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

The skills gap the project targets is real and well documented across the computing community. Quantum computers operate on fundamentally different principles than classical machines, manipulating quantum bits, or qubits, that exploit superposition and entanglement to represent and process information in ways no conventional processor can. Writing software for such devices requires an understanding of quantum mechanics, error-prone hardware behavior, and specialized algorithms, while extracting value from those algorithms in medicine additionally demands deep familiarity with molecular biology, clinical data, and drug discovery pipelines. Very few training programs ask learners to cross that entire bridge. Most quantum curricula are built by and for physicists and computer scientists, and most biomedical training barely touches quantum concepts at all. Jiang’s team is betting that the fastest route to a capable workforce is to build the bridge deliberately, in both directions, from the start.

According to the project description, the training program is domain-driven and organized around three integrated thrusts, each addressing a different layer of the expertise stack. The first thrust focuses on problem formulation: teaching participants to translate health care questions into forms that quantum algorithms can actually address. That includes quantum-compatible data representations, a subtle but crucial topic, because classical biomedical data, from imaging archives to genomic sequences, must be encoded into quantum states before any quantum speedup can even be contemplated. It also covers applications in molecular modeling and diagnostic analysis, two areas where researchers widely believe quantum hardware may eventually outperform classical methods, whether by simulating electronic structures of drug candidates more faithfully or by enhancing pattern recognition in high-dimensional clinical datasets.

The second thrust tackles a challenge that has plagued quantum software development since the field’s inception: productivity, portability, and reproducibility. Quantum programs written for one hardware platform often cannot run on another without substantial rework, because competing devices from different vendors use distinct qubit technologies, gate sets, and noise characteristics. The project’s modules therefore address cross-platform program translation, giving trainees the tools to move code between quantum systems rather than locking it to a single machine. Equally important is auditable workflow construction, a concept borrowed from the reproducibility movement in data science. In biomedical research, where results inform clinical decisions, being able to trace exactly how a computational result was produced is not optional. The curriculum also introduces uncertainty-aware evaluation, training participants to characterize and reason about noise, drift, and hardware variability, the persistent imperfections of today’s noisy intermediate-scale quantum devices that can silently corrupt results if left unexamined.

The third thrust is the most application-oriented of the three. Rather than stopping at theory or tooling, it guides participants through building complete quantum-classical workflows for concrete health care use cases. Among the named examples are protein structure prediction, a problem central to understanding disease and designing therapeutics, where quantum and classical processors would work in tandem, each handling the subtasks it performs best. The curriculum also covers quantum-ready dataset design, teaching learners how to prepare and structure biomedical data so that it can take advantage of quantum processing as hardware matures, and DNA and RNA therapeutics modeling, an area of intense pharmaceutical interest as nucleic-acid-based drugs move from laboratory curiosity to approved medicine. By working end to end, from raw problem to running workflow, trainees encounter the full friction of real-world quantum application development rather than an idealized classroom version.

What distinguishes the George Mason-led effort from many earlier workforce programs is its emphasis on sustainability and reuse. The project description states that the combined thrusts deliver a reusable and scalable training program, one that integrates rigorous instruction, hands-on practice, open-source resources, and access to both simulated and real quantum hardware. That last element matters enormously. Quantum hardware remains scarce, expensive, and geographically concentrated, and most students would never touch a genuine quantum processor without cloud-based access programs. By pairing simulators, which allow unlimited experimentation at the cost of classical computational overhead, with real devices, the program lets learners experience both the convenience of idealized models and the sobering realities of hardware noise. Open-source materials, meanwhile, mean the curriculum can outlive the grant period and be adopted by institutions far beyond the original collaborating states.

The dual-state collaboration referenced in the project title reflects the geographic spread of the investigator team, with George Mason University in Virginia anchoring a partnership that extends across state lines. Such multi-institutional structures are a deliberate feature of NSF’s CyberTraining program, which funds projects to grow and diversify the national cyberinfrastructure workforce. The logic is straightforward: workforce challenges are too large for any single campus to solve, and regional partnerships can pool complementary strengths, whether in quantum hardware expertise, biomedical research, or educational outreach. With four principal investigators sharing a $1 million budget, the project exemplifies the medium-scale implementation awards that CyberTraining uses to move promising training models from pilot scale to durable programs.

The timing of the initiative is hardly accidental. Federal and private investment in quantum information science has surged over the past decade, and national strategies in the United States and abroad explicitly identify workforce development as a bottleneck alongside hardware progress. Meanwhile, the biomedical sector is watching quantum simulation with growing interest, since molecules are themselves quantum objects and classical computers must approximate their behavior with computationally expensive methods. If quantum processors become reliable enough to simulate molecular interactions directly, the consequences for drug discovery, protein engineering, and personalized medicine could be profound. But that future depends on a generation of researchers who can stand with one foot in each world, and the supply of such people today is vanishingly small. Training programs like Jiang’s are an attempt to widen that pipeline before the hardware arrives rather than after.

