Qudit Implementation of the Rodeo Algorithm for Quantum Spectral Filtering

Abstract

Qudits, the multi-level generalization of qubits, provide a natural extension of the binary paradigm in quantum computation and offer new opportunities to enhance algorithmic performance. Beyond their direct applicability to the simulation of multi-level quantum systems, higher-dimensional ancillae can improve sampling efficiency in quantum algorithms by enabling the simultaneous implementation of multiple control operations, thereby reducing circuit complexity. In this work, we pursue three main objectives. First, we present a formulation of the Rodeo algorithm employing a general $d$-level ancilla qudit. Second, we introduce the concept of the \emph{Rodeo kernel}, defined as a two-frequency interferometer, which acts as a spectral filter in the energy domain. Finally, we propose a microcanonical protocol for the Rodeo algorithm. This protocol enables the estimation of entropic quantities through a single energy sweep and admits a natural interpretation as a Gaussian convolution of the density of states. To support the theoretical analysis, we perform numerical evaluations of the corresponding quantum circuit using ancilla qudits of dimensions three, four, and five. The simulations are performed for the one-dimensional Ising model, considering both spin-$\frac{1}{2}$ and spin-$1$ particles. The ancilla qutrit implementation exhibits an $18\%$ reduction in fluctuations compared to the qubit implementation. Our results show that the qudits provide a framework for spectral analysis and thermodynamic characterization of multi-level quantum systems.

Figures (6)

• Figure 1: (Color online) Circuit diagrams illustrating (a) the Rodeo algorithm, (b) a Ramsey interferometer, and (c) a Loschmidt-echo protocol. In the Rodeo implementation, the free-evolution arm of the Ramsey interferometer is replaced by a controlled time-evolution operator followed by a phase shift. As the phase shift constitutes an attempt to reverse the effect of the controlled evolution on the ancilla qudit, the Rodeo circuit admits a natural interpretation in terms of a Loschmidt echo.
• Figure 2: (Color online) Probability of measuring a $d$-level ancilla qudit in the $n$th state when the system is prepared in a Hamiltonian eigenstate [see eqn. (\ref{['eq:p_dn_x']})], where here $\Delta_x = \omega_xt d / 2$. (a) Probability distribution for $n = 0$ and $d = 2, 3,$ and $4$. (b) Probability distribution for $d = 3$ and $n = 0, 1,$ and $2$; the dotted red curve corresponds to the sum over all $n$.
• Figure 3: (Color online) Numerical evaluation of the Rodeo algorithm applied to the one-dimensional spin-$1/2$ Ising model with $N=5$ spins. The averages are over $500$ distinct evolution times per energy level, drawn from a Gaussian distribution with $\sigma = 5$. Panels (a) and (b) for system state $\ket{\psi} = \ket{0}$, while panels (c) and (d) $\ket{\psi} = 1/2\ket{1} + \sqrt{3}/2\ket{5}$. In panels (a) and (c), the ancilla qubit ($d=2$) implementation, and (b) and (d) the ancilla qutrit ($d=3$) is employed. The local maxima of $\overline{h}$ are interpreted as the probabilities of measuring the state $\ket{\psi}$ at energy $E$.
• Figure 4: Relative difference of the spectral amplitude obtained with a $d$-dimensional ancilla qudit with respect to the ancilla qubit case ($d=2$) for the one-dimensional spin-$\tfrac{1}{2}$ Ising model with input state $\ket{\psi}=\ket{0}$. The Gaussian distribution used in the averaging has $\sigma = 5$. Panel (a) shows the theoretical prediction for different ancilla dimensions (see eqn (\ref{['eq:relDiff_x']})). Panels (b)--(d) display the corresponding numerical results for a periodic chain of five spins, obtained from $500$ independent realizations of the evolution time for each energy value: (b) $d=3$, (c) $d=4$, and (d) $d=5$.
• Figure 5: Numerical evaluation of the Rodeo algorithm applied to the one-dimensional spin-$1$ Ising model with $N=3$ spins. The evolution times are sampled from a Gaussian distribution with $\sigma = 10$. Panel (a) shows the SA for the input state $\ket{\psi} = \ket{15}$, where the averages are taken over $500$ distinct evolution times for each energy value. Panel (b) presents the cumulative result obtained by summing the SA over all computational basis states, which reconstructs the degeneracy of the system.