Near-Hashing-Bound Multiple-Rate Quantum Turbo Short-Block Codes

Near-Hashing-Bound Multiple-Rate Quantum Turbo Short-Block Codes

Daryus Chandra, Zunaira Babar, Soon Xin Ng, and Lajos Hanzo The authors are with the School of Electronics and Computer Science, University of Southampton, Southampton, SO17 1BJ, UK (email: {dc2n14, zb2g10, sxn, lh}@ecs.soton.ac.uk).The financial support of the EPSRC under the grant EP/L018659/1 and the COALESCE project, that of the European Research Council, Advanced Fellow Grant QuantCom and that of the Royal Society’s Global Research Challenges Fund (GRCF) is gratefully acknowledged. Additionally, the authors acknowledge the use of the IRIDIS High Performance Computing Facility, and associated support services at the University of Southampton, in the completion of this work.
Abstract

Quantum stabilizer codes (QSCs) suffer from a low quantum coding rate, since they have to recover the quantum bits (qubits) in the face of both bit-flip and phase-flip errors. In this treatise, we conceive a low-complexity concatenated quantum turbo code (QTC) design exhibiting a high quantum coding rate. The high quantum coding rate is achieved by combining the quantum-domain version of short-block codes (SBCs) also known as single parity check (SPC) codes as the outer codes and quantum unity-rate codes (QURCs) as the inner codes. Despite its design simplicity, the proposed QTC yields a near-hashing-bound error correction performance. For instance, compared to the best half-rate QTC known in the literature, namely the QIrCC-QURC scheme, which operates at the distance of from the quantum hashing bound, our novel QSBC-QURC scheme can operate at the distance of . It is worth also mentioning that this is the first instantiation of QTCs capable of adjusting the quantum encoders according to the quantum coding rate required for mitigating the Pauli errors given the different depolarizing probabilities of the quantum channel.

quantum error detection codes, quantum error correction codes, concatenated codes, iterative decoding, topological codes, quantum turbo codes, short-block codes.
List of Acronyms
CNOT Controlled-NOT
CSS Calderbank-Shor-Steane
EXIT EXtrinsic Information Transfer
PCM Parity Check Matrix
QBCH Quantum Bose-Chaudhuri-Hocquenghem
QBER QuBit Error Ratio
QCC Quantum Convolutional Code
QEDC Quantum Error Detection Code
QECC Quantum Error Correction Code
QIrCC Quantum Irregular Convolutional Code
QSC Quantum Stabilizer Code
QSBC Quantum Short-Block Code
QTECC Quantum Topological Error Correction Code
QTC Quantum Turbo Code
QURC Quantum Unity-Rate Code
SBC Short-Block Code
SPC Single Parity Check

I Introduction

Quantum stabilizer codes (QSCs) [1, 2, 3, 4, 5, 6, 7] are capable of estimating both the number and the position of quantum bit (qubit) errors without collapsing the quantum state of physical qubits into their classical state. Hence, they can be viewed as the syndrome-based quantum error correction codes (QECCs). However, the QSCs suffer from a lower quantum coding rate than their classical counterparts, since they have to tackle not only the bit-flip errors but also the phase-flip errors [8]. Motivated by [9], where it has been demonstrated that concatenating a quantum linear block code with a unity-rate quantum convolutional code (QCC), which is also referred to as a quantum unity-rate code (QURC), facilitates the soft-decision-aided iterative decoding. Hence, a dramatic performance improvement may be attained without sacrificing the quantum coding rate. However, the proposal of [9] suffered from a relatively high error floor, despite relying on a low overall quantum coding rate of , when using a block code having a minimum distance of as the outer code. Additionally, in order to eliminate the error floor, this specific code construction required a doping mechanism for triggering the convergence of iterative decoding, since the QURC utilized a catastrophic encoder structure. Later in [10], it was shown that by carefully selecting the inner and the outer codes using an extrinsic information transfer (EXIT)-chart-guided code search, a half-rate quantum turbo code (QTC) can be conceived, which is capable of performing relatively close to the quantum hashing bound. Explicitly, this excellent error correction performance was achieved by a half-rate quantum irregular convolutional code (QIrCC) used as the outer code combined with a QURC as the inner code. This specific code is referred to as the QIrCC-QURC scheme.

