Short answer: The Knill-Laflamme conditions say when a quantum error-correcting code can correct a given set of error operators: the errors must not be able to distinguish states within the code subspace. They come from a 1996 arXiv paper by Emanuel Knill and Raymond Laflamme, which gives necessary and sufficient conditions for perfect recovery of an encoded state. This is established theory, and none of the pages used here describes hardware.
What the conditions say
MathWorld describes the conditions as characterizing when a quantum error-correcting code corrects a set of error operators. Writing P for the projection onto the code subspace, the condition is that P times the product of one error's adjoint with another error, times P, equals a scalar times P, for every pair of errors in the set. Equivalently, the error operators cannot distinguish states within the code subspace.
The original paper
The 1996 arXiv abstract (quant-ph/9604034) says quantum error correction will be necessary for preserving coherent states against noise and unwanted interactions in quantum computation and communication. The authors develop a general theory based on encoding states into larger Hilbert spaces subject to known interactions, and obtain necessary and sufficient conditions for perfect recovery of an encoded state after degradation by an interaction. They say the conditions depend only on the behavior of the logical states.
The abstract also relates the resulting recovery-operator-independent definition of a code to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information-theoretic identity. It introduces two notions of fidelity and error for imperfect recovery, one for pure states and one for entangled states, and says the entangled-state version is more appropriate for quantum memory and for teleportation in communication. The authors show that the error for entangled states is bounded linearly by the error for pure states.
A recovery argument from lecture notes
Lecture notes from a Berkeley course on quantum coding theory define a code as a subspace of a larger physical Hilbert space, and say a code corrects a set of errors if some recovery procedure restores every codeword after any error in the set. They model recovery as a unitary acting on the corrupted state and an ancilla, whose output register can be read as a syndrome recording which error occurred. Because the left side of that equation is linear in the encoded state, the notes argue that the syndrome term, scaled by the square root of the error probability, must be a constant that does not depend on the state.
Limits
The abstract text available for the 1996 paper is cut off before it finishes describing the qubit lower bounds, so this post does not summarize them beyond what is quoted above. The pages used are a reference page, an abstract and lecture notes, and none reports an experiment.
FAQ
Are the conditions necessary and sufficient?
The 1996 abstract says the authors obtain necessary and sufficient conditions for perfect recovery of an encoded state after degradation by an interaction.
Do the conditions depend on the logical state?
The abstract says they depend only on the behavior of the logical states. The Berkeley notes argue that the syndrome term must be independent of the state.
What does it mean that errors cannot distinguish codewords?
MathWorld puts it as: the error operators cannot distinguish states within the code subspace.
How are entangled states treated?
The 1996 paper introduces separate fidelity and error notions for pure and entangled states, says the entangled one suits quantum memory and teleportation, and shows the entangled-state error is bounded linearly by the pure-state error.
What is the syndrome register?
In the Berkeley notes, it is the ancilla output of the recovery unitary, which records what error was applied to the state.
All field notes