Abstract
Computational complexity is a quantum information concept that recently has found applications in the holographic understanding of the black hole interior. We consider quantum computational complexity for $n$ qubits using Nielsen's geometrical approach. In the definition of complexity there is a big amount of arbitrariness due to the choice of the penalty factors, which parametrizes the cost of the elementary computational gates. In order to reproduce desired features in holography, such as ergodicity and exponential maximal complexity for large number of qubits $n$, negative curvatures are required. With the simplest choice of penalties, this is achieved at the price of singular sectional curvatures in the large $n$ limit. We investigate a choice of penalties in which we can obtain negative curvatures in a smooth way. We also analyze the relation between operator and state complexities, framing the discussion with the language of Riemannian submersions. This provides a direct relation between geodesics and curvatures in the unitaries and the states spaces, which we also exploit to give a closed-form expression for the metric on the states in terms of the one for the operators. Finally, we study conjugate points for a large number of qubits in the unitary space and we provide a strong indication that maximal complexity scales exponentially with the number of qubits in a certain regime of the penalties space.
Highlights
An important problem in theoretical quantum computation is to determine the best quantum circuit to implement a desired unitary transformation
Part of it comes from the choice of the penalty factors for the Hermitian generators of the unitary transformations, which implement the physical concept that some operations can be harder than others to perform in a quantum circuit
II we review some results of [5] for the complexity geometry in the unitary space for an arbitrary number of qubits and we derive a useful explicit formula for sectional curvatures
Summary
An important problem in theoretical quantum computation is to determine the best quantum circuit to implement a desired unitary transformation. Even in the more traditional paradigm of Riemannian geometry, there is a lot of ambiguity in defining complexity Part of it comes from the choice of the penalty factors for the Hermitian generators of the unitary transformations, which implement the physical concept that some operations can be harder than others to perform in a quantum circuit. In this paper we study conjugate points of complexity metric both for one and for a large number of qubits From this analysis, we find an evidence that maximal complexity scales exponentially with n in the progressive model for large α.
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