Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

TREV: Python Library For Efficient Implementations Of Variational Quantum Algorithms For Optimization Using Tensor Networks

Keun Jun Park

Published Apr 28, 2026• 0 citations
Publisher

Abstract

Variational Quantum Algorithms (VQAs) are a central class of hybrid quantum-classical methods for optimization on noisy intermediate-scale quantum (NISQ) hardware. Their scalability is constrained by the cost of gradient evaluation via the parameter-shift rule, whose complexity grows with both the number of circuit parameters and Pauli terms in the Hamiltonian, quickly dominating runtime for deep circuits or large qubit counts. We present TREV (Tensor-Ring Evaluated Variational algorithms), a modular Python library that accelerates VQA simulation through batched parameter evaluation within tensor-ring state representations. By amortizing contraction and sampling costs across multiple parameter shifts, TREV exploits GPU-parallel tensor contractions to achieve substantial runtime reduction with tunable memory-throughput trade-offs. Implemented in PyTorch, the framework supports both deterministic contraction-based and stochastic sampling-based evaluations. Benchmarks on Max-Cut, Traveling Salesperson Problem (TSP), and molecular ground-state energy estimation (H2, H4, LiH, BeH2) show comparable accuracy and up to 97% runtime reduction over sequential evaluation, while outperforming PennyLane, TensorCircuit, and Qiskit in runtime scalability. The chemistry benchmarks further demonstrate that TREV handles arbitrary Pauli Hamiltonians beyond QUBO formulations, broadening its applicability to quantum chemistry.

DOI
10.25394/pgs.32085129

Related papers

TREV: Python Library for Efficient Implementations of Variational Quantum Algorithms for Optimization using Tensor Networks

Keun Jun Park, Dheeraj Peddireddy, Vaneet Aggarwal

Variational Quantum Algorithms (VQAs) are a central class of hybrid quantum–classical methods for optimization on noisy intermediate-scale quantum (NISQ) hardware. Their scalability is constrained by the cost of gradient evaluation via the parameter-shift rule, whose complexity grows with both the number of circuit parameters and Pauli terms in the Hamiltonian, quickly dominating runtime for deep circuits or large qubit counts. We present TREV (Tensor-Ring Evaluated Variational algorithms), a Python library that accelerates VQA simulation through batched parameter evaluation within tensor-ring state representations . By amortizing contraction and sampling costs across multiple parameter shifts, TREV exploits GPU-parallel tensor contractions to achieve substantial runtime reduction with tunable memory–throughput trade-offs. Implemented in PyTorch, the framework supports both deterministic contraction-based and stochastic sampling-based evaluations. Benchmarks on Max-Cut, Traveling Salesperson Problem (TSP), and molecular ground-state energy estimation (H 2 , H 4 , LiH, BeH 2 ) show comparable accuracy and up to 97% runtime reduction over sequential evaluation, while outperforming PennyLane, TensorCircuit, and Qiskit in runtime scalability. The chemistry benchmarks further demonstrate that TREV handles arbitrary Pauli Hamiltonians beyond QUBO formulations, broadening its applicability to quantum chemistry.

ACM Transactions on Quantum Computing•Jun 22, 2026•0 citations

Scalable Tensor Ring Methods for Variational Quantum Algorithms

Dheeraj Peddireddy

Variational quantum algorithms (VQAs) have emerged as a promising candidate for realizing quantum advantage in machine learning and optimization tasks on noisy intermediate-scale quantum (NISQ) devices. Despite significant theoretical promise, the practical impact of VQAs is constrained by the exponential growth in computational resources required for classical simulation, the onset of barren plateaus in parameterized landscapes, and the inherent difficulties of scaling classical optimization routines to circuits with many qubits and layers. Addressing these intertwined challenges is essential for both advancing quantum algorithm development and extracting actionable physical insights from modern quantum hardware.This dissertation develops, analyzes, and validates an efficient framework based on approximate tensor ring decompositions for the classical simulation, training, and benchmarking of variational quantum circuits and eigensolvers. The first component of this work formalizes the representation of quantum states and gates within a cyclic tensor ring structure, enabling the compression of parameterized quantum circuits to linear scaling in both qubit number and circuit depth, while preserving essential quantum correlations and entanglement structure. Gate operations are efficiently approximated via truncated singular value decomposition, and novel algorithms for forward propagation and gradient-based training in the tensor ring domain are developed. These methodological advances are systematically benchmarked on standard supervised learning tasks, including Iris and MNIST datasets, and compared to conventional statevector and matrix product state baselines, resulting in competitive accuracy and orders-of-magnitude reductions in simulation cost. Extending this framework, the dissertation also introduces a scalable method for gradient evaluation in variational quantum eigensolvers (VQE) using the approximate tensor ring representation. This approach enables the simulation and training of large circuits for combinatorial optimization problems, supporting practical qubit counts and circuit depths inaccessible to traditional simulation methods. Detailed experiments on the Maximum-Cut optimization problem demonstrate both the computational efficiency and robustness of the tensor ring framework in nontrivial optimization landscapes.Collectively, the results establish approximate tensor ring decomposition as a versatile and powerful framework for advancing quantum machine learning and discrete optimization, supporting both pre-deployment benchmarking on classical hardware and algorithm design for next-generation quantum devices. This work provides a foundation for the further exploration of scalable quantum-classical hybrid algorithms, with implications for quantum simulation, optimization, and beyond.

