Anticoncentration in Quantum Circuits

Updated 23 August 2025

Anticoncentration in quantum circuits is defined as the spread of output probabilities, ensuring most outcomes are near the uniform value instead of being peaked.
Techniques such as second moment analysis, collision probability, and inverse participation ratios quantify the transition to a Porter–Thomas-like distribution.
This property underpins complexity-theoretic proofs of quantum advantage and informs the impact of noise and circuit design on quantum benchmarking.

Anticoncentration properties of quantum circuits describe the statistical "spread" of output measurement probabilities in random, chaotic, or structurally complex quantum systems. Specifically, a family of quantum circuits is said to exhibit anticoncentration if, after acting on a simple input state (typically $|0^n\rangle$ ), the resulting distribution of outcome probabilities avoids being highly “peaked”: most probabilities are comparable to the uniform value $1/2^n$ , and few outcomes dominate. Anticoncentration is of central importance for complexity-theoretic proofs of quantum computational advantage, foundations of quantum benchmarking, and the fundamental dynamics of information scrambling and quantum chaos.

1. Mathematical Formulation of Anticoncentration

Anticoncentration quantifies the likelihood that a randomly chosen outcome $x$ after applying a random quantum circuit $U$ occurs with probability $p_U(x) = |\langle x|U|0\rangle|^2$ that is not exponentially smaller than the uniform value $1/2^n$ . A principal quantitative tool is the use of second moments (collision probability):

$Z = \mathbb{E}_U \left[ \sum_{x} p_U(x)^2 \right]$

For a Haar–random unitary on $n$ qudits ( $q$ levels per qudit), the Haar value is $Z_\text{Haar} = \frac{2}{q^n + 1}$ . A family of distributions $\{p_U(x)\}$ is said to “anticoncentrate” if $Z \leq O(q^{-n})$ ; equivalently, the probability of observing $p_U(x) \geq \alpha/q^n$ is lower bounded by a constant $(1-\alpha)^2/2$ (for relative $\alpha \in (0,1)$ ), as established by Paley–Zygmund–type inequalities and moment methods (Hangleiter et al., 2017).

A generalization applies to higher moments (inverse participation ratios, IPRs), participation entropy, and the full probability distribution of probabilities (PoP):

$\text{IPR}_k = \mathbb{E}_U \left[ \sum_x p_U(x)^k \right]$

$\mathcal{S}_k = \frac{1}{1-k} \log_2 \left( \sum_{x} p_U(x)^k \right)$

In the fully anticoncentrated (“random”) regime, the distribution of $p_U(x)$ follows the Porter–Thomas law: $P_{\text{Haar}}(p) \simeq N \exp(-N p)$ with $N = 2^n$ .

2. Mechanisms and Scaling Behavior in Random Circuit Ensembles

The emergence of anticoncentration in quantum circuits is closely tied to the approach of the circuit ensemble to an approximate unitary 2–design. A random circuit forms an $\varepsilon$ –approximate 2-design if its second-moment operator matches that of the Haar measure up to multiplicative $\varepsilon$ corrections:

$\| M_\mu^{(2)} - P^{(2)} \| \leq \epsilon/q^{4n}$

Anticoncentration is guaranteed when the ensemble is an exact or sufficiently precise approximate 2–design (Hangleiter et al., 2017, Dalzell et al., 2020, Yada et al., 10 Apr 2025). Practically:

In 1D (nearest-neighbor) circuits of qubits, a circuit size/depth scaling as $O(n \log n)$ gates (or $O(\log n)$ depth) suffices to achieve anticoncentration, with tight upper and lower bounds given in terms of collision probability and Ising/statistical-mechanics mappings (Dalzell et al., 2020).
In architectures with more general connectivities (e.g., complete-graph or higher-dimensional), similar scaling holds but with modified prefactors.
For Clifford circuits, random (stabilizer) circuits anticoncentrate within the stabilizer state space at logarithmic depth (Magni et al., 27 Feb 2025). Adding a polylogarithmic number of non-Clifford “magic” resources leads to full Porter–Thomas (Haar-like) statistics.

