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arXiv:2609.17457

One Gate at a Time: Complexity Growth in Random Quantum Circuits

Zhi Li

quant-phmath-phSubmitted Sep 15, 2026• Announced Sep 16, 2026• 0 citations
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Abstract

A random unitary quantum circuit is expected to be incompressible for exponentially long times. We show that the constant-error circuit complexity of a random unitary circuit grows almost linearly with time as $Ω(T/\log T)$. The bound holds for all $2\leq T\leq 4^n$ where $n$ is the system size, and involves no other $n$-dependence. This improves previous lower bounds derived from spectral gaps and unitary designs by a factor of $\mathrm{poly}(n)$. Drawing on insights from stochastic calculus, geometric functional analysis, and randomized linear algebra, our approach exploits the circuit's response to variations of individual gates and requires no control over convergence to high-order unitary designs.

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A random unitary quantum circuit is expected to be incompressible for exponentially long times. We show that the constant-error circuit complexity of a random unitary circuit grows almost linearly with time as Ω(T/log⁡T)Ω(T/\log T)Ω(T/logT). The bound holds for all 2≤T≤4n2\leq T\leq 4^n2≤T≤4n where nnn is the system size, and involves no other nnn-dependence. This improves previous lower bounds derived from spectral gaps and unitary designs by a factor of poly(n)\mathrm{poly}(n)poly(n). Drawing on insights from stochastic calculus, geometric functional analysis, and randomized linear algebra, our approach exploits the circuit's response to variations of individual gates and requires no control over convergence to high-order unitary designs.
The concept of quantum complexity has far-reaching implications spanning theoretical computer science, quantum many-body physics, and high-energy physics.The quantum complexity of a unitary transformation or quantum state is defined as the size of the shortest quantum computation that executes the unitary or prepares the state.It is reasonable to expect that the complexity of a quantum state governed by a chaotic many-body Hamiltonian grows linearly with time for a time that is exponential in the system size; however, because it is hard to rule out a shortcut that improves the efficiency of a computation, it is notoriously difficult to derive lower bounds on quantum complexity for particular unitaries or states without making additional assumptions.To go further, one may study more generic models of complexity growth.We provide a rigorous connection between complexity growth and unitary k-designs, ensembles that capture the randomness of the unitary group.This connection allows us to leverage existing results about design growth to draw conclusions about the growth of complexity.We prove that local random quantum circuits generate unitary transformations whose complexity grows linearly for a long time, mirroring the behavior one expects in chaotic quantum systems and verifying conjectures by Brown and Susskind.Moreover, our results apply under a strong definition of quantum complexity based on optimal distinguishing measurements.
We prove new lower bounds on the growth of robust quantum circuit complexity -- the minimal number of gates Cδ(U)C_δ(U)Cδ​(U) to approximate a unitary UUU up to an error of δδδ in operator norm distance. More precisely we show two bounds for random quantum circuits with local gates drawn from a subgroup of SU(4)SU(4)SU(4). First, for δ=Θ(2−n)δ=Θ(2^{-n})δ=Θ(2−n), we prove a linear growth rate: Cδ≥d/poly(n)C_δ\geq d/\mathrm{poly}(n)Cδ​≥d/poly(n) for random quantum circuits on nnn qubits with d≤2n/2d\leq 2^{n/2}d≤2n/2 gates. Second, for $ δ=Ω(1)$, we prove a square-root growth of complexity: Cδ≥d/poly(n)C_δ\geq \sqrt{d}/\mathrm{poly}(n)Cδ​≥d​/poly(n) for all d≤2n/2d\leq 2^{n/2}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 δδδ. 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 ttt-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.
Abstract The complexity of quantum states has become a key quantity of interest across various subfields of physics, from quantum computing to the theory of black holes. The evolution of generic quantum systems can be modelled by considering a collection of qubits subjected to sequences of random unitary gates. Here we investigate how the complexity of these random quantum circuits increases by considering how to construct a unitary operation from Haar-random two-qubit quantum gates. Implementing the unitary operation exactly requires a minimal number of gates—this is the operation’s exact circuit complexity. We prove a conjecture that this complexity grows linearly, before saturating when the number of applied gates reaches a threshold that grows exponentially with the number of qubits. Our proof overcomes difficulties in establishing lower bounds for the exact circuit complexity by combining differential topology and elementary algebraic geometry with an inductive construction of Clifford circuits.
