IEEE Transactions on Quantum Engineering

Encoding of Nonbinary Entanglement-Unassisted and Assisted Stabilizer Codes

Priya J. Nadkarni, Shayan Srinivasa Garani

Published Jan 1, 2021• 13 citations

Abstract

Quantum coding schemes over qudits using preshared entanglement between the encoder and decoder can provide better error correction capability than without it. In this article, we develop procedures for constructing encoding operators for entanglement-unassisted and entanglement-assisted qudit stabilizer codes over Fpk, with p prime and k ≥ 1 from first principles, generalizing prior works on qubit-based codes and codes that work over Fp. We also provide quantum encoding architectures based on the proposed encoding procedures using one and two qudit gates, useful toward realizing coded quantum computing and communication systems using qudits.

DOI: 10.1109/tqe.2021.3050848

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IEEE Transactions on Quantum Engineering

Encoding of Nonbinary Entanglement-Unassisted and Assisted Stabilizer Codes

Priya J. Nadkarni, Shayan Srinivasa Garani

Published Jan 1, 2021• 13 citations

Publisher

Abstract

DOI: 10.1109/tqe.2021.3050848

Related papers

Quantum error correction architecture for qudit stabilizer codes

Priya J. Nadkarni, Shayan Srinivasa Garani

Quantum communication channels benefit from nonbinary entanglement-assisted stabilizer codes using preshared entangled states for achieving better error correction capability compared to those that do not use preshared entanglement, making them indispensable for realizing large-scale quantum computing and communication systems over qudits. We provide a previously unreported design architecture of the syndrome computation unit for qudit stabilizer codes based on classical additive codes using discrete Fourier transform gates, $\mathrm{ADD}$ gates, and multiplication gates. The proposed syndrome computation circuit architectures are necessary toward the implementable realization of such entanglement-assisted and -unassisted qudit stabilizer codes within the quantum transceiver system. We further provide an equivalent design architecture of the syndrome computation unit that decomposes into two syndrome computation units based on X errors and Z errors separately for entanglement-assisted and -unassisted qudit CSS codes. The proposed quantum error correction architectures are useful for building high-density coded quantum memories for archival quantum storage.Quantum communication channels benefit from nonbinary entanglement-assisted stabilizer codes using preshared entangled states for achieving better error correction capability compared to those that do not use preshared entanglement, making them indispensable for realizing large-scale quantum computing and communication systems over qudits. We provide a previously unreported design architecture of the syndrome computation unit for qudit stabilizer codes based on classical additive codes using discrete Fourier transform gates, $\mathrm{ADD}$ gates, and multiplication gates. The proposed syndrome computation circuit architectures are necessary toward the implementable realization of such entanglement-assisted and -unassisted qudit stabilizer codes within the quantum transceiver system. We further provide an equivalent design architecture of the syndrome computation unit that decomposes into two syndrome computation units based on X errors and Z errors separately for entanglement-assisted and -unassisted qudit CSS codes. The proposed quantum error correction architectures are useful for building high-density coded quantum memories for archival quantum storage.

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Quantum error-correcting codes are essential for reliable quantum communication. The achievable parameters can be improved by using entanglement-assisted stabilizers at the cost of consuming pre-shared Bell pairs. If the availability of such Bell pairs is intermittent, the ability to adapt the code to available resources is beneficial. That is, to pick a single mother code that can be adapted to different entanglement budgets. To achieve this, we extend an existing puncturing construction for standard/unassisted stabilizers to the entanglement-assisted setting through a three-step procedure: Interpreting the code as an unassisted stabilizer, puncturing a receiver-side qubit, and recovering an entanglement-assisted representation. This procedure reduces the number of required Bell pairs by one while preserving the number of transmitted and logical qubits. We derive a bound on the resulting distance loss and establish conditions for preserving the original code distance. Finally, we analyse distinct, randomly generated, entanglement-assisted stabilizers with five to eight transmitted qubits. Of these codes, 14.75% have a greater distance than their associated unassisted representations. Among this subset, 81.4% admit at least one puncturing choice that preserves the original entanglement-assisted distance. These results illustrate how puncturing can reduce entanglement requirements while maintaining code distance.Quantum error-correcting codes are essential for reliable quantum communication. The achievable parameters can be improved by using entanglement-assisted stabilizers at the cost of consuming pre-shared Bell pairs. If the availability of such Bell pairs is intermittent, the ability to adapt the code to available resources is beneficial. That is, to pick a single mother code that can be adapted to different entanglement budgets. To achieve this, we extend an existing puncturing construction for standard/unassisted stabilizers to the entanglement-assisted setting through a three-step procedure: Interpreting the code as an unassisted stabilizer, puncturing a receiver-side qubit, and recovering an entanglement-assisted representation. This procedure reduces the number of required Bell pairs by one while preserving the number of transmitted and logical qubits. We derive a bound on the resulting distance loss and establish conditions for preserving the original code distance. Finally, we analyse distinct, randomly generated, entanglement-assisted stabilizers with five to eight transmitted qubits. Of these codes, 14.75% have a greater distance than their associated unassisted representations. Among this subset, 81.4% admit at least one puncturing choice that preserves the original entanglement-assisted distance. These results illustrate how puncturing can reduce entanglement requirements while maintaining code distance.

