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Purely Quantum Polar Codes

Frédéric Dupuis, Ashutosh Goswami, Mehdi Mhalla, Valentin Savin

Published Aug 25, 2019• 1 citation
OpenAlex

Abstract

We provide a purely quantum version of polar codes, achieving the coherent information of any quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where a two-qubit gate randomly chosen from the Clifford group is used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by preshared EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that quantum polarization can be achieved by choosing the channel combining Clifford operator randomly, from a much smaller subset of only 9 two-qubit Clifford gates. Subsequently, we show that a Pauli channel polarizes if and only if a specific classical channel over four symbol input set polarizes. We exploit this equivalence to prove fast polarization for Pauli channels, and to devise an efficient successive cancellation based decoding algorithm for such channels. Finally, we present a code construction based on chaining several quantum polar codes, which is shown to require a rate of preshared entanglement that vanishes asymptotically.

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Purely Quantum Polar Codes

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We provide a purely quantum version of polar codes, achieving the coherent information of any quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where random two-qubit Clifford gates are used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by sharing EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that a Pauli channel polarizes if and only if a specific classical channel over a four-symbol input set polarizes. We exploit this equivalence to prove fast polarization for Pauli channels, and to devise an efficient successive cancellation based decoding algorithm for such channels.

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We provide a purely quantum version of polar codes, achieving the symmetric coherent information of any qubit-input quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where a two-qubit gate randomly chosen from the Clifford group is used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by preshared EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that quantum polarization can be achieved by choosing the channel combining Clifford operator randomly, from a much smaller subset of only nine two-qubit Clifford gates. Subsequently, we show that a Pauli channel polarizes if and only if a specific classical channel over a four-symbol input set polarizes. We exploit this equivalence to prove fast polarization for Pauli channels, and to devise an efficient successive cancellation based decoding algorithm for such channels. Finally, we present a code construction based on chaining several quantum polar codes, which is shown to require a rate of preshared entanglement that vanishes asymptotically.

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Recently, a purely quantum version of polar codes has been proposed in~\cite{DGMS19} based on a quantum channel combining and splitting procedure, where a randomly chosen two-qubit Clifford unitary acts as a channel combining operation. Here, we consider the quantum polar code construction using the same channel combining and splitting procedure as in~\cite{DGMS19}, but with a fixed two-qubit Clifford unitary. For the family of Pauli channels, we show that polarization happens in multi-levels, where synthesized quantum virtual channels tend to become completely noisy, half-noisy, or noiseless. Further, we present a quantum polar code exploiting the multilevel nature of polarization, and provide an efficient decoding for this code. We show that half-noisy channels can be frozen by fixing their inputs in either the amplitude or the phase basis, which allows reducing the number of preshared EPR pairs compared to the construction in~\cite{DGMS19}. We provide an upper bound on the number of preshared EPR pairs, which is an equality in the case of the quantum erasure channel. To improve the speed of polarization, we propose an alternative construction, which again polarizes in multi-levels, and the previous upper bound on the number of preshared EPR pairs also holds. For a quantum erasure channel, we confirm by numerical analysis that the multilevel polarization happens relatively faster for the alternative construction.

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Quantum polar codes for arbitrary channels

Mark M. Wilde, Joseph M. Renes

We construct a new entanglement-assisted quantum polar coding scheme which achieves the symmetric coherent information rate by synthesizing “amplitude” and “phase” channels from a given, arbitrary quantum channel. We first demonstrate the coding scheme for arbitrary quantum channels with qubit inputs, and we show that quantum data can be reliably decoded by O(N) rounds of coherent quantum successive cancellation, followed by N controlled-NOT gates (where N is the number of channel uses). We also find that the entanglement consumption rate of the code vanishes for degradable quantum channels. Finally, we extend the coding scheme to channels with multiple qubit inputs. This gives a near-explicit method for realizing one of the most striking phenomena in quantum information theory: the superactivation effect, whereby two quantum channels which individually have zero quantum capacity can have a non-zero quantum capacity when used together.

