Variational Quantum Operator Simulation

Implementing time-evolution operators in shallow quantum circuits is important for quantum simulations. The standard method of Trotterization requires a large number of gates to achieve practical accuracy. Variational Quantum Simulation (VQS) is an algorithm that calculates the time evolution of a quantum state and can be executed with shallower circuits than Trotterization. However, the operator obtained by VQS evolves only a fixed initial state and is not the time evolution operator itself. In this paper, we propose Variational Quantum Operator Simulation (VQOS), a method to realize time evolution operators in shallow quantum circuits. This method is based on the variational principle for operators and does not require the implementation of the desired Trotter decomposition of the time evolution operator. We performed numerical simulations of the VQOS algorithm and successfully implemented the time evolution operator for closed systems in a quantum circuit that is up to 5 times shallower than the Trotterization. By providing a more practical way to implement time evolution operators, VQOS increases the applicability of near-term quantum computers.

💡 Research Summary

The paper introduces a novel algorithm called Variational Quantum Operator Simulation (VQOS) that enables the implementation of time‑evolution operators on near‑term quantum devices using significantly shallower circuits than conventional Trotterization. The authors begin by motivating the need for efficient operator synthesis: while Trotter product formulas can approximate the unitary (e^{-iHt}) they typically require a large number of gates to achieve acceptable fidelity, making them impractical on noisy intermediate‑scale quantum (NISQ) hardware. Variational Quantum Simulation (VQS), also known as Variational Quantum State Simulation (VQSS), offers a hybrid quantum‑classical approach that evolves a specific initial state via a parameterized circuit, but it does not produce the full evolution operator. Consequently, VQSS must be rerun for each new initial state and cannot be used directly in algorithms such as quantum phase estimation that rely on the operator itself.

VQOS overcomes this limitation by applying the variational principle directly to the operator. The method defines a parameterized unitary (\tilde U(\boldsymbol\theta(t)) = e^{i\theta_0(t)}U(\boldsymbol\theta(t))) and enforces the McLachlan‑type condition
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