Quantum State Tomography Techniques in Quantum Systems

Quantum state tomography (QST) and quantum process tomography (QPT) are indispensable tools for the characterisation, validation and benchmarking of quantum hardware. Traditional approaches rely on projective measurements combined with maximum-likelihood or least-squares estimation, but these scale exponentially with system size. Recent advances have exploited compressed sensing, tensor network representations and machine-learning algorithms to reduce the number of required measurements and computational overhead. Compressed sensing leverages low-rank structure in density matrices to reconstruct states from incomplete data sets, while tensor networks—such as matrix product states—capture the entanglement structure of many-body systems and enable scalable parameterisation of high-dimensional operators. Meanwhile, data-driven techniques inspired by unsupervised learning and variational ansätze offer adaptive and noise-resilient reconstruction protocols. Together, these developments have extended the applicability of QST and QPT to intermediate-scale quantum processors, trapped-ion simulators and photonic platforms, paving the way for routine diagnostics in quantum computing and communication technologies.

Research from Nature Portfolio

One study introduced a hybrid method combining tensor network representations with unsupervised machine learning to perform quantum process tomography on circuits up to ten qubits. By encoding the process matrix as a low-bond-dimension network and optimising against measurement data, the approach achieved process fidelities above 0.99 with orders of magnitude fewer measurement shots than conventional protocols. A second work demonstrated compressed sensing tomography on a seven-qubit trapped-ion system within a topological colour-code architecture. Using only 127 Pauli basis settings and exploiting low-rank structure, the experiment reconstructed the state with high fidelity despite statistical noise, illustrating the power of incomplete data in large quantum registers. A foundational contribution presented a linear regression estimation algorithm for state reconstruction, mapping tomography onto a parameter-estimation problem and providing analytical error bounds to guide optimal measurement selection. This method offered a clear computational advantage over maximum-likelihood estimation in high-dimensional systems.

Research from all publishers

A novel Fourier quantum process tomography scheme achieved robust characterisation of optical transformations using phase-retrieval concepts. By measuring in two conjugate spaces for varied state preparations, the method reconstructed process matrices with near-minimal measurement sets, reporting fidelities above 90 % and reduced post-processing cost. In another advance, a variational matrix product state ansatz enabled efficient reconstruction of highly entangled 20-qubit states produced by a trapped-ion quantum simulator. The protocol required data from only 27 bases and outperformed neural-network-based estimators in convergence and accuracy. A further development employed constrained gradient-descent on the Stiefel manifold to learn low-rank Kraus operators for both discrete- and continuous-variable processes. This technique combined the measurement efficiency of compressed sensing with the scalability of projected least squares, reconstructing multi-qubit processes with a minimal number of random measurements.

Quantum State Tomography Techniques in Quantum Systems publication trend

The graph below shows the total number of articles in quantum state tomography techniques in quantum systems across all publications each year (not limited to Nature Index journals).

Technical terms

Quantum state tomography: Reconstruction of an unknown quantum state’s density matrix from measurement data.

Quantum process tomography: Determination of an unknown quantum operation or channel by probing its action on known input states.

Tensor network: A factorised representation of high-dimensional tensors that captures entanglement structure efficiently.

Compressed sensing: A framework that exploits low-rank or sparse structure to recover signals from incomplete measurements.

Matrix product state: A one-dimensional tensor network ansatz that parametrises quantum many-body states with limited entanglement.

References

1. Quantum process tomography with unsupervised learning and tensor networks. Nature Communications (2023).
2. Experimental quantum compressed sensing for a seven-qubit system. Nature Communications (2017).
3. Quantum State Tomography via Linear Regression Estimation. Scientific Reports (2013).
4. Fourier Quantum Process Tomography. npj Quantum Information (2024).
5. Reconstructing Complex States of a 20-Qubit Quantum Simulator. PRX Quantum (2023).
6. Gradient-Descent Quantum Process Tomography by Learning Kraus Operators. Physical Review Letters (2023).