Frozen States Unlock Exponential Quantum Memory

Author: Denis Avetisyan

New research demonstrates a pathway to encoding vast numbers of stable qubits by harnessing the interplay of symmetry and fragmented quantum states.

A two-qubit logical space is defined by four symmetry-related frozen states interconnected via symmetry generators $X_A^{(B)}X_{A(B)}$, ensuring that any superposition of these states remains stable under the dynamics described by equation (2).

Combining symmetry enrichment with topologically stable Hilbert Space Fragmentation yields a robust platform for quantum information storage.

Maintaining robust quantum information storage remains a central challenge in quantum computing, often limited by environmental noise and decoherence. In the work ‘Profusion of Symmetry-Protected Qubits from Stable Ergodicity Breaking’, we demonstrate a pathway to encode an exponentially large number of topologically stable qubits by combining discrete symmetry with Hilbert space fragmentation. This approach yields qubits resilient to symmetry-respecting perturbations for extended periods, surpassing the robustness of prior non-topological fragmentation schemes. Could this synergy between symmetry enrichment and fragmentation unlock new possibilities for utilizing systems with fragmented Hilbert spaces as viable quantum memories?

The Fragile Dance of Ergodicity: Balancing Exploration and Preservation

The behavior of complex quantum systems, those comprised of many interacting particles, is often characterized by a principle called ergodicity. This means the system, over time, explores all states available to it with equal probability, effectively ‘sampling’ its entire possible configuration space. This thorough exploration isn’t merely a theoretical curiosity; it’s fundamentally linked to thermalization, the process by which a system reaches equilibrium and establishes a stable temperature. Without ergodicity, energy can become trapped in specific configurations, preventing the system from achieving a uniform distribution of energy and thus hindering the establishment of a predictable, stable state. The extent to which a system embodies ergodicity is therefore a key determinant of its thermal properties and its ability to reach a well-defined equilibrium, influencing a wide range of physical phenomena from the behavior of materials to the dynamics of chemical reactions.

The very property that allows many-body quantum systems to reach thermal equilibrium – ergodicity, or the tendency to explore all accessible states – presents a significant challenge for building stable quantum memories. A fully ergodic system, by its nature, facilitates the free flow of energy and information, meaning any quantum state used to encode data is susceptible to unwanted interactions and, consequently, decoherence. This leakage arises because ergodicity doesn’t discriminate between preserving the encoded information and allowing it to dissipate into the many other available states within the system. Essentially, the same mechanism that drives thermalization also erodes the delicate quantum states required for reliable data storage, demanding a careful re-evaluation of system design to prioritize information retention over complete state exploration.

The pursuit of robust quantum memories is driving research into systems that deliberately deviate from complete ergodicity. Traditional quantum systems, while facilitating thermalization, present a challenge for stable information storage because the free exploration of all accessible states allows encoded quantum information to readily dissipate through unwanted interactions. By engineering non-ergodic systems – those with constrained state accessibility – researchers aim to create “protected” quantum states where information leakage is suppressed. This involves designing interactions and environments that limit the system’s ability to explore its full state space, effectively isolating the encoded quantum information from disruptive influences. The creation of such systems represents a significant step towards realizing practical quantum technologies, offering the potential for long-lived quantum bits and reliable quantum computation.

Deconstructing the Hilbert Space: A New Topology of Quantum Control

Hilbert space fragmentation describes a breakdown of the connectivity within the many-body Hilbert space, resulting in distinct, isolated sectors. In a non-fragmented system, any quantum state can, in principle, be reached from any other through time evolution governed by the system’s Hamiltonian. However, fragmentation creates barriers to transitions between these sectors; the Hamiltonian does not allow for coupling between states residing in different disconnected regions of the Hilbert space. This effectively limits the dynamics of the system to within each individual sector, even though the total Hilbert space dimension may be substantial. The fragmentation is characterized by a proliferation of locally conserved quantities that prevent global mixing of states, and is not necessarily related to energy conservation, but rather to constraints imposed by symmetries or strong interactions within the system.

Hilbert space fragmentation is not a singular phenomenon but results from multiple physical mechanisms. Strong interactions between constituent particles can localize wavefunctions, effectively disconnecting different regions of the Hilbert space and preventing particle transport. Additionally, the imposition of specific symmetries, such as those arising from spatial constraints or conserved quantities, can further restrict the accessible states and induce fragmentation. These symmetries create selection rules that inhibit transitions between otherwise allowed states, leading to a partitioning of the total Hilbert space into disconnected sectors, each representing a localized and isolated subspace. The degree of fragmentation is directly related to the strength of these interactions and the specific nature of the imposed symmetries, influencing the system’s ability to explore the full range of possible quantum states.

