Quantum computing with anyons is fault tolerant

TL;DR

The paper demonstrates fault-tolerant, universal quantum computation using non-Abelian anyons in the $D(S_3)$ topological phase by developing an active error-correction scheme that operates during braiding and fusion measurements. Central to the approach are gauging and ungauging procedures that map to the stabilizer-based $D(\mathbb{Z}_3)$ code, just-in-time decoding to correct errors in real time, and domain-wall detectors that span between phases. The authors prove a threshold theorem under a local circuit-noise model, showing a nonzero threshold attainable on large devices and provide explicit bounds via a chunk-decomposition and linked-tree analysis. This work connects topological protection with active error correction to enable scalable, fault-tolerant topological quantum computation on contemporary hardware.

Abstract

In seminal work (arxiv:quant-ph/9707021) Alexei Kitaev proposed topological quantum computing (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:quant-ph/0001108, arXiv:0707.1889), whereby logic gates of a quantum computer are conducted by creating, braiding and fusing anyonic particles on a two-dimensional plane. Furthermore, he showed the proposal is inherently robust to local perturbations (arXiv:cond-mat/0010440, arxiv:quant-ph/9707021, arXiv:1001.0344, arXiv:1001.4363) when anyons are created as quasiparticle excitations of a topologically ordered lattice model prepared at zero temperature. Over the decades following this proposal there have been considerable technological developments towards the construction of a fault-tolerant quantum computer. Rather than maintaining some target ground state at zero temperature, a modern approach is to actively correct the errors a target state experiences, where we use noisy quantum circuit elements to identify and subsequently correct for deviations from the ideal state. We present an error-correction scheme that enables us to carry out robust universal quantum computation by braiding anyons. We show that our scheme can be carried out on a suitably large device with an arbitrarily small failure rate assuming circuit elements are below some threshold level of local noise. The error-corrected scheme we have developed therefore enables us to carry out fault-tolerant topological quantum computation using modern quantum hardware that is now under development.

Figures (10)

• Figure 1: Quantum computing with anyons displayed in spacetime where time runs upwards. The $D(S_3)$ phase is prepared by gauging an initial $D(\mathbb{Z}_3)$ state. Computational anyons (blue circles) are then prepared and braided while maintaining a very large separation. Errors, shown in red, acting on the topologically ordered lattice can be understood as string-like objects with erroneous anyons at their end points. Physical errors generate strings that lie perpendicular to the temporal direction, while measurement error strings run parallel to the time axis. When we are below the threshold error rate, errors form well-resolved clusters that, with high probability, are much smaller than the separation between computational anyons. We must correct errors in real time such that erroneous anyons cannot hide any important syndrome information. The just-in-time decoder takes syndrome information measured in the past and present timesteps to decide what correction to perform at each stage. When the decoder is confident a group of detection events should be mutually annihilated, the just-in-time-decoder commits to a correction. To make a correction, we ungauge the region supporting the anyons to reveal their collective fusion outcome and subsequently anihilate the erroneous anyons in the ungauged lattice before recovering the $D(S_3)$ phase. If the decoder is uncertain on how to correct errors, perhaps because they were detected recently, the correction can be deferred, as shown in red. Here, we correct the event with a measurement at the present time, moving the detection into the future. This allows us to delay correction. Ungauging can also be used to perform logical readout as shown at the top of the figure.
• Figure 2: a) Part of a computation by braiding. Computational anyons are separated from each other by at least $d$ in all directions; erroneous anyons must propagate between computational anyons via a sequence of $O(d)$ local errors to introduce a logical error. b) A low-weight local error may cause a logical error over time. A small error first creates a pair of anyons, shown in black, nearby a computational anyon. A second local error (red) propagates an anyon between the computational anyon and an erroneous anyon, such that the erroneous anyon hides the syndrome of the second error. If the first pair of erroneous anyons are not annihilated, a subsequent third small local error can propagate an anyon from the erroneous anyon pair and a second computational anyon (moved during the course of braiding), introducing a logical error. We conclude that we must promptly annihilate erroneous anyons. c) If the erroneous pair is annihilated quickly, the charge exchanged with the computational anyon will be revealed, therefore enabling us to suppress the logical error.
• Figure 3: a) Gauging and ungauging provide a dictionary between two codes, where each anyon has a counterpart. This allows us to construct meaningful detectors spanning the boundary. b) An error, contained in a local spacetime cluster (red) creates detection events (stars). The erroneous anyons are deferred for correction until a timestep when the decoder is confident to commit to a correction. To correct the error, we must use the properties of the anyon model. Local $D(S_3)$ stabilizer measurements alone cannot determine the fusion result of a configuration of erroneous anyons, as this is nonlocal information. Instead, the region is then ungauged, converting the nonlocal information into local data. After neutrality is evaluated, the $D(\mathbb{Z}_3)$ counterparts of the faults are fused. Finally, we gauge the lattice again to recover the $D(S_3)$ phase. c) These detectors are formed by taking pairs of stabilizers and their $D(\mathbb{Z}_3)$ counterparts in the ungauged phase.
• Figure 4: $D(S_3)$ lives on the lattice, with a local Hilbert space consisting of a qubit (unfilled circles) and a qutrit (shaded circles). A shaded circle with an unfilled circle above it indicates a qubit-control qutrit-target gate. a) Qubit vertex stabilizer $\alpha_u$ and b) qubit plaquette stabilizer $\beta_p$. c) Qutrit plaquette stabilizer $B_q$ and d) qutrit vertex stabilizer $A_v$.
• Figure 5: a) A logical qubit can be encoded in the fusion outcomes of four $\mu$ anyons: the two states are distinguished by the whether pairs fuse to identity or to $e$. The distance of the logical qubit is determined by the separation between the consistuent $\mu$ anyons, since logical operators must span at least two anyons forming the qubit. b) Logical operators can always be cleaned away from an ungauging region (shaded) containing only one anyon in the logical qubit, meaning the logical information is not disturbed by suitably localized ungauging operations. c) Ungauging is performed by measuring $\sigma^Z$ on every edge in a given region (highlighted in purple). The region without any $\mu$ anyons (bottom) will yield $\sigma^Z = -1$ outcomes that form closed loops, reflecting the fact that all $\beta_p$ stabilizers were satisfied prior to the measurement. A region with a $\mu$ anyon will have open strings of $-1$ outcomes, terminating at the locations of the $\mu$ anyons.

Theorems & Definitions (21)

• Definition 1: Chunk Decomposition
• Lemma 1: Nugget Separation Lemma
• Lemma 2
• proof
• Lemma 3: Linking
• proof
• Theorem 4: Linking properties
• proof
• Definition 2: Linked Tree
• Lemma 5