Quantum computing

A topological route to error correction

Quantum computing is plagued by noise and small errors. An approach based on topological techniques reduces the sensitivity to errors and boosts the prospects for building practical quantum computers. See Article p.489

Quantum computers have the potential to solve numerical problems that would be impossible on a classical computer. Roughly speaking, the superposition principle of quantum mechanics allows a quantum computer to perform many calculations simultaneously on a single processor, and entanglement (non-classical correlations) provides an exponential increase in its memory capacity. Unfortunately, the same properties that enhance the computational power of a quantum computer also make it sensitive to errors produced by interactions with the environment or by imperfect logic operations. In this issue, Yao et al.1 (page 489) describe the first experimental demonstration of a technique that uses topological effects to reduce the sensitivity of a quantum computer to errors.

The bits in a quantum computer, commonly referred to as qubits, can be represented by a two-state quantum system, such as the two quantized energy levels of an atom (Fig. 1a). One state represents a logical value of '0' and the other state represents a '1' — meaning that, like a classical computer, a quantum computer is a digital device. But unlike classical physics, quantum mechanics allows situations in which both possibilities (0 or 1) exist simultaneously. The probability of finding the system in the 0 or 1 state is equal to the square of a complex number known as the probability amplitude. As a result, the information stored in a qubit corresponds to a continuous range of probability amplitudes. According to quantum mechanics, the probability amplitudes of both the 0 and 1 states have wave-like properties, and their relative position in an oscillatory cycle corresponds to an additional degree of freedom known as their phase. In addition, the qubits can be entangled with one another in many different ways. Thus a qubit can contain much more information than a classical bit, which can have only a specific value of 0 or 1.

Figure 1: Measuring quantum bits.

a, Five qubits represented by the energy levels of five atoms (spheres). The ground state encodes the logical value '0', whereas the excited state encodes the value '1'. Because of their quantum nature, the atoms can be in both states simultaneously. The intensity of the spheres' colour denotes the probability that an energy level is occupied, and the colours indicate the phase of the oscillating probability amplitude associated with the energy level. b, After a measurement, the qubits collapse to classical states with a specific value of 0 or 1, and the phase information is destroyed, as illustrated by the black colour. c, A three-dimensional array of qubits can be used to implement topological error correction, which reduces the sensitivity of quantum computing to errors. The calculation consists of a series of measurements that proceeds from all of the qubits in the plane that forms the left side of the array through the adjacent planes to the right. Yao et al.1 demonstrated topological error correction using an ensemble of eight qubits.

Measuring the value of a qubit causes it to collapse to a specific value of 0 or 1, reducing it to a classical bit (Fig. 1b). Because measuring a qubit destroys its quantum-mechanical properties, it was not initially apparent whether there was any way to correct for errors in qubits without destroying them. It was subsequently shown2 that error correction was possible if a 'logical' qubit was constructed from a combination of multiple physical qubits (Fig. 1a). For example, the value of the logical qubit can be taken to be the parity of the ensemble of physical qubits, where parity is defined to be 0 if the sum of the qubit values is even and 1 if the sum is odd. But there are more efficient ways of encoding the logical information. Quantum logic operations on the qubit ensemble can be used to correct the errors in the individual qubits without measuring the value of the logical qubit, and thus without destroying the information it encodes. This allows the errors in a quantum computer to be made arbitrarily small — although additional errors will be introduced during the error-correction process itself, so the average error rate must be below a threshold on the order of 10−4 for conventional error-correction techniques.

In the type of topological error correction3,4 used by Yao et al.1, the logical qubits are distributed over a lattice of physical qubits in such a way that the information is automatically protected against most forms of error. This type of error correction is theoretically expected to increase the tolerance for errors to above 1%. The authors1 demonstrated topological error correction by combining topological techniques with cluster-state quantum computing5, in which a three-dimensional array, or cluster, of qubits is prepared with a carefully chosen form of entanglement between nearest neighbours in the array.

Their approach begins with measurement of all the qubits in the plane that forms the left side of the array (Fig. 1c). The results of those measurements are then used to decide what kind of measurements to perform on the next layer of adjacent qubits. No active logic operations are performed — instead, the calculation depends on choosing the measurements in such a way that the collapse of the quantum state produces the desired logical operations6,7. The calculation proceeds until the final layer of qubits on the far right side of the array — the values of which give the desired output of the calculation — is reached. The authors reduced the sensitivity of the calculations to environmental noise and small errors in the logical operations by optimizing the spatial arrangement, or topology, of the qubits and the measurements.

Yao and colleagues1 performed their experiment using an eight-qubit cluster, in which the value of each qubit (0 or 1) was represented by the polarization of a single photon (the direction of the photon's electric field). An optical route to quantum computing has the advantage that optical fibres can be readily used to transfer qubits from one location to another. The greatest challenge of an optical approach is developing an efficient mechanism for generating large numbers of entangled photons on demand. The authors were able to enhance the efficiency of their entangled-photon source using quantum-interference techniques, but further improvements in photon sources will be necessary.

Topological error correction could also be performed using qubits based on other physical systems, such as superconducting devices or trapped ions, which, like optical approaches, have allowed strong progress to be made in quantum computing. Other forms of topological error correction8 may be able to further reduce the sensitivity to experimental errors beyond that achieved by the authors. For example, it is possible to produce a change in the phase of a probability amplitude that depends only on the number of times that the trajectory of a quantum system circles a specific point in a complex mathematical space known as Hilbert space, regardless of the exact shape of the trajectory. Topological error correction can increase the tolerance for experimental errors to the point that it is consistent with experimental capabilities, and greatly increases the prospects for building large-scale quantum computers. The experiment by Yao et al. represents an essential first step in that direction.


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Correspondence to James D. Franson.

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Franson, J. A topological route to error correction. Nature 482, 479–480 (2012). https://doi.org/10.1038/482478a

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