# Optimized Implementation and Analysis of CHAM in Quantum Computing
## 1 Introduction  
In the recent years, we have seen a massive interest in research and development in quantum computing technology. There are many major players in the field, including international conglomerates such as IBM and Google, as well as universities and national research centers. They have been putting a lot of effort into the development of quantum computing technology. As a result, Google's research team, which had achieved quantum supremacy with its superconducting processor Sycamore in 2019 [1], recently demonstrated experimentally that errors can be reduced by increasing the number of logical qubits in their superconducting system [2]. In addition, the USTC research team demonstrated a quantum computational advantage with a 60-qubit superconducting quantum processor [3], and the Xanadu's team reported quantum computational advantages by showing that Gaussian Boson Sampling (GBS) tasks can be performed via a programmable photonic processor, Borealis [4]. However, the quantum computers--if they become functional--will massively disrupt a lot of the cryptographic systems that are presently deployed in various industrial applications. There are many facets, but in particular, the [PERSON]'s search algorithm [5] can be applied against any symmetric key ciphers [6] to reduce the security claim to the square root bound of what is expected against a classical computer. In addition to the [PERSON]'s algorithm, there are various quantum or search optimization algorithms such as Shor [7], Simon [8], and Particle Swarm Optimization algorithm [9, 10]. Other algorithms related to quantum computers include Classical-Quantum hybrid algorithms [11], which are mainly applied to quantum machine learning and quantum chemistry, unlike the quantum algorithms (i.e., Shor, Simon, and Grover) mentioned above. As such, althoughthere are various algorithms related to quantum computers, it is common to use [PERSON]'s algorithm in the symmetric-key cryptosystem. While it is acknowledged that a functional quantum computer has yet to see the light of the day, the researchers have been preemptively working against this emerging threat. There has also been a standardization effort by National Institute of Standards and Technology (NIST), a security organization run by the US government, to classify the quantum security level with respect of Advanced Encryption Standard (AES) for encryption and Secure Hash Algorithm (SHA) for hash [12]. In July 2022, four selected Post-Quantum Cryptography (PQC) algorithms have announced and created a new draft standard for the corresponding algorithms currently in progress [13]. Now, in order to apply the [PERSON]'s search algorithm, one needs to implement the cipher in quantum logic--the search complexity varies based on the quantum cost of the cipher implementation. Naturally, researchers have noted this, and have been working on finding an optimized quantum implementation of symmetric key ciphers (including [14, 15], etc.). In order to reduce the quantum cost, while the previous optimization research has focused only on reducing the number of qubits [16, 17], more recent research has further been considering reducing the overall depth and \(T\)-depth of quantum circuits [18, 19, 20]. Most of these studies also deal with estimating the attack cost of recovering the secret key of a symmetric key cipher via [PERSON]'s algorithm to assess security strength. At the same time, we have also witnessed a new design in the symmetric key ciphers, referred to as lightweight ciphers, which are specialized to work on tightly constrained devices [21]. Since lightweight block ciphers are essential to reduce costs with similar performance, methods of optimizing and implementing symmetric keys in a linear layer for more efficient use in lightweight environments have been steadily discussed. This includes studies [22, 23] that optimize methods using matrix decomposition, such as PLU factorization and Gaussian-Jordan elimination, by applying them to symmetric keys.