RESOURCE EFFICIENT DESIGN OF QUANTUM CIRCUITS FOR CRYPTANALYSIS AND SCIENTIFIC COMPUTING APPLICATIONS

Quantum computers offer the potential to extend our abilities to tackle computational problems in fields such as number theory, encryption, search and scientific computation. Up to a superpolynomial speedup has been reported for quantum algorithms in these areas. Motivated by the promise of faster computations, the development of quantum machines has caught the attention of both academics and industry researchers. Quantum machines are now at sizes where implementations of quantum algorithms or their components are now becoming possible. In order to implement quantum algorithms on quantum machines, resource efficient circuits and functional blocks must be designed. In this work, we propose quantum circuits for Galois and integer arithmetic. These quantum circuits are necessary building blocks to realize quantum algorithms. The design of resource efficient quantum circuits requires the designer takes into account the gate cost, quantum bit (qubit) cost, depth and garbage outputs of a quantum circuit. Existing quantum machines do not have many qubits meaning that circuits with high qubit cost cannot be implemented. In addition, quantum circuits are more prone to errors and garbage output removal adds to overall cost. As more gates are used, a quantum circuit sees an increased rate of failure. Failures and error rates can be countered by using quantum error correcting codes and fault tolerant implementations of universal gate sets (such as Clifford+T gates). However, Clifford+T gates are costly to implement with the T gate being significantly more costly than the Clifford gates. As a result, designers working with Clifford+T gates seek to minimize the number of T gates (T-count) and the depth of T gates (T-depth). In this work, we propose quantum circuits for Galois and integer arithmetic with lower T-count, T-depth and qubit cost than existing work. This work presents novel quantum circuits for squaring and exponentiation over binary extension fields (Galois fields of form GF(2 m )). The proposed circuits are shown to have lower depth, qubit and gate cost to existing work. We also present quantum circuits for the core operations of multiplication and division which enjoy lower T-count, T-depth and qubit costs compared to existing work. This work also illustrates the design of a T-count and qubit cost efficient design for the square root. This work concludes with an illustration of how the arithmetic circuits can be combined into a functional block to implement quantum image processing algorithms

Optimized Noise Suppression for Quantum Circuits

Quantum computation promises to advance a wide range of computational tasks. However, current quantum hardware suffers from noise and is too small for error correction. Thus, accurately utilizing noisy quantum computers strongly relies on noise characterization, mitigation, and suppression. Crucially, these methods must also be efficient in terms of their classical and quantum overhead. Here, we efficiently characterize and mitigate crosstalk noise, which is a severe error source in, e.g., cross-resonance based superconducting quantum processors. For crosstalk characterization, we develop a simplified measurement experiment. Furthermore, we analyze the problem of optimal experiment scheduling and solve it for common hardware architectures. After characterization, we mitigate noise in quantum circuits by a noise-aware qubit routing algorithm. Our integer programming algorithm extends previous work on optimized qubit routing by swap insertion. We incorporate the measured crosstalk errors in addition to other, more easily accessible noise data in the objective function. Furthermore, we strengthen the underlying integer linear model by proving a convex hull result about an associated class of polytopes, which has applications beyond this work. We evaluate the proposed method by characterizing crosstalk noise for a complete 27 qubit chip and leverage the resulting data to improve the approximation ratio of the Quantum Approximate Optimization Algorithm by up to 10 % compared to other established noise-aware routing methods. Our work clearly demonstrates the gains of including noise data when mapping abstract quantum circuits to hardware native ones

Efficient Arguments and Proofs for Batch Arithmetic Circuit Satisfiability

In this paper, we provide a systematic treatment for the batch arithmetic circuit satisfiability and evaluation problem. Building on the core idea which treats circuit inputs/outputs as a low-degree polynomials, we explore various interactive argument and proof schemes that can produce succinct proofs with short verification time. In particular, for the batch satisfiability problem, we provide a construction of succinct interactive argument of knowledge for generic log-space uniform circuits based on the bilinear pairing and common reference string assumption. Our argument has size in $O(poly(\lambda) \cdot (|\mathbf{w}| + d \log |C|))$, where $\lambda$ is the security parameter, $|\mathbf{w}|$ is the size of the witness, and $d$ and $|C|$ are the depth and size of the circuit, respectively. Note that the argument size is independent of the batch size. To the best of our knowledge, asymptotically it is the smallest among all known batch argument schemes that allow public verification. The batch satisfiablity problem simplifies to a batch evaluation problem when the circuit only takes in public inputs (i.e., no witness). For the evaluation problem, we construct statistically sound interactive proofs for various special yet highly important types of circuits, including linear circuits, and circuits representing sum of polynomials. Our proposed protocols are able to achieve proof sizes independent of the batch size. We also describe protocols optimized specifically for batch FFT and batch matrix multiplication which achieve desirable properties, including lower prover time and better composability. We believe these protocols are of interest in their own right and can be used as primitives in more complex applications

Quantum support vector data description for anomaly detection

Anomaly detection is a critical problem in data analysis and pattern recognition, finding applications in various domains. We introduce quantum support vector data description (QSVDD), an unsupervised learning algorithm designed for anomaly detection. QSVDD utilizes a shallow-depth quantum circuit to learn a minimum-volume hypersphere that tightly encloses normal data, tailored for the constraints of noisy intermediate-scale quantum (NISQ) computing. Simulation results on the MNIST and Fashion MNIST image datasets demonstrate that QSVDD outperforms both quantum autoencoder and deep learning-based approaches under similar training conditions. Notably, QSVDD offers the advantage of training an extremely small number of model parameters, which grows logarithmically with the number of input qubits. This enables efficient learning with a simple training landscape, presenting a compact quantum machine learning model with strong performance for anomaly detection.Comment: 14 pages, 5 figure