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Quantum Information and Computation

A comparison of decoherence-free subsystem/subspace for partially broken symmetry

S. Siddiqui, Julio R. Gea-Banacloche

Published Nov 1, 2005• 0 citations
Publisher

Abstract

We study the performance of the 3-qubit decoherence-free subsystem and the 4-qubit decoherence-free subspace in the presence of partially correlated noise. We characterize their performance in terms of the average and worst-case fidelity, as a function of the ratio of the interqubit distance to the correlation length of the noise, and find that, in general, more symmetric arrangements (triangles or squares) lead to better performance. Overall, we find the 3-qubit code to perform better than the 4-qubit code by about a factor of 2 in the average infidelity. We observe that this is related to the greater robustness of this code against uncorrelated (independent) errors.

DOI
10.26421/qic5.7-5

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We study the performance of the 3-qubit decoherence-free subsystem and the 4-qubit decoherence-free subspace in the presence of partially correlated noise. We characterize their performance in terms of the average and worst-case fidelity, as a function of the ratio of the interqubit distance to the correlation length of the noise, and find that, in general, more symmetric arrangements (triangles or squares) lead to better performance. Overall, we find the 3-qubit code to perform better than the 4-qubit code by about a factor of 2 in the average infidelity. We observe that this is related to the greater robustness of this code against uncorrelated (independent) errors.
We consider quantum error correction against correlated noise using simple and concatenated Calderbank-Shor-Steane codes as well as nnn-qubit repetition codes. We characterize the performance of various codes by means of the fidelity following the error correction of a single logical qubit in a quantum register. For concatenated codes we find a threshold in the single-qubit error rate below which the encoded qubit is perfectly protected. The threshold depends on the correlation strength of the noise and goes to zero for perfect correlation. Finally, we concatenate the traditional error correcting codes with a decoherence free subspace and evaluate the performance over the whole range from uncorrelated noise to perfectly correlated noise.
Quantum error-correcting codes (QECCs) and decoherence-free subspace (DFS) codes provide active and passive means, respectively, to address certain types of errors that arise during quantum computation. The latter technique is suitable to correct correlated errors with certain symmetries and the former to correct independent errors. The concatenation of a QECC and a DFS code results in a degenerate code that splits into actively and passively correcting parts, with the degeneracy impacting either part, leading to degenerate errors as well as degenerate stabilizer operators. The concatenation of the two types of code can aid universal fault-tolerant quantum computation when a mix of correlated and independent errors is encountered. In particular, we show that for sufficiently strongly correlated errors, the concatenation with the DFS as the inner code provides better entanglement fidelity, whereas for sufficiently independent errors, the concatenation with the QECC as the inner code is preferable. As illustrative examples, we examine in detail the concatenation of a two-qubit DFS code and a three-qubit repetition code or five-qubit Knill-Laflamme code, under independent and correlated errors.
Abstract We study the effectiveness of quantum error correction against coherent noise. Coherent errors (for example, unitary noise) can interfere constructively, so that in some cases the average infidelity of a quantum circuit subjected to coherent errors may increase quadratically with the circuit size; in contrast, when errors are incoherent (for example, depolarizing noise), the average infidelity increases at worst linearly with circuit size. We consider the performance of quantum stabilizer codes against a noise model in which a unitary rotation is applied to each qubit, where the axes and angles of rotation are nearly the same for all qubits. In particular, we show that for the toric code subject to such independent coherent noise, and for minimal-weight decoding, the logical channel after error correction becomes increasingly incoherent as the length of the code increases, provided the noise strength decays inversely with the code distance. A similar conclusion holds for weakly correlated coherent noise. Our methods can also be used for analyzing the performance of other codes and fault-tolerant protocols against coherent noise. However, our result does not show that the coherence of the logical channel is suppressed in the more physically relevant case where the noise strength is held constant as the code block grows, and we recount the difficulties that prevented us from extending the result to that case. Nevertheless our work supports the idea that fault-tolerant quantum computing schemes will work effectively against coherent noise, providing encouraging news for quantum hardware builders who worry about the damaging effects of control errors and coherent interactions with the environment.
In the scope of this work the coherence of quantum information, which is encoded into a qubit register, is analysed. The qubit register is modelled by a spin chain with finite inter-spin distance. In most physically relevant realisations this spin chain irreversibly interacts with a surrounding environment, such that a spin-boson model is used to describe the setting. Due to the interaction decoherence occurs among the qubits register and quantum information gets lost. Mechanisms to slow down this decoherence process are investigated. For that purpose, the techniques of encoding qubits into decoherence-reduced subspaces and quantum error correction are used. In both cases only a linear subspace of the complete available Hilbert space of the spin chain is used as quantum code. The stability of such a code against decoherence has to be evaluated. This evaluation is performed on average over all states within the code by a code fidelity.
When an environmental disturbance to a quantum system has a wavelength much larger than the system size, all qubits in the system are under the action of the same error operator. The noiseless subsystem and decoherence-free subspace are immune to such collective noise. We construct simple quantum circuits that implement these error-avoiding codes for a small number nnn of physical qubits. A single logical qubit is encoded with n=3n=3n=3 and 4, while two and three logical qubits are encoded with n=5n=5n=5 and 7, respectively. Recursive relations among subspaces employed in these codes play essential roles in our implementation.