Asymmetric Quantum Concatenated and Tensor Product Codes with Large $Z$-Distances

Quantum noise due to decoherence widely exists in quantum communication channels, quantum gates and quantum measurement. It is one of the biggest challenges in realizing large-scale quantum communication systems and fully fault-tolerant quantum computation. For a quantum state, the two main mechanisms of decoherence are population relaxation and dephasing. The level of noise is usually characterized by the relaxation time $\mathcal{T}_{1}$ and the dephasing time $\mathcal{T}_{2}$. Further, dephaisng usually generates a single phase flip error, while population relaxation generates a mixed bit-phase flip error. It is shown in almost all quantum systems that, the dephasing rate $1/\mathcal{T}_{2}$ is much faster than the relaxation rate $1/\mathcal{T}_{1}$, i.e., $\mathcal{T}_{1}\gg\mathcal{T}_{2}$ [1, 2]. For example, in the trapped ions [1, 3], the ratio $\mathcal{T}_{1}/\mathcal{T}_{2}$ can be larger than $10^{2}$ and, in quantum dots systems [4], it can be larger than $10^{4}$. Such large asymmetry between population relaxation and dephasing indicates that phase flip errors ($Z$-errors) happen much more frequently than bit flip errors ($X$-errors).

Steane first saw that prior knowledge of this asymmetry in errors could be leveraged for performance gains and, hence, proposed asymmetric quantum codes (AQCs) in [5]. In the years since, many AQCs have been designed to have a biased error correction towards $Z$-errors [1, 6, 7, 8]. For example, AQCs constructed from classical Bose-Chaudhuri-Hocquenghem (BCH) codes [9] and low-density parity-check (LDPC) codes [10, 11, 12, 13, 14] were proposed in [1, 7]. The BCH codes are used to correct $X$-errors and the more powerful LDPC codes are used to correct $Z$-errors. Another approach, devised by Galindo et al. [15], is to introduce some preshared entanglement [16, 17, 18, 19, 20, 21] to help construct AQCs. More recently, asymmetric errors have been explored as a way to help improve the fault-tolerant thresholds [6, 8], particularly, in topological quantum codes [2, 22, 23]. In [8], a family of asymmetric Bacon-Shor (ABS) codes with parameters $[[mn,1,m/n]]$, where $m$ and $n$ are positive integers, is used for fault-tolerant quantum computation against highly biased noise. For example, ABS codes with parameters $[[175,1,25/7]]$ and $[[315,1,35/9]]$ can achieve a very low logical error rate around $10^{-12}$ with much fewer physical two-qubit gates than symmetric quantum codes. In [22], surface codes definied on a $d\times d$ square lattice of qubits with $d=12,14,16,18,$ and $20$ have thresholds exceeding $5\%$ when the asymmetry between $Z$-errors and $X$-errors is around $100$. Even more, it is shown recently in [24] that thresholds for surface codes can exceed the zero-rate Shannon bound of Pauli channels when the asymmetry is properly large! These results reveal that the large asymmetry in quantum channels has a significant effect to quantum error correction and needs to be further exploited.

However, although there are many different constructions of AQCs in the literature, only a few are made on binary AQCs with a relatively large $Z$-distance $d_{Z}$. This is because the dual-containing constraint in CSS codes often makes constructing an AQC with a large minimum distance $d_{Z}$ difficult. Aly [25] and Sarvepalli et al. [7] derived families of binary asymmetric quantum Bose-Chaudhuri-Hocquenghemmds (QBCH) codes with minimum distances $d_{X}$ and $d_{Z}$, both upper bounded by the square root of the block length. Li et al. [26] were able to construct a few binary QBCH codes of length $n=2^{m}-1$ with a large minimum distance $d_{Z}$. Ezerman et al. [27] constructed some binary CSS-like AQCs of length $\leq 40$ with best-known parameters by exhaustively searching the database of MAGMA [28]. Additionally, several families of nonbinary AQCs with a large $d_{Z}$ have been developed, but all have a large field size [29, 30, 31].

