Analytical technique for simplification of the encoder–decoder circuit for a perfect five-qubit error correction

Simpler encoding and decoding networks are necessary for more reliable quantum error-correcting codes (QECCs). The simplification of the encoder–decoder circuit for a perfect five-qubit QECC can be derived analytically if the QECC is converted from its equivalent one-way entanglement purification protocol. In this work, the analytical method to simplify the encoder–decoder circuit is introduced and a circuit that is as simple as the existing simplest circuits is presented as an example. The encoder–decoder circuit presented here involves nine single- and two-qubit unitary operations, only six of which are controlled-NOT gates.


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Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT can be well derived so that it requires a simpler network for both encoding and decoding than the original one reported by Laflamme et al Bennett et al suggested, i.e. to use a Monte Carlo search program for deriving the QECC.
In realistic situations, to reduce the number of two-qubit gates necessary in the encoderdecoder circuit is significantly important for reliable five-qubit QECCs because two-qubit operations could be the more difficult ones to be implemented in a physical apparatus [10]. This work thus is motivated to derive five-qubit, single-error corrections which can be performed by using the least number of two-qubit operations in their encoder-decoder networks. The QECC presented as an example herein is derived analytically from the restricted 1-EPP proposed by Bennett et al [5] and its encoder-decoder network contains only six controlled-NOT (CNOT) gates and three single-qubit operations. The restricted 1-EPP therefore is depicted first in section 2. In section 3, we describe the systematic method for deriving 1-EPP in detail. A concrete example for the simplest quantum gate array then will be given to show the capacity of the present method. In section 4, we present the coding circuit which is converted directly from the 1-EPP and compare its efficiency with those of several existent encoder-decoder circuits. A conclusion is given in section 5.

The 5-EPR-pair single-error-correcting code 3
Suppose there exists a finite block-size 1-EPP which distills one good pair of spins in a specific Bell state from a block of five pairs, and no more than one of the five pairs is subjected to noise. When this 1-EPP is combined with a teleportation protocol, two parties, Alice and Bob, can transmit quantum states reliably from one to the other. The combination of the 1-EPP and teleportation protocol therefore is equivalent to a QECC. The 1-EPP considered herein is schematically depicted in figure 1. Suppose Alice is the encoder, Bob the decoder, and the Bell state + = (|00 + |11 )/ √ 2 is the good state to be purified. Alice and Bob are supposed to be provided with five pairs of spins in the state + by a quantum source (QS). However, they actually share five Bell states in which generic errors have or have not occurred on at most one Bell state due to the presence of noise N B in the quantum channel via which the pairs are transmitted. The noise models are assumed to be one-sided [5] and can cause the good Bell state + to become one of the incorrect Bell states The good Bell state + can become one of the erroneous Bell states expressed in (1) if it is subjected to either a phase error ( + → − ), an amplitude error ( + → + ), or both ( + → − ) [2,11]. When performing the 1-EPP, Alice and Bob have a total of 16 error syndromes to deal with. The collection of error syndromes includes the case that none of the five pairs has been subjected to errors and the 15 cases in which one of the five pairs has been subjected to one of the three types of error. The strategy of Alice and Bob is to perform a sequence of unilateral and bilateral unitary operations (as shown in figure 1, U 1 and U 2 performed by Alice and Bob, respectively) to transform the collection of the 16 error syndromes to another collection that can provide information about the errors subjected by their particles. Suppose the state of the first : the channel for the 1st pairs of entangled qubits : the channel for the 2nd -5th pairs of entangled qubits pair in the block is to be recovered. After performing the sequence of their operations (U 1 and U 2 , respectively), Alice and Bob should then perform local measurements on their respective halves of the second to fifth pairs. Alice sends her result via classical channels to Bob who then performs the Pauli operation U 3 to recover the original state of the first pair conditionally on both Alice's and his results. The ultimate requirement of these results of final measurement is that each and every one of them should be distinguishable from the others. In other words, there should be 16 distinct measurements obtained from the aforementioned transformation of the error syndrome. The main issue now is that the sequence of unilateral and bilateral unitary operations performed by the two parties to transform the error syndrome should be well designed so the requirement just mentioned can be fulfilled.
To arrange the sequence of operations, basic concepts of linear algebra are used. The four Bell states ± and ± are first labelled by two classical bits, namely, The right, low-order or amplitude bit identifies the / property of the Bell state, while the left, high-order or phase bit identifies the +/− property. Note that the combined result of the local measurements obtained by Alice and Bob on a Bell state is revealed by the Bell state's low or amplitude bit. In the representation of the high-low bits, each error syndrome thus is expressed as a ten-bit codeword, e.g., the error syndrome + − + + + is written as 00 11 00 00 00. Codewords of the error syndrome, denoted by e (i) r , i = 0, 1, . . . , 15, are listed in table 1. The effect of the sequence of unilateral and bilateral unitary operations performed by Alice and Bob is to map the codewords e (i) r onto another collection of ten-bit codewords w (i) . If both the codewords, e (i) r and w (i) are written as column vectors in the ten-dimensional Boolean-valued (∈ {0, 1}) space, then the mapping e (i) r → w (i) can be simply expressed by a Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Table 1. The correspondence among the error syndrome e (i) r (E (i) r ), the codeword w (i) (W (i) ), the measurement result v (i) and the Pauli operation U (i) 3 controlled by the measurement result in the restricted 1-EPP (five-qubit QECC) applying the encoder-decoder circuit shown in figure 3 (figure 4).
in accordance with relations (4) and (5). In the language of linear algebra, the action of the sequence of unilateral and bilateral unitary operations that accounts for the mapping e (i) r → w (i) is to perform a sequence of elementary row operations on the 10 × 10 identity matrix 1 to reduce it to the matrix M. In this spirit, Bennett et al [5] have undertaken a Monte Carlo numerical search program to find suitable solutions for matrix M and their corresponding encoder-decoder networks. Basically, the approach implemented by Bennett et al is a tedious numerical method of trial and error performing the transformation 1 → M subjected to a 'forward' sequence of local operations. In this work, we will present an analytical method for creating M implemented in the present QECC. The present method will be described in detail in section 3.

