On the problem of approximating the eigenvalues of undirected graphs in probabilistic logspace

We introduce the problem of approximating the eigenvalues of a given stochastic/symmetric matrix in the context of classical spacebounded computation. The problem can be exactly solved in DET ⊆ NC². Recently, it has been shown that the approximation problem can be solved by a quantum logspace algorithm. We show a BPL algorithm that approximates any eigenvalue with a constant accuracy. The result we obtain falls short of achieving the polynomially-small accuracy that the quantum algorithm achieves. Thus, at our current state of knowledge, we can achieve polynomially-small accuracy with quantum logspace algorithms, constant accuracy with probabilistic logspace algorithms, and no nontrivial result is known for deterministic logspace algorithms. The quantum algorithm also has the advantage of working over arbitrary, possibly non-stochastic Hermitian operators. Our work raises several challenges. First, a derandomization challenge, trying to achieve a deterministic algorithm approximating eigenvalues with some non-trivial accuracy. Second, a de-quantumization challenge, trying to decide whether the quantum logspace model is strictly stronger than the classical probabilistic one or not. It also casts the deterministic, probabilistic and quantum space-bounded models as problems in linear algebra with differences between symmetric, stochastic and arbitrary operators. We therefore believe the problem of approximating the eigenvalues of a graph is not only natural and important by itself, but also important for understanding the relative power of deterministic, probabilistic and quantum logspace computation.