Single-Shot Fault-Tolerant Quantum Error Correction

Conventional quantum error correcting codes require multiple rounds of measurements to detect errors with enough confidence in fault-tolerant scenarios. Here, I show that for suitable topological codes, a single round of local measurements is enough. This feature is generic and is related to self-correction and confinement phenomena in the corresponding quantum Hamiltonian model. Three-dimensional gauge color codes exhibit this single-shot feature, which also applies to initialization and gauge fixing. Assuming the time for efficient classical computations to be negligible, this yields a topological fault-tolerant quantum computing scheme where all elementary logical operations can be performed in constant time.

Popular Summary

Building a quantum computer is one of the greatest scientific and technological challenges of the present time. The key obstacle in this endeavor is noise, which appears, for example, in the form of quantum decoherence caused by interactions with the environment. Originally, it was thought that noise posed an insurmountable difficulty, but this belief was proven incorrect by the theoretical development of fault-tolerant computing techniques. Here, we demonstrate a technique that was once considered impossible: correcting errors using noisy information gathered locally in a given finite time. In order to reliably recover information about errors, the previous paradigm requires longer information-gathering operations as the intended computational precision increases.

We focus on systems containing multiple physical qubits on a lattice, and we investigate operations that can be thought of as containing multiple local operations. We assume that the qubits are linked (i.e., they can exhibit locality). While codes for correcting errors typically require many sequential measurements, we show that a single round of local measurements is sufficient for correcting errors. In achieving fault-tolerant quantum error correction, our new approach gives rise to the possibility of quantum computing with as little as possible (constant) time overhead due to fault tolerance. Moreover, it turns out that our new technique is connected with self-correction, an alternative approach to fault tolerance in which a quantum phase of matter is intrinsically robust against errors.

We expect that our results will pave the way for new fault-tolerant quantum computing techniques.

Figure 1

A quantum-local process involves a finite number of local operation rounds, each with a classical output that is globally processed to provide an input for the next round.

Figure 2

A quantum error-correcting code based on the 2D Ising model. Qubits sit at faces and check operators at edges. Three stages of error correction are depicted. (Left panel) When the qubits in the shadowed area are flipped, the check operators on the boundary detect the change. (Middle panel) Noisy measurements of the syndrome are performed. Measurements fail at some edges (red) providing a pseudosyndrome that is not closed and thus has end points (red circles). The failed measurements are estimated to correspond to a minimal set of edges with the same end points (dotted red) so that the effectively recovered syndrome is the boundary of the shaded area. (Right panel) After error correction is performed, the new syndrome corresponds to the set of edges with an effectively wrong measurement outcome.

Figure 3

Two stages in the extraction of the error syndrome. (Top) The measurements of the gauge generators provides a noisy gauge syndrome that does not satisfy color flux conservation at the points marked with a two-colored circle. The branching point marked in black satisfies flux conservation. (Bottom) The gauge syndrome is corrected by adding suitable edges. The error syndrome is the set of branching points of the net of fluxes, marked here as colored circles.

Figure 4

The set of edges with an effectively wrong measurement constitutes a gauge syndrome. If noise in measurements is below a threshold, this gauge syndrome is composed of small clusters. Thus, Each cluster contributes branching points that are thus close to each other. Moreover, they have overall neutral charge, i.e., there exists a local error with such a syndrome. This is the origin of confinement.