Computational and physical complexity of synthesizing random multi-qudit quantum states and unitary operators

TL;DR

This paper investigates the complexity of synthesizing random multi-qudit quantum states and operators, analyzing both computational and physical aspects using analytical and numerical methods.

Contribution

It provides a comparative analysis of the exponential computational complexity and the slower physical complexity in preparing random quantum states and unitaries.

Findings

Computational complexity scales exponentially with the number of qudits.

Physical complexity scales more slowly than computational complexity.

Numerical results support the difference in scaling behaviors.

Abstract

We analyze the complexity of synthesizing random states and unitary operators in a multi-qudit system in two paradigms. In one case, we consider the situation in which we manipulate the system by applying a sequence of one- and two-qudit quantum gates that constitute the elementary, and universal, gate set. The minimum number of gates required to perform the desired operation represents the computational complexity. In the other case, we consider the situation in which we manipulate the physical system using physical fields with optimized control pulses. The minimum time required to perform the desired operation represents the physical complexity. In both cases, we use analytical arguments in combination with optimal-control-theory numerical calculations to determine the complexity of random operations. We show that the computational complexity of random states or unitary operators scales exponentially with the number of qudits. Our numerical results suggest that the physical complexity of preparing random quantum states and unitary operators scales more slowly than the computational complexity. We discuss various implications of our results, especially concerning the relationship between random and pseudorandom states and unitary operators.