Long-range entangled criticality from coherent errors

In the quest to build a quantum computer, scientists face a fundamental problem: the delicate quantum states that hold information are easily corrupted by the environment. To protect this information, researchers use error-correcting codes, which spread a single piece of data across many physical particles so that if a few are damaged, the whole can still be recovered. This concept is deeply tied to the study of matter itself. Just as water can exist as ice or steam, quantum systems can exist in different "phases." Some of these phases are robust and can store information indefinitely, while others are fragile and lose it quickly. A key way to tell these phases apart is by looking at how information is shared across the system. In a healthy, protected phase, information is stored in a way that is deeply connected across long distances, a property known as long-range entanglement. When noise becomes too strong, this connection breaks, and the system falls into a short-range entangled state where the information is lost.

For years, scientists understood how random, unpredictable errors affect these systems. They knew there was a specific threshold of noise where the system would suddenly switch from a protected phase to a broken one. However, a different kind of error, known as a coherent error, has remained a mystery. Unlike random errors that flip bits like a coin toss, coherent errors are systematic, like a steady, unwanted rotation of every particle in the system. Because these errors are uniform and predictable, they behave differently than random noise, and it was unclear whether the system would simply break at a threshold or enter a strange, new state of matter. A team of researchers has now mapped out exactly what happens when a quantum computer suffers from these coherent errors, revealing a surprising new phase of matter that exists right at the edge of failure.

The researchers studied a specific design called the honeycomb surface code, which arranges qubits in a pattern resembling a honeycomb. They subjected this system to a uniform rotation of its particles, followed by a measurement process that checks for errors. By simulating this process on large systems, they discovered that the system does not simply fail when the rotation angle exceeds a certain limit. Instead, it undergoes a transition into a critical regime. In this new state, the system retains some ability to store quantum information, but it behaves very differently from the clean, protected state below the threshold. The researchers found that while the system still possesses long-range entanglement, the way information is correlated across the system changes dramatically. Below the threshold, correlations between distant parts of the system die out quickly, like a whisper fading in a large room. But above the threshold, these correlations do not fade away; instead, they persist and even grow slowly as the system gets larger, indicating a critical state where the system is balanced on a knife-edge between order and chaos.

To understand this new phase, the team developed two general principles that apply to many types of quantum systems. First, they proved that if a system preserves a specific type of symmetry while still holding onto some quantum information, it must be in a long-range entangled state. This means that even if the system is noisy, as long as it hasn't completely lost its quantum memory, the information is still deeply woven into the fabric of the system. Second, they created a method to determine if two different noisy states belong to the same phase of matter. They showed that if the correlations between different parts of the system remain short-ranged, one can transform the noisy state back into a clean one using only local operations. However, if these correlations become long-ranged, such a transformation is impossible, signaling that the system has entered a fundamentally different phase.

Applying these principles to their honeycomb model, the researchers found that the system crosses a sharp threshold at a rotation angle of approximately 0.09 times pi. Below this angle, the system is in a standard topological phase where errors can be fixed locally, and the system can be easily restored to its clean state. Above this angle, the system enters the critical regime they identified. In this regime, the system still holds onto a fraction of its quantum memory, but this memory is fragile in a new way. The researchers showed that this surviving memory cannot be recovered by local operations alone; it requires a global view of the entire system to decode. Furthermore, the system loses the ability to protect itself against local damage. If a small region of the system is erased, the information stored in the rest of the system cannot be recovered, unlike in the clean phase where the information can be routed around the damage.

The study also revealed that this critical regime is a unique form of symmetry breaking. In the clean phase, the system has a pair of symmetries that protect the information. In the critical regime, one of these symmetries remains exact, but its partner becomes "weak," meaning it no longer protects the information in a localized way. This results in a state where the system is long-range entangled but critical, with correlations that decay very slowly, following a power law rather than an exponential drop. This behavior is distinct from the usual breakdown of a quantum memory, where information is lost entirely. Instead, the system enters a state where information is preserved but is fundamentally harder to access and more sensitive to the global structure of the system.

The researchers also connected these findings to the physics of transport in a network of particles. They showed that the way correlations spread in the noisy quantum code is mathematically equivalent to how electricity flows through a network of wires. Below the threshold, the network acts like an insulator, blocking the flow of information. Above the threshold, the network becomes metallic, allowing information to flow freely across long distances. This connection explains why the system behaves so differently in the critical regime: the underlying network has become conductive, allowing correlations to persist over long distances. This insight provides a powerful new tool for understanding how quantum information behaves in the presence of systematic errors.

The discovery of this extended critical regime suggests that the landscape of quantum phases is richer than previously thought. It shows that a quantum system can survive beyond the point where it can no longer be perfectly corrected, entering a state where it retains some quantum properties but in a form that is fundamentally different from the clean state. This has important implications for the design of future quantum computers, as it suggests that even if a system is subjected to systematic errors, it may not simply fail but could instead settle into a new, stable state that still holds valuable information. The work provides a clear framework for identifying and characterizing these states, offering a new way to think about the resilience of quantum information in the face of noise. By mapping out the boundaries between these phases, the researchers have provided a roadmap for navigating the complex terrain of quantum error correction, showing that the path to a working quantum computer may involve not just avoiding errors, but understanding the new states of matter that emerge when they are present.