This paper explores the problem of quantum measurement complexity. In computability theory, the complexity of a problem is determined by how long it takes an effective algorithm to solve it. This complexity may be compared to the difficulty for a hypothetical oracle machine, the output of which may be verified by a computable function but cannot be simulated on a physical machine. We define a quantum oracle machine for measurements as one that can determine the state by examining a single copy. The complexity of measurement for a realizable machine will then be respect to the number of copies of the state that needs to be examined. A quantum oracle cannot perform simultaneous exact measurement of conjugate variables, although approximate measurement may be performed as circumscribed by the Heisenberg uncertainty relations. When considering the measurement of a variable, there might be residual uncertainty if the number of copies of the variable is limited. Specifically, we examine the quantum measurement complexity of linear polarization of photons that is used in several quantum cryptography schemes and we present a relation using information theoretic arguments. The idea of quantum measurement complexity is likely to find uses in measurements in biological systems.
This paper explores the problem of quantum measurement complexity. In computability theory, the complexity of a problem is determined by how long it takes an effective algorithm to solve it. This complexity may be compared to the difficulty for a hypothetical oracle machine, the output of which may be verified by a computable function but cannot be simulated on a physical machine. We define a quantum oracle machine for measurements as one that can determine the state by examining a single copy. The complexity of measurement for a realizable machine will then be respect to the number of copies of the state that needs to be examined. A quantum oracle cannot perform simultaneous exact measurement of conjugate variables, although approximate measurement may be performed as circumscribed by the Heisenberg uncertainty relations. When considering the measurement of a variable, there might be residual uncertainty if the number of copies of the variable is limited. Specifically, we examine the quantum measurement complexity of linear polarization of photons that is used in several quantum cryptography schemes and we present a relation using information theoretic arguments. The idea of quantum measurement complexity is likely to find uses in measurements in biological systems.
This paper explores the problem of quantum measurement complexity. In computability theory, the complexity of a problem is determined by how long it takes an effective algorithm to solve it. This complexity may be compared to the difficulty for a hypothetical oracle machine, the output of which may be verified by a computable function but cannot be simulated on a physical machine. We define a quantum oracle machine for measurements as one that can determine the state by examining a single copy. The complexity of measurement for a realizable machine will then be respect to the number of copies of the state that needs to be examined. A quantum oracle cannot perform simultaneous exact measurement of conjugate variables, although approximate measurement may be performed as circumscribed by the Heisenberg uncertainty relations. When considering the measurement of a variable, there might be residual uncertainty if the number of copies of the variable is limited. Specifically, we examine the quantum measurement complexity of linear polarization of photons that is used in several quantum cryptography schemes and we present a relation using information theoretic arguments. The idea of quantum measurement complexity is likely to find uses in measurements in biological systems.
This work presents a study of Kolmogorov complexity for general quantum states from the perspective of deterministic-control quantum Turing Machines (dcq-TM). We extend the dcq-TM model to incorporate mixed state inputs and outputs, and define dcq-computable states as those that can be approximated by a dcq-TM. Moreover, we introduce (conditional) Kolmogorov complexity of quantum states and use it to study three particular aspects of the algorithmic information contained in a quantum state: a comparison of the information in a quantum state with that of its classical representation as an array of real numbers, an exploration of the limits of quantum state copying in the context of algorithmic complexity, and study of the complexity of correlations in quantum systems, resulting in a correlation-aware definition for algorithmic mutual information that satisfies symmetry of information property.
Quantum computation - the use of quantum systems as bits, or qubits, to perform computation - has been proven to be exponentially faster than what is thought possible with classical computation for several tasks. Quantum computers have the unique ability to store an amount of information that is exponential in the number of constituent particles used to encode that information. This property, which is arguably where quantum computers derive their power, makes it very difficult to characterize quantum systems. In order to fully characterize an unknown quantum state, a series of measurements must be performed on it and the state deduced from the results of those measurements. This process, called quantum tomography, is known to require exponential resources in the number of qubits. Both the number of measurements - the measurement complexity - and the number of times each measurement must be repeated - the sample complexity - present serious challenges when performing tomography. In addition, the measurement apparatus itself, which is usually a quantum system, must be characterized leading to even more experimental complexity. Here, I present three experiments which aim to reduce the experimental complexity of quantum tomography. The first aims to reduce the number of measurements needed to locate the decoherence-free subspaces of a noisy quantum channel. The process of locating these subspaces, which would normally require full process tomography, is performed with quadratically fewer measurement settings. The second experiment aims to characterize both an uncalibrated measurement apparatus and an unknown quantum state at the same time. With a specific model of measurement, it is found that an uncalibrated parameter of the measurement can be directly extracted using a single measurement. In the third experiment, I perform an experiment in which the optimal sample complexity for quantum state tomography is obtained using adaptivity: the ability to change measurement settings part way through an experiment. All three experiments were performed using the polarization degree of freedom of photons generated by spontaneous parametric downconversion. The source of photon pairs and the measurement techniques used are described in detail, along with the theory of quantum distinguishability and its relevance to tomography.
