Efficient Quantum Measurement Tomography Reduces Computational Cost and Samples

Quantum measurement, a cornerstone of modern physics, relies on accurately characterising the measurements themselves, a process known as tomography. Achieving this efficiently is crucial for validating quantum devices and advancing quantum technologies. Leonardo Zambrano from ICFO – Institut de Ciencies Fotoniques, Sergi Ramos-Calderer from the Centre for Quantum Technologies, National University of Singapore, and colleagues detail a new protocol for fast quantum measurement tomography, achieving optimal sample complexity within the system’s dimensionality. Their work, entitled “Fast quantum measurement tomography with dimension-optimal error bounds”, presents a two-step estimation method that minimises classical computational cost while guaranteeing rigorous error bounds, demonstrated through both theoretical analysis and empirical studies on superconducting qubits.

Quantum measurement tomography (QMT) addresses the critical need to characterise quantum devices, enabling the reconstruction of how a measurement process transforms quantum states. As quantum technologies mature, accurate and efficient characterisation of these measurements becomes paramount for reliable experimentation and device validation. The process infers a positive operator-valued measure (POVM) from observed experimental statistics, essentially reverse-engineering the measurement itself. This is achieved by analysing data collected from a set of known quantum states, termed ‘probe states’, and then using classical computation to estimate the POVM. A POVM describes the probabilities of obtaining different outcomes when a quantum state is measured.

Traditional approaches to QMT often rely on optimisation techniques, such as maximum likelihood estimation, to find the POVM that best fits the observed data. While generally accurate, these methods become computationally demanding as the complexity of the quantum system increases. Consequently, there is a drive to develop methods that are both computationally efficient and provide reliable error bounds, even with limited data. Theoretical guarantees regarding the accuracy of the reconstructed POVM frequently hold only when a large number of measurements are taken, a condition not always practical in real-world scenarios. Reducing the ‘sample complexity’, which refers to the minimum number of measurements needed to reconstruct a measurement with a desired level of accuracy, is crucial for conserving resources, particularly when access to quantum devices is limited or measurements are time-consuming.

This research introduces a streamlined method for measurement tomography that significantly reduces the computational burden typically associated with analysing the data. The new protocol employs a two-stage approach, beginning with a least-squares estimation to generate an initial approximation of the POVM, followed by a projection step to ensure the result represents a physically valid measurement. Rigorous mathematical analysis demonstrates that this method achieves optimal sample complexity, requiring the minimum number of measurements necessary to accurately reconstruct the POVM.

Specifically, for a measurement acting on a d-dimensional quantum system, the protocol requires approximately d² samples to achieve a specified level of accuracy in the worst-case scenario, and slightly fewer samples for average-case performance. This represents a significant improvement over existing methods as the dimensionality of the quantum system increases. Researchers also establish lower bounds on the number of samples required by any measurement tomography protocol, confirming that their method is, within logarithmic factors, sample-optimal. The practicality of this approach is enhanced by its compatibility with specific probe state ensembles, namely global or local 2-designs, which simplify the mathematical analysis and allow for precise, non-asymptotic error guarantees.

To validate the theoretical findings, researchers conducted empirical studies using a superconducting quantum processor comprised of flux-tunable transmon qubits, implementing the tomography protocol on actual quantum hardware to assess its performance in a realistic setting. Results demonstrate that the method performs effectively even in the presence of noise and imperfections inherent in quantum devices, highlighting the importance of careful calibration of the quantum processor. Accurate device parameters, such as relaxation times and decoherence times, are crucial for achieving optimal performance. The availability of the code used in the experiments, utilising open-source libraries like Qibolab and Qibocal, further promotes reproducibility and allows other researchers to build upon these findings.

Researchers demonstrate that for a POVM operating on a d-dimensional quantum system, the protocol requires O(d²) samples to achieve a specified error in the worst-case distance, and O(d) samples for average-case distance, establishing almost matching lower bounds on the sample complexity required for any non-adaptive, single-copy POVM tomography protocol, proving the efficiency of the proposed method. Specifically, they demonstrate that any such protocol requires at least Ω(d²) samples, and Ω(d) samples, establishing that the proposed protocol is sample-optimal up to logarithmic factors, utilising global or local 2-designs as probe ensembles to allow for an analytic, closed-form solution, facilitating rigorous, non-asymptotic error guarantees.

To validate the theoretical findings, researchers conducted empirical tests on a noisy superconducting quantum processor utilising flux-tunable transmon qubits, confirming the protocol’s performance in a realistic setting and demonstrating its robustness to experimental imperfections. The results highlight the practical utility of the method for characterising quantum devices and improving the accuracy of quantum measurements, positioning it as a valuable tool for advancing quantum information processing and quantum technologies.

Future work will focus on extending this protocol to handle more complex measurement scenarios, including adaptive measurement strategies where subsequent measurements are informed by previous results. Investigating the performance of the protocol with different probe state ensembles and exploring techniques to further mitigate the effects of noise are also key areas for future research, with adapting the protocol for use with larger-scale quantum systems and exploring its potential applications in quantum state verification and quantum process tomography representing promising avenues for continued investigation.