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arXiv:1905.12053

Unitary designs from statistical mechanics in random quantum circuits

Nicholas Hunter-Jones

Published May 28, 2019• 51 citations
arXivHTMLPDF

Abstract

Random quantum circuits are proficient information scramblers and efficient generators of randomness, rapidly approximating moments of the unitary group. We study the convergence of local random quantum circuits to unitary $k$-designs. Employing a statistical mechanical mapping, we give an exact expression of the distance to forming an approximate design as a lattice partition function. In the statistical mechanics model, the approach to randomness has a simple interpretation in terms of domain walls extending through the circuit. We analytically compute the second moment, showing that random circuits acting on $n$ qudits form approximate 2-designs in $O(n)$ depth, as is known. Furthermore, we argue that random circuits form approximate unitary $k$-designs in $O(nk)$ depth and are thus essentially optimal in both $n$ and $k$. We can show this in the limit of large local dimension, but more generally rely on a conjecture about the dominance of certain domain wall configurations.

DOI
10.48550/arxiv.1905.12053

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Random quantum circuits are proficient information scramblers and efficient generators of randomness, rapidly approximating moments of the unitary group. We study the convergence of local random quantum circuits to unitary kkk-designs. Employing a statistical mechanical mapping, we give an exact expression of the distance to forming an approximate design as a lattice partition function. In the statistical mechanics model, the approach to randomness has a simple interpretation in terms of domain walls extending through the circuit. We analytically compute the second moment, showing that random circuits acting on nnn qudits form approximate 2-designs in O(n)O(n)O(n) depth, as is known. Furthermore, we argue that random circuits form approximate unitary kkk-designs in O(nk)O(nk)O(nk) depth and are thus essentially optimal in both nnn and kkk. We can show this in the limit of large local dimension, but more generally rely on a conjecture about the dominance of certain domain wall configurations.
Local random circuits scramble efficiently and, accordingly, have a range of applications in quantum information and quantum dynamics. With a global U(1) charge, however, the scrambling ability is reduced; for example, such random circuits do not generate the entire group of number-conserving unitaries. We establish two results using the statistical mechanics of k -fold replicated circuits. First, we show that finite moments cannot distinguish the ensemble that local random circuits generate from the Haar ensemble on the entire group of number-conserving unitaries. Specifically, the circuits form a k c -design with k c = O ( L d ) for a system in d spatial dimensions with linear dimension L . Second, for k < k c , we derive bounds on the depth τ required for the circuit to converge to an approximate k -design. The depth is lower bounded by diffusion k L 2 ln ( L ) ≲ τ . In contrast, without number conservation τ ∼ poly ( k ) L . The convergence of the circuit ensemble is controlled by the low-energy properties of a frustration-free quantum statistical model which spontaneously breaks k U(1) symmetries. We conjecture that the associated Goldstone modes set the spectral gap for arbitrary spatial and qudit dimensions, leading to an upper bound τ ≲ k L d + 2 .
Random many-body states and matrices are both a useful tool to model certain physical systems and an important asset for quantum computation. Realizing them, however, generally requires an exponential (in system size) amount of resources. Recent research has presented a way out by showing that one can generate random states and matrices, or more precisely a controlled approximation of them, by applying a quantum circuit built in terms of few-body unitary gates. Most of this research has been focussed on the case of quantum circuits composed by fully random unitary gates. Here we consider what happens for circuits that, instead, involve a minimal degree of randomness. Specifically, we concentrate on two different settings: (a) brickwork quantum circuits with a single one-qudit random matrix at a boundary; (b) brickwork quantum circuits with fixed interactions but random one-qudit gates everywhere. We show that, for any given initial state, (a) and (b) produce a distribution of states approaching the Haar distribution in the limit of large circuit depth. More precisely, we show that the moments of the distribution produced by our circuits can approximate the ones of the Haar distribution in a depth proportional to the system size. Interestingly we find that in both Cases (a) and (b) the relaxation to the Haar distribution occurs in two steps—this is in contrast with what happens in fully random circuits. Moreover, we show that choosing appropriately the fixed interactions, for example taking the local gate to be a dual-unitary gate with high enough entangling power, minimally random circuits produce a Haar random distribution more rapidly than fully random circuits. In particular, dual-unitary circuits with maximal entangling power—i.e. perfect tensors—appear to provide the optimal quantum state design preparation for any design number. Our results also suggest that the minimally random settings (a) and (b) give an optimal framework to prepare unitary designs, i.e., random unitary matrices.