As an important result in the classical domain, it was shown that an outer code exhibiting a minimum distance of is capable of guaranteeing the convergence of iterative decoding to a vanishingly low bit error ratio (BER) [11, 12], as exemplified by the family of single parity check codes (SPCs) or short-block codes (SBCs) [13, 14]. Furthermore, by exploiting the classical-to-quantum isomorphism [8, 15], we can indeed conceive the quantum version of the classical SBCs, which we referred to as quantum short-block codes (QSBCs). As an additional benefit, the QSBCs can also be viewed as quantum topological error correction codes (QTECCs). More specifically, the QTECC construction exhibits an inherent error detection and error correction capability, when the physical qubits are appropriately arranged on a lattice structure. However, most of the conventional techniques of constructing the QTECCs suffer from a low quantum coding rate as well as from the lack of flexibility, when choosing the number of logical qubits and also the quantum coding rate [15]. By contrast, when a similar approach relying on utilizing a lattice structure is invoked for constructing quantum error detection codes (QEDCs), instead of quantum error correction codes (QECCs), we found that the resultant topological QEDCs are flexible and exhibit high quantum coding rates.

Against this background, by amalgamating QURCs and QSBCs, we conceive the family of high-rate low-complexity serial QTCs, which are capable of operating at various quantum coding rates whilst relying on a flexible numbers of logical qubits. As a further benefit, they are capable of approaching the quantum hashing bound. More explicitly, our main contributions are as follows:

  • We present the general design of high-rate QSBCs exhibiting a minimum distance of , which constitute the family of QEDCs. Explicitly, the proposed QSBC can have a block length as short as four physical qubits () and it has a quantum coding rate of . As an additional benefit, the QSBCs have a scalable encoder structure and require only localized stabilizer measurements. In addition, we demonstrate that the associated stabilizer measurement can be localized by arranging the physical qubits on a polygon structure, such as a square, hexagon, octagon, etc. Hence, this is the first instantiation of high-rate, scalable, short-length, and high-rate QEDCs.

  • We amalgamate the QSBCs with the QURCs [10] for the sake of constructing soft-decision-aided QECCs without sacrificing the quantum coding rate, which results in a low-complexity high-rate QTC design. We refer to the resultant construction as a QSBC-QURC scheme. Despite having low complexity, the QSBC-QURC scheme is capable of operating close to the achievable quantum hashing bound. Quantitatively, for instance, our simulation results demonstrate that a half-rate QSBC-QURC operates at the distance of from the quantum hashing bound.

  • Finally, we conceive the first instantiation of a multi-rate scheme for serial QTCs relying on the flexible scalability of the QSBC-QURC construction. We determine the specific depolarizing probability values at which it is beneficial to switch the quantum coding rate based on the minimum QBER requirement of . Finally, we quantify the achievable goodput of the QSBC-QURC schemes conceived.

The rest of this treatise is organized as follows. In Section II, we present the general formulation of QSBCs in terms of their code construction, encoder, and stabilizer measurement. This is followed by Section III, where we propose on serial QTCs by utilizing QSBCs as the outer codes and a QURC as the inner code, which we refer to as the QSBC-QURC scheme. We analyze the convergence behavior of iterative-decoding-aided QSBC-QURC scheme using EXIT charts and evaluate its QBER and goodput in Section IV. Finally, we conclude the paper and present some possible directions for future research in Section V.

Ii Quantum Short-Block Codes

In this section, we introduce the quantum version of classical SBCs by describing their general construction, the structure of the quantum encoder, as well as the quantum circuit required for the stabilizer measurements. This section will also characterize both the flexibility and scalability of the QSBC encoders and the associated stabilizer measurements, as the natural evolution from their parity check matrix structure. Furthermore, we present a short tutorial on conducting the classical simulation for QSBCs. In order to avoid ambiguity, throughout the rest of this treatise the notation is used to denote the classical error correction code having codeword length of bits, information word bits, and a minimum distance of , while the notation is used for the quantum stabilizer code.

Ii-a Codes Construction

The classical SBCs are systematic linear binary block codes , whose generator matrix is defined by:

(1)

where is a -dimension identity matrix, is a -element binary matrix and is a matrix with all-one -element. Hence, the parity check matrix (PCM) of a systematic linear block code is encapsulated in

(2)

Therefore, the PCM of the classical SBCs is given by

(3)

which is an all-one element matrix. Finally, the resultant coding rate is given by

(4)

The minimum distance of the SBCs conceived is , hence this guarantees the convergence of iterative decoding to a vanishingly low BER, when they constitute the outer code [11, 12].

