The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine and continues to be of enormous benefit in the classical theory of computation. We propose that quantum computation, like its classical counterpart, may benefit from a version of the lambda calculus suitable for expressing and reasoning about quantum algorithms. In this paper we develop a quantum lambda calculus as an alternative model of quantum computation, which combines some of the benefits of both the quantum Turing machine and the quantum circuit models. The calculus turns out to be closely related to the linear lambda calculi used in the study of linear logic. We set up a computational model and an equational proof system for this calculus, and we argue that it is equivalent to the quantum Turing machine.
A seminar by Wehr @cite_41 suggests that linear logic may be useful in constructing a calculus for quantum computation within the mathematical framework of Chu spaces. However, the author stops short of developing such a calculus.
@cite_41
2006290558
Recently a great deal of attention has been focused on quantum computation following a sequence of results [Bernstein and Vazirani, in Proc. 25th Annual ACM Symposium Theory Comput., 1993, pp. 11--20, SIAM J. Comput., 26 (1997), pp. 1277--1339], [Simon, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 116--123, SIAM J. Comput., 26 (1997), pp. 1340--1349], [Shor, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 124--134] suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor's result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time, it is natural to ask whether all of @math can be efficiently solved in quantum polynomial time. In this paper, we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class @math cannot be solved on a quantum Turing machine (QTM) in time @math . We also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class @math cannot be solved on a QTM in time @math . The former bound is tight since recent work of Grover [in Proc. @math th Annual ACM Symposium Theory Comput. , 1996] shows how to accept the class @math relative to any oracle on a quantum computer in time @math .