The Straddling Gates Problem in Multi-partite Quantum Systems
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It is well known that an arbitrary n-qubit quantum state |Ψ⟩ can be prepared with Θ(2^n) two-qubit gates. In this work, we investigate the task in a "straddling gates" scenario: consider n qubits divided equally into two sets and gates within each set are free; what is the least cost of two-qubit gates straddling the sets (also known as the "binding complexity") for preparing an arbitrary quantum state, assuming no ancilla qubits allowed? In this work, we give an algorithm that fulfills the task with O(n^2 2^n/2) straddling gates, which nearly matches the lower bound to a lower order factor. We then prove any U(2^n) decomposition requires no more than O(2^n) straddling gates. This resolves an open problem posed by Vijay Balasubramanian, who was motivated by the "Complexity=Volume" conjecture in AdS/CFT theory. Furthermore, we extend our discussion to multi-partite systems, define a novel binding complexity class, the "Schmidt decomposable" states, and give a circuit construction explanation for its unique property. Lastly, we reveal binding complexity's significance, comparing it to Von Neumann entropy as an entanglement measure.
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