  Quantum random access codes (QRACs) provide a basic tool for demonstrating
the advantages of quantum resources and protocols, which have a wide range of
applications in quantum information processing tasks. However, the
investigation and application of high-dimensional $(d)$ multi-input $(n)$
$n^{(d)}\rightarrow1$ QRACs are still lacking. Here, we present a general
method to find the maximum success probability of $n^{(d)}\rightarrow1$ QRACs.
In particular, we give the analytical solution for maximum success probability
of $3^{(d)}\rightarrow1$ QRACs when measurement bases are mutually unbiased
bases (MUBs). Based on the analytical solution, we show the relationship
between MUBs and $n^{(d)}\rightarrow1$ QRACs. First, we provide a systematic
method of searching for the operational inequivalence of MUBs (OI-MUBs) when
the dimension $d$ is a prime power. Second, we theoretically prove that,
surprisingly, the commonly used Galois MUBs are not the optimal measurement
bases to obtain the maximum success probability of $n^{(d)}\rightarrow1$ QRACs,
which indicates a breakthrough according to the traditional conjecture
regarding the optimal measurement bases. Furthermore, based on high-fidelity
high-dimensional quantum states of orbital angular momentum, we experimentally
achieve two-input and three-input QRACs up to dimension 11. We experimentally
confirm the OI-MUBs when $d=5$. Our results open alternative avenues for
investigating the foundational properties of quantum mechanics and quantum
network coding.
