Approximating the Determinant of Well-Conditioned Matrices by Shallow Circuits

Abstract

The determinant can be computed by classical circuits of depth O(log 2 n), and therefore it can also be computed in classical space O(log 2 n). Recent progress by [24] implies a method to approximate the determinant of Hermitian matrices with condition number κ in quantum space O(logn + logκ). However, it is not known how to perform the task in less than O(log 2 n) space using classical resources only. In this work, we show that the condition number of a matrix implies an upper bound on the depth complexity (and therefore also on the space complexity) for this task: the determinant of Hermitian matrices with condition number κ can be approximated to inverse polynomial relative error with classical circuits of depth Õ(logn•logκ), and in particular one can approximate the determinant for sufficiently well-conditioned matrices in depth Õ(logn). Our algorithm combines Barvinok's recent complex-analytic approach for approximating combinatorial counting problems [2] with the depth-reduction theorem for low-degree arithmetic circuits [26].

Eliminating Intermediate Measurements in Space-Bounded Quantum Computation

“…Returning to an issue discussed in Section 1.1, Boix-Adserà, Eldar, and Mehraban [8] showed κ(n)-conditioned-DET ∈ DSPACE(log(n) log(κ(n))poly(log log n)). Moreover, they asked the question "is poly-conditioned-DET BQL-complete?"…”

“…To briefly explain the significance of the question posed by Boix-Adserà, Eldar, and Mehraban [8], note that their result shows poly-conditioned-DET ∈ DSPACE(log 2 (n)poly(log log n)). Therefore, as poly-conditioned-DET is BQL-complete, the statement BQL ⊆ DSPACE(log 2−ǫ n) would follow from either a small improvement in their result (i.e., proving a stronger upper bound on the needed amount of deterministic space in terms of κ(n)) or from a small improvement in our result (i.e., proving κ(n)-conditioned-DET remains BQL-hard for subpolynomial κ(n)).…”

“…Consider κ(n)-conditioned-DET, the problem of approximating, to within a multiplicative factor (1 + 1/poly(n)), the determinant of an n × n with condition number κ(n). Boix-Adserà, Eldar, and Mehraban [8] recently showed κ(n)-conditioned-DET ∈ DSPACE(log(n) log(κ(n))poly(log log n)). Furthermore, they raised the following question: is poly-conditioned-DET BQL-complete?…”