The key to construct an AQC is to find two classical linear codes that satisfy a certain dual-containing relationship. In classical codes, the two most useful combining methods for constructing linear codes from short constituent codes are: concatenated codes (CCs) [32] and tensor product codes (TPCs) [33, 34]. In general, CCs have a large minimum distance because the distances in the constituent codes are multiplied, while TPCs have a poor minimum distance but a better dimension as a trade-off. In [35], Maucher et al. show that generalized concatenated codes (GCCs) are equivalent to generalized tensor product codes (GTPCs).

It is not difficult to apply the concatenation method to the quantum realm, i.e., to construct concatenated quantum codes (CQCs) [36, 37] and quantum tensor product codes (QTPCs) [38, 39], including asymmetric QTPCs [40] and entanglement-assisted QTPCs [41]. CQCs and QTPCs also exhibit some similar characteristics to their classical counterparts. For example, CQCs have a large minimum distance but a relatively small dimension, which is seeing them play an important role in fault-tolerant quantum computation. And, like TPCs, QTPCs have a large dimension but a small minimum distance. However, it is worth noting that CQCs are not constructed from classical CCs directly, but rather by serially concatenating two constituent quantum codes. This means both the inner and outer constituent codes need to satisfy the dual-containing relationship, which limits their construction. The same does not apply to QTPCs, giving them a distinct advantage. But QTPCs usually have a poor minimum distance. Moreover, some CQCs are known to be degenerate codes [37], which is a unique phenomenon in quantum coding theory. Degenerate codes have an advantage in that they can correct more errors than non-degenerate codes, but, in general, they are difficult to decode (see [42]) with the classical decoding algorithms often failing outright.

Hence, in this paper, we propose a novel concatenation scheme called asymmetric quantum concatenated and tensor product codes (AQCTPCs) that combines both CCs and TPCs, where CCs are used to correct $Z$-errors, and TPCs are used to correct $X$-errors. Compared to the current methods, this new concatenation scheme has several advantages.

• 1)

  In AQCTPCs, only the outer constituent codes over the extension field need to satisfy the dual-containing constraint. The inner constituent codes can be any classical linear codes. Then we have much freedom in the choice of the constituent codes.

• 2)

  It is shown that AQCTPCs can be decoded efficiently provided that the classical constituent codes can be decoded efficiently. In addition, AQCTPCs are highly degenerate for correcting $X$-errors and they can correct many more $X$-errors beyond the error correction ability of the corresponding TPCs. Further, we show that the total inner decoding complexity of TPCs is reduced significantly from $\Omega(n_{2}a^{n_{1}})(a>1)$ to $O(n_{2})$ due to error degeneracy. To this end, we have developed a syndrome-based decoding algorithm specifically for AQCTPCs.

• 3)

  The AQCTPCs demonstrated in this paper are better than QBCH codes or asymmetric quantum algebraic geometry (QAG) codes as the block length goes to infinity. We construct a family of AQCTPCs with a very large $Z$-distance $d_{Z}$, of approximately half the block length. Meanwhile, the dimension and the $X$-distance $d_{X}$ continue increasing as the block length goes to infinity. If $d_{X}=2$, then the $Z$-distance $d_{Z}$ is larger than half the block length.

We compare the parameters of AQCTPCs to previous results, and provide a generalized AQCTPC concatenation scheme that uses GCCs and GTPCs. We list AQCTPCs with better parameters than the binary extension of asymmetric quantum Reed-Solomon (QRS) codes. We derive families of AQCTPCs with the largest $Z$-distance $d_{Z}$ compared to existed AQCs with comparable block length and $X$-distance $d_{X}$.

The rest of this paper is organized as follows. In Section II, we provide the basic notations and definitions needed for the construction of AQCTPCs. In Section III, we present the AQCTPC concatenation scheme and the decoding algorithms. Section IV provides detailed performance comparisons of AQCTPCs against previous constructions, and the discussions and conclusions follow in Section V.

In this section we first review some basic definitions and known results about stabilizer codes and AQCs, followed by the introduction of classical CCs and TPCs and their generalizations.

