Quantum Complexity Theory

Este trabajo presenta una demostración rigurosa de la monotonicidad de entropías estabilizadoras S_α(ρ)= (1/(1-α))log(Tr(ρ^α)) para parámetros α ≥ 2 bajo transformaciones completamente positivas quepreservan la traza (CPTP) y mantienen la estructura del grupo estabilizador. Esta contribución resuelveuna pregunta teórica fundamental en la teoría cuántica de recursos que ha permanecido abierta durantevarios años. La demostración utiliza propiedades de convexidad de funciones, teoría de matrices, yanálisis funcional para establecer que estas entropías constituyen monótonas válidas de recursoscuánticos. Los resultados tienen implicaciones directas para la optimización de protocolos de correcciónde errores cuánticos, destilación de estados mágicos, y el desarrollo de límites fundamentales encomputación cuántica tolerante a fallos. Se proporcionan aplicaciones computacionales que demuestranmejoras del 15-20% en límites de conversión de recursos comparado con métodos existentes.

This paper presents Universal Displacement Theory (UDT), a theoretical framework proposing that fundamental forces emerge from substrate displacement mechanics rather than field-based interactions. The theory suggests that particle collections create pressure variations in a quantum substrate field, with force manifestations determined by local particle density regimes.

Through dimensional analysis and Lagrangian formulations, we explore how gravitational, electromagnetic, nuclear, and weak interactions may theoretically derive from a single substrate field equation. The framework presents mathematical consistency across multiple orders of magnitude in particle density, though experimental validation remains an open question requiring further investigation.

G-Theory primary  contributions to science...

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This paper proposes a radical new **Unified Field Theory, G-Theory**, aiming to fundamentally simplify our understanding of physics. Its primary contribution lies in asserting that all known physical forces and phenomena can be explained through **just two foundational principles**:

1.  **Kiron Field Dynamics / Quantum Gravity:** Based on the Kiron, a fundamental unit of quantized kinetic energy, the theory suggests that gravity, nuclear forces, and even the origin of mass (replacing the Higgs mechanism) arise from interactions and dynamics within a universal Kiron field. This offers a unified explanation for mass, gravitational force as gyroscopic precession, and nuclear binding as short-range quantum gravity.

2.  **Entropy:** The weak nuclear force and radioactive decay are reinterpreted not as a separate fundamental force, but as emergent properties of entropy and statistical inefficiency within the Kiron field.

By integrating these concepts, G-Theory claims to **eliminate the need for several components of the Standard Model**, including gluons, W and Z bosons, quantum chromodynamics (QCD), and the electroweak theory. It offers novel explanations for quark confinement and asymptotic freedom, and purports to resolve the black hole information paradox.

The paper backs these claims with **quantitative predictions**, notably achieving over 98% accuracy in calculating nuclear binding energies for various nuclei, and high accuracy for fission/fusion energies and radioactive decay rates, all without relying on the eliminated fundamental forces or particles.

In essence, G-Theory presents a bold attempt at a **more elegant and unified framework for physics**, aiming to explain observed phenomena through emergent effects of a single fundamental entity and universal principles, thereby simplifying the standard model and reducing the number of fundamental interactions.

The classification and relationships between computational complexity classes represent one of the
most fundamental open problems in theoretical computer science and mathematics. While
significant progress has been made in understanding specific relationships (e.g., P vs NP), a
comprehensive unified theory that explains the complete landscape of complexity classes and their
interrelationships remains elusive.
This blueprint presents a novel approach to developing a complete theory of complexity classes
using the Advanced World Formula (AWF) framework. By applying dimensional analysis, binding
operators, and fractal patterns from the AWF, we establish a comprehensive theoretical foundation
that not only resolves outstanding questions about complexity class relationships but also connects
computational complexity theory to fundamental physics, creating a unified mathematical
framework.

Kolmogorov complexity is a measure of the information contained in a binary string. We investigate here the notion of quantum Kolmogorov complexity, a measure of the information required to describe a quantum state. We show that for any definition of quantum Kolmogorov complexity measuring the number of classical bits required to describe a pure quantum state, there exists a pure n-qubit state which requires exponentially many bits of description. This is shown by relating the classical communication complexity to the quantum Kolmogorov complexity. Furthermore we give some examples of how quantum Kolmogorov complexity can be applied to prove results in different fields, such as quantum computation and thermodynamics, and we generalize it to the case of mixed quantum states.

