Hybrid digital-analog protocols for simulating quantum multi-body interactions

TL;DR

A hardware-agnostic and scalable method that produces effective Hamiltonians with simultaneous non-commuting three- and four-body interactions that are generated non-perturbatively and without Trotter error -- capabilities not practically attainable on near-term hardware using purely digital or purely analog schemes.

Abstract

While quantum simulators promise to explore quantum many-body physics beyond classical computation, their capabilities are limited by the available native interactions in the hardware. On many platforms, accessible Hamiltonians are largely restricted to one- and two-body interactions, limiting access to multi-body Hamiltonians and to systems governed by simultaneous, non-commuting interaction terms that are central to condensed matter, quantum chemistry, and high-energy physics. We introduce and experimentally demonstrate a hybrid digital-analog protocol that overcomes these limitations by embedding analog evolution between shallow entangling-gate layers. This method produces effective Hamiltonians with simultaneous non-commuting three- and four-body interactions that are generated non-perturbatively and without Trotter error -- capabilities not practically attainable on near-term hardware using purely digital or purely analog schemes. We implement our scheme on a trapped-ion quantum processor and use it to realize a topological spin chain exhibiting prethermal strong zero modes persisting at high temperature, as well as models featuring three- and four-body interactions. Our hardware-agnostic and scalable method opens new routes to realizing complex many-body physics across quantum platforms.

Figures (12)

• Figure 1: Direct multi-body interactions beyond Trotterization Concept.a, Continuous analog evolution under native long-range couplings and local fields, sandwiched between two shallow digital gate layers, can generate concurrent, non-commuting multi-body interactions without Trotter discretization error. b, The resulting effective Hamiltonians are programmable, with tunable two-, three-, and four-body couplings, enabling simulation of diverse many-body models implemented without Trotterization. c, Example realizations include an open chain with a strong zero mode, a periodic topological loop, and a plaquette model with four-body interactions.
• Figure 2: Simulating topological properties of the cluster-field Hamiltonian.a, Adiabatic preparation of the cluster Hamiltonian ground state with open boundary conditions on a five-spin chain. The bulk fields $h_i$ ($i\neq1,L$) are ramped to zero while the bulk three-body couplings $g_i$ are ramped up; the edge fields $h_1,h_L$ are held constant. As a result, the bulk magnetization $|\Sigma_x|$ decreases, while the nonlocal string order parameter $O_s=\langle\hat{\sigma}_x^{(1)}\hat{\sigma}_y^{(2)}\Pi_{j=3}^{L-2}\hat{\sigma}_z^{(j+2)}\hat{\sigma}_y^{(L-1)}\hat{\sigma}_x^{(L)}\rangle$ between the string edges assumes a non-zero value, heralding the SPT phase. b, Dynamical transition from open to periodic boundary conditions by activating the three-body couplings $g_{1},\,g_{L}$ at the edges while simultaneously deactivating the edge fields $h_{1},\,h_{L}$. The left-right edge correlation $E_{1L}=\langle\hat{\sigma}_x^{(1)}\hat{\sigma}_x^{(L)}\rangle$ decays as edge-localized modes delocalize, while the boundary stabilizers $S_1=\braket{\hat{\sigma}_z^{(1)}\hat{\sigma}_x^{(2)}}$ and $S_L=\braket{\hat{\sigma}_x^{(L-1)}\hat{\sigma}_z^{(L)}}$ increase, consistent with uniform couplings around the ring at the final time. Throughout this transition $| O_s|$ stays near unity. The hybrid digital-analog protocol (\ref{['fig:Intro']}) enables continuous-time control of the effective Hamiltonian, free from Trotter error. Symbols show measured data, while solid lines show theoretical predictions; error bars denote $1\sigma$ binomial uncertainty.
• Figure 3: Signatures of strong zero modes.a, Schematic of the digital-analog protocol used to probe strong zero modes in a cluster-Ising Hamiltonian on a twelve-spin chain. The digital layer $\hat{D}_0$ prepares product states at varying energies to approximate the infinite-temperature ensemble, followed by a digital layer $\hat{D}_1$ and analog evolution under the transverse-field Ising Hamiltonian $\hat{H}_{\textrm{A}}=\sum_{ij}J_{ij}\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{x}^{(j)}+g\sum_{i}\hat{\sigma}_{z}^{(i)}$. A final digital layer $\hat{D}_2$ completes the sequence, enabling measurement of observables in rotated bases. XX gate between qubits $i$ and $j$ denotes the two-qubit operator $\exp(-i\tfrac{\pi}{4}\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{x}^{(j)})$. b, Effective Hamiltonian dynamics. Choosing $\hat{D}_2 \neq \hat{D}_1^\dagger$ implements evolution under the effective Hamiltonian $\hat{H}_{\textrm{eff}}$ in Eq. (\ref{['eq:H_cluster_Ising']}) while enabling efficient measurement of transformed observables $\langle \hat{V}^\dagger \hat{O} \hat{V} \rangle$ in place of $\langle \hat{O}\rangle$, where $\hat{V} = \hat{D}_2 \hat{D}_1$. c, Time evolution of the normalized edge correlator $\tilde{G}_{\textrm{left}}(t)$ at high temperature for varying interaction ratios $g/J$. Solid lines indicate ab-initio numerical predictions that include independently measured qubit dephasing. d, Long-time value of the unnormalized correlator associated with the stabilizer operator at site $j$ for $g/J=3.3$. For the ground state of the cluster Hamiltonian (blue diamonds), the correlator is near unity, consistent with topological protection of the ground state. For the high-temperature state (red circles), it is close to zero except at the edges, consistent with strong zero modes. See Methods for details. In panels c-d, symbols denote measured data, while lines show numerical simulations. Error bars denote the quadrature sum of the $1\sigma$ binomial uncertainties of each state and the $1\sigma$ standard error of the mean over all input states.
• Figure 4: Dynamics under a four-body interaction Hamiltonian.a, Effective Hamiltonian containing one-body (red arrows), two-body (purple bonds), and four-body (red-yellow rings) terms in a four-spin chain with periodic boundary conditions based on the digital-analog sequence in \ref{['fig:Four_body_protocol']}. b, Expectation values of individual Pauli strings, each obtained from single-qubit readout. c, Superpositions of multiple Pauli strings from single-qubit readout. The oscillations in c and d show coherent interference between contributions from different Hamiltonian terms (see Methods). Symbols represent measured data; solid lines show numerical simulations without free parameters. Error bars denote $1\sigma$ binomial uncertainties.
• Figure Extended Data Fig. 1: Cluster-Field Hamiltonian. a, An analog Hamiltonian composed of one-body terms is interleaved with digital layers containing two qubit gates between qubits $i$ and $j$ of the form $XX=\exp(-i\tfrac{\pi}{4}\hat{\sigma}_{x}^{(i)}\hat{\sigma}_{x}^{(j)})$ and $\overline{XX}=(XX)^{\dagger}$. Fluorescence imaging (left) identifies the middle five physical ions corresponding to each qubit/spin in a 15-ion chain. b, The resulting effective Hamiltonian contains one- and emergent three-body interactions not present in the original analog Hamiltonian.