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Unformatted text preview: Chapter 7 Quantum Error Correction 7.1 A Quantum Error-Correcting Code In our study of quantum algorithms, we have found persuasive evidence that a quantum computer would have extraordinary power. But will quantum computers really work? Will we ever be able to build and operate them? To do so, we must rise to the challenge of protecting quantum information from errors. As we have already noted in Chapter 1, there are several as- pects to this challenge. A quantum computer will inevitably interact with its surroundings, resulting in decoherence and hence in the decay of the quan- tum information stored in the device. Unless we can successfully combat decoherence, our computer is sure to fail. And even if we were able to pre- vent decoherence by perfectly isolating the computer from the environment, errors would still pose grave difficulties. Quantum gates (in contrast to clas- sical gates) are unitary transformations chosen from a continuum of possible values. Thus quantum gates cannot be implemented with perfect accuracy; the effects of small imperfections in the gates will accumulate, eventually leading to a serious failure in the computation. Any effective strategem to prevent errors in a quantum computer must protect against small unitary errors in a quantum circuit, as well as against decoherence. In this and the next chapter we will see how clever encoding of quan- tum information can protect against errors (in principle). This chapter will present the theory of quantum error-correcting codes. We will learn that quantum information, suitably encoded, can be deposited in a quantum mem- ory, exposed to the ravages of a noisy environment, and recovered without 1 2 CHAPTER 7. QUANTUM ERROR CORRECTION damage (if the noise is not too severe). Then in Chapter 8, we will extend the theory in two important ways. We will see that the recovery procedure can work effectively even if occasional errors occur during recovery. And we will learn how to process encoded information, so that a quantum computation can be executed successfully despite the debilitating effects of decoherence and faulty quantum gates. A quantum error-correcting code (QECC) can be viewed as a mapping of k qubits (a Hilbert space of dimension 2 k ) into n qubits (a Hilbert space of dimension 2 n ), where n > k . The k qubits are the logical qubits or encoded qubits that we wish to protect from error. The additional n- k qubits allow us to store the k logical qubits in a redundant fashion, so that the encoded information is not easily damaged. We can better understand the concept of a QECC by revisiting an ex- ample that was introduced in Chapter 1, Shors code with n = 9 and k = 1....
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