Theory
The unilateral and bilateral unitary operations performed in the 1-EPP in fact are their own inverse transformations, so if the sequence of operations is run in the reverse order, then the inverse transformations M → 1 are accomplished. In the spirit of inverse transformation, it thus allows us to derive all appropriate versions of M and the corresponding encoder-decoder networks by following an analytical way. More importantly, for a derived M, rearranging the sequence of row operations on the same inverse transformation M → 1 will help in constructing its simplest encoder-decoder network.
An elementary row operation corresponds to a basic unilateral or bilateral unitary operation. In the present protocol, Alice and Bob are confined to perform only three basic unitary operations because these operations are necessary and sufficient for the elementary row operations needed to achieve the mapping M → 1, and vice versa. These basic operations are: (i) a bilateral CNOT (BXOR), which performs the bit change where the subscripts S and T denote the source and target pairs, respectively; (ii) a bilateral π/2-rotation B y , which performs The unitary Pauli operation σ x performs a π-rotation of Alice or Bob's spin about the x-axis, while the bilateral operation B x (B y ) performs a π/2-rotation of both Alice and Bob's spins about the x (y)-axis. The unilateral operations are defined as those operators performed by Alice or Bob but not both. The bilateral operations are represented by a tensor product of one part of Bob and the same part of Alice. Note that the bilateral CNOT is performed such that the source qubits of Alice and Bob belong to a common pair, and the target qubits belong to another common pair.
The information obtained through local measurements and one-way communications can only deduce the low bit of a Bell pair, and the original state of the first Bell pair can only be recovered by the low-bit information. Then, for a successful 1-EPP, or its equivalent QECC, each and every measurement result v (i) is required to be distinguishable from the others, so 7 Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT the collection of v (i) in fact should contain all elements in the four-dimensional Booleanvalued space. To perform the aforementioned inverse transformation M → 1, the codewords of measurement result are first arranged according to relations (7) and the matrix M can be assumed as 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 a 10  b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9 It should be noted that the arrangement of the results of measurements shown in the above matrix is only one of the possible choices. By performing a sequence of row operations corresponding to the basic unitary operations, the assumed matrix M (8) actually is allowed to be reduced to one of all the alternatives akin to the identity matrix 1, and a suitable encoder-decoder network is constructed accordingly. The alternatives akin to the identity 1 are those obtained by (1) permuting column vectors within one of the five sets of two column vectors (x (3k−2) and x (3k−1) , k = 1, 2, . . . , 5), or (2) adding one column to the other within each of the groups, or (3) performing both actions. For example, an alternative could be When the derivation of M is done, the alternative akin to 1 is then converted back to the identity 1 by well rearranging its columns and the derived M is adjusted via the same column changes, in order to conform to equation (3). The procedure of reducing the matrix M to the alternative akin to the identity 1 is similar to the Gauss-Jordan elimination method for solving systems of linear equations. During the procedure of row operations, all the unknowns appearing in the assumed matrix M (8) are given or solved according to the structure of the alternative akin to 1. Details of the derivation can be found in [12].