By exploiting the classical-to-quantum isomorphism [8], this specific type of classical SBCs can be readily transformed into their quantum counterparts. To elaborate a little further, given a pair of classical codes and having PCMs and , respectively, a QSC having a binary PCM can be constructed from and so that will be used for mitigating the bit-flip errors and will be used for mitigating the phase-flip errors, where we have , , and . In general, there are two ways of constructing the binary PCM of a QSC given a pair of PCMs and . Firstly, we may construct a Calderbank-Steane-Shor (CSS) type quantum code, whose binary PCM is as follows [5]:

(5)

Secondly, we may also construct a non-CSS type quantum code, whose binary PCM is given by [6]

(6)

In order to conceive a valid PCM for a QSC , a pair of PCMs and has to satisfy the symplectic criterion, which is defined as [6]

(7)

Therefore, the symplectic criterion formulated for a CSS type quantum code can be reduced to

(8)

A specific case of CSS type quantum codes, where we have , is referred to as a dual-containing CSS quantum code. Therefore, the dual-containing CSS quantum codes can be instantly derived from the classical SBCs having the following PCMs:

(9)

where is an even number. This automatically satisfies the symplectic criterion of Eq. (8). Ultimately, the resultant PCM of QSBCs is given by:

(10)

Hence, for dual-containing CSS quantum codes, the relationship between the classical coding rate and the quantum coding rate can be described as follows [15, 16, 17]

(11)

Since the quantum coding rate has to be positive , the original classical code must exhibit a classical coding rate of .

Fig. 1: QSBCs can be arranged to match the qubit layout on a lattice structure. The blue circles represent the physical qubits, while the red circles denote the stabilizer operators. Each red square is utilized for measuring both and operator. Hence, the resultant quantum codes belong to the family of dual-containing CSS codes. The quantum coding rate for construction (a) is , for (b) is , and for (c) is . All of the QSBCs exhibit a minimum distance of .

For QSCs, the PCM is associated with stabilizer operators . For example, let us consider a classical SBC having a PCM of

(12)

exhibiting a classical coding rate of , which is associated with a QSBC of having a quantum coding rate of . The stabilizer operators of are given by111The shortened representation of stabilizer operators is used for simplifying the original representation of stabilizer operators. For example, the sorthened version of is used for simplifying , where the notation represents a tensor product. For the rest of the paper, we always use the shortened representation of stabilizer operators.

(13)

where and are the Pauli matrices. By exploiting the classical-to-quantum isomorphism, we arrive at the PCM of the dual-containing CSS quantum code formulated as

(14)

Similarly, we can readily extend the construction to a higher quantum coding rate, as exemplified by , which is derived from a classical SBC of . Hence, the stabilizer operators for are defined by

(15)

For this construction, the resultant quantum coding rate is . The same analogy can be used for constructing derived from a classical SBC . Therefore, the stabiizer operators are as follows:

(16)

The resultant QSBC exhibits a quantum coding rate of . The discussion on how the stabilizer operators can be invoked for detecting quantum errors will be discussed in subsection II-C.

Furthermore, the QSBCs can also be classified as a family of QTECCs, as shown in Fig. 1. Assuming that we can arrange the physical qubits on the vertices of a lattice structure, it inherently provides a localized stabilizer measurement property. Since the resultant QSC constructions are only capable of error detection, which is a consequnce of having a minimum distance of , the stabilizer measurements can only indicate the presence or the absence of quantum errors, but not the specific numbers or the position of the errors. However, we will show later that this error detection capability can be transformed into error correction capability by concatenating QSBCs with a carefully selected inner code.

Ii-B Quantum Encoder

In quantum domain, a -logical qubit information word in the state of can be transformed into a -physical qubit codeword in the state of , where , with the aid of auxiliary qubits initialized in the state of . This specific transformation, which is carried out by a unitary transformation referred to as a quantum encoder, can be formally described as follows222The superscript of in notation denotes the number of physical qubits, which is , given a quantum state . This notation will be used throughout this treatise.:

(17)

This process is reminiscent of the encoding process of classical error correction codes. To elaborate, in the classical domain, we can transform a -bit information word into an -bit codeword with the aid of the generator matrix , where the additional of bits are referred to as the redundant bits.

Based on the description of QSBCs in Subsection II-A, they can be classified as a member of the dual-containing CSS code family. For this specific class of quantum codes, the design of the quantum encoder can be readily derived from its classical PCMs [18, 19]. More specifically, the PCM of the classical code may be utilized to obtain the quantum encoder of a QSC , where , , and .

