Abstract
We consider quantum circuits consisting of randomly chosen two-local gates and study the number of gates needed for the distribution over measurement outcomes for typical circuit instances to be anti-concentrated, roughly meaning that the probability mass is not too concentrated on a small number of measurement outcomes. Understanding the conditions for anti-concentration is important for determining which quantum circuits are difficult to simulate classically, as anti-concentration has been in some cases an ingredient of mathematical arguments that simulation is hard and in other cases a necessary condition for easy simulation. Our definition of anti-concentration is that the expected collision probability, that is, the probability that two independently drawn outcomes will agree, is only a constant factor larger than if the distribution were uniform. We show that when the 2-local gates are each drawn from the Haar measure (or any two-design), at least $\Omega(n \log(n))$ gates (and thus $\Omega(\log(n))$ circuit depth) are needed for this condition to be met on an $n$ qudit circuit. In both the case where the gates are nearest-neighbor on a 1D ring and the case where gates are long-range, we show $O(n \log(n))$ gates are also sufficient, and we precisely compute the optimal constant prefactor for the $n \log(n)$. The technique we employ relies upon a mapping from the expected collision probability to the partition function of an Ising-like classical statistical mechanical model, which we manage to bound using stochastic and combinatorial techniques.
Highlights
Random quantum circuits (RQCs) are a crucial model for understanding a diverse set of phenomena in both quantum information and quantum many-body physics
The computing of transition amplitudes of RQCs has been shown to be just as difficult as for arbitrary quantum circuits [18–22], suggesting that classical simulation of RQCs should require exponential time. We focus on another property of RQCs called anticoncentration
We show that an (log(n)) lower bound on the depth needed for anticoncentration holds regardless of which RQC architecture we use, which refutes the conjecture from Ref. [32] that 2D RQCs anticoncentrate in O( log(n)) depth
Summary
Random quantum circuits (RQCs) are a crucial model for understanding a diverse set of phenomena in both quantum information and quantum many-body physics. The bit assignments can be interpreted as a Markov chain and the number of gates needed for anticoncentration translates into the time needed for certain expectation values to converge under the dynamics of the Markov chain This method yields sharp quantitative bounds but it produces an appealing qualitative explanation on how and why the collision probability reaches its limiting value, which allows for effective. The fact that anticoncentration occurs in (n log(n)) circuit size both in 1D and for the fully connected architecture—these being two opposite extremes of geometric locality—suggests that anticoncentration may require only (n log(n)) size for any reasonably well-connected architecture This comes in sharp contrast to the situation for unitary designs, where the scaling of the size needed with n is highly dependent on the architecture. The fact that we can prove tight upper and lower bounds suggests a broader utility for our method based on the correspondence between RQCs and statistical-mechanical partition functions
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