How much information can a qubit remember?

You can only extract one bit of information from a single qubit, but in some applications the ability to store a large amount of information and only later decide which portion is needed would still be useful.

If I have a large bitstring, can I store the entire string in the state of a single qubit (or small O(1)-sized collection of qubits) and still be able to retrieve any single bit?

• a qubit has only two states, but a multi-qubit system could store more than one bitstring. – ZeroTheHero Oct 28 '17 at 2:30
• Relevant terms: holevo bound and quantum advice. The short answer is that, for most intents and purposes, when it comes to storing retrievable information, qubits aren't better than bits. – Craig Gidney Oct 28 '17 at 20:20
• Not exactly the same, but perhaps relevant: You can transfer two bits by sending one qubit and already sharing a pair of entangled qubits <en.wikipedia.org/wiki/Superdense_coding>. – Sebastian Riese May 27 at 15:48

You have plenty of ways to "encode" classical data in a quantum systems. Given any data, encoded as a string of bits $\sigma$, you can:
• You can take the same $n$-length string, and create the state $$\otimes_{i=1}^n |\sigma_i\rangle$$.
Obviously, you can recover your data with $n$ different measurements. There are cases where this is what you want, but it's not always the case.
• As is often done in quantum algorithmics, you can create the state $$\sum_{i=1}^n \sigma_i |i\rangle$$ That uses just $\lceil log(n) \rceil$ qubits in order to have $2^n$ basis. There are plenty of situation where you can use a thing like this.
• You could also encode your data as a relative phase of $\sigma$ with $n$ bits of precision on a single qubit. (not much useful..) There is no chance to recover the data with just one state.