Computing non-equilibrium steady states of open quantum systems is a challenging task on conventional computers, motivating quantum algorithms for direct steady-state estimation. A natural route is to regard the steady state as the zero mode of the Liouvillian and to isolate this sector spectrally. We formulate this task as a known-zero-sector projection problem and implement the corresponding filter using the Rodeo algorithm, which performs stochastic spectral filtering through repeated controlled evolutions and measurement-conditioned filtering steps. In the steady-state setting, the filter can be centered directly at the known zero eigenvalue, avoiding the spectral search required in generic eigenstate preparation. Compared with a phase-estimation-based implementation of the same projection, the Rodeo approach enables restart on failure and reduces the target-error dependence of the filtering cost and controlled-evolution depth from power-law to logarithmic. This advantage becomes more pronounced as the spectral separation of the Hermitian Liouvillian embedding increases, allowing Rodeo filtering to outperform phase-estimation filtering already at modest controlled-evolution depths. Our results identify Rodeo filtering as a resource-efficient primitive for estimating steady-state observables in open quantum systems.
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35 pages, 11 figures, including Supplemental Material
Computing non-equilibrium steady states of open quantum systems is a challenging task on conventional computers, motivating quantum algorithms for direct steady-state estimation. A natural route is to regard the steady state as the zero mode of the Liouvillian and to isolate this sector spectrally. We formulate this task as a known-zero-sector projection problem and implement the corresponding filter using the Rodeo algorithm, which performs stochastic spectral filtering through repeated controlled evolutions and measurement-conditioned filtering steps. In the steady-state setting, the filter can be centered directly at the known zero eigenvalue, avoiding the spectral search required in generic eigenstate preparation. Compared with a phase-estimation-based implementation of the same projection, the Rodeo approach enables restart on failure and reduces the target-error dependence of the filtering cost and controlled-evolution depth from power-law to logarithmic. This advantage becomes more pronounced as the spectral separation of the Hermitian Liouvillian embedding increases, allowing Rodeo filtering to outperform phase-estimation filtering already at modest controlled-evolution depths. Our results identify Rodeo filtering as a resource-efficient primitive for estimating steady-state observables in open quantum systems.
We present a stochastic quantum computing algorithm that can prepare any eigenvector of a quantum Hamiltonian within a selected energy interval $[E\ensuremath{-}\ensuremath{\epsilon},E+\ensuremath{\epsilon}]$. In order to reduce the spectral weight of all other eigenvectors by a suppression factor $\ensuremath{\delta}$, the required computational effort scales as $O[|\mathrm{log}\ensuremath{\delta}|/(p\ensuremath{\epsilon})]$, where p is the squared overlap of the initial state with the target eigenvector. The method, which we call the rodeo algorithm, uses auxiliary qubits to control the time evolution of the Hamiltonian minus some tunable parameter E. With each auxiliary qubit measurement, the amplitudes of the eigenvectors are multiplied by a stochastic factor that depends on the proximity of their energy to E. In this manner, we converge to the target eigenvector with exponential accuracy in the number of measurements. In addition to preparing eigenvectors, the method can also compute the full spectrum of the Hamiltonian. We illustrate the performance with several examples. For energy eigenvalue determination with error $\ensuremath{\epsilon}$, the computational scaling is $O[(\mathrm{log}\ensuremath{\epsilon}{)}^{2}/(p\ensuremath{\epsilon})]$. For eigenstate preparation, the computational scaling is $O(\mathrm{log}\mathrm{\ensuremath{\Delta}}/p)$, where $\mathrm{\ensuremath{\Delta}}$ is the magnitude of the orthogonal component of the residual vector. The speed for eigenstate preparation is exponentially faster than that for phase estimation or adiabatic evolution.
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-$\sfrac{1}{2}$ and spin-1 particles. The ancilla qutrit implementation exhibits an 18% reduction in fluctuations compared to the qubit implementation. Our results show that qudits provide a framework for spectral analysis and thermodynamic characterization of multi-level quantum systems.
Simulating the dynamics and the non-equilibrium steady state of an open quantum system are hard computational tasks on conventional computers. For the simulation of the time evolution, several efficient quantum algorithms have recently been developed. However, computing the non-equilibrium steady state as the long-time limit of the system dynamics is often not a viable solution, because of exceedingly long transient features or strong quantum correlations in the dynamics. Here, we develop an efficient quantum algorithm for the direct estimation of averaged expectation values of observables on the non-equilibrium steady state, thus bypassing the time integration of the master equation. The algorithm encodes the vectorized representation of the density matrix on a quantum register, and makes use of quantum phase estimation to approximate the eigenvector associated to the zero eigenvalue of the generator of the system dynamics. We show that the output state of the algorithm allows to estimate expectation values of observables on the steady state. Away from critical points, where the Liouvillian gap scales as a power law of the system size, the quantum algorithm performs with exponential advantage compared to exact diagonalization.
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 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-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.
The rodeo algorithm has been proposed recently as an efficient method in quantum computing for projection of a given initial state onto a state of fixed energy for systems with discrete spectra. In the initial formulation of the rodeo algorithm, evolution times were chosen randomly via a Gaussian distribution with a fixed standard deviation. In this paper, it is shown that such a random approach for choosing times suffers from exponentially large fluctuations in the suppression of unwanted components: as the number of iterations gets large, the distribution of suppression factors obtained from random selection approaches a log-normal distribution, leading to remarkably large fluctuations. We note that by choosing times intentionally rather than randomly, such fluctuations can be avoided and strict upper bounds on the suppression can be obtained. Moreover, the average suppression using fixed computational cost can be reduced by many orders of magnitude relative to the random algorithm. A key to doing this is to choose times that vary over exponentially many timescales, starting from a modest maximum scale and going down to timescales exponentially smaller.
The density of states (DoS) encodes the thermodynamic and spectral properties of quantum many-body systems, yet its reconstruction becomes intractable for Hilbert spaces too large to diagonalize. Classically, the kernel polynomial method (KPM) addresses this by combining stochastic trace estimation with a smoothing kernel. Here we show that the Rodeo algorithm---one of the simplest eigenvalue-location protocols for near-term quantum hardware---provides a direct quantum analogue of this approach. Averaging the Rodeo response over Haar-random input states yields the DoS convolved with a spectral kernel fixed entirely by the distribution of evolution times: the random states play the role of stochastic trace estimation, and the temporal sampling distribution that of the damping kernel. The construction requires only the standard single-ancilla circuit, and quantum typicality suppresses the statistical error as the Hilbert-space dimension grows. We derive the estimator and its uncertainties, establish an explicit dictionary between signal-processing window functions and quantum reconstruction kernels, and validate the method on the one-dimensional transverse-field Ising and spin-1 models.