Anticoncentration persists under a broad class of gate sets, including non-Haar-local random gates, as long as the local spectral gap is nonvanishing; the time to form a unitary 2–design, and thus to reach anticoncentration, is at most a constant factor longer than for Haar-random gates (Yada et al., 10 Apr 2025).

3. Impact of Noise and Dissipative Processes

Noise fundamentally alters anticoncentration phenomena depending on its nature:

Unital Noise (e.g., depolarizing): In the presence of weak unital noise, the output distribution still anticoncentrates for moderate circuit depths; the distance from the uniform distribution decays exponentially in the depth, preserving substantial structure in the output for $d = O(\log n)$ up to $O(n)$ (Deshpande et al., 2021, Sauliere et al., 20 Aug 2025).
Non-unital Noise (e.g., amplitude damping): If non-unital noise is present, e.g., amplitude damping, collision probability grows exponentially with system size, leading to persistent concentration on low Hamming-weight outcomes—anticoncentration is never achieved at any depth (Fefferman et al., 2023).
Universal scaling: In the weak-noise regime, a universal form for the “probability-of-probabilities” (PoP) distribution arises. As a function of scaled inverse Thouless length $x = N/L(t)$ and noise parameter $\eta$ (total error budget), the $k$ th IPR follows

$I_k(x, \eta) = D^{1-k} \langle 1 | \exp{ \left[ x A - \eta Q \right] } | 1 \rangle$

and cross-entropy benchmarking is analytically given by

$\text{XEB} = 2 e^{-\eta/2} \left[ \cosh \theta(x, \eta) + \frac{x}{\theta(x,\eta)} \sinh \theta(x, \eta) \right] - 1$

with $\theta(x, \eta) = \sqrt{x^2 + (\eta/2)^2}$ (Sauliere et al., 20 Aug 2025).

Noise induces three scaling regimes (shallow, intermediate, deep) with transitions demarcated by $x \sim \sqrt{\eta}$ , and a universal quantum-to-classical crossover in PoP and XEB.

4. Universality and Finite-Size Corrections

Anticoncentration displays universal features across architectures when analyzed in the proper scaling limit. The participation entropy, IPRs, and overlap distributions all follow universal scaling formulas controlled by dimensionless parameters (ratio of system size $N$ to Thouless length or bond dimension, $x = N/w^t$ ), and corrections that scale as $O(1/N)$ or $O(1/\sqrt N)$ .

Explicitly, for inverse participation ratios in ensembles such as random matrix product states, Haar-random circuits, or brickwork circuits, the leading order and finite-size corrections (with parameters $\alpha$ , $\beta$ ) satisfy

$I_k = I_k^{\text{Haar}} \exp \left\{ \frac{k(k-1)}{2} \alpha \right\} \exp \left\{ - k(k-1) \left( k - \frac{1}{2} \right) \beta \right\} + O(\beta^2)$

where finite-size corrections $\beta$ encode the deviation from asymptotic Porter–Thomas statistics (Sauliere et al., 28 Feb 2025). The universal scaling collapse of numerical simulational data (up to $N=1024$ qudits) across a variety of architectures confirms these theoretical predictions.

5. Special Circuit Types: Clifford, Symplectic, and Bosonic Circuits

Clifford and Doped Clifford Circuits: Random Clifford circuits exhibit rapid (logarithmic in $N$ ) anticoncentration within the stabilizer subspace. Injection of $O(\log N)$ non-Clifford gates suffices to match Haar (Porter–Thomas) statistics in the IPR and participation entropy, establishing a “pseudo-magic” regime that bridges classical simulability and full quantum complexity (Magni et al., 27 Feb 2025, Magni et al., 2 Jun 2025).
Symplectic Circuits: Random circuits built from symplectic unitaries ( $\mathbb{SP}(d/2)$ ) have output Pauli measurement statistics governed by the Brauer algebra and display convergence to Gaussian processes. Anticoncentration (in computational basis probabilities) is achieved at logarithmic depth, analogous to unitary and orthogonal circuits; this holds for shallow circuits constructed from a universal (non-translationally invariant) generator set (García-Martín et al., 16 May 2024).
Bosonic Circuits (Gaussian Boson Sampling): The anticoncentration of the output probabilities undergoes a sharp transition as a function of the number $k$ of initially squeezed modes versus the number of measured photons $2n$. Sufficiently large $k$ (e.g., $k = \omega(n^2)$ ) yields weak anticoncentration (normalized second moment $\sim 1/\sqrt{n}$ ), while small $k$ leads to highly concentrated output (Ehrenberg et al., 2023).