•Jan 20, 2026•0 citations

Noisy tensor-ring approximation for computing gradients of a variational quantum eigensolver for combinatorial optimization

Dheeraj Peddireddy, Utkarsh Priyam, Vaneet Aggarwal

Variational quantum algorithms, especially quantum approximate optimization and the variational quantum eigensolver (VQE), have demonstrated their potential to provide computational advantage in the realm of combinatorial optimization. However, these algorithms suffer from classically intractable gradients limiting the scalability. This work addresses the scalability challenge for the VQE by proposing a classical gradient computation method which utilizes the parameter-shift rule but computes the expected values from the circuits using a tensor-ring approximation. The parametrized gates from the circuit transform the tensor ring by contracting the matrix along the free edges of the tensor ring. While the single-qubit gates do not alter the ring structure, the state transformations from the two-qubit rotations are evaluated by truncating the singular values, thereby preserving the structure of the tensor ring and reducing the computational complexity. This variation of the matrix-product-state approximation grows linearly in number of qubits and the number of two-qubit gates as opposed to the exponential growth in the classical simulations, allowing for a faster evaluation of the gradients on classical simulators.

Physical Review A•Oct 30, 2023•4 citations

A framework to evaluate the performance of Variational Quantum Algorithms

Ernesto Mamedaliev, Vladyslav Libov, Albert Nieto-Morales +2 more

Variational Quantum Algorithms (VQAs) are promising methods for solving combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) devices. However, benchmarking VQAs is difficult due to their stochastic behavior and the lack of standardized performance criteria. This work introduces a general framework for evaluating VQAs applied to Quadratic Unconstrained Binary Optimization (QUBO) problems. The framework uses three complementary metrics: feasibility, quality, and reproducibility. It also introduces a quality diagram that visualizes trade-offs between success probability and computational resources. Reproducibility is formalized using Shannon entropy, and a decision rule is defined for selecting algorithms under resource constraints. As a demonstration, the framework is applied to several VQAs using Conditional Value at Risk (CVaR) cost functions and different shot counts on a 16-qubit QUBO instance. The results show how the framework supports systematic benchmarking and provides a foundation for adaptive algorithm selection in hybrid quantum-classical workflows.

quant-pharXiv:2601.18812•Jan 22, 2026•0 citations

Noisy Tensor Ring approximation for computing gradients of Variational Quantum Eigensolver for Combinatorial Optimization

Dheeraj Peddireddy, Utkarsh Priyam, Vaneet Aggarwal

Variational Quantum algorithms, especially Quantum Approximate Optimization and Variational Quantum Eigensolver (VQE) have established their potential to provide computational advantage in the realm of combinatorial optimization. However, these algorithms suffer from classically intractable gradients limiting the scalability. This work addresses the scalability challenge for VQE by proposing a classical gradient computation method which utilizes the parameter shift rule but computes the expected values from the circuits using a tensor ring approximation. The parametrized gates from the circuit transform the tensor ring by contracting the matrix along the free edges of the tensor ring. While the single qubit gates do not alter the ring structure, the state transformations from the two qubit rotations are evaluated by truncating the singular values thereby preserving the structure of the tensor ring and reducing the computational complexity. This variation of the Matrix product state approximation grows linearly in number of qubits and the number of two qubit gates as opposed to the exponential growth in the classical simulations, allowing for a faster evaluation of the gradients on classical simulators.

arXiv:2307.03884•Jul 8, 2023•0 citations

Solving larger Travelling Salesman Problem networks with a penalty-free Variational Quantum Algorithm

