Simulating the quantum switch with quantum circuits is computationally hard.

Area of Science:

• Quantum Information Science
• Quantum Computing Theory

Background:

• Quantum channels are fundamental building blocks in quantum information processing.
• Indefinite causal order describes processes where the order of operations is not fixed.
• The quantum switch is a key example of a process exhibiting indefinite causal order.

Purpose of the Study:

• To investigate whether higher-order transformations in indefinite causal order, specifically the quantum switch, can be simulated by quantum circuits.
• To establish a quantitative measure of the difference in computational power between indefinite causal order and standard quantum circuits.

Main Methods:

• Theoretical analysis of quantum channel simulation.
• Proof of an exponential separation in quantum query complexity.
• Extension of results to probabilistic and approximate simulation scenarios.

Main Results:

• The quantum switch acting on two n-qubit channels cannot be simulated by a quantum circuit using k calls to one channel and one to the other if k < 2^n.
• An exponential separation in quantum query complexity is proven between indefinite causal order processes and quantum circuits.
• Simulation remains impossible even with one additional call to both input channels.

Conclusions:

• Processes with indefinite causal order possess computational power beyond that of standard quantum circuits.
• There is a fundamental, exponential gap in query complexity that cannot be bridged by simply increasing circuit depth.
• The findings have implications for the design and capabilities of future quantum information processing architectures.