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arXiv:2102.02494 · IEEE International Symposium on Information Theory (ISIT)

Permutation-invariant quantum coding for quantum deletion channels

Yingkai Ouyang

Published Jul 12, 2021• 24 citations
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Abstract

Quantum deletions, which are harder to correct than erasure errors, occur in many realistic settings. It is therefore pertinent to develop quantum coding schemes for quantum deletion channels. To date, not much is known about which explicit quantum error correction codes can combat quantum deletions. We note that any permutation-invariant quantum code that has a distance of$t+1$can correct$t$quantum deletions for any positive integer$t$in both the qubit and the qudit setting. Leveraging on coding properties of permutation-invariant quantum codes under erasure errors, we derive corresponding coding bounds for permutation-invariant quantum codes under quantum deletions. We focus our attention on a specific family of$N$-qubit permutation-invariant quantum codes, which we call shifted gnu codes. The main result of this work is that their encoding and decoding algorithms can be performed in$O(N)$and$O(N^{2})$.

DOI
10.1109/isit45174.2021.9518078

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Quantum deletions, which are harder to correct than erasure errors, occur in many realistic settings. It is therefore pertinent to develop quantum coding schemes for quantum deletion channels. To date, not much is known about which explicit quantum error correction codes can combat quantum deletions. We note that any permutation-invariant quantum code that has a distance oft+1t+1t+1can correcttttquantum deletions for any positive integertttin both the qubit and the qudit setting. Leveraging on coding properties of permutation-invariant quantum codes under erasure errors, we derive corresponding coding bounds for permutation-invariant quantum codes under quantum deletions. We focus our attention on a specific family ofNNN-qubit permutation-invariant quantum codes, which we call shifted gnu codes. The main result of this work is that their encoding and decoding algorithms can be performed inO(N)O(N)O(N)andO(N2)O(N^{2})O(N2).
We investigate Permutation-Invariant (PI) quantum error-correcting codes encoding a logical qudit of dimension dL\mathrm{d}_\mathrm{L}dL​ in PI states using physical qudits of dimension dP\mathrm{d}_\mathrm{P}dP​. We extend the Knill--Laflamme (KL) conditions for d−1d-1d−1 deletion errors from qubits to qudits and investigate numerically both qubit (dL=dP=2\mathrm{d}_\mathrm{L} = \mathrm{d}_\mathrm{P} = 2dL​=dP​=2) and qudit (dL>2\mathrm{d}_\mathrm{L} > 2dL​>2 or dP>2\mathrm{d}_\mathrm{P} > 2dP​>2) PI codes. We analyze the scaling of the block length nnn in terms of the code distance ddd, and compare to existing families of PI codes due to Ouyang, Aydin--Alekseyev--Barg (AAB) and Pollatsek--Ruskai (PR). Our three main findings are: (i) We conjecture that qubit PI codes correcting up to d−1d-1d−1 deletion errors have block length n(d)≥(3d2+1)/4n(d) \geq (3d^2 + 1) / 4n(d)≥(3d2+1)/4, which implies an upper bound d≤12n−3/3d \leq \sqrt{12n-3}/3d≤12n−3​/3 on their code distance, and that PR codes can saturate this bound. (ii) For qudit PI codes encoding a single qudit we numerically observe that increasing dP\mathrm{d}_\mathrm{P}dP​ results in nnn monotonically decreasing and approaching the quantum Singleton bound n(d)≥2d−1n(d) \geq 2d-1n(d)≥2d−1. (iii) We propose a semi-analytic extension of the qubit AAB construction to qudits that finds explicit solutions by solving a linear program. Our results therefore provide key insights into lower bounds on the block length scaling of both qubit and qudit PI codes, and demonstrate the benefit of increased physical local dimension in the context of PI codes.
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We present for the first time a general theory of error correction for permutation invariant (PI) codes. Using representation theory of the symmetric group we construct efficient algorithms that can correct any correctible error on any PI code. These algorithms involve measurements of total angular momentum, quantum Schur transforms or logical state teleportations, and geometric phase gates. For erasure errors, or more generally deletion errors, on certain PI codes, we give a simpler quantum error correction algorithm.
We investigate Permutation-Invariant (PI) quantum error-correcting codes encoding a logical qudit of dimension dL\mathrm{d}_\mathrm{L}dL​ in PI states using physical qudits of dimension dP\mathrm{d}_\mathrm{P}dP​. We extend the Knill--Laflamme (KL) conditions for d−1d-1d−1 deletion errors from qubits to qudits and investigate numerically both qubit (dL=dP=2\mathrm{d}_\mathrm{L} = \mathrm{d}_\mathrm{P} = 2dL​=dP​=2) and qudit (dL>2\mathrm{d}_\mathrm{L} > 2dL​>2 or dP>2\mathrm{d}_\mathrm{P} > 2dP​>2) PI codes. We analyze the scaling of the block length nnn in terms of the code distance ddd, and compare to existing families of PI codes due to Ouyang, Aydin--Alekseyev--Barg (AAB) and Pollatsek--Ruskai (PR). Our three main findings are: (i) We conjecture that qubit PI codes correcting up to d−1d-1d−1 deletion errors have block length n(d)≥(3d2+1)/4n(d) \geq (3d^2 + 1) / 4n(d)≥(3d2+1)/4, which implies an upper bound d≤12n−3/3d \leq \sqrt{12n-3}/3d≤12n−3​/3 on their code distance, and that PR codes can saturate this bound. (ii) For qudit PI codes encoding a single qudit we numerically observe that increasing dP\mathrm{d}_\mathrm{P}dP​ results in nnn monotonically decreasing and approaching the quantum Singleton bound n(d)≥2d−1n(d) \geq 2d-1n(d)≥2d−1. (iii) We propose a semi-analytic extension of the qubit AAB construction to qudits that finds explicit solutions by solving a linear program. Our results therefore provide key insights into lower bounds on the block length scaling of both qubit and qudit PI codes, and demonstrate the benefit of increased physical local dimension in the context of PI codes.
We construct a new family of permutationally invariant codes that correct t Pauli errors for any t≥1 . We also show that codes in the new family correct quantum deletion errors as well as spontaneous decay errors. Our construction contains some of the previously known permutationally invariant quantum codes as particular cases, which also admit transversal gates. In many cases, the codes in the new family are shorter than the best previously known explicit permutationally invariant codes for Pauli errors and deletions. Furthermore, our new code family includes a new ((4,2,2)) optimal single-deletion-correcting code. As a separate result, we generalize the conditions for permutationally invariant codes to correct t Pauli errors from the previously known results for t=1 to any number of errors. For small t , these conditions can be used to construct new examples of codes by computer.