Approximating the Determinant of Well-Conditioned Matrices by Shallow Circuits

12/09/2019
by   Enric Boix-Adserà, et al.
0

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 Ta-Shma [Ta13] implies a method to approximate the determinant of Hermitian matrices with condition number κ in quantum space O(log n + 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(log n ·logκ), and in particular one can approximate the determinant for sufficiently well-conditioned matrices in depth Õ(log n). Our algorithm combines Barvinok's recent complex-analytic approach for approximating combinatorial counting problems [Bar16] with the Valiant-Berkowitz-Skyum-Rackoff depth-reduction theorem for low-degree arithmetic circuits [Val83].

READ FULL TEXT

page 1

page 2

page 3

page 4

research
08/17/2020

Bounds on the QAC^0 Complexity of Approximating Parity

QAC circuits are quantum circuits with one-qubit gates and Toffoli gates...
research
11/06/2019

Interactive shallow Clifford circuits: quantum advantage against NC^1 and beyond

Recent work of Bravyi et al. and follow-up work by Bene Watts et al. dem...
research
07/11/2019

Optimal Space-Depth Trade-Off of CNOT Circuits in Quantum Logic Synthesis

Due to the decoherence of the state-of-the-art physical implementations ...
research
03/29/2020

A Polynomial Degree Bound on Defining Equations of Non-rigid Matrices and Small Linear Circuits

We show that there is a defining equation of degree at most 𝗉𝗈𝗅𝗒(n) for ...
research
04/10/2018

On top fan-in vs formal degree for depth-3 arithmetic circuits

We show that over the field of complex numbers, every homogeneous polyno...
research
06/05/2021

Complexity of Modular Circuits

We study how the complexity of modular circuits computing AND depends on...
research
09/06/2018

Quantum algorithms and approximating polynomials for composed functions with shared inputs

We give new quantum algorithms for evaluating composed functions whose i...

Please sign up or login with your details

Forgot password? Click here to reset