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arXiv:2505.00701

A log-depth in-place quantum Fourier transform that rarely needs ancillas

Gregory D. Kahanamoku–Meyer, John Blue, Thiago Bergamaschi, Craig Gidney, Isaac L. Chuang

Published May 1, 2025• 0 citations
arXivHTMLPDF

Abstract

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 $O(\log (n / ε))$ for tunable error parameter $ε$, uses $n$ 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 $ε$ on all input states except an $O(ε)$-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 $2n + O(n/\log n)$ 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 $O(\log (n/ε))$ with a sublinear number of ancilla qubits. The reduction uses long-range gates but no measurements.

DOI
10.48550/arxiv.2505.00701

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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.
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.
Quantum computation is traditionally expressed in terms of quantum bits, or qubits. In this work, we instead consider three-level qu trits . Past work with qutrits has demonstrated only constant factor improvements, owing to the log 2 (3) binary-to-ternary compression factor. We present a novel technique, intermediate qutrits, to achieve sublinear depth decompositions of the Generalized Toffoli and other arithmetic circuits using no additional ancilla—a significant improvement over linear depth for the best qubit-only equivalents. For example, our Generalized Toffoli construction features a 70× improvement in two-qudit gate count over a qubit-only decomposition. This results in circuit cost reductions for important algorithms like quantum neurons, Grover search, and even Shor’s algorithm. Using a previously developed simulator with near-term noise models, we demonstrate for these models over 90% mean reliability (fidelity) for the Toffoli construction, versus under 30% for the qubit-only baseline. For our other constructions, such as the Incrementer, the A + B adder and the +K adder, we demonstrate the power of intermediate qutrits in producing asymptotic depth improvements with no additional ancilla. Together, these results suggest qutrits offer a promising path toward scaling quantum computation.
Unitary synthesis underlies universal quantum computation by translating unitary transformations into circuits of elementary gates. We show that for any budget m≥0m\geq 0m≥0 of ancilla, every nnn-qubit unitary can be exactly implemented using a quantum circuit consisting of O(4n)O(4^n)O(4n) CNOT and single-qubit gates in O(n+4n/(n+m))O(n+4^n/(n+m))O(n+4n/(n+m)) layers attaining the asymptotically optimal depth--ancilla tradeoff while simultaneously achieving optimal circuit size. The key idea is to encode computational basis states as wavefunctions sharing a Gaussian envelope, converting unitary action on amplitudes into coordinate transformations that admit shallow circuit implementations. The continuous encoding is then discretized, with approximation errors removed by exact residual correction. The depth-ancilla tradeoff follows by recursively applying the quantum Shannon decomposition, synthesizing its smaller unitary factors, and reusing ancillary qubits.
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.
A promising avenue for scaling quantum computing is to connect quantum processing units (QPUs) by generating entanglement between them. This requires circuit partitioning: partially rewriting quantum circuits to run on a distributed quantum system using quantum teleportation protocols, while preserving the unitary operation implemented by the circuit. The key metric to minimize when partitioning is the e-bit count, defined as the number of maximally entangled qubit pairs that must be generated between QPUs. We focus on partitioning the quantum Fourier transform (QFT) circuit, which is widely used as a subroutine in quantum algorithms such as quantum phase estimation and arithmetic circuits. Specifically, we present an optimal gate-packing-based partitioning scheme under bounded communication ancilla constraints, as well as a novel detached-gate-based partitioning scheme that achieves optimal e-bit consumption when ancilla availability is unrestricted. We compare these schemes against prior analytical partitioning approaches for the QFT and evaluate them against partitions produced by general-purpose circuit partitioning algorithms. We further validate our approach by implementing the bounded-ancilla circuits on quantum hardware.