 Research
 Open access
 Published:
Efficient quantum state estimation with lowrank matrix completion
EPJ Quantum Technology volume 11, Article number: 50 (2024)
Abstract
This paper introduces a novel and efficient technique for quantum state estimation, coined as lowrank matrixcompletion quantum state tomography for characterizing pure quantum states, as it requires only nonentangling bases and \(2n + 1\) local Pauli operators. This significantly reduces the complexity of the process and increases the accuracy of the state estimation, as it eliminates the need for the entangling bases, which are experimentally difficult to implement on quantum devices. The required minimal postprocessing, improved accuracy and efficacy of this matrixcompletionbased method make it an ideal benchmarking tool for investigating the properties of quantum systems, enabling researchers to verify the accuracy of quantum devices, characterize their performance, and explore the underlying physics of quantum phenomena. Our numerical results demonstrate that this method outperforms contemporary techniques in its ability to accurately reconstruct multiqubit quantum states on real quantum devices, making it an invaluable contribution to the field of quantum state characterization and an essential step toward the reliable deployment of intermediate and largescale quantum devices.
1 Introduction
Noisy intermediatescale quantum (NISQ) devices consisting of hundreds of qubits will be available soon [1]. Several technologies for these NISQ devices are being pursued such as superconducting quantum circuits [2], trapped ions [3], quantum dot [4], cold atoms [5], and photonic platforms [6]. These NISQ computers are expected to perform tasks that surpass the capability of the most powerful classical computers available today [7–9]. However, noisy quantum gates and decoherence limits the desired level of improvement over the existing current classical computing devices for complex algorithms. Bearing this in mind, the development of NISQ devices has posed a paradox: how to validate that the device produces the desired result? For characterization, certification, and benchmarking [10] of these noisy devices, quantum state tomography comes into play, which is the gold standard for the reconstruction of a quantum state [11]. Reconstruction of the quantum state from measurements on the replica of an unknown given quantum state is termed as the quantum state tomography. Quantum state tomography generally consists of a twostep process: the first is collecting the data from optimized measurement design on quantum systems; the second is the classical postprocessing of the gathered data. Quantum state tomography problem becomes intractable due to an exponential growth of the system size with increase in the number of qubits under consideration [12, 13].
A conventional state standard tomography employs \(d^{2}\) or more measurement settings, where \(d=2^{n}\) for n qubits [14–17]. A simple counting argument suggests that only \(O (d )\) measurement settings are sufficient to perform the tomography of pure quantum states [18, 19]. Compressed sensing and lowrank matrix recovery from sparse matrix algorithms have been developed to achieve this bound [20–22]. In [23, 24], the authors proposed \(d+1\) mutually unbiased bases measurements for full quantum state reconstruction. The states with special properties, for instance, matrix product states [25] or permutationally invariant quantum states [26] also lead to a significant reduction in the number of quantum measurement settings. The pure quantum state can also be retrieved with only five and three measurement settings given in [27–29]. Recently, noise resilient and robust selfguided quantum state tomography is proposed [30–32].
The use of entangled bases in most of the aforementioned tomography algorithms is common, however, their implementation on NISQ devices is difficult due to errors introduced by the controlledNOT (CNOT) gate and the high circuit depth leading to qubit decoherence. The CNOT gate is an essential transformation required for implementing these sets of entangled measurements. Hence, these entangled bases significantly reduce the performance of state estimation task. The introduction of local basis measurement for state reconstruction can be a massive catalyst for the benchmarking of quantum devices. For this purpose, local basis measurements are introduced in [33]. They have demonstrated the task of tomography of pure quantum state by using \(kn+1\) measurement bases, where \(k \geq 2\). Their algorithm is based on solving several systems of linear equations. The required number of bases increases (\(k > 2\)) if the determinant of the linear system of equations vanishes. The performance is quite low as compared to the numerical results of entangled measurement algorithms. Given the rank of a quantum state, the projected least squares can be used to achieve higher accuracy with the minimum number of copies. However, this comes at the cost of the local Pauli basis measurement settings increasing exponentially. The success of tomography algorithms, therefore, relies on the efficient use of resources to balance accuracy, efficiency, and fidelity.
