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Quantum Computation and Error Correction

Quantum Computation and Error Correction. Ali Soleimani. 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. Example:. This makes many problems hard .

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Quantum Computation and Error Correction

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  1. Quantum ComputationandError 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 ~ 2N 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 2N-dimensional vector space. For 2 qubits, a |0>Ä|0> + b |1>Ä|0> + c|0>Ä|1> + d|1>Ä|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 ~2N values. So we can check ~2N 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 7N 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/9705031 • http://www.theory.caltech.edu/people/preskill/ph229/ • Nielsen & Chuang, Quantum Computation, 2000

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