The RIKEN Pioneering Project “Mathematical Foundation of Quantum Information” aims to integrate three theoretical pillars — operator algebras, quantum resource theory, and fault-tolerant quantum computing — into a unified theory that reveals the fundamental structure of quantum information under noise. Through a common language spanning mathematics, physics, and information theory, the project also aims to make foundational contributions to the theory of quantum gravity.
Research Overview — Toward the von Neumann of the 21st Century
John von Neumann left a series of achievements that laid the foundations of modern quantum science: Mathematical Foundations of Quantum Mechanics (1932), the founding of operator algebra theory (1929), and the von Neumann architecture for digital computers (1945). More recently, theoretical physicist Edward Witten has pointed out that operator algebras are essential to understanding quantum gravity, leading many to say that “von Neumann was a hundred years ahead of his time.” This project carries forward the tradition of operator algebra theory to build an unexplored, unified framework connecting quantum information, quantum error correction, and fundamental physics.
Toward a Unified Theory of Quantum Information
Fault tolerance (codes, anyons, fermions), operator algebras (subfactors, symmetry, phases), and resource theory (entanglement, capacity, protocols) are three closely related areas that together constitute a single theory of quantum information. This project integrates these three theoretical pillars around the following unifying goals:
- Elucidating the fundamental structure of quantum information under noise
- Building a common language spanning mathematics, physics, and information theory
Expected outcomes include a general theoretical model for fault-tolerant quantum computation and noisy quantum information processing, an understanding of topological order through operator algebras, an integration of category theory and tensor networks, and an operator-algebraic foundation for information loss and error recovery — areas where current theory relies on unrealistic, idealized assumptions, or where existing methods offer only very limited understanding.
Vision, Research Themes, and Goals
This project breaks down the overarching goal of a “unified theory of quantum information under noise” into three layers:
- (A) Mathematical foundations (abstract level): Structural formulation via operator algebras, subfactors, and category theory; a unified description of quantum resources, entanglement, and symmetry
- (B) Intermediate theory (connecting level): Operator-algebraic description of noisy quantum channels and open systems; integration of quantum resource theory and error correction (non-asymptotic, finite-size regimes)
- (C) Applications and physical connections (concrete level): Evaluating the fundamental limits of fault-tolerant quantum computation; elucidating the correspondence between topological phases, quantum codes, and noisy quantum dynamics
The first half of the project develops (A) and (B), leading into the development of (C) in the second half. Each postdoctoral researcher engages with two or more of these layers, working to connect them.
Goals
- Establishing a unified mathematical framework for quantum information under noise
- Presenting the fundamental limits and design principles of fault-tolerant quantum computation
- Building a common language spanning mathematics, physics, and information theory
Purpose of the Research Plan
This project aims for long-term leadership through mathematically grounded basic research in quantum information science — an approach made possible by RIKEN, where researchers in mathematics, physics, and information science are gathered together.
The foundations of quantum error-correcting codes, quantum computation, and quantum algorithms share structures closely resembling gauge theory in particle physics, and similar mathematical structures also appear in topological condensed matter physics. In recent years, methods from algebraic topology in particular have proven effective across disciplines, and are also connected to research on quantum gravity and black holes. The project also advances mathematical research on open quantum systems (decoherence, quantum information thermodynamics, and quantum resources).
Progress in quantum information and quantum computing is extremely rapid, and mathematically grounded basic research offers a valuable counter-strategy. In industry, Microsoft (Station Q) has invested heavily in mathematical foundational research typified by topological quantum computing, and Quantinuum’s research follows a similar direction.
Overview of the Research Plan
This project aims to carry forward the tradition of operator algebra research while building an unexplored, unified framework connecting quantum information, quantum error correction, and fundamental physics.
Specifically, we aim to elucidate the limits of quantum resource manipulation and communication under realistic noise through new mathematical developments in quantum information theory, and to derive implementable protocols and their ultimate performance limits using adaptive learning and representation-theoretic methods. We also aim to uncover the essential origin of fault tolerance, which is indispensable for large-scale quantum computation, by systematically studying the relationships among topological phases, quantum code structures, and noise dynamics. Through analysis of topological order in mixed states, fermionic and Majorana systems, and next-generation quantum LDPC (Low-Density Parity-Check) codes, we aim to establish design principles for intrinsically protected, low-overhead quantum systems — contributing to the foundations of fault-tolerant quantum computation and quantum gravity theory.
Research Plans by Team
Kawahigashi Team — Mathematics
Operator algebra theory has developed into a powerful mathematical framework underlying quantum mechanics, quantum field theory, and quantum statistical mechanics, yet its integration with quantum information theory remains largely unexplored. The Kawahigashi team works to bridge this gap and build a new mathematical foundation. In recent years, the importance of mathematical perspectives has grown across disciplines — from non-invertible symmetries in high-energy physics, to tensor networks in condensed matter physics, to topological quantum computing. This project’s distinctive strength lies in bringing together leading researchers in mathematics and physics to reconnect fields that had been growing apart. By using operator algebras as a common language to reconnect quantum information with modern theoretical physics, the team expects outcomes ranging from fault-tolerant quantum computing to a new mathematical understanding of quantum gravity.
Furusaki Team — Condensed Matter Physics
One of the biggest obstacles to scalable quantum computing is the enormous overhead required for fault-tolerant quantum error correction — prohibitively large in the case of the surface code. The Furusaki team aims to elucidate the physical mechanisms that intrinsically stabilize quantum information, and to use them to design low-overhead quantum computing architectures. This involves: (1) focusing on topological phases to clarify how overall structure, symmetry, and entanglement suppress local errors; (2) targeting next-generation quantum codes to build a unified theoretical foundation for understanding the interplay of geometric structure, noise, and computational resources with better scaling potential; and (3) introducing fermionic quantum computation as a complementary approach that avoids the costs associated with qubit mappings and simplifies operations — together aiming toward protected, scalable quantum computation.
Regula Team — Quantum Information
Understanding how to optimally transform and manipulate quantum resources is a central challenge in quantum information and quantum computing, yet conventional theoretical frameworks often rely on idealized assumptions chosen for analytical convenience, which tend to diverge from real implementations. The Regula team extends the scope of conventional methods for quantum resource manipulation and quantum communication, introducing and systematizing new approaches that reshape how quantum resources themselves are understood. This involves: (1) establishing a new framework for constructing implementable quantum information protocols using adaptive learning methods that avoid the bottlenecks of existing approaches; (2) using representation-theoretic methods to analyze the optimal performance and ultimate limits of general protocols, yielding practical performance benchmarks; and (3) precisely understanding realistic quantum information processing under non-asymptotic, near-term conditions through advances in probabilistic methods such as Rényi divergences.