Let us now embark on creating the quantum encoder of dual-containing CSS codes derived from the classical codes . The PCM of the classical code can be represented by a full-rank matrix having elements. Naturally, every PCM of linear block codes can be transformed into the corresponding systematic form of

(18)

by using row operations and column permutations, where is a -dimension identity matrix, and the matrix has elements. For the next step, we may further reduce the matrix into another systematic form of

(19)

where is another -dimension identity matrix and the matrix has elements.

Ultimately, the quantum encoder of a dual-containing CSS code can be described as a two-stage encoder. The first stage of the quantum encoder is used for initializing a set of codewords , which must not have a difference exactly corresponding to a specific legitimate codewords of , where is the dual code of . Therefore, it can be utilized for generating the unique cosets of relative to . Next, the second stage of quantum encoder is invoked for generating the code space of according to the PCM . Hence, the resultant states after the first and the second stage are constituted by the superposition of all the codewords of generated by the second stage added to the codewords of generated in the first stage. The more general method of constructing the quantum encoder for all types of QSCs, including the non-dual-containing CSS codes and non-CSS codes, can be found in [20, 21].

Fig. 2: The quantum encoder can be described as a two-stage encoder. For the QSBCs, the CNOT connections of the first stage are defined by matrix of Eq. (22), while those of the second stage are defined by matrix of Eq. (21).

For gleaning a clearer idea about the two-stage quantum encoder , let us consider the QSBC and construct its quantum encoder based on the classical PCM derived from a classical code as follows:

(20)

Fortunately, the PCM has already a systematic structure, hence we have . From the PCM in Eq. (20), matrix is readily given by

(21)

Consequently, given that , we obtain the matrix as follows:

(22)
(a) The quantum encoder for .
(b) The quantum encoder for .
(c) The quantum encoder for .
Fig. 3: The QSBC encoders for various quantum coding rates illustrating the flexibility and scalability of the code constructions. Figure (a) depicts the quantum encoder of the QSBC . By incorporating additional gates and connections denoted by the dashed blue lines in Fig. (b) into the quantum encoder (a), we obtain the quantum encoder of the QSBC . Similarly, by adding more gates and connections denoted by a dashed-dotted red lines to the quantum encoder (b), we can readily create the quantum encoder of the QSBC . In other words, we can construct the quantum encoder of QSBCs for a high quantum coding rate, which will simultaneously contain the quantum encoder of the lower quantum coding rate. Therefore, it is no longer necessary to build more than one quantum encoder for various quantum coding rate QSBCs because the quantum encoder of QSBCs exhibit a self-contained structure. This is reminiscent of the classical rate-compatible punctured codes proposed in [22].

Based on Eq. (21) and (22), the first stage and the second stage of the quantum encoder of the QSBC is denoted by and , respectively, in Fig. 2. To elaboreate a little further, the CNOT connections between the logical qubits in the state of and the first auxiliary qubits are defined by the matrix . More specifically, given the element of the matrix , if the value of , it means that the -th logical qubit controls the CNOT connection of the -th auxiliary target qubit. As an example, based on the matrix in Eq. (22), we can see that at the first stage of the quantum encoder the first auxiliary qubit is controlled by all four logical qubits in the state of , since the matrix contains all 1 elements.

Consequently, the first stage of the quantum encoder transforms the first auxiliary qubits, which are initialized to the state of , according to the logical qubits . Explicitly, given that , where is a -bit binary string, the first stage of the quantum encoder transforms the auxiliary qubits into the state of . Therefore, the state of physical qubits at the output of the first stage , namely , created by the action of the first stage of quantum encoder of Fig. 2 can be expressed as follows:

(23)

where . Hence, as we have mentioned earlier that the first stage of quantum encoder creates a set of codewords , which must not have a difference exactly corresponding to a specific legitimate codeword in . For example, based on the PCM of a classical code given in Eq. (20), we can construct the code space of the dual code as follows:

(24)

For a given -bit binary string of , the binary string of is indeed a codeword in . Let us denote the set of all the -bit binary string of as , which basically creates a subspace of , i.e. . More explicitly, the code space of based on matrix of Eq. (22) is given as follows:

(25)

It is clear that none of the codeword in differs by one element in , i.e. adding any codeword from the non-zero codeword in to the one of the codeword in does not yield another codeword in , since the last bits of contains all 0 element, while the non-zero codeword of has 1s in the last bits. Hence, each -bit binary string of creates a unique coset in relative to the .