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A systematic scenario example
There are so many solutions for the assumed M which are all suitable for the 1-EPP; however, only one of them has been adjusted and presented as: 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 Let us show the systematic scenario for accomplishing the transformation M 1 → 1 by one of the simplest networks. The matrix M 1 can be rephrased as where the matrix elements m αβ denote the 2 × 2 matrices: and so forth. The next step of our method is a procedure of elementary row operations on the matrix M 1 (10) subjected to a suitable sequence of the basic operations. When the assumed matrix M 1 is transformed into the identity matrix 1 under the series of row operations, the unknowns a r , b r , . . . , f r will be solved stepwise in accordance with the structure of 1. It is easy to show that a sequence of row operations can do the transformation on two Bell states α and β in a group enumerated by γ, namely, provided that det(m αγ ) = 1 and det(m βγ ) = 0. Here I denotes the 2 × 2 identity matrix. For example, the consecutive transformation Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT can be accomplished if the operation B y is first performed on Bell state β, then a σ x B x is performed on Bell state α followed by a BXOR performed on both states, as Bell state α being the source and Bell state β being the target. It can be found in what follows that the unknowns assumed in the matrix M 1 either will be given based on the requirement for the transformation described in (13), or will be determined according to the unique structure of the identity matrix 1.
In the first stage of row operations, we are confined to performing a transformation of the matrix M 1 (11a) such that m 44 → I and m 4k , m k4 → 0, for k = 1, 2, 3 and 5, according to the structure of 1. Let det(m 44 ) = 1 and det(m 14 ) = · · · = det(m 54 ) = 0, which imply Clearly, there are totally 640 solutions for the unknowns appearing in (10) to be considered in this stage (ten for the condition a 7 b 8 ⊕ a 8 b 7 = 0, two for each of the six arbitrary Boolean valued unknowns, and thus totally 10 × 2 6 = 640 solutions). To illustrate the simplest way of creating Boolean functions, however, only one among these 640 cases is considered. Let us consider the case in which Then, by performing the operations shown in figure 2(a), we have the transformation M 1 → M 1 , in which we have chosen the following setting for the unknowns: Let us proceed to apply the second series of operations, as depicted in figure 2(b), to perform the transformations m 22 → I and m 2k , m k2 → 0, for k = 1, 3 and 5. As a result, we have Finally, if the matrix M 1 is transformed through additional two BXOR and one σ x B x operations, as shown in figure 2(c), it results in the identity matrix 1. In this stage, we have set the rest of the unknowns to be one of the alternatives: a 1 = 1, a 2 = 0, a 5 = 1, a 6 = 0, a 9 = 0 and a 10 = 0. The whole sequence of basic operations, as shown in figure 3, is obtained by combining the three subsequences as shown in figures 2(a)-(c). It will transform the matrix M 1 into the identity matrix 1. This network is the simplest one since it involves only six BXORs, M 1 1 Performed by this network, the correspondence between the error syndromes e (i) r and the combined measurement results v (i) r is also listed in table 1. Referring to table 1, or the matrix M 1 , when Bob obtains the measurement result v (2) (= 0110), for example, he knows the pair to be purified is in the state + (= 01) and thus simply performs the Pauli operation U (2) 3 = σ x to recover it to the good state + .

