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arXiv:2407.19561 · Leibniz international proceedings in informatics

Anti-Concentration for the Unitary Haar Measure and Applications to Random Quantum Circuits

Bill Fefferman, Ghosh, Soumik, Zhan, Wei

Published Jul 28, 2024• 0 citations
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Abstract

We prove a Carbery-Wright style anti-concentration inequality for the unitary Haar measure, by showing that the probability of a polynomial in the entries of a random unitary falling into an $\varepsilon$ range is at most a polynomial in $\varepsilon$. Using it, we show that the scrambling speed of a random quantum circuit is lower bounded: Namely, every input qubit has an influence that is at least inverse exponential in depth, on any output qubit touched by its lightcone. Our result on scrambling speed works with high probability over the choice of a circuit from an ensemble, as opposed to just working in expectation. As an application, we give the first polynomial-time algorithm for learning log-depth random quantum circuits with Haar random gates up to polynomially small diamond distance, given oracle access to the circuit. Other applications of this new scrambling speed lower bound include: $\bullet$ An optimal $Ω(\log \varepsilon^{-1})$ depth lower bound for $\varepsilon$-approximate unitary designs on any circuit architecture; $\bullet$ A polynomial-time quantum algorithm that computes the depth of a bounded-depth circuit, given oracle access to the circuit. Our learning and depth-testing algorithms apply to architectures defined over any geometric dimension, and can be generalized to a wide class of architectures with good lightcone properties.

