The Nature and Correction of Diabatic Errors in Anyon Braiding

Topological phases of matter are a potential platform for the storage and processing of quantum information with intrinsic error rates that decrease exponentially with inverse temperature and with the length scales of the system, such as the distance between quasiparticles. However, it is less well understood how error rates depend on the speed with which non-Abelian quasiparticles are braided. In general, diabatic corrections to the holonomy or Berry’s matrix vanish at least inversely with the length of time for the braid, with faster decay occurring as the time dependence is made smoother. We show that such corrections will not affect quantum information encoded in topological degrees of freedom, unless they involve the creation of topologically nontrivial quasiparticles. Moreover, we show how measurements that detect unintentionally created quasiparticles can be used to control this source of error.

Popular Summary

Quantum computation is focused on processing information quantum mechanically to solve problems that are difficult or impossible to investigate using a classical computer. Of the many approaches to quantum computing, topological quantum computation is particularly appealing because it stores and manipulates information nonlocally, thereby drastically reducing susceptibility to the environmentally induced errors that plague conventional quantum-computing systems. However, researchers are still investigating whether topological quantum computation is feasible in practice. Here, we address this question by studying how a topological system is affected by the precise details of its time evolution.

Quantum information can be encoded in the nonlocal state space of non-Abelian quasiparticles, which are particlelike objects with exotic exchange statistics such as Majorana zero modes occurring in topological superconducting nanowires. Computational gates can be implemented in such systems by exchanging the positions of these quasiparticles, forming “braids” of their worldline trajectories in $2+1D$ spacetime. When the braid exchange is conducted infinitely slowly and smoothly, errors in the system are exponentially small in the ratios of the system size to the correlation length and the energy gap to the temperature. We analyze the diabatic corrections to quasiparticle braiding (i.e., the errors resulting from a finite operation time). Our results demonstrate that diabatic errors are generally not exponentially suppressed and are accordingly a serious issue for topological quantum computation. We identify how these errors arise in quasiparticle exchange and propose a method of error correction. We additionally describe an experimental implementation of our error-correction proposal for systems of Majorana zero modes that could be realized with current technology.

We anticipate that our findings will be of interest to both theorists and experimentalists in fields spanning condensed matter physics, atomic physics, and quantum information.

Figure 1

In the ${\tau }_z=1$ sector, the instantaneous eigenstates of $H(t)$ trace out an octant of the Bloch sphere, shown above as the contour $C$. At times $t=0$, $t_1$, $t_2$, $t_{\mathrm{op}}$, only one of the ${\mathrm{\Delta }}_i$ is nonzero. At these times, ${\sigma }_i$ commutes with the Hamiltonian, and the corresponding point on the contour is one of the corners or “turning points”. The holonomic phase at the end of the evolution is half the solid angle traced out by the contour $C$, $[\mathrm{\Omega }(C)/2]=-(\pi /4)$.

Figure 2

With dissipation (red solid line), transition probability $P_{\mathrm{G}\to \mathrm{E}}$ vs the gap multiplied by the total evolution time $\mathrm{\Delta }{t}_{\mathrm{op}}$, due to diabatic effects for $k=0$ (top panel) and $k=1$ (bottom panel). The long time tail is fitted to $c_0/(\mathrm{\Delta }{t}_{\mathrm{op}}{)}^x$ with $x\approx 2$ (brown dashed line). We choose the cutoff ${\omega }_c=10\mathrm{\Delta }$, Ohmic bath at low temperature $T=1/\beta =0.001\mathrm{\Delta }$, and system-bath coupling ${\lambda }_1={\lambda }_2={\lambda }_3=0.01\mathrm{\Delta }$. The black solid line shows the results without dissipation, and the envelope function for long times is fitted to $c_0^{\prime }/(\mathrm{\Delta }{t}_{\mathrm{op}}{)}^x$ with $x\approx 2k+2$ (blue dashed line).

Figure 3

The deviation of the density matrix after projection onto the ground state, $\parallel {\rho }_{\mathrm{G}}(t_{\mathrm{op}})-{\rho }_A\parallel$, vs the gap multiplied by the total evolution time, $\mathrm{\Delta }{t}_{\mathrm{op}}$. The parameters are the same as in Fig. 2.

Figure 4

Schematic of the braiding process at a $T$ junction. Each dot represents a MZM, and the lines connecting dots indicate which MZMs are in a definite fusion channel at a given time. This sequence of states can be obtained as the ground states of a Hamiltonian with nonzero coupling between the pair connected by a line at each step and by adiabatically tuning the Hamiltonian from one step to the next. This sequence effectively braids the MZMs labeled by red and blue dots.

Figure 5

The effect of a diabatic error, which transfers a neutral fermion from the qubit to the ancillas, on the braiding. For Ising anyons, it turns a counterclockwise braiding into a clockwise one, up to an overall phase.

Figure 6

The measurement-only protocol for braiding. The resulting operation depends on the fusion channels $s_1$ and $s_2$ of the intermediate measurements (which can take the values $i$ or $\psi$). If $s_1=s_2$, the result is a counterclockwise braid; otherwise it is the inverse braid.

