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Tight bounds on depth-2 QAC-circuits computing parity

Stephen Fenner, Daniel Grier, Daniel Padé, Thomas Thierauf

TL;DR

The paper proves that depth-$2$ $ ext{QAC}$ circuits cannot compute parity on more than three non-target qubits, even with arbitrary ancilla, establishing a tight nonexistence result. It introduces an algebraic framework that uses irreducible multivariate polynomials and an Entanglement Lemma for the $ extup{C}_S Z$ gate to separate parity functionality from shallow quantum circuits. This nonasymptotic depth-2 lower bound complements existing asymptotic bounds for constant-depth quantum circuits, offering a distinct technique set based on algebraic and entanglement analyses. The work also outlines a path to extend these ideas to higher depths and to address broader topologies in quantum circuit complexity.

Abstract

We show that the parity of more than three non-target input bits cannot be computed by QAC-circuits of depth-2, not even uncleanly, regardless of the number of ancilla qubits. This result is incomparable with other recent lower bounds on constant-depth QAC-circuits by Rosenthal [ICTS~2021,arXiv:2008.07470] and uses different techniques which may be of independent interest: 1. We show that all members of a certain class of multivariate polynomials are irreducible. The proof applies a technique of Shpilka & Volkovich [STOC 2008]. 2. We give a tight-in-some-sense characterization of when a multiqubit CZ gate creates or removes entanglement from the state it is applied to. The current paper strengthens an earlier version of the paper [arXiv:2005.12169].

Tight bounds on depth-2 QAC-circuits computing parity

TL;DR

The paper proves that depth- circuits cannot compute parity on more than three non-target qubits, even with arbitrary ancilla, establishing a tight nonexistence result. It introduces an algebraic framework that uses irreducible multivariate polynomials and an Entanglement Lemma for the gate to separate parity functionality from shallow quantum circuits. This nonasymptotic depth-2 lower bound complements existing asymptotic bounds for constant-depth quantum circuits, offering a distinct technique set based on algebraic and entanglement analyses. The work also outlines a path to extend these ideas to higher depths and to address broader topologies in quantum circuit complexity.

Abstract

We show that the parity of more than three non-target input bits cannot be computed by QAC-circuits of depth-2, not even uncleanly, regardless of the number of ancilla qubits. This result is incomparable with other recent lower bounds on constant-depth QAC-circuits by Rosenthal [ICTS~2021,arXiv:2008.07470] and uses different techniques which may be of independent interest: 1. We show that all members of a certain class of multivariate polynomials are irreducible. The proof applies a technique of Shpilka & Volkovich [STOC 2008]. 2. We give a tight-in-some-sense characterization of when a multiqubit CZ gate creates or removes entanglement from the state it is applied to. The current paper strengthens an earlier version of the paper [arXiv:2005.12169].

Paper Structure

This paper contains 15 sections, 18 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Algebraic Preliminaries
  4. Quantum Circuit Preliminaries
  5. Representing Quantum States by Polynomials
  6. Irreducibility Results
  7. The Entanglement Lemma
  8. Case (1):
  9. Pure Parity States
  10. Quantum Circuit Lower Bounds
  11. Case 1: $C$'s target does not encounter a multiqubit ${\textup{C}_{}Z}$-gate on layer $d$.
  12. Case 2: $C$'s target is pass-through.
  13. Case 1: $G_0^{(2)}$ does not touch one of the qubits $1$, $2$, or $3$.
  14. Case 2: $G_0^{(2)}$ touches all of the qubits $1$, $2$, and $3$ (i.e., not Case 1).
  15. Further Research

Key Result

Lemma 3.1

Let $f \in \mathbb{F}[{\boldsymbol x}]$ be a polynomial and let ${\boldsymbol a}\in\mathbb{F}^n$ be a justifying assignment of $f$. Then $I\subseteq [n]$ satisfies $f({\boldsymbol a}) \cdot f \equiv f|_{{\boldsymbol x}_I={\boldsymbol a}}\cdot f|_{{\boldsymbol x}_{[n]\setminus I}={\boldsymbol a}}$, i

Figures (2)

  • Figure 1: The most general partitioning of $[r]$ into intersections of the sets $A,B,C,D,S$. Some intersections may be empty.
  • Figure 2: The portion of a typical circuit $C$ before layer $2$. The top line is qubit $0$ (the target). $\ket{\psi_b}$ on qubits $1$ and $2$ turns $G_0^{(1)}$ off. Here, $G_0^{(1)}$ is depicted as touching all qubits, but this need not be the case.

Theorems & Definitions (50)

  • Conjecture 1.1
  • Definition 2.2: Justifying assignment
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 3.1: SV:indecomposable
  • Corollary 3.2
  • proof
  • ...and 40 more