We study random quantum circuits with symmetry, where the local 2-site unitaries are drawn from a quotient or subgroup of the full unitary group $U(d)$. Random quantum circuits are minimal models of local quantum chaotic dynamics and can be used to study operator growth and the emergence of diffusive hydrodynamics. We derive the transition probabilities for the stochastic process governing the growth of operators in four classes of symmetric random circuits. We then compute the butterfly velocities and diffusion constants for a spreading operator by solving a simple random walk in each class of circuits.
We study random quantum circuits with symmetry, where the local 2-site unitaries are drawn from a quotient or subgroup of the full unitary group U(d). Random quantum circuits are minimal models of local quantum chaotic dynamics and can be used to study operator growth and the emergence of diffusive hydrodynamics. We derive the transition probabilities for the stochastic process governing the growth of operators in four classes of symmetric random circuits. We then compute the butterfly velocities and diffusion constants for a spreading operator by solving a simple random walk in each class of circuits.
Random quantum circuits yield minimally structured models for chaotic quantum dynamics, which are able to capture, for example, universal properties of entanglement growth.We provide exact results and coarse-grained models for the spreading of operators by quantum circuits made of Haar-random unitaries.We study both 1 þ 1D and higher dimensions and argue that the coarse-grained pictures carry over to operator spreading in generic many-body systems.In 1 þ 1D, we demonstrate that the out-of-time-order correlator (OTOC) satisfies a biased diffusion equation, which gives exact results for the spatial profile of the OTOC and determines the butterfly speed v B .We find that in 1 þ 1D, the "front" of the OTOC broadens diffusively, with a width scaling in time as t 1=2 .We address fluctuations in the OTOC between different realizations of the random circuit, arguing that they are negligible in comparison to the broadening of the front within a realization.Turning to higher dimensions, we show that the averaged OTOC can be understood exactly via a remarkable correspondence with a purely classical droplet growth problem.This implies that the width of the front of the averaged OTOC scales as t 1=3 in 2 þ 1D and as t 0.240 in 3 þ 1D (exponents of the Kardar-Parisi-Zhang universality class).We support our analytic argument with simulations in 2 þ 1D.We point out that, in two or higher spatial dimensions, the shape of the spreading operator at late times is affected by underlying lattice symmetries and, in general, is not spherical.However, when full spatial rotational symmetry is present in 2 þ 1D, our mapping implies an exact asymptotic form for the OTOC, in terms of the Tracy-Widom distribution.For an alternative perspective on the OTOC in 1 þ 1D, we map it to the partition function of an Ising-like statistical mechanics model.As a result of special structure arising from unitarity, this partition function reduces to a random walk calculation which can be performed exactly.We also use this mapping to give exact results for entanglement growth in 1 þ 1D circuits.
Random unitary circuits have become a model system to investigate information scrambling in quantum systems. In the literature, mostly random circuits with Haar-distributed gate operations have been considered. In this work, we investigate operator spreading in random unitary circuits in which the elementary gate operations are drawn from general unitary-invariant ensembles, which include the well-studied Haar-distributed random unitary circuits as a special case. Similar to the Haar-distributed case, the long-time behavior of operator spreading with the more general unitary-invariant gate distribution is governed by drift-diffusion equations characterized by the butterfly velocity v B and a diffusion constant D . Differences with the Haar-random case are (i) that it takes a finite time τ b until ensemble-averaged Pauli-string weights take a “binary” form, in which they depend only on whether Pauli operators inside the support of the Pauli strong are equal to the identity matrix, and (ii) that the operator spreading is characterized by a finite “domain-wall width” n DW separating regions with a random-matrix-like Pauli-string distribution. To illustrate these findings, we perform explicit calculations for random unitary circuits distributed according to the Poisson kernel, which interpolates between the trivial and Haar-distributed circuits.
We investigate operator spreading in random quantum circuits with gates drawn from orthogonal-invariant or symplectic-invariant ensembles, revealing several key distinctions from the well-studied unitary-invariant case. We find that the ensemble-averaged Pauli-string weights relax to a ternary-valued structure, instead of the binary structure of unitary-invariant circuits. For orthogonal- or symplectic-invariant circuits, the domain wall separating trivial and scrambled regions has a finite width even for Haar-random gates, whereas domain walls are sharp for Haar-distributed random unitary circuits. We further find a fundamental dichotomy between random circuits with two-qubit gates from the two disconnected components of the orthogonal group: While the butterfly velocity for the special orthogonal ensemble lies between zero and the Haar value, the negative-determinant sector exhibits a non-zero lower bound for any gate distribution. Moreover, for qudit size q=2, the butterfly velocity can exceed that of the Haar-random ensemble.