Daniel Goldsmith, Xing Liang, Dimitrios Makris +1 more

The Travelling Salesman Problem (TSP) is a well-known NP-hard combinatorial optimisation problem with important industrial applications. Although extensively studied, quantum solutions for instances beyond a dozen locations remain uncommon. We present a hybrid penalty-free Variational Quantum Algorithm (VQA) that significantly reduces both the qubit count and the runtime through a compact penalty-free formulation scaling as $\mathcal{O}(nlog_2 n)$ qubits compared to conventional formulations with $\mathcal{O}(n^2)$ scaling. The VQA achieves solution qualities exceeding 90\% on Rigetti quantum hardware for instances of up to 17 locations. To the best of our knowledge, this represents the largest TSP instance solved on gate-based hardware. We systematically evaluate multiple encoding and optimisation strategies, and show that warm-start initialisation improves solution quality, while runtime is reduced by nearly two orders of magnitude through the use of gradient-free optimisation methods and cost-function caching. We additionally introduce a novel VQA inspired classical machine learning model (VQAi-CML) and benchmark the VQA against VQAi-CML, a greedy heuristic, a Monte Carlo baseline, and a strong classical optimisation baseline. Across the networks evaluated, the VQA consistently achieves superior solution quality relative to VQAi-CML, greedy heuristic, and Monte Carlo baselines, but not the classical solver

