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Quantum Algorithms for Computing Maximal Quantum $f$-divergence and Kubo-Ando means

Trung Hoa Dinh, Nhat A. Nghiem

TL;DR

This work develops two quantum algorithms for computing fundamental quantum information measures by embedding matrices and operations into quantum circuits via block-encoding and Quantum Singular Value Transformation (QSVT). It first targets maximal quantum $f$-divergences $D_f^{\max}(\rho\|\sigma)$ for operator-convex functions $f$, deriving procedures that block-encode density operators, form normalized composites like $\gamma$, and approximate $f(\gamma)$ (notably $\gamma\log\gamma$ for $f(x)=x\log x$) using direct QSVT or Stieltjes/Pade-type representations; trace estimation then yields the divergence with explicit complexity bounds tied to condition numbers. The second major thrust computes Kubo--Ando means $\sigma_f$ of PD operators, using harmonic–mean mixtures and Löwner–Stieltjes representations to realize $A\sigma_f B$ as block-encodings of sums or resolvents, respectively, with quadrature approximations over representation measures. By unifying entropic quantities and matrix means under a single block-encoding framework, the paper provides universal, programmable quantum subroutines that can adapt to the specific $f$ or mean of interest and offers explicit resource scaling in terms of block-encoding costs, conditioning, and desired accuracy. The approach promises practical impact for quantum information geometry, state analysis, and related optimization tasks, and opens pathways for applying quantum speedups to a broad class of operator-valued functionals.

Abstract

The development of quantum computation has resulted in many quantum algorithms for a wide array of tasks. Recently, there is a growing interest in using quantum computing techniques to estimate or compute quantum information-theoretic quantities such as Renyi entropy, Von Neumann entropy, matrix means, etc. Motivated by these results, we present quantum algorithms for computing the maximal quantum $f$-divergences and the operator-theoretic matrix Kubo--Ando means. Both of them involve Renyi entropies, matrix means as special cases, thus implying the universality of our framework.

Quantum Algorithms for Computing Maximal Quantum $f$-divergence and Kubo-Ando means

TL;DR

This work develops two quantum algorithms for computing fundamental quantum information measures by embedding matrices and operations into quantum circuits via block-encoding and Quantum Singular Value Transformation (QSVT). It first targets maximal quantum -divergences for operator-convex functions , deriving procedures that block-encode density operators, form normalized composites like , and approximate (notably for ) using direct QSVT or Stieltjes/Pade-type representations; trace estimation then yields the divergence with explicit complexity bounds tied to condition numbers. The second major thrust computes Kubo--Ando means of PD operators, using harmonic–mean mixtures and Löwner–Stieltjes representations to realize as block-encodings of sums or resolvents, respectively, with quadrature approximations over representation measures. By unifying entropic quantities and matrix means under a single block-encoding framework, the paper provides universal, programmable quantum subroutines that can adapt to the specific or mean of interest and offers explicit resource scaling in terms of block-encoding costs, conditioning, and desired accuracy. The approach promises practical impact for quantum information geometry, state analysis, and related optimization tasks, and opens pathways for applying quantum speedups to a broad class of operator-valued functionals.

Abstract

The development of quantum computation has resulted in many quantum algorithms for a wide array of tasks. Recently, there is a growing interest in using quantum computing techniques to estimate or compute quantum information-theoretic quantities such as Renyi entropy, Von Neumann entropy, matrix means, etc. Motivated by these results, we present quantum algorithms for computing the maximal quantum -divergences and the operator-theoretic matrix Kubo--Ando means. Both of them involve Renyi entropies, matrix means as special cases, thus implying the universality of our framework.

Paper Structure

This paper contains 13 sections, 7 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Overview of objectives, techniques and main results
  3. Quantum algorithms for maximal quantum $f$-divergence
  4. Case $f(x)=x\log x$
  5. General operator-convex $f$: Padé-type (resolvent) approximations
  6. Quantum algorithms for Kubo-Ando means
  7. Kubo--Ando representation (harmonic–mean mixture)
  8. Löwner--Stieltjes (resolvent) representation
  9. Conclusion
  10. Preliminaries
  11. Block-Encodings
  12. Quantum singular value transformation
  13. Operations involving block-encoded operators

Key Result

Lemma 2.1

The block-encoding of density matrix $\rho \in \mathbb{C}^{N \times N}$ of interested can be obtained via the following ways.

Theorems & Definitions (10)

