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arXiv:2208.14249

Optimizing the number of CNOT gates in one-dimensional nearest-neighbor quantum Fourier transform circuit

Byeongyong Park, Doyeol Ahn

Published Aug 30, 2022• 0 citations
arXivHTMLPDF

Abstract

The physical limitations of quantum hardware often require nearest-neighbor qubit structures, in which two-qubit gates are required to construct nearest-neighbor quantum circuits. However, two-qubit gates are considered a major cost of quantum circuits because of their high error rate as compared with single-qubit gates. The controlled-not (CNOT) gate is the typical choice of a two-qubit gate for universal quantum circuit implementation together with the set of single-qubit gates. In this study, we construct a one-dimensional nearest-neighbor circuit of quantum Fourier transform (QFT), which is one of the most frequently used quantum algorithms. Compared with previous studies on n-qubit one-dimensional nearest-neighbor QFT circuits, it is found that our method reduces the number of CNOT gates by ~60%. Additionally, we showed that our results for the one-dimensional nearest-neighbor circuit can be applied to quantum amplitude estimation.

DOI
10.48550/arxiv.2208.14249

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The physical limitations of quantum hardware often require nearest-neighbor qubit structures, in which two-qubit gates are required to construct nearest-neighbor quantum circuits. However, two-qubit gates are considered a major cost of quantum circuits because of their high error rate as compared with single-qubit gates. The controlled-not (CNOT) gate is the typical choice of a two-qubit gate for universal quantum circuit implementation together with the set of single-qubit gates. In this study, we construct a one-dimensional nearest-neighbor circuit of quantum Fourier transform (QFT), which is one of the most frequently used quantum algorithms. Compared with previous studies on n-qubit one-dimensional nearest-neighbor QFT circuits, it is found that our method reduces the number of CNOT gates by ~60%. Additionally, we showed that our results for the one-dimensional nearest-neighbor circuit can be applied to quantum amplitude estimation.
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In the paper, we consider quantum circuits for the Quantum Fourier Transform (QFT) algorithm. The QFT algorithm is a very popular technique used in many quantum algorithms. We present a generic method for constructing quantum circuits for this algorithm implementing on quantum devices with restrictions. Many quantum devices (for example, based on superconductors) have restrictions on applying two-qubit gates. These restrictions are presented by a qubit connectivity graph. Typically, researchers consider only the linear nearest neighbor (LNN) architecture of the qubit connection, but current devices have more complex graphs. We present a method for arbitrary connected graphs that minimizes the number of CNOT gates in the circuit for implementing on such architecture. We compare quantum circuits built by our algorithm with existing quantum circuits optimized for specific graphs that are Linear-nearest-neighbor (LNN) architecture, ``sun'' (a cycle with tails, presented by the 16-qubit IBMQ device) and ``two joint suns'' (two joint cycles with tails, presented by the 27-qubit IBMQ device). Our generic method gives similar results with existing optimized circuits for ``sun'' and ``two joint suns'' architectures, and a circuit with slightly more CNOT gates for the LNN architecture. At the same time, our method allows us to construct a circuit for arbitrary connected graphs.
The quantum Fourier transform (QFT) is a crucial subroutine in many quantum algorithms. In this paper, we study the exact lower bound problem of CNOT gate complexity for fault-tolerant QFT. First, we consider approximating the ancilla-free controlled-RkR_kRk​ in the QFT logical circuit with a standard set of universal gates, aiming to minimize the number of T gates. Various single-qubit gates are generated in addition to CNOT gates when the controlled-RkR_kRk​ is decomposed in different ways, we propose an algorithm that combines numerical and analytical methods to exactly compute the minimum T gate count for approximating any single-qubit gate with any given accuracy. Afterwards, we prove that the exact lower bound problem of T gate complexity for the QFT is NP-complete. Furthermore, we provide the transversal implementation of universal quantum gates and prove that it has the minimum number of CNOT gates and analyze the minimum CNOT count for transversally implementing the T gate. We then exactly compute the exact lower bound of CNOT gate complexity for fault-tolerant QFT with the current maximum fault-tolerant accuracy 10^{-2}. Our work can provide a reference for designing algorithms based on active defense in a quantum setting.
In the paper, we consider quantum circuits for the Quantum Fourier Transform (QFT) algorithm. We present a generic method for constructing quantum circuits for QFT on quantum devices with restrictions. Many quantum devices (for example, based on superconductors) have restrictions on applying two-qubit gates presented by a qubit connectivity graph. Typically, researchers consider only the linear nearest neighbor (LNN) architecture, but current devices have more complex graphs. We present a method for arbitrary connected graphs that minimizes the number of CNOT gates in the circuit for implementing on such architecture.
Multi-controlled single-target (MC) gates are some of the most crucial building blocks for varied quantum algorithms. How to implement them optimally is thus a pivotal question. To answer this question in an architecture-independent manner, and to get a worst-case estimate, we should look at a linear nearest-neighbor (LNN) architecture, as this can be embedded in almost any qubit connectivity. Motivated by the above, here we describe a method which implements MC gates using no more than ∼4k+8n\sim 4k+8n∼4k+8n CNOT gates -- up-to 60%60\%60% reduction over state-of-the-art -- while allowing for complete flexibility to choose the locations of nnn controls, the target, and a dirty ancilla out of kkk qubits. More strikingly, in case k≈nk \approx nk≈n, our upper bound is ∼12n\sim 12n∼12n -- the best known for unrestricted connectivity -- and if n=1n = 1n=1, our upper bound is ∼4k\sim 4k∼4k -- the best known for a single long-range CNOT gate over kkk qubits -- therefore, if our upper bound can be reduced, then the cost of one or both of these simpler versions of MC gates will be immediately reduced accordingly. In practice, our method provides circuits that tend to require fewer CNOT gates than our upper bound for almost any given instance of MC gates.
Quantum circuits should comply with the Nearest Neighbor Architecture (NNA), which primarily supports only two-qubit operations between adjacent qubits. Typically, con-verting a quantum circuit to an NNA -compliant circuit involves the insertion of SWAP gates. However, employing Gaussian Elimination often results in a smaller NNA -compliant quantum circuit. If we fix the order of qubits to apply Gaussian Elimination (GE), it appears there is no room to improve the GE-based synthesis method. This paper reveals that we can enhance the GE-based method by inserting CNOT gates before and/or after a target circuit in many cases. As a result, we achieve a reduction of approximately 16% in the number of CNOT gates by inserting CNOT gates into the target circuits compared to the original GE-based method. Additionally, we utilize a Simulated Annealing (SA)-based method to achieve an optimal circuit, resulting in an 18% reduction in the number of CNOT gates by inserting CNOT gates, compared to the original GE-based method.