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arXiv:quant-ph/0606107 · Physical Review A

Asymmetric quantum error-correcting codes

L. B. Ioffe, Marc Mézard

Published Mar 29, 2007• 170 citations
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Abstract

The noise in physical qubits is fundamentally asymmetric: in most devices, phase errors are much more probable than bit flips. We propose a quantum error-correcting code that takes advantage of this asymmetry and shows good performance at a relatively small cost in redundancy, requiring less than a doubling of the number of physical qubits for error correction. This code is particularly adapted for building an efficient quantum memory.

DOI
10.1103/physreva.75.032345

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The noise in physical qubits is fundamentally asymmetric: in most devices, phase errors are much more probable than bit flips. We propose a quantum error-correcting code that takes advantage of this asymmetry and shows good performance at a relatively small cost in redundancy, requiring less than a doubling of the number of physical qubits for error correction. This code is particularly adapted for building an efficient quantum memory.
Contrary to the assumption that most quantum error-correcting codes (QECC) make, it is expected that phase errors are much more likely than bit errors in physical devices. By employing the entanglement-assisted stabilizer formalism, we develop a new kind of error-correcting protocol which can flexibly trade error correction abilities between the two types of errors, such that high error correction performance is achieved both in symmetric and in asymmetric situations. The characteristics of the QECCs can be optimized in an adaptive manner during information transmission. The proposed entanglement-assisted QECCs require only one ebit regardless of the degree of asymmetry at a given moment and can be decoded in polynomial time.
In many physical systems it is expected that environmental decoherence will exhibit an asymmetry between dephasing and relaxation that may result in qubits experiencing discrete phase errors more frequently than discrete bit errors. In the presence of such an error asymmetry, an appropriately asymmetric quantum code---that is, a code that can correct more phase errors than bit errors---will be more efficient than a traditional, symmetric quantum code. Here we construct fault tolerant circuits to convert between an asymmetric subsystem code and a symmetric subsystem code. We show that, for a moderate error asymmetry, the failure rate of a logical circuit can be reduced by using a combined symmetric asymmetric system and that doing so does not preclude universality.
Recently, quantum error-correcting codes were proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit flip and phase flip errors. An example for a channel which exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit flips and phase flips can be related to relaxation and dephasing time, respectively. We give systematic constructions of asymmetric quantum stabilizer codes that exploit this asymmetry. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.
In most practical quantum mechanical systems, quantum noise due to decoherence is highly biased towards dephasing. The quantum state suffers from phase flip noise much more seriously than from the bit flip noise. In this work, we construct new families of asymmetric quantum concatenated codes (AQCCs) to deal with such biased quantum noise. Our construction is based on a novel concatenation scheme for constructing AQCCs with large asymmetries, in which classical tensor product codes and concatenated codes are utilized to correct phase flip noise and bit flip noise, respectively. We generalize the original concatenation scheme to a more general case for better correcting degenerate errors. Moreover, we focus on constructing nonbinary AQCCs that are highly degenerate. Compared to previous literatures, AQCCs constructed in this paper show much better parameter performance than existed ones. Furthermore, we design the specific encoding circuit of the AQCCs. It is shown that our codes can be encoded more efficiently than standard quantum codes.
Quantum error correction comes from the marriage of quantum mechanics with the theory of error-correcting codes, which is still at the rudimentary stage tackling only 1 qubit to date. As, unlike the classical world, the extent of error in the quantum world is substantially high and unavoidable, scaling up the quantum error correction from 1 qubit to higher is of utmost significane. Therefore, to achive the scaling up, we present a general quantum error correction code that can be adapted for any number of quantum errors (bit-flip error to be specific in our case). This generalized code for bit-flip error correction realizes a delicate blending of using controlled-NOT gates with the notion of quantum entanglement. We augment the blending with majority voting in the final stage of the error correction. We demonstrate efficacy of our proposed error correction code through real experimentation in IBM Qiskit. Our experimental results up to 3-qubit bit-flip errors in the available Qiskit infrastructure confirm a total escape from errors using our proposed error correction code.
Recently, quantum error-correcting codes have been proposed that capitalize on the fact that many physical error models lead to a significant asymmetry between the probabilities for bit- and phase-flip errors. An example for a channel that exhibits such asymmetry is the combined amplitude damping and dephasing channel, where the probabilities of bit and phase flips can be related to relaxation and dephasing time, respectively. We study asymmetric quantum codes that are obtained from the Calderbank–Shor–Steane (CSS) construction. For such codes, we derive upper bounds on the code parameters using linear programming. A central result of this paper is the explicit construction of some new families of asymmetric quantum stabilizer codes from pairs of nested classical codes. For instance, we derive asymmetric codes using a combination of Bose–Chaudhuri–Hocquenghem (BCH) and finite geometry low-density parity-check (LDPC) codes. We show that the asymmetric quantum codes offer two advantages, namely to allow a higher rate without sacrificing performance when compared with symmetric codes and vice versa to allow a higher performance when compared with symmetric codes of comparable rates. Our approach is based on a CSS construction that combines BCH and finite geometry LDPC codes.