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Robert Lin

Publications and source records attributed to Robert Lin.

6 recordsLinked to original sources
arXiv2024

Positive Conserved Quantities in the Klein-Gordon Equation

We introduce an embedding of the Klein-Gordon equation into a pair of coupled equations that are first-order in time. The existence of such an embedding is based on a positivity property exhibited by the Klein-Gordon equation. These coupled equations provide a more satisfactory reduction of the Klein-Gordon equation to first-order differential equations in time than the Schrodinger equation. Using this embedding, we show that the ``negative probabilities" associated with the Klein-Gordon equation do not need to be resolved by introducing matrices as Dirac did with his eponymous equation. For the case of the massive Klein-Gordon equation, the coupled equations are equivalent to a forward Schrodinger equation in time and a backward Schrodinger equation in time, respectively, corresponding to a particle and its antiparticle. We show that there are two positive integrals that are conserved (constant in time) in the Klein-Gordon equation and thus provide a concrete resolution of the historical puzzle regarding the previously supposed lack of a probabilistic interpretation for the field governed by the Klein-Gordon equation. A significant consequence is that the Schrodinger equation is given a relativistic formulation, which does not require creation and annihilation operators, i.e. quantum fields. Physically, this corresponds to a theory in which the positive and negative energy parts do not directly interact, hence there will be no annihilation events--for example, particle-antiparticle collisions which do not result in photon emission. Thus, one practical consequence of this relativistically consistent theory is a simple explanation for dark matter.

quant-ph↗
arXiv2024

A New Class of Algorithms for Finding Short Vectors in Lattices Lifted from Co-dimension $k$ Codes

We introduce a new class of algorithms for finding a short vector in lattices defined by codes of co-dimension $k$ over $\mathbb{Z}_P^d$, where $P$ is prime. The co-dimension $1$ case is solved by exploiting the packing properties of the projections mod $P$ of an initial set of non-lattice vectors onto a single dual codeword. The technical tools we introduce are sorting of the projections followed by single-step pairwise Euclidean reduction of the projections, resulting in monotonic convergence of the positive-valued projections to zero. The length of vectors grows by a geometric factor each iteration. For fixed $P$ and $d$, and large enough user-defined input sets, we show that it is possible to minimize the number of iterations, and thus the overall length expansion factor, to obtain a short lattice vector. Thus we obtain a novel approach for controlling the output length, which resolves an open problem posed by Noah Stephens-Davidowitz (the possibility of an approximation scheme for the shortest-vector problem (SVP) which does not reduce to near-exact SVP). In our approach, one may obtain short vectors even when the lattice dimension is quite large, e.g., 8000. For fixed $P$, the algorithm yields shorter vectors for larger $d$. We additionally present a number of extensions and generalizations of our fundamental co-dimension $1$ method. These include a method for obtaining many different lattice vectors by multiplying the dual codeword by an integer and then modding by $P$; a co-dimension $k$ generalization; a large input set generalization; and finally, a "block" generalization, which involves the replacement of pairwise (Euclidean) reduction by a $k$-party (non-Euclidean) reduction. The $k$-block generalization of our algorithm constitutes a class of polynomial-time algorithms indexed by $k\geq 2$, which yield successively improved approximations for the short vector problem.

cs.CR↗
arXiv2022

Kraus-Like Decompositions

We introduce a new decomposition of quantum channels acting on group algebras, which we term Kraus-like (operator) decompositions. We motivate this decomposition with a general nonexistence result for Kraus operator decompositions in this setting. Given a length function which is a class function on a finite group, we construct a corresponding Kraus-like decomposition. We prove that this Kraus-like decomposition is \textit{convex} (meaning its coefficients are nonnegative and satisfy a sum rule) if and only if the length is conditionally negative definite. For a general finite group, we prove a stability condition which shows that the existence of a convex Kraus-like decomposition for all $t>0$ small enough necessarily implies existence for all time $t>0$. Using the stability condition, we show that for a general finite group, conditional negativity of the length function is equivalent to a set of semidefinite linear constraints on the length function. Our result implies that in the group algebra setting, a semigroup $P_t$ induced by a length function which is a class function is a quantum channel for all $t\geq 0$ if and only if it possesses a convex Kraus-like decomposition for all $t>0$.

math-ph↗
arXiv2021

A Graphical Calculus for Quantum Computing with Multiple Qudits using Generalized Clifford Algebras

In this work, we develop a graphical calculus for multi-qudit computations with generalized Clifford algebras, building off the algebraic framework developed in our prior work. We build our graphical calculus out of a fixed set of graphical primitives defined by algebraic expressions constructed out of elements of a given generalized Clifford algebra, a graphical primitive corresponding to the ground state, and also graphical primitives corresponding to projections onto the ground state of each qudit. We establish many properties of the graphical calculus using purely algebraic methods, including a novel algebraic proof of a Yang-Baxter equation and a construction of a corresponding braid group representation. Our algebraic proof, which applies to arbitrary qudit dimension, also enables a resolution of an open problem of Cobanera and Ortiz on the construction of self-dual braid group representations for even qudit dimension. We also derive several new identities for the braid elements, which are key to our proofs. Furthermore, we demonstrate that in many cases, the verification of involved vector identities can be reduced to the combinatorial application of two basic vector identities. Additionally, in terms of quantum computation, we demonstrate that it is feasible to envision implementing the braid operators for quantum computation, by showing that they are 2-local operators. In fact, these braid elements are almost Clifford gates, for they normalize the generalized Pauli group up to an extra factor $\zeta$, which is an appropriate square root of a primitive root of unity.

quant-ph↗
arXiv2021

An Algebraic Framework for Multi-Qudit Computations with Generalized Clifford Algebras

In this article, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies these axioms. Strengths of our algebraic framework include the minimality of its assumptions, and the readiness by which one may give an explicit construction satisfying these assumptions. In terms of applications, this algebraic framework provides a solid foundation which opens the way for developing a graphical calculus for multi-qudit representations of generalized Clifford algebras using purely algebraic methods, which is addressed in a follow-up paper.

quant-ph↗