On the complexity of solving linear congruences and computing nullspaces modulo a constant

We consider the problems of determining the feasibility of a linear congruence, producing a solution to a linear congruence, and finding a spanning set for the nullspace of an integer matrix, where each problem is considered modulo an arbitrary constant k>1. These problems are known to be complete for the logspace modular counting classes {Mod_k L} = {coMod_k L} in special case that k is prime (Buntrock et al, 1992). By considering variants of standard logspace function classes --- related to #L and functions computable by UL machines, but which only characterize the number of accepting paths modulo k --- we show that these problems of linear algebra are also complete for {coMod_k L} for any constant k>1. Our results are obtained by defining a class of functions FUL_k which are low for {Mod_k L} and {coMod_k L} for k>1, using ideas similar to those used in the case of k prime in (Buntrock et al, 1992) to show closure of Mod_k L under NC^1 reductions (including {Mod_k L} oracle reductions). In addition to the results above, we briefly consider the relationship of the class FUL_k for arbitrary moduli k to the class {F.coMod_k L} of functions whose output symbols are verifiable by {coMod_k L} algorithms; and consider what consequences such a comparison may have for oracle closure results of the form {Mod_k L}^{Mod_k L} = {Mod_k L} for composite k.Comment: 17 pages, one Appendix; minor corrections and revisions to presentation, new observations regarding the prospect of oracle closures. Comments welcom

A linearized stabilizer formalism for systems of finite dimension

The stabilizer formalism is a scheme, generalizing well-known techniques developed by Gottesman [quant-ph/9705052] in the case of qubits, to efficiently simulate a class of transformations ("stabilizer circuits", which include the quantum Fourier transform and highly entangling operations) on standard basis states of d-dimensional qudits. To determine the state of a simulated system, existing treatments involve the computation of cumulative phase factors which involve quadratic dependencies. We present a simple formalism in which Pauli operators are represented using displacement operators in discrete phase space, expressing the evolution of the state via linear transformations modulo D <= 2d. We thus obtain a simple proof that simulating stabilizer circuits on n qudits, involving any constant number of measurement rounds, is complete for the complexity class coMod_{d}L and may be simulated by O(log(n)^2)-depth boolean circuits for any constant d >= 2.Comment: 25 pages, 3 figures. Reorganized to collect complexity results; some corrections and elaborations of technical results. Differs slightly from the version to be published (fixed typos, changes of wording to accommodate page breaks for a different article format). To appear as QIC vol 13 (2013), pp.73--11