The project also carries an explicitly ethical dimension. In its framing, the program positions participants to contribute to the national quantum workforce and to the responsible use of quantum computing in biomedical research. That phrase, responsible use, signals an awareness that powerful computational tools in medicine raise questions of privacy, equity, and validation that purely technical curricula often ignore. Auditable workflows and uncertainty-aware evaluation, both named components of the training, are as much about scientific integrity as about engineering practice. Teaching future practitioners to quantify and communicate the uncertainty in quantum-assisted results, and to document their methods so others can verify them, builds accountability into the workforce itself rather than bolting it on later.

For George Mason University, Virginia’s largest public research university, the award extends a growing portfolio in quantum and computing research at an institution that enrolls more than 40,000 students from 130 countries and all 50 states. For the broader community, the project’s real legacy may be its template: a domain-driven, three-thrust curriculum that others can adapt to pair quantum computing with fields beyond health care, from finance to materials science. Funding runs through late July 2030, giving the team nearly four years to refine the modules, train cohorts of participants, and demonstrate that dual expertise can be taught systematically rather than acquired by rare accident. If the model works, the quantum computers of the 2030s may find a workforce already waiting for them, fluent in qubits and clinical data alike, and ready to put one of the century’s most consequential technologies to work on some of medicine’s hardest problems.

Subject of Research: Workforce training at the intersection of quantum computing and health care through a NSF CyberTraining collaboration

Article Title: Jiang building dual-expertise workforce in quantum computing & health care through dual-state collaboration

Article References: Jiang building dual-expertise workforce in quantum computing & health care through dual-state collaboration. (n.d.). Original publication

Image Credits: AI Generated

DOI: Not provided

Keywords: quantum computing, health care, workforce development, CyberTraining, National Science Foundation, George Mason University, quantum algorithms, biomedical research, protein structure prediction, quantum software, molecular modeling, dual-state collaboration

Encoding is one of the first and most consequential decisions in designing a quantum program. Just as classical computers must decide how to represent integers or characters as bits, quantum algorithms must map problem variables—whether they denote colors, routes, or cluster assignments—onto the quantum states of a register. Over the past decade, researchers have adapted classical schemes such as one-hot, binary, and Gray codes to qubits, and have invented quantum-native alternatives like angle encoding, amplitude encoding, and Schmidt decomposition. Each approach embodies a trade-off between the number of qubits consumed and the complexity of the circuits needed to operate on them. One-hot encoding, in which one bit is reserved for every possible value a variable can take and exactly one of those bits is set to 1, is particularly popular because it is simple to encode, decode, and check for validity—even though it is wasteful in qubit count. A variable with four possible values, for instance, needs four qubits under one-hot rules, yielding the four allowed states 0001, 0010, 0100, and 1000.

The insight behind Multi-Unary is disarmingly simple. Instead of requiring that exactly one bit be set to 1, the scheme requires only that at least one bit be set to 1, with the single forbidden state being the all-zero string. Under this relaxation, the four-qubit register above can represent not just four values but fifteen—the full set of non-zero bit strings. Because more than one value can live in the same group of qubits, the Spanish team describes Multi-Unary as a “dense” encoding scheme, and, to their knowledge, it is the first encoding in the quantum computing literature with this property. The consequence is profound for algorithms like Grover’s: a single measured bit string can now encode several valid assignments at once, so the quantum state effectively contains and delivers multiple solutions in a single shot.

Grover’s algorithm, first proposed in 1996, searches an unstructured space of size N containing M solutions with a complexity of O(√(N/M)), a quadratic speedup over any classical method. The algorithm begins by placing the input qubits into an equal superposition of all possible states. It then applies an oracle—a quantum subroutine that recognizes valid solutions and flips their phase by π radians—followed by a diffusion operation that inverts the amplitudes about their mean, amplifying the marked states. Each oracle-plus-diffusion cycle is one Grover iteration, and the probability of measuring a solution after i iterations is given by P = sin²((2i+1)·θg), where θg = arcsin(√(M/N)). The optimal number of iterations scales as R ≤ ⌈(π/4)√(N/M)⌉. Crucially, both the success probability and the required iteration count depend on the ratio M/N: the more states qualify as solutions, the better.