The second stage of the quantum encoder is started by initializing the remaining auxiliary qubits in the state of states by using Hadamard gates, which can be expressed mathematically as follows:

(26)

Explicitly, Eq. (26) basically represents a sum of an equal superposition of all possible states over all binary strings of length . Next, we combine the initialized equal superposition of Eq. (26) with the output of the first stage . Let us assume that string denotes the string of . The effect of this operation is to add rows of PCM to the binary string . Hence, for a given -bit binary input string of , the final state of the physical qubits after the first and the second stage of the quantum encoder can be expressed as

(27)

Finally, if the state of the logical qubits is expressed in the form of the superposition the binary strings as follows:

(28)

then the output state of the physical qubits can be formulated as

(29)

Readers who might be interested in different examples of creating encoders for various dual-containing CSS codes exemplified by Steane’s code of [3] and by the quantum Bose-Chaudhuri-Hocquenghem (QBCH) code of [23], please refer to [18].

The resultant quantum encoders conceived for the QSBCs , , and can be seen in Fig. 3, where it shows that the QSBC encoders exhibit the natural design for flexibility. The inverse encoder is implemented with the aid of the same exact quantum circuit, where the input and output qubits are reversed.

Ii-C Stabilizer Measurement for QSBCs

The QSCs are capable of predicting both the number and the position of errors without actually observing the states of the physical qubits. In order to achieve this, a syndrome-decoding-like method was introduced [1]. Instead of observing the information within the physical qubits, which would collapse the superposition state to a classical state, a set of auxiliary qubits are prepared for observing the syndrome of the physical qubits using the so-called stabilizer operators. A stabilizer operator belonging to the stabilizer group is an -tuple Pauli operator, which stabilizes the state of the encoded physical qubits may be formulated as follows:

(30)
(a) The -stabilizer measurements for QSBCs.
(b) The -stabilizer measurement for QSBCs.
Fig. 4: The stabilizer measurements of QSBCs for various quantum coding rates of QSBCs illustrating the flexibility and scalability of the encoder constructions. Even though the QSBCs can be defined as illustrated in Fig. 1, from implementation perspective the QSBCs can be arranged to be more scalable. For example, all the physical qubits for QSBCs having quantum coding rate of can be arranged on an octagon.

An error operator inflicted by the quantum channel on the encoded state of physical qubits transforms the legitimate state into the contaminated received state , which can be expressed as follows:

(31)

The syndrome values can be obtained by performing eigenvalue-based measurement of the received state assisted by the auxiliary qubits, which can be defined as follows:

(32)

The eigenvalues attained from the stabilizer measurement act similarly to the values of the classical syndrome measurements. Hence, they also can be used for inferring both the number and the position of errors without actually observing the state of the physical qubits.

As we briefly discussed, the eigenvalue measurements require an extra auxiliary qubit for each stabilizer measurement. As for QSBCs, the stabilizer measurement can be implemented using the circuits seen in Fig. 4. More specifically, Fig. 4(a) depicts the -stabilizer measurement, while Fig 4(b) portrays the -stabilizer measurement. It can be clearly observed from Fig. 4 that for both and stabilizer measurements, the circuit constructed for realizing the stabilizer measurements of a QSBC having a higher quantum coding rate inherently contains the stabilizer measurements required for a QSBC having a lower quantum coding rate. To elaborate a little further, in Fig. 4, the circuit implementation of the stabilizer measurements of a -rate QSBC is highlighted using black solid lines. In case we want to employ another QSBC having a higher quantum coding rate, for example a -rate QSBC, we can simply incorporate the stabilizer measurement from the -rate scheme and add further gates, which are highlighted using blue dashed lines in Fig. 4, without changing the stabilizer measurement circuit. A similar approach is applicable when we want to employ a -rate QSBC. We incorporate the stabilizer measurements of the -rate QSBC and then add more gates, which are highlighted using red dashed lines. Ultimately, we have shown that the nature of the PCMs from the QSBCs leads to a very convenient design for their quantum encoders and for their stabilizer measurements, which are capable of supporting multiple quantum coding rates of using a single quantum circuit implementation.

Ii-D Classical Simulation for QSBCs

The quantum encoder of a QSBC and its inverse encoder are composed of quantum Clifford gates. This implies that they can be conveniently simulated using classical computers [24]. An -tuple Pauli operator can be represented by a -element binary vector, where each of the -element binary vectors constitutes a Pauli and a Pauli component. The mapping of the Pauli matrix to the associated binary vector can be formulated as follows:

(33)

The evolution of the Pauli operator over quantum Clifford gates can be described using the conjugation operation. Explicitly, the conjugation of a unitary operator under the unitary transformation is the unitary transformation , which is defined as [24]

(34)

For instance, based on Eq. (34), the conjugation of the Pauli matrix and over Hadamard gate is given by

(35)

Additionally, we can also describe the conjugation of a Pauli matrix and over a two-qubit quantum gate, as exemplified by a CNOT gate, as follows:

(36)

where the first Pauli matrix is applied to the control qubit, while the second Pauli matrix is applied to the target qubit.

