Quantum algorithms are often represented in terms of quantum circuits operating on ideal (logical) qubits. However, the practical implementation of these algorithms poses significant challenges. Many quantum algorithms require a substantial number of logical qubits, and the inherent susceptibility to errors of quantum computers require quantum error correction. The integration of error correction introduces overhead in terms of both space (physical qubits required) and runtime (how long the algorithm needs to be run for). This paper addresses the complexity of comparing classical and quantum algorithms, primarily stemming from the additional quantum error correction overhead. We propose a comprehensive framework that facilitates a direct and meaningful comparison between classical and quantum algorithms. By acknowledging and addressing the challenges introduced by quantum error correction, our framework aims to provide a clearer understanding of the comparative performance of classical and quantum computing approaches. This work contributes to understanding the practical viability and potential advantages of quantum algorithms in real-world applications. We apply our framework to quantum cryptanalysis, since it is well known that quantum algorithms can break factoring and discrete logarithm based cryptography and weaken symmetric cryptography and hash functions. In order to estimate the real-world impact of these attacks, apart from tracking the development of fault-tolerant quantum computers it is important to have an estimate of the resources needed to implement these quantum attacks. This analysis provides state-of-the art snap-shot estimates of the realistic costs of implementing quantum attacks on these important cryptographic algorithms, assuming quantum fault-tolerance is achieved using surface code methods, and spanning a range of potential error rates. These estimates serve as a guide for gauging the realistic impact of these algorithms and for benchmarking the impact of future advances in quantum algorithms, circuit synthesis and optimization, fault-tolerance methods and physical error rates.

While mapping a quantum circuit to the physical layer one has to consider the numerous constraints imposed by the underlying hardware architecture. Connectivity of the physical qubits is one such constraint that restricts two-qubit operations, such as CNOT, to “connected” qubits. SWAP gates can be used to place the logical qubits on admissible physical qubits, but they entail a significant increase in CNOT-count. In this article, we consider the problem of reducing the CNOT-count in Clifford+T circuits on connectivity-constrained architectures, like noisy intermediate-scale quantum (NISQ) computing devices. We “slice” the circuit at the position of Hadamard gates and “build” the intermediate$\{\text {CNOT},{T}\}$subcircuits using Steiner trees, significantly improving on previous methods. We compared the performance of our algorithms while mapping different benchmark and random circuits to some well-known architectures, such as 9-qubit square grid, 16-qubit square grid, Rigetti 16-qubit Aspen, 16-qubit IBM QX5, and 20-qubit IBM Tokyo. Our methods give less CNOT-count compared to Qiskit and TKET transpiler as well as using SWAP gates. Assuming most of the errors in an NISQ circuit implementation are due to CNOT errors, then our method would allow circuits with a few times more CNOT gates be reliably implemented than the previous methods would permit.