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

arXiv:2012.14494 · New Journal of Physics

Quantum state tomography as a numerical optimization problem

Violeta N. Ivanova-Rohling, Guido Burkard, Niklas Rohling

Published Nov 23, 2021• 1 citation
arXivHTMLPDF

Abstract

Abstract We present a framework that formulates the quest for the most efficient quantum state tomography (QST) measurement set as an optimization problem which can be solved numerically, where the optimization goal is the maximization of the information gain. This approach can be applied to a broad spectrum of relevant setups including measurements restricted to a subsystem. To illustrate the power of this method we present results for the six-dimensional Hilbert space constituted by a qubit–qutrit system, which could be realized e.g. by the 14N nuclear spin-1 and two electronic spin states of a nitrogen-vacancy center in diamond. Measurements of the qubit subsystem are expressed by projectors of rank three, i.e. projectors on half-dimensional subspaces. For systems consisting only of qubits, it was shown analytically that a set of projectors on half-dimensional subspaces can be arranged in an informationally optimal fashion for QST, thus forming so-called mutually unbiased subspaces. Our method goes beyond qubits-only systems and we find that in dimension six such a set of mutually-unbiased subspaces can be approximated with a deviation irrelevant for practical applications.

DOI
10.1088/1367-2630/ac3c0e

Related papers

Optimal quantum state tomography with noisy gates

Violeta N. Ivanova-Rohling, Niklas Rohling, Guido Burkard

Abstract Quantum state tomography (QST) represents an essential tool for the characterization, verification, and validation (QCVV) of quantum processors. Only for a few idealized scenarios, there are analytic results for the optimal measurement set for QST. E.g., in a setting of non-degenerate measurements, an optimal minimal set of measurement operators for QST has eigenbases which are mutually unbiased. However, in other set-ups, dependent on the rank of the projection operators and the size of the quantum system, the optimal choice of measurements for efficient QST needs to be numerically approximated. We have generalized this problem by introducing the framework of customized efficient QST. Here we extend customized QST and look for the optimal measurement set for QST in the case where some of the quantum gates applied in the measurement process are noisy. To achieve this, we use two distinct noise models: first, the depolarizing channel, and second, over- and under-rotation in single-qubit and to two-qubit gates (for further information, please see Methods). We demonstrate the benefit of using entangling gates for the efficient QST measurement schemes for two qubits at realistic noise levels, by comparing the fidelity of reconstruction of our optimized QST measurement set to the state-of-the-art scheme using only product bases.

arXiv:2203.05677•EPJ Quantum Technology•Jun 28, 2023•14 citations

Qutrit state tomography via optimal unambiguous state discrimination and mutually unbiased measurements

Hao Yuan, Xuan Shi, Lian-Fang Han

We propose an efficient scheme for qutrit state tomography based on optimal unambiguous discrimination of three nonorthogonal pure qutrit states. In our scheme, we first introduce an ancillary qubit initially prepared in a known state and apply a joint unitary transformation on the whole system composed of a qutrit state and an ancillary qubit state. Afterwards, we make a local von Neumann measurement (VNM) on the ancillary qubit. If it is projected onto the state |0〉, we finally perform mutually unbiased measurements (MUMs) on the collapsed qutrit state arbitrarily chosen from a complete set of four mutually unbiased bases (MUBs). With only six properly-chosen projectors, the real and imaginary parts of non-diagonal elements can be entirely obtained. Together with the diagonal elements obtained directly from one VNM in the computational bases, the density matrix of an arbitrary unknown qutrit state can be fully determined. Compared with the previous work (R. Salazar and A. Delgado 2012), our scheme consumes much less resources. In comparison with MUBs-based tomography scheme (H. Yuan, et al., 2016), our scheme requires more measurements and less projectors, but pays the price of a loss of certain fraction identically prepared states. Our proposal can be extended to the case of other d-dimensional quantum states if their d MUBs are exactly known.

Results in Physics•Jan 22, 2022•1 citation

Optimal choice of state tomography quorum formed by projection operators

Violeta N. Ivanova-Rohling, Niklas Rohling

A minimal set of measurement operators for quantum state tomography has in the nondegenerate case ideally eigenbases which are mutually unbiased. This is different for the degenerate case. Here we consider the situation where the measurement operators are projections on individual pure quantum states. This corresponds to maximal degeneracy. We present numerically optimized sets of projectors and find that they significantly outperform those which are taken from a set of mutually unbiased bases.

arXiv:1810.09484•Physical Review A•Sep 24, 2019•7 citations

Heuristically optimizing, synthesizing, and prioritizing measurement settings for quantum state tomography

