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arXiv:2012.05259 · Physical Review A

Improved spectral gaps for random quantum circuits: Large local dimensions and all-to-all interactions

Jonas Haferkamp, Nicholas Hunter-Jones

Published Aug 17, 2021• 30 citations
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Abstract

Random quantum circuits are a central concept in quantum information theory with applications ranging from demonstrations of quantum computational advantage to descriptions of scrambling in strongly interacting systems and black holes. The utility of random quantum circuits in these settings stems from their ability to rapidly generate quantum pseudorandomness. In a seminal paper by Brand\~ao, Harrow, and Horodecki [Commun. Math. Phys. 346, 397 (2016)] it was proven that the $t\mathrm{th}$ moment operator of local random quantum circuits on $n$ qudits with local dimension $q$ has a spectral gap of at least $\mathrm{\ensuremath{\Omega}}({n}^{\ensuremath{-}1}{t}^{\ensuremath{-}5\ensuremath{-}3.1/ln(q)})$, which implies that they are efficient constructions of approximate unitary designs. As a first result, we use Knabe bounds for the spectral gaps of frustration-free Hamiltonians to show that one-dimensional random quantum circuits have a spectral gap scaling as $\mathrm{\ensuremath{\Omega}}({n}^{\ensuremath{-}1})$, provided that $t$ is small compared to the local dimension: ${t}^{2}\ensuremath{\le}O(q)$. This implies a (nearly) linear scaling of the circuit depth in the design order $t$. Our second result is an unconditional spectral gap bounded below by $\mathrm{\ensuremath{\Omega}}[{n}^{\ensuremath{-}1}{ln}^{\ensuremath{-}1}(n){t}^{\ensuremath{-}\ensuremath{\alpha}(q)}]$ for random quantum circuits with all-to-all interactions. This improves both the $n$ and $t$ scaling in design depth for the nonlocal model. We show this by proving a recursion relation for the spectral gaps involving an auxiliary random walk. Lastly, we solve the smallest nontrivial case exactly and combine with numerics and Knabe bounds to improve the constants involved in the spectral gap for small values of $t$.

DOI
10.1103/physreva.104.022417

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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.
The spectral gap of local random quantum circuits is a fundamental property that determines how close the moments of the circuit's unitaries match those of a Haar random distribution. When studying spectral gaps, it is common to bound these quantities using tools from statistical mechanics or via quantum information-based inequalities. By focusing on the second moment of one-dimensional unitary circuits where nearest-neighboring gates act on sets of qudits (with open and closed boundary conditions), we show that one can exactly compute the associated spectral gaps. Indeed, having access to their functional form allows us to prove several important results, such as the fact that the spectral gap for closed boundary condition is exactly the square of the gap for open boundaries, as well as improve on previously known bounds for approximate design convergence. Finally, we verify our theoretical results by numerically computing the spectral gap for systems of up to 70 qubits, as well as comparing them to gaps of random orthogonal and symplectic circuits.
Random reversible and quantum circuits form random walks on the alternating group Alt(2n) and unitary group SU(2n), respectively, with each random gate as one step of the walk. Existing bounds on the spectral gap for the t-th moment of these random walks have inverse-polynomial dependence in both n and t. We prove that the gap for random reversible circuits is Ω(n−3) for all t≥1, and the gap for random quantum circuits is Ω(n−3) for t ≤ Θ(2n/2).Importantly, these gaps are independent of t in the respective regimes. We can further improve both gaps to n−1/polylog(n, t) for t ≤ 2Θ(n), which is tight up to polylog factors in n and t. Our spectral gap results have a number of consequences:1)Random reversible circuits with O(n4t)\mathcal{O}\left( {{n^4}t} \right)O(n4t) gates form multiplicative-error t-wise independent (even) permutations for all t ≥ 1; for t ≤ Θ(2n/6.1), we show that O~(n2t)\tilde {\mathcal{O}}\left( {{n^2}t} \right)O~(n2t) gates suffice.2)Random quantum circuits with O(n4t)\mathcal{O}\left( {{n^4}t} \right)O(n4t) gates form multiplicative-error unitary t-designs for t ≤Θ(2n/2); for t ≤ Θ(22n/5), we show that O~(n2t)\tilde {\mathcal{O}}\left( {{n^2}t} \right)O~(n2t) gates suffice.3)The robust quantum circuit complexity of random quantum circuits grows linearly for an exponentially long time, proving the robust Brown–Susskind conjecture [1], [2]. We also show an analogous result for random reversible circuits.Our spectral gap bounds are proven by reducing random quantum circuits to a more structured walk: a modification of the "PFC ensemble" from [3] together with an expander on the alternating group due to Kassabov [4], for which we give an efficient implementation using reversible circuits. In our reduction, we approximate the structured walk with local random circuits without losing the gap, which uses tools from the study of frustration-free Hamiltonians.
A fundamental question is understanding the rate at which random quantum circuits converge to the Haar measure. One quantity which is important in establishing this rate is the spectral gap of a random quantum ensemble. In this work we establish a new bound on the spectral gap of the t-th moment of a one-dimensional brickwork architecture on N qudits. This bound is independent of both t and N, provided t does not exceed the qudit dimension q. We also show that the bound is nearly optimal. The improved spectral gaps gives large improvements to the constant factors in known results on the approximate t-design depths of the 1D brickwork, of generic circuit architectures, and of specially-constructed architectures which scramble in depth O(log N). We moreover show that the spectral gap gives the dominant epsilon-dependence of the t-design depth at small epsilon. Our spectral gap bound is obtained by bounding the N-site 1D brickwork architecture by the spectra of 3-site operators. We then exploit a block-triangular hierarchy and a global symmetry in these operators in order to efficiently bound them. The technical methods used are a qualitatively different approach for bounding spectral gaps and and have little in common with previous techniques.
We prove constant lower bounds for the spectral gap of the following random walks on unitary groups SU(2n)\mathsf{SU}(2^n)SU(2n) on nnn qubits. (i) Random Pauli Rotation: choose an nnn-qubit Pauli operator PPP and an angle θ∈[0,2π)θ\in [0,2π)θ∈[0,2π), both uniformly at random, and apply eiθPe^{\mathrm i θP}eiθP. (ii) Brickwork Random Unitary Circuit: choose n−1n-1n−1 unitaries UiU_{i}Ui​ uniformly at random from SU(4)\mathsf{SU}(4)SU(4) independently, and apply U2j−1U_{2j-1}U2j−1​ on two qubits 2j−1,2j2j-1, 2j2j−1,2j and then U2jU_{2j}U2j​ on two qubits 2j,2j+12j, 2j+12j,2j+1. Importantly, the spectral gap lower bounds are independent of nnn and apply for all finite dimensional unitary representations of SU(2n)\mathsf{SU}(2^n)SU(2n) uniformly, including those that appear in unitary ttt-designs. We also prove analogous constant-gap results for orthogonal and Clifford groups; the latter is indispensable for our result on the Brickwork Random Unitary Circuit.
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.