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# Decoherence

Of course, there is a catch or two to all of this glorious quantum future: perfect quantum computers are very hard to build.

Any tiny imperfection in our ability to control or measure the qubits, or any unintended interaction between the qubits and the outside environment, damages the quantum state. This decoherence not only affects whether our qubit is zero or one, it also affects the phase of the qubit, and our ability to properly create the interference that is vital to the operation of quantum algorithms.

## Control

The technology necessary to control and measure individual electrons, atoms, or photons must be precise and sensitive. Often, it must also be complex; signals that control a superconducting qubit might be carefully constructed pulses of microwave radiation, those that control individual atoms might be precisely timed pulses of laser light. For example, if the laser is left on a smidgen too long, the single-qubit rotation may go a little too far; instead of an intended $\pi$ rotation, you might get $1.03\pi$ instead.

This results in a certain probability of an error if you measure the qubit right away, or worse, the error in one qubit can propagate to other qubits as you continue on with your algorithm.

## Isolation

Qubits must be properly isolated from the outside environment: given that we use light, or microwaves, or magnetic fields to control qubits, any stray signals can damage our state. The basic way to isolate something from radio waves is to put it inside a Faraday cage, a shell of conductive metal that absorbs the signal. Even with our best effort, though, some radio waves can leak through, and some quantum systems are so sensitive that interference caused by electric trains running a kilometer away from the laboratory can be detected!

Without proper isolation from the environment, it is also possible information leaks out of the system, causing our qubits to be measured inadvertently by the environment, or become entangled with the environment. In the worst case, this will destroy the superposition or entanglement of our qubits, leaving us with nothing but noisy, random classical data.

Different physical technologies for making the qubits require different control technology and hence are susceptible to different interference from the outside. Fundamentally, it’s a tradeoff: qubits that are easy to control intentionally are also easily affected unintentionally.

## Getting the Vocabulary Right: Pure and Mixed States

Getting the vocabulary right when dealing with errors will keep the confusion to a minimum. Physicists always mean something precise when they use a particular word, but these meanings do not always line up exactly with daily use.

We have talked already about superposition states and entangled states. Until this discussion of decoherence, everything we have talked about has assumed that the quantum computer works perfectly. Without errors, all of the states we have presented are pure states. A state that might have errors, on the other hand, is called a mixed state. Note that it’s very tempting to call a superposition or an entangled state “mixed”, but that’s incorrect use of the term.

Mixed states are described, in part, in terms of their fidelity. The fidelity of a quantum state is the probability that the state we have created is identical to the state we think we have created, regardless of whether that state is a simple state such as $|0\rangle$ or a superposition such as $(|0\rangle + |1\rangle)/\sqrt{2}$.

The fidelity $F$ is a number between 0.0 and 1.0. $F = 1.0$ implies that our quantum computer works perfectly. For a single qubit, $F = 0.5$ means that the state has a $50\%$ probability of being the state we expect, and a $50\%$ probability of being the wrong state; its state is completely random. Such a state is referred to as completely mixed.

For more than two qubits, $F$ is the probability that all of them are behaving the way we expect, based on the algorithm we’ve run so far. For two qubits, the completely mixed state is $F = 0.25$, for three qubits, it’s $F = 0.125$, and so on.

We will talk more about decoherence and error correction later in the course.