Kolmogorov complexity is a measure of the information contained in a binary string. We investigate here the notion of quantum Kolmogorov complexity, a measure of the information required to describe a quantum state. We show that for any definition of quantum Kolmogorov complexity measuring the number of classical bits required to describe a pure quantum state, there exists a pure n-qubit state which requires exponentially many bits of description. This is shown by relating the classical communication complexity to the quantum Kolmogorov complexity. Furthermore, we give some examples of how quantum Kolmogorov complexity can be applied to prove results in different fields, such as quantum computation and thermodynamics, and we generalize it to the case of mixed quantum states.

We give a definition for the Kolmogorov complexity of a pure quantum state. In classical information theory, the algorithmic complexity of a string is a measure of the information needed by a universal machine to reproduce the string itself. We define the complexity of a quantum state by means of the classical description complexity of an (abstract) experimental procedure that allows us to prepare the state with a given fidelity. We argue that our definition satisfies the intuitive idea of complexity as a measure of "how difficult" it is to prepare a state. We apply this definition to give an upper bound on the algorithmic complexity of a number of known states. Furthermore, we establish a connection between the entanglement of a quantum state and its algorithmic complexity.

We define the algorithmic complexity of a quantum state relative to a given precision parameter, and give upper bounds for various examples of states. We also establish a connection between the entanglement of a quantum state and its algorithmic complexity.

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. • BQP is low for PP, i.e., PP BQP = PP. • There exists a relativized world where P = BQP and the polynomial-time hierarchy is infinite. • There exists a relativized world where BQP does not have complete sets. • There exists a relativized world where P = BQP but P = UP ∩ coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.

At the threshold where light surrenders to gravity's embrace lies a mathematical structure of profound elegance. This paper unveils a formalisation of black hole physics through the crystalline lens of information geometry, where the dance between entropy and curvature reveals itself in perfect mathematical harmony. By constructing a statistical manifold upon which probability distributions flow along Fisher-Rao geodesics, the thermodynamic properties of black holes emerge not as mysterious anomalies but as inevitable consequences of the underlying geometric structure. The Bekenstein-Hawking entropy formula, S = k_B A/4ℏG, materialises naturally from the volume of accessible states in this statistical manifold, while Hawking radiation becomes the whisper of maximum entropy under the constraints of energy conservation. A precise proof establishes the exact correspondence between quantum Fisher information and the temperature of radiation, resolving the information paradox through the subtle interplay of geometric invariants. This succeeds the Schwarzschild case to encompass rotating and charged black holes, offering experimentally testable predictions and revealing profound connections to quantum chaos and holography. In the geometry of information it may be found, in the description of black holes, a subtle glimpse into the fundamental nature of spacetime itself.

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. • BQP is low for PP, i.e., PP BQP = PP. • There exists a relativized world where P = BQP and the polynomial-time hierarchy is infinite. • There exists a relativized world where BQP does not have complete sets. • There exists a relativized world where P = BQP but P = UP ∩ coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.

Computing the ground-state energy of interacting electron (fermion) problems has recently been shown to be hard for QMA, a quantum analogue of the complexity class NP. Fermionic problems are usually hard, a phenomenon widely attributed to the so-called sign problem occurring in Quantum Monte Carlo simulations. The corresponding bosonic problems are, according to conventional wisdom, tractable. Here, we discuss the complexity of interacting boson problems and show that they are also QMA-hard. In addition, we show that the bosonic version of the so-called N -representability problem is QMA-complete, as hard as its fermionic version. As a consequence, these problems are unlikely to have efficient quantum algorithms.

We describe Kitaev's result from 1999, in which he defines the complexity class QMA, the quantum analog of the class NP, and shows that a natural extension of 3−SAT, namely local Hamiltonians, is QMA complete. The result builds upon the classical Cook-Levin proof of the NP completeness of SAT , but differs from it in several fundamental ways, which we highlight. This result raises a rich array of open problems related to quantum complexity, algorithms and entanglement, which we state at the end of this survey. This survey is the extension of lecture notes taken by Naveh for Aharonov's quantum computation course, held in Tel Aviv University, 2001.