DOI
10.4230/lipics.itcs.2026.57

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We prove a Carbery-Wright style anti-concentration inequality for the unitary Haar measure, by showing that the probability of a polynomial in the entries of a random unitary falling into an ε\varepsilonε range is at most a polynomial in ε\varepsilonε. Using it, we show that the scrambling speed of a random quantum circuit is lower bounded: Namely, every input qubit has an influence that is at least inverse exponential in depth, on any output qubit touched by its lightcone. Our result on scrambling speed works with high probability over the choice of a circuit from an ensemble, as opposed to just working in expectation. As an application, we give the first polynomial-time algorithm for learning log-depth random quantum circuits with Haar random gates up to polynomially small diamond distance, given oracle access to the circuit. Other applications of this new scrambling speed lower bound include: ∙\bullet∙ An optimal Ω(log⁡ε−1)Ω(\log \varepsilon^{-1})Ω(logε−1) depth lower bound for ε\varepsilonε-approximate unitary designs on any circuit architecture; ∙\bullet∙ A polynomial-time quantum algorithm that computes the depth of a bounded-depth circuit, given oracle access to the circuit. Our learning and depth-testing algorithms apply to architectures defined over any geometric dimension, and can be generalized to a wide class of architectures with good lightcone properties.
We give an efficient algorithm for learning kkk-dimensional brickwork random quantum circuits using only copies of the output state obtained by applying UUU to the all-zero input. For a depth-ddd circuit on nnn sites with random 2ℓ2\ell2ℓ-qubit gates, the algorithm learns the original circuit UUU with high probability in poly(n,2ℓd)\text{poly}(n,2^{\ell d})poly(n,2ℓd) time for every constant dimensional lattice. In particular, the algorithm is polynomial time as long as ℓd=O(log⁡n)\ell d = O(\log n)ℓd=O(logn). In one dimension, this reaches the natural boundary suggested by pseudorandomness: pseudorandom states require ℓd=ω(log⁡n)\ell d=ω(\log n)ℓd=ω(logn), and structured cryptographic constructions suggest that this scale may be achievable from above. In higher dimensions, it remains plausible that the same ℓd=ω(log⁡n)\ell d=ω(\log n)ℓd=ω(logn) scale remains the threshold for ancilla-free pseudorandomness, and our results help clarify the conditions under which pseudorandomness can arise in this setting. The main idea behind our algorithm is a local correlation criterion that identifies gates in the final layer without learning their entire backward light cones, avoiding a bottleneck in the previous approaches. A key technical ingredient is a Carbery-Wright type anticoncentration inequality for low-degree polynomials of Haar random unitaries whose small-ball exponent is independent of the matrix dimension. The dimension-independent exponent is crucial for handling gates of growing locality. This anticoncentration result may also be of independent interest.
Unitary designs provide finite-moment approximations to Haar-random unitaries, with wide-ranging applications across physics and quantum information, from scrambling and black-hole dynamics to foundational primitives in quantum algorithms. Strong unitary designs capture a more demanding operational notion of approximation, requiring indistinguishability from Haar randomness even for quantum algorithms that may access a unitary not only in the forward direction, but also through its inverse, transpose, and complex conjugate. Motivated by the physical requirement that scrambling arise within the system itself, Schuster, Ma, Lombardi, Brandão, and Huang (arXiv:2509.26310) left open whether strong unitary designs can be generated in logarithmic depth using only the system qubits. For every fixed design order kkk and measurable-error tolerance, we construct strong approximate unitary kkk-designs in optimal Θ(log⁡n)Θ(\log n)Θ(logn) all-to-all circuit depth using only the nnn original system qubits. Our new ingredient is a logarithmic-depth Pauli-mixing bound for the perfect-matching ensemble, whose layers pair the qubits uniformly at random and apply independent random two-qubit gates. This bound controls the mixed forward-reverse two-query case, which we combine with existing design and gluing results to obtain strong unitary designs of arbitrary fixed order.
Abstract We prove that  poly (t)⋅n1/D{{\,\textrm{poly}\,}}(t) \cdot n^{1/D}poly(t)⋅n1/D poly ( t ) · n 1 / D -depth local random quantum circuits with two qudit nearest-neighbor gates on a D -dimensional lattice with n qudits are approximate t -designs in various measures. These include the “monomial” measure, meaning that the monomials of a random circuit from this family have expectation close to the value that would result from the Haar measure. Previously, the best bound was  poly (t)⋅n{{\,\textrm{poly}\,}}(t)\cdot npoly(t)⋅n poly ( t ) · n due to Brandão–Harrow–Horodecki (Commun Math Phys 346(2):397–434, 2016) for D=1D=1D=1 D = 1 . We also improve the “scrambling” and “decoupling” bounds for spatially local random circuits due to Brown and Fawzi (Scrambling speed of random quantum circuits, 2012). One consequence of our result is that assuming the polynomial hierarchy (  PH {{\,\mathrm{\textsf{PH}}\,}}PH PH ) is infinite and that certain counting problems are #P\#{\textsf{P}}#P # P -hard “on average”, sampling within total variation distance from these circuits is hard for classical computers. Previously, exact sampling from the outputs of even constant-depth quantum circuits was known to be hard for classical computers under these assumptions. However the standard strategy for extending this hardness result to approximate sampling requires