Quantum Error Correction, an informal introduction. Start. <-- [prev] . [next] --> |
2. The basic idea
Before we present the basic idea of quantum error correction, it is worth considering in slightly more detail the problem stated at the end of the last chapter. On the one hand we need measurement and dissipation to achieve stability, but on the other we must avoid measurement and dissipation in order to avoid uncontrolled disturbance.It is sometimes stated that quantum superposition or quantum entanglement is itself fragile. However this statement is misleading because it is not true that entangled states are especially fragile compared to non-entangled ones (at least for quantum computing purposes, where any sort of uncontrolled state-change is bad). The reason qubits are unstable is really more mundane: it is simply because they are small. For small entities, small external influences are enough to perturb them. The problem of quantum error correction is not that quantum things are less stable than one might expect; the problem is that standard active stabilisation methods cannot be applied.
It might be suggested, why not just work with larger things? The trouble is that the world simply does not work like that. The fundamental scale is set by Planck's constant, h. The physical dimensions of this scale are (energy)×(time), or (distance)×(momentum). The result is that, for example, if one uses two states having a comfortably large energy gap between them, one finds that the time scale on which the system is evolving is that much faster. You can trade off energy differences for required timing precision, but you can't find well-separated energies with modest evolution times. Similarly, superpositions of states that are well-separated in position will be only narrowly distinguished in momentum, etc.
Quantum error correction finds a way through these difficulties by clever use of structure and entanglement. The best way to understand it is in information terms, and that will be presented in the following sections. The basic ideas are as follows.
- Carefully designed states of multiple qubits. Instead of using physical qubits (such as single atoms, single electrons, single photons) one at a time, we use them in groups. In this sense we do use "larger" things. However, the joint states are not well-separated in physical properties such as energy or angular momentum (indeed, they may be degenerate). They differ only in the pattern which they express. For example, the pattern 1010101 differs from 1011010.
- Measurements that observe errors without observing information. QEC does make use of measurement and dissipation. However, the measurements are cleverly designed so that they "ask" only very restricted questions of the qubits. These measurements allow one to observe the errors (i.e. state changes) without observing the stored quantum information (i.e. the states themselves). One never measures the state of any single qubit. Instead, joint measurements are made which ask about joint properties, such as "how many 1's are there in the pattern?".
- Phase information as well as bit information must be guarded. As well as protecting qubits from flipping from "up" to "down" states (like 0 to 1) we have to protect them from rotating in the horizontal plane, i.e. preserve the phase in superpositions such as |0> + exp(i f) |1> . This is an important difference from the case of classical bits, and requires a basic new insight.