Therefore, using the conjugation definition of Eq. (34), we can keep track of the evolution of any -tuple Pauli operator owing to unitary operations carried out by quantum Clifford gates, such as the QSBC encoders and also its inverse encoder illustrated in Fig. 3. Since the quantum encoders of QSBCs are only composed of Hadamard and CNOT gates, we can create a -element binary matrix for classically simulating the evolution of the Pauli operator over the quantum encoder . As an example, let us consider the quantum encoder of the QSBC seen in Fig. 5.

Fig. 5: The quantum encoder of the QSBC .

We commence by initializing , where is a -dimensional identity matrix formulated as follows:

(37)

The -th and the -th column of matrix are associated with the evolution of the Pauli matrices and , respectively, on the -th qubit.

The first unitary operation in the QSBC encoder of Fig. 5 is the unitary operation , where the notation means that the -th qubit controls the -th qubit. Now, based on Eq. (36), the CNOT unitary transformation propagates the Pauli matrix from the control qubit to the target qubit and by contrast, propagates the Pauli matrix from the target qubit to the control qubit. Therefore, in the matrix , the unitary transformation can be carried out by replacing the -th column with the modulo-2 addition between the -th column and the -th column then replacing the -th column with the modulo-2 addition between the -th column and the -th column. In the case of , the unitary transformation can simply be viewed as copying the value from the -rd column to the -st column then copying the value from -th column to the -th column. Hence, the unitary transformation transforms the matrix into the matrix as follows:

(38)

We indicate the matrix elements involved in the associated transformation using bold red fonts. The unitary transformation is followed by the second unitary transformation taking place in the QSBC encoder of Fig. 5, namely the . Following the same method as that used for obtaining the matrix , the action of applied to the matrix yields the matrix as follows:

(39)

The third unitary transformation taking place within the QSBC encoder of Fig 5 is the Hadamard transformation . Based on Eq. (35), the Hadamard transformation modifies the Pauli matrix into and vice versa. Therefore, in a matrix , a Hadamard transformation of can be interpreted as swapping the value of the -th column and the -th column. In case of the matrix , the action of the unitary transformation swaps the value of -th column and the -th column, hence resulting in the matrix as follows:

(40)

These operations are then followed by the unitary transformation , which yields the matrix as follows:

(41)

The action of upon the matrix gives us the matrix as follows:

(42)

Finally, applying the transforms the matrix into the final matrix as follows:

(43)

The resultant matrix is the classical analogue of the quantum encoder of the QSBC seen in Fig. 5. In order to obtain the matrix of the quantum inverse encoder , the same method is invoked. The only difference is that we apply the transformation for each step by reading the quantum circuit from right to the left.

In order to show that the matrix can be used for classical simulation, let us consider a Pauli operator as follows:

(44)

By using the Pauli-to-binary mapping of Eq. (33), the Pauli operator given in Eq. (44) can be transformed into its classical analogue as follows:

(45)

The resultant Pauli operator due to the QSBC encoder can be obtained by modulo-2 multiplication of the vector and the matrix , which gives us

(46)

From the resultant vector of Eq. (46) and also from the Pauli-to-binary mapping of Eq. (33), we can map back the binary vector into its corresponding Pauli operator, which gives us as follows:

(47)

In order to simplify the expression of matrix , often the seed transformation representation is used. The element of the seed transformation is the decimal representation of the -th row of the binary matrix . Therefore, based on the matrix of Eq. (43), the associated seed transformation is given by

(48)

Finally, the seed transformation for all the QSBC encoders of Fig. 3 is provided in Table. I.

Iii Quantum Turbo Code Design Using QSBCs

QSBCs on their own are only capable of detecting the presence of errors in the state of physical qubits, but not correcting them. This limited capability is due to the minimum distance of inherited from the classical SBCs. In this treatise, we invoke the QTC scheme utilized in [25, 16] as the foundation of developing QSBC-based QTCs. The QSBCs are invoked as the outer codes, while a non-catastrophic and non-recursive QURC is used as the inner code, which we will refer to as the QSBC-QURC scheme for the rest of this treatise.