The paper proposes a novel approach to quantum state tomography called lowrank matrixcompletion (LRMC) quantum state tomography. This approach allows for the estimation of pure quantum states with only a minimum of \(2n + 1\) local Pauli measurements for an nqubit state. The algorithm makes use of experimental measurements to provide the necessary density matrix elements and then uses singular value shrinkage to fill in the missing entries. The method outperforms the existing techniques, as illustrated by simulations and experiments, and is capable of accurately detecting pure states with computational basis and local Pauli measurements.
2 Method
The method used in this paper for pure quantum state tomography involves a matrix filling technique involving \(2n +1\) local measurements for an nqubit system. This technique utilizes the algorithm for lowrank matrix completion [34]. The algorithm begins by experimentally acquiring enough entries of the pure state’s density matrix. This data is then used by the matrix completion algorithm to obtain the complete density matrix of the desired state using a rank1 approximation.
Filling in the missing entries of a rank1 matrix is directly connected to the problem of finding cycles in a graph [34]. The rank1 matrix of dimension \(d\times d\) can be completely reconstructed by knowing only \(2d1\) entries of the matrix. These \(2d1\) elements form a spanning tree in the rowcolumn graph, as demonstrated in Fig. 1. The successful reconstruction of the missing entries of the given matrix depends on the tree reaching all nodes without forming a loop. If a loop exists in the rowcolumn graph, then it is not possible to reconstruct the rank1 matrix. However, if the \(2d1\) entries of the matrix do not form a cycle, then we can optimally impute all the missing entries using matrix completion techniques.
In this paper, we use the rank shrinkage procedure known as the EckartYoung theorem [35] to compute the rank1 density matrix from less measurement setting. This theorem states that the best lowrank approximation of a given matrix can be achieved by taking its singular value decomposition (SVD) and preserving only the largest singular values and their respective singular vectors. For a matrix A, the rankk approximation of \(A_{k}\) is obtained by retaining only the k largest singular values and their corresponding singular vectors:
where \(\lambda _{i}\) are the singular values, \(u_{i}\) and \(v_{i}\) are the corresponding left and right singular vectors. For any matrix A, its best rankk approximation is given as
where \(\\cdot \ _{F}\) is the Frobenius norm and B any rankk matrix. This remarks that SVD also gives the best low rank approximation in spectral norm [36]. This technique find applications in data compression, noise reduction, and dimensionality reduction. In the context of quantum state tomography, the theorem facilitates the reconstruction of best lowrank quantum states from incomplete measurements settings.
In our algorithm, we perform local Pauli measurements on each subsystem of multipartite pure quantum state. We first employ the computational basis measurement to obtain the diagonal elements of the unknown density matrix. Next, we measure nqubit systems in Pauli measurement setting
Here, i denotes the qubits’s index and j denotes the Pauli operator index, which can be either 1 or 2. The index i ranges \(i=1,2,\ldots ,n\), where \(\sigma _{j}^{i} \in \{\sigma _{1} ,\sigma _{2} \}\) represents Pauli X (\(\sigma _{1}\)) and Pauli Y (\(\sigma _{2}\)) matrices.
We begin by assuming that we have access to N copies of a nqubit multipartite unknown pure state . In the simulation presented in the following section, for the sake of simplicity, we allocate an equal number of copies for all measurement settings. Specifically, we measure each of the \(2n+1\) bases using \(N/(2n + 1)\) shots. In the first step of our algorithm, we obtain the diagonal elements of ρ by measuring \(N/(2n + 1)\) of its copies in the computational basis of the ddimensional Hilbert space. From the measurement results, diagonal elements of ρ can be directly obtained as
where \(\mathbb{Q}_{d} = \{0, 1, \ldots , d  1 \}\), \(N_{i}\) is the number of times measurement result corresponding to was obtained in the first step and ρ̂ denotes an estimate of ρ.
The offdiagonal element \(\rho _{j, k}\) of ρ can be obtained by measuring it in the eigenbasis of Pauli operators pair \(\Lambda _{1}^{i}\) and \(\Lambda _{2}^{i}\). This can be seen by first noticing that the eigenvectors of \(\Lambda _{1}^{i}\) and \(\Lambda _{2}^{i}\) are
where \(\nu \in \mathbb{Q}_{\frac{d}{2^{i}}}\), \(\zeta \in \mathbb{Q}_{2^{i  1}}\) and the basis sets are constructed from all possible combinations of values in ν, ζ and i qubits.