6. Physical and Complexity-Theoretic Implications

Anticoncentration is a cornerstone in average-case complexity arguments for the classical hardness of random quantum circuit sampling and related schemes (e.g., IQP circuits, Boson sampling). The three general ingredients in such hardness proofs are: (1) anticoncentration, (2) worst-case #P-hardness for output amplitudes, and (3) conjectured average-case hardness. Anticoncentration ensures that many output probabilities are of order $1/N$, so classical simulation cannot approximate all probabilities to within small additive error unless the Polynomial Hierarchy collapses (Hangleiter et al., 2017).

Conversely, lack of anticoncentration (e.g., under strong noise, insufficient circuit depth, or unstructured architectures) enables trivial or efficient classical algorithms to approximate or even simulate the output distribution, nullifying potential “quantum advantage.” For instance, non-unital noise induces persistent concentration, invalidating standard stockmeyer-based reductions (Fefferman et al., 2023).

Anticoncentration’s emergence—at $O(\log N)$ depth for random/chaotic circuits, or sharply at certain transition points in sparse-graph IQP or bosonic circuits—signals where complexity transitions from classical tractability to regimes supporting robust quantum speedup (Park et al., 2022, Ehrenberg et al., 2023, Magni et al., 27 Feb 2025). In both separations and quantum benchmarking, explicit knowledge of finite-size corrections and universal scaling laws is indispensable for making rigorous statements about near-term devices (Sauliere et al., 20 Aug 2025, Sauliere et al., 28 Feb 2025).

7. Connections to Magic and Complexity Growth in Ergodic Dynamics

Anticoncentration dynamics are often intertwined with the growth (“spreading”) of magic—the resource quantifying non-stabilizerness/quantum complexity. In random circuits and digital Floquet systems, both magic (as measured by stabilizer entropy or CSS entropy) and participation entropy (anticoncentration) equilibrate at timescales $t_\mathrm{sat} \propto \log N$ , as both quantities rapidly approach their Haar-random (maximally complex) value (Turkeshi et al., 4 Jul 2024, Tirrito et al., 13 Dec 2024). In contrast, Hamiltonian (energy-conserving) systems exhibit much slower (linear-in- $N$ ) relaxation toward complexity and display multifractal (incomplete) anticoncentration (Tirrito et al., 13 Dec 2024).

A unified picture thus emerges: in generic, locally chaotic quantum circuits and ergodic Floquet systems, anticoncentration and magic generation are simultaneous, rapid, and highly universal. Their breakdown—via conservation laws, structural constraints, or strong noise—manifests as slower relaxation, lack of robust quantum advantage, and limited complexity growth.

Table 1: Anticoncentration Scaling in Different Circuit Classes

Circuit Class	Depth/Resource for Anticoncentration	Output Distribution
Haar-random/local random (unitary)	$O(\log N)$	Porter–Thomas law
Clifford circuits (no magic)	$O(\log N)$	Stabilizer overlap stats
Clifford + $O(\log N)$ magic gates	$O(\log N)$ + $O(\log N)$ resources	Porter–Thomas law
Symplectic circuits	$O(\log N)$	Gaussian process statistics
Noisy (unital) circuits	$O(\log N)$ (weak noise regime)	Modified Porter–Thomas / universal PoP
Noisy (non-unital) circuits	No anticoncentration at any depth	Strongly peaked / non-uniform
Gaussian Boson Sampling	$k = \omega(n^2)$ (modes vs. photons)	Weak anticoncentration

Anticoncentration theorems and their scaling laws form a rigorous foundation for much of random circuit quantum information science, connecting the physical “delocalization” of quantum states in Hilbert space with computational intractability and resource-theoretic complexity. The universality and robustness of the phenomenon are essential both for theoretical progress and for interpreting the performance of experimental quantum devices (Hangleiter et al., 2017, Dalzell et al., 2020, Sauliere et al., 28 Feb 2025, Sauliere et al., 20 Aug 2025).