A log-depth in-place quantum Fourier transform that rarely needs ancillas (2505.00701v2)

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

Optimistic Quantum Circuits and Their Application to the Quantum Fourier Transform

This paper, authored by Gregory D. Kahanamoku-Meyer et al., explores the concept of optimistic quantum circuits—a class of quantum circuits designed to approximate a desired unitary operation well on most, but not necessarily all, input states. This approach draws inspiration from classical computing techniques where average-case performance is prioritized over worst-case scenarios. In quantum computing, where conditional execution based on quantum states is challenging due to superposition, the paper proposes a novel method to circumvent this limitation.

Optimistic Quantum Circuits

The authors introduce optimistic quantum circuits as circuits providing good approximations of desired unitaries on most bases states but potentially underperforming on certain inputs. These circuits exploit an average-case error, allowing for efficient resource usage by not being overly meticulous with low-probability input errors. The concept is formalized mathematically, with error measurements being basis-independent and favorably smaller over a vast majority of the Hilbert space.

A significant contribution of this work is the proposed methodology to convert optimistic circuits into circuits with guaranteed performance on all inputs. The approach uses unitary 1-designs to achieve randomized reductions, effectively transforming worst-case inputs into average-case scenarios without compromising depth efficiency or computational overhead.

Application to Quantum Fourier Transform

The quantum Fourier transform (QFT) serves as a potent demonstration of optimistic quantum circuits. Quantum phase estimation (QPE)—a critical operation in quantum computing—is leveraged in parallel among qubit blocks to manage dependencies. The proposed optimistic QFT operates with $O(\log(n/\epsilon))$ depth, requires no ancillas, operates with logarithmic locality in 1D-arrangements, and maintains measurement-free mechanics. These attributes are notable improvements in depth and resource optimization compared to traditional approximate QFT constructions.

The simplicity and efficiency of this approach facilitate promising applications in fast arithmetic constructions, among which Shor's algorithm for factoring is a prime beneficiary. Employing optimistic QFTs, this algorithm achieves significant reductions in computational overhead and qubit economy, exemplifying the practical utility of optimistic circuits in streamlined quantum computation tasks.

Implications and Future Directions

The implications of this research are substantial, impacting both practical and theoretical quantum computing landscapes. On a practical level, the adoption of optimistic circuits can lead to considerable resource savings, improved implementation of algorithms such as factoring, and potentially broaden the scope and feasibility of quantum applications significantly.

Theoretically, the framework laid down offers a new lens through which quantum algorithm design can be evaluated, effectively offering pathways to novel constructions that improve upon worst-case scenario designs. Further exploration is likely to expand the utility of optimistic circuits across different domains in quantum computing and inspire parallel advancements in quantum algorithm efficiency.

Future developments can delve into optimization strategies for these circuits, application invariants across quantum operating environments, and the exploration of more complex quantum operations beyond QFT that might benefit from optimistic circuit design.

In conclusion, the paper's analytical and constructive approach adds a robust layer to quantum circuit design strategies, emphasizing optimization of average-case performance and resource efficiency. This paradigm is expected to evolve, leading to innovative applications and potentially bridging gaps between classical and quantum algorithm design methodologies.