Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation, and in particular, on problems with an algebraic flavor.
Our previous work about algebraic-geometric invariants of the mixed states are extended and a stronger separability criterion is given. We also show that the Schmidt number of pure states in bipartite quantum systems, a classical concept, is actually an algebraic-geometric invariant.
We introduce an algebraic measure of correlations in bipartite quantum systems. The proposed quantity, called maximal mutual correlation, provides the information how much a given state differs from the product state of its marginals. In contrast to the entanglement or quantum discord, maximal mutual correlation can be non - zero for some classically correlated quantum states.
In this work, we study the Codeword Stabilized Quantum Codes (CWS codes) a generalization of the stabilizers quantum codes using a new approach, the algebraic structure of modules, a generalization of linear spaces. We show then a new result that relates CWS codes with stabilizer codes generalizing results in the literature.
We provide algebraic criteria for the unitarity of linear quantum cellular automata, i.e. one dimensional quantum cellular automata. We derive these both by direct combinatorial arguments, and by adding constraints into the model which do not change the quantum cellular automata's computational power. The configurations we consider have finite but unbounded size.
Products and tensor products are linked by a universal property. Imposing the invariance of the laws of Nature under tensor composition along with Leibniz identity determines quantum and classical mechanics algebraic structure through the interplay between products, coproducts, and the tensor product. Violations of Bell's inequalities distinguishes quantum from classical mechanics.
In this paper we present the fundamentals of the so-called algebraic approach to propositional quantum logics. We define the set of formulas describing quantum reality as a free algebra freely generated by the set of quantum propositional variables. We define the general notion of logic as a structural consequence operation. Next we introduce the concept of logical matrices understood as a models of quantum logics. We give the definitions of two quantum consequence operations defined in these models.
We present quantum query complexity bounds for testing algebraic properties. For a set S and a binary operation on S, we consider the decision problem whether $S$ is a semigroup or has an identity element. If S is a monoid, we want to decide whether S is a group. We present quantum algorithms for these problems that improve the best known classical complexity bounds. In particular, we give the first application of the new quantum random walk technique by Magniez, Nayak, Roland, and Santha that improves the previous bounds by Ambainis and Szegedy. We also present several lower bounds for testin…
It's argued that Information-Theoretical restrictions for the systems selfdescription are important for Quantum Measurement Problem. They are described by O information system restricted states R formalism by Breuer and can be obtained also in Algebraic QM considering Segal Algebra of O observables. From Segal theorem it's shown that R describes the random measurement outcomes in the individual events.
Operator quantum error-correction is a technique for robustly storing quantum information in the presence of noise. It generalizes the standard theory of quantum error-correction, and provides a unified framework for topics such as quantum error-correction, decoherence-free subspaces, and noiseless subsystems. This paper develops (a) easily applied algebraic and information-theoretic conditions which characterize when operator quantum error-correction is feasible; (b) a representation theorem for a class of noise processes which can be corrected using operator quantum error-correction; and (c)…
We investigate the relation between multilinear mappings and multipartite states. We show that the isomorphism between multilinear mapping and tensor product completely characterizes decomposable multipartite states in a mathematically well-defined manner.
We propose a quantum algorithm to solve systems of nonlinear algebraic equations. In the ideal case the complexity of the algorithm is linear in the number of variables $n$, which means our algorithm's complexity is less than $O(n^{3})$ of the corresponding classical ones when the set of equations has many variables and not so high orders. Different with most popular way of representing the results by state vector, we get the results in computation basis which is readable directly.
The noncontextuality condition states that a value of any observable is independent of which other compatible observable is measured jointly with it. Klyachko, Can, Binicio{ğ}lu, and Shumovsky have introduced an inequality which holds if there is a noncontextual hidden variable theory. It is called KCBS inequality, which is state-dependent. Its violation shows a contradiction between predictions of quantum theory and noncontextual hidden variable theories. In the present paper, it is shown that there is a state which does not violate KCBS inequality in the case of quantum mechanics of finite d…
The biological hierarchy and the differences between living and non-living systems are considered from the standpoint of quantum mechanics. The hierarchical organization of biological systems requires hierarchical organization of quantum states. The construction of the hierarchical space of state vectors is presented. The application of similar structures to quantum information processing is considered.
