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Quantum Information and Computation

The quantum Fourier transform on a linear nearest neighbor architecture

Yasuhiro Takahashi, Noboru Kunihiro, Kazuo Ohta

Published May 1, 2007• 37 citations
Publisher

Abstract

We show how to construct an efficient quantum circuit for computing a good approximation of the quantum Fourier transform on a linear nearest neighbor architecture. The constructed circuit uses no ancillary qubits and its depth and size are $O(n)$ and $O(n\log n)$, respectively, where $n$ is the length of the input. The circuit is useful for decreasing the size of Fowler et al.'s quantum circuit for Shor's factoring algorithm on a linear nearest neighbor architecture.

DOI
10.26421/qic7.4-7

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We show how to construct an efficient quantum circuit for computing a good approximation of the quantum Fourier transform on a linear nearest neighbor architecture. The constructed circuit uses no ancillary qubits and its depth and size are O(n)O(n)O(n) and O(nlog⁡n)O(n\log n)O(nlogn), respectively, where nnn is the length of the input. The circuit is useful for decreasing the size of Fowler et al.'s quantum circuit for Shor's factoring algorithm on a linear nearest neighbor architecture.
Quantum circuits for elementary arithmetic operations are important not only for implementing Shor's factoring algorithm on a quantum computer but also for understanding the computational power of small quantum circuits, such as linear-size or logarithmic-depth quantum circuits. This paper surveys some recent approaches to constructing efficient quantum circuits for elementary arithmetic operations and their applications to Shor's factoring algorithm. It covers addition, comparison, and the quantum Fourier transform used for addition.
Physical limitations of quantum hardware often necessitate nearest-neighbor (NN) architecture. When synthesizing quantum circuits using the basic gate library, which consists of CNOT and single-qubit gates, CNOT gates are required to convert a quantum circuit into one suitable for an NN architecture. In the basic gate library, CNOT gates are considered the primary cost of quantum circuits due to their higher error rates and longer execution times compared to single-qubit gates. In this paper, we propose a new linear NN (LNN) circuit design for quantum Fourier transform (QFT), one of the most versatile subroutines in quantum algorithms. Our LNN QFT circuit has only about 40% of the number of CNOT gates compared to previously known LNN QFT circuits. Subsequently, we input both our QFT circuits and conventional QFT circuits into the Qiskit transpiler to construct QFTs on IBM quantum computers, which necessitate NN architectures. Consequently, our QFT circuits demonstrate a substantial advantage over conventional QFT circuits in terms of the number of CNOT gates. This outcome implies that the proposed LNN QFT circuit design could serve as a novel foundation for developing QFT circuits implemented in quantum hardware that demands NN architecture.
In this paper, we investigate how quantum architectures affect the efficiency of the execution of the quantum Fourier transform (QFT) and linear transformations, which are essential parts of the stabilizer and Clifford group circuits. In particular, we show that in most common and realistic physical architectures including the linear nearest neighbor, two-dimensional lattice, and bounded degree graph (containing a chain of length nnn), nnn-qubit QFT and nnn-qubit stabilizer circuits can be parallelized to linear depth using no auxiliary qubits. We construct lower bounds that show the efficiency of our approach.
We show how to construct a fast quantum circuit for computing the sum of two nnn-bit binary numbers with few qubits. The constructed circuit uses O(n/log⁡n)O(n/\log n)O(n/logn) ancillary qubits and its depth and size are O(log⁡n)O(\log n)O(logn) and O(n)O(n)O(n), respectively. The number of ancillary qubits is asymptotically less than that in Draper et al.'s quantum carry-lookahead adder, and the depth and size are asymptotically the same as those of Draper et al.'s. Moreover, we show that the circuit is useful for constructing an efficient quantum circuit for Shor's factoring algorithm.
When designing quantum circuits for a given unitary, it can be much cheaper to achieve a good approximation on most inputs than on all inputs. In this work we formalize this idea, and propose that such "optimistic quantum circuits" are often sufficient in the context of larger quantum algorithms. For the rare algorithm in which a subroutine needs to be a good approximation on all inputs, we provide a reduction which transforms optimistic circuits into general ones. Applying these ideas, we build an optimistic circuit for the in-place quantum Fourier transform (QFT). Our circuit has depth O(log⁡(n/ε))O(\log (n / ε))O(log(n/ε)) for tunable error parameter εεε, uses nnn total qubits, i.e. no ancillas, is local for input qubits arranged in 1D, and is measurement-free. The circuit's error is bounded by εεε on all input states except an O(ε)O(ε)O(ε)-sized fraction of the Hilbert space. The circuit is also rather simple and thus may be practically useful. Combined with recent QFT-based fast arithmetic constructions [arXiv:2403.18006], the optimistic QFT yields factoring circuits of nearly linear depth using only 2n+O(n/log⁡n)2n + O(n/\log n)2n+O(n/logn) total qubits. Additionally, we apply our reduction technique to yield an approximate QFT with well-controlled error on all inputs; it is the first to achieve the asymptotically optimal depth of O(log⁡(n/ε))O(\log (n/ε))O(log(n/ε)) with a sublinear number of ancilla qubits. The reduction uses long-range gates but no measurements.