Dynamic Concatenation of Quantum Error Correction in Integrated Quantum Computing Architecture

Resource overhead problem caused by concatenation in quantum error correction (QEC) is of significant importance for the realization of fault-tolerant quantum computation (FTQC). To attack this problem, we propose a novel scheme by considering integrated FTQC architecture where the concatenation level is controlled dynamically; i.e., less (or more) concatenation levels are imposed by good (or poor) performance gates—we call this scheme “dynamic concatenation” in this sense. Such a dynamic concatenation is realizable in an integrated structure of FTQC, as the information of the concatenation can be communicated between classical system elements (e.g., compiler and system organizer) and the logical qubits in real-time. We derive the effective lower and upper bounds of the length of gate decomposition in order to achieve the practical advantage, namely of reduction of the overall operation time. By considering two non-trivial examples, it is shown that the aforementioned advantage can indeed be achieved in the presented scheme. Our result also provides an important scientific message, i.e., the interplay between “classical” and “quantum” can be helpful in QEC.

As mentioned in the main text, when the concatenation level of the logical qubits is lowered, the decrease in the total fidelity should be tolerated. First, we approximate the upper bound Eq. (2) of N. ( where F are the performance of T , H, S gate at l − s level and N T , N H , N S are the number of T , H, S gate. DC-level s is the number of concatenations lowered by our scheme. ε τ is the maximum tolerable error rate obtained by the quantum compiler through algorithm analysis. All operations in computing should have an error rate lower than ε τ . We calibrated ε τ to a factor of γ to get a more conservative approach to the results(Here, we set γ = 10).
The performance of single-qubit gates is approximated to T gate because it is expected to have the worst performance among single-qubit gates. Hereafter, we denote F which means the fidelity of a singe-qubit gate. Also, the sum of N T , N H , and N S is denoted by N, which means the length of a single qubit gates sequence.
The product of the en/decoding circuits' fidelities at each level can be approximated by the product of the highest level en/decoding circuits' fidelities. Note that these en/decoding circuits for Steane code are not fault-tolerant.
Therefore, Eq. 2 is approximated as follows.
In most cases, the en/decoding circuits consist of Hadamard gates and CNOT gates. The fidelities of the en/decoding circuits are determined by the fidelities of Hadamard gates and CNOT gates and it can be approximated with the worst one. We assume that the performance of a two-qubit gate is worse than that of a single-qubit gate, so each fidelity can be approximated as follows.
where p D , p E,H , and p E,CNOT are factors related to the fidelities of the encoding and decoding circuits and p E is sum of p E,H and p E,CNOT . There are many ways to obtain these values, the method used in the main text is explained in chapter 2. Fidelities of en/decoding circuits can be approximated about F where p E,D is sum of p D and p E . Hereafter, we set p E,D = 53 Then Eq. (4) is approximated as follows.
We assume that F CNOT and the formula can be approximated as follows.
Then we can get the upper bound of Eq. (9) by taking logarithm on both sides of the Eq. (8).
Now we approximate the lower bound Eq. (10) of N. 1Q is l level single-qubit gate operating time. Eq. 10 can be rewritten as: The product of the en/decoding circuits' fidelities can be approximated by the product of the highest level en/decoding circuits' fidelities. Since the performance of the single-qubit gates at any level is close to one, we will approximate N by one. Then Eq. (11) can be rewritten as: Because the operating time at the highest level will be the longest, we approximate can be ignored. Therefore we can obtain the final form of the lower bound.
2/6 Fidelity factor of en/decoding circuit We use a factor p E,D in Eq. (9) and its value is related to the fidelities of the en/decoding circuits. We assume that the whole fidelity will be degraded as the gate fidelity by multiplying the fidelity of the gate to the whole fidelity. We will briefly show the en/decoding circuits of the Steane code we used and the p E,D obtained from it. Each circuit is not fault-tolerant, but it is assumed that each qubit is a logical qubit so it can tolerate some errors in the en/decoding process. We consider that the influence of each gate on fidelity propagates through a CNOT gate. p E,H and p E,CNOT are 5 and 30, respectively. Finally, the value of p E is 35.
In fig. 1, the number before the comma is p E,H associated with the single-qubit gates and the number after it is p E,CNOT associated with the two-qubit gates. The red number represents the final p value of that qubit line at the very end. We define the largest number of the red numbers as the p value of the whole circuit. In the case of fig. 1, p E,H is 5, p E,CNOT is 30, and p E is 35. We also consider that the influence of each gate on the fidelity is propagated through the CNOT gate in the decoding circuit. We do not consider the correction through the ancilla qubit measurement at the final stage, thus we omit the several CNOT and Hadarmard gates in this circuit. Finally, the value of p D is 18.
In the case of fig. 2, p D is 18. Then the p E,D is obtained by adding p E and p D , in this case 53.

Estimation of N for the dynamic concatenation
With Eq. (9), we have calculated the example of N with the certain ε τ , F (l−1) 1Q when s = 1 and p E,D = 53. Fig. 3. and Table 1. represent N with these parameters. 1Q ) on Logarithmic scales by averaging the values in Table 1 1Q ) is 10 5 , a single-qubit gate sequence with a length of 9863 can be operated with the concatenation level being one step lower.
We can check that N is almost the same depending on the ratio of ε τ to 1 − F (l−1) 1Q . If 1 − F (l−1) 1Q and ε τ differ by 10 −5 times, it can be seen that N is close to 10,000, and the above values are expected that there will be no meaning in a practical case.
In such a situation, we can lower the concatenation level further, rather than use the meaninglessly long N. Table 2. represents how the concatenation level can be lowered for ε τ = 1.0 × 10 −6 , 1.0 × 10 −10 ≤ 1 − F (l−1) 1Q ≤ 1.0 × 10 −6 . The DC may seem impractical because we only deal with too large N or s. At the logical qubit level, however, we can get sufficient gain by lowering the concatenation level by one or two steps. This simple example will provide how much gain can be achieved by lowering the concatenation level one step. When R Z ( π 8 ) is decomposed and executed, R Z ( π 8 ) can be decomposed into approximately 250 other single gates as follows. HTHTSHTHTSHTSHTHTHTHTHTSHTSHTSHTHTSHTSHTSHTSHTHTSHTSHTSHTHTHTSHTSHTSHTHTHTH THTHTSHTSHTHTHTSHTSHTSHTSHTSHTSHTSHTSHTHTHTHTSHTSHTHTSHTSHTHTHTHTSHTSHTSHTH TSHTSHTHTHTHTHTSHTHTSHTHTHTSHTSHTSHTSHTHTHTSHTHTSHTSHTHTHTHTSHTSHTHTSHTSHTH TSHTSHTSHTSHTHTSHTHTHTSHTH.
If the concatenation level can be lowered by one, we can obtain a gain of approximately 22 times in the operating time with the condition of table 1. in main text. is 10 −3 times smaller than ε τ . As mentioned above, this is the maximum N when the concatenation level is lowered by one step. Thus, by lowering the concatenation level further, we can get more reasonable N and s.