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arXiv:2303.16944

On the moments of random quantum circuits and robust quantum complexity

Jonas Haferkamp

Published Mar 29, 2023• 1 citation
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Abstract

We prove new lower bounds on the growth of robust quantum circuit complexity -- the minimal number of gates $C_δ(U)$ to approximate a unitary $U$ up to an error of $δ$ in operator norm distance. More precisely we show two bounds for random quantum circuits with local gates drawn from a subgroup of $SU(4)$. First, for $δ=Θ(2^{-n})$, we prove a linear growth rate: $C_δ\geq d/\mathrm{poly}(n)$ for random quantum circuits on $n$ qubits with $d\leq 2^{n/2}$ gates. Second, for $ δ=Ω(1)$, we prove a square-root growth of complexity: $C_δ\geq \sqrt{d}/\mathrm{poly}(n)$ for all $d\leq 2^{n/2}$. Finally, we provide a simple conjecture regarding the Fourier support of randomly drawn Boolean functions that would imply linear growth for constant $δ$. 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 $t$-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.

DOI
10.48550/arxiv.2303.16944

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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) δ=Ω(1)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.
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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 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.
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.
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 ttht\mathrm{th}tth moment operator of local random quantum circuits on nnn qudits with local dimension qqq 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 ttt 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 ttt. 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 nnn and ttt 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 ttt.
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.