Statistics of Partial Measurements in Random Quantum Circuits

The bit-string probabilities of a random pure quantum state are known to follow an exponential distribution when all qubits are measured. In this work, we extend this result to the case of partial measurements on random states. We demonstrate that the bit-string probability distribution arising from partial measurements follows a scaled beta distribution, with shape parameters determined by the Hilbert-space dimension of the unmeasured subsystem. In the limit of large system size, this distribution converges to a gamma distribution and reduces to the exponential form under full measurement. When only a small number of qubits are measured, the distribution becomes approximately Gaussian, sharply peaked around the mean probability. We further derive the corresponding distributions for random states subject to depolarizing noise and show that depolarization scales and shifts the distribution in proportion to its strength. These analytical results establish a unified statistical framework for partial-measurement outcomes in random quantum circuits and enable efficient validation of random-circuit sampling using partial measurement data.