Integrability of Goldilocks quantum cellular automata

TL;DR

This work proves that a subclass of Goldilocks QCA, including the QCA implemented experimentally, map to free fermions and therefore can be simulated classically and yields a parametric quantum circuit with tunable integrability properties useful for testing quantum hardware.

Abstract

Goldilocks quantum cellular automata (QCA) have been simulated on quantum hardware and produce emergent small-world correlation networks. In Goldilocks QCA, a single-qubit unitary is applied to each qubit in a one-dimensional chain subject to a balance constraint: a qubit is updated if its neighbors are in different computational-basis states. We prove that a subclass of Goldilocks QCA, including the QCA implemented experimentally, map to free fermions and therefore can be simulated classically. We support this claim with two proofs, one involving a Jordan-Wigner transformation and one mapping the integrable six-vertex model to QCA. We compute local conserved quantities of these QCA and predict experimentally measurable expectation values. These calculations can be applied to test large digital quantum computers. In contrast, typical Goldilocks QCA have equilibration properties and quasienergy-level statistics that suggest nonintegrability. Still, each of the latter QCA conserves one quantity useful for error mitigation. Our work yields a parametric quantum circuit with tunable integrability properties useful for testing quantum hardware.

Figures (4)

• Figure 1: Goldilocks-QCA circuit. Each half-black-half-white circle represents one control of a generalized XOR constraint. The orange box represents the single-qubit unitary $\hat{V}$. The initial state is a tilted ferromagnet with a polar angle $\theta$ and an azimuthal angle $\phi$. Each dangling control gate wraps around to the other side of the system, obeying periodic boundary conditions. The purple box marks the time-step gate $\hat{U}$, which consists of two layers, $\hat{G}(\hat{V}, q)$, wherein $q = 0, 1$.
• Figure 2: Six-vertex model's allowed vertices. Thick, blue lines represent downward-oriented edges (equivalently, occupied edges). Thin, black lines represent upward-oriented edges (equivalently, empty edges). The net flux at each vertex vanishes, according to the ice condition. The allowed vertex configurations have classical Boltzmann weights (or quantum transition amplitudes) $a_i$, $b_i$, and $c_i$, wherein $i=1,2$.
• Figure 3: Expectation-value dynamics and level statistics. (a) Dynamics generated by integrable Goldilocks QCA $\hat{U} \bm{(} \hat{V}_{\rm free}(\pi/4, 0,-) \bm{)}$, for a system size $L=256$. The initial (Gaussian) state is the $y$-ferromagnet. (b) Median expectation value from 100 realizations of generic Goldilocks QCA $\hat{U}{\bm (}\hat{V}(a,b){\bm )}$ (solid curves) and one outlying realization [$96^{\rm th}$ percentile of $\mathcal{F}(9;z,3)$, dot-dashed curves]. Darker curves signal larger system sizes ($L=18, 22, 26$). The initial state is the tilted ferromagnet $\ket{\Psi_0(0.56, 3.7)}$. In (a)–(b), horizontal dashed curves mark the thermodynamic limits predicted by the appropriate (truncated) GGE. (c) Distribution $P(r)$ over level-spacing ratios. The solid black curve shows the median distribution, calculated from 100 realizations of $\hat{V}(a,b)$ at $L=16$, from the $(K{=}1, \, q_1{=}0)$ sector's 3,200 levels. The faint curves show individual realizations. The black dashed line shows the median distribution for the same 100 $\hat{V}$ realizations but a smaller system ($L=14$) and the $(K{=}1, \, q_1{=}2)$ sector's 858 levels. The Wigner-Dyson (dashed, cyan) and Poisson (dot-dashed, red) distributions serve as references.
• Figure 4: QCA Venn diagram. QCA compatible with both the six-vertex model and the Goldilocks constraint are free-fermionic.