🔬 physics

Realization of decoherence-induced average symmetry-protected topological phases on quantum processors

This paper demonstrates the realization and characterization of decoherence-induced average symmetry-protected topological (ASPT) phases on programmable quantum processors by engineering sublattice-selective dephasing to preserve ensemble-level symmetry while diagnosing the resulting mixed-state topological order through replica string correlators and entanglement spectra.

Original authors: Ruizhe Shen, Xuejia Yu, Ching Hua Lee

Published 2026-10-22

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Ruizhe Shen, Xuejia Yu, Ching Hua Lee

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the quantum world, matter can organize itself in ways that defy our everyday intuition. One such arrangement is known as a symmetry-protected topological phase. Imagine a material that looks ordinary on the inside but holds a special, unbreakable pattern on its edges, a pattern that only exists because the system obeys specific rules of balance, or symmetry. For decades, scientists studied these phases in perfect, isolated systems where the rules were absolute and the state of the matter was pure. However, the real world is rarely perfect. Quantum systems are often open to their environment, leading to a process called decoherence, where the delicate quantum information gets scrambled by noise. This usually destroys the special patterns scientists are trying to study. A newer idea suggests that even in this messy, noisy environment, a form of order can survive if we look at the system not as a single snapshot, but as a collection of many possible outcomes averaged together. This concept, known as an average symmetry-protected topological phase, proposes that while individual moments of noise might break the rules, the collective behavior of the system can still hold onto its topological identity.

A team of researchers led by Ruizhe Shen, Xuejia Yu, and Ching Hua Lee has now taken this theoretical idea and brought it to life on actual quantum computers. Working with programmable processors, they set out to create a specific type of quantum state that starts as a perfect, ordered pattern and then deliberately introduces a controlled amount of noise to see what happens. They began with a one-dimensional chain of quantum bits, arranged in a configuration known as a cluster state, which is famous for its robust topological properties. In this perfect state, the system is protected by two distinct types of symmetry, one acting on the odd-numbered positions in the chain and another on the even-numbered positions. The researchers then engineered a process to selectively disrupt only the even-numbered positions. They did this by attaching extra helper bits to the even spots and measuring them in a way that randomly flips the state of those spots, effectively simulating a specific kind of environmental noise.

The result was a hybrid state where the rules of the game had changed. The symmetry protecting the odd-numbered positions remained strong and unbroken, but the symmetry for the even-numbered positions was transformed. Instead of being a strict rule that applied to every single moment, it became an average rule that only held true when looking at the entire collection of possible outcomes. The researchers confirmed this transformation by measuring a quantity called the symmetry charge, which acts like a score for how well the system follows the rules. As they increased the amount of noise, this score for the even positions dropped, signaling that the strict, moment-to-moment protection was gone. However, the system still behaved as if the symmetry existed when viewed as a whole.

The most challenging part of the experiment was proving that the special topological order survived this transition. In a normal, noisy system, the usual way to detect this order—a specific pattern of connections across the chain—would vanish because the random flips would cancel each other out. To get around this, the team used a clever trick involving two identical copies of the noisy system. By measuring how these two copies interacted in a specific way, they could access a type of information that is hidden from single measurements. They found that while the standard signal for the topological order disappeared, this new, two-copy measurement remained strong and clear. This proved that the topological structure was not lost; it had simply moved into a different layer of the system's description, one that only becomes visible when looking at the density of the entire ensemble of possibilities.

To further verify their findings, the researchers reconstructed the internal energy landscape of the system's edge. In a perfect topological state, the edge behaves in a very specific way, with energy levels that come in identical pairs. Even after the noise was introduced, the researchers observed that these pairs remained intact. The noise did not break the pairing; instead, it simply shifted the weight of the system, moving some of the probability into different internal sectors while leaving the fundamental edge structure untouched. This experiment demonstrates that decoherence, often seen as the enemy of quantum states, can actually be used as a tool to engineer new types of matter. By carefully controlling how noise is applied, scientists can create and study complex phases of matter that exist only in the messy, mixed state of the real world, opening a new path for understanding quantum materials beyond the idealized conditions of the past.