These orthonormal bases build a complete projector system with elements
where \(x\in \{1, 2 \}\), \(j=2^{i} \nu +\zeta \) and \(k=2^{i} \nu +\zeta +2^{i1}\). Let \(p_{x, m}^{j, k}\) be the probability of obtaining the measurement outcome corresponding to the projector \(\Pi _{x, m}^{j, k}\). Then, we have
where is the element of ρ at \((j, k)\)th index. From the hermiticity of ρ, we have \(\rho _{k, j} = \rho _{j, k}^{*}\). Using this fact, we get \(\rho _{j, k} + \rho _{k, j} = 2 \operatorname{Re} \{ \rho _{j, k} \}\) and \(\rho _{j, k}  \rho _{k, j} = 2 \operatorname{Im} \{ \rho _{j, k} \}\). By subtracting equation (9) from (10) and equation (11) from (12), and then combining the results while utilizing the facts about the real and imaginary components of \(\hat{\rho }_{j, k}\), we can derive the value of \(\hat{\rho }_{j, k}\) as
From \(\hat{\rho}_{j,k}\), we can easily get \(\hat{\rho}_{k,j}=\hat{\rho}_{j,k}^{*}\). This setup provides us with a sufficient number of elements to fill in the missing entries of the density matrix. Let ρ̂ be \(d \times d\) matrix containing all entries obtained from measurement operators along with missing entries at \((i,j ) \in \Delta \), i.e.,
Several methods exist for imputing missing entries in a density matrix, such as substituting them with zeros or calculating the average of the respective column or row. However, the choice of initial imputation is less critical since our algorithm will update only these missing values while preserving the values obtained from measurement operators [37, 38]. Let \(\hat{\rho }_{:,j}\) denotes the jth column of the density matrix ρ̂. We calculate the missing entries by taking the average of the each column in the density matrix as
We can also detect if an unknown quantum state is pure using our algorithm without the need for additional measurements. A density matrix ρ is considered pure if the equation
holds for every \(i,j=1,2,\ldots ,d\). We can verify the purity of the unknown state by using Equation (16) on the measured structure from Fig. 2 and applying it to elements of the constructed density matrix.
The complete algorithm for constructing the density matrix of a pure quantum state ρ is as follows:

1.
Measure ρ in the computational basis to obtain the diagonal entries of the density matrix through Equation (4).

2.
Obtain entries of the density matrix \(\hat{\rho}_{j,k}\). These entries are obtained by measuring in all Pauli operators measurement setting of Equation (3) and using Equation (13) as described above.

3.
Fill all the missing entries \(\hat{\rho}_{i,j}\) using (15).

4.
Perform the singular value decomposition of the density matrix
$$\begin{aligned} \hat{ \rho }=L\Sigma R^{\dagger }, \end{aligned}$$(17)where Σ is a diagonal matrix containing singular values entries on its diagonal as \(\lambda _{1},\lambda _{2}, \ldots ,\lambda _{d}\) and L (and R) is the left (and right) orthogonal matrix of dimension d.

5.
Shrink the singular values of ρ̂ by setting singularvalues as
$$\begin{aligned} \hat{\lambda }_{i}= \textstyle\begin{cases} \lambda _{i}\lambda _{r+1}, & \text{for } i\leq r \\ 0 , & \text{for } i> r \end{cases}\displaystyle \end{aligned}$$(18)where r is the rank of the density matrix and we obtain the new diagonal matrix Σ̂ with entries \(\hat{\lambda }_{1},\hat{\lambda }_{2}, \ldots ,\hat{\lambda }_{d}\). Since a pure state in quantum mechanics has a density matrix with only one nonzero eigenvalue, its rank is equal to 1, meaning \(r=1\).

6.
Replace diagonal matrix Σ with Σ̂ to obtain
$$\begin{aligned} \hat{\sigma }=L\hat{\Sigma } R^{\dagger }. \end{aligned}$$(19) 
7.
