Short remarks on shallow unitary circuits

TL;DR

The paper demonstrates that shallow symmetric quantum circuits cannot emulate global Haar-random unitaries when a local conserved charge exists, requiring depth linear in system size for accurate symmetric unitary designs. It then presents a constructive shallow-circuit learning framework based on Arrighi’s identity, showing how to extract the action of a shallow unitary on local Pauli operators via quantum tomography, with extensions to quantum cellular automata (QCA) and considerations of locality/spread. Finally, it provides a new proof that translation-invariant QCAs in any dimension admit staircase implementations with only $O(n)$ local gates, using invertible subalgebra pumping and a dimensional induction, and discusses the implications for QCA triviality and circuit-depth tradeoffs. Collectively, the work clarifies when shallow circuits can mimic randomness under symmetry constraints, offers practical learning protocols for shallow unitaries, and establishes linear gate-count realizations for QCAs across dimensions, highlighting open problems in efficient conversions between QCAs and staircase circuits.

Abstract

(i) We point out that every local unitary circuit of depth smaller than the linear system size is easily distinguished from a global Haar random unitary if there is a conserved quantity that is a sum of local operators. This is always the case with a continuous onsite symmetry or with a local energy conservation law. (ii) We explain a simple algorithm for a formulation of the shallow unitary circuit learning problem and relate it to an open question on strictly locality-preserving unitaries (quantum cellular automata). (iii) We show that any translation-invariant quantum cellular automaton in $D$-dimensional lattice of volume $V$ can be implemented using only $O(V)$ local gates in a staircase fashion using invertible subalgebra pumping.

Figures (4)

• Figure 1: Any circuit $W$ of depth smaller than the linear system size is a product of $W_{AB}$ and $W_{BC}$.
• Figure 2: Comb connectivity. A qudit is placed on every tip and spine vertex. A two-qudit gate may only act on an edge. A discrete symmetry acts on the tip qudits.
• Figure 3: The QCA $\zeta_c$ acts as $\zeta$ on $z < c - r'$ but as identity on $z \ge c$. The composition $\tau_{c,c'} = \zeta_{c'} \circ \zeta_{c}^{-1}$ is then supported on the region $c - r' \le z < c'$ which is compact along $z$-direction but otherwise not. This is the equality (a). Since $\tau_{c,c'}$ acts by $\zeta$ on the interior of the region $c - r' \le z < c'$, a composition $\mu_1$ of $\zeta$ and a product of separated $\tau$'s gives a product of separated QCA, each of which can be regarded as a QCA in one dimension lower. This is the equality (b).
• Figure 4: Implementation of $\beta_{\mathcal{A}}$ that pumps an invertible subalgebra ${\mathcal{B}}$ using a swap-like QCA $\gamma$. The blue triangle on the right diagram is ${\mathcal{A}}$ and the orange is ${\mathcal{B}}$. By the staircase, only ${\mathcal{B}}$ is pumped to the right, which is exactly the action of $\beta_{\mathcal{A}}$.