To appreciate why this matters, it helps to understand what these quantities measure. Divergences are the workhorses of information theory: they quantify how distinguishable two probability distributions, or in the quantum setting two quantum states, are from one another. The famous quantum relative entropy, the Rényi divergences used in cryptography and quantum channel discrimination, and the fidelity-like measures that certify quantum memories are all members of the f-divergence family. The maximal quantum f-divergence, in particular, is defined through an optimization over measurements and captures the ultimate statistical distinguishability of two quantum states. Computing it exactly is generally intractable on classical machines because it involves functions of matrices whose dimensions grow exponentially with the number of qubits. The Kubo–Ando means, introduced by Fumio Kubo and Tsuyoshi Ando in 1980, form the complete family of two-variable matrix means that respect the right symmetries and monotonicity properties, generalizing the arithmetic, geometric, and harmonic means from numbers to matrices.

The technical engine behind the new algorithms is the block-encoding framework, which has become one of the most powerful unifying tools in quantum computing. A block encoding embeds an arbitrary matrix A into a larger unitary operator, the only kind of operation a quantum computer can natively perform. Formally, a unitary U acting on the system qubits plus a set of ancilla qubits is called an (α, a, ε)-block encoding of A if the top-left block of U, after subtracting out the ancilla register, equals A scaled by a normalization factor α, up to error ε. Once a matrix is encoded this way, an entire toolbox of quantum algorithms can act on it with provable complexity guarantees. Matrix inversion, Hamiltonian simulation, eigenvalue estimation, and principal component analysis can all be expressed in this language, which is precisely why the authors describe block encoding as a foundational bridge that transforms mathematical objects into quantum operations.

Sitting on top of block encoding is the quantum singular value transformation, or QSVT, developed by András Gilyén and collaborators in 2019. QSVT takes a block-encoded matrix and applies a polynomial transformation to its singular values: given a polynomial p of degree d bounded by one on the interval from minus one to one, a quantum circuit of length proportional to d, built from alternating applications of the block-encoding unitary and carefully chosen single-qubit phase rotations, implements a block encoding of p applied to the matrix. The consequence is striking. Smooth analytic functions such as the exponential can be approximated with circuit depth growing only logarithmically in the inverse precision, while even discontinuous functions like the sign function require depth scaling as one over the precision. For the quantities in this paper, the relevant functions are the operator monotone functions, a special class that includes the square root, the logarithm, and the power functions x raised to any exponent between zero and one.

Here the authors deploy a piece of classical mathematics that dates back to the early twentieth century. Operator monotone functions on the positive real line admit a representation, due to Löwner and Pick and Nevanlinna, as an integral of simple rational functions of the form x divided by x plus λ, weighted by a positive measure. This structure means that such functions extend analytically to Stieltjes transforms, and a classical theorem on the convergence of Padé approximants guarantees that rational functions of this form converge uniformly to the target function on any compact interval away from the singularities. The upshot, formalized in the paper as a positive rational approximation lemma, is that any operator monotone function can be approximated on a relevant interval by a sum of positive rational terms with an error that shrinks exponentially in the number of terms. Each rational term can itself be implemented efficiently, because it requires only the matrix A and the resolvent-like operation of adding a scaled identity, both of which are straightforward within the block-encoding paradigm.

With these ingredients assembled, the construction of the algorithms becomes transparent, at least in outline. For the Kubo–Ando means, the key observation is that every such mean of two positive definite matrices A and B can be written using an operator monotone function applied to a ratio-like combination of the two matrices. The authors first build block encodings of the individual matrices, then use composition lemmas to form products and linear combinations of block-encoded operators, and finally apply the QSVT machinery with the rational approximation of the relevant operator monotone function. Along the way, they rely on established results for handling matrix powers: negative powers, which amount to matrix inversion, cost complexity proportional to the condition number times the block-encoding cost times the logarithm of the inverse precision, while fractional positive powers between zero and one carry a similar logarithmic dependence on precision. The end result is a quantum circuit that block-encodes the desired Kubo–Ando mean with complexity scaling polynomially in the condition number and logarithmically in the target accuracy.

The computation of maximal quantum f-divergences follows a parallel logic. Because these divergences are built from operator monotone functions of the ratio between two density matrices, the same rational-approximation-plus-QSVT pipeline applies. The generality of the framework is one of its most attractive features: since the Rényi entropies and the standard matrix means appear as special cases, the paper’s results immediately subsume and generalize a string of earlier algorithms, including quantum algorithms for estimating Rényi entropies of quantum states, algorithms for estimating quantum entropies more broadly, and a 2025 npj Quantum Information paper on quantum algorithms for matrix geometric means. Rather than treating each of these quantities with a dedicated procedure, the new work folds them into a single modular architecture, in the spirit of how block encoding itself unified quantum linear algebra.

The complexity analysis reveals both the power and the honest limitations of the approach. The algorithms inherit the standard caveats of block-encoding-based methods: they require coherent access to the input matrices through efficient block-encoding oracles, and their running times depend on normalization factors and condition numbers that can in the worst case be large. The dependence on precision, however, is only logarithmic for the smooth functions at the heart of the construction, which is exponentially better than naive approaches based on Taylor series or direct sampling. This logarithmic dependence traces directly back to the exponentially convergent Padé and Gauss-type quadrature approximations of Stieltjes functions, a beautiful example of classical approximation theory doing heavy lifting inside a quantum circuit. For practitioners, the message is that whenever the relevant matrices admit efficient block encodings, which is the case for many structured families arising in quantum information and physics, these divergences and means can be estimated to high accuracy with resources that scale gently in the desired precision.

Beyond the immediate results, the paper signals a broader trend in which quantum algorithms are being extended from computational tasks like solving linear systems to the evaluation of the mathematical quantities that populate theoretical physics and information theory. Recent preprints by the same authors on estimating nonlinear physical quantities by measuring ancillas and on refined algorithms for principal component analysis suggest a sustained research program along these lines. As quantum hardware matures, tools of this kind could become standard instruments for characterizing entanglement, benchmarking quantum channels, and computing geometric distances between quantum states, including the Bures–Wasserstein distance that has attracted attention in both quantum information and machine learning. By showing that entire families of divergences and means, not just isolated examples, can be handled within one framework, Dinh and Nghiem have provided a template that future algorithm designers can instantiate again and again, turning some of the most elegant structures of matrix analysis into runnable quantum circuits.

Subject of Research: Quantum algorithms for computing maximal quantum f-divergences and Kubo–Ando operator means

Article Title: Quantum algorithms for computing maximal quantum f-divergence and Kubo–Ando means

Article References: Dinh, T. H., & Nghiem, N. A. (2026). Quantum algorithms for computing maximal quantum f-divergence and Kubo–Ando means. Quantum Information Processing, 25(10), Article 322. https://doi.org/10.1007/s11128-026-05309-8

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05309-8

Keywords: quantum algorithms, quantum information, f-divergence, Kubo–Ando means, block encodings, quantum singular value transformation, Rényi entropy, matrix means, operator monotone functions, quantum computing, entropy estimation, Padé approximation