Update only the missing entries while keeping the entries obtained from measurement unchanged
$$\begin{aligned} \hat{\rho } (\Delta )=\hat{\sigma } (\Delta ) , \quad \forall (i,j) \in \Delta . \end{aligned}$$(20) 
8.
Repeat steps 47 until \(\\hat{\rho }_{\mathrm{iter}}\hat{\rho }_{\mathrm{iter}1}\_{F} \leq \epsilon \), where ϵ is some tolerence, is reached either after p times or after K iterations.
A schematic diagram of the experimentally obtained density matrix is shown in Fig. 2 where
and the complete procedure is outlined in Algorithm 1. The convergence of this iterative approach is discussed in [37–39], which addresses the norm minimization problem. Here, the iterative process is governed by two main stopping criteria:

Tolerance: A predefined threshold for the Frobenius norm of the difference between successive density matrix estimates, ensuring that iterations continue until the estimate stabilizes within this threshold.

Patience: A limit on the number of iterations to perform without significant improvement, preventing endless iteration in cases of slow convergence.
Our proposed algorithm demonstrates effective performance across a broad spectrum of quantum states. However, the algorithm struggles to reconstruct excessively sparse quantum states. It can accurately reconstruct GreenbergerHorneZeilinger (GHZ) states of the form , but it encounters difficulties with states like , failing to account for the imaginary component. We obtain the same density matrix from the reconstruction of the and states.
3 Results
To gauge the accuracy of state tomography, we use the common figure of merit, infidelity, which characterizes the distance between these states. Infidelity is defined as
where \(F (\rho ,\sigma )\) is the fidelity between ρ and σ.
Our proposed algorithm is compared to two existing techniques, the fivebasis method, and the scalable estimation method, in order to evaluate its performance. The fivebasis method employs entangled basis measurements to reconstruct pure quantum states, while the scalable estimation method relies on local basis measurements. To assess the accuracy of each method, we generated 10^{2} randomly selected pure quantum states according to the Haar distribution, with n varying from 2 to 5, and plotted the median infidelity as a function of the number of shots for each. The comparison of these experimental simulated results is demonstrated in Fig. 3 and shows that our algorithm outperforms both the fivebasis and scalable estimation methods, as it exhibits lower median infidelity.
Our technique has been experimentally verified using a 7qubit IBMQJakarta superconducting quantum computer [40]. This device, with average measurement and CNOT errors of \(2.220e^{2}\) and \(7.511e^{3}\) respectively, is used to implement our algorithm. The circuit design, shown in Fig. 2, is used to generate 10^{2} Haarrandom pure threequbit states on the real IBMQJakarta device. The median fidelity results are then plotted as a kernel density estimate (KDE) in Fig. 4, along with the least square with known rank one projection. These numerical results demonstrate the precision of our algorithm in benchmarking cloudbased IBM quantum devices, as compared to the projected least square.
We observe that the state preparation error associated with generating random quantum states on a real quantum computer becomes increasingly dominant as the number of qubits increases. To overcome this challenge, we utilized GHZ states, which are a class of entangled quantum states that are easy to prepare on real quantum hardware. An nqubit GHZ state is shown as
In Fig. 5, we employ a different device, IBMQManila, for our experiment. The KDE of the median fidelity for 11 GHZ states with \(n=3,4,5\) qubits is plotted, using 2^{13} shots. This device has an average readout and CNOT error of \(2.110e^{2}\) and \(6.33e^{3}\), respectively. Our algorithm is able to reconstruct GHZ states with a median fidelity of over 86% for up to five qubits on real IBM devices.
4 Conclusion
The proposed algorithm in this paper offers a significant advancement in the estimation of pure states on NISQ devices. By relying on local measurements and local gates, it provides a computationally efficient solution that is less susceptible to errors and decoherence. The implementation of local Pauli bases with simple gates such as H and S contributes to the improved accuracy of the algorithm. Additionally, the reduced circuit depth of the algorithm can have a crucial impact in mitigating the negative effects of gate errors and decoherence that are prevalent in real quantum devices. Overall, this algorithm for estimating pure states has the potential to play a crucial role in improving the accuracy of estimation and benchmarking tasks on NISQ devices. Moreover, its possible extension to mixed states through carefully designed measurements and higher rank approximations could further enhance its applicability in quantum computing. Thus, this algorithm holds great promise for advancing the capabilities of NISQ devices and bringing us closer to practical quantum computing applications.
