Accelerating Tensor Ring VQE with Batched Contraction

The Variational Quantum Eigensolver (VQE) is a central method in quantum machine learning, with applications across disciplines such as finance, biostatistics, and related fields. Despite its impact, VQE suffers from limited scalability due to the exponential space complexity \(O(2^N)\) required to store quantum states and compute expectation values, where N is the number of qubits. To address this, the Matrix Product State (MPS) VQE was developed, reducing storage complexity to \(O(n\chi ^2)\) , where \(\chi \) is the bond dimension. However, the bond dimension typically grows with entangling operations such as CNOT or SWAP, leading to large memory demands in highly entangled circuits. The Tensor Ring (TR) VQE improves upon this by fixing the bond dimension and connecting the first and last qubits, thereby avoiding unbounded memory growth and enabling more complex circuits. Nevertheless, TR VQE still requires exponential space complexity \(O(2^N)\) to evaluate expectation values, which restricts its scalability. In this work, we propose a space-efficient TR VQE method for combinatorial optimization. Our key contribution is a batched expectation value calculation that leverages MPS-style contraction to evaluate multiple Pauli terms simultaneously. This batching improves efficiency while reducing expectation evaluation space complexity to \(O(n\chi ^4)\) . Moreover, it increases only linear space complexity with respect to the number of Pauli terms, providing a scalable path toward practical quantum optimization.