# The logical qubit and the correction of quantum errors

## Introduction: Physical Qubit vs Logical Qubit

As discussed in the previous article, the unit of information processed by a quantum computer is generically called Qubit or Quantum Bit.

However, for a deeper understanding of the problems related to the realization of a quantum computer, it is necessary to introduce and distinguish the notions of physical qubit and logical qubit.

Inside a quantum processor, each physical qubit is in correspondence one by one with the quantum state of a particle or, more generally, of a microscopic phenomenon that encodes it. There are various types of physical qubits depending on the coding technology. As an example, among the best known, we cite the physical qubits with superconductors, trapped ions and photons.

The physical qubit is an unstable entity. In fact, any attempt to conduct the physical qubit to the desired computations through external actions can cause the alteration of the quantum state, thus nullifying the goodness of the encoded information.

To mitigate this problem, several physical qubits are made to interact with the aim of keeping a single quantum state stable over time on which it is possible to act in a non-destructive way. A set of physical qubits with this characteristic is called a logical qubit. The logical qubit is a theoretical entity corresponding to the real unit of quantum information that a quantum processor is able to control and the number of logical qubits identifies the actual associated computing capacity. To realize the logical qubit, the application of quantum error correction codes (Quantum Error Correction) is used, in a similar way to what happens for the transmission errors of classical bits.

The physical-logical qubit dualism is still an unclear subject in the state of the art, the fundamental question to be addressed is the precise estimate of the number of physical qubits necessary to create a logical one. This estimate depends, first of all, on quantum technology intended both as a state of technological advancement and as a technology for coding physical qubits. At the moment there is no reason to think that a possible estimate could be valid at the same time for different technologies, indeed it is reasonable to imagine that systems of different nature exhibit very different behaviors in this sense. For example, superconducting physical qubits are much more unstable than trapped ion qubits, although they offer other implementation benefits. Second, the complexity of the quantum circuit you want to implement also affects this estimate. The longer the running time of the circuit, the more the time in which the logical qubit remains stable must be extended and, consequently, the more physical qubits will be needed for this purpose.

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### The authors

**Edoardo Signorini**, graduated in Mathematics with a curriculum in cryptography at the University of Trento, in 2020 carried out the Master’s thesis in the field of Post-Quantum Cryptography (PQC) at Telsy. He currently holds the role of cryptographer within the Telsy research group and carries out an industrial research doctorate in Pure and Applied Mathematics at the Politecnico di Torino. His activity focuses on PQC research and on the analysis and development of cryptographic protocols.

**Francesco Stocco**, a master’s degree in Mathematics at the University of Padua and the Université de Bordeaux attending the course of study “Algebra Geometry And Number Theory” (ALGANT), joined the Telsy research group in Cryptography at end of 2020 focusing in particular on issues related to quantum technologies.