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arXiv:1803.04933 · npj Quantum Information

Approximate quantum Fourier transform with O(n log(n)) T gates

Yunseong Nam, Yuan Su, Dmitri A Maslov

Published Mar 13, 2020• 93 citations
arXivHTMLPDF

Abstract

Abstract The ability to implement the Quantum Fourier Transform (QFT) efficiently on a quantum computer facilitates the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, solving systems of linear equations, and phase estimation, to name a few. The standard fault-tolerant implementation of an n-qubit unitary QFT approximates the desired transformation by removing small-angle controlled rotations and synthesizing the remaining ones into Clifford+T gates, incurring the T-count complexity of $$O(n\,{\mathrm{log}}^{2}\,(n))$$ O ( n log 2 ( n ) ) . In this paper, we show how to obtain approximate QFT with the T-count of $$O(n\,{\mathrm{log}}\,(n))$$ O ( n log ( n ) ) . For brevity, the above figures omit the dependence on the approximation error ε, assuming the error is fixed. Our approach relies on quantum circuits with measurements and feedforward, and on reusing a special quantum state that induces the phase gradient transformation. We report asymptotic analysis as well as concrete circuits, demonstrating significant advantages in both theory and practice.

DOI
10.1038/s41534-020-0257-5

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Abstract The ability to implement the Quantum Fourier Transform (QFT) efficiently on a quantum computer facilitates the advantages offered by a variety of fundamental quantum algorithms, such as those for integer factoring, computing discrete logarithm over Abelian groups, solving systems of linear equations, and phase estimation, to name a few. The standard fault-tolerant implementation of an n-qubit unitary QFT approximates the desired transformation by removing small-angle controlled rotations and synthesizing the remaining ones into Clifford+T gates, incurring the T-count complexity of O(n log2 (n))O(n\,{\mathrm{log}}^{2}\,(n))O(nlog2(n)) O ( n log 2 ( n ) ) . In this paper, we show how to obtain approximate QFT with the T-count of O(n log (n))O(n\,{\mathrm{log}}\,(n))O(nlog(n)) O ( n log ( n ) ) . For brevity, the above figures omit the dependence on the approximation error ε, assuming the error is fixed. Our approach relies on quantum circuits with measurements and feedforward, and on reusing a special quantum state that induces the phase gradient transformation. We report asymptotic analysis as well as concrete circuits, demonstrating significant advantages in both theory and practice.
The quantum Fourier transform (QFT) is a fundamental component in various quantum algorithms, including Shor's factoring algorithm and the Harrow-Hassidim-Lloyd (HHL) algorithm for solving systems of linear equations. Efficient implementation of the QFT is essential for the practical realization of large-scale quantum algorithms, especially in fault-tolerant quantum computing. In fault-tolerant implementations, the Clifford + T gate library is the standard choice for building quantum circuits. As the most resource-intensive component within this framework, the T gate's associated cost poses a significant challenge to the efficient implementation of the QFT and its dependent algorithms. While approximate QFT (AQFT) circuits reduce this cost, state-of-the-art implementations still require a T-count of [Formula: see text] and a T-depth of [Formula: see text]. Although these results represent a notable achievement, the associated resource cost remains a primary bottleneck for practical, large-scale quantum algorithms, motivating further optimization. To address this bottleneck, this paper introduces two novel [Formula: see text]-qubit AQFT circuits with an approximation error of [Formula: see text]. Our first design, AQFT Circuit 1, halves the T-count to [Formula: see text] by constructing inverse phase gradient transformation (PGT) circuits without using additional non-Clifford gates and by implementing the inverse PGTs using quantum adders. Our second design, AQFT Circuit 2, reduces the T-depth to [Formula: see text] through parallelization of the inverse PGTs that add only [Formula: see text] additional T gates. For both AQFT circuits, the state-of-the-art linear-depth quantum adder is employed. We demonstrate that employing the linear-depth quantum adder provides advantages over the currently known logarithmic-depth quantum adder, not only in terms of T-count but also in T-depth optimization for the AQFT, particularly within the range [Formula: see text], which encompasses practical system sizes.
The quantum Fourier transform (QFT) is a ubiquitous quantum operation that is used in numerous quantum computing applications. The major obstacle to constructing a QFT circuit is that numerous elementary gates are required. Among the elementary gates, T gates dominate the cost of fault-tolerant implementation. Currently, the smallest-known T-count required to construct an n-qubit QFT circuit approximated to error O(\varepsilon) is ~8nlog_2(n/\varepsilon). Moreover, the depth of T gates (T-depth) in the approximate QFT circuit is ~2nlog_2(n/\varepsilon). This approximate QFT circuit was constructed using Toffoli gates and quantum adders. In this study, we present a new n-qubit QFT circuit approximated to error O(\varepsilon). Our approximate QFT circuit shows a T-count of ~4nlog_2(n/\varepsilon) and a T-depth of ~nlog_2(n/\varepsilon). Toffoli gates, which account for half of the T-count in the approximate QFT circuit reported in the previous study, are unnecessary in our construction. Quantum adders, which dominate the leading order term of T-depth in our approximate QFT circuit, are arranged in parallel to reduce T-depth.