Data availability
The source code of the numerical and experimental generated data that support the findings of this study are available from the corresponding author upon reasonable request.
References
Preskill J. Quantum computing in the NISQ era and beyond. Quantum. 2018;2:79.
Kandala A, Mezzacapo A, Temme K, Takita M, Brink M, Chow JM, Gambetta JM. Hardwareefficient variational quantum eigensolver for small molecules and quantum magnets. Nature. 2017;549(7671):242–6.
Zhang J, Pagano G, Hess PW, Kyprianidis A, Becker P, Kaplan H, Gorshkov AV, Gong ZX, Monroe C. Observation of a manybody dynamical phase transition with a 53qubit quantum simulator. Nature. 2017;551(7682):601–4.
Loss D, DiVincenzo DP. Quantum computation with quantum dots. Phys Rev A. 1998;57:120–6.
Bernien H, Schwartz S, Keesling A, Levine H, Omran A, Pichler H, Choi S, Zibrov AS, Endres M, Greine M, et al.. Probing manybody dynamics on a 51atom quantum simulator. Nature. 2017;551(7682):579–84.
Reck M, Zeilinger A, Bernstein HJ, Bertani P. Experimental realization of any discrete unitary operator. Phys Rev Lett. 1994;73:58–61.
Terhal BM. Quantum supremacy, here we come. Nat Phys. 2018;14(6):530–1.
Madsen LS, Laudenbach F, Askarani MF, Rortais F, Vincent T, Bulmer JF, Miatto FM, Neuhaus L, Helt LG, Collins MJ, et al.. Quantum computational advantage with a programmable photonic processor. Nature. 2022;606(7912):75–81.
Arute F, Arya K, Babbush R, Bacon D, Bardin JC, Barends R, Biswas R, Boixo S, Brandao FG, Buell DA, et al.. Quantum supremacy using a programmable superconducting processor. Nature. 2019;574(7779):505–10.
Eisert J, Hangleiter D, Walk N, Roth I, Markham D, Parekh R, Chabaud U, Kashefi E. Quantum certification and benchmarking. Nat Rev Phys. 2020;2(7):382–90.
Torlai G, Melko RG. Machinelearning quantum states in the NISQ era. Annu Rev Condens Matter Phys. 2020;11(1):325–44.
Shabani A, Kosut RL, Mohseni M, Rabitz H, Broome MA, Almeida MP, Fedrizzi A, White AG. Efficient measurement of quantum dynamics via compressive sensing. Phys Rev Lett. 2011;106:100401.
Lloyd S, Mohseni M, Rebentrost P. Quantum principal component analysis. Nat Phys. 2014;10(9):631–3.
Banaszek K, D’Ariano GM, Paris MGA, Sacchi MF. Maximumlikelihood estimation of the density matrix. Phys Rev A. 1999;61:010304.
Opatrný T, Welsch DG, Vogel W. Leastsquares inversion for densitymatrix reconstruction. Phys Rev A. 1997;56:1788–99.
Qi B, Hou Z, Li L, Dongi D, Xiang G, Guo G. Quantum state tomography via linear regression estimation. Sci Rep. 2013;3:3496.
BlumeKohout R. Optimal, reliable estimation of quantum states. New J Phys. 2010;12(4):043034.
Dodonov VV, Man’ko VI. Positive distribution description for spin states. Phys Lett A. 1997;229(6):335–9.
Amiet JP, Weigert S. Reconstructing a pure state of a spinsthrough three SternGerlach measurements. J Phys A, Math Gen. 1999;32(15):2777–84.
Steffens A, Riofrío CA, McCutcheon W, Roth I, Bell BA, McMillan A, Tame MS, Rarity JG, Eisert J. Experimentally exploring compressed sensing quantum tomography. Quantum Sci Technol. 2017;2(2):025005.
Kueng R, Rauhut H, Terstiege U. Low rank matrix recovery from rank one measurements. Appl Comput Harmon Anal. 2017;42(1):88–116.
