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Inductive Graph Convolutional Quantum Process Tomography: A Scalable Geometric Deep Learning Approach

Imed Bouaker

Published Mar 24, 2026• 0 citations
Publisher

Abstract

Abstract Quantum process tomography (QPT) is a fundamental task in quantum information science, yet it suffers from exponential scaling in the number of qubits. We introduce an inductive graph convolutional approach, IGC-QPT, that overcomes this limitation by exploiting the geometric structure of Choi matrices. The method constructs a graph based on a fast low-rank approximation of the Bures distance, then learns a GraphSAGE encoder to map measurement data to low-dimensional embeddings that preserve the local geometry of the process space. This enables efficient and accurate reconstruction of unknown quantum processes from limited measurements for systems up to seven qubits. Theoretical analysis provides sample complexity bounds with complete proofs, including a detailed analysis of the low-rank approximation error. Extensive numerical experiments on 2–7 qubit systems, with full statistical reporting, demonstrate that IGC-QPT significantly outperforms standard QPT (LS), compressed sensing (CS), and neural network methods (NN), achieving high fidelity with orders of magnitude speedup. A detailed discussion of assumptions, limitations, and practical considerations such as transfer learning for reducing training requirements is provided, along with robustness checks.

DOI
10.21203/rs.3.rs-9118683/v1

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Abstract Quantum process tomography (QPT) is a fundamental task in quantum information science, yet it suffers from exponential scaling in the number of qubits. We introduce an inductive graph convolutional approach, IGC-QPT, that overcomes this limitation by exploiting the geometric structure of Choi matrices. The method constructs a graph based on a fast low-rank approximation of the Bures distance, then learns a GraphSAGE encoder to map measurement data to low-dimensional embeddings that preserve the local geometry of the process space. This enables efficient and accurate reconstruction of unknown quantum processes from limited measurements for systems up to seven qubits. Theoretical analysis provides sample complexity bounds with complete proofs, including a detailed analysis of the low-rank approximation error. Extensive numerical experiments on 2–7 qubit systems, with full statistical reporting, demonstrate that IGC-QPT significantly outperforms standard QPT (LS), compressed sensing (CS), and neural network methods (NN), achieving high fidelity with orders of magnitude speedup. A detailed discussion of assumptions, limitations, and practical considerations such as transfer learning for reducing training requirements is provided, along with robustness checks.
We perform quantum process tomography (QPT) for both discrete- and continuous-variable quantum systems by learning a process representation using Kraus operators. The Kraus form ensures that the reconstructed process is completely positive. To make the process trace preserving, we use a constrained gradient-descent (GD) approach on the so-called Stiefel manifold during optimization to obtain the Kraus operators. Our ansatz uses a few Kraus operators to avoid direct estimation of large process matrices, e.g., the Choi matrix, for low-rank quantum processes. The GD-QPT matches the performance of both compressed-sensing (CS) and projected least-squares (PLS) QPT in benchmarks with two-qubit random processes, but shines by combining the best features of these two methods. Similar to CS (but unlike PLS), GD-QPT can reconstruct a process from just a small number of random measurements, and similar to PLS (but unlike CS) it also works for larger system sizes, up to at least five qubits. We envisage that the data-driven approach of GD-QPT can become a practical tool that greatly reduces the cost and computational effort for QPT in intermediate-scale quantum systems.
Quantum process tomography (QPT) protocols are often designed by lifting quantum state tomography (QST) protocols through the Choi-Jamiolkowski isomorphism, at the cost of an extra step enforcing the trace-preserving (TP) constraint. Existing approaches fall into two classes. The folklore partial normalization, which sandwiches the density estimate between inverse square roots of its input marginal, has a closed form but is used as a heuristic, its only analyses being asymptotic and losing a factor polynomial in the dimension. Projection-based methods are principled but have no closed forms and scale poorly with dimension. We resolve this dichotomy by showing that the partial normalization is the exact projection onto the Choi states of quantum channels in fidelity (equivalently in Bures or purified distance). Consequently, any QST algorithm with guarantees in fidelity lifts to a QPT algorithm that inherits its sample complexity, through a single closed-form congruence that preserves the Choi rank and is computed in Kraus form without any eigendecomposition of Choi dimension. We lift two single-copy protocols and prove that the non-adaptive one (Fidelity Projected Least Squares) recovers the Choi rank and reaches infidelity ε\varepsilonε from a number of channel uses proportional to 1/ε1/\varepsilon1/ε. Numerically, for channels of low Choi rank, it improves on the state-of-the-art projected least squares in accuracy, and by orders of magnitude in the time and memory of the TP-regularization step and of the returned estimate.
