Trigonometric continuous-variable gates and hybrid quantum simulations

TL;DR

The paper introduces trigonometric continuous-variable gates as a complementary universality for hybrid qubit-qumode quantum computing, enabling Fourier-like representations of non-polynomial bosonic operators. It develops a deterministic ancilla-based construction to implement unitary and non-unitary gates of the form e^{-i t cos A} and e^{-i t sin A}, applicable to arbitrary Hermitian A, and demonstrates their utility by simulating the lattice sine-Gordon model. Through quantum imaginary-time evolution, real-time dynamics, and non-perturbative observables such as time-dependent vertex correlators and kink profiles, the work shows how these gates naturally access solitonic and topological features on NISQ-era hardware. The approach complements existing polynomial CV gate sets and paves the way for broader applications in condensed matter, quantum chemistry, and biological models, while highlighting hardware-compatibility and future extensions to higher dimensions and more complex field theories.

Abstract

Hybrid qubit-qumode quantum computing platforms provide a natural setting for simulating interacting bosonic quantum field theories. However, existing continuous-variable gate constructions rely predominantly on polynomial functions of canonical quadratures. In this work, we introduce a complementary universality paradigm based on trigonometric continuous-variable gates, which enable a Fourier-like representation of bosonic operators and are particularly well suited for periodic and non-perturbative interactions. We present a deterministic ancilla-based method for implementing unitary and non-unitary trigonometric gates whose arguments are arbitrary Hermitian functions of qumode quadratures. As a concrete application, we develop a hybrid qubit-qumode quantum simulation of the lattice sine-Gordon model. Using these gates, we prepare ground states via quantum imaginary-time evolution, simulate real-time dynamics, compute time-dependent vertex two-point correlation functions, and extract quantum kink profiles under topological boundary conditions. Our results demonstrate that trigonometric continuous-variable gates provide a physically natural framework for simulating interacting field theories on near-term hybrid quantum hardware, while establishing a parallel route to universality beyond polynomial gate constructions. We expect that the trigonometric gates introduced here to find broader applications, including quantum simulations of condensed matter systems, quantum chemistry, and biological models.

Figures (8)

• Figure 1: Circuit that implements the exponentiation of any Pauli string $P = P^\dagger$.
• Figure 2: Circuit that implements the exponentiation of the hybrid qubit-qumode operator $\Sigma = \Sigma^\dagger$. Here, the qubit (qumode) is represented by a single (thick or triple) wire.
• Figure 3: Circuit implementing the cosine gate as a function of the position quadrature $\hat{x}$ up to errors of order $\mathcal{O}(t^2)$, see Eq. (\ref{['eq:cosinegate']}).
• Figure 4: Circuit that implements the non-unitary prescription in Eq. \ref{['eq:exp_Pauli_non_unitary']}, by exploiting the initialization and measurement of the ancilla qubit $c$.
• Figure 5: Survival probability of the free vacuum $\ket{n=0}^{\otimes L}$ for $L=3$ lattice sites as a function of time for representative model parameters. For the classical simulations, we choose different cutoffs for the local Hilbert space as indicated in the figure. The markers indicate Trotter steps.