Noise-Assisted Feedback Control of Open Quantum Systems for Ground State Properties

TL;DR

This work presents a method for simulating open quantum system dynamics on a quantum computer, including negative dissipation rates in the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, and develops a quantum algorithm for calculating ground-state properties that exploits feedback-controlled, noise-assisted dynamics.

Abstract

Intrinsic noise in pre-fault-tolerant quantum devices poses a major challenge to the reliable realization of unitary dynamics in quantum algorithms and simulations. To address this, we present a method for simulating open quantum system dynamics on a quantum computer, including negative dissipation rates in the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation. Our approach lies beyond the standard Markovian approximation, enabling the controlled study of non-Markovian processes within a quantum simulation framework. Using this method, we develop a quantum algorithm for calculating ground-state properties that exploits feedback-controlled, noise-assisted dynamics. In this scheme, Lyapunov-based feedback steers the system toward a target virtual state under engineered noise conditions. This framework offers a promising strategy for harnessing current quantum hardware and advancing robust control protocols based on open system dynamics.

Figures (7)

• Figure 1: Schematic diagram of FQA in ideal, FQA in NISQ, and NAFQA in NISQ devices. (a) Ideal implementation of FQAs in a noiseless quantum computer consisting of $s$ Trotter layers. (b) Actual implementation of FQAs in NISQ devices where the ideal layer containing $U_p$ and $U_d(\beta_i)$ is interleaved with the inherent noise channel $\Lambda_s$. The dashed line represents the intrinsic device noise, over which the user generally has no control. (c) Our NAFQA algorithm is demonstrated to utilize noise to drive an OQS simulation by introducing an additional noise map $\Lambda(\Gamma_s)$ based on the feedback law. Note that the noise channels $\Lambda$ represent non-unitary evolutions and can be obtained via the stochastic unravelings of the GKSL master equation.
• Figure 2: Numerical results for the 5-qubit Maxcut problem. (a) The approximation ratio $r$, (b) the control terms $\beta(t)$ and $\Gamma(t)$, (c) the success probability $\phi$ as a function of the Trotter step with a fixed time step $dt = 0.07$. The inset in (b) shows $\Gamma(t)$ for the Pauli error term $IIYII$, whereas all the other Pauli terms' error probabilities are set to zero. (d) Relative error as a function of the number of layers in the double logarithm scale with several numbers of trajectories, $M$. (e) $r$ and $\phi$ at large times demonstrate the non-monotonic behavior. (f) Time averaged relative error as a function of $M$.
• Figure 3: Time evolution of approximate ratio $r$ and success probability $\phi$ with different parameters involved in the NAFQA protocol. We considered several cases: (a) varying the time steps $dt$ with a fixed sample size, $M = 12000$, (b) different threshold values of $\Gamma(t)$ from $\text{th} = -0.05$ to $\text{th} = -0.25$ with $dt = 0.07$ and $M = 12000$. The dotted and solid lines indicate the noisy and NAFQA evolutions, respectively. (c) Various numbers of samples $M$ are used for a fixed time step $dt = 0.07$ and threshold value of $\text{th} = -0.15$ .
• Figure 4: Performance of FQA in the closed and open quantum system framework. Approximation ratio $r$ for the 5-qubit Maxcut problem with varying the (a) control field $\beta$ and (b) Trotter-step $dt$. Panel (a) is shown for the fixed Trotter step $dt = 0.07$, whereas panel (b) is illustrated for $\beta = - A$. The dotted and solid lines represent the noisy and ideal evolutions, respectively. The inset shows the purity $\mathcal{P}$ of the noisy evolutions as a function of Trotter steps.
• Figure 5: The performance of NAFQA against the noisy FQA with error probabilities given in Fig. \ref{['second_err_probs']}. The mean and standard error (a) approximation ratio $r$, (b) control field $\beta$, and (c) success probability $\phi$ for the 3-qubit spin glass Hamiltonians over 25 instances are illustrated. The numerical simulations are carried out with a time step of $dt = 0.005$, yielding NAFQA results from 5000 trajectories. (d) error probabilities $\Gamma_k(t)$ for $k\in{IYI, ZYI, XII, IXZ, IXI}$.