Quantum error correction architecture for qudit stabilizer codes

Priya J. Nadkarni and Shayan Srinivasa Garani

Phys. Rev. A 103, 042420 – Published 22 April 2021

Abstract

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, $ADD$ 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.

Received 20 October 2020
Revised 9 March 2021
Accepted 7 April 2021

DOI:https://doi.org/10.1103/PhysRevA.103.042420

Physics Subject Headings (PhySH)

Quantum error correction

Quantum Information, Science & Technology

Authors & Affiliations

Priya J. Nadkarni and Shayan Srinivasa Garani ^*

Department of Electronic Systems Engineering, Indian Institute of Science, Bengaluru, 560012 India

^*shayangs@iisc.ac.in

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Vol. 103, Iss. 4 — April 2021

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Images

Figure 1
Information flow in coded quantum systems: The maximally entangled state $| η 〉$ is generated by performing ${DFT}_{p}^{- 1}$ and ${ADD}_{p}^{- 1}$ operations on ${| 00 〉}_{p}$ , where ${| 00 〉}_{p}$ are two subqudits implemented with two different sets of qudits. One set of qudits is sent to the transmitter and the other to the receiver, where they are stored in quantum registers. In the transmitter, using encoding operation $E$ , the $n_{m}^{'}$ subqudit unencoded quantum information $| ϕ 〉$ is encoded to the codeword $| ψ 〉$ using $n_{a}^{'}$ ancilla subqudits in state ${| 0 〉}_{p}$ and the transmitter end subqudits of $n_{e}^{'}$ maximally entangled states. The transmitter end qudits of $| ψ 〉$ are transmitted through a quantum communication or storage channel that introduces an error $U$ . Error correction is performed over $(U \otimes I_{p}^{\otimes n_{e}^{'}}) | ψ 〉$ using $n_{s}^{'}$ syndrome subqudits initially in state ${| 0 〉}_{p}$ by computing the syndrome $| s 〉$ using the syndrome computation circuit, followed by error deduction and recovery. The decoding operation $E^{- 1}$ is performed on the first $n$ qudits of the codeword to obtain the estimated unencoded quantum information $| ϕ_{e} 〉$ .
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Figure 2
The syndrome computation operator based on stabilizer $S_{1}$ : (a) The scaled addition operation ${ScAdd}_{p} (l)$ over subqudits is obtained using the multiplication gates $M_{l}$ and $M_{l^{- 1}}$ and the ${ADD}_{p}$ gate, where $l \in F_{p}$ . (b) Using the scaled addition operators over subqudits, a scaled addition operator $ScAdd (κ)$ over a qudit is obtained, where $κ = \sum_{j = 0}^{m - 1} κ_{j} α^{j}$ . For $ScAdd (κ)$ the control subqudit is a single subqudit and the target is either $k$ subqudits or a qudit. (c) Using $DFT$ operations over qudits and subqudits, and the scaled addition operators over qudits with the syndrome subqudit as the control and the codeword qudits as the target, the syndrome computation is performed. Post syndrome computation, the syndrome qudits are in state $| i 〉$ , where $i = Symp (S_{1}, E)$ .
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Figure 3
Syndrome computation circuit schematic for qudit stabilizer codes: The ${DFT}_{p}$ operations are first performed on the $ρ$ syndrome subqudits, followed by $S_{D}$ on the codeword subqudits along with the syndrome subqudits. Finally, the ${DFT}_{p}^{†}$ operation is performed on the $ρ$ syndrome subqudits to obtain the syndrome $| l_{1} \dots l_{ρ} 〉$ over the syndrome subqudits.
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Figure 4
Equivalent syndrome computation circuit schematic for qudit stabilizer codes. (a) The operator $X_{β}^{'}$ is computed using scaled addition operations from Fig. 2 based on $b_{i} s$ , where $β = \sum_{i = 0}^{m - 1} b_{i} α^{i}$ . (b) The operator $X (β_{y}, y)$ is obtained from a series of $X_{β}^{'}$ operations. $X_{P} (γ_{y}, y)$ can be obtained similarly. (c) The operator $S_{y}$ is obtained from $X (β_{y}, y)$ , $X_{P} (γ_{y}, y)$ and ${DFT}_{p}$ operations. (d) The syndrome computation circuit involves a series of $S_{y}$ operations, where $y \in {1, \dots, ρ}$ , on the erroneous state $E | ψ 〉$ along with the syndrome subqudits in state ${| 0 〉}_{p}$ each to obtain $E | ψ 〉$ and the syndrome state $| l_{1} l_{2} \dots l_{ρ} 〉$ .
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Figure 5
Equivalent syndrome computation circuit schematic for CSS codes over qudits: The syndrome computation circuit involves two parts: (a) Syndrome computation operator for the $X^{(p^{m})}$ errors using a series of $X_{P} (γ_{y}, y)$ operations, where $y \in {m ρ_{1} + 1, \dots, ρ}$ ; and (b) Syndrome computation operator for the $Z^{(p^{m})}$ errors using ${DFT}_{p}$ operations, a series of $X (β_{y}, y)$ operations, where $y \in {1, \dots, m ρ_{1}}$ , and ${DFT}_{p}^{†}$ operations. After syndrome computation, the syndrome subqudits are in syndrome state $| s 〉$ , while the codeword subqudits remain in state $E | ψ 〉$ .
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