The XY interaction can only flip spins in pairs, so it never changes the number of excitations by an odd amount — the parity of that count is conserved. That single symmetry cleaves the whole dynamics into two independent halves, and the algebra that generates it splits with them. Build those halves exactly below, then watch a symmetric noise channel bleed probability between them — the even baseline against which the hardware's measured asymmetry stands out.
Write the parity operator $P=\prod_i Z_i$. A computational basis state $\lvert x\rangle$ is its eigenstate with value $(-1)^{\operatorname{popcount}(x)}$ — plus one if it has an even number of ones, minus one if odd. Because the XY term swaps $\lvert 01\rangle\leftrightarrow\lvert 10\rangle$ it leaves that count's parity fixed, so $[H_{XY},P]=0$. The Hilbert space therefore breaks into an even and an odd sector of equal size $2^{n-1}$, and the dynamical Lie algebra generated by the Trotter terms breaks with it.
The two blocks are the two parity sectors; nothing the Hamiltonian does connects them. That is the exact statement the panel constructs — the sector each basis state belongs to, the equal counts, and the algebra's dimension — for any qubit number you pick.
The left grid is every computational basis state, coloured by the sector its excitation-count parity assigns it; the two colours come out in exactly equal number, $2^{n-1}$ each. The right panel starts a state in the even sector and, separately, in the odd sector, then applies an independent bit-flip of probability $p$ to each qubit: the leaked fraction is $\tfrac12\bigl(1-(1-2p)^n\bigr)$, and — because the channel treats every qubit alike — it is identical for the two sectors. A symmetric model gives no even–odd preference.
A conserved parity detects errors for free, but to protect a logical qubit you go one step further: encode it so every codeword lives entirely inside one parity sector. A repetition code of odd distance $d$ does exactly that — each logical bit becomes $d$ physical copies, and because $d$ is odd the block's $Z$‑parity equals the logical bit. The whole logical word therefore sits in a definite physical‑parity sector, and any single physical bit‑flip flips that parity, so the check flags it with certainty. The panel encodes one representative block, lets you inject a flip, and reads out the surface‑code resources the $N$‑oscillator register would cost.
Here $b\in\{0,1\}$ is the logical bit, $q_{\text{surf}}$ the flat rotated‑surface‑code qubit count for $N$ logical oscillators at distance $d$, and $p_L$ the sub‑threshold logical error rate with threshold $p_{\text{th}}=0.01$ and prefactor $A=0.1$. The step‑fidelity figure is a deliberately conservative planning bound, $\exp(-N d\,p_L)$: one logical‑error opportunity per QEC round for every oscillator.
| flat surface-code qubits | — |
| repetition scaffold qubits | — |
| QEC rounds / Trotter step | — |
| wall-clock µs / step | — |
| logical error rate / round | — |
| step-fidelity bound | — |
Rows are integer or closed-form — recomputed here, verified bit-exact against the library's resource model.
A conserved parity is a free error check: any weight that crosses between sectors could only have come from noise, so the leakage is a built-in decoherence probe that needs no extra measurement. And if one sector really is hardier than the other, encoding the information there is a hardware-level advantage for nothing. The platform's DLA-parity work is the apparatus for finding and quantifying exactly that.
| Quantity | Value | Meaning |
|---|---|---|
| Parity operator | $P=\prod_i Z_i$ | excitation count mod 2 |
| Sector size | $2^{n-1}$ each | even and odd, equal |
| DLA | $\mathfrak{su}(2^{n-1})\oplus\mathfrak{su}(2^{n-1})$ | two independent blocks |
| DLA dimension | $2^{2n-1}-2$ | generators of the dynamics |
| Parity leakage | out-of-sector weight | decoherence fingerprint |
Evidence boundary: this panel computes exact, checkable facts — the parity sectors, their equal sizes, the DLA dimension $2^{2n-1}-2$, and the symmetric-channel leakage — live in your browser, verified against direct enumeration. It deliberately does not compute the measured even–odd asymmetry: that is a hardware observation, with its data on the results page. The symmetric model here is the null baseline the observation departs from.