Theory of quantum error-correcting codes

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. *

Quantum computation in quantum-Hall systems

“…It has been realized that quantum computers can be faster than classical computers for some problems [1][2][3][8][9][10][11][12][13]. The analog nature of errors and possible error correction schemes have been explored [6,7,9,[13][14][15][16][17][18][19][20][21]. There have also been several proposals for actual realizations of quantum information processing [4,5,13,[22][23][24][25][26][27][28][29][30][31].…”

“…Theoretically, the most striking recent development has been the formulation of the fault-tolerant error correction schemes [7,13,17,[19][20][21]. Correction of analog errors inherent in quantum computation due to the superpositionof-states property (which in turn is central to the speed-up of some calculations) means an uphill battle against the second law of thermodynamics.…”

“…In all the proposals for quantum computation [1][2][3][4][5][6][7]13,[22][23][24][25][26][27][28][29][30][31], reading off the final spin states by measuring, and also the measurement processes that are required for error correction [7,13,17,[19][20][21], are most challenging to realize. This is because direct interaction of a microscopic system with any macroscopic system for the purpose of measurement is disruptive and difficult to carry out in an orderly fashion for all the individual spins in the system.…”

“…The NMR variant [29][30][31] measures instead ensemble averages (expectation values). Certain "fault-tolerant" error correction schemes [7,13,17,[19][20][21] actually also require measurements of some of the qubits during the computation.…”

“…Correction of analog errors inherent in quantum computation due to the superpositionof-states property (which in turn is central to the speed-up of some calculations) means an uphill battle against the second law of thermodynamics. These error-correction schemes [7,13,17,[19][20][21] aim at calculations that can go on indefinitely provided the overall error rate at each qubit is small enough.…”