Quantum complexity theory is concerned with the amount of elementary quantum resources needed to build a quantum system or a quantum operation. The fundamental question in quantum complexity is to define and quantify suitable complexity measures. This non-trivial question has attracted the attention of quantum information scientists, computer scientists, and high energy physicists alike. In this paper, we combine the approach in [LBK + 22] and well-established tools from noncommutative geometry [Con90, Rie99, CS03] to propose a unified framework for resource-dependent complexity measures of general quantum channels, also known as Lipschitz complexity. This framework is suitable to study the complexity of both open and closed quantum systems. The central class of examples in this paper is the so-called Wasserstein complexity introduced in [LBK + 22, PMTL21]. We use geometric methods to provide upper and lower bounds on this class of complexity measures [Nie06, NDGD06a, NDGD06b]. Finally, we study the Lipschitz complexity of random quantum circuits and dynamics of open quantum systems in finite dimensional setting. In particular, we show that generically the complexity grows linearly in time before the return time. This is the same qualitative behavior conjecture by Brown and Susskind [BS18, BS19]. We also provide an infinite dimensional example where linear growth does not hold.

State transformation problems such as compressing quantum information or breaking quantum commitments are fundamental quantum tasks. However, their computational difficulty cannot easily be characterized using traditional complexity theory, which focuses on tasks with classical inputs and outputs. To study the complexity of such state transformation tasks, we introduce a framework for unitary synthesis problems, including notions of reductions and unitary complexity classes. We use this framework to study the complexity of transforming one entangled state into another via local operations. We formalize this as the Uhlmann Transformation Problem, an algorithmic version of Uhlmann's theorem. Then, we prove structural results relating the complexity of the Uhlmann Transformation Problem, polynomial space quantum computation, and zero knowledge protocols. The Uhlmann Transformation Problem allows us to characterize the complexity of a variety of tasks in quantum information processing, including decoding noisy quantum channels, breaking falsifiable quantum cryptographic assumptions, implementing optimal prover strategies in quantum interactive proofs, and decoding the Hawking radiation of black holes. Our framework for unitary complexity thus provides new avenues for studying the computational complexity of many natural quantum information processing tasks.

Inspired by random walk on graphs, diffusion map (DM) is a class of unsupervised machine learning that offers automatic identification of low-dimensional data structure hidden in a highdimensional dataset. In recent years, among its many applications, DM has been successfully applied to discover relevant order parameters in many-body systems, enabling automatic classification of quantum phases of matter. However, classical DM algorithm is computationally prohibitive for a large dataset, and any reduction of the time complexity would be desirable. With a quantum computational speedup in mind, we propose a quantum algorithm for DM, termed quantum diffusion map (qDM). Our qDM takes as an input N classical data vectors, performs an eigen-decomposition of the Markov transition matrix in time O(log 3 N), and classically constructs the diffusion map via the readout (tomography) of the eigenvectors, giving a total expected runtime proportional to N 2 polylog N. Lastly, quantum subroutines in qDM for constructing a Markov transition matrix, and for analyzing its spectral properties can also be useful for other random walk-based algorithms.

The creation complexity of a quantum state is the minimum number of elementary gates required to create it from a basic initial state. The creation complexity of quantum states is closely related to the complexity of quantum circuits, which is crucial in developing efficient quantum algorithms that can outperform classical algorithms. A major question unanswered so far is what quantum states can be created with a number of elementary gates that scales polynomially with the number of qubits. In this work, it is first shown that for an entirely general quantum state it is exponentially hard (requires a number of steps that scales exponentially with the number of qubits) to determine if the creation complexity is polynomial. Then, it is shown that it is possible for a large class of quantum states with polynomial creation complexity to have common coefficient features such that, given any candidate quantum state, an efficient coefficient sampling procedure can be designed to determine if the state belongs to the class or not with arbitrarily high success probability. Consequently, partial knowledge of a quantum state's creation complexity is obtained, which can be useful for designing quantum circuits and algorithms involving such a state.

In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing. The approach proposed in (Marzuoli A and Rasetti M 2002, 2005a) relies on the (re)coupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on 'qubits' and Boolean gates. Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of values as well as continuous (angular) variables. Such a framework is an ideal playground to discuss discrete (digital) and analogic computational processes, together with their relationships occuring when a consistent semiclassical limit takes place on discrete quantum gates. When working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum finite states-machines which in turn represent discrete versions of topological quantum computation models. We argue that our model embodies a sort of unifying paradigm for computing inspired by Nature and, even more ambitiously, a universal setting in which suitably encoded quantum symbolic manipulations of combinatorial, topological and algebraic problems might find their 'natural' computational reference model.

