Fractons in flatland

—by Dominic Williamson

EQUS researchers have shown that unconventional subsystem symmetries applied to two-dimensional topological phases of matter enforce mobility constraints on anyonic quasiparticle excitations that facilitate new types of symmetry fractionalisation and increase quantum memory times for topological quantum codes experiencing biased noise.

Topological phases of matter are analogous to the well-known classical phases—solid, liquid, gas and plasma—but are driven by quantum rather than thermal fluctuations and are characterised by quantum entanglement rather than classical correlations.  Two-dimensional topological phases are characterised by anyonic quasiparticles, so-named because they have exchange statistics that interpolate between the +1 of bosons and −1 of fermions.  The exchange of anyons can even generate quantum gates capable of universal topologically protected quantum computation.

In three dimensions, quasiparticles were previously expected to all be either bosons or fermions, because the path of one particle around another deforms smoothly to a trivial operation.  Recently, however, topological phases of matter have been discovered that contain quasiparticles with restricted, or even no, mobility.  This discovery has revealed a landscape of new possibilities collectively referred to as fracton topological phases.  These phases have attracted attention as candidates for good quantum hard drives, because the immobility of fracton quasiparticles means errors diffuse slowly, resulting in long quantum memory lifetimes.  However, the existence of fractons in two-dimensional topological phases has been argued to be impossible.

In collaboration with researchers in the US, I have been exploring scenarios in which the addition of unconventional symmetries constrain anyons to behave like fractons under symmetry-respecting operations.  Such behaviour is potentially important for improving the performance of quantum error-correcting codes under biased noise in situations relevant to cutting-edge experimental implementations (with tens to hundreds of qubits).

We showed that the dynamics of anyon excitations in a modified topological colour code under extremely biased noise—relevant for experimental platforms such as superconducting Kerr-cat qubits—becomes that of immobile fractons.  We used a classical cellular automaton to perform measurements and feedback operations that effectively shape the dynamics under random biased noise into the slow thermal dynamics of fractons.  This allowed us to harness the increased memory times of fracton topological orders in only two dimensions.  We observed an increase in memory time even when the relevant symmetry is not strictly enforced, meaning costly global decoding steps need to be performed less often, potentially loosening the resource requirements for quantum error correction with the colour code under highly biased noise.

We also discovered new types of symmetry fractionalisation on anyons under symmetries that act on line- or fractal-shaped subsystems.  Fractionalisation was one of the earliest discoveries associated with topological phases of matter.  It famously occurs in the fractional quantum Hall effect, whereby electrons split into quasiparticle excitations each with a fraction of the total electron charge.  The fractional quantum Hall effect loosens the conventional quantisation constraint (that charge is an integer) to the constraint that charge is rational, with the fraction depending on the topological phase.  In our work, a constraint that all subsystem symmetries over the whole system (such as on lines and columns of a square grid) multiply to 1 is loosened once an anyon is introduced to being ±1 (that is, the product of the symmetries only has to square to 1).  This result required us to get around a no-go argument, based on the two-dimensional mobility of anyons, that rules out nontrivial actions of subsystem symmetry on anyons in two dimensions.  To do so, we demonstrated that anyons become symmetry-enforced fractons, with restricted mobility under symmetry-preserving operations.

Our results suggest an interplay between unconventional symmetry and topology in two-dimensional topological codes, with potential application in designing better-performing quantum codes under physically motivated noise models.

Dominic Williamson is a Research Fellow at the University of Sydney.  This work forms part of EQUS’ 1kQubit Flagship research program.