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Birkhoff's theorem tells that any doubly stochastic matrix can be decomposed as a weighted sum of permutation matrices. A similar theorem reveals that any unitary matrix can be decomposed as a weighted sum of complex permutation matrices.…
We present a novel framework for Linear Combination of Unitaries (LCU)-style decomposition tailored to structured sparse matrices, which frequently arise in the numerical solution of partial differential equations (PDEs). While LCU is a…
Herein, we introduce a strategy to decompose an arbitrary square matrix into a linear combination of non-unitaries (LCNU) where each non-unitary term is embedded into a unitary matrix. The result is a linear combination of unitaries (LCU)…
Data centers are increasingly using high-speed circuit switches to cope with the growing demand and reduce operational costs. One of the fundamental tasks of circuit switches is to compute a sparse collection of switching configurations to…
The classic Birkhoff- von Neumann theorem states that the set of doubly stochastic matrices is the convex hull of the permutation matrices. In this paper, we study a generalisation of this theorem in the type $II_1$ setting. Namely, we…
The iterative method of Sinkhorn allows, starting from an arbitrary real matrix with non-negative entries, to find a so-called 'scaled matrix' which is doubly stochastic, i.e. a matrix with all entries in the interval (0, 1) and with all…
Sinkhorn proved that every entry-wise positive matrix can be made doubly stochastic by multiplying with two diagonal matrices. In this note we prove a recently conjectured analogue for unitary matrices: every unitary can be decomposed into…
We consider different Linear Combination of Unitaries (LCU) decompositions for molecular electronic structure Hamiltonians. Using these LCU decompositions for Hamiltonian simulation on a quantum computer, the main figure of merit is the…
We investigate ancilla-free linear combination of unitaries (LCU) as a framework for approximating complex quantum circuits. This is particularly effective for quantum optimization algorithms, where candidate solutions can be evaluated…
The Birkhoff polytope (the convex hull of the set of permutation matrices) is frequently invoked in formulating relaxations of optimization problems over permutations. The Birkhoff polytope is represented using $\Theta(n^2)$ variables and…
The randomized linear combination of unitaries (LCU) method with many applications to early fault-tolerant quantum computing algorithms has been proposed. This quantum algorithm computes the same expectation values as the original, fully…
Swap mapping is a quantum compiler optimization that, by introducing SWAP gates, maps a logical quantum circuit to an equivalent physically implementable one. The physical implementability of a circuit is determined by the fulfillment of…
Hyper-Connections (HC) generalizes residual connections by introducing dynamic residual matrices that mix information across multiple residual streams, accelerating convergence in deep neural networks. However, unconstrained residual…
The Linear Combination of Unitaries (LCU) method is a powerful scheme for the block encoding of operators but suffers from high overheads. In this work, we discuss the parallelisation of LCU and in particular the SELECT subroutine of LCU…
A highly anticipated use of quantum computers is the simulation of complex quantum systems including molecules and other many-body systems. One promising method involves directly applying a linear combination of unitaries (LCU) to…
Consider the problem of minimizing an expected logarithmic loss over either the probability simplex or the set of quantum density matrices. This problem includes tasks such as solving the Poisson inverse problem, computing the…
The success of Hyper-Connections (HC) in neural networks (NN) has also highlighted issues related to its training instability and restricted scalability. The Manifold-Constrained Hyper-Connections (mHC) mitigate these challenges by…
Entropic regularization provides a simple way to approximate linear programs whose constraints split into two or more tractable blocks. The resulting objectives are amenable to cyclic Kullback-Leibler (KL) Bregman projections, with…
Discrete circular convolution over $\mathbb{Z}/N\mathbb{Z}$ is a linear operator and can be implemented on quantum hardware within the linear-combination-of-unitaries (LCU) framework. In this work, we make this connection explicit through…
One limitation of the variational quantum eigensolver algorithm is the large number of measurement steps required to estimate different terms in the Hamiltonian of interest. Unitary partitioning reduces this overhead by transforming the…