We study the scrambling of local quantum information in chaotic many-body systems in the presence of a locally conserved quantity like charge or energy that moves diffusively.The interplay between conservation laws and scrambling sheds light on the mechanism by which unitary quantum dynamics, which is reversible, gives rise to diffusive hydrodynamics, which is a slow dissipative process.We obtain our results in a random quantum circuit model that is constrained to have a conservation law.We find that a generic spreading operator consists of two parts: (i) a conserved part which comprises the weight of the spreading operator on the local conserved densities, whose dynamics is described by diffusive charge spreading; this conserved part also acts as a source that steadily emits a flux of (ii) nonconserved operators.This emission leads to dissipation in the operator hydrodynamics, with the dissipative process being the slow conversion of operator weight from local conserved operators to nonconserved, at a rate set by the local diffusion current.The emitted nonconserved parts then spread ballistically at a butterfly speed, thus becoming highly nonlocal and, hence, essentially nonobservable, thereby acting as the "reservoir" that facilitates the dissipation.In addition, we find that the nonconserved component develops a power-law tail behind its leading ballistic front due to the slow dynamics of the conserved components.This implies that the out-of-time-order commutator between two initially separated operators grows sharply upon the arrival of the ballistic front, but, in contrast to systems with no conservation laws, it develops a diffusive tail and approaches its asymptotic late-time value only as a power of time instead of exponentially.We also derive these results within an effective hydrodynamic description which contains multiple coupled diffusion equations.
Optimal Scaling of Unitary Design Formation in U(1)-Symmetric Random Circuits: A Bottleneck Slower than Charge Transport
Understanding randomness formation under conservation laws is important both for characterizing chaotic phenomena in isolated quantum systems and for symmetry-constrained quantum information processing. U(1)-symmetric random circuits on an N-qubit system provide a minimal tractable setting for this problem, in which the formation rate of unitary designs was conjectured to be governed by charge transport in the underlying circuit geometry. However, whether charge transport indeed sets the slowest relaxation mode has remained an open problem, due to the difficulty in deriving a tight lower bound on the symmetric design formation rate. In this work, we resolve this problem by determining the optimal scaling of the unitary k-design formation rate for k≤O(logN) across a wide range of circuit geometries, including lattices, expander graphs, and all-to-all interactions. In particular, for k≥2, we find that the formation rate is governed by a two-particle encounter rate---the rate at which two random walkers encounter each other in a given circuit geometry. This encounter rate is parametrically smaller than the single-particle transport rate in many geometries, thereby disproving the previous conjecture. We further establish that this encounter mechanism indeed sets the design formation rate by deriving its matching lower bound with a novel proof technique based on an auxiliary subgroup ensemble. Although this encounter mode imposes a robust bottleneck on design formation in U(1)-symmetric random circuits, we also show that this bottleneck can be circumvented by using symmetry-breaking local gates and provide an explicit protocol that achieves faster design formation. These results establish a new picture of randomness formation under conservation laws and provide guiding principles for characterizing and efficiently generating symmetry-constrained randomness.
We prove new lower bounds on the growth of robust quantum circuit complexity -- the minimal number of gates Cδ(U) to approximate a unitary U up to an error of δ in operator norm distance. More precisely we show two bounds for random quantum circuits with local gates drawn from a subgroup of SU(4). First, for δ=Θ(2−n), we prove a linear growth rate: Cδ≥d/poly(n) for random quantum circuits on n qubits with d≤2n/2 gates. Second, for $ δ=Ω(1)$, we prove a square-root growth of complexity: Cδ≥d/poly(n) for all d≤2n/2. Finally, we provide a simple conjecture regarding the Fourier support of randomly drawn Boolean functions that would imply linear growth for constant δ. While these results follow from bounds on the moments of random quantum circuits, we do not make use of existing results on the generation of unitary t-designs. Instead, we bound the moments of an auxiliary random walk on the diagonal unitaries acting on phase states. In particular, our proof is comparably short and self-contained.