Sumukh S. Moudghalya, Anton Frisk Kockum, Akshay Gaikwad

A key task in many quantum-computing applications, e.g., quantum simulation and quantum state tomography (QST), is to partition an arbitrary set of operators into mutually commuting subsets for efficient measurements. However, brute-force approaches to this task quickly become intractable as the number and dimensionality of operators grow. Here, we reformulate operator partitioning as a graph-coloring (GC) problem and develop an efficient computational framework to solve it, balancing accuracy and efficiency. Our framework enables leveraging a range of GC algorithms, which we benchmark for operator partitioning. Then, we demonstrate their utility in optimizing QST experiments, where determining non-overlapping data acquisition settings for QST is a major challenge, and prioritizing among these settings, i.e., selecting the experiments that provide the most information. We further show how to perform these experiments by synthesizing Clifford circuits for joint measurement of commuting Pauli operators in multi-qubit systems. We validate our framework across multi-qubit (up to five qubits), multi-qutrit (up to three qutrits), and hybrid qubit-qutrit systems. Our results show that heuristic GC methods substantially reduce the number of required measurement settings for QST and enable priority-based scheduling that maximizes the information gain per experiment. The optimization converges within minutes on a student-grade laptop, providing speedups of several orders of magnitude over brute-force methods already for these relatively small quantum systems. This demonstrates the potential of GC heuristics as a scalable and practical tool for characterization of noisy intermediate-scale quantum devices. We have made the Python implementation of our GC framework to optimize and schedule QST experiments publicly available at https://github.com/ssm8015/QST_GT.git.

quant-pharXiv:2609.02633•Sep 3, 2026•0 citations

Quantum Optical Implementation of Adaptive Quantum State Tomography

Phillip Cornett, Albert B. Dinkins V, Lucas Vogeli +2 more

Quantum state tomography (QST) estimates an unknown quantum state from a measurement process typically performed on identical copies of the target state and plays a core role in characterizing quantum information systems. However, because quantum measurement outcomes are probabilistic, QST methods often require many shots per basis to achieve high-fidelity estimates, which hinders scalability to many-qubit applications and slows the transmission of quantum information. A more efficient procedure is therefore required for modern-day quantum operations in quantum computing, communications, and sensing. We address this need with an adaptive QST approach that measures a single-qubit state in a wide set of bases. We include bases beyond the Pauli triad {X,Y,Z} by leveraging the infinite-dimensional Hilbert space of quantum mechanics, increasing the likelihood of producing highly polarized measurement outcomes and finding an optimally informative basis. With this basis, the state can be identified with fewer measurements by focusing on directions that most reduce uncertainty of the state. In addition to theoretical innovation in [9], we experimentally verify in a simplified photonic realization using 13 bases that having many more available measurement bases than the Pauli triad benefits QST. This engineering implementation demonstrates that adaptive QST is both efficient in performance and practical to implement.

International Symposium on Wireless Personal Multimedia Communications (WPMC)•Nov 9, 2025•1 citation

Optimal quantum tomography of permutationally invariant qubits

A. B. Klimov, Gunnar G. Bjork, Luis Lorenzo Sánchez-Soto

Mutually unbiased bases determine an optimal set of measurements to extract complete information about the quantum state of a system. However, quite often a priori information about the state exists, making some of the measurement bases superfluous. This is, for example, the case when multiqubit states belong to the permutationally invariant subspace. In this paper we derive the minimal sets of mutually unbiased bases needed to tomographically reconstruct such states.

