Quantum Invariance

Vasil Dinev Penchev

Mingsheng Ying, Shenggang Ying, Xiaodi Wu

Program invariant is a fundamental notion widely used in program verification and analysis. The aim of this paper is twofold: (i) find an appropriate definition of invariants for quantum programs; and (ii) develop an effective technique of invariant generation for verification and analysis of quantum programs. Interestingly, the notion of invariant can be defined for quantum programs in two different ways -- additive invariants and multiplicative invariants -- corresponding to two interpretations of implication in a continuous valued logic: the Lukasiewicz implication and the Godel implication. It is shown that both of them can be used to establish partial correctness of quantum programs. The problem of generating additive invariants of quantum programs is addressed by reducing it to an SDP (Semidefinite Programming) problem. This approach is applied with an SDP solver to generate invariants of two important quantum algorithms -- quantum walk and quantum Metropolis sampling. Our examples show that the generated invariants can be used to verify correctness of these algorithms and are helpful in optimising quantum Metropolis sampling. To our knowledge, this paper is the first attempt to define the notion of invariant and to develop a method of invariant generation for quantum programs.

S. L. Wu, X. Y. Zhang, X. X. Yi

For a closed quantum system, a dynamical invariant is defined as an operator whose expectation value is a constant. In this paper, we extend the concept of dynamical invariants from closed systems to open systems. A dynamical equation for invariants (the dynamical invariant condition) is derived for Markovian dynamics. Different from dynamical invariants of closed quantum systems, the time evolution of dynamical invariants of open quantum systems is no longer unitary, and eigenvalues of any invariant are time dependent in general. Since any Hermitian operator which can fulfill the dynamical invariant condition is a dynamical invariant, we construct a type of special dynamical invariant of which a part of the eigenvalues is still constant. The dynamical invariants in the subspace spanned by these eigenstates thus evolve unitarily.

Donald Salisbury

Alon E. Faraggi, Marco Matone

Mario E. Vélez, Juan Ospina

An emergent trend in quantum computation is the topological quantum computation (TQC). Briefly, TQC results from the application of quantum computation with the aim to solve the problems of quantum topology such as topological invariants for knots and links (Jones polynomials, HOMFLY polynomials, Khovanov polynomials); topological invariants for graphs (Tutte polynomial and Bollobás-Riordan polynomial); topological invariants for 3-manifolds (Reshetiskin-Turaev, Turaev-Viro and Turaer-Viro-Ocneanu invariants) and topological invariants for 4-manifolds( Crane-Yetter invariants). In a few words, TQC is concerned with the formulation of quantum algorithms for the computation of these topological invariants in quantum topology. Given that one of the fundamental achievements of quantum topology was the discovery of strong connections between monoidal categories and 3-dimensional manifolds, in TQC is possible and necessary to exploit such connections with the purpose to formulate universal quantum algorithms for topological invariants of 3-manifolds. In the present work we make an exploration of such possibilities. Specifically we search for universal quantum algorithms for generalized Turaev-Viro invariants of 3-manifolds such as the Turaev-Viro-Ocneanu invariants, the Kashaev-Baseilhac-Benedetti invariants of 3-manifolds with links and the Geer-Kashaev-Turaev invariants of 3-manifolds with a link and a principal bundle. We also look for physical systems (three dimensional topological insulators and three-dimensional gravity) over which implement the resulting universal topological quantum algorithms.