Bolduc E, Knee GC, Gauger EM, Leach J. Projected gradient descent algorithms for quantum state tomography. npj Quantum Inf. 2017;3(1):1–9.
Wootters WK, Fields BD. Optimal statedetermination by mutually unbiased measurements. Ann Phys. 1989;191(2):363–81.
Adamson RBA, Steinberg AM. Improving quantum state estimation with mutually unbiased bases. Phys Rev Lett. 2010;105:030406.
Cramer M, Plenio MB, Flammia ST, Somma R, Gross D, Bartlett SD, LandonCardinal O, Poulin D, Liu YK. Efficient quantum state tomography. Nat Commun. 2010;1(1):1–7.
Tóth G, Wieczorek W, Gross D, Krischek R, Schwemmer C, Weinfurter H. Permutationally invariant quantum tomography. Phys Rev Lett. 2010;105:250403.
Goyeneche D, Cańas G, Etcheverry S, Gómez ES, Xavier GB, Lima G, Delgado A. Five measurement bases determine pure quantum states on any dimension. Phys Rev Lett. 2015;115:090401.
Carmeli C, Heinosaari T, Kech M, Schultz J, Toigo A. Stable pure state quantum tomography from five orthonormal bases. Europhys Lett. 2016;115(3):30001.
Zambrano L, Pereira L, Martínez D, Cańas G, Lima G, Delgado A. Estimation of pure states using three measurement bases. Phys Rev Appl. 2020;14:064004.
Rambach M, Qaryan M, Kewming M, Ferrie C, White AG, Romero J. Robust and efficient highdimensional quantum state tomography. Phys Rev Lett. 2021;126:100402.
Ferrie C. Selfguided quantum tomography. Phys Rev Lett. 2014;113:190404.
Chapman RJ, Ferrie C, Peruzzo A. Experimental demonstration of selfguided quantum tomography. Phys Rev Lett. 2016;117:040402.
Pereira L, Zambrano L, Delgado A. Scalable estimation of pure multiqubit states. npj Quantum Inf. 2022;8(1):1–12.
Strang G. In: Linear algebra and learning from data 1(1). 2019. p. 413–28.
Eckart C, Young G. The approximation of one matrix by another of lower rank. Psychometrika. 1936;1(3):211–8.
Kumar NK, Schneider J. Literature survey on low rank approximation of matrices. Linear Multilinear Algebra. 2017;65(11):2212–44.
Troyanskaya O, Cantor M, Sherlock G, Brown P, Hastie T, Tibshirani R, Botstein D, Altman RB. Missing value estimation methods for DNA microarrays. Bioinformatics. 2001;17(6):520–5.
Cho M. Imputation of missing values by low rank matrix approximation. US Bureau of Labor Statistics 2021.
Cai JF, Cand‘es EJ, Shen Z. A singular value thresholding algorithm for matrix completion. SIAM J Optim. 2010;20(4):1956–82.
7qubit backend: IBM Q team, IBM Q 7 Jakarta backend specification v1.2.5. (Accessed: Jan 2023).
Acknowledgements
We acknowledge the use of the IBM Q for this work. The views expressed are those of the authors and do not reflect the official policy or position of IBM or the IBM Quantum team.
Funding
This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2022R1A4A3033401) and by the MSIT (Ministry of Science and ICT), Korea, under the ITRC (Information Technology Research Center) support program (IITP2021002046) supervised by the IITP (Institute for Information & Communications Technology Planning & Evaluation).
Author information
Authors and Affiliations
Contributions
ST and AF contributed the idea. ST, AF, and JR developed the theory and wrote the manuscript. TQD and HS improved the manuscript and supervised the research. All the authors contributed in analyzing and discussing the results and improving the manuscript. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons AttributionNonCommercialNoDerivatives 4.0 International License, which permits any noncommercial use, sharing, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if you modified the licensed material. You do not have permission under this licence to share adapted material derived from this article or parts of it. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/byncnd/4.0/.
About this article
Cite this article
Tariq, S., Farooq, A., Rehman, J.U. et al. Efficient quantum state estimation with lowrank matrix completion. EPJ Quantum Technol. 11, 50 (2024). https://doi.org/10.1140/epjqt/s4050702400261x
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjqt/s4050702400261x