The QTC scheme was first introduced by Poulin et. al in [25]. The proposed QTC utilized two QCCs, which were concatenated in a serial manner. Each of the QCC components exhibits a quantum coding rate and hence, the final quantum coding rate is . To the best of our knowledge, the QTCs operating closest to the quantum hashing bound rely on the construction presented in [26, 16] in the open literature. The near-hashing-bound performance was attained by utilizing QIrCCs both as the outer and the inner codes. Furthermore, the weighting factors of the QIrCC component codes were optimized by invoking EXIT-chart-based heuristic search [27, 26]. Readers who are interested to delve deeper into QTCs and into near-hashing-bound constructions, please refer to [16].

Fig. 6: The general schematic of serial QTCs utilizing QURC as the inner code. The decoding process, in general, can be separated into two parallel processes, namely the quantum processing part and the classical processing part. The quantum part is represented by the green dashed lines, while the classical part is represented by red the dashed lines.

Iii-a Encoding Process

In this section, we will describe the proposed QSBC-QURC scheme, whose general schematic can be seen in Fig. 6. The outer encoder in Fig. 6 is a QSBC encoder, which is already shown in Fig. 3. It maps logical qubits into physical qubits with the aid of auxiliary qubits according to the following transformation:

(49)

The output of is fed to the interleaver , which can be represented mathematically as a permutation matrix and can be realized physically as a series of quantum SWAP gates. The interleaving process can be formally written as

(50)

where we have , since the interleaver does not alter the number of physical qubits, only rearranges the position of the qubits indices in the quantum state. Hence, the Hamming weight of the state of physical qubits is not changed after this process. Next, the output of the interleaver is fed into the inner encoder , which carries out the following transformation:

(51)

The encoder maps the state of logical qubits into the state of physical qubits with the aid of auxiliary qubits. Since, we are employing the QURCs as the inner codes, the transformation in Eq. (51) can be further simplified as

(52)

The main difference between the interleaver and the QURC is that the state of physical qubits after the QURC may experience Hamming weight alterations.

Iii-B Quantum Depolarizing Channel

After the encoding process, the encoded state of physical qubits may experience quantum decoherence. In this study, we use the quantum depolarizing channel [28]. This depolarizing channel models the imperfection of quantum gates, as well as the coherence taking place in the quantum memory, and even the actual quantum transmission channel through free space or optical fiber channels. The quantum decoherence is represented by the -tuple Pauli operator and its action imposed upon the encoded state of physical qubits can be expressed as

(53)

The error operator is characterized by the depolarizing probability . To elaborate a little further, the error operator is an -tuple Pauli operator, where each qubit may independently experience a bit-flip () error, a phase-flip () error as well as a simultaneous bit-flip and phase flip () error. The probability of each qubit experiencing an , , and error is denoted by , , and , respectively. Under the assumption that and , this quantum channel is referred to as symmetric quantum depolarizing channel [28]. Needless to say, it is always possible to create a model where we have the assumption that , which can be deeemed to be more realistic [29]. However, choosing the value such as will provide us with the worst-case scenario, because we have to provide the same level of protection for different types of errors without favoring only one specific type of error, which can result in quantum coding rate or QBER improvements. A more detailed discourse on QSC design for asymmetric quantum depolarizing channels, enthusiastic readers might like to refer to [30, 31, 32, 29].

Iii-C Decoding Process

Generally speaking, the decoding process of any QSC relies on the conjunction of two parts, namely the quantum information processing part and the classical information processing part. In Fig. 6, the quantum processing part is marked by the components bounded by the green dashed lines, while the classical processing part is represented by the components bounded by the red dashed lines. First, let us describe the quantum processing part. The corrupted state of physical qubits is fed to the inverse encoder of Fig. 6, which represents the conjugate transpose of encoder . Physically, they can be implemented identically with the only difference is that the input and output of the inverse encoder is in the reverse position compared to the encoder . Since the quantum encoder and its inverse encoder are composed by the quantum Clifford gates and the error operator is an -tuple Pauli operator , the act of inverse encoders will decompose the error operator into two error components as follows:

(54)

where is the error operator on logical qubits and is the error operator on auxiliary qubits. The auxiliary qubits are then measured in the relevant computational basis, where the value can be treated as a syndrome in classical error correction codes, which is forwarded to the classical processing part. However, since the inverse encoder of is an inverse encoder of a QURC, Eq. (54) can be further simplified to

(55)

since we have . Next, the output of the inverse encoder is passed trough the deinterleaver . This transformation can be formally expressed as

(56)

The output of the deinterleaver is then processed as the input of , which is subjected to an identical transformation as . This can be expressed as follows:

(57)

In this QSBC-QURC scheme, the inverse encoder of is constituted by the quantum inverse encoder of the QSBC, which is implemented by flipping the input and ouput of the quantum encoder seen in Fig. 3.