We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of 'knot invariants', among which the Jones polynomial plays a prominent role, since it can be associated with observables in topological quantum field theory. Although the problem of computing the Jones polynomial is intractable in the framework of classical complexity theory, it has been recently recognized that a quantum computer is capable of approximating it in an efficient way. The quantum algorithms discussed here represent a breakthrough for quantum computation, since approximating the Jones polynomial is actually a 'universal problem', namely the hardest problem that a quantum computer can efficiently handle.

In this article we introduce a new complexity class called PQMA log (2). Informally, this is the class of languages for which membership has a logarithmic-size quantum proof with perfect completeness and soundness which is polynomially close to 1 in a context where the verifier is provided a proof with two unentangled parts. We then show that PQMA log (2) = NP. For this to be possible, it is important, when defining the class, not to give too much power to the verifier. This result, when compared to the fact that QMA log = BQP, gives us new insight on the power of quantum information and the impact of entanglement.

In this article we introduce a new complexity class called PQMA log (2). Informally, this is the class of languages for which membership has a logarithmic-size quantum proof with perfect completeness and soundness which is polynomially close to 1 in a context where the verifier is provided a proof with two unentangled parts. We then show that PQMA log (2) = NP. For this to be possible, it is important, when defining the class, not to give too much power to the verifier. This result, when compared to the fact that QMA log = BQP, gives us new insight on the power of quantum information and the impact of entanglement.

In this review article, we are interested in the detailed analysis of complexity aspects of both time and space that arises from the implementation of a quantum algorithm on a quantum based hardware. In particular, some steps of the implementation, as state preparation and readout processes, in most of the cases can surpass the complexity aspects of the algorithm itself. We present the complexity involved in the full implementation of quantum algorithms for solving linear systems of equations and linear system of differential equations, from state preparation to the number of measurements needed to obtain good statistics from the final states of the quantum system, in order to assess the overall complexity of the processes.

In this work, we are interested in the detailed analysis of complexity aspects of both time and space that arises from the implementation of a quantum algorithm on a quantum based hardware. In particular, some steps of the implementation, as state preparation and readout processes, in most of the cases can surpass the complexity aspects of the algorithm itself. We present the complexity involved in the full implementation of quantum algorithms for solving linear systems of equations and linear system of differential equations, from state preparation to the number of measurements needed to obtain good statistics from the final states of the quantum system, in order to assess the overall complexity of the processes.

To which extent the whole of a system cannot be reduced into the sum of its parts? Apart from complexity theory, this question stands on the core of composite quantum states structure, given the intrinsic structural properties of Hilbert spaces' tensor product [1]. This arises in both cases due to correlations playing a role in the overall observed dynamics, or statistical properties. Those correlations give rise to regularities on those objects, being spatial, temporal or both. As a way to study them, information-theoretic and minimal-description methods are commonly used. Apart from the well known information measures, in the quantum setting there are also some proposed quantum informational minimal description-oriented approaches [2], based on a quantum extension of Kolmogorov's complexity. Although in general among them a quantification is achieved, there is no clear structural characterization. This being the main goal of the hierarchical estimation of correlations [3], called here hierarchical information. Those tools have been used in classical systems as Cellular Automata and Coupled Tent Maps [3], and later formalized [4]. This is an approach essentially based on quantifying the degree of interaction from few to increasingly more parts of one system, in a fashion that as many parts contribute to the system dynamics, higher orders of interaction rise, giving room for richer dynamics, richer statistics. Hierarchical information (HI) generalizes

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. • BQP is low for PP, i.e., PP BQP = PP. • There exists a relativized world where P = BQP and the polynomial-time hierarchy is infinite. • There exists a relativized world where BQP does not have complete sets. • There exists a relativized world where P = BQP but P = UP ∩ coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.

In this note we show that all languages in NP have very short (logarithmic size) quantum proofs which can be verified provided that two unentangled copies are given. We thus introduce a new complexity class QMA log (2) and show that NP ⊆ QMA log (2). This gives a new perspective when compared to the previously known result QMA log = BQP.

In this note we show that all languages in NP have very short (logarithmic size) quantum proofs which can be verified provided that two unentangled copies are given. We thus introduce a new complexity class QMA log (2) and show that NP ⊆ QMA log (2). This gives a new perspective when compared to the previously known result QMA log = BQP.

