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arXiv:quant-ph/0403071

Improved Bounds for the Approximate QFT

Donny Cheung

Published Mar 9, 2004• 11 citations
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Abstract

It has previously been established that the logarithmic-depth approximate quantum Fourier transform (AQFT) provides a suitable replacement for the regular QFT in many quantum algorithms. Since the AQFT is less accurate by definition, polynomially many more applications of the AQFT are required to achieve the original accuracy. However, in many quantum algorithms, the smaller size of the AQFT circuit yields a net improvement over using the QFT. This paper presents a more thorough analysis of the AQFT circuit, which leads to the surprising conclusion that for sufficiently large input sizes, the difference between the QFT and the logarithmic-depth AQFT is negligible. In effect, the AQFT can be used as an direct replacement for the QFT, yielding improvements in any application which does not require exact quantum computation.

DOI
10.48550/arxiv.quant-ph/0403071

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It has previously been established that the logarithmic-depth approximate quantum Fourier transform (AQFT) provides a suitable replacement for the regular QFT in many quantum algorithms. Since the AQFT is less accurate by definition, polynomially many more applications of the AQFT are required to achieve the original accuracy. However, in many quantum algorithms, the smaller size of the AQFT circuit yields a net improvement over using the QFT. This paper presents a more thorough analysis of the AQFT circuit, which leads to the surprising conclusion that for sufficiently large input sizes, the difference between the QFT and the logarithmic-depth AQFT is negligible. In effect, the AQFT can be used as an direct replacement for the QFT, yielding improvements in any application which does not require exact quantum computation.
The approximate quantum Fourier transform (AQFT) on nnn qubits can be implemented in logarithmic depth using 8n8n8n qubits with all-to-all connectivity, as shown in [Hales, PhD Thesis Berkeley, 2002]. However, realizing the required all-to-all connectivity can be challenging in practice. In this work, we use dynamic circuits, i.e., mid-circuit measurements and feed-forward operations, to implement the AQFT in logarithmic depth using only 4n4n4n qubits arranged on a line with nearest-neighbor connectivity. Furthermore, for states with a specific structure, the number of qubits can be further reduced to 2n2n2n while keeping the logarithmic depth and line connectivity. As part of our construction, we introduce a new implementation of an adder with logarithmic depth on a line, which allows us to improve the AQFT construction of Hales.
We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n+log log(1//spl epsiv/)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2/sup n/ with error bounded by /spl epsiv/. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n//spl epsiv/)). As an application of this depth bound, we show that P. Shor's (1997) factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomial size, in combination with classical polynomial-time pre- and postprocessing. Next, we prove an /spl Omega/(log n) lower bound on the depth complexity of approximations of the QFT with constant error. This implies that the above upper bound is asymptotically tight (for a reasonable range of values of /spl epsiv/). We also give an upper bound of O(n(log n)/sup 2/ log log n) on the circuit size of the exact QFT modulo 2/sup n/, for which the best previous bound was O(n/sup 2/). Finally, based on our circuits for the QFT with power-of-2 moduli, we show that the QFT with respect to an arbitrary modulus m can be approximated with accuracy /spl epsiv/ with circuits of depth O((log log m)(log log 1//spl epsiv/)) and size polynomial in log m+log(1//spl epsiv/).
We present an asymptotically improved algorithm for implementing the Quantum Fourier Transform (QFT) in both the exact and approximate settings. Historically, the approximate QFT has been implemented in Θ(nlog⁡n)Θ(n \log n)Θ(nlogn) gates, and the exact in Θ(n2)Θ(n^2)Θ(n2) gates. In this work, we show that these costs can be reduced by leveraging a novel formulation of the QFT that recurses on two partitions of the qubits. Specifically, our approach yields an Θ(n(log⁡log⁡n)2)Θ(n(\log \log n)^2)Θ(n(loglogn)2) algorithm for the approximate QFT using Θ(log⁡n)Θ(\log n)Θ(logn) ancillas, and an Θ(n(log⁡n)2)Θ(n(\log n)^2)Θ(n(logn)2) algorithm for the exact QFT requiring Θ(n)Θ(n)Θ(n) ancillas.
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.
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.