The Quantum Fourier Transform (QFT)andQuantumPhaseEstimation (QPE)are twofundamental algorithmic frameworks in quantum computation, and they provide the mathematical basis of many exponential-speedup quan tum algorithms, such as Shor’s factoring algorithm and quantum simulation techniques. In this work we present a detailed theoretical and computational study of the Quantum Fourier Transform and its integration in the Quan tum Phase Estimation. The discussion begins with the linear algebraic structure of the Hilbert spaces, the unitary transformations, and the complex vector representations that enable quantum interference and superposition. The QFT is studied as the unitary transform of computational basis states in the amplitudes in the frequency domain by phase rotations and tensor-product decompositions. The efficient circuit implementation in terms of Hadamard and controlled phase gates of this transform is discussed. The advantage of the transform in terms of polynomial time complexity over the classical discrete Fourier transform in some quantum scenarios is discussed.The QFT is implemented and the Quantum Phase Estimation algorithm is proposed to obtain the eigenphase information of the unitary operators. In the report the mathematical representation of QPE is derived by using the controlled unitary evolution, the inverse Fourier transforms and the measurement theory in finite dimensional Hilbert spaces. We discuss the probabilistic interpretation of phase extraction in detail and the convergence properties of the bi nary phase approximation. We also discuss the relation of QPE to eigenvalue problems in quantum mechanics and its applications to Hamiltonian simulation, quantum chemistry, cryptography, and hidden subgroup prob lems.Implementation aspects of these algorithms are studied using quantum circuit representations and oracle based computation models. We also discuss error sensitivity, qubit precision requirements, decoherence limita tions and practical constraints of Noisy Intermediate-Scale Quantum (NISQ) hardware. The work synthesizes the mathematical foundations, computational complexity analysis and physical realizations of the QFT and the QPE, demonstrating how the Fourier-based quantum transformations provide the essential mechanism underlying modern quantum algorithmic speedups and scalable quantum information processing.
The inseparable coupling of a quantum system with its environment adds noise at resonant frequencies, causing energy fluctuations and losses entanglement of the fragile quantum state, informally known as decoherence. By truncating Fourier wiggles with the fault-tolerant features, a robust quantum Fourier transform (QFT) scheme can mitigate the decoherence effect. However, a low qubit-count and circuit-depth-based quantum circuit is desirable because of decoherence rate scale up linearly with qubit-count, shortening the coherence time of a fragile quantum state. Indeed, due to the high latency of 1-qubit phase-gates, phase-depth becomes a more dominant factor in circuit-depth and pays a high price for scalability. The above issues are addressed in this manuscript by implementing a fault-tolerant approximate quantum Fourier-transform (AQFT) in linear phase-depth with zero ancillary cost and O(nlog⁡n)\mathcal {O}(n\log n)O(nlogn) phase-count using the Clifford+ ZNZ_NZN​ -group. By truncating lower rotated phase operators, a robust fault-tolerant AQFT is achieved with linear phase-depth and zero ancillary cost, as well as 75%, 87.6%, 94.22%, and 26.99% average improvements in phase-depth, phase-count, CNOT-count, and ancillary cost respectively over the present-state-of-the-arts 1 . The proposed AQFT maintains polynomially small approximation error, achieving a strong balance between accuracy and efficiency that makes it practical for most algorithms and well suited for NISQ hardware.
We propose a quantum analogue of Bluestein's algorithm (QBA) that implements an exact NNN-point Quantum Fourier Transform (QFT) for arbitrary NNN. Our construction factors the NNN-dimensional QFT unitary into three diagonal quadratic-phase gates and two standard radix-2 QFT subcircuits of size M=2mM = 2^mM=2m (with M≥2N−1M \ge 2N - 1M≥2N−1). This achieves asymptotic gate complexity O((log⁡N)2)O((\log N)^2)O((logN)2) and uses O(log⁡N)O(\log N)O(logN) qubits, matching the performance of a power-of-two QFT on mmm qubits while avoiding the need to embed into a larger Hilbert space. We validate the correctness of the algorithm through a concrete implementation in Qiskit and classical simulation, confirming that QBA produces the exact NNN-point discrete Fourier transform on arbitrary-length inputs.
Quantum Fourier transforms (QFTs) are essential primitives in quantum algorithms. While abelian groups admit efficient QFT circuits, with circuit size polynomial in the logarithm of the group order, efficient constructions are known for relatively few non-abelian families. We develop two new approaches to QFT circuit construction, based on Mackey theory and Clifford theory, respectively, and use them to show exponential improvement in circuit cost for specific group families. Using the Mackey-theoretic approach, we obtain explicit quantum circuits for the QFT over GL2(Fq)\mathrm{GL}_2(F_q)GL2​(Fq​) that scale polynomially in log⁡q\log qlogq, rather than polynomially in qqq. Using the Clifford-theoretic approach, we obtain QFT circuits for wreath products F≀SnF\wr S_nF≀Sn​, whose cost depends on the cost of a QFT over FFF and the size of its representation registers. This removes the restriction ∣F∣=poly⁡(n)|F|=\operatorname{poly}(n)∣F∣=poly(n) required by previous generic constructions and can yield exponential improvements when FFF itself has an efficient QFT. Together, these methods provide new systematic tools to construct QFTs for broad classes of finite groups.