In this thesis we address the question, what is the resource, or property, that enables the advantage of quantum computers?The theory of quantum computers dates back to the eighties, so one would think there already is an answer to this question.There are several proposed solutions, but to this date, there is no consensus on an answer.Primarily, the advantage of quantum computers is characterized by a speedup for certain computational problems.This speedup is measured by comparing quantum algorithms with the bestknown classical algorithms.For some algorithms we assume access to an object called oracle.The oracle computes a function, and the complexity of the oracle is of no concern.Instead, we count the number of queries to the oracle needed to solve the problem.Informally, the question we ask using an oracle is: if we can compute this function efficiently, what else could we then compute.However, using oracles while measuring a quantum speedup, we assume access to vastly different oracles residing in different models of computation.For our investigation of the speedup, we introduce a classical simulation framework that imitates quantum algorithms.The simulation suggests that the property enabling the potential quantum speedup is the ability to store, process, and retrieve information in an additional degree of freedom.We then theoretically verified that this is true for all problems that can be efficiently solved with a quantum computer.In parallel to this, we also see that quantum oracles sharply specify the information we can retrieve from the additional degree of freedom, while regular oracles do not.A regular oracle does not even allow for an extra degree of freedom.We conclude that comparing quantum with classical oracle query complexity bounds does not provide conclusive evidence for a quantum advantage.I want to thank Jan-Åke Larsson, without whom this research would not have been possible.If there is any magic in the world, it shows through Jan-Åke's ability to simultaneously be a good supervisor, colleague, and friend.I would also like to thank my co-supervisor Peter Jonsson, for many valuable discussions -not only restricted to theoretical computer science.To my friends and collogues that have challenged me to become better, especially Jonathan Jogenfors
Forty years ago, Wiesner proposed using quantum states to create money that is physically impossible to counterfeit, something that cannot be done in the classical world. However, Wiesner's scheme required a central bank to verify the money, and the question of whether there can be unclonable quantum money that anyone can verify has remained open since. One can also ask a related question, which seems to be new: can quantum states be used as copy-protected programs, which let the user evaluate some function f, but not create more programs for f? This paper tackles both questions using the arsenal of modern computational complexity. Our main result is that there exist quantum oracles relative to which publicly-verifiable quantum money is possible, and any family of functions that cannot be efficiently learned from its input-output behavior can be quantumly copy-protected. This provides the first formal evidence that these tasks are achievable. The technical core of our result is a "Complexity-Theoretic No-Cloning Theorem," which generalizes both the standard No-Cloning Theorem and the optimality of Grover search, and might be of independent interest. Our security argument also requires explicit constructions of quantum t-designs. Moving beyond the oracle world, we also present an explicit candidate scheme for publicly-verifiable quantum money, based on random stabilizer states; as well as two explicit schemes for copy-protecting the family of point functions. We do not know how to base the security of these schemes on any existing cryptographic assumption. (Note that without an oracle, we can only hope for security under some computational assumption).
This paper explores the application of quantum information entropy to the theory of computational complexity. We posit that quantum information entropy can serve as a novel metric for quantifying the complexity of quantum algorithms and their associated resource requirements. The core argument is that traditional measures of computational complexity, such as circuit depth or quantum gate count, often fail to fully capture the inherent challenges posed by quantum systems. By leveraging entropy, we aim to provide a more nuanced and potentially more accurate assessment of the difficulty of quantum computations. This work introduces a framework for analyzing quantum algorithms through the lens of information entropy, exploring relationships between algorithmic complexity, entropy generation, and the resources needed to execute these algorithms. Specifically, we examine how entropy can be used to characterize the 'quantumness' of an algorithm and its impact on the overall computational cost. The ultimate goal is to develop a more robust and predictive model for quantum computation, leading to improved algorithm design and resource allocation. ---