Theory of Quantum Error Correction for General Noise

Abstract

Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition 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. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-errorcorrecting quantum codes, provided that the interaction is dominated by an identity component. *

Encoding one logical qubit into six physical qubits

“…The essential insight of Knill et al was that the most general way to encode quantum information is into a subsystem rather than a subspace [12]. In the case when the information is encoded in a single subsystem, the Hilbert space decomposes as H = (H A ⊗ H B ) ⊕ H C where the subsystem H A stores the protected information.…”

“…[14] that this subsystem structure of a Hilbert space is useful for active quantum error-correction as well (Knill et al did not explicitly recognize this ability in Ref. [12].) See Ref.…”

An Approach by Representation of Algebras for Decoherence-Free Subspaces

“…In this scheme, these subspaces may be appear without spatial symmetry. A more general approach using the concept of interaction algebra was performed by Knill et al [16]. Decoherence-free subspaces can be recognized as special case of general decomposition for appropriate graded interaction algebra.…”

Self-protected quantum algorithms based on quantum state tomography

“…While the active strategies to prevent errors, such as quantum error correcting codes [17,18,19,20] may, in principle, be universal as claimed, passive prevention methods have hardware resource advantages. For example, decoherence-free subspaces (DFS) and noiseless subsystems (NS) [21,22,23,24] are based on the symmetry of the system-bath interaction, so do not require active detection and correction of errors. Another passive technique, holonomic quantum computation, is robust against stochastic errors in the control process [25,26].…”