Recently, the dynamics of quantum systems that involve both unitary evolution and quantum measurements have attracted attention due to the exotic phenomenon of measurement-induced phase transitions. The latter refers to a sudden change in a property of a state of n qubits, such as its entanglement entropy, depending on the rate at which individual qubits are measured. At the same time, quantum complexity emerged as a key quantity for the identification of complex behaviour in quantum many-body dynamics. In this work, we investigate the dynamics of the quantum state complexity in monitored random circuits, where n qubits evolve according to a random unitary circuit and are individually measured with a fixed probability at each time step. We find that the evolution of the exact quantum state complexity undergoes a phase transition when changing the measurement rate. Below a critical measurement rate, the complexity grows at least linearly in time until {saturating to a value e Ω ( n ) .} Above, the complexity does not exceed poly ⁡ ( n ) . In our proof, we make use of percolation theory to find paths along which an exponentially long quantum computation can be run below the critical rate, and to identify events where the state complexity is reset to zero above the critical rate. We lower bound the exact state complexity in the former regime using recently developed techniques from algebraic geometry. Our results combine quantum complexity growth, phase transitions, and computation with measurements to help understand the behavior of monitored random circuits and to make progress towards determining the computational power of measurements in many-body systems.
We study the growth of entanglement and circuit complexity in random passive linear optical networks as a function of the circuit depth. For entanglement dynamics, we start with an initial Gaussian state with all nnn modes equally squeezed. For random brickwork circuits, we show that entanglement, as measured by the Rényi-2 entropy, grows diffusively as a function of the depth, in contrast to the linear growth for random qubit brickwork circuits. In the other direction, for arbitrary circuit geometries we prove bounds on depths which ensure the average-case subsystem entanglement reaches within a constant factor of the maximum value in all subsystems, and bounds which ensure closeness of the random linear optical unitary to a Haar random unitary in L2L^2L2 Wasserstein distance. We also consider robust circuit complexity for random one-dimensional brickwall circuits, as measured by the minimum number of gates required in any circuit that approximately implements the linear optical unitary. Viewing this as a function of the number of modes and the circuit depth, we show the robust circuit complexity for random one-dimensional brickwall circuits scales at most diffusively in the depth with high probability, again in contrast to the linear scaling for certain random qubit circuits. The corresponding Gaussian unitary U~\tilde{\mathcal U}U~ for the approximate implementation retains high output fidelity ∣⟨ψ∣U†U~∣ψ⟩∣2|\langleψ|\mathcal U^\dagger \tilde{\mathcal U}|ψ\rangle|^2∣⟨ψ∣U†U~∣ψ⟩∣2 for pure states ∣ψ⟩|ψ\rangle∣ψ⟩ with constrained expected photon-number.
Abstract Many quantum information protocols require the implementation of random unitaries. Because it takes exponential resources to produce Haar-random unitaries drawn from the full n -qubit group, one often resorts to t -designs. Unitary t -designs mimic the Haar-measure up to t -th moments. It is known that Clifford operations can implement at most 3-designs. In this work, we quantify the non-Clifford resources required to break this barrier. We find that it suffices to inject O(t4log⁡2(t)log⁡(1/ε))O(t^{4}\log ^{2}(t)\log (1/\varepsilon ))O(t4log2(t)log(1/ε)) O ( t 4 log 2 ( t ) log ( 1 / ε ) ) many non-Clifford gates into a polynomial-depth random Clifford circuit to obtain an ε\varepsilon ε ε -approximate t -design. Strikingly, the number of non-Clifford gates required is independent of the system size – asymptotically, the density of non-Clifford gates is allowed to tend to zero. We also derive novel bounds on the convergence time of random Clifford circuits to the t -th moment of the uniform distribution on the Clifford group. Our proofs exploit a recently developed variant of Schur-Weyl duality for the Clifford group, as well as bounds on restricted spectral gaps of averaging operators.