Linear Growth of Circuit Complexity from Brownian Dynamics

TL;DR

The paper demonstrates that Brownian quantum many-body systems—spin and fermionic clusters with time-dependent all-to-all interactions—generate approximate unitary $k$-designs in times that scale linearly with $kN$, by mapping the Frame Potential to an effective statistical-mechanics problem and solving it via path integrals and mean-field Hamiltonians. It provides explicit results for Brownian spin clusters and Brownian SYK fermions, including ground-state degeneracies and spectral gaps that control the approach to Haar randomness, and extends the analysis to time-independent chaotic Hamiltonians perturbed by random noise. The work connects quantum chaos, replica methods, and holographic ideas (via replica wormholes) to a concrete, analytically tractable picture of linear growth of circuit complexity. Overall, it establishes a robust framework showing that simple Brownian dynamics can realize fast, linearly-scaling design generation with clear quantitative bounds, with potential implications for quantum randomness, benchmarking, and holographic complexity.

Abstract

We calculate the frame potential for Brownian clusters of $N$ spins or fermions with time-dependent all-to-all interactions. In both cases the problem can be mapped to an effective statistical mechanics problem which we study using a path integral approach. We argue that the $k$th frame potential comes within $ε$ of the Haar value after a time of order $t \sim k N + k \log k + \log ε^{-1}$. Using a bound on the diamond norm, this implies that such circuits are capable of coming very close to a unitary $k$-design after a time of order $t \sim k N$. We also consider the same question for systems with a time-independent Hamiltonian and argue that a small amount of time-dependent randomness is sufficient to generate a $k$-design in linear time provided the underlying Hamiltonian is quantum chaotic. These models provide explicit examples of linear complexity growth that are also analytically tractable.

Figures (5)

• Figure 1: Linear growth of complexity in Brownian spin and fermion clusters. The cluster evolves under time-dependent unitary dynamics $U$ (a) composed of Brownian 2-body interactions (spins) or Brownian 4-body interactions (fermions). To compute the Frame Potential $F^{(k)}$ we consider $2k$ replicas (b) labelled by $r,\overline{r} = 1,\ldots,k$ for forward and backward time evolution, respectively. Following disorder averaging, the $2k$-replica system is governed by an effective Hamiltonian $H_{\mathrm{eff}}$ whose ground state manifold (c) is $k!$-fold degenerate for spins and $2^k k!$-fold degenerate for fermions. For spins, each ground state corresponds to one of $k!$ possible pairings (red) between the $r,\overline{r}$ replicas, labelled by elements $\pi$ of the symmetric group $S_k$. For fermions there is an extra factor of $2^k$ coming from $\pm$ signs attached to each pairing. In both cases, a finite gap $\Delta$ to the excited state manifold guarantees that the clusters form $k$-designs in a time $t \sim k N$ linear in both $k$ and $N$ given $k \le e^N$.
• Figure 2: Imaginary part of eigenvalues of $H_{\text{eff}}$ for $D=32$ and $k=1$ with $H$ a GOE random matrix, a single $O$ given by a diagonal matrix with equal numbers of $\pm1$s, and $g=0.2$. The standard deviation of terms in $H$ is taken to be $\log(D)/\sqrt{D}$ to mimic an extensive qubit Hamiltonian with spectrum in $[-\log(D),\log(D)]$. The dashed horizontal lines indicate $1/D$ variance around $-g$, and the rightmost red circle indicates the dark state.
• Figure 3: The distribution of eigenvalues of $M$. The figure shows exact diagonalization for $D=256$ from $50$ samples. The total density is normalized to one.
• Figure 4: Average decay gap of $H_{\text{eff}}$ for $k=1$ as a function of Hilbert space dimension $D$. $H$ is a GOE random matrix, $O$ is a diagonal matrix with equal numbers of $\pm1$, and $g=0.2$. The standard deviation of terms in $H$ is taken to be $\log(D)/\sqrt{D}$. For $D=10,12...,24$ we average over 2000 samples, and for $D=26,...,32$ we average over 1000 samples. The blue curve is a fit to function $a + b/\sqrt{D} + c/D$, where the fit value of $a$ is $0.2006$ which is very close to $g=0.2$.
• Figure 5: Distinguishing a unitary channel $\mathcal{U}$ from the depolarizing channel $\mathcal{D}$. To distinguish $\mathcal{U}$ from $\mathcal{D}$, Bob is allowed to use an ancilla system $R$, along with arbitrary pre- and post-preparation circuits of depth $r'+r" = r$. Bob cannot reliably distinguish a $k$-design $\mathcal{U}$ from $\mathcal{D}$ unless he has circuits of depth at least $r \sim k N / \log N$.