quantum gate directory

Number-preserving (or excitation-preserving) gates conserve the total number of $|1\rangle$s: they never create or destroy excitations, only move them around and adjust phases. Formally, they are the unitaries commuting with the total number operator $\hat{N} = \sum_i (I - Z_i)/2$ — the gates with a global $\mathsf{U}(1)$ symmetry.

$$\mathcal{N}_n := \left\{ U \in \mathsf{U}(2^n) \mid U \hat{N} U^\dag = \hat{N} \right\}$$

Equivalently, $U$ is block-diagonal with respect to Hamming weight: it decomposes as $\bigoplus_{k=0}^{n} U_k$ with $U_k$ acting on the $\binom{n}{k}$-dimensional weight-$k$ sector.

Properties

• The whole diagonal group is contained here: a gate that fixes every basis state certainly fixes its Hamming weight. The interesting members are the ones that move excitations — SWAP, iSWAP, the XY interaction, Givens rotations, FSim.
• On two qubits, a number-preserving gate is a phase on $|00\rangle$, a phase on $|11\rangle$, and an arbitrary $\mathsf{U}(2)$ rotation of the single-excitation subspace — exactly the FSim family and its relatives. The contrast case is the $XX - YY$ interaction, which coherently converts $|00\rangle \leftrightarrow |11\rangle$ and breaks the symmetry.

Why it matters

• Chemistry and fermions. Molecular Hamiltonians conserve electron number, so chemistry ansätze are built from number-preserving blocks: ladders of Givens rotations tiled across a register can prepare any state of fixed particle number.
• Hardware. Energy-conserving interactions (exchange couplings, beam splitters) are natively number-preserving, which is why the iSWAP/FSim families are what superconducting hardware actually implements.
• Error mitigation. If a circuit should conserve $\hat{N}$, measuring the total excitation number and discarding violating shots (symmetry verification) catches any error that changes the weight — bit flips in particular.