This is exactly where Multi-Unary shines. By enlarging the set of bit strings that count as legitimate solutions, the scheme increases M, which the team proves mathematically raises the success probability P and lowers the required number of Grover iterations R. They derive the partial derivatives of both quantities with respect to M, showing rigorously that P grows monotonically and R shrinks monotonically over the entire operating range. The practical payoff comes in three forms: a higher chance that any given measurement yields a solution, fewer oracle calls needed to maximize that chance, and measured outputs that can bundle multiple solutions into one value, reducing the total number of circuit executions—or “shots”—needed to recover a desired fraction of all solutions.

To demonstrate the idea concretely, the researchers applied Multi-Unary to the graph coloring problem, a canonical combinatorial challenge that asks for the assignment of k colors to the nodes of a graph such that no two adjacent nodes share a color. Graph coloring matters well beyond theory—it underpins scheduling, register allocation, and resource assignment tasks—and it has become a standard benchmark for quantum algorithms, tackled previously with quantum annealing, variational approaches, and Grover-based oracles using both binary and one-hot encodings. In the Spanish team’s demonstration, a six-node, six-edge graph is colored with three colors, the minimum possible.

Their oracle design rests on quantum counters, built from multi-controlled X gates, which tally constraint violations. The logic proceeds in three steps. First, for each node, the counter is incremented if all of that node’s color qubits are zero—penalizing strings that violate the Multi-Unary rule that every variable must carry at least one value. Second, for every edge and every color, the counter is incremented if both endpoints display that same color simultaneously, penalizing genuine coloring conflicts. Third, only strings for which the counter reads zero—strings satisfying every constraint—have their phases rotated by a multi-controlled Z gate, marking them for amplification. A single ancilla qubit acts as a trigger that fires each counter increment and is uncomputed immediately afterward, while the remaining ancillas store the running count. The design also incorporates a subtle optimization: because early counter increments only touch the least significant bits of the count, the number of controls on successive increment gates can be grown gradually, trimming expensive multi-controlled gates.

The resource analysis for the example circuit is instructive. The complete oracle comprises 367 gates, of which 284 are controlled X gates—and 188 of those, fully 66.2 percent, implement the counter alone, underscoring how central counting is to this style of oracle construction. The circuit depth is 287 layers, dominated by the trigger qubit through which every increment must pass. Simulating the full circuit with 36 Grover iterations—the number that maximizes success probability—yielded a 99.88 percent chance of measuring a solution, and a histogram of 1,000 shots in Qiskit’s Aer simulator recovered 126 distinct solution values, with only two spurious non-solutions appearing once each, a rate consistent with statistical expectations.

The headline result is the sheer expansion of the solution space. Under a conventional one-hot oracle, the example graph has exactly 48 valid colorings. With Multi-Unary, the number of solution values jumps to 126—78 additional bit strings that are also valid, thanks to nodes that can carry more than one compatible color. This flexibility arises naturally in real graphs: leaf nodes and nodes with low connectivity often admit several admissible colors, and Multi-Unary lets the quantum state express that freedom directly. Decoding is straightforward: any measured variable whose bits contain more than one 1 is expanded into as many one-hot candidates as there are set bits, each assigning a single value while preserving the rest. In one worked example, the output string 011 010 001 100 010 101—where two nodes each hold two colors—unpacks into four distinct, fully valid colorings of the graph. A single measurement delivered four answers.

The authors are careful to position their contribution honestly. Binary encoding still wins on raw qubit count—a 50-node graph with 10 colors needs 200 binary-encoded input qubits versus 500 for one-hot or Multi-Unary—but binary circuits must then explicitly reject the invalid states corresponding to out-of-range values, adding gates and ancillas that Multi-Unary avoids by construction. Indeed, validity checking is where Multi-Unary beats its closest cousin outright: the only forbidden state under Multi-Unary is the all-zero string, detectable with a simple NOR gate, whereas one-hot verification requires a counter to confirm that exactly one bit is set. Because encoding and decoding work identically in both schemes, the team argues that Multi-Unary could outright replace one-hot encoding, with migration requiring changes only to the oracle’s validity-check portion—and a simpler oracle as a reward. The advantage does taper off for densely connected graphs, where fewer nodes admit multiple colors and the scheme converges back toward plain one-hot behavior.

The implications reach beyond graph coloring. The researchers point to shortest-path, graph coverage, and clustering problems as natural targets, and note that their encoding could slot into emerging frameworks like the discrete quantum intermediate representation proposed for encoding-independent optimization. For today’s noisy, resource-starved quantum hardware, where every ancilla qubit and every additional Grover iteration carries a real cost, a scheme that squeezes multiple solutions into the same qubits—and trims the iteration count as a bonus—offers a rare kind of free lunch. If quantum computing’s near-term future depends on getting the most out of every qubit, Multi-Unary suggests that sometimes the smartest move is not a new algorithm, but a new way of writing the problem down.