In this thesis, we study the interplay of constraints and complexity in quantum estimation. We investigate three inference problems, where additional structure in the form of constraints is exploited to reduce the sample and/or computational complexity. The first example is concerned with uncertainty quantification in quantum state estimation, where the positive-semidefinite constraint is used to construct more powerful, that is smaller, error regions. However, as we show in this work, doing so in an optimal way constitutes a computationally hard problem, and therefore, is intractable for larger systems. The second inference problem deals with phase retrieval and its application to characterizing linear optical circuits. The main chal- lenge here is the fact that the measurements are insensitive to complex phases, and hence, reconstruction requires deliberate utilization of interference. We propose a reconstruction algorithm based on ideas from low-rank matrix recovery. More specifically, we map the problem of reconstruction from phase-insensitive measurements to the problem of recovering a rank-one matrix from linear measurements. For the efficient solution of the latter it is crucial to exploit the rank-one constraint. Finally, we investigate low-rank tensor recovery – the problem of reconstructing a low-complexity tensor embedded in an exponentially large space. We provide evidence that with high probability succesful recovery is feasible by Gaussian product tensors with the number of measurements only depending on the target’s intrinsic complexity, and hence, scaling polynomially in the order of tensor. Therefore, the low-rank constraint can be exploited to dramatically reduce the sample complexity of the problem. Additionally, the use of measurement tensors with an efficient representation is necessary for computational efficiency.
In this thesis, we study the interplay of constraints and complexity in quantum estimation. We investigate three inference problems, where additional structure in the form of constraints is exploited to reduce the sample and/or computational complexity. The first example is concerned with uncertainty quantification in quantum state estimation, where the positive-semidefinite constraint is used to construct more powerful, that is smaller, error regions. However, as we show in this work, doing so in an optimal way constitutes a computationally hard problem, and therefore, is intractable for larger systems. The second inference problem deals with phase retrieval and its application to characterizing linear optical circuits. The main chal- lenge here is the fact that the measurements are insensitive to complex phases, and hence, reconstruction requires deliberate utilization of interference. We propose a reconstruction algorithm based on ideas from low-rank matrix recovery. More specifically, we map the problem of reconstruction from phase-insensitive measurements to the problem of recovering a rank-one matrix from linear measurements. For the efficient solution of the latter it is crucial to exploit the rank-one constraint. Finally, we investigate low-rank tensor recovery – the problem of reconstructing a low-complexity tensor embedded in an exponentially large space. We provide evidence that with high probability succesful recovery is feasible by Gaussian product tensors with the number of measurements only depending on the target’s intrinsic complexity, and hence, scaling polynomially in the order of tensor. Therefore, the low-rank constraint can be exploited to dramatically reduce the sample complexity of the problem. Additionally, the use of measurement tensors with an efficient representation is necessary for computational efficiency.
Existing quantum state tomography methods are limited in scalability due to their high computation and memory demands, making them impractical for the recovery of large quantum states. In this work, we address these limitations by reformulating the maximum likelihood estimation problem using the Burer--Monteiro factorization, resulting in a nonconvex but low-rank parametrization of the density matrix. We derive a fully unconstrained formulation by analytically eliminating the trace-one and positive-semidefinite constraints, thereby avoiding the need for projection steps during optimization. Furthermore, we determine the Lagrange multiplier associated with the unit-trace constraint a priori, reducing computational overhead. The resulting formulation is amenable to scalable first-order optimization, and we demonstrate its tractability using the limited-memory Broyden-Fletcher-Goldfarb-Shanno method. Importantly, we also propose a low-memory version of the above algorithm to fully recover certain large quantum states with Pauli-based positive-operator-value measurements. Our low-memory algorithm avoids explicitly forming any density matrix, and does not require the density matrix to have a matrix product state or other tensor structure. For a fixed number of measurements and fixed rank, our algorithm requires just O(dlog2d) complexity per iteration to recover a $d\ifmmode\times\else\texttimes\fi{}d$ density matrix. Additionally, we derive a useful error bound that can be used to give a rigorous termination criterion. We numerically demonstrate that our method is competitive with state-of-the-art algorithms for moderately sized problems, and then demonstrate that our method can solve a 20-qubit problem on a laptop in under 5 hours.
