In principle you could think about long-term storage of qubits (which are the storage units in a quantum computer instead of bits). However, it's a fanciful thought for several reasons. First, the state of the art is to figure out how to make useful qubits last for more than a few milliseconds. Getting them to last for seconds would already be a huge advance; getting them to last for years would probably require layers of sophisticated quantum error correction. It's hard to imagine that there would ever be any entirely passive technology to store qubits that is similar to flash memory or anything like that. Second, and this is a related point, it is mathematically impossible to both preserve a qubit and read its value. That is, you can physically preserve a qubit after reading it, but the act of reading it changes its state. So even if you did have long-term qubits, their storage would be a time capsule. You wouldn't be able to read them without using them up! This could be useful somehow, but I'm not sure how. I don't know that anyone is working on this, even hypothetically.
The closest that I can think of is that light travelling through the universe carries quantum states that you could call "qubits". These qubits can last millions or even billions of years. They can't be used in a quantum computer, but I have heard that very-long-distance quantum entanglement has been detected in physical measurements. I.e., it's interesting as astronomy rather than as computer science.