arXiv:1201.2906•IEEE International Symposium on Information Theory (ISIT)•Jul 1, 2012•26 citations
We provide a purely quantum version of polar codes, achieving the coherent information of any quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where a two-qubit gate randomly chosen from the Clifford group is used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by preshared EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that quantum polarization can be achieved by choosing the channel combining Clifford operator randomly, from a much smaller subset of only 9 two-qubit Clifford gates. Subsequently, we show that a Pauli channel polarizes if and only if a specific classical channel over four symbol input set polarizes. We exploit this equivalence to prove fast polarization for Pauli channels, and to devise an efficient successive cancellation based decoding algorithm for such channels. Finally, we present a code construction based on chaining several quantum polar codes, which is shown to require a rate of preshared entanglement that vanishes asymptotically.
We provide a purely quantum version of polar codes, achieving the coherent information of any quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where random two-qubit Clifford gates are used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by sharing EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that a Pauli channel polarizes if and only if a specific classical channel over a four-symbol input set polarizes. We exploit this equivalence to prove fast polarization for Pauli channels, and to devise an efficient successive cancellation based decoding algorithm for such channels.
We provide a purely quantum version of polar codes, achieving the symmetric coherent information of any qubit-input quantum channel. Our scheme relies on a recursive channel combining and splitting construction, where a two-qubit gate randomly chosen from the Clifford group is used to combine two single-qubit channels. The inputs to the synthesized bad channels are frozen by preshared EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. We further show that quantum polarization can be achieved by choosing the channel combining Clifford operator randomly, from a much smaller subset of only nine two-qubit Clifford gates. Subsequently, we show that a Pauli channel polarizes if and only if a specific classical channel over a four-symbol input set polarizes. We exploit this equivalence to prove fast polarization for Pauli channels, and to devise an efficient successive cancellation based decoding algorithm for such channels. Finally, we present a code construction based on chaining several quantum polar codes, which is shown to require a rate of preshared entanglement that vanishes asymptotically.
We provide a generalization of quantum polar codes to quantum channels with qudit-input, achieving the symmetric coherent information of the channel. Our scheme relies on a channel combining and splitting construction, where a two-qudit unitary randomly chosen from a unitary 2-design is used to combine two instances of a qudit-input channel. The inputs to the synthesized bad channels are frozen by sharing EPR pairs between the sender and the receiver, so our scheme is entanglement assisted. Using the fact that the generalized two-qudit Clifford group forms a unitary 2-design, we conclude that the channel combining operation can be chosen from this set. Moreover, we show that polarization also happens for a much smaller subset of two-qudit Cliffords, which is not a unitary 2-design. Finally, we show how to decode the proposed quantum polar codes on Pauli qudit channels.
Recently, a purely quantum version of polar codes has been proposed in~\cite{DGMS19} based on a quantum channel combining and splitting procedure, where a randomly chosen two-qubit Clifford unitary acts as a channel combining operation. Here, we consider the quantum polar code construction using the same channel combining and splitting procedure as in~\cite{DGMS19}, but with a fixed two-qubit Clifford unitary. For the family of Pauli channels, we show that polarization happens in multi-levels, where synthesized quantum virtual channels tend to become completely noisy, half-noisy, or noiseless. Further, we present a quantum polar code exploiting the multilevel nature of polarization, and provide an efficient decoding for this code. We show that half-noisy channels can be frozen by fixing their inputs in either the amplitude or the phase basis, which allows reducing the number of preshared EPR pairs compared to the construction in~\cite{DGMS19}. We provide an upper bound on the number of preshared EPR pairs, which is an equality in the case of the quantum erasure channel. To improve the speed of polarization, we propose an alternative construction, which again polarizes in multi-levels, and the previous upper bound on the number of preshared EPR pairs also holds. For a quantum erasure channel, we confirm by numerical analysis that the multilevel polarization happens relatively faster for the alternative construction.
Polar coding, introduced 2008 by Arıkan, is the first (very) efficiently encodable and decodable coding scheme whose information transmission rate provably achieves the Shannon bound for classical discrete memoryless channels in the asymptotic limit of large block sizes. Here, we study the use of polar codes for the transmission of quantum information. Focusing on the case of qubit Pauli channels and qubit erasure channels, we use classical polar codes to construct a coding scheme that asymptotically achieves a net transmission rate equal to the coherent information using efficient encoding and decoding operations and code construction. Our codes generally require preshared entanglement between sender and receiver, but for channels with a sufficiently low noise level we demonstrate that the rate of preshared entanglement required is zero.
We construct a new entanglement-assisted quantum polar coding scheme which achieves the symmetric coherent information rate by synthesizing “amplitude” and “phase” channels from a given, arbitrary quantum channel. We first demonstrate the coding scheme for arbitrary quantum channels with qubit inputs, and we show that quantum data can be reliably decoded by O(N) rounds of coherent quantum successive cancellation, followed by N controlled-NOT gates (where N is the number of channel uses). We also find that the entanglement consumption rate of the code vanishes for degradable quantum channels. Finally, we extend the coding scheme to channels with multiple qubit inputs. This gives a near-explicit method for realizing one of the most striking phenomena in quantum information theory: the superactivation effect, whereby two quantum channels which individually have zero quantum capacity can have a non-zero quantum capacity when used together.