In this section, we present the AQCTPC concatenation framework, where CCs are used to correct $Z$-errors and TPCs are used to correct $X$-errors. In our construction, the dimension of the inner codes of CCs needs to be equal to the number of parity checks of the inner codes of TPCs. Let $C_{1}=[n_{1},k_{1},d_{1}]_{q}$ denote an arbitrary $q$-ary linear code and $C_{2}=[n_{2},k_{2},d_{2}]_{q^{k_{1}}}$ and $C_{3}=[n_{2},k_{3},d_{3}]_{q^{k_{1}}}$ denote two linear codes over the extension field $\mathbb{F}_{q^{k_{1}}}$. Let $\mathcal{C}_{C}=C_{3}\otimes_{C}C_{1}$ be the CC of $C_{1}$ and $C_{3}$, and let $\mathcal{C}_{T}=C_{2}\otimes_{T}C_{1}^{\bot}$ be the TPC of $C_{1}^{\bot}$ and $C_{2}$. Then we have the following dual-containing relationship between CCs and TPCs.

By combining $\mathcal{C}_{C}=C_{3}\otimes_{C}C_{1}$ and $\mathcal{C}_{T}=C_{2}\otimes_{T}C_{1}^{\bot}$, we have the construction of AQCTPCs as follows.

The AQCTPC concatenation scheme has several advantages over the current methods. First, only the outer constituent codes $C_{2}$ and $C_{3}$ over the extension field need to satisfy the dual-containing constraint. Then we have much freedom in the choice of the outer codes. It is generally believed that certain families of linear codes over the extension field can easily satisfy the dual-containing constraint. For example, the dual-containing relationship of $\ell$-ary MDS codes has been determined for all possible dimensions and block length less than $\ell+2$, see e.g., in [62, 63, 64], where $\ell$ is a power of a prime. We can let $C_{2}$ and $C_{3}$ be two MDS codes that satisfy the dual-containing constraint and with reasonable block length. Second, in the following proof we show that AQCTPCs are highly degenerate in that they can correct more $X$-errors than a corresponding classical TPC.

In the proof of Theorem III.1, we have given the decoding of AQCTPCs for correcting $X$-errors. We summarize and provide the whole decoding process in Algorithm 1.

On the other hand, like the serial decoding of classical CCs, the decoding of $Z$-errors in AQCTPCs can also be done serially, i.e., an inner decoding followed by an outer decoding. However, the decoding algorithm for classical CCs can not be used to decode $Z$-errors directly. Instead, a modified version of syndrome-based decoding is needed, as explained next.

Before performing the decoding, the ancilla needs to be measured first to determine the syndrome information. Denote the encoded quantum basis states of the AQCTPC $\mathcal{Q}$ by

where $u\in\mathcal{C}_{T}$. Suppose that a $Z$-error $e_{Z}$ happens to the encoded sate (49), then

First we take the syndrome measurement using the inner parity check matrix $H_{c_{1}}$ to get the inner syndrome information

where $w=(w_{1},\ldots,w_{n_{2}})$ and $e_{Z}=(e_{Z_{1}},\ldots,e_{Z_{n_{2}}})$. The inner decodings are done first according to the inner syndrome information $\Psi_{\textrm{I}_{i}}(1\leq i\leq n_{2})$. They result in $n_{2}$ decoded error sequences $\overline{e}_{Z_{i}}(1\leq i\leq n_{2})$, each of length $n_{1}$. Denote by $\overline{e}_{Z}=(\overline{e}_{Z_{1}},\ldots,\overline{e}_{Z_{n_{2}}})$. We add the decoded result $\overline{e}_{Z}$ to (50), and then perform the measurement using the parity check matrix $[H_{c_{3}}]\otimes(I_{k_{1}},\textbf{0})$ to get the outer syndrome information

Discarding the zero part in $\Psi_{\textrm{O}}$ due to the 0 sub-block in $(I_{k_{1}},\textbf{0})$, the punctured $\Psi_{\textrm{O}}$ is then mapped into a sequence $\overline{\Psi}_{\textrm{O}}$ with elements over field $\mathbb{F}_{q^{k_{1}}}$.

