Computational advantage from quantum-controlled ordering of gates

Abstract: It is usually assumed that a quantum computation is performed by applying gates in a specific order. One can relax this assumption by allowing a control quantum system to switch the order in which the gates are applied. This provides a more general kind of quantum computing, that allows transformations on blackbox quantum gates that are impossible in a circuit with fixed order. Here we show that this model of quantum computing is physically realizable, by proposing an interferometric setup that can implement such a quantum control of the order between the gates. We show that this new resource provides a reduction in computational complexity: we propose a problem that can be solved using $O(n)$ blackbox queries, whereas the best known quantum algorithm with fixed order between the gates requires $O(n^2)$ queries. Furthermore, we conjecture that solving this problem in a classical computer takes exponential time, which may be of independent interest.

• The paper introduces quantum-controlled ordering of gates to achieve a clear computational advantage through reduced query complexity.
• It employs a model using superpositions of gate orders to solve a unitary matrix problem in O(n) queries versus O(n^2) for fixed-order circuits.
• The work conjectures that classical approaches require exponential time, marking a paradigm shift in quantum computation.

Computational Advantage from Quantum-Controlled Ordering of Gates

The paper "Computational advantage from quantum-controlled ordering of gates" introduces a novel model of quantum computation that allows for the quantum control of the order in which quantum gates are applied. This control results in a more generalized approach to quantum computing, expanding beyond the traditional fixed-order quantum circuit framework. Central to this model is the concept of "superpositions of orders," which can achieve tasks previously deemed impossible within the standard quantum circuit framework.

Key Contributions

The main contribution of this paper is the demonstration that a quantum computation model incorporating dynamic order control of gates can yield a computational advantage. Specifically, the authors present a problem concerning sets of unitary matrices, where the quantum control of gate order enables a solution with significantly reduced query complexity.

• Problem Definition: The problem involves a set of $n$ unitary matrices, alongside a promise that these matrices satisfy one of $n!$ specific properties. The task is to identify which property is satisfied using blackbox queries to the matrices. The structure of the problem is tightly linked to the permutation of unitary operators, implicating a combinatorial nature catered towards the proposed quantum model.
• Query Complexity Reduction: The authors provide an algorithm employing a quantum-controlled order of gate application, achieving a query complexity of $O(n)$ for solving the problem. In the traditional fixed-order quantum circuit model, the best-known algorithm demands $O(n^2)$ queries. This emphasizes the concrete computational benefit attained from controlling the order of gate application.
• Conjectured Classical Intractability: Further, it is conjectured that classical approaches would require exponential time to solve the same problem, underscoring a potential exponential separation in efficiency between quantum and classical computation within the proposed scenario.

Theoretical and Practical Implications

Theoretically, this research suggests that quantum circuit models may benefit from extending beyond fixed-order constraints, potentially broadening the scope of complexity classes accessible via quantum computation. The approach presents a paradigm shift where the underlying architecture of computation itself—the order of operations—becomes a resource to be optimized and controlled.

Practically, the paper proposes an interferometric setup for realizing these computations, indicating the feasibility of experimental implementations. For lower dimensions (e.g., a 2-switch setup), realizations with current quantum optics techniques are plausible. However, scaling such setups presents significant challenges, necessitating further advancements in quantum control and engineering.

Future Directions

The research opens avenues for investigating other computational problems under this new paradigm, potentially exploring more substantial complexity reductions or uncovering novel quantum algorithms leveraging order control. Additionally, further study into experimental implementations could lead to practical quantum circuits that exploit dynamic order control, bridging the gap between theoretical models and real-world quantum computing applications.

Conclusively, while maintaining a polynomial complexity reduction, the proposed model hints at fertile ground for future exploration, inviting inquiry into the intersection of computational order properties and quantum information science advances.