Related papers: Identity Testing for Circuits with Exponentiation …
Given a random $n \times n$ symmetric matrix $\boldsymbol W$ drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form $\boldsymbol x^\top \boldsymbol…
We present an accreditation protocol for the outputs of noisy intermediate-scale quantum devices. By testing entire circuits rather than individual gates, our accreditation protocol can provide an upper-bound on the variation distance…
We consider recent works on the simulation of quantum circuits using the formalism of matrix product states and the formalism of contracting tensor networks. We provide simplified direct proofs of many of these results, extending an…
Solitude verification is arguably one of the simplest fundamental problems in distributed computing, where the goal is to verify that there is a unique contender in a network. This paper devises a quantum algorithm that exactly solves the…
Fully Homomorphic Encryption (FHE) is one of the most promising technologies for privacy protection as it allows an arbitrary number of function computations over encrypted data. However, the computational cost of these FHE systems limits…
This work addresses integrating probabilistic propositional logic constraints into the distribution encoded by a probabilistic circuit (PC). PCs are a class of tractable models that allow efficient computations (such as conditional and…
Probabilistic verification problems of neural networks are concerned with formally analysing the output distribution of a neural network under a probability distribution of the inputs. Examples of probabilistic verification problems include…
Two polynomials $f, g \in \mathbb{F}[x_1, \ldots, x_n]$ are called shift-equivalent if there exists a vector $(a_1, \ldots, a_n) \in \mathbb{F}^n$ such that the polynomial identity $f(x_1+a_1, \ldots, x_n+a_n) \equiv g(x_1,\ldots,x_n)$…
This article proposes a formalism which unifies Hamiltonian simulation techniques from different fields. This formalism leads to a competitive method to construct the Hamiltonian simulation with a comprehensible, simple-to-implement circuit…
Circuit compilation, a crucial process for adapting quantum algorithms to hardware constraints, often operates as a ``black box,'' with limited visibility into the optimization techniques used by proprietary systems or advanced open-source…
In this paper, we initiate the study of deterministic PIT for $\Sigma^{[k]}\Pi\Sigma\Pi^{[\delta]}$ circuits over fields of any characteristic, where $k$ and $\delta$ are bounded. Our main result is a deterministic polynomial-time black-box…
Rational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits,…
We give quantum circuits that simulate an arbitrary two-qubit unitary operator up to global phase. For several quantum gate libraries we prove that gate counts are optimal in worst and average cases. Our lower and upper bounds compare…
Algebraic independence is an advanced notion in commutative algebra that generalizes independence of linear polynomials to higher degree. Polynomials {f_1, ..., f_m} \subset \F[x_1, ..., x_n] are called algebraically independent if there is…
A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be…
Zero-noise extrapolation provides a practical means of suppressing gate errors in current noisy intermediate-scale quantum hardware. The accuracy of the zero-noise estimate depends sensitively on the fidelity of the assumed noise model to…
We give a nontrivial algorithm for the satisfiability problem for cn-wire threshold circuits of depth two which is better than exhaustive search by a factor 2^{sn} where s= 1/c^{O(c^2)}. We believe that this is the first nontrivial…
As state-of-the-art neural networks are deployed on reasoning and algorithmic tasks, exactness guarantees become increasingly important. However, high average-case accuracy can still mask inconsistent behaviors. This motivates exact…
We show that noiseless random quantum circuits pass the linear cross-entropy benchmark (LXEB) test with high probability. If the circuits are linear depth, and thus form unitary 4-designs, the LXEB test is passed with probability…
Probabilistic pushdown automata (pPDA) are a standard model for discrete probabilistic programs with procedures and recursion. In pPDA, many quantitative properties are characterized as least fixpoints of polynomial equation systems. In…