Topological Hilbert space fragmentation arises from the system’s global topological invariants, providing inherent stability against local perturbations. Unlike fragmentation caused by energetic disorder or many-body interactions, topological fragmentation is protected by these invariants, meaning the disconnected sectors of the Hilbert space remain isolated even with changes to local parameters. This protection stems from the fact that altering the fragmentation pattern would require a change in the global topology, which is energetically unfavorable or impossible without fundamentally altering the system’s structure. Specifically, these invariants, often described by $Z_n$ or other topological groups, dictate the allowed number of disconnected sectors and prevent transitions between them, ensuring robustness of the fragmented state.

The CZp Model: A Playground for Frozen Quantum States

The CZp model, a lattice system with specific interactions and symmetries, demonstrates Hilbert space fragmentation where the overall Hilbert space breaks down into disconnected sectors. This fragmentation arises from the combination of a $Z_2$ symmetry associated with particle number conservation and the interactions within the model. Crucially, local operators that act on one fragmented sector do not generally commute with operators in other sectors, preventing transitions between them. This topological fragmentation is not due to a physical gap in the spectrum, but rather a consequence of the model’s symmetry and interaction structure, leading to exponentially many locally stable, disconnected states.

The CZp model demonstrates a specific number of stable, localized states, termed ‘frozen states’, resulting from its fragmented Hilbert space. The total count of these frozen states is determined by the lattice size, $L$, and is calculated as $2^{L+2} – 8$. This formula indicates an exponential increase in the number of frozen states with increasing lattice size, offset by a constant value. These states are stable due to the topological constraints imposed by the model’s symmetry and interactions, preventing decay to lower energy configurations.

Frozen states within the CZp model are characterized by specific loop configurations on the square lattice. These loops, formed by the model’s interactions and symmetry constraints, demarcate boundaries between fragmented Hilbert space sectors. A given frozen state is uniquely identified by the arrangement of these loops; loops that wrap around the lattice, or intersect in specific patterns, define the topological properties of that state. The total number of frozen states, $2^{L+2} – 8$, arises from the possible arrangements and combinations of these loops, with each unique loop configuration corresponding to a distinct, stable frozen state within the fragmented system.

The transversal CNOT gate, as defined in Eq. (6), operates on a set of frozen states established in Figure 1 to perform a logical CNOT operation.

Encoding Resilience: Fractured Space and the Promise of Robust Quantum Memory

Quantum memory, a cornerstone of scalable quantum computation, faces the persistent challenge of decoherence – the loss of quantum information due to environmental interactions. Researchers are exploring innovative strategies to shield qubits, and a promising approach involves encoding a vast number of physical qubits within the highly stable, ‘frozen’ states of the complex $CZp$ model. This isn’t simply about redundancy; the specific structure of these frozen states offers intrinsic protection. By distributing quantum information across an exponentially large Hilbert space defined by these states, the system becomes remarkably resilient to local disturbances. Effectively, any error affecting a small number of physical qubits is diluted and doesn’t corrupt the encoded logical qubit, paving the way for robust and long-lived quantum storage.

A significant challenge in building practical quantum computers lies in preserving the delicate quantum states of qubits long enough to perform meaningful calculations. Encoding quantum information in a fractured space, as proposed by the CZp model, offers a pathway to address this through the application of transversal logical gates. These gates operate directly on the encoded qubits without probing the underlying fragile quantum state, effectively shielding the encoded information from decoherence. This is crucial because standard quantum gates can easily disrupt the encoded state, leading to errors; transversal gates, however, circumvent this issue by acting only on the collective properties of the qubits. The ability to perform computation via these gates unlocks the potential for fault-tolerant quantum computation, enabling complex algorithms to be run reliably and paving the way for scalable quantum technologies.

The promise of robust quantum computation within this fractured space is tempered by the limitations imposed by the Eastin-Knill theorem. This fundamental principle dictates that not all quantum gates can be implemented transversally – meaning without directly acting on the encoded qubits themselves, but instead on the physical qubits constituting the encoded information. Consequently, meticulous design of quantum operations is essential; any attempt to apply a non-transversal gate risks collapsing the encoded quantum state and losing the information it holds. Researchers must therefore carefully select and construct gates that adhere to these constraints, often necessitating the decomposition of complex operations into a series of permissible, transversal steps. This careful choreography ensures that quantum computations can proceed without compromising the integrity of the encoded qubits, safeguarding the fragile quantum information against errors and decoherence.

