonian. See Appendix \ref{['app:calibration_sequences']} for details on the calibration of the Stark drives.
• Figure 4: Two qutrit analog-digital experiment. (a) Two-qutrit protocol. Evolution of the simulation is achieved by tiling evolution periods of a fixed duration $T_E=\qty{800}{\ns}$ interleaved with 60$\pi_{02}$ pulses on both transmons. Data points in \ref{['fig:analog_data_double']} are measured after $2N$ evolution periods. The $\pi_{02}$ pulses are performed by driving both $\ket{0}\leftrightarrow\ket{1}$ and $\ket{1}\leftrightarrow\ket{2}$ transitions on resonance simultaneously forney_multifrequency_2010champion_multi-frequency_2024. During each evolution period, in addition to the detuned drives described in Sec. \ref{['sec:single_qutrit_analog']}, a Stark tone $\Omega_S(t)$ is applied to each transmon, which modifies the entangling Hamiltonian $\tilde{H}_\mathrm{ck}$ such that the average interaction after two steps is approximately $\tilde{H}_\mathrm{ck}^\mathrm{dd}\approx\tilde{z}_{11}L_z \!\otimes\! L_z$. (b) Expectation values of $L_z$ (top) and $L_z^2$ (bottom) for two qutrits during the analog-digital protocol. Dotted lines show exact theory evolution, solid lines are master equation simulations assuming independently measured dephasing and relaxation times. The scale frequency was fixed so that $t_s^{-1}=\tilde{z}_{11}/2\pi\approx\qty{110}{\kHz}$. The initial transmon state was $\ket{\psi_\mathrm{init}}_{\!AB}=\ket{21}$.
• Figure 5: Exact diagonalization (Exact), device data with randomized compilation (Device (RC) ) and error-mitigated device results (Purified) of field operators $\langle L_{z}\rangle$ and $\langle L_{z}^{2} \rangle$ at (a,b) $\kappa/2\pi=\chi/2\pi=1.0$ for the single qutrit experiment and (c,d) $\kappa/2\pi=\beta/2\pi=\chi/2\pi=1.0$ for the two-qutrit experiment. For a discussion on the impact of different types of error mitigation schemes see Appendix \ref{['app_digital_mitigation']}. The dynamics start from the initial state $\ket{\psi_\mathrm{init}}_{\!AB}=\ket{21}$.