[Paper Review] Quantum circuits of c -- Z and SWAP gates optimization and entanglement

Marc Bataille, Jean-Gabriel Luque|arXiv (Cornell University)|Oct 2, 2018

Quantum Computing Algorithms and Architecture22 references4 citations

TL;DR

This paper investigates quantum circuits composed solely of controlled-Z (c-Z) and SWAP gates, proving they form a finite group under composition. It introduces an efficient simplification algorithm based on the algebraic structure of this group and analyzes how such circuits generate entanglement, particularly through SLOCC equivalence, revealing that specific entangled states like V₃, V₉, and V₁₄ can be generated from the |0000⟩ state using only these gates.

ABSTRACT

We have studied the algebraic structure underlying the quantum circuits composed by $\\cZ$ and $\\Swap$ gates. Our results are applied to optimize the circuits and to understand the emergence of entanglement when a circuit acts on a fully factorized state.

Motivation & Objective

• To understand the algebraic and combinatorial structure of quantum circuits composed exclusively of c-Z and SWAP gates.
• To develop an efficient circuit simplification algorithm based on the finite group formed by these gates.
• To investigate how entanglement emerges when such circuits act on a factorized input state.
• To determine whether different entanglement classes—especially SLOCC-inequivalent ones—can be reached using only c-Z and SWAP gates.
• To explore the potential of these gates for generating complex entangled states such as V₃, V₉, and V₁₄ from the |0000⟩ state.

Proposed method

• Prove that the set of circuits generated by c-Z and SWAP gates forms a finite group, denoted cZS_k, under composition.
• Use group-theoretic techniques, including conjugacy classes and representation theory, to analyze the structure of cZS_k.
• Apply Dehn’s algorithm, suggested by a collaborator, to simplify quantum circuits by reducing gate count.
• Employ SLOCC (Stochastic Local Operations and Classical Communication) equivalence to classify entanglement types generated by the circuits.
• Analyze specific entangled states like |GHZ_k⟩, |W_k⟩, and V₃, V₉, V₁₄ to determine their reachability via c-Z and SWAP operations.
• Use geometric and algebraic invariants such as L, M, N, and Q_i to characterize the entanglement structure of generated states.

Experimental results

Research questions

• RQ1 Can the circuit simplification problem for c-Z and SWAP gates be solved efficiently using the underlying group structure?
• RQ2 What is the algebraic and combinatorial structure of the group generated by c-Z and SWAP gates on k qubits?
• RQ3 Which entanglement classes—particularly SLOCC-inequivalent ones—can be generated from a factorized input state using only c-Z and SWAP gates?
• RQ4 Can complex entangled states such as V₃, V₉, and V₁₄ be synthesized from |0000⟩ using only c-Z and SWAP operations?
• RQ5 Is there a polynomial or Hopf algebraic structure underlying the group cZS_k that could support deeper theoretical insights?

Key findings

• The set of quantum circuits composed of c-Z and SWAP gates forms a finite group, cZS_k, which enables systematic analysis and simplification.
• An efficient circuit simplification algorithm is derived from the group structure, reducing gate count and improving reliability.
• The group cZS_k contains elements that generate entangled states such as |GHZ_k⟩ and |W_k⟩, demonstrating the capability to create high-depth entanglement.
• States V₃ and V₁₄ belong to the third secant variety σ₃(ℙ¹×ℙ¹×ℙ¹×ℙ¹), indicating a specific algebraic-geometric entanglement structure.
• The state V₉ belongs to σ₃(ℙ³×ℙ³), showing a more general entanglement class than V₃ and V₁₄, with L ≠ 0 but M = N = 0.
• The paper demonstrates that generically entangled states like V₃, V₉, and V₁₄ can be generated from the |0000⟩ state using only c-Z and SWAP gates, confirming their reachability under SLOCC.