The outer decoding is done with a syndrome-based decoding of $C_{3}$ according to the outer syndrome information $\overline{\Psi}_{\textrm{O}}$. If the outer decoding is successful, we can obtain a decoded sequence $\mathbf{e}^{\prime}_{Z}=(\mathbf{e}^{\prime}_{Z_{1}},\ldots,\mathbf{e}^{\prime}_{Z_{n_{2}}})$ with elements over $\mathbb{F}_{q^{k_{1}}}$. Then we map the sequence $\mathbf{e}^{\prime}_{Z}$ back to the basis field $\mathbb{F}_{q}$, and derive a decoded error sequence $e^{\prime}_{Z}=(e^{\prime}_{Z_{1}},\ldots,e^{\prime}_{Z_{n_{2}}})$ with elements over $\mathbb{F}_{q}$, where $e^{\prime}_{Z_{i}}(1\leq i\leq n_{2})$ is the sub-sequence of length $k_{1}$. But notice that $e^{\prime}_{Z}$ is incomplete due to the 0 sub-block in $(I_{k_{1}},\textbf{0})$. In order to derive the fully-decoded error sequence, we only need to do some operations according to the inner syndrome information in (51). Denote by $\widetilde{e}_{Z}=(\widetilde{e}_{Z_{1}},\ldots,\widetilde{e}_{Z_{n_{2}}})$ and $\widetilde{e}_{Z_{i}}=(e^{\prime}_{Z_{i}},f_{Z_{i}})(1\leq i\leq n_{2})$, where $f_{Z_{i}}$ denotes the unknown errors in $\widetilde{e}_{Z_{i}}$ and is of length $r_{1}$. Suppose that $H_{c_{1}}=(P_{1},P_{2})$, where $P_{1}$ is of size $r_{1}\times k_{1}$ and $P_{2}$ is an invertible $r_{1}\times r_{1}$ matrix, then we have $\Psi_{\textrm{I}_{i}}=H_{c_{1}}\widetilde{e}_{Z_{i}}^{T}=P_{1}e^{\prime T}_{Z_{i}}+P_{2}f_{Z_{i}}^{T}$ and then $f_{Z_{i}}=[P_{2}^{-1}(\Psi_{\textrm{I}_{i}}-P_{1}e^{\prime T}_{Z_{i}})]^{T}$, where $1\leq i\leq n_{2}$.

Similar to classical CCs, no matter how many $Z$-errors happen in each sub-block of length $n_{1}$, the outer decoding will not be affected provided that the total number of erroneous sub-blocks does not exceed the error correction ability of the outer code $C_{3}$. A summary of the full decoding process is provided in Algorithm 2. A complexity analysis of the whole decoding process follows.

In terms of decoding $X$-errors with the TPC, we first need to map the outer decoding sequence from $\mathbb{F}_{q^{k_{1}}}$ to $\mathbb{F}_{q}$ whose running time complexity is $O(n_{2})$ (see Algorithm 1, lines $11$-$15$). And it is easy to see that the complexity of the inner syndrome decoding of $C_{1}^{\bot}=[n_{1},r_{1}]$ is $O(1)$ since we just need to do $\widetilde{e}_{X_{l}}=(\Phi_{\textrm{I}_{l}},\textbf{0})$, for $1\leq l\leq n_{2}$. Therefore, the inner decoding complexity (IDC) of the TPC is $O(n_{2})$. Recall that the IDC of classical TPCs is $\Omega(n_{2}a^{n_{1}})(a>1)$ by using the maximum likelihood (ML) decoding in general, which is enormous if $n_{1}$ is large. Even though the inner codes can be efficiently decoded, see, e.g., [52, 34], the IDC of TPCs is still $\Omega(n_{2}n_{1}^{b})(b>0)$. Here, in quantum cases, we consider error degeneracy in the inner decoding and significantly reduce the IDC of TPCs to $O(n_{2})$ in general.