Since the complexity of quantum state tomography (QST) scales exponentially with system size, exploiting priors such as low-rankness, tensor-network structures, and neural-network representations is essential for scalable QST in terms of sample complexity and parameter complexity. In this paper, we introduce a unified framework, termed structured factorization, that builds on BurerMonteiro-type factorization by parametrizing the density matrix as FF†, where the factor F is constrained to belong to a structured model class. This factorization guarantees physical validity by construction while allowing a broad range of structural priors to be incorporated directly through the choice of the factor space, ranging from the generic Cholesky decomposition to low-rank matrices, matrix product operators, and neural density operators based on multilayer perceptron and transformer architectures. Building on this structured factorization framework, we formulate QST as an optimization problem over the factor space from measurement data. We first develop a unified statistical analysis of the sample complexity of least-squares estimation for a broad class of structured quantum states. We then propose a projected gradient descent method that operates directly on the factor space and accommodates a wide range of structural parametrizations and reconstruction objectives. To further exploit the geometry of the maximum-likelihood estimation formulation and the constraints on the factors, we derive a power method that yields a step-size-free algorithm with fast convergence, recovering Covers method as a special case when the factor is unconstrained.
Quantum sensors promise measurement sensitivities that can scale at the Heisenberg limit, but in practice their performance is often degraded by noise, finite sampling, and implementation imperfections. In this work we present a general framework for improving parameter estimation in such settings by exploiting intrinsic structural constraints of time-domain correlation functions. Our approach builds on the observation of Kemper et al. [PRL 132, 160403 (2024)] that two-time correlation functions of Hermitian observables generate Gram matrices that are positive semidefinite, a property that can be violated in experimentally acquired data. We formulate signal reconstruction as a convex optimization problem that enforces positive semidefiniteness, Toeplitz structure, and low-rank priors motivated by the underlying dynamics. We show analytically that, under suitable conditions, the ground-truth signal can be uniquely identified in the noiseless case and recovered stably in the presence of noise. We further demonstrate numerically, in a GHZ-based magnetometry protocol, that enforcing these physical constraints can significantly improve frequency estimation from sparse and noisy data. In particular, we observe a clear advantage in the data-starved regime, where only a small number of time samples are available and standard spectral estimation methods, including matrix pencil techniques, provide limited or unstable improvement over direct fitting. While the reconstructed signals do not in general reach the shot-noise-limited performance, the proposed approach consistently reduces estimation error and recovers much of the underlying structure of the signal. These results indicate that incorporating universal physical constraints into data analysis can enhance the practical performance of quantum sensing protocols without requiring additional hardware resources or calibration.
This thesis investigates three distinct facets of the theory of quantum information. The first two, quantum state estimation and quantum process estimation, are closely related and deal with the question of how to estimate the classical parameters in a quantum mechanical model. The third attempts to bring quantum theory as close as possible to classical theory through the formalism of quasi-probability. \n \nBuilding a large scale quantum information processor is a significant challenge. First, we require an accurate characterization of the dynamics experienced by the device to allow for the application of error correcting codes and other tools for implementing useful quantum algorithms. The necessary scaling of computational resources needed to characterize a quantum system as a function of the number of subsystems is by now a well studied problem (the scaling is generally exponential). However, irrespective of the computational resources necessary to just write-down a classical description of a quantum state, we can ask about the experimental resources necessary to obtain data (measurement complexity) and the computational resources necessary to generate such a characterization (estimation complexity). These problems are studied here and approached from two directions. \n \nThe first problem we address is that of quantum state estimation. We apply high-level decision theoretic principles (applied in classical problems such as, for example, universal data compression) to the estimation of a qubit state. We prove that quantum states are more difficult to estimate than their classical counterparts by finding optimal estimation strategies. These strategies, requiring the solution to a difficult optimization problem, are difficult to implement in practise. Fortunately, we find estimation algorithms which come close to optimal but require far fewer resources to compute. Finally, we provide a classical analog of this quantum mechanical problem which reproduces, and gives intuitive explanations for, many of its features, such as why adaptive tomography can quadratically reduce its difficulty. \n \nThe second method for practical characterization of quantum devices takes is applied to the problem of quantum process estimation. This differs from the above analysis in two ways: (1) we apply strong restrictions on knowledge of various estimation and control parameters (the former making the problem easier, the latter making it harder); and (2) we consider the problem of designing future experiments based on the outcomes of past experiments. We show in test cases that adaptive protocols can exponentially outperform their off-line counterparts. Moreover, we adapt machine learning algorithms to the problem which bring these experimental design methodologies to realm of experimental feasibility. \n \nIn the final chapter we move away from estimation problems to show formally that a classical representation of quantum theory is not tenable. This intuitive conclusion is formally borne out through the connection to quasi-probability -- where it is equivalent to the necessity of negative probability in all such representations of quantum theory. In particular, we generalize previous no-go theorems to arbitrary classical representations of quantum systems of arbitrary dimension. We also discuss recent progress in the program to identify quantum resources for subtheories of quantum theory and operational restrictions motivated by quantum computation.