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Encoder Circuit Optimization for Nonbinary Quantum Error Correction Codes in Prime Dimensions: An Algorithmic Framework

Aditya Sodhani, Keshab K. Parhi

Quantum computers are a revolutionary class of computational platforms with applications in combinatorial and global optimization, machine learning, and other domains involving computationally hard problems. While these machines typically operate on qubits—quantum information elements that can occupy superpositions of the basis $|0\rangle$ and $|1\rangle$ states—recent advances have demonstrated the practical implementation of higher-dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher-order states are less stable and thus remain coherent for a shorter duration than the basis $|0\rangle$ and $|1\rangle$ states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (d-level quantum systems). Previous works have typically established generating gate sets for higher-dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13–44% reduction in encoder circuit gate count for the qutrit (d = 3) $[[9,5,3]]_{3}$ , $[[5,1,3]]_{3}$ , and $[[7,1,3]]_{3}$ codes, and a 9–21% reduction for the ququint (d = 5) $[[10,6,3]]_{5}$ code when compared to prior work. We also achieved circuit depth reductions up to 42%.Quantum computers are a revolutionary class of computational platforms with applications in combinatorial and global optimization, machine learning, and other domains involving computationally hard problems. While these machines typically operate on qubits—quantum information elements that can occupy superpositions of the basis$|0\rangle$and$|1\rangle$states—recent advances have demonstrated the practical implementation of higher-dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher-order states are less stable and thus remain coherent for a shorter duration than the basis$|0\rangle$and$|1\rangle$states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (d-level quantum systems). Previous works have typically established generating gate sets for higher-dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13–44% reduction in encoder circuit gate count for the qutrit (d = 3)$[[9,5,3]]_{3}$,$[[5,1,3]]_{3}$, and$[[7,1,3]]_{3}$codes, and a 9–21% reduction for the ququint (d = 5)$[[10,6,3]]_{5}$code when compared to prior work. We also achieved circuit depth reductions up to 42%.

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Entanglement-assisted fault-tolerant encoding and decoding for qudit stabilizer codes

Abhi Kumar Sharma, Shayan Srinivasa Garani

We propose fault-tolerant quantum encoders for Calderbank-Shor-Steane (CSS) and non-CSS codes over qudits by utilizing $g$ -partite maximally entangled states. The qudits are grouped into $g$ contiguous blocks such that entangled states are shared across the blocks, and multiqudit gates are applied locally to restrict the error propagation within each block. We also propose a transversal fault-tolerant block decoder to decode the blocks individually and recover the complete encoded quantum information. The optimum value of $g$ to obtain an improvement in the probability of error propagation of the proposed encoders against non-fault-tolerant schemes is analytically determined for application in the encoding circuit. The proposed fault-tolerant scheme with entanglement assistance scales linearly with the number of $g$ -partite entangled states and overcomes the limitations of diminishing coding rates for higher levels of concatenation within the concatenated coding scheme. We validate the proposed ideas by constructing fault-tolerant encoding circuits for the non-CSS ${[[5,1,3]]}_{3}$ code and the ${[[7,1,3]]}_{3}$ CSS code through illustrative examples. We reduced the probability of error propagation from 0.248 to 0.212 for the ${[[5,1,3]]}_{3}$ code and from 0.207 to 0.174 for the ${[[7,1,3]]}_{3}$ code. For example, over ${[[100,70]]}_{2}$ quantum code without any fault tolerance, using three-partite entangled pairs, one can get two orders of improvement (1e-2) in the probability of error propagation compared to the non-fault-tolerant encoder by encoding over 190 physical qubits with the same number of stabilizers.We propose fault-tolerant quantum encoders for Calderbank-Shor-Steane (CSS) and non-CSS codes over qudits by utilizing $g$-partite maximally entangled states. The qudits are grouped into $g$ contiguous blocks such that entangled states are shared across the blocks, and multiqudit gates are applied locally to restrict the error propagation within each block. We also propose a transversal fault-tolerant block decoder to decode the blocks individually and recover the complete encoded quantum information. The optimum value of $g$ to obtain an improvement in the probability of error propagation of the proposed encoders against non-fault-tolerant schemes is analytically determined for application in the encoding circuit. The proposed fault-tolerant scheme with entanglement assistance scales linearly with the number of $g$-partite entangled states and overcomes the limitations of diminishing coding rates for higher levels of concatenation within the concatenated coding scheme. We validate the proposed ideas by constructing fault-tolerant encoding circuits for the non-CSS ${[[5,1,3]]}_{3}$ code and the ${[[7,1,3]]}_{3}$ CSS code through illustrative examples. We reduced the probability of error propagation from 0.248 to 0.212 for the ${[[5,1,3]]}_{3}$ code and from 0.207 to 0.174 for the ${[[7,1,3]]}_{3}$ code. For example, over ${[[100,70]]}_{2}$ quantum code without any fault tolerance, using three-partite entangled pairs, one can get two orders of improvement (1e-2) in the probability of error propagation compared to the non-fault-tolerant encoder by encoding over 190 physical qubits with the same number of stabilizers.

