Quantum codes satisfy certain properties
A successful scheme for Quantum error correction satisfies certain properties.
Suppose we encode a k-qubit system in a larger system with n qubits.
n k o o o o o o o o o o o o o o o o --> o o o o o o o o o o o o o o o o o o
- The k-qubit system has 2ᵏ basis states. The code has 2ᵏ basis “codewords” corresponding to the original basis states.
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A linear combination of unencoded basis states corresponds to a linear combination of encoded basis codewords.
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lemma temp : ... := sorry
“So the space T of valid codewords (the coding space) is a Hilbert space in its own right. It’s a subspace of the full 2ⁿ-dimensional Hilbert space.”
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“If we can correct errors E and F then we can correct aE + bF. So we only need to consider whether the code can correct a basis of errors.”
- A convenient basis is the set of tensor products of {X, Y, Z, I}. The carrier set of the Pauli group;
“To correct two errors E₁ and E₂ we must always be able to distinguish error E₁ acting on one basis codeword |ψᵢ⟩ from error E₂ acting on a different basis codeword |ψⱼ⟩; E.g, an error on one qubit from nothing happening to another.
We can only be sure of doing this if E₁|ψᵢ⟩ is orthogonal to E₂|ψⱼ⟩:
⟨ψᵢ|E₁†E₂|ψⱼ⟩ = 0.”
“When we make a measurement to find out about about the error, we must learn nothing about the state of the code within the code space; Otherwise we disturb a superposition. Therefore, measuring the operator E₁†E₂ to learn about the error must return the same distribution for all the basis codewords:
⟨ψᵢ|E₁†E₂|ψᵢ⟩ = ⟨ψⱼ|E₁†E₂|ψⱼ⟩.”
These two conditions can be combined into one:
⟨ψᵢ|E₁†E₂|ψⱼ⟩ = C₁₂δᵢⱼ,
where C₁₂ is independent of i and j.
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“A simple way to satisfy the error-correcting conditions is just to demand that every pair of errors in E be such that
E₁†E₂ ∉ normalizer(S).- In such a case, each error is assigned a unique syndrome, and codes along with an error set satisfying this property are known as nondegenerate codes. Codes with a corresponding error set not satisfying this are known as degenerate codes.”
Two codes can satisfy Properties if any quantum code but have different Code efficiency;