Figure 7

The above flowchart outlines the two methods of using forced measurement or its generalizations for recovery protocols, as discussed in the text. The one-directional arrows (black) indicate a process that yields a desired or acceptable outcome for which we do not apply a recovery protocol. The bidirectional arrows (red) indicate a process that yields an undesired or unacceptable outcome for which we apply a recovery protocol. We can schematically think of the recovery protocol as backtracking and trying the process again, with a new probability of obtaining a desired outcome. The simple method applies a recovery protocol whenever a turning point measurement outcome indicates that a diabatic transition occurred to an excited state. The procrastination method will accept either measurement outcome at the first turning point. However, when the first measurement outcome is $s_1=-1$, if we procrastinate, we must require the second turning point to have measurement outcome $s_2=-1$ because we need two wrongs to make a right.

Figure 8

In the $T$ junction shown here, anyons $a_1$, $a_2$, and $a_3$ have topological charge $a$, and anyon $a_0$ has topological charge $\overline{a}$. Similar to the protocol for braiding MZMs, at times $t=0$ and $t=t_{\mathrm{op}}$, anyons $a_0$ and $a_2$ (both in black) form the ancillary pair of anyons. We exchange the positions of the red and blue anyons using a protocol identical to that shown in Fig. 4, with the labels ${\gamma }_i$ replaced by $a_i$.

Figure 9

The left side shows the braiding diagram corresponding to the diabatic error correction protocol addressing the creation of a bulk quasiparticle. The anyon $a$ can emit an anyon $c$, which we need to detect, trap, and then fuse back together with $a$. The right side shows that this process is equivalent to the intended braid, up to an unimportant overall phase. Note that the $b$ line is not actually necessary for this statement.

Figure 10

Top panel: The simplest flux-tunable MZM braiding setup, following Refs. [42, 43]. The black lines are nanowires hosting MZMs (red dots) at their end points. We tune the Josephson junctions’ flux values $|{\mathrm{\Phi }}_i|$ between 0 and ${\mathrm{\Phi }}_{\max }<\frac{1}{2}{\mathrm{\Phi }}_0$ to change the strength of the Coulomb coupling between the MZM pairs ${\gamma }_i$ and ${\gamma }_i^{\prime }$. Bottom panel: Flux values at the four turning points. When $|{\mathrm{\Phi }}_i|={\mathrm{\Phi }}_{\max }$, ${\mathrm{\Delta }}_i$ is maximized and ${\gamma }_i$ is coupled to the center MZM ${\gamma }_0$, formed out of a linear combination of ${\gamma }_1^{\prime }$, ${\gamma }_2^{\prime }$, and ${\gamma }_3^{\prime }$.

Figure 11

The $\pi$ junction proposed by Ref. [43]. The MZM system sits inside a superconducting qubit formed by the bus and ground islands. The topological qubit is embedded into a transmission-line resonator to allow read-out of the qubit state.

Figure 12

Top panel: Modified $T$-junction structure designed to allow for parity measurements at each turning point. Bottom table: Flux values for measurement at the turning points. Note that a junction with $\mathrm{\Phi }=0$ maximizes the Josephson energy and phase locks its neighboring superconductors, while a junction with $|\mathrm{\Phi }|={\mathrm{\Phi }}_{\max }$ minimizes its Josephson energy and essentially decouples the neighboring superconductors. At time $t_1$, islands 2 and 3 are phase locked to the ground and decoupled from the bus, while island 1 is phase locked to the bus and decoupled from the ground.

Figure 13

A $\pi$ junction designed to allow fermion parity measurements at each turning point.

Figure 14

Transition probability $P_{G\to E}$ vs braiding period $t_{\mathrm{op}}$ without dissipation. Here, $k=0$, 1, 2, 3 refers to the number of vanishing time derivatives of the Hamiltonian at each turning point.

Figure 15

Phase error $\parallel {\rho }_{\mathrm{G}}(t_{\mathrm{op}})-{\rho }_A\parallel$ against braiding period $t_{\mathrm{op}}$ without dissipation. Here, $k=0$, 1, 2, 3, 4 refers to the number of vanishing time derivatives of the Hamiltonian at each turning point.

Figure 16

Low-energy MZM picture.

Figure 17

$T$-junction architecture with parity-check ability. Every MZM island can be connected to the bus (ground) with the other two MZM islands connected to the ground (bus). No MZM islands share a split Josephson junction; thus, the Coulomb couplings for different MZM pairs are independent of each other.

Figure 18

Schematic illustration of the transmon qubit [53]. The qubit (lower-middle) is embedded inside a coplanar waveguide (two ground planes with a feed line running through the center). The short section of feed line traps a standing wave (shown in red) and can thus be thought of as a resonator. The qubit (in this case, two superconductors connected by a split Josephson junction) is capacitively coupled to the resonator and slightly shifts the resonator’s frequency. A wave traveling down the feed line will experience an impedance mismatch due to the resonator or qubit. This will result in a transmission amplitude $S_{21}$ whose value depends on the state of the qubit.