Physical Review ADec 31, 20242 citations

\mathrm{ADD}

gates, and multiplication gates. The proposed syndrome computation circuit architectures are necessary toward the implementable realization of such entanglement-assisted and -unassisted qudit stabilizer codes within the quantum transceiver system. We further provide an equivalent design architecture of the syndrome computation unit that decomposes into two syndrome computation units based on X errors and Z errors separately for entanglement-assisted and -unassisted qudit CSS codes. The proposed quantum error correction architectures are useful for building high-density coded quantum memories for archival quantum storage.

We generalize the stabilizer formalism for entanglement-assisted quantum error-correcting codes with noisy ebits (EAQECCs-Ne) from the binary case to the general

q

-ary case, where

q

is a prime power. By leveraging the structure of the generalized Pauli group over

\mathbb{F}_q

and symplectic geometry over

\mathbb{F}_q^{2n}

, we establish a unified framework for constructing EAQECCs-Ne for qudit systems. Equivalent formulations in terms of symplectic geometry over

\mathbb{F}_q

and additive codes over

\mathbb{F}_q^{2n}

are derived. We further construct several families of

q

-ary EAQECCs with noise ebits and analyze their performance compared to optimal stabilizer codes. Our results demonstrate that under certain noise conditions, the proposed EAQECCs-Ne can outperform standard stabilizer codes with equivalent error-correcting capability, offering a promising approach for fault-tolerant quantum computation in high-dimensional quantum systems.

|0\rangle

and

|1\rangle

states—recent advances have demonstrated the practical implementation of higher-dimensional quantum systems (qudits) across various hardware platforms. In these hardware realizations, the higher-order states are less stable and thus remain coherent for a shorter duration than the basis

|0\rangle

and

|1\rangle

states. Moreover, formal methods for designing efficient encoder circuits for these systems remain underexplored. This limitation motivates the development of efficient circuit techniques for qudit systems (d-level quantum systems). Previous works have typically established generating gate sets for higher-dimensional codes by generalizing the methods used for qubits. In this work, we introduce a systematic framework for optimizing encoder circuits for prime-dimension stabilizer codes. This framework is based on novel generating gate sets whose elements map directly to efficient Clifford gate sequences. We demonstrate the effectiveness of this method on key codes, achieving a 13–44% reduction in encoder circuit gate count for the qutrit (d = 3)

[[9,5,3]]_{3}

[[5,1,3]]_{3}

, and

[[7,1,3]]_{3}

codes, and a 9–21% reduction for the ququint (d = 5)

[[10,6,3]]_{5}

code when compared to prior work. We also achieved circuit depth reductions up to 42%.

We propose fault-tolerant quantum encoders for Calderbank-Shor-Steane (CSS) and non-CSS codes over qudits by utilizing

g

-partite maximally entangled states. The qudits are grouped into

g

contiguous blocks such that entangled states are shared across the blocks, and multiqudit gates are applied locally to restrict the error propagation within each block. We also propose a transversal fault-tolerant block decoder to decode the blocks individually and recover the complete encoded quantum information. The optimum value of

g

to obtain an improvement in the probability of error propagation of the proposed encoders against non-fault-tolerant schemes is analytically determined for application in the encoding circuit. The proposed fault-tolerant scheme with entanglement assistance scales linearly with the number of

g

-partite entangled states and overcomes the limitations of diminishing coding rates for higher levels of concatenation within the concatenated coding scheme. We validate the proposed ideas by constructing fault-tolerant encoding circuits for the non-CSS

{[[5,1,3]]}_{3}

code and the

{[[7,1,3]]}_{3}

CSS code through illustrative examples. We reduced the probability of error propagation from 0.248 to 0.212 for the

{[[5,1,3]]}_{3}

code and from 0.207 to 0.174 for the

{[[7,1,3]]}_{3}

code. For example, over

{[[100,70]]}_{2}

quantum code without any fault tolerance, using three-partite entangled pairs, one can get two orders of improvement (1e-2) in the probability of error propagation compared to the non-fault-tolerant encoder by encoding over 190 physical qubits with the same number of stabilizers.