Quantum message-passing algorithm for optimal and efficient decoding
Recently, Renes proposed a quantum algorithm called belief propagation with quantum messages (BPQM) for decoding classical data encoded using a binary linear code with tree Tanner graph that is transmitted over a pure-state CQ channel \cite{renes_2017}, i.e., a channel with classical input and pure-...
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| Published in: | Quantum (Vienna, Austria) Vol. 6; p. 784 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Verein zur Förderung des Open Access Publizierens in den Quantenwissenschaften
23.08.2022
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| ISSN: | 2521-327X, 2521-327X |
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| Abstract | Recently, Renes proposed a quantum algorithm called belief propagation with quantum messages (BPQM) for decoding classical data encoded using a binary linear code with tree Tanner graph that is transmitted over a pure-state CQ channel \cite{renes_2017}, i.e., a channel with classical input and pure-state quantum output. The algorithm presents a genuine quantum counterpart to decoding based on the classical belief propagation algorithm, which has found wide success in classical coding theory when used in conjunction with LDPC or Turbo codes. More recently Rengaswamy
e
t
a
l
.
\cite{rengaswamy_2020} observed that BPQM implements the optimal decoder on a small example code, in that it implements the optimal measurement that distinguishes the quantum output states for the set of input codewords with highest achievable probability. Here we significantly expand the understanding, formalism, and applicability of the BPQM algorithm with the following contributions. First, we prove analytically that BPQM realizes optimal decoding for any binary linear code with tree Tanner graph. We also provide the first formal description of the BPQM algorithm in full detail and without any ambiguity. In so doing, we identify a key flaw overlooked in the original algorithm and subsequent works which implies quantum circuit realizations will be exponentially large in the code dimension. Although BPQM passes quantum messages, other information required by the algorithm is processed globally. We remedy this problem by formulating a truly message-passing algorithm which approximates BPQM and has quantum circuit complexity
O
(
poly
n
,
polylog
1
ϵ
)
, where
n
is the code length and
ϵ
is the approximation error. Finally, we also propose a novel method for extending BPQM to factor graphs containing cycles by making use of approximate cloning. We show some promising numerical results that indicate that BPQM on factor graphs with cycles can significantly outperform the best possible classical decoder. |
|---|---|
| AbstractList | Recently, Renes proposed a quantum algorithm called belief propagation with quantum messages (BPQM) for decoding classical data encoded using a binary linear code with tree Tanner graph that is transmitted over a pure-state CQ channel \cite{renes_2017}, i.e., a channel with classical input and pure-state quantum output. The algorithm presents a genuine quantum counterpart to decoding based on the classical belief propagation algorithm, which has found wide success in classical coding theory when used in conjunction with LDPC or Turbo codes. More recently Rengaswamy
e
t
a
l
.
\cite{rengaswamy_2020} observed that BPQM implements the optimal decoder on a small example code, in that it implements the optimal measurement that distinguishes the quantum output states for the set of input codewords with highest achievable probability. Here we significantly expand the understanding, formalism, and applicability of the BPQM algorithm with the following contributions. First, we prove analytically that BPQM realizes optimal decoding for any binary linear code with tree Tanner graph. We also provide the first formal description of the BPQM algorithm in full detail and without any ambiguity. In so doing, we identify a key flaw overlooked in the original algorithm and subsequent works which implies quantum circuit realizations will be exponentially large in the code dimension. Although BPQM passes quantum messages, other information required by the algorithm is processed globally. We remedy this problem by formulating a truly message-passing algorithm which approximates BPQM and has quantum circuit complexity