Quantum process tomography is an experimental technique to fully characterize an unknown quantum process. Standard quantum process tomography suffers from exponentially scaling of the number of measurements with the increasing system size. In this work, we put forward a quantum machine learning algorithm which approximately encodes the unknown unitary quantum process into a relatively shallow depth parametric quantum circuit. We demonstrate our method by reconstructing the unitary quantum processes resulting from the quantum Hamiltonian evolution and random quantum circuits up to eight qubits. Results show that those quantum processes could be reconstructed with high fidelity, while the number of input states required are at least two orders of magnitude less than required by the standard quantum process tomography.
Quantum state and process tomography constitute essential diagnostic tools in quantum information science, yet their standard formulations suffer from prohibitive computational scaling as the number of qubits grows. In this work, we introduce a physics-constrained conditional generative adversarial network that bypasses iterative constrained inversion by directly learning a forward generative mapping conditioned on Pauli expectation values. The generator embeds a differentiable Cholesky layer at its output, which enforces Hermiticity, positive semidefiniteness, and unit trace by construction. Our experiments reveal that the strength of the L1L^1L1 penalty critically governs the emergence of GHZ coherence during training: an excessively large penalty postpones the coherence onset and yields a prolonged low-fidelity plateau, whereas an intermediate value enables the fastest stable convergence. Moreover, for high-temperature thermal states, an over-complete measurement basis proves necessary to prevent sustained late-stage fluctuations. By extending the same Cholesky constraint to the Choi-matrix representation, the framework naturally accommodates quantum process tomography. For systems with n≥6n \ge 6n≥6 qubits, the exponential growth of the underlying 2n×2n2^n \times 2^n2n×2n density matrix remains the fundamental bottleneck; we discuss how integrating tensor-network structures can contain the per-iteration cost while preserving reconstruction fidelity. Altogether, these results suggest that physically constrained generative learning offers a scalable and amortizable pathway toward data-driven tomography for noisy intermediate-scale quantum devices.
Abstract Efficient and accurate characterization of an experimental setup is a critical requirement in any physical setting. In the quantum realm, the characterization of an unknown operator is experimentally accomplished via Quantum Process Tomography (QPT). This technique combines the outcomes of different projective measurements to reconstruct the underlying process matrix, typically extracted from maximum-likelihood estimation. Here, we exploit the logical correspondence between optical polarization and two-level quantum systems to retrieve the complex action of structured metasurfaces within a QPT-inspired context. In particular, we investigate a deep-learning approach that allows for fast and accurate reconstructions of space-dependent SU(2) operators by only processing a minimal set of measurements. We train a convolutional neural network based on a scalable U-Net architecture to process entire experimental images in parallel. Synthetic processes are reconstructed with average fidelity above 90%. The performance of our routine is experimentally validated in the case of space-dependent polarization transformations acting on a classical laser beam. Our approach further expands the toolbox of data-driven approaches to QPT and shows promise in the real-time characterization of complex optical gates.
Quantum process tomography is a fundamental and critical benchmarking and certification tool that is capable of fully characterizing an unknown quantum process. Standard quantum process tomography suffers from an exponentially scaling number of measurements and complicated data post-processing due to the curse of dimensionality. On the other hand, non-unitary operators are more realistic cases. In this work, we put forward a variational quantum process tomography method based on the supervised quantum machine learning framework. It approximates the unknown non-unitary quantum process utilizing a relatively shallow depth parametric quantum circuit and fewer input states. Numerically, we verified our method by reconstructing the non-unitary quantum mappings up to eight qubits in two cases: the weighted sum of the randomly generated quantum circuits and the imaginary time evolution of the Heisenberg XXZ spin chain Hamiltonian. Results show that those quantum processes could be reconstructed with high fidelities (>99%) and shallow depth parametric quantum circuits (d≤8), while the number of input states required is at least two orders of magnitude less than the demands of the standard quantum process tomography. Our work shows the potential of the variational quantum process tomography method in characterizing non-unitary operators.