[Paper Review] Reassessing the computational advantage of quantum-controlled ordering of gates

Martin J. Renner, Časlav Brukner|arXiv (Cornell University)|Feb 22, 2021

Quantum Computing Algorithms and ArchitectureComputer Science36 references14 citations

TL;DR

This paper re-evaluates the quantum advantage of indefinite causal order in solving the Fourier Promise Problem (FPP), demonstrating that causal quantum algorithms can achieve significantly better query complexity than previously thought. By exploiting structured decomposition of permutation phases and recursive block processing, the authors present a causal algorithm solving the specific FPP in O(n log n) queries and the general FPP in O(n√n) queries—reducing the presumed quantum advantage of the quantum-n-switch from O(n²) to O(n log n) or O(n√n).

ABSTRACT

Research on indefinite causal structures is a rapidly evolving field that has a potential not only to make a radical revision of the classical understanding of space-time but also to achieve enhanced functionalities of quantum information processing. For example, it is known that indefinite causal structures provide exponential advantage in communication complexity when compared to causal protocols. In quantum computation, such structures can decide whether two unitary gates commute or anticommute with a single call to each gate, which is impossible with conventional (causal) quantum algorithms. A generalization of this effect to $n$ unitary gates, originally introduced in M. Ara\'ujo et al., Phys. Rev. Lett. 113, 250402 (2014) and often called Fourier promise problem (FPP), can be solved with the quantum-$n$-switch and a single call to each gate, while the best known causal algorithm so far calls $O(n^2)$ gates. In this work, we show that this advantage is smaller than expected. In fact, we present a causal algorithm that solves the only known specific FPP with $O(n \log(n))$ queries and a causal algorithm that solves every FPP with $O(n\sqrt{n})$ queries. Besides the interest in such algorithms on their own, our results limit the expected advantage of indefinite causal structures for these problems.

Motivation & Objective

• To re-evaluate the claimed computational advantage of quantum-controlled ordering of gates via the quantum-n-switch for solving the Fourier Promise Problem (FPP).
• To develop efficient causal quantum algorithms that match or surpass the query complexity of the quantum-n-switch for FPP instances.
• To demonstrate that the exponential or quadratic advantage of indefinite causal structures is overestimated, by constructing causal alternatives with near-optimal query scaling.

Proposed method

• Proposes a recursive block decomposition of permutations into sub-blocks of size ˆn = n/2, enabling phase accumulation across blocks.
• Uses a control system with ˆk = n/ˆn = 2 control qubits per block to encode permutation orderings and manage phase accumulation.
• Introduces a dual-target system approach: one system |Ψk⟩ processes the forward permutation, and another |Φk⟩ processes the reversed permutation to cancel out unwanted relative phases.
• Employs the factoradic number system to label permutations and map them to control qubit states, enabling systematic phase tracking.
• Derives a key identity (Eq. B.15) showing that the relative phase between a permuted block and the base permutation is inverted in the reversed block, enabling cancellation.
• Applies this phase cancellation mechanism across multiple blocks to preserve the total phase of the full permutation while minimizing gate calls.

Experimental results

Research questions

• RQ1 Can causal quantum algorithms achieve query complexity competitive with the quantum-n-switch for the Fourier Promise Problem?
• RQ2 What is the minimal number of queries required to solve the FPP using only causal quantum circuits?
• RQ3 Does the presumed exponential or quadratic advantage of indefinite causal order in the FPP hold when more efficient causal algorithms are considered?
• RQ4 Can structured decomposition of permutations and phase cancellation reduce the need for superpositions over gate orders?
• RQ5 How does the query complexity of causal algorithms scale with n for both specific and general FPP instances?

Key findings

• A causal algorithm solves the specific FPP with O(n log n) queries, significantly improving upon the previously assumed O(n²) lower bound for causal circuits.
• A general causal algorithm solves every FPP in O(n√n) queries, demonstrating that the quantum-n-switch's advantage is bounded and not exponential.
• The paper proves that the total phase of a permutation can be decomposed into pairwise phases αij, enabling systematic phase tracking and cancellation.
• The phase cancellation mechanism between forward and reversed blocks ensures that the net phase of the full permutation is preserved while reducing gate calls.
• For n = 8, the algorithm uses only 46 black-box unitary calls (down from 56) by omitting redundant target systems and control qubits, showing practical efficiency.
• The results imply that the computational advantage of indefinite causal structures for the FPP is smaller than previously believed, limiting their expected benefit in quantum advantage scenarios.