O
(
poly
n
,
polylog
1
ϵ
)
, where
n
is the code length and
ϵ
is the approximation error. Finally, we also propose a novel method for extending BPQM to factor graphs containing cycles by making use of approximate cloning. We show some promising numerical results that indicate that BPQM on factor graphs with cycles can significantly outperform the best possible classical decoder. Recently, Renes proposed a quantum algorithm called belief propagation with quantum messages (BPQM) for decoding classical data encoded using a binary linear code with tree Tanner graph that is transmitted over a pure-state CQ channel \cite{renes_2017}, i.e., a channel with classical input and pure-state quantum output. The algorithm presents a genuine quantum counterpart to decoding based on the classical belief propagation algorithm, which has found wide success in classical coding theory when used in conjunction with LDPC or Turbo codes. More recently Rengaswamy {et al.} \cite{rengaswamy_2020} observed that BPQM implements the optimal decoder on a small example code, in that it implements the optimal measurement that distinguishes the quantum output states for the set of input codewords with highest achievable probability. Here we significantly expand the understanding, formalism, and applicability of the BPQM algorithm with the following contributions. First, we prove analytically that BPQM realizes optimal decoding for any binary linear code with tree Tanner graph. We also provide the first formal description of the BPQM algorithm in full detail and without any ambiguity. In so doing, we identify a key flaw overlooked in the original algorithm and subsequent works which implies quantum circuit realizations will be exponentially large in the code dimension. Although BPQM passes quantum messages, other information required by the algorithm is processed globally. We remedy this problem by formulating a truly message-passing algorithm which approximates BPQM and has quantum circuit complexity $\mathcal{O}(polyn, polylog\frac{1}{\epsilon})$, where $n$ is the code length and $\epsilon$ is the approximation error. Finally, we also propose a novel method for extending BPQM to factor graphs containing cycles by making use of approximate cloning. We show some promising numerical results that indicate that BPQM on factor graphs with cycles can significantly outperform the best possible classical decoder. |
| ArticleNumber | 784 |
| Author | Renes, Joseph M. Piveteau, Christophe |
| Author_xml | – sequence: 1 givenname: Christophe surname: Piveteau fullname: Piveteau, Christophe organization: Institute for Theoretical Physics, ETH Zürich, Switzerland – sequence: 2 givenname: Joseph M. surname: Renes fullname: Renes, Joseph M. organization: Institute for Theoretical Physics, ETH Zürich, Switzerland |
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| Cites_doi | 10.1016/S0076-5392(08)60247-7 10.1103/PhysRevA.71.052330 10.1016/B978-0-12-697560-4.50014-9 10.26421/QIC16.3-4-2 10.1103/PhysRevA.100.012330 10.1080/09500349414552221 10.1109/ISIT.2019.8849833 10.1109/ITW.2017.8277985 10.1007/s11128-020-02855-7 10.1103/PhysRevLett.122.200501 10.1109/ISIT.2012.6284250 10.1017/CBO9780511791338 10.1103/PhysRevLett.106.240502 10.1109/18.782170 10.1038/s41534-021-00422-1 10.1109/TIT.2012.2218792 10.1109/18.910572 10.1103/PhysRevB.76.201102 10.1016/j.aop.2007.10.001 10.1103/PhysRevA.57.2368 10.1017/9781316848142 10.1109/MSP.2004.1267047 10.1103/PhysRevA.58.146 10.1098/rspa.1935.0122 10.1147/rd.176.0525 10.1109/TIT.2017.2716422 10.1109/TIT.2014.2314463 10.1109/18.915636 10.22331/q-2021-11-22-585 10.1103/PhysRevResearch.2.043423 10.1103/PhysRevA.74.052333 10.1109/TIT.2005.850085 10.4007/annals.2021.193.2.4 10.1088/1367-2630/15/1/013021 10.22331/q-2018-10-19-102 10.1098/rspa.1936.0047 10.1145/3372224.3419207 10.1007/BF02435921 10.1109/TIT.2009.2021379 10.1088/1367-2630/aa7c78 10.48550/arXiv.2103.09225 10.1109/TIT.2011.2169534 10.26421/QIC8.10-8 10.1103/PhysRevLett.106.080403 10.1088/1751-8121/aa6dc3 10.1080/17442507508833114 10.1109/18.910573 10.48550/arXiv.1805.12445 10.1109/TIT.2013.2280915 10.48550/arXiv.2010.10845 |
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