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arXiv:1302.3247 · Physical Review A

Permutation-invariant quantum codes

Yingkai Ouyang

Published Dec 10, 2014• 91 citations
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Abstract

A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system. When the physical model is the Heisenberg ferromagnet in the absence of an external magnetic field, the corresponding ground space contains all permutation-invariant states. We use techniques from combinatorics and operator theory to construct families of permutation-invariant quantum codes. These codes have length proportional to ${t}^{2}$; one family of codes perfectly corrects arbitrary weight $t$ errors, while the other family of codes approximately correct $t$ spontaneous decay errors. The analysis of our codes' performance with respect to spontaneous decay errors utilizes elementary matrix analysis, where we revisit and extend the quantum error correction criterion of Knill and Laflamme, and Leung, Chuang, Nielsen and Yamamoto.

DOI
10.1103/physreva.90.062317

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A quantum code is a subspace of a Hilbert space of a physical system chosen to be correctable against a given class of errors, where information can be encoded. Ideally, the quantum code lies within the ground space of the physical system. When the physical model is the Heisenberg ferromagnet in the absence of an external magnetic field, the corresponding ground space contains all permutation-invariant states. We use techniques from combinatorics and operator theory to construct families of permutation-invariant quantum codes. These codes have length proportional to t2{t}^{2}t2; one family of codes perfectly corrects arbitrary weight ttt errors, while the other family of codes approximately correct ttt spontaneous decay errors. The analysis of our codes' performance with respect to spontaneous decay errors utilizes elementary matrix analysis, where we revisit and extend the quantum error correction criterion of Knill and Laflamme, and Leung, Chuang, Nielsen and Yamamoto.
A permutation-invariant code on mmm qubits is a subspace of the symmetric subspace of the mmm qubits. We derive permutation-invariant codes that can encode an increasing amount of quantum information while suppressing leading-order spontaneous decay errors. To prove the result, we use elementary number theory with prior theory on permutation-invariant codes and quantum error correction.
We construct a new family of permutationally in-variant codes that correcttttPauli errors for anyt⩾1t\geqslant 1t⩾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. 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. This is an extended abstract of the preprint [1].
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.
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 introduce a generalisation of quantum error correction, relaxing the requirement that a code should identify and correct a set of physical errors on the Hilbert space of a quantum computer exactly, instead allowing recovery up to a pre-specified admissible set of errors on the code space. We call these quantum error transmuting codes. They are of particular interest for the simulation of noisy quantum systems, and for use in algorithms inherently robust to errors of a particular character. Necessary and sufficient algebraic conditions on the set of physical and admissible errors for error transmutation are derived, generalising the Knill-Laflamme quantum error correction conditions. We demonstrate how some existing codes, including fermionic encodings, have error transmuting properties to interesting classes of admissible errors. Additionally, we report on the existence of some new codes, including low-qubit and translation invariant examples.
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.