Related papers: Identity Testing for +-Regular Noncommutative Arit…
Matrix-product unitaries (MPUs) are many-body unitary operators that, as a consequence of their tensor-network structure, preserve the entanglement area law in 1D systems. However, it is unknown how to implement an MPU as a quantum circuit…
We show that there is a randomized algorithm that, when given a small constant-depth Boolean circuit $C$ made up of gates that compute constant-degree Polynomial Threshold functions or PTFs (i.e., Boolean functions that compute signs of…
Let $T$ be a matrix whose entries are linear forms over the noncommutative variables $x_1, x_2, \ldots, x_n$. The noncommutative Edmonds' problem (NSINGULAR) aims to determine whether $T$ is invertible in the free skew field generated by…
In this paper, we show exponential lower bounds for the class of homogeneous depth-$5$ circuits over all small finite fields. More formally, we show that there is an explicit family $\{P_d : d \in \mathbb{N}\}$ of polynomials in…
Sampling from the output distributions of quantum computations comprising only commuting gates, known as instantaneous quantum polynomial (IQP) computations, is believed to be intractable for classical computers, and hence this task has…
It has recently been shown that the problem of testing global convexity of polynomials of degree four is {strongly} NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global…
Polynomial threshold gates are basic processing units of an artificial neural network. When the input vectors are binary vectors, these gates correspond to Boolean functions and can be analyzed via their polynomial representations. In…
We show that the parity of more than three non-target input bits cannot be computed by QAC-circuits of depth-2, not even uncleanly, regardless of the number of ancilla qubits. This result is incomparable with other recent lower bounds on…
The $N$th power of a polynomial matrix of fixed size and degree can be computed by binary powering as fast as multiplying two polynomials of linear degree in~$N$. When Fast Fourier Transform (FFT) is available, the resulting complexity is…
We study the computational complexity of decomposing finite discrete dynamical systems (FDDSs) in terms of the semiring operations of alternative and synchronous execution, which is useful for the analysis of discrete phenomena in science…
The work in this paper is to initiate a theory of testing monomials in multivariate polynomials. The central question is to ask whether a polynomial represented by certain economically compact structure has a multilinear monomial in its…
Inspired by the concept of fault tolerance quantum computation, this article proposes a framework dubbed Exact Homomorphic Encryption, EHE, enabling exact computations on encrypted data without the need for pre-decryption. The introduction…
Probabilistic circuits compute multilinear polynomials that represent multivariate probability distributions. They are tractable models that support efficient marginal inference. However, various polynomial semantics have been considered in…
We develop a family of quantum accreditation protocols for quantum circuits with non-Clifford two-qubit gates. The latter includes families of gates such as the fSim and XY families of gates, native to existing hardwares. We provide…
Polymorphic circuits are a special kind of circuits which possess some different build-in functions and these functions are activated by environment parameters, like light and VDD. Some theories have been proposed to guide the design of…
Assuming that the Permanent polynomial requires algebraic circuits of exponential size, we show that the class VNP does not have efficiently computable equations. In other words, any nonzero polynomial that vanishes on the coefficient…
We study the Identity Problem, the problem of determining if a finitely generated semigroup of matrices contains the identity matrix; see Problem 3 (Chapter 10.3) in ``Unsolved Problems in Mathematical Systems and Control Theory'' by…
We formalize a framework of algebraically natural lower bounds for algebraic circuits. Just as with the natural proofs notion of Razborov and Rudich for boolean circuit lower bounds, our notion of algebraically natural lower bounds captures…
We design polynomial size, constant depth (namely, $\mathsf{AC}^0$) arithmetic formulae for the greatest common divisor (GCD) of two polynomials, as well as the related problems of the discriminant, resultant, B\'ezout coefficients,…
The identification of an unknown quantum gate is a significant issue in quantum technology. In this paper, we propose a quantum gate identification method within the framework of quantum process tomography. In this method, a series of pure…