IEEE International Conference on Quantum Control, Computing and Learning (qCCL)

Real Symmetric Matrices Larger Than a Hermitian Matrix: Applications to Gaussian Quantum Information Processing*

Shan Ma, Yuchen Xiong, Lihan Zhou, Shibei Xue

Published Jun 25, 2025• 0 citations

Abstract

In this paper, we investigate the set of real symmetric matrices that are larger than a given Hermitian matrix in the semidefinite order, with particular attention to its minimal element. We explore the applications of this result in Gaussian quantum information processing. Furthermore, we propose potential research directions, especially the extension of this result to the set of real symmetric matrices that are larger than multiple given Hermitian matrices. These topics are of interest in both matrix theory and quantum information processing.

DOI: 10.1109/qccl65142.2025.11158248

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IEEE International Conference on Quantum Control, Computing and Learning (qCCL)

Real Symmetric Matrices Larger Than a Hermitian Matrix: Applications to Gaussian Quantum Information Processing*

Shan Ma, Yuchen Xiong, Lihan Zhou, Shibei Xue

Published Jun 25, 2025• 0 citations

Publisher

Abstract

DOI: 10.1109/qccl65142.2025.11158248

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2^n\times 2^n

unitary matrices, which describe operations on a closed

n

-qubit system. We define a set of simple quantum gates, called controlled single qubit gates, and their associated operational cost. We then present a recurrence scheme to decompose a general

2^n\times 2^n

unitary matrix to the product of no more than

2^{n-1}(2^n-1)

single qubit gates with small number of controls. In chapter 3, we address the problem of finding a specific element

\Phi

among a given set of quantum channels

\mathcal{S}

that will produce the optimal value of a scalar function

D(\rho_1,\Phi(\rho_2))

, on two fixed quantum states

\rho_1

and

\rho_2

. Some of the functions we considered for

D(\cdot,\cdot)

are the trace distance, quantum fidelity and quantum relative entropy. We discuss the optimal solution when

\mathcal{S}

is the set of unitary quantum channels, the set of mixed unitary channels, the set of unital quantum channels, and the set of all quantum channels. In chapter 4, we focus on the spectral properties of qubit-qudit bipartite states with a maximally mixed qudit subsystem. More specifically, given positive numbers

a_1\geq\ldots\geq a_{2n}\geq 0

, we want to determine if there exist a

2n\times 2n

density matrix

\rho

having eigenvalues

a_1,\ldots,a_{2n}

and satisfying $\tr_1(\rho)=\frac{1}{n}I_n$. This problem is a special case of the more general quantum marginal problem. We give the minimal necessary and sufficient conditions on

a_1,\ldots,a_{2n}

for

n\leq 6

and state some observations on general values of

n

. In chapter 5, we discuss the numerical method of alternating projections and illustrate its usefulness in: (a) constructing a quantum channel, if it exists, such that

\Phi(\rho^{(1)})=\sigma^{(1)},\ldots, \Phi(\rho^{(k)})=\sigma^{(k)}

for given

\rho^{(1)},\ldots,\rho^{(k)}\in \mathcal{D}_n

and

\sigma^{(1)},\ldots,\sigma^{(k)}\in \mathcal{D}_m

, (b) constructing a multipartite state

\rho

having a prescribed set of reduced states

\rho_1,\ldots, \rho_r

r

of its subsystems, (c) constructing a multipartite state

\rho

having prescribed reduced states and additional properties such as having prescribed eigenvalues, prescribed rank or low von Neuman entropy; and (d) determining if a square matrix

A

can be written as a product of two positive semidefinite contractions. In chapter 6, we examine the shape of the Minkowski product of convex subsets

K_1

and

K_2

of $\IC$ given by

K_1K_2 = \{ab: a \in K_1, b\in K_2\}

, which has applications in the study of the product numerical range and quantum error-correction. In \citep{Karol}, it was conjectured that

K_1K_2

is star-shaped when

K_1

and

K_2

are convex. We give counterexamples to show that this conjecture does not hold in general but we show that the set

K_1K_2

is star-shaped if

K_1

is a line segment or a circular disk.

m

constraint matrices, each of dimension

n

and rank

r

, our algorithm can compute any entry and efficient descriptions of the spectral decomposition of the solution matrix. The algorithm runs in time

O(m\cdot\mathrm{poly}(\log n,r,1/\varepsilon))

given access to a sampling-based low-overhead data structure for the constraint matrices, where

\varepsilon

is the precision of the solution. In addition, we apply our algorithm to a quantum state learning task as an application. Technically, our approach aligns with 1) SDP solvers based on the matrix multiplicative weight (MMW) framework by Arora and Kale [TOC '12]; 2) sampling-based dequantizing framework pioneered by Tang [STOC '19]. In order to compute the matrix exponential required in the MMW framework, we introduce two new techniques that may be of independent interest:

\bullet

Weighted sampling: assuming sampling access to each individual constraint matrix

A_{1},\ldots,A_τ

, we propose a procedure that gives a good approximation of

A=A_{1}+\cdots+A_τ

\bullet

Symmetric approximation: we propose a sampling procedure that gives the \emph{spectral decomposition} of a low-rank Hermitian matrix

A

. To the best of our knowledge, this is the first sampling-based algorithm for spectral decomposition, as previous works only give singular values and vectors.