Technical Summary: Realization of Decoherence-Induced Average Symmetry-Protected Topological Phases on Quantum Processors

Problem Statement
Symmetry-protected topological (SPT) phases are traditionally defined for pure states protected by exact symmetries. However, open quantum systems naturally evolve into mixed states due to decoherence. In these ensembles, symmetries can manifest differently: a "strong" symmetry constrains the density matrix under separate left and right actions, whereas a "weak" symmetry only requires invariance under conjugation. A specific class of weak symmetries, known as "average symmetries," arises when individual quantum trajectories break the symmetry, but the ensemble average restores it. While theoretical frameworks for Average SPT (ASPT) phases exist, establishing their order at larger scales and directly resolving their nonlocal string order and projective edge structures at the density-matrix level on programmable hardware remains a significant challenge.

Methodology
The authors implement a decoherence-induced ASPT phase on IBM Quantum processors using a one-dimensional cluster-state SPT as the starting point. The methodology involves three primary stages:

State Preparation and Engineered Decoherence:
- A 1D cluster state $|\psi_{cl}\rangle$ is prepared using Hadamard gates and nearest-neighbor CZ gates.
- To convert the even-sublattice symmetry from strong to average, the authors engineer local Pauli-Z dephasing exclusively on even sites. This is achieved via ancilla-assisted quantum circuits: an ancilla qubit is prepared in a superposition state $R_y(2\theta)|0\rangle$ , followed by a system-ancilla CZ gate.
- Measuring the ancilla outcomes implements a probabilistic Pauli-Z error ( $Z_j$ ) with probability $p = \sin^2\theta$ or the identity with probability $1-p$ . Discarding the ancilla record realizes the mixed state $\rho_{dec} = \sum_\eta P(\eta) Z^{(\eta)} \rho_{cl} Z^{(\eta)}$ , where $\eta$ represents the dephasing trajectory.
Symmetry Characterization:
- The loss of strong symmetry on the even sublattice is diagnosed by measuring the symmetry charge $Q_{even} = \text{Tr}(\rho_{dec} U_{even})$ , which is expected to decay as $(1-2p)^{L/2}$ .
- The fragility of conventional single-copy string order is tested by measuring the string correlator $\langle O_{i,k} \rangle$ under both engineered dephasing and additional random rotations to distinguish controlled suppression from hardware noise.
Probing Mixed-State Topology:
- To recover topological information lost in linear ensemble averages, the authors measure a normalized Rényi-2 string correlator: $C^{(2)}_{i,k} = \text{Tr}(\rho_{dec} O_{i,k} \rho_{dec} O_{i,k}) / \text{Tr}(\rho_{dec}^2)$ .
- This requires preparing two independent replicas of the decohered state ( $\rho_A$ and $\rho_B$ ) and performing pairwise destructive SWAP readouts. The numerator involves inserting the string operator $O_{i,k}$ on one replica before the SWAP measurement.
- Additionally, the half-chain reduced density matrix is reconstructed via quantum state tomography to analyze the entanglement spectrum.

Key Contributions and Results

Realization of ASPT Order: The experiment successfully generates a mixed state with a hybrid symmetry structure: the odd-sublattice symmetry remains strong, while the even-sublattice symmetry survives only as an average symmetry.
Decay of Strong Symmetry Signatures: Measurements confirm the predicted decay of the even-sublattice symmetry charge and the suppression of the conventional single-copy string correlator as the dephasing probability $p$ and system size $L$ increase. This suppression is attributed to the cancellation of trajectory-dependent signs.
Recovery of Topological Order via Nonlinear Probes: Crucially, the two-replica Rényi-2 string correlator remains nontrivial (close to the ideal value of 1) even when the single-copy signal vanishes. This demonstrates that the topological order is encoded in the density-matrix structure and is robust against the specific type of engineered decoherence used.
Entanglement Spectrum Analysis: Reconstructed half-chain spectra for system sizes up to $L=8$ show that while dephasing redistributes spectral weight among different reduced-stabilizer sectors, the characteristic twofold degeneracy associated with the projective edge degrees of freedom is preserved.

Significance
The paper establishes a gate-based route to engineering and probing ASPT order on programmable quantum hardware. By demonstrating that structured decoherence can serve as a programmable resource for realizing mixed-state topological quantum matter, the work bridges the gap between theoretical ASPT frameworks and experimental realization. The authors claim that their results provide complementary signatures—symmetry charge decay, nonlinear string order, and spectral pairing—that collectively confirm the existence of decoherence-induced ASPT phases. This framework opens avenues for exploring mixed-state topological transitions in higher dimensions and interacting SPT states beyond the exactly solvable cluster limit.

Technical Summary: Realization of Decoherence-Induced Average Symmetry-Protected Topological Phases on Quantum Processors

More like this