Divide-and-Conquer Simulation of Open Quantum Systems

TL;DR

This work tackles the challenge of simulating open quantum systems (OQS) on quantum hardware by addressing non-unitary dynamics that hinder deterministic implementations. It introduces a divide-and-conquer mixed-state preparation to coherently combine outputs from non-unitary Kraus-dilation subcircuits (Sz.-Nagy or SVD), and a grouping strategy that expands Kraus operators into expanded operators to balance circuit depth, CNOT count, and qubit resources. The approach achieves a full quantum dynamical output with depth scaling as $O(d^2 + \log m)$ and a success probability of $P_0 = 1/m$, improving over the $O(md^2)$ depth of Stinespring while still producing the full state on hardware. Computational experiments and a proof-of-concept FMO simulation on current NISQ devices illustrate the trade-offs and practical viability, highlighting the method’s potential for reservoir engineering and quantum information tasks that rely on open-system dynamics.

Abstract

One of the promises of quantum computing is to simulate physical systems efficiently. However, the simulation of open quantum systems - where interactions with the environment play a crucial role - remains challenging for quantum computing, as it is impossible to implement deterministically non-unitary operators on a quantum computer without auxiliary qubits. The Stinespring dilation can simulate an open dynamic but requires a high circuit depth, which is impractical for NISQ devices. An alternative approach is parallel probabilistic block-encoding methods, such as the Sz.-Nagy and Singular Value Decomposition dilations. These methods result in shallower circuits but are hybrid methods, and we do not simulate the quantum dynamic on the quantum computer. In this work, we describe a divide-and-conquer strategy for preparing mixed states to combine the output of each Kraus operator dilation and obtain the complete dynamic on quantum hardware with a lower circuit depth. The work also introduces a balanced strategy that groups the original Kraus operators into an expanded operator, leading to a trade-off between circuit depth, CNOT count, and number of qubits. We perform a computational analysis to demonstrate the advantages of the new method and present a proof-of-concept simulation of the Fenna-Matthews-Olson dynamic on current quantum hardware.

Figures (9)

• Figure 1: Implementation of a Stinespring dilation as a quantum circuit. The implementation utilizes $\lceil \log m \rceil$ auxiliary qubits for the dilation, that are traced out at the end to obtain the state $\Lambda (\ket{\psi} \bra{\psi})$ for the $n$ qubit system.
• Figure 2: Implementation of a Sz.-Nagy dilation as a quantum circuit. The auxiliary qubit is measured, and if the measurement outcome is $\ket{0}$, we obtain ${M}_k\ket{\psi}$ in the $n$ system qubits.
• Figure 3: Circuit implementation of the Singular Value Decomposition dilation for a non-unitary operator ${M}_k$. The non-unitary operator is decomposed into ${M}_k = U_k \Sigma_k V^{\dagger}_k$, where $U_k, V_k$ are unitaries and $\Sigma_k$ is a non-unitary diagonal. $\Sigma_k$ is then dilated into a unitary $U_{\Sigma_k} = \Sigma_{k}^{(+)} \oplus \Sigma_{k}^{(-)}$, which acts on the system qubits and in an ancilla qubit.
• Figure 4: Example of a circuit that prepares the mixed state $\rho_f = \sum^4_{i=1} \ket{\psi_i}\bra{\psi_i}$, composed of $4$ pure states.
• Figure 5: Comparison of Depth and CNOT count for the Sz.-Nagy dilations for different group sizes ${l}$. The Depth for each group size is in red, while the CNOT count is in blue. The dashed/dotted orange/black line corresponds to the Stinespring dilation depth/CNOT count.

Theorems & Definitions (1)

• Theorem 1