Variational Quantum Algorithm for Unitary Dilation

TL;DR

A hybrid quantum-classical framework for efficiently implementing approximate unitary dilations of non-unitary operators with enhanced noise resilience and robustness against device noise is introduced, achieving high-fidelity simulation.

Abstract

We introduce a hybrid quantum-classical framework for efficiently implementing approximate unitary dilations of non-unitary operators with enhanced noise resilience. The method embeds a target non-unitary operator into a subblock of a unitary matrix generated by a parameterized quantum circuit with universal expressivity, while a classical optimizer adjusts circuit parameters under the global unitary constraint. As a representative application, we consider the non-unitary propagator of a Lindbladian superoperator acting on the vectorized density matrix, which is relevant for simulating open quantum systems. We further validate the approach experimentally on superconducting devices in the Quafu quantum cloud computing cluster. Compared with standard dilation protocols, our method significantly reduces quantum resource requirements and improves robustness against device noise, achieving high-fidelity simulation. Its generality also enables compatibility with non-Markovian dynamics and Kraus-operator-based evolutions, providing a practical pathway for the noise-resilient simulation of non-unitary processes on near-term quantum hardware.

Figures (4)

• Figure 1: The VQAUD allows the parameterized quantum circuit to achieve universal matrix expressivity for any given dimension. By leveraging quantum operator properties while preserving global unitary constraints, we define the cost function $C(\boldsymbol{\theta}) = \| U^{(l)}(\boldsymbol{\theta}) - \alpha M \|_F$ and employ classical optimization algorithms to minimize it, such that the upper-left submatrix of the quantum circuit's unitary matrix approximates the target non-unitary operator $\alpha M$. This formally establishes a unitary dilation for the non-unitary operator $\alpha M$ under the given dimensional constraints.
• Figure 2: VQAUD-based simulations of Lindblad dynamics for multiple two-level systems. (a) Cost functions obtained from BFGS optimization at different circuit depths, demonstrating the implementation accuracy of the non-unitary propagator. (b) Population dynamics of a Markovian driven two-level open quantum system with $H = \frac{\Omega}{2} (|0\rangle\langle 1| + |1\rangle\langle 0|)$, decay rate $\gamma = \Omega/10$, and pure dephasing rate $\gamma_{\mathrm{dp}} = \Omega/50$, starting from the initial state $|0\rangle$. (c) Population dynamics of a non-Markovian detuned damped Jaynes--Cummings model with $\lambda = \gamma_0/5$ and $\Delta = 8\lambda$, initialized in $\rho(0) = (2|0\rangle\langle 0| + 3|1\rangle\langle 1|)/5$. The dashed lines and hollow circles represent numerical simulation results from the master equation and VQAUD, respectively, while solid circles with plus and cross markers denote results from VQAUD and the standard Sz.-Nagy dilation algorithm implemented on the Baihua superconducting quantum computing platform. Each time-step point is estimated from $2^{15}$ projective measurement shots.
• Figure 3: Numerical simulation results of steady states ($t\to\infty$) of (a) three-level and (b) four-level open systems ($\gamma=\Omega/10$) from initial states in uniform superposition via three approaches (VQAUD, standard Sz.- Nagy dilation algorithm and unitary decomposition algorithm) using quantum devices noise parameters $\lambda=2\times10^{-3}$ and $\omega=1\times10^{-3}$. The dashed lines indicate the ideal reference from the master equation simulation.
• Figure 4: Single-optimization VQAUD simulation of time-independent Lindblad dynamics for a driven two-level Markovian open quantum system. The inset shows the upper bound of the 2-norm of the remainder in the Taylor expansion of the Liouvillian propagator at different approximation orders.