By using the Lewis-Riesenfeld theory and the invariant-related unitary transformation formulation, the exact solutions of the {\it time-dependent} Schrödinger equations which govern the various Lie-algebraic quantum systems in atomic physics, quantum optics, nuclear physics and laser physics are obtained. It is shown that the {\it explicit} solutions may also be obtained by working in a sub-Hilbert-space corresponding to a particular eigenvalue of the conserved generator ({\it i. e.}, the {\it time-independent} invariant) for some quantum systems without quasi-algebraic structures. The global …
No-cloning theorem says that there is no unitary operation that makes perfect clones of non-orthogonal quantum states. The objective of the present paper is to examine whether an imperfect cloning operation exists or not in a C*-algebraic framework. We define a universal $ε$-imperfect cloning operation which tolerates a finite loss $ε$ of fidelity in the cloned state, and show that an individual system's algebra of observables is Abelian if and only if there is a universal $ε$-imperfect cloning operation in the case where the loss of fidelity is less than 1/4. Therefore, in this case no univer…
Einstein introduced the locality principle which states that all physical effect in some finite space-time region does not influence its space-like separated finite region. Recently, in algebraic quantum field theory, Rédei captured the idea of the locality principle by the notion of operational separability. The operation in operational separability is performed in some finite space-time region, and leaves unchanged the state in its space-like separated finite space-time region. This operation is defined with a completely positive map. In the present paper, we justify using a completely posit…
Okay, Roberts, Bartlett and Raussendorf recently introduced a new cohomological approach to contextuality in measurement based quantum computation. We give an abstract description of their obstruction and the algebraic structure it exploits, using the sheaf theoretic framework of Abramsky and Brandenburger. At this level of generality we contrast their approach to the Cech cohomology obstruction of Abramsky, Mansfield and Barbosa and give a direct proof that Cech cohomology is at least as powerful.
We introduce algebraic approach for superoperators that might be useful tool for investigation of quantum (bosonic) multi-mode systems and its dynamics. In order to demonstrate potential of proposed method we consider multi-mode Liouvillian superoperator that describes relaxation dynamics of a quantum system (including thermalization and intermode coupling). Considered algebraic structure of superoperators that form Liouvillian and their algebraic properties allows us to diagonilize multi-mode Liouvillian to find its spectrum. Also it allows to derive linear by mean number of thermal (environm…
Quantum computers can execute algorithms that sometimes dramatically outperform classical computation. Undoubtedly the best-known example of this is Shor's discovery of an efficient quantum algorithm for factoring integers, whereas the same problem appears to be intractable on classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article will review the cu…
This paper outlines the common ground between the motivations lying behind Hans Primas' algebraic approach to quantum phenomena and those lying behind David Bohm's approach which led to his notion of implicate/explicate order. This connection has been made possible by the recent application of orthogonal Clifford algebraic techniques to the de Broglie-Bohm approach for relativistic systems with spin.
We relate SLOCC equivalence classes of qudit states to moduli spaces of Calabi-Yau manifolds equipped with a collection of line bundles. The cases of 3 qutrits and 4 qubits are also related to noncommutative algebraic geometry.
A consistent realization of the quantum operators corresponding to the canonically conjugate phase and number variables is proposed, resorting to the irreducible unitary representations of the Lie algebra su(1,1), as proposed by Kastrup.
We study the properties of quantum cusp and butterfly catastrophes from an algebraic viewpoint. The analysis employs an interacting boson model Hamiltonian describing quantum phase transitions between specific quadrupole shapes by interpolating between two incompatible dynamical symmetry limits. The classical properties are determined by using coherent states to construct the complete phase diagrams associated with Landau potentials exhibiting such catastrophes.The quantum properties are determined by analyzing the spectra, transition rates and symmetry character of the eigenstates of critical…
We investigate the algebraic reasoning of quantum programs inspired by the success of classical program analysis based on Kleene algebra. One prominent example of such is the famous Kleene Algebra with Tests (KAT), which has furnished both theoretical insights and practical tools. The succinctness of algebraic reasoning would be especially desirable for scalable analysis of quantum programs, given the involvement of exponential-size matrices in most of the existing methods. A few key features of KAT including the idempotent law and the nice properties of classical tests, however, fail to hold …
We give an algebraic formulation based on Clifford algebras and algebraic spinors for quantum information. In this context, logic gates and concepts such as chirality, charge conjugation, parity and time reversal are introduced and explored in connection with states of qubits. Supersymmetry and M-superalgebra are also analysed with our formalism. Specifically we use extensively the algebras $Cl_{3,0}$ and $Cl_{1,3}$ as well as tensor products of Clifford algebras.