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.
Random quantum circuits aim to efficiently reproduce the statistical properties of ideal random quantum evolution. One approach is to construct approximate unitary designs, which match the moments of Haar-random unitaries up to a prescribed order with controlled error. In this work, we establish quantitative guarantees for how accurately these designs reproduce the full distributions of individual output probabilities. We show that, for every strong ε\varepsilonε-approximate unitary kkk-design on nnn qubits, the distribution of each individual output probability is within O(2nk(k+2n)+ε)O\left(\sqrt{\frac{2^n}{k(k+2^n)}}+\varepsilon\right)O(k(k+2n)2n​​+ε) in Kolmogorov distance of the finite-dimensional Porter-Thomas distribution, a beta distribution with parameters 111 and 2n−12^n-12n−1. This bound is optimal up to constant factors, implying that a substantially better uniform bound requires additional structure. We further show that local invariance yields an exponential improvement in the dependence on kkk, with guarantees also in the stronger metric of total variation: every strong ε\varepsilonε-approximate unitary kkk-design invariant under unitary translations acting on log⁡k+O(1)\log k+O(1)logk+O(1) qubits achieves error 2−Ω(k)+O(ε)2^{-Ω(k)}+O(\varepsilon)2−Ω(k)+O(ε) in Kolmogorov distance and 2−Ω(k)+O(εlog⁡(2/ε))2^{-Ω(k)}+O(\varepsilon\log(2/\varepsilon))2−Ω(k)+O(εlog(2/ε)) in total variation distance. These results identify both the distributional accuracy guaranteed by moment matching and the additional structure that substantially improves it. As an application, we show that if the marginal distribution of each output probability of a strong approximate design is within Kolmogorov distance ηηη of its Haar counterpart, then the expected Shannon entropy of the output distribution differs from its Haar value by O(η)O(\sqrtη)O(η​).
We consider random quantum circuits (RQC) on arbitrary connected graphs whose edges determine the allowed 222-qudit interactions. Prior work has established that such nnn-qudit circuits with local dimension qqq on 1D, complete, and DDD-dimensional graphs form approximate unitary designs, that is, they generate unitaries from distributions close to the Haar measure on the unitary group U(qn)U(q^n)U(qn) after polynomially many gates. Here, we extend those results by proving that RQCs comprised of O(poly(n,k))O(\mathrm{poly}(n,k))O(poly(n,k)) gates on a wide class of graphs form approximate unitary kkk-designs. We prove that RQCs on graphs with spanning trees of bounded degree and height form kkk-designs after O(∣E∣n poly(k))O(|E|n\,\mathrm{poly}(k))O(∣E∣npoly(k)) gates, where ∣E∣|E|∣E∣ is the number of edges in the graph. Furthermore, we identify larger classes of graphs for which RQCs generate approximate designs in polynomial circuit size. For k≤4k \leq 4k≤4, we show that RQCs on graphs of certain maximum degrees form designs after O(∣E∣n)O(|E|n)O(∣E∣n) gates, providing explicit constants. We determine our circuit size bounds from the spectral gaps of local Hamiltonians. To that end, we extend the finite-size (or Knabe) method for bounding gaps of frustration-free Hamiltonians on regular graphs to arbitrary connected graphs. We further introduce a new method based on the Detectability Lemma for determining the spectral gaps of Hamiltonians on arbitrary graphs. Our methods have wider applicability as the first method provides a succinct alternative proof of [Commun. Math. Phys. 291, 257 (2009)] and the second method proves that RQCs on any connected architecture form approximate designs in quasi-polynomial circuit size.
Random unitary circuits are powerful minimal models for chaotic nonequilibrium evolution. Here, the authors establish an exact mapping between dynamical observables in random circuits and a statistical mechanical model of interacting spins, and use it to study systematically the entanglement generated by a random circuit consisting of local gates. A domain wall line tension in this spin model sets the entanglement growth rate. By a careful analysis of the interaction between domain walls, the authors derive explicitly the Kardar-Parisi-Zhang equation that governs the evolution of second Renyi entropy and identify a phase transition of the growth-rate function.