Finally, based on the information obtained from the classical information processing part, the error recovery operator is applied to the output of the inverse encoder in order to obtain the predicted logical qubit state as follows:

(58)

If , we obtain , which completes our decoding process.

Let us now take a step back to elaborate a little further on the classical processing part of the decoding process. The classical decoder part for a QTC is very similar to that of classical turbo codes. It consists of two soft-input soft-output (SISO) decoders, an interleaver, and a deinterleaver.

As seen in Fig. 6, the classical processing is started by obtaining the quantum depolarizing probability of the quantum channel associated with the error operator . In this work, we assume that we have perfect knowledge of the quantum depolarizing probability . The depolarizing probability value and the a priori information obtained from the outer SISO decoder are used by the inner SISO decoder for calculating the extrinsic information . For the first iteration, the depolarizing probability is the only input value used by the inner SISO decoder. Hence, the value of is initialized to be equiprobable. Next, the extrinsic information is interleaved in order to obtain the a priori information for the outer SISO decoder. By combining the a priori information and the syndrome value , the outer SISO decoder calculates the extrinsic information . The extrinsic value is then deinterleaved to yield which is fed into the inner SISO decoder. This process is performed iteratively until one of the following conditions is satisfied: the converged mutual information is attained or the maximum affordable number of iterations is reached. On the final iteration, the outer SISO decoder will produce , which is the most likely error pattern, given the value of and provided by the quantum processing part. The value of is obtained for performing error recovery, as detailed in Eq. (58). A more rigorous treatment on the classical processing part of QTCs can be found in [33, 16]

Iv Results and Analysis

In this section, firstly, we analyze the performance of the QSBC-QURC conceived using EXIT charts [34, 35, 27]. In the classical domain, EXIT charts constitute a powerful tool, which is often used for guiding the design of near-capacity iterative error correction and for predicting their performance. An initial encouraging effort conducted in [26] invoked EXIT charts for predicting the performance of iterative QTCs demonstrating that EXIT charts can be indeed extended to the quantum domain. Secondly, we proceed by characterizing the performance of the QSBC-QURC scheme in terms of its quantum bit error ratio (QBER), which we obtained from our Monte Carlo simulations. Thirdly, we translate the QBER performance to the distance from the quantum hashing bound, which directly corresponds to the efficiency of quantum channel utilization, which is also related to the goodput. Finally, we use the goodput metric for determining the depolarizing probability at which switching to different quantum coding rate becomes beneficial for conceiving a multi-rate QSBC-QURC scheme.

Iv-a EXIT Chart

In the classical domain, the encoders exhibiting recursive and non-catastrophic properties are highly desirable for conceiving near-capacity turbo codes. Unfortunately, in the quantum domain, the QCCs cannot be simultaneously recursive and non-catastrophic [36]. The recursive structure of QCCs is required for ensuring the convergence of iterative decoding to a vanishingly low QBER. Additionally, the QCCs exhibiting catastrophic structure require a doping mechanism or entanglement-assisted solution in order to substantially benefit from iterative decoding, since catastrophic QCCs provide zero a priori information [37, 33]. These two solutions are beyond the scope of our discussions in this paper. Fortunately, a non-recursive and non-catastrophic QCCs can still be designed for striking an attractive compromise, since they can achieve beneficial iteration gains even if the inner decoder EXIT curve terminates at the point for , provided that it only intersects with the outer decoder EXIT curves near  [10]. Based on these conditions, an exhaustive EXIT-chart-based heuristic search has been conducted to find a “good” QURC. The resultant seed transformation for such a QURC is given by

(59)

In this treatise, our QSBC-QURC scheme utilized a specific a QURC whose seed transformation is given in Eq. (59) and the QSBCs of , and were used as the outer codes. The seed transformations of the QSBCs are given in Table I. As a benchmark, we use the the QIrCC-QURC scheme presented in [10], where the QIrCCs are optimized using EXIT-chart-aided method specified in [26, 16]. The seed transformation of the QIrCC component codes is given in Table II. As we have described briefly in Subsection II-D, the seed transformation is the decimal representation used for describing the quantum gate connections amongst the physical qubits within the quantum encoder . Also, it can be used for simulating the QSCs classically.

Quantum coding rate Seed transformation
TABLE I: The seed transformation associated with QSBCs having various quantum coding rates .
Quantum coding rate Seed transformation