Quantum state tomography (QST) is a fundamental task in quantum information science that aims to reconstruct unknown quantum states from measurement data. However, the exponential growth of Hilbert-space dimension with system size makes full tomography of general quantum states statistically and computationally prohibitive. This challenge has motivated extensive research on structured quantum state tomography, where prior structure, such as low-rankness, tensor-network representations, shallow quantum circuits, and neural quantum states, can substantially reduce the effective degrees of freedom and enable scalable recovery. In this review, we provide a unified perspective on QST for structured quantum states through three closely related themes: compact state representations, measurement design, and computational algorithms. After reviewing common models for structured quantum states, we survey existing work on geometric preservation properties of measurement frameworks, ranging from informationally complete POVMs to randomized measurements, and their implications for sample complexity. On the algorithmic side, we review optimization methods for reconstructing structured quantum states from empirical measurements. By connecting QST with broader principles from compressive sensing, matrix sensing, and structured inverse problems, this survey highlights common theoretical foundations underlying sample complexity, measurement efficiency, and scalable recovery.
Optimization is a fundamental area in mathematics and computer science, with many real-world applications. In this thesis we study the efficiency with which we can solve certain optimization problems from two different perspectives. Firstly, we study it from the perspective of matrix factorization ranks, which comes with connections to quantum information theory. Secondly, we study it from the perspective of quantum computing. This thesis is accordingly divided into two parts, where we take these perspectives. In the first part of this thesis we study several matrix factorization ranks and their connections to quantum information theory. In Chapter 5, we first give a unified approach to lower bounding matrix factorization ranks, using polynomial optimization techniques. In Chapter 6, we exploit the connection between one particular factorization rank, the completely positive semidefinite rank, and quantum correlations to provide an explicit family of matrices with a large completely positive semidefinite factorization rank. In Chapter 8, we use the same connection to study quantum versions of classical graph parameters from the perspective of polynomial optimization (albeit in noncommutative variables). In Chapter 7 we propose and study a new measure for the amount of entanglement needed to realize quantum correlations. We can approximate this new measure using our approach to matrix factorization ranks, i.e., through polynomial optimization in noncommutative variables. In the second part of this thesis we turn our attention to the question: Can we solve optimization problems faster on a quantum computer? First, in Chapter 10, we look at the problem of evaluating a Boolean function. We give a new semidefinite programming characterization of the minimum number of quantum queries to the input that are needed to determine the corresponding function value. In Chapter 11 we revisit the semidefinite programming problem that plays a crucial role in the rest of the thesis; we provide a quantum algorithm for solving semidefinite programs. Finally, in Chapter 12, we study the general problem of solving convex optimization problems in the oracle model. We provide both upper and lower bounds on the efficiency of various reductions in the quantum setting. In particular, we show that quantum computers are more efficient than classical computers for the task of answering separation queries when access to the convex problem is given through membership queries.