Quantum Error-Correcting Codes Based on Orthogonal Arrays.

In this paper, by using the Hamming distance, we establish a relation between quantum error-correcting codes ((N,K,d+1))s and orthogonal arrays with orthogonal partitions. Therefore, this is a generalization of the relation between quantum error-correcting codes ((N,1,d+1))s and irredundant orthogonal arrays. This relation is used for the construction of pure quantum error-correcting codes. As applications of this method, numerous infinite families of optimal quantum codes can be constructed explicitly such as ((3,s,2))s for all si≥3, ((4,s2,2))s for all si≥5, ((5,s,3))s for all si≥4, ((6,s2,3))s for all si≥5, ((7,s3,3))s for all si≥7, ((8,s2,4))s for all si≥9, ((9,s3,4))s for all si≥11, ((9,s,5))s for all si≥9, ((10,s2,5))s for all si≥11, ((11,s,6))s for all si≥11, and ((12,s2,6))s for all si≥13, where s=s1⋯sn and s1,…,sn are all prime powers. The advantages of our approach over existing methods lie in the facts that these results are not just existence results, but constructive results, the codes constructed are pure, and each basis state of these codes has far less terms. Moreover, the above method developed can be extended to construction of quantum error-correcting codes over mixed alphabets.

Resource analysis and modifications of quantum computing with noisy qubits for elliptic curve discrete logarithms.

We estimate the number of physical qubits and execution time by decomposing an implementation of Shor's algorithm for elliptic curve discrete logarithms into universal gate units at the logical level when surface codes are used. We herein also present modified quantum circuits for elliptic curve discrete logarithms and compare our results with those of the original quantum circuit implementations at the physical level. Through the analysis, we show that the use of more logical qubits in quantum algorithms does not always lead to the use of more physical qubits. We assumed using rotated surface code and logical qubits with all-to-all connectivity. The number of physical qubits and execution time are expressed in terms of bit length, physical gate error rate, and probability of algorithm failure. In addition, we compare our results with the number of physical qubits and execution time of Shor's factoring algorithm to assess the risk of attack by quantum computers in RSA and elliptic curve cryptography.

Experimental exploration of five-qubit quantum error-correcting code with superconducting qubits.

Quantum error correction is an essential ingredient for universal quantum computing. Despite tremendous experimental efforts in the study of quantum error correction, to date, there has been no demonstration in the realisation of universal quantum error-correcting code, with the subsequent verification of all key features including the identification of an arbitrary physical error, the capability for transversal manipulation of the logical state and state decoding. To address this challenge, we experimentally realise the [5, 1, 3] code, the so-called smallest perfect code that permits corrections of generic single-qubit errors. In the experiment, having optimised the encoding circuit, we employ an array of superconducting qubits to realise the [5, 1, 3] code for several typical logical states including the magic state, an indispensable resource for realising non-Clifford gates. The encoded states are prepared with an average fidelity of [Formula: see text] while with a high fidelity of [Formula: see text] in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilisers. We further implement logical Pauli operations with a fidelity of [Formula: see text] within the code space. Finally, we realise the decoding circuit and recover the input state with an overall fidelity of [Formula: see text], in total with 92 gates. Our work demonstrates each key aspect of the [5, 1, 3] code and verifies the viability of experimental realisation of quantum error-correcting codes with superconducting qubits.