Perfect Codes Over Induced Subgraphs of Unit Graphs of Ring of Integers Modulo n

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes

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 $57.1(3)\%$ while with a high fidelity of $98.6(1)\%$ in the code space. Then, the arbitrary single-qubit errors introduced manually are identified by measuring the stabilizers. We further implement logical Pauli operations with a fidelity of $97.2(2)\%$ within the code space. Finally, we realise the decoding circuit and recover the input state with an overall fidelity of $74.5(6)\%$, in total with 92 gates. Our work demonstrates each key aspect of the [[5, 1, 3]] code and verifies the viability of experimental realization of quantum error correcting codes with superconducting qubits.

Non-Ideal Pragmatic Responses to the Problem of Error

Having discovered that no ideal Pragmatic Responses to the problem of error are acceptable, Chapter 6 explores the more modest non-ideal Pragmatic Response. This response advocates seeking a moral code that may fall short of complete error-freedom but that achieves a greater degree of error-freedom, and thus a higher degree of extended usability, than rival moral codes. According to this view, if a code’s extended usability value is higher than that of another code, the first code is better than the second. To implement this strategy requires introducing key new concepts, such as concepts of the bare and code-weighted usability of a principle, the deontically perfect code, the deontic merit of a code, the weight merit of a code, and usability value of a moral code. Assessment of the strategy’s success is taken up in Chapter 7.