In this paper we present the computational model underlying the one-way quantum computer which we introduced recently [Phys. Rev. Lett. {\bf{86}}, 5188 (2001)]. The one-way quantum computer has the property that any quantum logic network can be simulated on it. Conversely, not all ways of quantum information processing that are possible with the one-way quantum computer can be understood properly in network model terms. We show that the logical depth is, for certain algorithms, lower than has so far been known for networks. For example, every quantum circuit in the Clifford group can be performed on the one-way quantum computer in a single step.

We define the algorithmic complexity of a quantum state relative to a given precision parameter, and give upper bounds for various examples of states. We also establish a connection between the entanglement of a quantum state and its algorithmic complexity.

We use the powerful tools of counting complexity and generic oracles to help understand the limitations of the complexity of quantum computation. We show several results for the probabilistic quantum class BQP. • BQP is low for PP, i.e., PP BQP = PP. • There exists a relativized world where P = BQP and the polynomial-time hierarchy is infinite. • There exists a relativized world where BQP does not have complete sets. • There exists a relativized world where P = BQP but P = UP ∩ coUP and one-way functions exist. This gives a relativized answer to an open question of Simon.

Motivated by the result that an 'approximate' evaluation of the Jones polynomial of a braid at a 5 th root of unity can be used to simulate the quantum part of any algorithm in the quantum complexity class BQP, and results relating BQP to the counting class GapP, we introduce a form of additive approximation which can be used to simulate a function in BQP. We show that all functions in the classes #P and GapP have such an approximation scheme under certain natural normalisations. However we are unable to determine whether the particular functions we are motivated by, such as the above evaluation of the Jones polynomial, can be approximated in this way. We close with some open problems motivated by this work.

This handout was created in the context of a talk we gave at the \Graduate Seminar on Topics in Quantum Computation" by Prof. Nitin Saxena at the University of Bonn in the Summer Semester 2011. It is heavily based on the lecture notes [AAR] and the book [NC].

In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing. The approach proposed in (Marzuoli A and Rasetti M 2002, 2005a) relies on the (re)coupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on `qubits' and Boolean gates. Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of values as well as continuous (angular) variables. Such a framework is an ideal playground to discuss discrete (digital) and analogic computational processes, together with their relationships occuring when a consistent semiclassical limit takes place on discrete quantum gates. When working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum finite sta...

Previouslywe noted in numerical calculations that a certain unitary ninej coefficient U9j =((jj) 2j (jj) 2j | (jj) 2j (jj) (2j−2)) I decreases with increasing j and for small I the decrease is of the form C j m e −αj. The exponential decay dominates. It is here shown , using the Stirling approximation, that α= 4 ln(2), However the simplest version of this approximation gives the wrong m dependence and the inclusion of the "small" term is necessary.

In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing. The approach proposed in (Marzuoli A and Rasetti M 2002, 2005a) relies on the (re)coupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on 'qubits' and Boolean gates. Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of values as well as continuous (angular) variables. Such a framework is an ideal playground to discuss discrete (digital) and analogic computational processes, together with their relationships occuring when a consistent semiclassical limit takes place on discrete quantum gates. When working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum finite states-machines which in turn represent discrete versions of topological quantum computation models. We argue that our model embodies a sort of unifying paradigm for computing inspired by Nature and, even more ambitiously, a universal setting in which suitably encoded quantum symbolic manipulations of combinatorial, topological and algebraic problems might find their 'natural' computational reference model.

In the past few years there has been a tumultuous activity aimed at introducing novel conceptual schemes for quantum computing. The approach proposed in Rasetti M 2002, 2005a) relies on the (re)coupling theory of SU(2) angular momenta and can be viewed as a generalization to arbitrary values of the spin variables of the usual quantum-circuit model based on 'qubits' and Boolean gates. Computational states belong to finite-dimensional Hilbert spaces labelled by both discrete and continuous parameters, and unitary gates may depend on quantum numbers ranging over finite sets of values as well as continuous (angular) variables. Such a framework is an ideal playground to discuss discrete (digital) and analogic computational processes, together with their relationships occuring when a consistent semiclassical limit takes place on discrete quantum gates. When working with purely discrete unitary gates, the simulator is naturally modelled as families of quantum finite states-machines which in turn represent discrete versions of topological quantum computation models. We argue that our model embodies a sort of unifying paradigm for computing inspired by Nature and, even more ambitiously, a universal setting in which suitably encoded quantum symbolic manipulations of combinatorial, topological and algebraic problems might find their 'natural' computational reference model.