The encoder-decoder circuit for a perfect five-qubit error correction
The 1-EPP depicted above can be directly converted to a five-qubit QECC whose encoderdecoder circuit has the same configuration as the one shown in figure 4. However, in the language of QECC, the classical high-low or phase-amplitude bits used to code the Bell state in the 1-EPP are now used to code operators belonging to the Pauli group, namely, I = 00, σ x = 01, σ z = 10, σ y = 11. When acting on a single qubit, the Pauli operator produces either no error (by I), a bit flip error (by σ x ), a phase flip error (by σ z ), or a bit-phase flip error (by σ y ). Therefore, such a code is convenient because the codewords e (i) r are now replaced by E (i) r , which represent the 16 error syndromes described by five-Pauli-operartor tensor products. Furthermore, the transformation described by the matrix equation (3) is now replaced by the similarity transformation of operators described as: W (i) = UE (i) r U + , where U (U + ) represents the sequence of the basic operations performed in the decoder (encoder) circuit. Clearly, both the encoder and decoder circuits have exactly the same quantum gate arrangement but they  should be run in opposite orders. In order to perform the transformation mentioned above, this time the single-qubit Hadamard transformation: H = H + = (σ x + σ z )/ √ 2, is used to perform the bit change H(x, y)H + → (y, x), the single-qubit transformation: Q = Q + = (σ y + σ z )/ √ 2, is used to perform Q(x, y)Q + → (x, x ⊕ y), and the two-qubit CNOT gate is used to perform (CNOT)(x S , y S )(x T , y T )(CNOT) + → (x S ⊕ x T , y S )(x T , y S ⊕ y T ), respectively. That is, in the five-qubit QECC to be presented, the basic single-and two-qubit operations needed to be implemented are H, Q and CNOT.
For the present five-qubit QECC, the correspondence between the codewords W (i) and E (i) r is exactly the same as that between the derived matrix M 1 given in (9) and the identity 1. The QECC is performed as follows. If a state |φ = α |0 + β |1 is to be protected in a quantum computation, it is first accompanied with four extra qubits in the state |0 . Then the five-qubit state |φ |0 |0 |0 |0 is encoded by the performance of U + . After the encoded state is subjected to E (i) r , the erroneous state then is decoded by the implementation of U. The resulting state turns out to be where U (i) 3 is the single-qubit Pauli operation acting on the first qubit and is dependent on the measurement result on the four extra qubits. When the extra qubits are measured in Institute of Physics ⌽ DEUTSCHE PHYSIKALISCHE GESELLSCHAFT Table 2. Three efficiency criteria and the corresponding costs for four circuits have been presented. Circuit 1 is given by Bennett et al (figure 18 in [5]) and is unoptimized. The optimized circuit of Bennett et al denoted by Circuit 2, mentioned in [5], consists of six two-qubit controlled-NOT gates only. Since the number of laser pulses depends on the detailed structure of the circuit, it is not shown here for laking the detailed information. Circuit 3 is the simplification of the coding circuit of Laflamme et al proposed by Braunstein and Smolin (figure 1 in [8]). One can find that the original circuit of Laflamme et al ( figure 1 in [4]) is more complicated and requires 41 laser pulses. Circuit 4 denotes the simpest circuit which has been found by computer search (figure 3 in [8]) and by the systematic method presented in this work.

Conclusion
This work has presented a rather simple encoder-decoder circuit to perform the five-qubit, singleerror correction protocol. The QECC derived herein is converted directly from the restricted 1-EPP depicted above, so a major part of this work is dedicated to the depiction of the 1-EPP. The present encoder-decoder circuit is the simplest one corresponding to the derived matrix M 1 given in (20), which is derived via an analytical approach [12]. This analytical approach, as shown, can help in deriving not only the suitable matrix M for the five-qubit QECC but also the simplest version of encoder-decoder network corresponding to the derived matrix. However, many possible matrices M suitable for the QECC remain to be discovered analytically and, thus, so many candidates of encoder-decoder circuit that require only six CNOTs. The simplest network that is even simpler than the present one and the Braunstein and Smolin circuit [8] might not be found from these candidates. However, a more convincing proof which could be a numerical approach based on the analytical approach introduced in [12] is required in future work.