In this paper is discussed description of some algebraic structures in quantum theory by using formal recursive constructions with "complex Poincaré group" ISO(4,C).
We present a general framework of quantum error-correcting codes (QECCs) as a subspace of a complex Hilbert space and the corresponding error models. Then we illustrate how QECCs can be constructed using techniques from algebraic coding theory. Additionally, we discuss secondary constructions for QECCs, leading to propagation rules for the parameters of QECCs.
We investigate prime factorization from two perspectives: quantum annealing and computational algebraic geometry, specifically Gröbner bases. We present a novel scalable algorithm which combines the two approaches and leads to the factorization of all bi-primes up to just over $200 \, 000$, the largest number factored to date using a quantum processor.
We discuss the alternative algebraic structures on the manifold of quantum states arising from alternative Hermitian structures associated with quantum bi-Hamiltonian systems. We also consider the consequences at the level of the Heisenberg picture in terms of deformations of the associative product on the space of observables.
Here we deal in a pedagogical way with an approach to construct an algebraic structure for the Quantum Mechanical measurement processes from the concept of \emph{Measurement Symbol}. Such concept was conceived by Julian S. Schwinger and constitutes a fundamental piece in his variational formalism and its several applications.
The GNS representation construction is considered in a general case of topological involutive algebras of quantum systems, including quantum fields, and inequivalent state spaces of these systems are characterized. We aim to show that, from the physical viewpoint, they can be treated as classical fields by analogy with a Higgs vacuum field.
Quantum synchronizable codes are quantum error-correcting codes that can correct the effects of quantum noise as well as block synchronization errors. We improve the previously known general framework for designing quantum synchronizable codes through more extensive use of the theory of finite fields. This makes it possible to widen the range of tolerable magnitude of block synchronization errors while giving mathematical insight into the algebraic mechanism of synchronization recovery. Also given are families of quantum synchronizable codes based on punctured Reed-Muller codes and their ambie…
We analyze the effects of disorder on the correlation functions of one-dimensional quantum models of fermions and spins with long-range interactions that decay with distance $\ell$ as a power-law $1/\ell^α$. Using a combination of analytical and numerical results, we demonstrate that power-law interactions imply a long-distance algebraic decay of correlations within disordered-localized phases, for all exponents $α$. The exponent of algebraic decay depends only on $α$, and not, e.g., on the strength of disorder. We find a similar algebraic localization for wave-functions. These results are in …
Algebraic approach to quantum non - separability is applied to the case of two qubits. It is based on the partition of the algebra of observables into independent subalgebras and the tensor product structure of the Hilbert space is not exploited. Even in this simple case, such general formulation has some advantages. Using algebraic formalism we can explicitly show the relativity of the notion of entanglement to the observables measured in the system and characterize separable and non - separable pure states. As a universal measure of non - separability of pure states we propose to take so cal…
The first construction of strongly secure quantum ramp secret sharing by Zhang and Matsumoto had an undesirable feature that the dimension of quantum shares must be larger than the number of shares. By using algebraic curves over finite fields, we propose a new construction in which the number of shares can become arbitrarily large for fixed dimension of shares.
Université de Bourgogne Franche-Comté（勃艮第-弗朗什孔泰大学）；Auburn University（奥本大学）
We review supervised learning and deep neural network design for learning membership on algebraic varieties. We demonstrate that these trained artificial neural networks can predict the entanglement type for quantum states. We give examples for detecting degenerate states, as well as border rank classification for up to 5 binary qubits and 3 qutrits (ternary qubits).
Many-body entanglement is studied within the algebraic approach to quantum physics in systems made of Majorana fermions. In this framework, the notion of separability stems from partitions of the algebra of observables and properties of the associated correlation functions, rather than on particle tensor products. This allows obtaining a complete characterization of non-separable Majorana fermion states. These results may find direct applications in quantum metrology: using Majorana systems, sub-shot noise accuracy in parameter estimations can be achieved without preliminary, resource consumin…
