Straddling-gates problem in multipartite quantum systems

TL;DR

This paper investigates the minimal number of 'straddling' two-qubit gates needed to prepare quantum states in multipartite systems, revealing bounds and implications for quantum complexity and distributed computation.

Contribution

It establishes bounds on straddling gate complexity for state preparation and extends results to multipartite states, addressing an open problem in quantum complexity theory.

Findings

Theta(2^{k_1}) straddling gates suffice without entanglement assumptions

Any unitary on 2^n qubits can be implemented with Theta(4^{k_1}) straddling gates

Multipartite Schmidt decomposable states have linear binding complexity in the number of parties

Abstract

We study a variant of quantum circuit complexity, the binding complexity: Consider a $n$-qubit system divided into two sets of $k_1$, $k_2$ qubits each ($k_1\leq k_2$) and gates within each set are free; what is the least cost of two-qubit gates ''straddling'' the sets for preparing an arbitrary quantum state, assuming no ancilla qubits allowed? Firstly, our work suggests that, without making assumptions on the entanglement spectrum, $\Theta(2^{k_1})$ straddling gates always suffice. We then prove any $\text{U}(2^n)$ unitary synthesis can be accomplished with $\Theta(4^{k_1})$ straddling gates. Furthermore, we extend our results to multipartite systems, and show that any $m$-partite Schmidt decomposable state has binding complexity linear in $m$, which hints its multi-separable property. This result not only resolves an open problem posed by Vijay Balasubramanian, who was initially motivated by the ''Complexity=Volume'' conjecture in quantum gravity, but also offers realistic applications in distributed quantum computation in the near future.