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Thermodynamics of QEC

diagram showing overall movement of noise when quantum error correction
is viewed as a type of thermodynamic heat-engine This picture shows an overall view of quantum error correction, considered from a thermodynamic point of view.

The correction process works is like a heat engine, using work to pump heat out of the noisy quantum register back into the "hot" environment. The "quantum register" is a set of physical qubits. The "environment" refers to all other physical systems, including for example neighbouring atoms, fluctuating electromagnetic fields, and imprecision in the control pulses that are used to manipulate the qubits. Since the environment is hotter than the register, heat naturally flows from it into the register, corrupting the qubits. The yellow circle represents the heat engine itself, that is, the complicated sequence of quantum gates and measurements used to perform error correction. These operations are themselves imprecise, and this introduces a further difficulty: the engine must, overall, extract more noise than it introduces. To accomplish this feat, it turns out to be very useful to introduce a further set of qubits, called an "ancilla" or "helper". The main job of the heat engine is to make this ancilla cold. It does this by constructing, amongst the ancilla qubits, a highly structured (and therefore low entropy) quantum state. If the quantum register is like some water we want to keep cold, than the ancilla state is like an "ice crystal". Heat naturally flows from the register to the ancilla when the two are coupled.

The special shape of the arrows from register to ancilla, and from ancilla to heat engine, is alluding to an important feature of the way the heat (or entropy) moves. It is useful to identify the heat as formed of essentially two components: random rotations of qubits about one axis (x), and random rotations about another axis (z). The issue is that the engine can't manage to make the ancilla cold in both senses at once. However, when the register and ancilla are allowed to couple (using quantum gate operations), the physical interaction has a highly useful property: it allows only a restricted kind of heat flow, where x errors move in one direction, z errors move in the other. It is only necessary to make the ancilla colder than the register in one sense (say x) at any given time. One alternates the type of coupling between register and ancilla, first allowing x noise to flow out of the register, then z noise, always re-cooling the ancilla in between.