Prethermal Strong Zero Modes and Topological Qubits

We prove that quantum information encoded in some topological excitations, including certain Majorana zero modes, is protected in closed systems for a time scale exponentially long in system parameters. This protection holds even at infinite temperature. At lower temperatures, the decay time becomes even longer, with a temperature dependence controlled by an effective gap that is parametrically larger than the actual energy gap of the system. This nonequilibrium dynamical phenomenon is a form of prethermalization and occurs because of obstructions to the equilibration of edge or defect degrees of freedom with the bulk. We analyze the ramifications for ordered and topological phases in one, two, and three dimensions, with examples including Majorana and parafermionic zero modes in interacting spin chains. Our results are based on a nonperturbative analysis valid in any dimension, and they are illustrated by numerical simulations in one dimension. We discuss the implications for experiments on quantum-dot chains tuned into a regime supporting end Majorana zero modes and on trapped ion chains.

Popular Summary

Quantum computation, which relies on some of the nonintuitive behaviors of subatomic particles, is widely seen as a potentially powerful platform for solving computational problems that are too complex for traditional computers. Researchers are making a considerable effort, both theoretically and experimentally, to develop new methods of preserving quantum coherence—that is, enabling quantum effects to be maintained for times long enough to exploit them for computations. One promising approach is to exploit topological invariants, which guarantee the properties of the system remain robust against many types of errors. However, such topological quantum information is not completely protected. One notable issue is quasiparticle poisoning, where thermal effects create quasiparticles that destroy quantum coherence. We develop a method, which applies to systems in any number of dimensions, for keeping such thermal effects at bay for exponentially long times, thus preserving the quantum coherence.

Using a combination of mathematical analysis and numerical simulations, we show precisely how to provide additional protection for the Majorana zero modes being intensively investigated as a platform for topological quantum computation. We explain how our mathematical and numerical results can be seen by experiments on superconducting-semiconductor heterostructures and on trapped ion chains.

Our results show that, because of nonequilibrium effects, topological quantum information may be more strongly protected than previously realized.

Figure 1

(a) A 1D chain of a topological superconductor, with Majorana zero modes at the edges. (b) At finite temperature, mobile quasiparticles can annihilate on the Majorana zero modes, decohering the quantum information stored. (c) With strong disorder, the chain can be made to be MBL, such that the localized quasiparticles are not able to annihilate on the boundaries. (d) In a suitable “prethermal” regime, the quasiparticles are mobile but are prevented from annihilating on the boundary by an approximate conservation law.

Figure 2

The decay time (on a logarithmic scale) of a MZM at $T=\infty$ for $L=8–14$ sites, as a function of $h$ and $h_2$, as obtained by exact diagonalization. The data collapse onto the form $\ln \tau J=J/h_{2}f(h/h_2)+\text{constant}$, as expected, with deviations attributable to finite-size effects when the lifetime is long. The decay time of the spin at the center of the $L=14$ chain (dashed line) is not prethermalization protected and is much shorter.

Figure 3

The decay of a MZM at low energies, as obtained by TEBD for systems of length $L=40$, which shows that a long-lived MZM persists even in a much larger system. Red: Decay of the strong zero mode Green’s function $⟨\mathrm{\Psi }|{\gamma }_1^A(t){\gamma }_1^A(0)|\mathrm{\Psi }⟩$ for an initial state $|\mathrm{\Psi }⟩$ containing a single quasiparticle close to the left edge (i.e., $-{\gamma }_3^B{\gamma }_4^A=-1$ and ${\gamma }_i^B{\gamma }_{i+1}^A=1$ for $i\ne 3$), for $L=40$ and $h/J=h_2/J=0.075$. This is equivalent in the corresponding transverse-field Ising chain to the spin correlation $⟨\mathrm{\Psi }|{\sigma }_1^z(t){\sigma }_1^z(0)|\mathrm{\Psi }⟩$. Blue: Decay of the bulk spin correlation $⟨\mathrm{\Psi }|{\sigma }_{L/2}^z(t){\sigma }_{L/2}^z(t)|\mathrm{\Psi }⟩$ in the transverse-field Ising chain.

Figure 4

(a) Prethermalization does not protect against qubit errors at the boundaries between “rough” and “smooth” edges of the toric code on a surface with boundary. Here, an $e$ particle (blue dot) is absorbed by the rough boundary (top edge), while an $m$ particle (red cross) is emitted by the neighboring smooth boundary (right edge), as described in the text. (b) When a qubit is encoded in a loop that carries “Cheshire charge” in 3D, errors associated with a magnetic flux loop encircling (sequence of dotted red lines, whose growth and subsequent shrinkage are indicated by the green arrows) the Cheshire loop are suppressed by the size of the Cheshire loop, since the magnetic flux loops have a line tension at temperatures below the bulk phase transition temperature. There is no such protection against the emission or absorption of a pointlike electric charge (blue circles), even below the phase transition temperature into the topological phase. However, prethermalization can suppress such processes exponentially.

Figure 5

The decay time (on a logarithmic scale) of a MZM in the transverse-field Ising model with long-ranged interactions at $T=\infty$ for $L=8–14$ sites and $h=0.2$, as obtained by exact diagonalization. It is plotted as a function of $\alpha$, the power of the decay of the long-ranged interaction. As in Fig. 2, the decay time saturates in $L$ for larger values of the perturbing couplings. The decay times for ferromagnetic and antiferromagnetic interactions are the same at $T=\infty$; it is only at lower temperatures where the effect of the long-ranged order for $\alpha <2$ in the ferromagnetic case becomes important.