We are a long way away from VISA level transactions however there are some methods for Bitcoin to deal with the data on the scale that is used by VISA (2K transactions per second).
Remote blockchain. Currently every client downloads a copy of the block chain. There is no reason for a client to store a copy of the blockchain IF there is an online entity they trust which has a complete copy of the blockchain. If transaction volume even got to 1% of VISA most clients would partner with a "block chain providers" possibly for a minimal monthly fee and simply query that provider about various transactions. Before someone cries "centralization" there would be no official blockchain provider but instead numerous independent entities offering a service of allowing fast and secure blockchain "lookups". If someone is paranoid they could use a client which queries two different block chain providers to ensure they return same data.
GPU Acceleration of transaction verification. GPU are currently used to find a hash to sign the block as this is cryptographically intensive and any speed up pays huge dividends. The bitcoind uses non openCL CPU to verify transactions simply because currently the required CPU load is much lower. The paper linked by author indicates it would take 50 modern CPU to handle VISA level transaction volumes. That stat is misleading. A high end GPU is roughly 20x as powerful as high end CPU for cryptographic hashing. Furthermore even with say 50% annual transaction volume growth we are two decades away from VISA level volume. Moore's law would indicate that average CPU would be more than 1000x as powerful a current CPU is today and a GPU in 20 year closer to 20,000x current top of the line CPU. In other words transaction volume will grow SLOWER than computation power. That combined with the ability to use OpenCL to acceleration transaction verification means the hardware will be a non-issue by the time transaction volume reaches that point.
Actual transaction volumes. Bitcoin is digital cash. Many people today don't like to use cash for online and offline transactions. As a result payment systems will eventually be built on-top of bitcoin and bitcoin will only be used for large settlements between providers. Thus even if bitcoin (and higher level systems) eventually reached transaction volumes equal to VISA the amount added to bitcoin blockchain would be significantly lower. Also the "VISA level" is more an end game scenario. A more likely interim goal (say over next 2 decades) would be a PayPal level transaction volume which is much more modest at a mere 100 transactions per second (vs 4000 for VISA).