Beyond Fragmentation: Charting New Landscapes for Quantum Control

An alternative to the commonly studied CZp model for understanding quantum many-body fragmentation lies in the Quad-Flip framework. This model leverages the concept of “clock spins” – operators that cycle through states – and organizes the system’s Hilbert space into Krylov sectors of prime size, denoted as $m$. Unlike approaches focusing on conserved quantities, the Quad-Flip model emphasizes the dynamics between these sectors, revealing fragmentation patterns driven by non-conservation of certain operators. By analyzing how quantum information spreads within and between these prime-sized sectors, researchers gain a complementary perspective on the emergence of fragmented phases, potentially unlocking new strategies for stabilizing and manipulating quantum states in complex systems and offering insights beyond those provided by traditional symmetry-based analyses.

The practical realization of Hilbert space fragmentation – a state where a quantum system breaks into independently controllable subsystems – often faces challenges due to environmental noise and imperfections. Researchers are actively developing symmetry enrichment techniques to bolster the robustness of these fragmented states. These methods involve leveraging inherent or engineered symmetries within the quantum system, effectively creating a ‘protective shield’ around the fragmented Hilbert space. By exploiting these symmetries, the system becomes less susceptible to perturbations that would otherwise destroy the fragmentation. This is achieved by mapping the original, fragile fragmentation pattern onto a larger, symmetry-protected subspace, increasing the number of states that exhibit fragmentation and thereby making it more resilient. Consequently, these advancements pave the way for more stable and reliable quantum memories and computational architectures, where isolated quantum bits can be maintained and manipulated with greater fidelity, even in noisy environments.

The deliberate engineering of quantum fragmentation patterns and symmetry configurations presents a powerful pathway towards realizing advanced quantum technologies. Researchers are discovering that carefully controlling how a quantum system breaks down into disconnected pieces – its fragmentation – allows for the creation of highly robust quantum memories, where information is encoded across multiple, isolated sectors, protecting it from local disturbances. Moreover, manipulating the symmetries within these fragmented Hilbert spaces can unlock novel computational paradigms, potentially enabling algorithms tailored to exploit the system’s inherent structure. This approach goes beyond traditional qubit-based architectures, offering possibilities for distributed quantum computation and fault-tolerant quantum information processing, where the system’s very fragmentation becomes a resource for preserving quantum coherence and enhancing computational power. The exploration of diverse fragmentation landscapes, therefore, is not merely a theoretical exercise, but a crucial step towards building practical and scalable quantum devices.

The pursuit of robust quantum memory, as detailed in this work, mirrors a fundamental principle of resilient systems. This research demonstrates a method for creating exponentially many stable qubits through symmetry enrichment and Hilbert Space Fragmentation. It suggests that complexity doesn’t necessarily equate to fragility; rather, carefully constructed limitations-the ‘breaking of ergodicity’-can yield surprising stability. As Albert Einstein once observed, “It is the theory which decides what can be observed.” This study elegantly proves this; by theoretically constructing a system with specific symmetries and fragmentation, the researchers have opened a path to observing-and potentially harnessing-remarkably stable quantum states. If the system survives on duct tape, it’s probably overengineered; this approach suggests the opposite – elegant simplicity through constrained complexity.

Beyond Frozen States

The proliferation of symmetry-protected qubits from engineered ergodicity breaking offers a compelling, if somewhat paradoxical, path toward robust quantum memory. The elegance lies in leveraging fragmentation – deliberately restricting the Hilbert space – to create order. Yet, this approach begs the question of scalability. While the theoretical construction demonstrates exponential capacity, the practical realization necessitates precise control over many-body interactions, a challenge not easily dismissed. The cost of topological protection is, inevitably, a reduction in accessible Hilbert space; the true art will be in minimizing that cost while maximizing the number of effectively isolated qubits.

Future investigations must address the interplay between symmetry enrichment and disorder. The current work suggests stability arises from a specific balance; but what happens when that balance is perturbed? Are there inherent limits to the level of fragmentation tolerable before the encoded information becomes unrecoverable? Furthermore, exploring alternative symmetries-beyond those examined here-could unlock new avenues for qubit design and control. The apparent simplicity of the frozen states belies a deep complexity, hinting at a rich landscape of possible configurations.

Ultimately, the field faces a familiar tension: the pursuit of robustness often demands sacrificing flexibility. This work provides a fascinating example of how to navigate that trade-off, but it is merely a starting point. The next step is not simply to build more qubits, but to understand the fundamental limits of their stability-and to find clever ways to circumvent them.

Original article: https://arxiv.org/pdf/2512.20393.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/