Saturation and recurrence of quantum complexity in random quantum circuits

Abstract

Quantum complexity is a measure of the minimal number of elementary operations required to approximately prepare a given state or unitary channel. Recently, this concept has found applications beyond quantum computing-in studying the dynamics of quantum many-body systems and the long-time properties of AdS black holes. In this context Brown and Susskind [1] conjectured that the complexity of a chaotic quantum system grows linearly in time up to times exponential in the system size, saturating at a maximal value, and remaining maximally complex until undergoing recurrences at doubly-exponential times. In this work we prove the saturation and recurrence of the complexity of quantum states and unitaries in a model of chaotic time-evolution based on random quantum circuits, in which a local random unitary transformation is applied to the system at every time step. Importantly, our findings hold for quite general random circuit models, irrespective of the gate set and geometry of qubit interactions. Our results advance an understanding of the long-time behaviour of chaotic quantum systems and could shed light on the physics of black hole interiors. From a technical perspective our results are based on establishing new quantitative connections between the Haar measure and high-degree approximate designs, as well as the fact that random quantum circuits of sufficiently high depth converge to approximate designs.

Matrix concentration inequalities and efficiency of random universal sets of quantum gates

“…Unitaries constituting δ-approximate t-design form -nets for t d 5/2 and δ 3/2 d d 2 [13]. As a direct consequence of property (1) δ-approximate t-designs find numerous applications throughout quantum information, including randomized benchmarking [16,17], information transmission [18], quantum state discrimination [19], criteria for universality of quantum gates [10] and complexity growth [7,[20][21][22]. It is also known that the constant A(S) from the Solovay-Kitaev theorem is inversely proportional to 1 − δ(ν S ), where δ(ν S ) := sup t δ(ν S , t), whenever δ(ν S ) < 1 [15].…”

Concentration of quantum equilibration and an estimate of the recurrence time

“…A given Hamiltonian should also have other later state-independent recurrences. For instance, the recent result for random circuits [51] suggests that a recurrence in complexity of e −i t H might still doubly exponential, but with a larger exponent that Eq. ( 36).…”