1. Quantum Computation and Error Correction Ali Soleimani

2. Classical computation Digital: each bit can be on or off, ie in {0,1} Storage scales linearly: if we want to store two numbers, need twice the number of bits.

3. Example: This makes many problems hard. Usually when # possible solutions grows exponentially. Factoring: this is the basis of cryptography. N = p1^? * p2^? * p3^? * … * pk^? We have ~ 2 N possible factorizations to check.

4. Quantum Computation Is analog: –Each bit (‘qubit’) lives in a 2-dim Hilbert space –We call the basis {|0>, |1>} –So states are not only |0> and |1>, but also include many like (|0> + |1>)/√ 2 and (2 |0> - i |1>)/ √5 –We use two important quantum properties: entanglement and superposition

5. Entanglement Uses quantum property of superposition of states. –N-qubit system is a 2 N -dimensional vector space. For 2 qubits, a |0>  |0> + b |1>  |0> + c  |0>  |1> + d  |1>  |1> Stores an exponential amount of data!

6. Superposition Since O(|A>+|B>) = O(|A>) + O(|B>), we can superpose multiple states of interest, operate on the superposition, and get the operation done on both states at once. Allow us to work in parallel on ~2 N values. So we can check ~2 N possible solutions to a problem all at once.

7. Uses Can solve previously difficult problems easily. –Factorization of large integers [Shor]. –Database searching. –Probably will not do P=NP. Some ramifications: –Cryptography becomes insecure. –Can use previously slow algorithms in optimization.

8. Difficulties Lots of them! Construction – right now we are at ~4 qubits Gate construction –Infinitude of unitary transformations, must approximate by discrete ones Decoherence –Must isolate from environment.

9. Sources of error Interaction with the environment unavoidable Gates –Continuum of transforms that might be needed. –Can be at best approximations. Errors add up –Not binary –Small errors will combine into larger ones

10. Errors are difficult to fix Can’t just measure and correct! –This destroys superposition –Must detect and correct error without measuring the data. Infinitude of possible errors to fix

11. Error-Correcting Codes Encode one qubit in block of many Hash block before measuring –Shows error and location only, no data Measurement places system into one of two states: –Large fixed error occurred in known position. –No error occurred. –This fixes both measurement/small error problems.

12. Concatened codes Idea: apply error-correction to “elementary” qubits. –For two layers, need 2 errors twice to fail. –N layers: represent one qubit as 7 N qubits. Proven: if elementary error probability is low enough (10 -4 ), concatenated error probability goes quickly to zero as number of layers increase.

13. More Information… arXiv:quant-ph/ –e.g., http://arxiv.org/quant-ph/9705031http://arxiv.org/quant-ph/9705031 http://www.theory.caltech.edu/people/preskill/ph229/ Nielsen & Chuang, Quantum Computation, 2000