the quantum circuits to have a property called “anti-concentration”, meaning roughly that the output has near-maximal entropy. Unitary 2-designs have the desired anti-concentration property. Our result improves the required depth for this level of anti-concentration from linear depth to a sub-linear value, depending on the geometry of the interactions. This is relevant to a recent experiment by the Google Quantum AI group to perform such a sampling task with 53 qubits on a two-dimensional lattice (Arute in Nature 574(7779):505–510, 2019; Boixo et al. in Nate Phys 14(6):595–600, 2018) (and related experiments by USTC), and confirms their conjecture that O(n)O(\sqrt{n})O(n​) O ( n ) depth suffices for anti-concentration. The proof is based on a previous construction of t -designs by Brandão et al. (2016), an analysis of how approximate designs behave under composition, and an extension of the quasi-orthogonality of permutation operators developed by Brandão et al. (2016). Different versions of the approximate design condition correspond to different norms, and part of our contribution is to introduce the norm corresponding to anti-concentration and to establish equivalence between these various norms for low-depth circuits. For random circuits with long-range gates, we use different methods to show that anti-concentration happens at circuit size O(nln⁡2n)O(n\ln ^2 n)O(nln2n) O ( n ln 2 n ) corresponding to depth O(ln⁡3n)O(\ln ^3 n)O(ln3n) O ( ln 3 n ) . We also show a lower bound of Ω(nln⁡n)\Omega (n \ln n)Ω(nlnn) Ω ( n ln n )
The applications of random quantum circuits range from quantum computing and quantum many-body systems to the physics of black holes. Many of these applications are related to the generation of quantum pseudorandomness: Random quantum circuits are known to approximate unitary t -designs. Unitary t -designs are probability distributions that mimic Haar randomness up to t th moments. In a seminal paper, Brandão, Harrow and Horodecki prove that random quantum circuits on qubits in a brickwork architecture of depth O(nt10.5) are approximate unitary t -designs. In this work, we revisit this argument, which lower bounds the spectral gap of moment operators for local random quantum circuits by Ω(n−1t−9.5) . We improve this lower bound to Ω(n−1t−4−o(1)) , where the o(1) term goes to 0 as t→∞ . A direct consequence of this scaling is that random quantum circuits generate approximate unitary t -designs in depth O(nt5+o(1)) . Our techniques involve Gao's quantum union bound and the unreasonable effectiveness of the Clifford group. As an auxiliary result, we prove fast convergence to the Haar measure for random Clifford unitaries interleaved with Haar random single qubit unitaries.
Motivated by the recent experimental demonstrations of quantum supremacy, proving the hardness of the output of random quantum circuits is an imperative near term goal. We prove under the complexity theoretical assumption of the non-collapse of the polynomial hierarchy that approximating the output probabilities of random quantum circuits to within exp⁡(−Ω(mlog⁡m))\exp(-\Omega(m\log m))exp(−Ω(mlogm)) additive error is hard for any classical computer, where mmm is the number of gates in the quantum computation. More precisely, we show that the above problem is #P\#\mathsf{P}#P-hard under BPPNP\mathsf{BPP}^{\mathsf{NP}}BPPNP reduction. In the recent experiments, the quantum circuit has nnn-qubits and the architecture is a two-dimensional grid of size n×n\sqrt{n}\times\sqrt{n}n​×n​. Indeed for constant depth circuits approximating the output probabilities to within 2−Ω(nlog⁡n)2^{-\Omega(n\log{n})}2−Ω(nlogn) is hard. For circuits of depth log⁡n\log{n}logn or n\sqrt{n}n​ for which the anti-concentration property holds, approximating the output probabilities to within 2−Ω(nlog⁡2n)2^{-\Omega(n\log^2{n})}2−Ω(nlog2n) and 2−Ω(n3/2log⁡n)2^{-\Omega(n^{3/2}\log n)}2−Ω(n3/2logn) is hard respectively. We made an effort to find the best proofs and proved these results from first principles, which do not use the standard techniques such as the Berlekamp--Welch algorithm, the usual Paturi's lemma, and Rakhmanov's result.
Approximate unitary designs reproduce the statistics of Haar-random unitaries to a given order and accuracy. Recent constructions realize such designs with logarithmic-depth circuits, enabling shallow measurement protocols for various quantum state-learning tasks while preserving performance. These results raise the question of whether approximate designs can replace exact designs more generally. We show that even exponentially small design error need not preserve the learning guarantees of exact designs. This motivates deriving learning guarantees directly from the measurement circuit structure. For observable estimation with classical shadows, we prove that logarithmic-depth two-layer Clifford circuits yield an unbiased estimator matching the global Clifford variance scaling for every state and Hermitian observable. Beyond observable estimation, this bound allows shallow measurements to retain global Clifford guarantees for other tasks, including setting-efficient tomography, mixed-state metrology, and stabilizer structure learning. We complement these statistical guarantees with a compact, exact tensor-network representation of the inverse shadow channel. When measurement bases are reused, approximate unitary designs of arbitrarily high order need not uniformly match the Haar variance in the many-shot limit. For the all-to-all random two-local circuits considered, doing so requires nearly linear depth. Together, our results reveal the capabilities and limitations of shallow quantum state learning, highlighting the distinction between approximating Haar randomness and reproducing its learning guarantees.