  • Lemma 2.1: Block-encoding density operator gilyen2019quantumnghiem2025new1
  • Lemma 2.2: Block-encoding accessible matrices
  • Lemma 2.3: Approximation of $\log x$
  • Theorem 2.1: Estimating quantum $f$-divergence
  • Theorem 2.2: Kubo-Ando means
  • Definition 4.1: Operator mean kubo1980means
  • Definition A.1: Block-encoding childs2017quantumgilyen2019quantum
  • Remark A.1: Properties of block-encoding unitary
  • Theorem A.1: Quantum Singular Value Transformation gilyen2019quantum
  • Lemma A.1: Product of block-encoded matrices [Lemma 30 of gilyen2019quantum] If $U$ is an $(\alpha, a, \delta)$-block-encoding of a matrix $A$ and $V$ is a $(\beta, b, \varepsilon)$-block-encoding of a matrix $B$, then there exists a unitary $W$ that is an $(\alpha\beta, a+b, \alpha\varepsilon+\beta\delta)$-block-encoding of $AB$. Moreover, $W$ can be implemented by one query to each of $U$ and $V$. Let $\alpha, \beta > 0$ be constants, and let $\vec{\gamma} \in \mathbb{R}^m$ be a vector such that $\|\vec{\gamma}\|_1 \leq 1$. Suppose that each $U_j$ is a $(1, a, \varepsilon)$-block-encoding of a matrix $A_j$ for $j=1,\dots,m$. Then there exists a unitary $U$ that is a $(\eta, a+O(\log m), \varepsilon)$-block-encoding of $\sum_{j=1}^m \gamma_j A_j$, where $\eta = \|\vec{\gamma}\|_1$. Moreover, $U$ can be implemented by one query to each $U_j$ and $O(m)$ additional gates. The lemmas below are direct consequences of Lemma \ref{['lemma: qsvt']} with appropriate choice of function $p(x)$ (in Lemma \ref{['lemma: qsvt']}). Given a block encoding of a positive matrix $\frac{\mathcal{M}}{\gamma}$ such that $\frac{\mathbb{I}}{\kappa_M} \leq \frac{\mathcal{M}}{\gamma}\leq \mathbb{I}.$ then we can implement an $\epsilon$-approximated block encoding of $\mathcal{M}^{-c}/(2\kappa_M^c)$ in complexity $\mathcal{O}( \kappa_M T_M (1+c) \log( \frac{\gamma }{\epsilon} ) )$ where $T_M$ is the complexity to obtain the block encoding of $\mathcal{M}$. In principle, the lemma above can be used to invert a block-encoded matrix, by choosing $c=1$. However, there exists a more efficient way to do so, by using another choice of function within Lemma \ref{['lemma: qsvt']}. Given a block encoding of some matrix $A$ with operator norm $||A|| \leq 1$ and block-encoding complexity $T_A$, then there is a quantum circuit producing an $\epsilon$-approximated block encoding of ${A^{-1}}/{\kappa}$ where $\kappa$ is the conditional number of $A$. The complexity of this quantum circuit is $\mathcal{O}\left( \kappa T_A \log \left({1}/{\epsilon}\right)\right)$. Given a block encoding of a positive matrix $\mathcal{M}/\gamma$ such that $\frac{\mathbb{I}}{\kappa_M} \leq \frac{\mathcal{M}}{\gamma} \leq \mathbb{I}.$ Let $c \in (0,1)$. Then we can implement an $\epsilon$-approximated block encoding of $(\mathcal{M}/\gamma)^c/2$ in time complexity $\mathcal{O}( \kappa_M T_M \log (\frac{ 1}{\epsilon}) )$, where $T_M$ is the complexity to obtain the block encoding of $\mathcal{M}/\gamma$. Given the block-encoding of some matrix $A$, the block-encoding of $A/p$ where $p > 1$ can be prepared with an extra $\mathcal{O}(1)$ cost. Let $U$, $\Pi$, $\widetilde{\Pi} \in {\rm End}(\mathcal{H}_U)$ be linear operators on $\mathcal{H}_U$ such that $U$ is a unitary, and $\Pi$, $\widetilde{\Pi}$ are orthogonal projectors. Let $\gamma>1$ and $\delta,\epsilon \in (0,\frac{1}{2})$. Suppose that $\widetilde{\Pi}U\Pi=W \Sigma V^\dagger=\sum_{i}\varsigma_i\ket{w_i}\bra{v_i}$ is a singular value decomposition. Then there is an $m= \mathcal{O} (\frac{\gamma}{\delta} \log \left(\frac{\gamma}{\epsilon} \right))$ and an efficiently computable $\Phi\in\mathbb{R}^m$ such that \left(\bra{+}\otimes\widetilde{\Pi}_{\leq\frac{1-\delta}{\gamma}}\right)U_\Phi \left(\ket{+}\otimes\Pi_{\leq\frac{1-\delta}{\gamma}}\right)=\sum_{i\colon\varsigma_i\leq \frac{1-\delta}{\gamma} }\tilde{\varsigma}_i\ket{w_i}\bra{v_i} , \text{ where } |\!|\frac{\tilde{\varsigma}_i}{\gamma\varsigma_i}-1 |\!|\leq \epsilon. Moreover, $U_\Phi$ can be implemented using a single ancilla qubit with $m$ uses of $U$ and $U^\dagger$, $m$ uses of C$_\Pi$NOT and $m$ uses of C$_{\widetilde{\Pi}}$NOT gates and $m$ single qubit gates. Here, C$_\Pi$NOT$:=X \otimes \Pi + I \otimes (I - \Pi)$ and a similar definition for C$_{\widetilde{\Pi}}$NOT; see Definition 2 in gilyen2019quantum,$U_\Phi$: alternating phase modulation sequence; see Definition 15 in gilyen2019quantum,$\Pi_{\leq \delta}$, $\widetilde{\Pi}_{\leq \delta}$: singular value threshold projectors; see Definition 24 in gilyen2019quantum. To extend to operator monotone functions, we need some approximation to it. of donoghue2012monotone), (Eq. (5.62) of bhatia2006riemannianbhatia2009positive),donoghue2012monotonebhatia2019buresbhatia2006riemannianhansen2013fast A function $f:(0,\infty)\to\mathbb{R}$ is operator monotone if and only if it admits the integral representation $f(x)=\alpha+\beta x+\int_{0}^{\infty}\frac{x}{x+\lambda}\,d\mu(\lambda),$ for some $\alpha\in\mathbb{R}$, $\beta\ge 0$, and a finite positive measure $\mu$ on $[0,\infty)$ for which the integral converges. This is a standard consequence of approximating a Stieltjes transform via positive quadrature and Padé approximants for Stieltjes (Markov) functions. In particular: Let $f(z)=\int_0^\infty \frac{d\mu(t)}{z-t}$ be a Stieltjes transform of a positive measure $\mu$ with finitely many poles. Then the diagonal Padé approximants $[n/n]_f(z)$ converge uniformly to $f(z)$ on compact subsets of the complex plane away from the support of $\mu$. 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