Programmable Open Quantum Systems

TL;DR

The paper develops a framework to program the dynamics of open quantum systems by coupling fixed quantum processors with time-varying program states, focusing on Lindbladian semigroups. It identifies symmetry-driven classes (e.g., covariant Lindbladians, Pauli-Lindbladians) that are quantum-programmable with finite program dimension and provides necessary conditions ruling out broad nonprogrammable cases, including coherent generators and amplitude damping. For non-CPTP dynamics, the authors construct explicit HPTP/quasi-quantum protocols and analyze their resource costs using a diamond-norm-based programming cost, proving properties like faithfulness and monotonicity. They also connect programmable channel theory with open-system dynamics, offering symmetry-based compression, port-based strategies, and SDP-driven cost estimates to guide practical semigroup simulation and control in noisy quantum devices.

Abstract

Programmability is a unifying paradigm for enacting families of quantum transformations via fixed processors and program states, with a fundamental role and broad impact in quantum computation and control. While there has been a shift from viewing open systems solely as a source of error to treating them as a computational resource, their programmability remains largely unexplored. In this work, we develop a framework that characterizes and quantifies the programmability of Lindbladian semigroups by combining physically implementable retrieval maps with time varying program states. Within this framework, we identify quantum programmable classes enabled by symmetry and stochastic structure, including covariant semigroups and fully dissipative Pauli Lindbladians with finite program dimension. We further provide a necessary condition for physical programmability that rules out coherent generators and typical dissipators generating amplitude damping. For such nonphysically programmable cases, we construct explicit protocols with finite resources. Finally, we introduce an operational programming cost, defined via the number of samples required to program the Lindbladian, and establish its core structural properties, such as continuity and faithfulness. These results provide a notion of programming cost for Lindbladians, bridge programmable channel theory and open system dynamics, and yield symmetry driven compression schemes and actionable resource estimates for semigroup simulation and control in noisy quantum technologies.

Figures (6)

• Figure 1: Overview diagram of programming open-system dynamics. By injecting the program $\pi_t$ at time $t$, the processor ${\cal P}$ can reproduce the action of $e^{t{\cal L}}$ onto an arbitrary initial state $\rho_0$.
• Figure 2: Examination of the accuracy of the quasi-sampling method for programming the SWAP–dephasing Lindbladian with $\lambda=0.5$. The solid line represents the exact analytical result, while the scattering points denote quasi-sampled estimates for each time point. The simulation is performed over the time interval $T\in[0,10]$, demonstrating excellent agreement between quasi-sampling and the exact dynamics.
• Figure 3: Numerical estimations on the port-based $\epsilon$-error programming cost $\gamma_{\epsilon}$ of two Lindbladians with respect to the $\epsilon\in[0,0.2]$. The blue and red curves correspond to programming the damping Lindbladian and the damping Lindbladian supplemented with a $Z$ rotation, respectively. The time interval is fixed to $[0,10]$.
• Figure S4: An $\operatorname{HPTP}$-programming protocol of swap-dephasing channel based on $2$ measurement outcomes of the control qubit. The measurement in the programming channel is taken with respect to the computational basis $|j\rangle$ and the outcome $j$ decides which ${\cal P}_j$ is applied. The quasi-decomposition coefficients are attached on the LHS, in front of each operation circuit.
• Figure S5: Circuit implementation for realizing the amplitude damping channel.

Theorems & Definitions (41)

• Theorem 1
• Corollary 2
• Proposition 3
• Proposition 4
• Proposition 5
• Proposition 6
• Definition S1: quantum dynamical semigroup
• Definition S2: $Q$-matrix Norris1998markov
• Lemma 1
• Theorem S7