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arXiv:2406.01350

Realization of permutation groups by quantum circuit

Junchi Liu, Yangyang Ren, Yan Cao, Hanyi Sun, Lin Chen

Published Jun 3, 2024• 1 citation
arXivHTMLPDF

Abstract

In this paper, we exclusively utilize CNOT gates for implementing permutation groups generated by more than two elements. In Lemma 1, we recall that three CNOT gates are both necessary and sufficient to execute a two-qubit swap gate operation. Subsequently, in Lemma 2, we show that the maximum number of CNOT gates needed to carry out an n-qubit substitution operation is 3(n-1). Moving forward, our analysis in Section 3 reveals that utilizing five or fewer CNOT gates is insufficient for implementing a three-qubit swap gate corresponding to the permutation element (123). Hence six CNOT gates are both necessary and sufficient for implementing (123). This is done by employing a graph-theoretic approach to rigorously validate the results in terms of at most five CNOT gates. Using computational tools, we exhaustively explore all valid circuit diagrams containing exactly six CNOT gates to successfully execute the swap gate for (123), by explaining the equivalence classes in Remark 6 and Table 2. We conclude them in Theorem 7.To extend our analysis to the multiqubit scenario, we present the reducible and irreducible permutation elements in Definition 8. We clarify the equivalence between rows in the multi-qubit space and provide an approximate upper bound for multi-qubits to perform the aforementioned operations in Theorem 9. The comprehensive exploration of this paper aims to pave the way for further advancements in understanding quantum circuit optimization via multiple use of a specific two-qubit gate.