quant-pharXiv:2512.06523•Dec 9, 2025•0 citations
Variational Quantum Algorithms (VQAs) are a central class of hybrid quantum-classical methods for optimization on noisy intermediate-scale quantum (NISQ) hardware. Their scalability is constrained by the cost of gradient evaluation via the parameter-shift rule, whose complexity grows with both the number of circuit parameters and Pauli terms in the Hamiltonian, quickly dominating runtime for deep circuits or large qubit counts. We present TREV (Tensor-Ring Evaluated Variational algorithms), a modular Python library that accelerates VQA simulation through batched parameter evaluation within tensor-ring state representations. By amortizing contraction and sampling costs across multiple parameter shifts, TREV exploits GPU-parallel tensor contractions to achieve substantial runtime reduction with tunable memory-throughput trade-offs. Implemented in PyTorch, the framework supports both deterministic contraction-based and stochastic sampling-based evaluations. Benchmarks on Max-Cut, Traveling Salesperson Problem (TSP), and molecular ground-state energy estimation (H2, H4, LiH, BeH2) show comparable accuracy and up to 97% runtime reduction over sequential evaluation, while outperforming PennyLane, TensorCircuit, and Qiskit in runtime scalability. The chemistry benchmarks further demonstrate that TREV handles arbitrary Pauli Hamiltonians beyond QUBO formulations, broadening its applicability to quantum chemistry.
Variational Quantum Algorithms (VQAs) are a central class of hybrid quantum–classical methods for optimization on noisy intermediate-scale quantum (NISQ) hardware. Their scalability is constrained by the cost of gradient evaluation via the parameter-shift rule, whose complexity grows with both the number of circuit parameters and Pauli terms in the Hamiltonian, quickly dominating runtime for deep circuits or large qubit counts. We present TREV (Tensor-Ring Evaluated Variational algorithms), a Python library that accelerates VQA simulation through batched parameter evaluation within tensor-ring state representations . By amortizing contraction and sampling costs across multiple parameter shifts, TREV exploits GPU-parallel tensor contractions to achieve substantial runtime reduction with tunable memory–throughput trade-offs. Implemented in PyTorch, the framework supports both deterministic contraction-based and stochastic sampling-based evaluations. Benchmarks on Max-Cut, Traveling Salesperson Problem (TSP), and molecular ground-state energy estimation (H 2 , H 4 , LiH, BeH 2 ) show comparable accuracy and up to 97% runtime reduction over sequential evaluation, while outperforming PennyLane, TensorCircuit, and Qiskit in runtime scalability. The chemistry benchmarks further demonstrate that TREV handles arbitrary Pauli Hamiltonians beyond QUBO formulations, broadening its applicability to quantum chemistry.
Variational quantum algorithms (VQAs) have emerged as a promising candidate for realizing quantum advantage in machine learning and optimization tasks on noisy intermediate-scale quantum (NISQ) devices. Despite significant theoretical promise, the practical impact of VQAs is constrained by the exponential growth in computational resources required for classical simulation, the onset of barren plateaus in parameterized landscapes, and the inherent difficulties of scaling classical optimization routines to circuits with many qubits and layers. Addressing these intertwined challenges is essential for both advancing quantum algorithm development and extracting actionable physical insights from modern quantum hardware.This dissertation develops, analyzes, and validates an efficient framework based on approximate tensor ring decompositions for the classical simulation, training, and benchmarking of variational quantum circuits and eigensolvers. The first component of this work formalizes the representation of quantum states and gates within a cyclic tensor ring structure, enabling the compression of parameterized quantum circuits to linear scaling in both qubit number and circuit depth, while preserving essential quantum correlations and entanglement structure. Gate operations are efficiently approximated via truncated singular value decomposition, and novel algorithms for forward propagation and gradient-based training in the tensor ring domain are developed. These methodological advances are systematically benchmarked on standard supervised learning tasks, including Iris and MNIST datasets, and compared to conventional statevector and matrix product state baselines, resulting in competitive accuracy and orders-of-magnitude reductions in simulation cost. Extending this framework, the dissertation also introduces a scalable method for gradient evaluation in variational quantum eigensolvers (VQE) using the approximate tensor ring representation. This approach enables the simulation and training of large circuits for combinatorial optimization problems, supporting practical qubit counts and circuit depths inaccessible to traditional simulation methods. Detailed experiments on the Maximum-Cut optimization problem demonstrate both the computational efficiency and robustness of the tensor ring framework in nontrivial optimization landscapes.Collectively, the results establish approximate tensor ring decomposition as a versatile and powerful framework for advancing quantum machine learning and discrete optimization, supporting both pre-deployment benchmarking on classical hardware and algorithm design for next-generation quantum devices. This work provides a foundation for the further exploration of scalable quantum-classical hybrid algorithms, with implications for quantum simulation, optimization, and beyond.
Variational quantum algorithms, especially quantum approximate optimization and the variational quantum eigensolver (VQE), have demonstrated their potential to provide computational advantage in the realm of combinatorial optimization. However, these algorithms suffer from classically intractable gradients limiting the scalability. This work addresses the scalability challenge for the VQE by proposing a classical gradient computation method which utilizes the parameter-shift rule but computes the expected values from the circuits using a tensor-ring approximation. The parametrized gates from the circuit transform the tensor ring by contracting the matrix along the free edges of the tensor ring. While the single-qubit gates do not alter the ring structure, the state transformations from the two-qubit rotations are evaluated by truncating the singular values, thereby preserving the structure of the tensor ring and reducing the computational complexity. This variation of the matrix-product-state approximation grows linearly in number of qubits and the number of two-qubit gates as opposed to the exponential growth in the classical simulations, allowing for a faster evaluation of the gradients on classical simulators.
Variational Quantum Algorithms (VQAs) are promising methods for solving combinatorial optimization problems on noisy intermediate-scale quantum (NISQ) devices. However, benchmarking VQAs is difficult due to their stochastic behavior and the lack of standardized performance criteria. This work introduces a general framework for evaluating VQAs applied to Quadratic Unconstrained Binary Optimization (QUBO) problems. The framework uses three complementary metrics: feasibility, quality, and reproducibility. It also introduces a quality diagram that visualizes trade-offs between success probability and computational resources. Reproducibility is formalized using Shannon entropy, and a decision rule is defined for selecting algorithms under resource constraints. As a demonstration, the framework is applied to several VQAs using Conditional Value at Risk (CVaR) cost functions and different shot counts on a 16-qubit QUBO instance. The results show how the framework supports systematic benchmarking and provides a foundation for adaptive algorithm selection in hybrid quantum-classical workflows.
Variational Quantum algorithms, especially Quantum Approximate Optimization and Variational Quantum Eigensolver (VQE) have established their potential to provide computational advantage in the realm of combinatorial optimization. However, these algorithms suffer from classically intractable gradients limiting the scalability. This work addresses the scalability challenge for VQE by proposing a classical gradient computation method which utilizes the parameter shift rule but computes the expected values from the circuits using a tensor ring approximation. The parametrized gates from the circuit transform the tensor ring by contracting the matrix along the free edges of the tensor ring. While the single qubit gates do not alter the ring structure, the state transformations from the two qubit rotations are evaluated by truncating the singular values thereby preserving the structure of the tensor ring and reducing the computational complexity. This variation of the Matrix product state approximation grows linearly in number of qubits and the number of two qubit gates as opposed to the exponential growth in the classical simulations, allowing for a faster evaluation of the gradients on classical simulators.
The Travelling Salesman Problem (TSP) is a well-known NP-hard combinatorial optimisation problem with important industrial applications. Although extensively studied, quantum solutions for instances beyond a dozen locations remain uncommon. We present a hybrid penalty-free Variational Quantum Algorithm (VQA) that significantly reduces both the qubit count and the runtime through a compact penalty-free formulation scaling as O(nlog2n)\mathcal{O}(nlog_2 n)O(nlog2​n) qubits compared to conventional formulations with O(n2)\mathcal{O}(n^2)O(n2) scaling. The VQA achieves solution qualities exceeding 90\% on Rigetti quantum hardware for instances of up to 17 locations. To the best of our knowledge, this represents the largest TSP instance solved on gate-based hardware. We systematically evaluate multiple encoding and optimisation strategies, and show that warm-start initialisation improves solution quality, while runtime is reduced by nearly two orders of magnitude through the use of gradient-free optimisation methods and cost-function caching. We additionally introduce a novel VQA inspired classical machine learning model (VQAi-CML) and benchmark the VQA against VQAi-CML, a greedy heuristic, a Monte Carlo baseline, and a strong classical optimisation baseline. Across the networks evaluated, the VQA consistently achieves superior solution quality relative to VQAi-CML, greedy heuristic, and Monte Carlo baselines, but not the classical solver