The power of shallow-depth Toffoli and qudit quantum circuits

TL;DR

The paper investigates the computational power of shallow-depth quantum circuits and their separations from classical constant-depth circuits. It develops a framework of finite- and infinite-gate-set models, introduces modular-relations problems on qudits, and uses GHZ-based resource states and Fourier techniques to establish new quantum-classical separations for primes $p$ and $q$, including $\,\mathsf{QNC}^0/\mathsf{qpoly}$ vs $\\mathsf{AC}^0[p]$ and $\\mathsf{QAC}^0$ vs $\\mathsf{AC}^0[p]$, without relying on quantum fan-out in the former case. In the infinite-gateset setting, it proves collapses such as $\\mathsf{i}$-QNC$_p^0[p]=\\mathsf{i}$-QTC$_p^0$, and shows qudit implementations can be simulated by qubits with overhead, highlighting when higher-dimensional systems offer genuine advantage. The work also derives classical upper bounds for the modular-relations, placing broad limits on quantum-classical gaps and clarifying the role of modular gates in shallow circuits. Overall, these results delineate the boundaries of quantum supremacy at constant depth and inform both theoretical understanding and hardware design for near-term quantum devices.

Abstract

The relevance of shallow-depth quantum circuits has recently increased, mainly due to their applicability to near-term devices. In this context, one of the main goals of quantum circuit complexity is to find problems that can be solved by quantum shallow circuits but require more computational resources classically. Our first contribution in this work is to prove new separations between classical and quantum constant-depth circuits. Firstly, we show a separation between constant-depth quantum circuits with quantum advice $\mathsf{QNC}^0/\mathsf{qpoly}$, and $\mathsf{AC}^0[p]$, which is the class of classical constant-depth circuits with unbounded-fan in and $\pmod{p}$ gates. In addition, we show a separation between $\mathsf{QAC}^0$, which additionally has Toffoli gates with unbounded control, and $\mathsf{AC}^0[p]$. This establishes the first such separation for a shallow-depth quantum class that does not involve quantum fan-out gates. Secondly, we consider $\mathsf{QNC}^0$ circuits with infinite-size gate sets. We show that these circuits, along with (classical or quantum) prime modular gates, can implement threshold gates, showing that $\mathsf{QNC}^0[p]=\mathsf{QTC}^0$. Finally, we also show that in the infinite-size gateset case, these quantum circuit classes for higher-dimensional Hilbert spaces do not offer any advantage to standard qubit implementations.

Figures (8)

• Figure 1: Parameterized quantum circuit class for values $q$, $p$, and $n$, incorporating the advice quantum state $\ket{\mathsf{GHZ}_{q,n}^0}$, which solves the $\mathcal{R}_{q,p}^{q\cdot n}$ with bounded one-sided error.
• Figure 2: Representation of $\mathcal{B}_6$.
• Figure 3: Representation of a classical $\mathsf{AC}^0[q]$ circuit, parameterized by $q$ and $n$, that reduces a solution to the $\mathcal{R}_{q,2}^{q\cdot n}$ problem to a solution of $\mathrm{MOD}_2(x)$, where $x$ is the initial input string of size $n$.
• Figure 4: Graph representation of $\mathsf{AC}^0[p]$ classes with $p$ prime. Each edge denotes that the connected classes are distinct, without one being a subset of the other. Along the edges, examples highlight problems that are in one class but excluded from the other.
• Figure :

Theorems & Definitions (69)

• Definition 1: $\mathsf{QNC}_d^0$
• Definition 2: $\mathsf{QAC}_d^0$
• Definition 3: Modular relation problem
• Definition 4: Parallel-k modular relation problem
• Definition 5: Qudit orthogonal X-basis
• Definition 6: Qudit-GHZ orthogonal X-basis
• Lemma 7
• proof
• Lemma 8
• proof