Tight bounds on depth-2 QAC-circuits computing parity

2025-04-08

AI Summary

This work investigates the computational limits of depth-2 quantum approximate circuits (QACs) for computing parity functions. **Problem:** Can such circuits—whether “clean” or not—compute the parity of more than three input bits? **Method:** The authors develop a novel analytical framework that integrates algebraic complexity theory (specifically, the Shpilka–Volkovich factorization), irreducibility analysis of multivariate polynomials, and dynamical characterization of entanglement generation induced by CZ gates. **Contribution/Results:** They establish the first tight lower bound, rigorously proving that no depth-2 QAC can compute the parity function on *n* > 3 bits—regardless of the number of ancillary qubits. This result is independent of prior constant-depth lower bounds and substantially strengthens earlier work. The framework is the first to leverage polynomial irreducibility in conjunction with gate-level entanglement dynamics for QAC analysis, yielding a robust and broadly applicable technique for reasoning about shallow quantum circuits.

Technology Category

• Machine Learning: Quantum Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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].

Problem

• Proving depth-2 QAC-circuits cannot compute parity of >3 input bits
• Characterizing irreducibility of multivariate polynomials in quantum circuits
• Analyzing entanglement effects of multiqubit CZ gates

Innovation

• Proves depth-2 QAC cannot compute multi-bit parity
• Uses multivariate polynomial irreducibility techniques
• Characterizes entanglement effects of multiqubit CZ gates