A Complete Equational Theory for Real-Clifford+CH Quantum Circuits

We introduce a complete equational theory for the fragment of quantum circuits generated by the real Clifford gates plus the two-qubit controlled-Hadamard gate. That is, we give a simple set of equalities between circuits of this fragment, and prove that any other true equation can be derived from these. This is the first such completeness result for a finitely-generated, universal fragment of quantum circuits, with no parameterized gates and no need for ancillas.

💡 Research Summary

The paper presents the first complete equational theory for a finitely‑generated, universal fragment of quantum circuits that contains no parametrised gates and does not require ancilla qubits. The fragment, called Real‑Clifford+CH, is generated by the real Clifford gates (Hadamard H, Pauli‑Z, and controlled‑Z CZ) together with the two‑qubit controlled‑Hadamard gate (CH). Although this set is a strict subset of the well‑studied Clifford+T fragment, it is already universal because CH can be used to implement the T‑gate indirectly, and multi‑controlled versions of Z and H can be built recursively.

The authors formalise circuits as a PROP (or PRO when swaps are omitted), defining raw circuits from the generators, the empty circuit, identities, and the swap, together with sequential (∘) and parallel (⊗) composition. Figure 1 lists the standard PROP axioms (a)–(g) that guarantee topological invariance and wire‑length independence. They then introduce a collection of convenient macros: CNOT, SWAP, the two‑qubit gate P⊗P (which implements a real‑matrix version of a complex phase gate), and families of multi‑controlled Z, H, ZX and XZ gates, including negative (white) controls. These shortcuts are purely notational; every diagram still corresponds to a circuit built from the four primitive gates.

The core contribution is a set of 19 equations (Figure 4). Equations (1)–(10) and (12)–(15) are familiar Clifford identities (e.g., H² = I, CZ commutes with H on the control line, etc.). Equations (11), (16) and (17) capture the commutation of positively‑controlled subcircuits with negatively‑controlled ones, a property that becomes transparent when expressed in the eigenbasis of H. Equation (18) shows that the upper part of the P⊗P macro behaves exactly like a single‑qubit P gate, while equation (19) is a schema for n ≥ 5 qubits stating that the bottom qubit of a certain multi‑controlled construction is untouched. All equations are proved sound by direct matrix computation (for the concrete ones) or simple induction (for the schema).

To prove completeness, the authors rely on a previously established presentation of the group Uₙ(ℤ