arXiv:1301.2123•Physical Review A•Jan 10, 2013•25 citations
Abstract We present a framework that formulates the quest for the most efficient quantum state tomography (QST) measurement set as an optimization problem which can be solved numerically, where the optimization goal is the maximization of the information gain. This approach can be applied to a broad spectrum of relevant setups including measurements restricted to a subsystem. To illustrate the power of this method we present results for the six-dimensional Hilbert space constituted by a qubit–qutrit system, which could be realized e.g. by the 14N nuclear spin-1 and two electronic spin states of a nitrogen-vacancy center in diamond. Measurements of the qubit subsystem are expressed by projectors of rank three, i.e. projectors on half-dimensional subspaces. For systems consisting only of qubits, it was shown analytically that a set of projectors on half-dimensional subspaces can be arranged in an informationally optimal fashion for QST, thus forming so-called mutually unbiased subspaces. Our method goes beyond qubits-only systems and we find that in dimension six such a set of mutually-unbiased subspaces can be approximated with a deviation irrelevant for practical applications.
Abstract Quantum state tomography (QST) represents an essential tool for the characterization, verification, and validation (QCVV) of quantum processors. Only for a few idealized scenarios, there are analytic results for the optimal measurement set for QST. E.g., in a setting of non-degenerate measurements, an optimal minimal set of measurement operators for QST has eigenbases which are mutually unbiased. However, in other set-ups, dependent on the rank of the projection operators and the size of the quantum system, the optimal choice of measurements for efficient QST needs to be numerically approximated. We have generalized this problem by introducing the framework of customized efficient QST. Here we extend customized QST and look for the optimal measurement set for QST in the case where some of the quantum gates applied in the measurement process are noisy. To achieve this, we use two distinct noise models: first, the depolarizing channel, and second, over- and under-rotation in single-qubit and to two-qubit gates (for further information, please see Methods). We demonstrate the benefit of using entangling gates for the efficient QST measurement schemes for two qubits at realistic noise levels, by comparing the fidelity of reconstruction of our optimized QST measurement set to the state-of-the-art scheme using only product bases.
We propose an efficient scheme for qutrit state tomography based on optimal unambiguous discrimination of three nonorthogonal pure qutrit states. In our scheme, we first introduce an ancillary qubit initially prepared in a known state and apply a joint unitary transformation on the whole system composed of a qutrit state and an ancillary qubit state. Afterwards, we make a local von Neumann measurement (VNM) on the ancillary qubit. If it is projected onto the state |0〉, we finally perform mutually unbiased measurements (MUMs) on the collapsed qutrit state arbitrarily chosen from a complete set of four mutually unbiased bases (MUBs). With only six properly-chosen projectors, the real and imaginary parts of non-diagonal elements can be entirely obtained. Together with the diagonal elements obtained directly from one VNM in the computational bases, the density matrix of an arbitrary unknown qutrit state can be fully determined. Compared with the previous work (R. Salazar and A. Delgado 2012), our scheme consumes much less resources. In comparison with MUBs-based tomography scheme (H. Yuan, et al., 2016), our scheme requires more measurements and less projectors, but pays the price of a loss of certain fraction identically prepared states. Our proposal can be extended to the case of other d-dimensional quantum states if their d MUBs are exactly known.
A key task in many quantum-computing applications, e.g., quantum simulation and quantum state tomography (QST), is to partition an arbitrary set of operators into mutually commuting subsets for efficient measurements. However, brute-force approaches to this task quickly become intractable as the number and dimensionality of operators grow. Here, we reformulate operator partitioning as a graph-coloring (GC) problem and develop an efficient computational framework to solve it, balancing accuracy and efficiency. Our framework enables leveraging a range of GC algorithms, which we benchmark for operator partitioning. Then, we demonstrate their utility in optimizing QST experiments, where determining non-overlapping data acquisition settings for QST is a major challenge, and prioritizing among these settings, i.e., selecting the experiments that provide the most information. We further show how to perform these experiments by synthesizing Clifford circuits for joint measurement of commuting Pauli operators in multi-qubit systems. We validate our framework across multi-qubit (up to five qubits), multi-qutrit (up to three qutrits), and hybrid qubit-qutrit systems. Our results show that heuristic GC methods substantially reduce the number of required measurement settings for QST and enable priority-based scheduling that maximizes the information gain per experiment. The optimization converges within minutes on a student-grade laptop, providing speedups of several orders of magnitude over brute-force methods already for these relatively small quantum systems. This demonstrates the potential of GC heuristics as a scalable and practical tool for characterization of noisy intermediate-scale quantum devices. We have made the Python implementation of our GC framework to optimize and schedule QST experiments publicly available at https://github.com/ssm8015/QST_GT.git.
Quantum state tomography (QST) estimates an unknown quantum state from a measurement process typically performed on identical copies of the target state and plays a core role in characterizing quantum information systems. However, because quantum measurement outcomes are probabilistic, QST methods often require many shots per basis to achieve high-fidelity estimates, which hinders scalability to many-qubit applications and slows the transmission of quantum information. A more efficient procedure is therefore required for modern-day quantum operations in quantum computing, communications, and sensing. We address this need with an adaptive QST approach that measures a single-qubit state in a wide set of bases. We include bases beyond the Pauli triad {X,Y,Z} by leveraging the infinite-dimensional Hilbert space of quantum mechanics, increasing the likelihood of producing highly polarized measurement outcomes and finding an optimally informative basis. With this basis, the state can be identified with fewer measurements by focusing on directions that most reduce uncertainty of the state. In addition to theoretical innovation in [9], we experimentally verify in a simplified photonic realization using 13 bases that having many more available measurement bases than the Pauli triad benefits QST. This engineering implementation demonstrates that adaptive QST is both efficient in performance and practical to implement.