DOI
10.48550/arxiv.2406.01350

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In this paper, we exclusively utilize CNOT gates for implementing permutation groups generated by more than two elements. In Lemma 1, we recall that three CNOT gates are both necessary and sufficient to execute a two-qubit swap gate operation. Subsequently, in Lemma 2, we show that the maximum number of CNOT gates needed to carry out an n-qubit substitution operation is 3(n-1). Moving forward, our analysis in Section 3 reveals that utilizing five or fewer CNOT gates is insufficient for implementing a three-qubit swap gate corresponding to the permutation element (123). Hence six CNOT gates are both necessary and sufficient for implementing (123). This is done by employing a graph-theoretic approach to rigorously validate the results in terms of at most five CNOT gates. Using computational tools, we exhaustively explore all valid circuit diagrams containing exactly six CNOT gates to successfully execute the swap gate for (123), by explaining the equivalence classes in Remark 6 and Table 2. We conclude them in Theorem 7.To extend our analysis to the multiqubit scenario, we present the reducible and irreducible permutation elements in Definition 8. We clarify the equivalence between rows in the multi-qubit space and provide an approximate upper bound for multi-qubits to perform the aforementioned operations in Theorem 9. The comprehensive exploration of this paper aims to pave the way for further advancements in understanding quantum circuit optimization via multiple use of a specific two-qubit gate.
Quantum computers offer enormous speed advantages over their classical counterparts. Still, optimization on quantum circuits is necessary to further increase their potential. Additionally, physical realizations of quantum computers place restrictions on quantum circuits, regarding the available quantum gates. In order to satisfy these restrictions, non-native gates need to be expressed as an equivalent cascade of natively available quantum gates which induces a mapping overhead. Two complementary approaches to this problem are to move around the qubits (using SWAP gates) or to apply so-called remote gates, i.e. pre-computed cascades of native gates which keep the qubit placement.In this paper, we explore how combinations of movements and remote gates can be employed to reduce the required overhead regarding the number of native gates as well as the circuit depth. We also discuss ways to find out which qubits to address with the movements in order to optimize these metrics. Our general evaluation is supplemented by evaluations on two IBM quantum computer architectures to show how quantum circuits can be optimized by the presented patterns.
IBM architectures impose some restrictions the quantum circuits that can be implemented. Only gates from Clifford+T gate library can be used and not all of the CNOT gates are available. Some of the CNOT need to be mapped into a sequence of gates. In this paper we present a set of mappings to conform to the restrictions imposed by IBM's architectures. These mappings require fewer gates than SWAP gates. It is well known, that permuting the qubits will yield circuits with different number of gates. The design in this paper uses efficient mappings with qubit permutations to obtain circuits with a reduced number of gates. Results have shown that the proposed approach reduces circuits by up to 64% compared with Qiskit and up to 42% compared with another mapping algorithm.
We perform optimal-control-theory calculations to determine the minimum number of two-qubit controlled-not (cnot) gates needed to perform quantum state preparation and unitary operator synthesis for few-qubit systems. By considering all possible gate configurations, we determine the maximum achievable fidelity as a function of quantum circuit size. This information allows us to identify the minimum circuit size needed for a specific target operation and enumerate the different gate configurations that allow a perfect implementation of the operation. We find that there are a large number of configurations that all produce the desired result, even at the minimum number of gates. We also show that the number of entangling gates can be reduced if we use multiqubit entangling gates instead of two-qubit cnot gates, as one might expect based on parameter counting calculations. In addition to treating the general case of arbitrary target states or unitary operators, we apply the numerical approach to the special case of synthesizing the multiqubit Toffoli gate. This approach can be used to investigate any other specific few-qubit task and provides insight into the tightness of different bounds in the literature.
Despite rapid advances in quantum computing technologies, the qubit connectivity limitation remains to be a critical challenge. Both near-term NISQ quantum computers and relatively long-term scalable quantum architectures do not offer full connectivity. As a result, quantum circuits may not be directly executed on quantum hardware, and a quantum compiler needs to perform qubit routing to make the circuit compatible with the device layout. During the qubit routing step, the compiler inserts SWAP gates and performs circuit transformations. Given the connectivity topology of the target hardware, there are typically multiple qubit routing candidates. The state-of-the-art compilers use a cost function to evaluate the number of SWAP gates for different routes and then select the one with the minimum number of SWAP gates. After qubit routing, the quantum compiler performs gate optimizations upon the circuit with the newly inserted SWAP gates.In this paper, we observe that the aforementioned qubit routing is not optimal, and qubit routing should not be independent on subsequent gate optimizations. We find that with the consideration of gate optimizations, not all of the SWAP gates have the same basis-gate cost. These insights lead to the development of our qubit routing algorithm, NASSC (Not All Swaps have the Same Cost). NASSC is the first algorithm that considers the subsequent optimizations during the routing step. Our optimization-aware qubit routing leads to better routing decisions and benefits subsequent optimizations. We also propose a new optimization-aware decomposition for the inserted SWAP gates. Our experiments show that the routing overhead compiled with our routing algorithm is reduced by up to 69.30% (21.30% on average) in the number of CNOT gates and up to 43.50% (7.61% on average) in the circuit depth compared with the state-of-the-art scheme, SABRE.
In this study, we delve into optimizing optical quantum swap gate designs, which are crucial in the advancement of quantum computing. Using qINTERCONNECT, we conduct a comparative analysis of swap gate designs, including those based on equivalent circuits like CNOT gates and others utilizing dedicated architectures. Our aim is to compare the balance between operational efficiency and error tolerance in these varying designs, so as to measure their potential for scalability. Through simulations grounded in real-world fabrication parameters, we assess the fidelity and success probabilities of each design approach. Our findings indicate that while equivalent circuit-based designs are traditionally favored in other fields of quantum computing, dedicated swap gate architectures may offer enhanced robustness and scalability in optics. This study provides a comparative understanding of different swap gate designs, elucidating the relative merits and drawbacks of current designs, and offering insights that are relevant for practical applications in quantum computing.
Analysis of converting C0\mathfrak{C^{0}}C0-circuit into C∗\mathfrak{C^{*}}C∗-circuit
A C∗\mathfrak{C^{*}}C∗ -circuit, which was proposed in Asiacypt 2022 by Huang and Sun (Advances in cryptology – ASIACRYPT 2022, pp. 614–644, 2022), can directly perform calculations with the existing quantum states, thereby reducing the use of quantum resources in quantum logic synthesis. We theoretically prove how to convert a C0\mathfrak{C^{0}}C0 -circuit into the corresponding C∗\mathfrak{C^{*}}C∗ -circuit through two lemmas and one theorem. The first lemma proves the interchangeability of CNOT gates and NOT gates by using the equivalence of quantum circuits. The second lemma proves that adding CNOT gates to the front of a quantum circuit whose initial states are all $|0\rangle $ s will not change the output states of the circuit. The theorem is used to describe what kind of C0\mathfrak{C^{0}}C0 -circuit can be transformed into C∗\mathfrak{C^{*}}C∗ -circuit, and the correctness of this transformation is proved. Our work will provide a theoretical basis for converting C0\mathfrak{C^{0}}C0 -circuit to C∗\mathfrak{C^{*}}C∗ -circuit. Then applying the theoretical analysis results to the multiplication over GF(28)\text{GF}(2^{8})GF(28) , the constructed quantum circuit needs 27 Toffoli gates and 118 CNOT gates, which is 15 fewer Toffoli gates and 43 CNOT gates than the current best result. This shows that the method of constructing quantum circuits by using the conversion of C0\mathfrak{C^{0}}C0 -circuit to C∗\mathfrak{C^{*}}C∗ -circuit is very efficient.